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Functional consequences of network architecture in rat hippocampus: a computational study

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Functional consequences of network architecture in rat hippocampus: a computational study

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Content Functional Consequences of Network Architecture in Rat Hippocampus A Computational Study Gene Jong Yu Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY BIOMEDICAL ENGINEERING Los Angeles, California, USA December 2019 Dissertation Committee: Theodore W. Berger, PhD - Chair, David Packard Chair in Engineering and Professor of Biomedical Engineering, USC Gianluca Lazzi, PhD - Provost Professor of Ophthalmology and Electrical Engineering, USC Vasilis Z. Marmarelis, PhD - Dean's Professor of Biomedical Engineering and Professor of Biomedical Engineering, USC Bartlett W. Mel, PhD - Associate Professor of Biomedical Engineering, USC Dong Song, PhD - Research Associate Professor of Biomedical Engineering, USC Abstract In the past decades, the combination of both the capability to simultaneously record multiple neurons and computational methods for studying the activity of neuronal populations, i.e. multiple neurons, have given weight to a hypothesis that may be added to the neuron doctrine: Though neurons are undisputedly the fun- damental, discrete units that comprise a neural region, the function of a neural region emerges as a consequence to the activity of groups or ensembles of neurons rather than single neurons. The overall theme of this work is to understand how a large population of discrete elements, i.e. neurons, and the connectivity of these ele- ments, in uence the dynamical response of a neural system and achieve a collective function. The hippocampus is an ideal neural region to study the relation between con- nectivity and function because its molecular, cellular, and anatomical properties have been particularly well studied. Furthermore, the hippocampus expresses a dis- tinct and topographically organized connectivity between each of its subelds. The function of the hippocampus also gathers signicant interest as the hippocampus is crucial to memory formation. However, the exact functions of the hippocampus and the purpose of the hippocampal connectivity, and the other properties that comprise a hippocampus, are not known. To investigate the relation between connectivity and hippocampal function, I constructed a computational neuronal network model, designed with a three-dimensional geometry and a numerical scale approaching the full number of neurons and synapses of a single hemisphere of a rat hippocampus, which include the dentate gyrus, CA3, and CA1 subelds. Each subeld includes the principal neuron and at least one interneuron, the basket cell. This model was used to investigate the role of topo- graphic connectivity on the emergent spatio-temporal dynamics that were generated by the hippocampal neurons and its in uence on the encoding of spatial information. The thesis will be divided into four major sections: a description of the large- scale model and the method of estimating anatomical connectivity, investigations of the large-scale model using white noise input, the development of a MIMO model framework to quantify the spatio-temporal patterns that the hippocampal subelds perform, and investigations on the role of connectivity in processing spatial infor- mation using a physiologically realistic and behaviorally relevant input, i.e. grid cell activity. i Acknowledgments What is there to say? Still, I can't be silent. No matter the individual, one must consider those who came before and credit their strength through which the individual became. Therefore, I'd like to take the opportunity to give recognition and gratitude. To Dr. Berger. It is impossible to begin to describe the might of this man, my mentor. I only hope that I can smuggle out even a mote of this might. Thank you for all the wisdom that you had bestowed to me throughout the years. To Dong and Jean-Marie. There was an interminable distance between me and the man at the top in which I would have truly been lost. If it weren't for the beacons you had axed to light my way. Thank you for all of your guidance. To Dr. Marmarelis, Dr. Mel, Dr. Lazzi, and Dr. Fraser. As my committee members, it was you who rst gave me recognition. Thank you for dedicating the time to nurture and bear witness to me and my research. To the people from the lab: Brian, Huijing, Sushmita, Viviane, Uldric, Aaron, Penning, Rosa, Clay, Adam, Eric, Ali, Mickey, Phillip, Pallavi, Zane, Xiwei, Wenx- uan, Bryan, Jonathan... I was but a lump of rock in a sea of rocks, and together through the grind we went. Thank you for allowing me to polished by your presences. To Charles, David, Don, Robert, and Doris. Who supported me with invisible strands. And special thanks to NIH Grant U01 EB025830 and ONR Grant N00014-13-1- 0211 which supported my work, and the USC High-Performance Computing Center for giving life to it. ii Table of Contents Abstract i Acknowledgments ii Chapter 1 Introduction 1 1.1 Multi-Scale Neural System Modeling . . . . . . . . . . . . . . . . . . 2 1.2 \Intermediate" Levels of System Function . . . . . . . . . . . . . . . 3 1.3 Connectivity on Higher Level Function . . . . . . . . . . . . . . . . . 4 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2 Large-Scale Model of Rat Hippocampus 9 2.1 Overview of Rat Hippocampus . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Anatomical Description of Hippocampal Structure . . . . . . . 11 2.1.2 Description of Main Neuron Types in Model . . . . . . . . . . 14 2.2 Large-Scale Model of Rat Hippocampus . . . . . . . . . . . . . . . . . 16 2.2.1 Formation of Anatomical Maps . . . . . . . . . . . . . . . . . 16 2.2.2 Compartmental Models of Neurons . . . . . . . . . . . . . . . 17 2.2.3 Generation of Dendritic Morphologies . . . . . . . . . . . . . . 18 2.2.4 Specication of Passive and Active Properties . . . . . . . . . 18 2.2.5 Synaptic Density . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.6 Synaptic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Anatomically-Constrained Connectivity . . . . . . . . . . . . . . . . . 26 2.3.1 Topography of Entorhinal Projection to Hippocampus . . . . 27 2.3.2 Topography of Mossy Fibers . . . . . . . . . . . . . . . . . . . 30 2.3.3 Topography of CA3 Associational System . . . . . . . . . . . 32 2.3.4 Topography of Projections within the Hippocampal Subelds . 35 2.3.5 Conduction Velocity of Action Potentials . . . . . . . . . . . . 36 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 iii TABLE OF CONTENTS Chapter 3 Spatio-Temporal Patterns of the Large-Scale Model 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Emergence of Spatio-Temporal Clusters . . . . . . . . . . . . . . . . . 47 3.2.1 Spatio-Temporal Clusters as an Emergent Property . . . . . . 47 3.2.2 Removal of Extrinsic and Intrinsic Sources of Inhibition . . . . 50 3.2.3 Topographic Connectivity as a Source of Spatial Correlation . 51 3.2.4 Sources of Temporal Correlation . . . . . . . . . . . . . . . . . 53 3.2.5 Modulation of Clusters via Interneurons . . . . . . . . . . . . 54 3.2.6 Clusters as a Higher Level of Functional Organization . . . . . 57 3.3 Propagation and Transformation of Clusters . . . . . . . . . . . . . . 58 3.3.1 Entorhinal-CA3 Perforant Path Projection . . . . . . . . . . . 58 3.3.2 Dentate-CA3 Mossy Fiber Projection . . . . . . . . . . . . . . 60 3.3.3 Combined Entorhinal- and Dentate-CA3 Projections . . . . . 62 3.3.4 Associational System . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.5 Interactions between the Associational System and Feedfor- ward/Feedback Inhibition . . . . . . . . . . . . . . . . . . . . 64 3.3.6 Summary of CA3 Work . . . . . . . . . . . . . . . . . . . . . . 73 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 4 Spatio-Temporal Filters for Population Dynamics 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.1 Neural Mass and Population Spiking Rate . . . . . . . . . . . 79 4.2.2 Computational Framework . . . . . . . . . . . . . . . . . . . . 80 4.2.3 Optimization of Basis Function Parameters . . . . . . . . . . . 83 4.2.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 84 4.2.5 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.1 Metaparameter Optimization . . . . . . . . . . . . . . . . . . 86 4.3.2 Model Prediction Performance . . . . . . . . . . . . . . . . . . 86 4.3.3 Spatio-Temporal Filters . . . . . . . . . . . . . . . . . . . . . 88 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 iv TABLE OF CONTENTS Chapter 5 Connectivity-Dependent Information Processing 92 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.1 Large-Scale Model . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2.2 Grid Cell Activity . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2.3 Place Field Calculation . . . . . . . . . . . . . . . . . . . . . . 99 5.2.4 Spatial Information Score . . . . . . . . . . . . . . . . . . . . 100 5.2.5 Recursive Point Process Filter for Decoding . . . . . . . . . . 100 5.2.6 Lower Bound of Mutual Information Encoded by Network . . 102 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.1 Emergence of Place Fields . . . . . . . . . . . . . . . . . . . . 102 5.3.2 Evaluating Gradients in Dentate Gyrus . . . . . . . . . . . . . 103 5.3.3 Place Field Area Gradient Depends on Axonal Anatomy . . . 104 5.3.4 Spatial Information Score Depends on Axonal Anatomy . . . . 106 5.3.5 Multi-Resolution Inputs Aect Spatial Information Score . . . 107 5.3.6 Spatial Information Score and Multi-Resolution Representa- tions of Space Predict Decoding Performance . . . . . . . . . . 107 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.1 Large-Scale Modeling and the Incorporation of Biological Con- straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dorso-Ventral Gradient in Dentate Gyrus . . . . . . . . . . . . 111 5.4.3 Transverse Gradient in Dentate Gyrus . . . . . . . . . . . . . 111 5.4.4 High-Level Constraints Predict Function of Lower-Levels . . . 112 5.4.5 Additional Contributions to Spatial Encoding . . . . . . . . . 113 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 v Chapter 1 Introduction 1 CHAPTER 1. INTRODUCTION 1.1 Multi-Scale Neural System Modeling Possibly more than any other brain area, interest in the hippocampus has gen- erated vast amounts of experimental data through the eorts of the neuroscience community. This has led to the accumulation of large amounts of quantitative information describing the properties of the hippocampus, including anatomical, morphological, biophysical, biochemical, and synaptic transmission levels of anal- ysis. As with other brain areas, however, this wealth of structural and functional detail has not guaranteed insight into higher levels of system operation. The eld continues, and rightfully so, to struggle to understand how all of the cellular and network properties of the hippocampus dynamically interact to produce the global functional properties of the larger system. Multiple theories and hypotheses have been put forward to contextualize and interpret subsets of the huge amount of hippocampal data. These works have at- tempted to provide possible explanations for various behavioral and cognitive func- tions believed to be subserved by the hippocampus (Marr et al., 1991; McNaughton and Morris, 1987; Levy, 1989; Treves and Rolls, 1994; McClelland et al., 1995; Has- selmo, 2005; Solstad et al., 2006; Myers and Scharfman, 2011). Although these studies are highly admirable and have helped guide the eld in its thinking about the cellular and network bases of hippocampal memory, their results cannot provide a complete answer given the limited amount of available quantitative experimental data on which they are based, and given the multiple levels of neural organization that various models and theories must \reach over" to account for system/behavioral phenomena. There are few computational frameworks (see Morgan and Soltesz, 2010) that integrate at least a signicant portion of the quantitative cellular and anatomical data of hippocampus for its multiple subelds. As a consequence, there are no models to date that successfully integrate such data and extend those \lower level" properties to putative explanations of \higher level" function, be it at the system level or the cognitive and behavioral levels. If there is any one brain structure that provides an opportunity for understanding integration from molecular to systems neural function, it is the rat hippocampus. One basis for this argument has already been mentioned, namely, the large body of quantitative information already collected about its molecular, cellular, and network structure and function. The second basis is the nature and relative simplicity of the structural organization of the hippocampus. 2 CHAPTER 1. INTRODUCTION There is a highly well-dened organization to the hippocampal formation con- sisting of a predominantly feedforward architecture wherein activity is propagated from the entorhinal cortex to the dentate gyrus, to the CA3/4 pyramidal zones, and nally to the CA1/2 pyramidal regions (Andersen et al., 1971; Swanson et al., 1978). The projections between each of the subelds exhibit unique organizational principles, e.g., targets and topography (Swanson et al., 1978; Ishizuka et al., 1990; Freund and Buzs aki, 1996; Dolorfo and Amaral, 1998). These studies reveal the incomplete (in the sense that any one neuron within a subeld does not project to all neurons in the next subeld) and non-random nature of hippocampal connec- tivity. Topography determines the structural connectivity of a neural system which organizes individual neurons into collections of neurons and oers a bridge between lower level cellular dynamics and higher level population dynamics. 1.2 \Intermediate" Levels of System Function One of the fundamental issues that has arisen in using the multi-scale model of hippocampus described in this work is that there is a need to identify what might be termed \intermediate" levels of hippocampal function. There are generally well- understood interpretations of \synaptic function" (presynaptic release, postsynaptic current and/or potential, etc.) and \cellular function" (metabolism, action poten- tial generation, etc.). Although there are multiple possible denitions for each level of functionality, there is general agreement on what are the small number of pos- sibilities in each case. But once \molecular," \synaptic," and "cellular function" are accounted for, what is the denition of "multi-cellular" or \system" function, upon which there is generally little if any universal agreement? Even more di- cult to address is the identication of levels of functions that lie between cellular, multi-cellular, behavioral, or cognitive functions. Behavioral and inferred cognitive functions for the hippocampus have a theoretically rich and experimentally broad history (O'Keefe and Nadel, 1978; Berger et al., 1986; Squire, 1986; Cohen et al., 1993; Nadel and Moscovitch, 1997). How hippocampal cognitive and behavioral functions derive from cellular and molecular dynamics still remains a mystery. For insights to be reached on this issue, we must identify levels of system or subsystem function that lie between the cellular and the behavioral. One of the main objectives in developing the multi-scale model of hippocampus was so that 3 CHAPTER 1. INTRODUCTION it would be possible to \observe" simulated spiking activity of large numbers of neurons simultaneously. The model represents the activity of many more neurons than had ever been observed previously, either computationally or experimentally. If there were regularities in neuronal ring that became apparent beyond the level of tens or hundreds of neurons, i.e., beyond the levels typically used in previous studies identifying behavioral or cognitive population \correlates" of hippocampal cell activity (Berger et al., 1983; Berger and Weisz, 1987; Eichenbaum et al., 1989; Hampson et al., 1999; Berger et al., 2011; Krupic et al., 2012; McKenzie et al., 2014), such \regularities" in population ring of hippocampal neurons would indicate the existence of some kind of higher-level \structure" in the organization of the hippocampal system. The computational studies reviewed here have revealed one such organizational structure: the \clusters" of neural ring revealed by the present analyses indicate correlated levels of excitability distributed both in space and time throughout the dorso-ventral extent of the hippocampus in response to low levels of random entorhinal cortical activity. Because such correlated, clustering of neuronal ring is expressed to a low level of relatively even entorhinal input, and does not require a radically, strong, and/or rhythmic bursting from entorhinal cortex, the work in the thesis hypothesizes that the neural clusters represent a preferential space- time ltering by a neural system that constitutes a rst order level of processing by the multiple stages of hippocampal circuitry. 1.3 Connectivity on Higher Level Function As mentioned previously, the spatial extent of connectivity, i.e. the topographic connectivity, between and within a neural region is nite; a presynaptic neuron does not project to every neuron in the postsynaptic population. The spatial extent of the connectivity is limited by anatomical features such as the size of the presy- naptic neuron's axon collaterals. In constructing a neuronal network model which contains large numbers of neurons, an anatomically realistic connectivity for the network becomes crucial because, as the extent of the neural system being modeled is expanded beyond the size of an axon collateral, the model must then preserve the anatomical limitations in the connectivity. Seldom have computational models represented the full three-dimensional anatomical extent of rat hippocampus, and as a consequence even fewer computational models have attempted to consider the 4 CHAPTER 1. INTRODUCTION eects of topographic connectivity in hippocampus. The connectivity is the rst level of processing that occurs from the cellular level to the multi-cellular level as it relates the activity of one neuron to all the neurons to which it connects. As such, we further hypothesize that the spatial component of the space-time ltering performed by the neural system is mediated by the connectivity and that it further in uences system function. 1.4 Summary The thesis is organized in the following manner. First, the computational plat- form used to develop a three-dimensional, large-scale, computational neuronal net- work model of rat hippocampus will be described. The large-scale model represents the entorhinal cortex, dentate gyrus, and CA3/4 regions. The initial results of the model that include the discovery of clusters of activity and the relations between features of the clusters to properties of the neural system in the entorhinal-dentate network will be presented. Next, the transformation of the clusters due to their propagation from the dentate gyrus to the CA3 will be described. To quantify the transformation and the space-time properties of each hippocampal subregion, the population dynamics will be modeled as a set of interconnected neural mass models using a generalized linear model framework with spatial Chebyshev basis functions and temporal Laguerre basis functions. Finally, the eect of connectivity on the processing of spatial information will be explored using grid cell input activity and point-process neural decoding framework. 5 CHAPTER 1. INTRODUCTION 1.5 References Andersen, P., Bliss, T., and Skrede, K. (1971). Lamellar organization of hippocampal pathways. Experimental brain research, 13(2):222|238. Berger, T. W., Berry, S. D., and Thompson, R. F. (1986). Role of the Hippocampus in Classical Conditioning of Aversive and Appetitive Behaviors, pages 203{239. Springer US, Boston, MA. Berger, T. W., Hampson, R. E., Song, D., Goonawardena, A., Marmarelis, V. Z., and Deadwyler, S. A. (2011). A cortical neural prosthesis for restoring and enhancing memory. Journal of Neural Engineering, 8(4):046017. Berger, T. W., Rinaldi, P. C., Weisz, D. J., and Thompson, R. F. (1983). Single- unit analysis of dierent hippocampal cell types during classical conditioning of rabbit nictitating membrane response. Journal of Neurophysiology, 50(5):1197{ 1219. PMID: 6644367. Berger, T. W. and Weisz, D. J. (1987). chapter Single unit analysis of hippocampal pyramidal and granule cells and their role in classical conditioning of the rabbit nictitating membrane response., pages 217{253. Lawrence Erlbaum Associates, Inc, Hillsdale, NJ, US. Cohen, P., Cohen, N. J., Eichenbaum, H., and Eichenbaum, U. (1993). Memory, Amnesia, and the Hippocampal System. A Bradford book. MIT Press. Dolorfo, C. L. and Amaral, D. G. (1998). Entorhinal cortex of the rat: Topographic organization of the cells of origin of the perforant path projection to the dentate gyrus. Journal of Comparative Neurology, 398(1):25{48. Eichenbaum, H., Wiener, S., Shapiro, M., and Cohen, N. (1989). The organization of spatial coding in the hippocampus: a study of neural ensemble activity. Journal of Neuroscience, 9(8):2764{2775. Freund, T. and Buzs aki, G. (1996). Interneurons of the hippocampus. Hippocampus, 6(4):347{470. Hampson, R. E., Simeral, J. D., and Deadwyler, S. A. (1999). Distribution of spatial and nonspatial information in dorsal hippocampus. Nature, 402(6762):610{614. 6 CHAPTER 1. INTRODUCTION Hasselmo, M. E. (2005). What is the function of hippocampal theta rhythm?|linking behavioral data to phasic properties of eld potential and unit recording data. Hippocampus, 15(7):936{949. Ishizuka, N., Weber, J., and Amaral, D. G. (1990). Organization of intrahippocam- pal projections originating from ca3 pyramidal cells in the rat. Journal of Com- parative Neurology, 295(4):580{623. Krupic, J., Burgess, N., and O'Keefe, J. (2012). Neural representations of location composed of spatially periodic bands. Science, 337(6096):853{857. Levy, W. B. (1989). A computational approach to hippocampal function. In Hawkins, R. D. and Bower, G. H., editors, Computational Models of Learning in Simple Neural Systems, volume 23 of Psychology of Learning and Motivation, pages 243 { 305. Academic Press. Marr, D., Willshaw, D., and McNaughton, B. (1991). Simple Memory: A Theory for Archicortex, pages 59{128. Birkh auser Boston, Boston, MA. McClelland, J. L., McNaughton, B. L., and O'Reilly, R. C. (1995). Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionist models of learning and memory. Psychological Review, 102(3):419{457. McKenzie, S., Frank, A. J., Kinsky, N. R., Porter, B., Rivi ere, P. D., and Eichen- baum, H. (2014). Hippocampal representation of related and opposing memo- ries develop within distinct, hierarchically organized neural schemas. Neuron, 83(1):202 { 215. McNaughton, B. and Morris, R. (1987). Hippocampal synaptic enhancement and in- formation storage within a distributed memory system. Trends in Neurosciences, 10(10):408 { 415. Myers, C. E. and Scharfman, H. E. (2011). Pattern separation in the dentate gyrus: A role for the ca3 backprojection. Hippocampus, 21(11):1190{1215. Nadel, L. and Moscovitch, M. (1997). Memory consolidation, retrograde amnesia and the hippocampal complex. Current Opinion in Neurobiology, 7(2):217 { 227. 7 CHAPTER 1. INTRODUCTION O'Keefe, J. and Nadel, L. (1978). The Hippocampus as a Cognitive Map. Oxford: Clarendon Press. Solstad, T., Moser, E. I., and Einevoll, G. T. (2006). From grid cells to place cells: A mathematical model. Hippocampus, 16(12):1026{1031. Squire, L. (1986). Mechanisms of memory. Science, 232(4758):1612{1619. Swanson, L. W., Wyss, J. M., and Cowan, W. M. (1978). An autoradiographic study of the organization of intrahippocampal association pathways in the rat. Journal of Comparative Neurology, 181(4):681{715. Treves, A. and Rolls, E. T. (1994). Computational analysis of the role of the hip- pocampus in memory. Hippocampus, 4(3):374{391. 8 Chapter 2 Large-Scale Model of Rat Hippocampus 9 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Large-scale modeling of neural systems is a fairly new eld and can be divided into large-scale models using simplied neurons, e.g. leaky integrate and re neu- rons, and large-scale models using detailed neuron models, e.g. compartmental neuron models. Large-scale, in terms of numbers of neurons, will be restricted to networks that contain at least 10,000 neurons. Large-scale neuronal network models using compartmental neuron models is a fairly recent eld, and there are few exam- ples of these types of models that are being developed. The rst major publication that proposed the feasibility in developing large-scale, biologically realistic models of neural systems was printed in 2006 (Markram, 2006). In hippocampus, the rst large-scale dentate gyrus model with compartmental models was published in 2007 by the Soltesz group which contained 50,000 neurons arranged along a one-dimensional line and did not contain topographic connectivity (Dyhrfjeld-Johnsen et al., 2007). The same group published a full-scale model of CA1 in 2013 which contained 350,000 CA1 pyramidal cells and 8 interneuron types arranged along a two-dimensional geometry (Bezaire and Soltesz, 2013). However, though these models included excitatory inputs from aerent regions, they did not incorporate the proper topographic connectivity for the excitatory inputs, e.g. en- torhinal to dentate and CA3 to CA1. These limitations and their lack of CA3 region prevent these models from being connected to construct a complete hippocampal model. In 2015, a large-scale, compartmental model of the cortical column was pub- lished (Markram et al., 2015). These models represent the only large-scale neuronal networks with compartmental neuron models to date. We are proposing a computational framework that is able to integrate the ma- jority of available, quantitative structural and functional information at various levels of organization to generate a large-scale, biologically realistic, neural network model with the goal of representing all of the major neurons and neuron types, and the synaptic connectivity, found in one hemisphere of the rat hippocampus. In this approach, detailed neuron models are constructed using a multi-compartment approach (on the order of hundreds of compartments per neuron) which are then ge- ometrically arranged in three-dimensional space to encompass the entire anatomical extent of the hippocampus, and synaptically connected using proper topographical constraints on the connectivity. The model was developed following the order that activity propagates through the trisynaptic circuit: dentate gyrus, CA3, and CA1. The present model includes the entorhinal cortex and two of three major subelds 10 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS of hippocampus: the dentate gyrus and CA3 subeld. The connectivity from be- tween all regions were constrained using anatomical data to represent a topographic connectivity. Each hippocampal subeld contains their principal neuron and one major interneuron. The excitatory synaptic connections between principal neurons contain AMPA and NMDA receptor channels. Inhibitory synaptic connections are mediated by GABA-A receptor channels. Though much data on the hippocampus exists, the search, quantication, and implementation of these data into a comprehensive model is problematic, considering the large variety of techniques that can be applied to characterize any particular hippocampal feature and the number of features that require investigation. Below, we review the experimental work that has been incorporated into this version of the model and the methods we used to extract and implement the results of those studies. 2.1 Overview of Rat Hippocampus Before a description of the model can be made, some background on the anatomy of the hippocampus and the neuron types used in the model must be presented. In this section, a brief summary of the properties important to the model is made. 2.1.1 Anatomical Description of Hippocampal Structure The three-dimensional structure of the hippocampus resembles a curved cylinder (see Fig. 2.1) with a single homogeneous granule cell layer and a pyramidal cell layer that traditionally has been divided into four subsections (Lorente De N o, 1934; Cajal, 1968). A cross-section of the hippocampus can reveal its principal internal structure in the form of two interlocking C-shapes. One of the structures is known as the dentate gyrus, and the other is known as the cornus ammonis (CA) which is commonly divided into 2 main subelds, the CA3 and the CA1. Though designations for the CA2 and CA4 exist, the CA4 in rat is dened as the hilus in dentate gyrus, and the CA2, due to its small size, had typically been given a lower emphasis in the trisynaptic circuit. However, there is now increasing evidence that the CA2 cannot be ignored (Jones and McHugh, 2011). 11 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.1: Schematic representation of the rat hippocampus. (Left) Location of the hippocampus (yellow) relative to the rest of the rat brain (neocortex removed). (Middle) Depiction of how transverse slices typically are obtained relative to the septo-temporal axis of the hippocampus. (Right) The classical trisynaptic circuit of the hippocampus where the entorhinal cortex (EC) projects its inputs to the dentate gyrus (DG), the dentate projects to the CA3/4 regions, the CA3/4 projects to the CA1/2 regions, and the CA1 provides the output of the hippocampus to other cortical structures. Not shown here are entorhinal projections to the distal dendrites of CA3 (from layer II) and to the distal dendrites of CA1 (from layer III). Granule cells are the principal neurons of the dentate gyrus. The principal neurons of the CA3 and CA1 are called pyramidal cells, but there are distinct dierences between the pyramidal cells of the CA3 and CA1. The trisynaptic circuit describes the feedforward excitatory circuit which begins with the entorhinal cortex and proceeds to the dentate gyrus, CA3, and then the CA1. The entorhinal cortex, which lies outside the hippocampus (but is formally de- ned as part of the hippocampal formation), contributes signicantly to the input of the hippocampus. Layer II cells of the entorhinal cortex send axons to the outer two- thirds of the molecular layer of the dentate gyrus and CA3/4 subeld and synapse along the infra- and supra-pyramidal blades as well as extending into and forming synapses within the stratum lacunosum-moleculare within the CA3 region (Hjorth- Simonsen and Jeune, 1972; Yeckel and Berger, 1990; Witter, 2007). Layer III cells of the entorhinal project to the stratum lacunosum-moleculare within the CA1 region (Yeckel and Berger, 1995). Thus, the entorhinal cortex provides feedforward exci- tatory input to all major hippocampal subelds. However, due to the anatomical organization of the projection, entorhinal activity, in terms of latency, rst activates the dentate gyrus, then the CA3, and nally the CA1. The dentate gyrus is divided into two blades, the upper and lower half of its C-shape. The suprapyramidal blade, also known as the enclosed or dorsal blade, refers to the half of the dentate that is encapsulated by the CA regions. The remain- 12 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.2: Division of the dentate gyrus into the molecular, granule cell, and polymorphic layers. The molecular layer contains the dendrites of the granule cells. The densely packed cell bodies of the granule cells form the granule cell layer. The polymorphic layer is comprised of inhibitory and excitatory interneurons. Granule cell axons collateralize within the polymorphic layer to provide input to the interneurons. Granule cells also send axons through the polymorphic layer to synapse with CA3/4 pyramidal cells ing half is labeled the infrapyramidal blade, also known as the exposed or ventral blade, and the crest refers to the region where the infrapyramidal and suprapyra- midal blades join. Furthermore, the dentate gyrus is divided into three layers. The outermost layer is the molecular layer, the middle layer is the granule cell layer, and the nal layer is the hilus, or the polymorphic layer (see Fig. 2.2). Dentate granule cells project mossy bers into the CA3 which form excitatory connections with the pyramidal cells. The CA3 forms half of the second C-shape, called the cornu ammonis, which comprises a hippocampal cross-section and is enclosed by the suprapyramidal and infrapyramidal blades of the dentate gyrus. The CA1 forms the other half of the C- shape. The CA2 refers to a small region that serves as a boundary between the CA3 and CA1. The CA3 subeld is further subdivided into the CA3c, CA3b, and CA3a 13 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS based on their proximity to the dentate gyrus along the transverse axis. The CA3c is most proximal to the dentate gyrus, and the CA3a is the most distal to the dentate gyrus. The CA3b lies between the CA3c and CA3a. The CA3 pyramidal cells give rise to dense axon collaterals, called Schaer collaterals, that synapse extensively within both the CA3 and CA1 subelds. Due to their projections within CA3, the Schaer collaterals form a strong excitatory recurrent pathway, or associational system, within the CA3 subeld. In addition, the CA3 Schaer collaterals form a strong feedforward excitatory connection with the CA1 subeld. The axons of the CA1 then form the output of the hippocampal formation. 2.1.2 Description of Main Neuron Types in Model Dentate Gyrus The two main dentate neuron types that were included in the model were the granule cells and basket cells. Granule cells are the principal neurons of the dentate gyrus and are situated with their somata in the granule cell layer. Their apical den- drites extend into and span the molecular layer. Granule cells receive the majority of the entorhinal inputs which are excitatory. They receive the entorhinal inputs in a laminar manner, with lateral entorhinal neurons sending axons distally to the outer third of the molecular layer and with medial entorhinal neurons sending axons more proximally to the middle third of the molecular layer. The granule cell axons, while arborizing within the hilus, project a single primary axon, known as the mossy ber, to cells in the CA3 region. Basket cells are interneurons within the granule cell layer that provide inhibitory input to granule cells (Gamrani et al., 1986). Parvalbumin-positive, pyramidal bas- ket cells are a subset of basket cells that have apical dendrites that extend into the molecular layer from which they receive excitatory input from the entorhinal cortex, in the same laminarly organization as granule cells, and basal dendrites that arborize within the hilus from which they receive excitatory input from granule cells (Ser- ess and Ribak, 1983; Ribak and Seress, 1983; Zipp et al., 1989; Ribak et al., 1990; Acs ady et al., 2000). Basket cell axons collateralize extensively in the granule cell layer and the innermost regions of the molecular layer where they form GABAergic synapses with granule cells, primarily on their cell bodies and the initial segments of their axons (Seress and Ribak, 1983). Thus, synaptic arrangements exist that provide the basis for both feedforward and feedback inhibition (Fig. 2.3). 14 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.3: Schematic of the neural circuits in the dentate gyrus model. The feedback inhibition circuit is formed by granule and basket cells. Feedforward inhibition is achieved by the entorhinal activation of basket cells. Mossy cells form the associational system and provide excitatory feedback to granule cells. Mossy cells also disynaptically inhibit granule cells by activating basket cells CA3 The two main CA3 neuron types that were included in the model were the CA3 pyramidal cells and basket cells. The soma of CA3 pyramidal cells are located in the stratum pyramidale. Their basal dendrites extend into the stratum oriens and receive input predominantly from recurrent CA3 axon collaterals. The apical dendrites span several layers. Most proximal to the stratum pyramidale is the stratum lucidum which receives mossy ber input from dentate granule cells. Next is the stratum radiatum which also receives a signicant amount of input from reurrent CA3 axon collaterals. The most distal layer is the stratum lacunosum-moleculare which receives input entorhinal cortex. The entorhinal inputs are a continuation of the collaterals that initial pass through the middle and outer molecular layers of the dentate gyrus and remain similarly laminar. Apical dendrites in the stratum lacunosum-moleculare that are more proximal to the soma receive input from the medial entorhinal cortex, and the apical dendrites that are more distal receive input from lateral entorinal cortex. Parvalbumin-positive, pyramidal basket cells are also present in the CA3. Their morphology is distributed in a manner similar to that of CA3 pyramidal cells. Fur- thermore, they receive inputs from similar neuron types in the same strata as the CA3 pyramidal cells. Basket cells were included as they represent the largest in- terneuron population in CA3 that participate in feedback inhibition circuits (Pelkey et al., 2017). 15 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS 2.2 Large-Scale Model of Rat Hippocampus Though much data on the hippocampus exists, the search, quantication, and implementation of these data into a comprehensive model is problematic, considering the large variety of techniques that can be applied to characterize any particular hippocampal feature and the number of features that require investigation. Below, we review the experimental work that has been incorporated into this version of the model and the methods we used to extract and implement the results of those studies. 2.2.1 Formation of Anatomical Maps Using the terminology that describes a cylinder, the geometry of rat hippocampus can be described as having a "height" that is much larger than its "radius". The "height" can be called the longitudinal axis and is also known as the septo-temporal axis, the dorso-ventral axis, and the rostro-caudal axis for rat. The "radius" of the hippocampus is not used to describe its secondary axis. Rather, it is most common that the two interlocking C's of from the cross-section of hippocampus, the dentate gyrus and CA region, are unfurled along their longitudinal axes to form a pair of two-dimensional surfaces: one representing dentate gyrus and one representing the CA region. Swanson et al. (1978) pioneered the method to \unfold" the hippocampus to create the two-dimensional, attened representations of its subelds (this attened representation can be extended to the entorhinal cortex as well), that well preserves most of the relative anatomical geometry that exists in the original three-dimensional structure. Much of the data involving topography and the distributions of cellular populations is presented using such two-dimensional maps or, in some cases due to technical constraints, along a one-dimensional axis. The axes describing the attened maps can be used to easily project such two-dimensional data onto a proper three-dimensional hippocampal structure. In this work, the longer axis will be referred to as the longitudinal axis, and the secondary axis will be referred to as the transverse axis throughout the thesis. The work of Gaarskjaer (1978) was used in the model to create a more detailed anatomical map of the dentate gyrus due to its inclusion of both length measure- ments and neuron density measurements along the extents of both the suprapyra- 16 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS midal and infrapyramidal blades of the dentate gyrus as well as for the CA3 region. The basket cell distribution within the dentate has been less completely investigated, but ratios of granule cells and interneurons have been reported with septo-temporal and infra- and supra-pyramidal dierences. The granule cell:basket cell ratio in the suprapyramidal blade is approximately 100:1 at the septal end and approximately 150:1 at the temporal end while the ratio in the infrapyramidal blade is 180:1 septally and 300:1 temporally (Seress and Pokorny, 1981). The ratios were interpolated to provide a complete distribution of basket cell densities along the dentate gyrus. The distribution of basket cells in CA3 was taken from the work of Cz eh et al. (2015). The total numbers of the relevant neurons that are in the entorhinal-dentate-CA3 system of the rat are listed in 2.1. Cell Number Lateral entorhinal cortical cells 46,000 Medial entorhinal cortical cells 66,000 Dentate granule cells 1,200,000 Dentate basket cells 4,500 CA3 pyramidal cells 250,000 CA3 basket cells 8,000 Table 2.1: Cell numbers in the large-scale model 2.2.2 Compartmental Models of Neurons The large-scale modeling platform uses a compartmental modeling approach for simulating neuronal activity. Compartmental modeling can preserve the three- dimensional morphological features of neurons by dividing the morphology into dis- crete compartments. The size of the compartments can be changed to control the granularity and complexity of the neuronal representation. Each compartment then represents neural dynamics at a subcellular level and is accomplished by modeling the dynamics as an analog electrical circuit with discrete circuit components. For example, membrane processes, such as active ion channels, can be modeled as vari- able resistors that have a dependence on physiologically relevant quantities such as ion concentration or membrane potential. 17 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS The eld of compartmental models of neurons and the representation of mem- brane processes are well established and will not be elaborated further. The remain- ing text in this section will be used to describe the cell types that were included in the model. 2.2.3 Generation of Dendritic Morphologies Neurons are commonly classied, in part, based on stereotypical morphological features that describe the branching of their dendrites. The diversity of morphologi- cal types has lead many neuro-scientists to investigate the functional role of dierent branching characteristics. The dendritic morphology of neurons has been shown to greatly in uence several factors during input processing such as the propagation and attenuation of postsynaptic potentials and the linear or nonlinear integration of multiple inputs (Krueppel et al., 2011). Given this morphological diversity and its functional importance, the database NeuroMorpho.org was used to obtain three- dimensional reconstructions of granule cell morphologies which were then used to generate the distributions of the relevant parameters using L-Measure (Rihn and Claiborne, 1990; Ascoli et al., 2007; Scorcioni et al., 2008)). The parameters were used by a software tool called L-NEURON to generate unique dendritic morpholo- gies for each granule cell in the network (Ascoli and Krichmar, 2000; Hendrickson et al., 2016). The parameters provide the geometrical points at which a bifurcation can occur, the number of branches, their angles, etc. (see 2.2) This methodology was applied to dentate granule cells. CA3 pyramidal cells were represented by a single representative morphology obtained from the NeuroMorpho.org database. The dendritic morphology of basket cells vary as a function of cell location and the shape of the curvature of the hippocampus at that location. Due to the limited sample size of reconstructions, proper morphologies were not considered for these cell types. Due to the lack of information and to decrease the computational load of the simulations, basket cells for this level of analysis were represented using a single somatic compartment. 2.2.4 Specication of Passive and Active Properties The specication of morphology accounts for some of the passive propagation of electrical activity, i.e., the electrotonic response, but to create a complete model of the dendritic processing of granule cells, the parameters for passive properties 18 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Distribution Mean nMin Std. Dev nMax Soma diameter Gaussian 9.0 2.0 Number of stems Uniform 2 4 Stem initial diameter Gaussian 1.51 0.39 Branching diameter Gaussian 0.49 0.28 IBF branch length Gaussian 10.7 8.4 Terminal branch length Gaussian 10.7 8.4 Daughter ratio Uniform 1 2 Taper ratio Gaussian 0.10 0.08 Rall power Constant 1.5 - Bifurcation amplitude Gaussian 42 13 Tree elev. (narrow) Gaussian 10 2 Tree elev. (medium) Gaussian 42 2 Tree elev. (wide) Gaussian 75 2 Table 2.2: Morphological parameters for granule cells needed to be augmented by active dendritic properties due to voltage-dependent channels also found in the dendritic regions (Krueppel et al., 2011). Dentate Gyrus The discretization of dendritic morphologies into compartments, the embedding of passive and active mechanisms into the compartments, and the simulation of the resulting model was performed using the NEURON simulation environment v7.3 and scripted using Python v2.7 (Carnevale and Hines, 2006; Oliphant, 2007; Hines et al., 2009). The passive and active properties can be set to match experimental data (see Fig. 2.4), much of which has been pioneered by previous groups. The works of these groups are the basis of the granule cell and basket cell models (Yuen and Durand, 1991; Aradi and Holmes, 1999; Aradi and Soltesz, 2002; Santhakumar et al., 2005). The active and passive properties used are summarized in 2.5. The resulting heterogeneous distribution of ion channel densities and the similarly het- erogeneous nature of the morphologies then were able to closely approximate the electrophysiological responses of these neuron types. 19 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Any additional parameter changes that were necessary to properly recreate the original results after being converted to the NEURON simulation environment were performed manually. CA3 The CA3 pyramidal cell models were derived on the work of Hemond et al. (2008). In their work, they identied three dierent classes of CA3 pyramidal cells based on their response due to current clamp input. The bursting type responded with a burst of spikes that began with the onset of the current clamp before terminating and remaining silent. The strong frequency adaptation type responded with a series of spikes with interspike intervals that grew progressively longer after each spike. The weak frequency adaptation type responded with a constant interspike interval during the duration of the current clamp. 2.2.5 Synaptic Density The inputs to a postsynaptic neuron for each possible presynaptic cell type were determined by calculating the pairwise distance between the postsynaptic neuron and all of the presynaptic neurons and computing the connection probability us- ing an appropriate probability distribution. Inputs were randomly selected until a threshold number of inputs, determined by the convergence value for the presynap- tic cell type, was satised. The convergence value denotes the number of aerent connections that a postsynaptic neuron receives from a given presynaptic neuron type and was estimated by considering the number of synapses that were available for a presynaptic neuron type. The determination of the convergence is detailed below. Dentate Granule Cell Hama et al. (1989) quantied the spine density of granule cells by performing analyses on electron microscopy images of the granule cell dendrites. Though Hama et al. did not separately quantify the synaptic density based on the location of the granule cells with respect to the suprapyramidal and infrapyramidal blades of the dentate gyrus, Desmond and Levy (1985) reported signicant dierences between the blades. However, the methodology of Hama et al. was preferred as they used higher 20 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.4: Granule cell electrophysiology. Simulation results are on the top row. Experimental data are on the bottom row. (Top left) When current is injected at the soma, the granule cell responds by ring an action potential with a latency of approximately 100 ms. (Top right) When the current amplitude is just over the threshold required to elicit a second action potential, its latency is approximately 350 ms. This matches experimental data (bottom, reproduced from Spruston and Johnston 1992 21 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.5: Three spiking behaviors observed in recordings of CA3 pyramidal cells and the recreation of those behaviors in simulation. (Left) Bursting type. (Middle) Strongly adapting type. (Right) Weakly adapting type. resolution electron microscopy imaging to perform the counting rather than light microscopy. Using the ratio of the spine densities between the blades as reported by Desmond and Levy, the spine densities for the infrapyramidal blade based on the work of Hama et al. were estimated. Furthermore, Crain et al. (1973) were able to identify asymmetric synapses in only a certain proportion of spines in the distal and middle dendrites, signifying excitatory synapses presumably from perforant path input. Claiborne et al. (1990) characterized the dendritic lengths of axons that lie in the various strata. With a total mean length of 3,478 m for suprapyramidal granule cells and 2,793 m for infrapyramidal granule cells, and a mean of 30% of the dendrites in the middle third of the molecular layer and 40% in the distal third, the mean numbers of synapses available for the lateral and medial perforant path were computed as 2,417 and 2,117 for suprapyramidal granule cells and 1,480 and 1,253 for infrapyramidal cells, respectively. Halasy and Somogyi (1993) reported that 7-8% of synapses on granule cell dendrites in the molecular layer are GABA-immunopositive, and these dendritic synapses represent 75% of the inhibitory synapses on granule cells with the remain- ing 25% located in the granule cell layer. Given total spine counts of 8,695 and 5,533 for suprapyramidal and infrapyramidal granule cells, respectively, the num- ber of inhibitory inputs in the molecular layer would be 652 and 415. This would 22 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS then leave 217 and 138 synapses in the granule cell layer. Basket cells send their axon collaterals predominantly to the granule cell layer, but they are not the only interneurons to do so. Chandelier cells, not included in this study, are hilar cells that also provide inhibitory input to the granule cell layer (Soriano and Frotscher, 1989; Buhl et al., 1994). Given the average number of boutons between basket cells and chandelier cells, 11,400 and 3,800, approximately 75% of the synapses in the granule cell layer should be dedicated for basket cell input (S k et al., 1997). The convergence of basket cells onto granule cells should then be 174 and 110 for the suprapyramidal and infrapyramidal granule cells. However, parvalbumin-positive basket cells only make up 62% of the basket cell population, so the convergence is appropriately shifted to 108 and 68, respectively (Buckmaster and Dudek, 1997). Dentate Basket Cell The total dendritic length of dentate basket cells has been reported to be 4,530 m (Zhang and Buckmaster, 2009). Of this length, the basal dendrites receive input from the granule cells. The proportion of dendrite that lies in the molecular layer (apical dendrites) versus the hilus (basal dendrites) was estimated from measure- ments of surface area with an apical surface area of 7,600 m 2 and a basal surface area of 2,200 m 2 (Vida, 2010). The synaptic density for the dentate basket cell was taken from an estimate made by Patton and McNaughton (1995) which was 1 synapse/m. Another study reported that approximately 10% of synapses in CA1 basket cells are GABAergic (Guly as et al., 1999). Using these data, the mean num- ber of granule cell inputs for a basket cell was estimated to be 915. Assuming that the distribution of basket cell dendrites in the molecular layer was similar to that of granule cells, the number of lateral and medial entorhinal inputs for a basket cell was calculated to be 1,045 and 783, respectively. CA3 Pyramidal Cell Data concerning the spine densities of the CA3 pyramidal cells' dendrites across the dierent strata are scarce. The precedent for estimating the total number of synapses is to use the spine densities that have been measured from CA1 pyramidal cell dendrites. However, the distribution of CA3 pyramidal cell dendrites across the strata has been well characterized. There is a gradient with respect to the transverse axis along which the amount of dendrite has been shown to vary (Fig. 2.6). 23 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS The data was tted using a linear regression to capture this gradient. The number of synapses that each CA3 pyramidal cell received were then estimated by multiplying the synaptic densities from each stratum by the appropriate dendritic lengths. CA3 Basket Cell Basket cells within the CA3 have a dendritic morphology that is similary dis- tributed throughout the CA3 strata as the CA3 pyramidal cells. The basket cells have basal dendrites in the stratum oriens and apical dendrites that extend from the stratum pyramidale to the stratum lacunosum-moleculare and also receive sim- ilar aerent inputs. Therefore, CA3 basket cells participate in both feedforward and feedback inhibition circuits. Similar to the CA3 pyramidal cells, data for the synaptic density of CA3 basket cells are scarce, so CA1 basket cells is used instead. However, not only are spine densities for CA3 basket cells absent, their dendritic distribution is also poorly studied. The following estimates of synapse numbers are based entirely on CA1 basket cell data. They receive 307 and 560 inputs from the medial and lateral entorhinal cortices. They receive 8796 and 3681 inputs in strata radiatum and oriens. 2.2.6 Synaptic Model In the currently described large-scale network, synapses were the exclusive mech- anism through which neuron-to-neuron communication was mediated. The synapse was phenomenologically and deterministically represented so, upon being triggered by an action potential, the synaptic conductance would follow a time-course dic- tated by a double exponential function (Equation 2.1) according to the Exp2Syn mechanism in NEURON which would then be used to calculate the synaptic cur- rent (Equation 2.2). Inhibitory GABAergic synapses for the present model were restricted to the GABA A subtype and also were modeled using the Equations (2.1) and (2.2). NMDA receptor dynamics were modeled using the Equations (2.3) and which includes the voltage-dependent dynamics mediated by the magnesium ion in the denominator. The parameters of synapses between the various cell type pairs are summarized in 2.3 and 2.4. 24 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.6: The distribution of total dendritic length of CA3 pyramidal cells across all the strata and as a function of proximodistal location from dentate gyrus were measured. Linear regressions were performed to quantify the relations. 25 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS I Exp2Syn (t) =g(t)(V m E ions ) (2.1) g(t)/e t 2 e t 1 ; 2 > 1 (2.2) I NMDA (t;V ) = I Exp2Syn (t) 1 + [Mg] 2+ K 0 exp( 0:001zFV RT ) (2.3) Granule Cell Basket Cell Medial perforant path g max (S) 1.17e-5 4.21e-6 1 (ms) 1.05 1.05 2 (ms) 5.75 5.75 Lateral perforant path g max (S) 1.50e-5 4.21e-6 1 (ms) 1.05 1.05 2 (ms) 5.75 5.75 Granule cell g max (S) - 1.13e-4 1 (ms) - 0.1 2 (ms) - 0.49 Presynaptic neurons are in the rst column. Postsynaptic neurons are in the rst row. The reversal potentials for AMPA synapses were 0 mV. Table 2.3: Parameters for AMPA receptors 2.3 Anatomically-Constrained Connectivity With the anatomical map and the neurons dened, the next step in completing the large-scale neural network was to connect the neuron models to each other. Due to the extensive work in extracting the connectivity, it is described in its own 26 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS section here. The connectivity methods used in this work are largely derived from the work of Patton and McNaughton (1995), who compiled an extensive amount of information concerning the connectivity of the dentate gyrus and described a method of distance-based probabilistic connectivity. Connectivity in the model was estimated using probability distributions that were constrained by experimental data. The main concerns were, given the origin of the axon, the postsynaptic region to which the axon is sent, and, once the axon arrives at the postsynaptic region, the spatial distribution of the axon terminals. The next challenge after nding such data was the quantication of the work which was a non-trivial task due to the qualitative manner in which a majority of the works were presented. The key works that were used to constrain each projection are detailed below. Granule Cell Basket Cell Basket cell g max (S) 1.24e-3 - 1 (ms) 0.1 - 2 (ms) 12.35 - Presynaptic neurons are in the rst column. Postsynaptic neurons are in the rst row. The reversal potentials for GABA A synapses were -75 mV. Table 2.4: Parameters for GABA A receptors 2.3.1 Topography of Entorhinal Projection to Hippocampus A major division of the entorhinal cortex is its separation into the medial and lateral regions. The projection of the axons from the entorhinal cortex to the hip- pocampus is termed the perforant path. An important topographical distinction between the medial and lateral entorhinal cortex is that upon reaching the dentate gyrus, the lateral perforant path terminates within the outer third of molecular layer, and the medial perforant path terminates within the middle third (Hjorth- Simonsen and Jeune, 1972; Witter, 2007). This anatomical feature is preserved in the present model by limiting the respective connections to the appropriate regions of the granule cell morphologies. 27 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.7: Summary of the image-processing pipeline used to quantify the connectivity of entorhinal cortical projections to the dentate gyrus. Not all data are shown. (1) The data in the anatomical subject maps are digitized and grouped according to injection location. (2) The maps are projected onto a standard coordinate space. (3) The sets are averaged. (4) The averaged group data are projected onto an average anatomical map. The compass represents the rostro-caudal and mediolateral axes A signicant study by Dolorfo and Amaral (1998) was used to guide our models of the regional mappings from entorhinal cortex to the dentate gyrus. By inject- ing retrograde dye tracers in the dentate gyri of rats, the entorhinal origins of the cells projecting to those injection sites in the dentate were revealed. Injections were performed along the entire septo-temporal, or longitudinal, extent of the dentate, creating a thorough topographic map of the organization of entorhinal-dentate pro- jections. Each injection was performed in a separate rat, and the result of the injection was presented as a grayscale heat map overlaid on a two-dimensional, at- tened representation of the entorhinal cortex of the rat. The grayscale heat map represented the density of entorhinal neurons that projected to the injection site (Fig. 2.7). Quantifying the data to use in the model proved a challenge due to the quali- tative presentation of the results and the dissimilarity of brain shapes for each rat. To address this, a processing work ow was developed (Fig. 2.7). In the rst step, the results of each injection were digitized and the unique shape of the entorhinal cortex was extracted. Next, the individual entorhinal maps were transformed into a standard map, and the standardized maps were grouped based on the region of pro- jection. The grouped, standardized maps were averaged, and the resulting averaged maps were transformed back to a representative entorhinal map that was created by calculating the average of all of the entorhinal maps. The nal grayscale heat maps 28 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.8: Connectivity mapping from the lateral and medial entorhinal cortices (LEC and MEC) to the dentate gyrus (DG). Like colors between the entorhinal cortex and dentate gyrus correspond to the origin and destination of a projection, e.g., red entorhinal regions project to red dentate regions, blue entorhinal regions project to blue dentate regions, etc. were used to determine, given the origin of the neuron in the entorhinal cortex, the probabilistic location within the dentate gyrus to which the axon was connected. The nal mapping depicts a medio-lateral gradient in the lateral entorhinal cortex that projects along the longitudinal axis of the dentate gyrus. In the medial entorhi- nal cortex, there is a dorso-ventral gradient that projects along the longitudinal axis of the dentate gyrus. A summary of the mapping is shown in Figure 2.8. The above study informed the regional mapping of the entorhinal-dentate pro- jection, but it did not describe the morphology of the entorhinal axonal arbors. At the cellular level, Tamamaki and Nojyo (1993) produced some of the few reported single entorhinal neuron perforant path axon terminal eld reconstructions. Based on their work, the septo-temporal extent of the axon terminals was constrained to be in the range of 1 to 1.5 mm. Given that the axon terminals cover the entire transverse extent of the dentate gyrus, the entorhinal axons in the model were rep- resented using Gaussian distributions with a standard deviation 0.167 mm which corresponds to approximately 1 mm being covered within three standard deviations from the center of the axon terminal eld. These distributions determined the con- nectivity between the entorhinal neurons and the granule cells, providing the basis for feedforward excitation in this system. 29 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.9: The staining of the perforant path shows that the axons travel beyond the dentate gyrus and move into the CA3 region in stratum lacunosum-moleculare. Because the distribution of perforant path axons within CA3 is an extension from their location within the dentate gyrus, the entorhinal-dentate topography was applied to the entorhinal-CA3 projection (Fig. 2.9). 2.3.2 Topography of Mossy Fibers The axons of the DG granule cells that project to the CA3 are known as mossy bers. The trajectory of a mossy ber has been shown to follow a very characteristic course through the CA3. From the granule cell's septo-temporal position, the mossy ber travels in a predominantly transverse direction across the CA3c and CA3b, but upon reaching the CA3a, the mossy ber turns toward the temporal pole of the hippocampus. Swanson et al. (1978) reported the aggregate mossy ber trajectories originating from dierent septo-temporal positions along the DG by injecting the granule cells with a tracer (Fig. 2.10). They then estimated the changes in individual trajectories as a function of the granule cell's septo-temporal position. Using 17 equally-spaced sampling points along the transverse axis, the mossy bers originating from a given septo-temporal level were parameterized as the change in septo-temporal position with respect to the septo-temporal origin. The deviation 30 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.10: Anterograde tracer injections within dentate gyrus resulted in a laminar labeling in the CA3 (top left). Injections were performed at various septo-temporal levels and a summary gure was determined. Digital reconstructions of mossy bers revealed that they course through the CA3 in an arc (bottom left). A depiction of how the data were quantied is shown on the right. The osets of a ber relative to the septo-temporal position of the granule cell from which it originated were measured. 31 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS as a function of septo-temporal location was interpolated using a 4th-order polyno- mial function resulting in 17 polynomial functions for each of the samples along the transverse axis. Given a granule cell's septo-temporal position in the DG, 17 points corresponding to the mossy ber's trajectory could be computed. Variability was added to the points by multiplying a factor to the septo-temporal deviation that followed a random uniform distribution between 0.7 and 1.3. The mossy ber was nally computed using a cubic-spline interpolation through the 17 points. In a dierent study, the inter-synapse distance along the mossy bers were re- ported with signicant dierences between the 3 transverse subregions of the CA3: 162 12.6 um in the CA3c, 223 19.3 um in the CA3b, and 345 27.5 um in the CA3a (Acs ady et al., 1998). Using this information, potential synapse locations were derived for each granule cell by creating an algorithm that would create synaptic locations along the generated mossy bers using the inter-synapse distance distri- butions for each of the transverse CA3 subregions. The mossy bers travel through specic layers (or strata) of the CA3 corresponding depending on whether the mossy bers originate from the infrapyramidal blade or suprapyramidal blade of the DG. The suprapyramidal mossy bers travel exclusively through the apical stratum lu- cidum, but the infrapyramidal mossy bers follow a dierent route. Through the CA3c, the infrapyramidal mossy bers travel through the basal layer but cross to the stratum lucidum at the CA3c-CA3b interface. The infrapyramidal mossy bers then remain in the stratum lucidum. This was implemented in our topography by having the infrapyramidal mossy bers by simply stepping the mossy bers from the basal layer to the stratum lucidum once they reach the CA3c-CA3b interface. 2.3.3 Topography of CA3 Associational System Although the previous topographies described excitatory feedforward connec- tions, one of the unique features of the hippocampus compared to the organization of the other cortical structures is the extensive amount of excitatory feedback that is present from the CA3 to itself. The most complete study of the distribution of CA3 axon collaterals was performed by Ishizuka et al. (1990). In their work, horseradish peroxidase was injected into CA3 pyramidal cells that were located at nine separate septo-temporal and transverse sections of the CA3 which resulted in the ipsilateral labeling of their axon collaterals (Fig. 2.11). The data presented the resulting septo- temporal and transverse distributions of the axon collaterals in the CA3, features 32 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.11: Data collected in Ishizuka et. al, 1990 to reveal the topographic organization of the CA3 associational system. The black spot signies the injection site. The dot stipling indicates the relative amount of labeling found in a region. A heavy stipling corresponds to heavy labeling and vice versa. The diagonal hatched marks correspond to the CA3 axons that were traveling from CA3 to CA1 and subsequently did not synapse with CA3 or CA1 pyramidal cells. important to understanding the topographic projection of the CA3 associational sys- tem, which have yet to be repeated in rat with more advanced techniques. To use the data to constrain our model of connectivity, there were three issues that needed to be addressed: The axonal distributions in CA1 were reported qualitatively with two levels of axonal density, the CA1 brain regions for each experiment had dierent shapes, and the experiments did not span the entire CA3 region. We developed a work ow for digitizing the data, standardizing the data, tting the distributions to a parameterized equation that could vary as a function of CA3 pyramidal cell location (Yu et al., 2014). In the digitizing step, masks were drawn over the published gures to label the entirety of the CA3 region, to obtain the map boundaries, and the density levels of the axon collaterals. In the standardization step, a standard space with dimensions much greater than the original data (10,000 x 10,000 pixels) was chosen to ensure the subject map boundaries would be preserved. The ratios between the rows and columns of the individual subject space maps and the standard map were obtained to redraw the data in a standard space. In the 33 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS parameterization step, the parameters for a modied two-dimensional skew normal equation was optimized using the least-squares method. (x) = 1 p 2 e x 2 2 (2.4) (x) = 1 2 [1 +erf( x p 2 )] (2.5) f(x) = 2 ! ( x ! )(( x ! )) (2.6) x 0 =xcos() +ysin() (2.7) y 0 =xsin() +ycos() (2.8) f(x;y) = 4 ! 1 ! 2 ( x 0 1 ! 1 )( 1 ( x 0 1 ! 1 ))( y 0 2 ! 2 )( 2 ( y 0 2 ! 2 )) (2.9) Equation (2.4) is the probability distribution function of a normal distribution, (2.5) is the cumulative distribution function of a normal distribution, and (2.6) describes a skew normal function where controls the skew, is an oset, and ! is the standard deviation. An additional parameter was used to control the rotation of the function using Equations (2.7) and (2.8). Equation (2.9) shows the nal two-dimensional skew normal function. 34 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Figure 2.12: A comparison between the generated CA3 axon distributions and the original data. On the left, the plots are organized by row and column in the same manner as the experimental data, but the plots also included interpolated axon distributions that were not present in the original data. The plots use the same color map ranges to demonstrate the dierences in density based on the septo-temporal location within the CA3. The dimensions of the standard map were reduced to 250 x 250 pixels for the op- timization to decrease computational time. Afterwards, the parameters were linearly interpolated as a function of septo-temporal location within the CA3 to estimate the shape of the axon distribution between the available data. The parameters were then extrapolated beyond the locations covered by available data by using the slope of the parameters obtained by the linear interpolation. In all cases, the slope needed to be multiplied by a scalar factor to reduce the magnitude of the slope so that the skew normal functions did not return abnormal values and so that the mode of the function remained within the CA3 map boundaries. The results of the optimized parameters compared to the original data are shown in Fig. 2.12. For each injection, the skew normal distributions were able to capture the general shape and spread of the original data and preserve the relation between the axon distribution and the CA3 pyramidal cell's proximo-distal origin. 2.3.4 Topography of Projections within the Hippocampal Subelds Upon identifying a neuron and lling it with dye, septo-temporal cross sections of the hippocampus can be made, and the total length of axon that exists in the cross sections can be quantied. Such experiments have yielded histograms of the 35 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS total axon length of a neuron as a function of distance away from the cell body. The histograms were tted to Gaussian functions to extract the standard deviations that parameterize the spatial distributions of axon terminal elds of dentate granule cells and basket cells. The septo-temporal and transverse standard deviations for granule cells were estimated to be 0.152 mm and 0.333 mm, respectively (Patton and McNaughton, 1995; Buckmaster and Dudek, 1999). The corresponding standard deviations for basket cells were estimated to be 0.215 mm and 0.150 mm (Han et al., 1993; S k et al., 1997). The resulting two-dimensional Gaussian distributions described the probability of connectivity between granule cells and basket cells. 2.3.5 Conduction Velocity of Action Potentials The present large-scale model makes a critical assumption about the function of axons in that it assumes that an axon acts merely as a propagator of action potentials from generation near the soma to the end of the terminal. Though studies exist that catalogue the role of axons in modulating synaptic transmission, the present model did not explicitly model axon morphologies and compartments. Rather, axons were functionally represented by incorporating the delay associated with the propagation of the action potential from the soma to the corresponding presynaptic terminal. This was calculated using the physical distance between the soma and the synapse and reported action potential propagation velocities. To account for the time delay between the generation of an action potential and its arrival at the presynaptic terminal where it triggers neurotransmitter release, conduction velocity values, taken from the literature, and the Euclidean distance between the neurons were used. For the entorhinal conduction delays, a bifurcation point was assigned at the crest of the dentate gyrus at a longitudinal location ac- cording to topographic rules described earlier. The distance between the perforation point and the postsynaptic neuron was used to calculate the delay. The conduction velocity was estimated to be 0.3 m/s Andersen et al. (1978). 2.4 Discussion The culmination of all of these steps resulted in the formation of a neuronal network approximating the entorhinal-dentate-CA3 system of the hippocampus in which spiking activity from the entorhinal cortex was projected to the dentate gyrus 36 CHAPTER 2. LARGE-SCALE MODEL OF RAT HIPPOCAMPUS and spatially distributed according to the topographical rules that determined con- nectivity, and the activity was converted into postsynaptic potentials (PSPs) in the corresponding neuron models based on the equations that specied the appropri- ate synaptic receptor channel dynamics. The PSPs were propagated through the dendritic morphologies, interacting with PSPs arising from other input timings and activating voltage-gated ion channels, resulting in a nonlinear transformation of the PSPs as they traveled towards and were integrated at the soma. Upon reaching a threshold, which is consequently determined by the ion channel composition, the soma generates an action potential which activates additional synapses based on the local topography of network and activates a similar sequence of events for all postsynaptically coupled neurons. The spatial distribution of action potential events throughout the hippocampal network and their temporal intervals form the basis of the large-scale population dynamics that are analyzed in this thesis. This large-scale model represents one of only a few large-scale models with bi- ologically realistic morphologies that have been published. In hippocampus, the rst large-scale dentate gyrus model with compartmental models was published in 2007 by the Soltesz group which contained 50,000 neurons arranged along a one- dimensional line and did not contain topographic connectivity (Dyhrfjeld-Johnsen et al., 2007). The model had 4 cell types. The same group published a full-scale model of CA1 in 2013 which contained 350,000 CA1 pyramidal cells and 8 interneu- ron types arranged along a two-dimensional geometry (Bezaire and Soltesz, 2013). 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LARGE-SCALE MODEL OF RAT HIPPOCAMPUS Mechanism Granule Cell GCL Inner Third Middle Third Outer Third Basket Cell C m (m=cm 2 ) 9.8 9.8 15.68 15.68 15.68 1.4 Ra ( cm) 210 210 210 210 210 100 Sodium (S=cm 2 ) 0.84 0.126 0.091 0.056 - 0.12 Slow delayed rectier K + (S=cm 2 ) 6e-3 6e-3 6e-3 6e-3 8e-3 - Fast delayed rectier K + (S=cm 2 ) 0.036 9e-3 9e-3 2.25e-3 2.25e-3 0.013 A-Type K + (S=cm 2 ) 0.108 - - - - 1.5e-4 L-Type Ca 2+ (S=cm 2 ) 2.5e-3 7.5e-3 7.5e-3 5e-4 - 5e-3 N-Type Ca 2+ (S=cm 2 ) 7.35e-4 2.2e-3 7.35e-4 7.35e-4 7.35e-4 8e-4 T-Type Ca 2+ (S=cm 2 ) 1e-3 4e-4 2e-4 - - 2e-6 Ca-Dependent K + (S=cm 2 ) 1e-3 4e-4 2e-4 - - 2e-6 Ca- and V-Dependent K + (S=cm 2 ) 1.2e-4 1.2e-4 2e-4 4.8e-4 4.8e-4 2e-4 Leak (S=cm 2 ) 2.9e-4 2.9e-4 4.6e-4 4.6e-4 4.6e-4 1.8e-3 Membrane Time Constant (ms) 10 10 10 10 10 10 Steady-State Intracellular Ca 2+ (mM) 5e-6 5e-6 5e-6 5e-6 5e-6 5e-6 Table 2.5: Distribution of passive and active properties in dentate neuron models 44 Chapter 3 Spatio-Temporal Patterns of the Large-Scale Model 45 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL 3.1 Introduction The initial hypothesis that drove the work was if a neuronal network was con- structed to scale, then emergent properties would appear; the result of the neuronal network would be greater than the sum of its parts. The concept of emergence is almost as nascent as the concept of the neuron, rst being used in scientic litera- ture in 1875 (Goldstein, 1999). However, due to its colloquial use, a strict denition has not been accepted, and its meaning is still an on-going conversation to this day. De Wolf and Holvoet (2005) proposed a semantic denition of emergence: \A sys- tem exhibits emergence when there are coherent emergents at the macro-level that dynamically arise from the interactions between the parts at the micro-level. Such emergents are novel w.r.t the individual parts of the system." From this denition, macro-level refers to the system as a whole, and the micro- level refers to the individual entities that comprise the system. Emergents are generally dened as properties, behaviors, structures, or patterns that lie at the macro-level and arise due to interactions at the micro-level. Emergents must be novel in that the collective behavior is not readily understood from the observation of the individual parts. Coherence, in one denition, refers to a correlation between components that \. . . separate lower level components into a higher level unity." Emergent properties that are described in the thesis will be assessed using this de- nition. An information-theoretic denition of emergence has also been proposed but has not been applied to the work. In modeling, emergence is a concept that is in direct contrast to reductionism which states that a system can be reduced to the sum of its parts. Under emergence, a system cannot be studied by observing the individual parts but by studying each of the parts in the context of a system as a whole. In conceptualizing the computational framework upon which the large-scale model was built, the neuron was considered the micro-scale with respect to the denition above; it is the part, the individual. The hippocampus was considered the macro-scale, the whole. Upon the construction of the large-scale model, it was uncertain what the emer- gents could be or how the system could be perturbed to observe these emergents. Therefore, an approach commonly used in system identication was used to dene the input. The large-scale model was treated as a black box and a white-noise input was generated to uniformly test the response to a large range of frequencies. 46 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL The chapter is divided into two sections. First, the dentate response to white- noise input from the entorinal cortex will be presented. Then, the CA3 response will be described. 3.2 Emergence of Spatio-Temporal Clusters For our initial simulation study to characterize dentate granule cell responses to entorhinal input, the dentate network was driven using independent, identically dis- tributed Poisson point processes which generated inter-stimulus intervals (ISIs) at a mean frequency of 3 Hz. The 3 Hz was chosen to represent a baseline of spontaneous activity for the entorhinal cortex. A Poisson process was used to generate ISIs that would perturb the synapses at a broad range of frequencies approximating a white noise input with which to investigate the entorhinal-dentate system. The resulting entorhinal activity was uncorrelated spatially and temporally. Once the inputs were generated, the same ISIs were used to perturb the network for all of the simulations that are described in this work. Initial simulations were performed at the full number of neurons in the entorhinal cortex and dentate gyrus (Table 1). To explore the various phenomena that were observed at the full scale network, subsequent simulations were performed at a reduced scale with a tenth of the number of neurons to decrease the simulation times. Simulations performed at the reduced scale continued to exhibit the relevant phenomena seen at the full scale. Simulations were performed on a computing cluster (hpc.usc.edu) using 125 dual quad-core 2.33 Ghz Intel-based nodes with 16 GB of RAM per node for a total of 1,000 compute cores and 2 TB of RAM. The nodes were connected by a 10G Myrinet networking backbone. At full scale with 112,000 entorhinal cortex cells, 1,200,000 granule cells, and 4,000 basket cells with a simulation time of 4,000 ms, simulations required approximately 87 hours to complete. Reduced scale simulations required approximately 9 hours. 3.2.1 Spatio-Temporal Clusters as an Emergent Property The spiking activity of the network is depicted using raster plots with time on the x-axis. The entorhinal activity is sorted by cell ID which demonstrates the uncorrelated properties of its spatio-temporal ring pattern. For neurons in the 47 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.1: Simulation result for topographically constrained entorhinal-dentate network with feedback inhibition at full scale with 1,200,000 granule cells. (Top) Entorhinal activity was generated by homogeneous Poisson process and was spatially and temporally uncorrelated. Medial entorhinal activity is in red and lateral entorhinal activity is in blue. (Middle) Granule cell activity displays spatiotemporal clustering with local regions of dense activity. (Bottom) Basket cell activity was also clustered, being driven, and activated in a feedback manner by granule cell activity. dentate, the longitudinal location of a spike within the dentate is plotted on the y-axis. The basket cell activity is plotted similarly. The initial expectation was that spatio-temporally uncorrelated input from the entorhinal cortex would result in spatio-temporally uncorrelated output of granule cells. Contrary to that hypothesis, the dentate system responded with localized regions of spatially and temporally dense activity that were interspersed with periods of reduced activity. The dense activity spanned a spatial extent of 1-3 mm and persisted for approximately 50-75 ms with periods of reduced activity lasting 50-100 ms. Regions of dense activity were called \clusters" (Fig. 3.1). The clustered activity was not a transient response but was the steady state response after approximately one second of simulation had passed. The transient response was characterized by synchronized, oscillatory behavior (not shown). Dur- ing this phase, the entire extent of the dentate gyrus alternated between periods of activity and inactivity before evolving into clustered activity which persisted indef- initely for the rest of the simulation. 48 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.2: DENCLUE analysis of granule cell-spiking activity which identies clusters based on the local density. Each identied cluster is plotted with a separate color. The cluster analysis shows that clusters are organized by the transverse axis in addition to the longitudinal and temporal axes. Clusters persisted even after the network was scaled to one tenth of the full scale (Fig. 3.3). The mean ring rate of the granule cells was 1.28 Hz. A density-based clustering algorithm, DENCLUE 2.0 (Hinneburg and Gabriel, 2007), was used to detect clusters for more in-depth characterization (Fig. 3.2). The mean number of spikes that contributed to each cluster in the reduced network was 156. Clusters had a mean temporal width of 18 ms and spatial extent of 0.90 mm. The mean density of the clusters was 12 spikes msmm 2 , and the inter-centroid time between the clusters was 11 ms. Clusters were not formed due to bursts of spikes by individual granule cells. Rather, clusters were a result of increased population activity. The clusters are an expression of a spatio-temporal correlation in the system. To test the robustness of this correlation, spatio-temporal correlation maps were computed in which the cross-correlations between cell pairs from the network were computed and the longitudinal distance between the cells was used to sort the cor- relations (Fig. 3.3). The spike times were sorted using a time bin of 5 ms, and the resolution of the cell distance was set to 0.05 mm. A uniform random sampling of 10,000 cells was performed, and the correlations were calculated with all unique cell pair combinations from this sampling. The resulting maps capture the average 49 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.3: Raster plots of activity from an entorhinal-dentate network at 1/10th of the full scale while cumula- tively removing extrinsic and intrinsic sources of inhibition as well as topographical connectivity constraints. The corresponding correlation maps are to the right of each raster plot. (a) Clustered activity persists in a network that is scaled down. The correlation map exhibits spatial and temporal correlation that matches the size and extent of the dentate clusters. (b) Basket cells are removed as a source of extrinsic inhibition, but clustered activity re- mains. (c) Extrinsic and intrinsic sources of inhibition are eliminated by reducing the AHP amplitude and removing basket cells. Background activity increases, but clusters are still present. (d) In the absence of basket cells and AHP, entorhinal cortical cells and granule cells are randomly connected to eliminate topography. It is only after topographical connectivity constraints are removed that the clusters disappear features of the clusters that are apparent visually and veries the existence of a spatio-temporal correlation in the dentate population which is not present in the entorhinal population. 3.2.2 Removal of Extrinsic and Intrinsic Sources of Inhibi- tion To identify the anatomical properties that were responsible for the spatio-temporal correlation, physiological components were successively removed until the clusters were substantially modied or no longer detected. The initial hypothesis was that the clusters were formed due to sources of inhibition. First, basket cells were consid- ered as an extrinsic source of inhibition due to the inhibitory feedback they provide 50 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL to granule cells. The dynamics of inhibition and spatial distribution of bers should generally correspond to the inter-cluster timing and cluster size. The removal of basket cells from the network increased the level of background activity or \noise" in the system but changed the shape of clusters only subtly (Fig. 3.3). Next, an intrinsic source of inhibition was investigated, the afterhyperpolarization (AHP) of granule cells. The reduced spiking during the AHP could contribute to the reduced inter-cluster activity. The amplitude of the fast AHP of the granule cells was reduced by half and the half-height width was reduced by one-third by removing the calcium- dependent potassium conductances (the small conductance calcium-activated potas- sium channel, SK, and the large conductance calcium-activated potassium channel, BK). In the absence of basket cell inhibition and the AHP, a greater amount of noise was present making clusters more dicult to detect visually, but the correla- tion maps continued to demonstrate the presence of a substantial spatio-temporal correlation (Fig. 3.3). Both of these simulations indicated that clusters were not a result of neurobiological mechanisms that contribute to the inhibition of granule cell activity. 3.2.3 Topographic Connectivity as a Source of Spatial Cor- relation The next simulation sought to eliminate topography. With topography, entorhi- nal cortical cells exhibited an axon terminal eld that spanned a longitudinal extent of 1 mm, constraining their postsynaptic targets to granule cells within this extent. Topography was removed by using a random connectivity; entorhinal cortical cells were allowed to synapse with any dentate granule cell with equal probability. Each neuron received the same number of connections as before, but the potential ori- gin of the inputs was entirely random. The result of removing the topography of the entorhinal projection, in the absence of basket cells and a reduced AHP, was the elimination of clustered activity (Fig. 3.3). This result demonstrated that the spatial correlation and clusters are primarily dependent on the topography of the entorhinal-dentate system. Given that both the spatial extent of the clusters and the span of the axon terminal eld were approximately 1 mm, it was presumed that the axon terminal eld could be acting as a spatial lter that controlled the spatial correlation in the population activity. To test this hypothesis, the axon terminal eld of the entorhinal- 51 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.4: Eect of increasing the axon terminal eld extent in the septo-temporal direction. Raster plots of the granule cell activity are in the left column. Correlation maps are plotted on the right. As the terminal eld extent is increased, the cluster size is increased, and this is further re ected by the expanding spatial extent of the correlation. 52 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.5: Granule cell activity if the time course of the entorhinal-dentate EPSP is expanded. Fewer clusters were seen in the granule cell activity, but the temporal width of the clusters increased. This is also seen in the correlation map. dentate projection was varied from 0.5 mm to 5 mm (Fig. 3.4). Basket cells were not included in these simulations, and the granule cell models were restored to their original ion channel composition, i.e. amplitudes of AHPs were not reduced. The simulations veried that the extent of the axon terminal eld determines the spatial extent of the clustered activity. However, the temporal extent of the clusters was not aected by the axon terminal eld. 3.2.4 Sources of Temporal Correlation Though extrinsic and intrinsic sources of inhibition, i.e. basket cells and AHP, were not found to be the source of clusters, they were able to modulate the ap- pearance of the clusters and the temporal aspect of the correlation maps (Fig. 3.3). In particular, they aected the regions of negative correlation that appeared on ei- ther sides of the positive-correlation lobes. However, neither processes signicantly changed the width of the lobes. We hypothesized that by changing the temporal properties of the EPSP, the temporal width of both the clusters and correlation could be manipulated. The second time constant 2 of the double exponential equation that describes the synaptic dynamics primarily in uences the width of the resulting PSP. To extend the temporal width of the PSP, 2 was increased by a factor of ten. The rst time constant 1 primarily aects the rise time of the PSP and was not manipulated for this simulation. In order to maintain similar levels of activity, the amplitude of the EPSP was altered such that its integral remained unchanged with respect to the original, experimentally-based EPSP waveform. The 53 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL population responded with wider but fewer clusters that appeared with a dierent spatio-temporal pattern than the control. The simulations veried that the tempo- ral extent of the clusters and correlation were proportional to the temporal extent of the EPSP (Fig. 3.5). 3.2.5 Modulation of Clusters via Interneurons The model of the dentate gyrus used in the present studies was comprehensive, particularly with respect to the morphologies of granule cells, topographic orga- nization of perforant path bers, biophysical properties of granule cell bodies and dendrites, relative numbers of granule cells and basket cell inhibitory interneurons, total numbers of neurons included in the model, and other prominent features known to be characteristic of the hippocampal entorhinal-dentate system in the rat. In ad- dition, and relevant to the current volume, in the present model we have investigated the role of basket cells forming feedforward and feedback inhibitory pathways, and mossy cells contributing feedback from the hilus to granule cells and inhibitory in- terneurons (Hendrickson et al., 2015, 2016). Positive feedback was mediated by mossy cells which monosynaptically provided excitatory input to granule cells but also disynaptically provoked an inhibitory eect by activating basket cells. The role of feedback inhibition was investigated by increasing the strength of basket cell activation by granule cells and observing the granule cell activity. As the coupling strength was increased, the clustered activity began to align temporally (Fig. 3.6). At higher coupling strengths, the aligned clusters became joined into a single vertical band, and the granule cell activity appeared as an oscillation between periods of activity and inactivity. The oscillation frequency of the synchronous activity was evaluated using Fourier analysis which exhibited a primary peak at 22 Hz and, at the higher coupling strengths, a resonant peak at 45 Hz. Feedforward inhibition was investigated by increasing the strength of basket cell activation by entorhinal cortical cells and observing granule cell activity. Alone, feedforward inhibition acted to dampen activity and eventually ceased any granule cell activity from occurring at higher coupling strengths (gure not shown). The interactions between feedforward and feedback inhibition then were explored (Fig. 3.7). 54 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.6: Results of increasing the strength of feedback inhibition. As feedback inhibition increased, synchrony increased and oscillatory behavior became apparent. The Fourier transforms depict a strengthening of oscillation at 22 Hz. 55 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.7: Results of increasing the strength of feedforward inhibition while the network was in an oscillatory state due to feedback inhibition. As feedforward inhibition was increased, the oscillation frequency increased, but beyond a certain level, oscillatory activity was dampened, and the granule cells begin to exhibit clustered activity 56 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL The strength of feedback inhibition was set such that granule cells exhibited a 22 Hz oscillation. In this state, the entorhinal-basket cell coupling strength was steadily increased. At lower coupling strengths, the 22 Hz oscillation was shifted toward higher frequencies, but beyond a certain level, the peak oscillation was weakened and eventually eliminated, reverting to clustered activity. 3.2.6 Clusters as a Higher Level of Functional Organization The large scale model introduced here oers one of the few insights into in vivo population dynamics by incorporating an anatomically-derived connectivity and large numbers of detailed, biologically realistic neuron models. The emergence of spatially and temporally nite clusters in the spiking activity, indicative of a spatio-temporal correlation in the network, is a unique discovery that has yet to be validated experimentally but remains an hypothesis highly suggestive of how population activity could be organized at a higher level in the hippocampal dentate system. Topography, terminal eld size, and synaptic communication are integral properties that underlie all neural systems and were found to signicantly in uence the shape of the clusters. The fundamental nature of topography, terminal elds, and synaptic transmission suggests that clusters could be found in many neural systems and that clusters may act as a basic unit of neuronal activity at the population level. Though the correlation maps were able to be computed through averaging, the magnitude of the correlations were relatively low for all simulation cases. Clusters could be visualized or detected only because of the scale of the network in terms of the geometry and the number of neurons. A network that is restricted in scale to that of a typical 400 m hippocampal in vitro slice would not exhibit clustered activity because the spatial extent of the system would be insucient to observe the spatial boundaries of a cluster which are approximately 1 mm. Without having constructed a network of the magnitude reported here, the number of observable spikes would not be sucient to calculate the correlations. As described above, the clusters themselves are sparse, with an average of 156 spikes per cluster. This is not to suggest that the clusters represent a minor aspect of granule cell activity but rather to emphasize the phenomena that would otherwise be overlooked had a large-scale network not been constructed. 57 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL 3.3 Propagation and Transformation of Clusters By extending the model beyond the dentate gyrus and incorporating the CA3 subeld, the propagation and transformation of the granule cell clusters can be in- vestigated (Yu et al., 2015). Because the topography of the projections between each of the subelds is unique, multiple connectivity schemes will be tested and compared. The mossy bers, which are composed of the axonal projection from the dentate granule cells to the CA3 pyramidal cells, originate from a large presynaptic population and has a low divergence on the postsynaptic population, i.e. each gran- ule cell contacts 11-15 CA3 pyramidal cells (Acs ady et al., 1998). Conversely, the entorhinal projection to CA3 have high divergence values similar to the entorhinal- dentate projection. Within the CA3 region, there is an extensive, widely diverging associational system which provides strong recurrent excitatory activation of CA3 pyramidal cells which is balanced by strong feedback inhibition via the activation of interneurons. This section will describe the individual and successive addition of the aerent inputs to the CA3 and their interaction with the associational sys- tem. Thus, the transformation of the granule cell clusters due to these additional topographic connectivity schemes will be explored. 3.3.1 Entorhinal-CA3 Perforant Path Projection Though clustered activity was found in the dentate granule cells, it was not known whether the clusters were a unique condition due to the specic electrophys- iological properties of granule cells in combination with a topographically-organized connectivity. The magnitude of the afterhyperpolarization on cluster emergence had been tested in granule cells and found to aect the background activity between clusters, but it did not eliminate the spatio-temporal correlations of the network activity. Because the axons of the entorhinal-CA3 perforant path projection were an en passante extension of the entorhinal-dentate perforant path projection, we had hy- pothesized, given that dierences in cellular biophysics did not entirely eliminate spatio-temporal correlations in the network, that clusters would form in CA3 pyra- midal cells when stimulated by the entorhinal-CA3 perforant path. In the absence of extrinsic inhibition, the CA3 pyramidal cell network activity became organized into clusters (Fig. 3.8). However, there were high levels of background activity 58 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.8: (Top) Raster plot of CA3 activity in response to entorhinal input. (Bottom) CA3 activity ltered using spatio-temporal correlation to emphasize clusters. between clusters. The spatio-temporal correlation analysis veried the presence of clusters, though the peak correlation was lower. A two-dimensional correlation was computed between the spatio-temporal correlation and the CA3 pyramidal cell spik- ing activity to lter the noise and reveal the locations of the clusters. The high level of background activity was attributed to the more depolarized resting membrane potentials of the CA3 pyramidal cells compared to granule cells and the unique properties of the three CA3 pyramidal cell types, i.e. bursting, adapting, and non- adapting. Bursting CA3 pyramidal cells generated more spikes, contributing to the background activity. Furthermore, the adapting and non-adapting CA3 pyramidal cells had lower afterhyperpolarization amplitudes than granule cells. 59 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL 3.3.2 Dentate-CA3 Mossy Fiber Projection Mossy ber projections from dentate granule cells are characterized as en pas- sante axonal processes with trajectories that span the proximodistal axis of the CA3. A ber stays within the same septo-temporal position as the granule cell from which it originates for approximately two-thirds of the proximodistal axis. In the most distal third of the CA3, the ber turns downward toward the temporal pole. Spiking dentate activity was propagated to the CA3 through the mossy bers. We had previously shown that the mossy ber system preserves the spatio-temporal correlations present in the granule cells, e.g., if there are no spatio-temporal correla- tions in the granule cell activity, then the mossy ber activation of CA3 pyramidal cells will generate CA3 network activity with no spatio-temporal correlation. The dentate activity used to activate the CA3 network was organized into spatio- temporal clusters of activity (Fig. 3.9). There were 486 clusters that appeared over 4 seconds of activity. There was an average of 447 spikes per cluster. The average temporal width was 19 ms, and the average longitudinal span was 1.36 mm. A dentate granule spike did not spike more than once per cluster, so each spike in a cluster came from a unique granule cell. The average ring rate of the granule cells was 0.62 Hz, so a single granule cell could only contribute to 1 to 3 clusters. The resulting CA3 pyramidal cell activity preserved, to a large extent, the spatio- temporal pattern that was present in the dentate gyrus (Fig. 3.9). Specically, some of the clusters, approximately 13% as identied by the DENCLUE algorithm, that appeared in the dentate gyrus also appeared in the CA3. The CA3 clusters had an average delay of 9 ms, as computed by cross-correlation, compared to the dentate gyrus but also had a slightly dierent shape. In addition to a roughly circular cluster shape, the CA3 clusters also exhibited a "tail" which extended downwards towards the temporal pole. Some clusters in the dentate gyrus did not appear in the CA3 if the density of the dentate clusters was insucient to cause a spiking response in the CA3. When the synaptic weight of the dentate-CA3 projection was increased by a factor of ve, then approximately 80% of dentate clusters were successfully expressed in the CA3 (Fig. 3.9). These CA3 clusters also had an average delay of 9 ms. The successful preservation of the dentate spatio-temporal pattern in the CA3 was due to the predominantly lamellar trajectory of the mossy bers. Dentate activity at a longitudinal position would be propagated to a corresponding area in 60 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.9: (Top) Clusters from dentate granule cells in response to entorinal input. (Middle) Clusters from CA3 pyramidal cells in response to solely mossy ber input at normal strength. (Bottom) Clusters from CA3 pyramidal cells in response to solely mossy ber input at 5 times strength. 61 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.10: (Top) Clusters from dentate granule cells in response to entorhinal input. (Bottom) Clusters from CA3 pyramidal cells in response to combined entorhinal and mossy ber input at normal strength. the CA3 at a similar longitudinal position. Thus, the dentate clusters would appear as clusters in the CA3. The downward turn of the mossy bers in the distal portion of the CA3 caused the downward "tail" of the CA3 clusters. Another interpretation of these results is that the mossy bers largely preserve the spatio-temporal correlations that are present in the dentate gyrus. 3.3.3 Combined Entorhinal- and Dentate-CA3 Projections In this set of simulations, entorhinal activity was propagated to the CA3 both monosynaptically and disynaptically. The direct perforant path projection from en- torhinal cortex to CA3 was a monosynaptic projection. To clarify the disynaptic projection, the same entorhinal activity was propagated to the dentate gyrus. The dentate gyrus then transformed this activity and propagated the transformed ac- tivity to the CA3 via the mossy bers. Thus, the CA3 received both the "white" entorhinal input and the clustered/spatio-temporally correlated dentate input. 62 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL The spatio-temporal pattern of the CA3 due to the simultaneous inputs exhibited a combination of the characteristics seen in the responses due to each input individ- ually (Fig. 3.10). First, the dentate clusters were preserved in CA3 due to the mossy bers. Second, there was prominent background activity as seen by the response to the perforant path projection. However, the key dierence is that whereas the response due to the mossy bers resulted in 100% of the clusters being preserved, the response due to the combined entorhinal- and dentate-CA3 projection resulted in all the dentate clusters being present in the CA3. The delay between dentate and CA3 clusters increased from 9 ms, as seen from clusters due solely by mossy ber input, to 15 ms. Furthermore, the CA3 cluster "tails" became thicker resulting in generally larger clusters. Finally, some of the patterns seen in the response to the perforant path can also be observed in the combined response. In the context of spatio-temporal pattern propagation, these results demonstrate that the perforant path projection acts to enhance the pattern carried by the mossy bers to enable them to better perpetuate the dentate pattern in CA3. A spatio-temporal cross-correlation was performed between the dentate activity and the CA3 activity in response to entorhinal and dentate input. The maximum cross-correlation was 0.0047. In contrast, the maximum cross-correlation between the CA3 activity in response to entorhinal input and the CA3 activity in response to entorhinal and dentate input was 0.00038 which is 8% of the dentate-CA3 cross- correlation. The above analysis indicates that the CA3 response to entorhinal and dentate input expresses both of the spatial-temporal patterns due to the eect of the individual entorhinal and dentate inputs and in approximately the ratio measured by the cross-correlation. 3.3.4 Associational System In the next set of simulations, the CA3 associational system was included. The CA3 was activated by both entorhinal and dentate input, and the synaptic strength of the associational input was varied starting from the synaptic strength that resulted in EPSP amplitudes that matched experimental data. The synaptic strength was decreased logarithmically to observe how the associational system would transform the spatio-temporal patterns elicited by the combined entorhinal and dentate input. These simulations do not yet have basket cells present. 63 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.11: Transformation of clusters due to entorhinal and mossy ber input by associational system at varying synaptic strengths and no basket cells. The CA3 activity with no associational system, i.e., associational synaptic strength of zero, was used as the base state for the comparisons. Without associational in- put but including the entorhinal and dentate inputs the CA3 network responded with clusters that were initiated by the dentate input and a background level of activity attributed to the entorhinal input (Fig. 3.11). Adding a very weak associa- tional input, i.e., a synaptic strength that is 1000x smaller than the experimentally- constrained value, the clusters remain at the same spatio-temporal position, but the background level of activity increases. As the strength of the associational input is increased, a rhythmicity develops between clusters. However, the clusters remain a part of the spatio-temporal pattern and have a higher density of spiking than the inter-cluster rhythmicity. The rhythmicity does not represent a synchronization across the entire network. Rather, the rhythmicity tends to appear as continuing "echoes" or wave fronts of a cluster which are generated after the appearance of a cluster. 3.3.5 Interactions between the Associational System and Feedforward/Feedback Inhibition In the following simulations, pyramidal basket cells had been added. Pyramidal basket cells have dendrites in all CA3 strata and receive entorhinal input through the 64 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL stratum lacunosum-moleculare, granule cell input from the stratum lucidum, and CA3 pyramidal cell input from the stratum radiatum and stratum oriens. Using a CA3 pyramidal cell network that received entorhinal, dentate, and associational input, the eect of feedforward inhibition, via entorhinal and dentate input, and feedback inhibition, via the associational system, and their interactions were inves- tigated. Feedforward Inhibition Feedforward inhibition is known to dampen activity, and we had previously shown that it could also reduce oscillations and synchrony. Thus, we investigated how the dierent feedforward inhibition circuits, activated by entorhinal and dentate activity, aected the highly active CA3 network that had resulted from associational connections at full synaptic strength. In other words, the following sections inves- tigated how dierent feedforward inhibition circuits interacted with an excitatory associational system. The networks simulated in this section did not yet have feed- back inhibition, i.e., basket cells did not receive input from CA3 pyramidal cells. Entorhinal Feedforward Inhibition At an experimentally-constrained synaptic strength, the entorhinal feedforward inhibition prevented any CA3 pyramidal cell activity from being generated (Fig. 3.12). The synaptic strength was then decreased until CA3 pyramidal cell activity could be seen. At lower synaptic strengths, i.e., at 0.05 and 0.1 times the origi- nal strength, the network exhibited an oscillatory behavior in which a higher 70 Hz rhythmicity was modulated by a lower 14 Hz rhythmicity. When the synap- tic strength was reduced to 0.15 times the original strength, the 14 Hz and 70 Hz rhythmicity disappeared, and a spatio-temporal pattern consisting of asynchronous clusters appeared. Above 0.15, i.e., at 0.2 and above, the feedforward inhibition would eliminate all CA3 activity (not shown). These results demonstrate that the balance of entorhinal feedforward inhibition and the associational system can sig- nicantly change the state of the network. The 14 Hz and 70 Hz rhythmicity arose due to the interaction between feedfor- ward inhibition and the excitatory feedback provided by the associational system. To demonstrate this, a spectral analysis of the CA3 activity was performed for three conditions: 1) both the associational system and entorhinal feedforward inhibition 65 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.12: CA3 activity due to entorhinal and mossy ber input, the associational system, and varying levels of feedforward inhibition mediated by the perforant path Figure 3.13: Spiking rhythmicity due to the interactions between the associational system and the entorhinal-basket cell feedforward inhibition circuit. Raster activity is on the left, and Fourier transforms are on the right. 66 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.14: CA3 activity due to the interactions between the associational system and feedforward inhibition mediated by mossy bers. were present, 2) the associational system was present but the entorhinal feedfor- ward inhibition was removed, and 3) the associational system was disconnected but entorhinal feedforward inhibition was present (Fig. 3.13). The analysis revealed that the associational system was the source of the high frequency activity, and the entorhinal feedforward inhibition was the source of the low frequency activity. Dentate Feedforward Inhibition Given a densely active CA3 network, the mossy ber feedforward inhibition resulted in absence clusters, meaning that there were spatially and temporally nite regions in which there was an absence of activity (Fig. 3.14). There were two main causes to this striking dierence in pattern. First is the fact that the dentate input was clustered. Second is the lamellar nature of the mossy ber projection. In terms of activation of CA3 pyramidal cells, we had shown that this resulted in a continuation of the clustered spatio-temporal pattern in CA3 pyramidal cells. In addition, this resulted in a similar clustered spatio-temporal pattern in the basket cells. The basket cells would then inhibit the CA3 pyramidal cells following the same spatio-temporal pattern. The absence clusters appeared in the same locations as the clusters seen in section 3.2. Reducing the synaptic strength of the mossy 67 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.15: Basket cell activity within CA3 due to the interactions between the associational system and feedforward inhibition mediated by mossy bers. ber-basket cell projection decreased the spatial and temporal size of the absence clusters. Interaction of Entorhinal and Dentate Feedforward Inhibition In this set of simulations, both feedforward inhibition circuits were activated (Fig. 3.16). With entorhinal feedforward inhibition at 0.05 and 0.1 and without dentate feedforward inhibition, i.e., the left-most column, a rhythmicity appeared. When the dentate feedforward inhibition was added, the absence clusters that it generated were superimposed on the rhythmicity caused by the entorhinal feedfor- ward inhibition, and the relatively synchronized activity became destabilized. At a lower entorhinal feedforward inhibition of 0.05 and between the absence clusters, the 70 Hz rhythmicity was still present. At an entorhinal feedforward inhibition of 0.1, the 70 Hz rhythmicity between the absence clusters became diminished. When the entorhinal feedforward inhibition was set to 0.15 and without dentate feedforward inhibition, there was no global rhythmicity but rather an asynchronous clustered activity with a low ring rate. The presence of dentate feedforward inhi- bition further imposed absence clusters upon this pattern. 68 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.16: CA3 activity due to interactions between the associational system and feedforward inhibition by both the perforant path and mossy bers. In general, the combination of the two feedforward inhibition systems removed synchronized activity, added clustered activity, and sparsied the CA3 pyramidal cell activity. Feedback Inhibition In this set of simulations, the eect of feedback inhibition was investigated. Feedforward inhibition from the entorhinal and dentate projections was removed. The variables that were changed were the synaptic strength of the associational system activation of the basket cells and the synaptic strength of the basket cell inhibition of the CA3 pyramidal cells. Feedback inhibition was able to switch the activity of CA3 pyramidal cells into three, possibly four, dierent states. When the strength of the associational system activation of basket cells was low, the network entered an oscillatory state that was synchronized throughout the entire longitudinal axis and can be described visually as bands of activity. As the associational system activation of basket cells became stronger, the bands became discontinuous in space. There is some semblance of 69 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.17: CA3 activity due to the interactions between the associational system and feedback inhibtion. temporal synchrony, but the bands have become separated spatially. At the higher extreme of associational activation, the density of the bands decreases, and the bands become further separated spatially, i.e., the extent of spatial correlation is reduced. Finally, there were conditions of feedback inhibition under which the CA3 net- work activity became similar to as if there was no feedback inhibition at all. There are two reasons for this. Both reasons contribute to an over-activation of basket cells by the associational system (Fig. 3.18). One condition is that the synaptic strength between the associational system and basket cells became too strong, re- sulting in an over-activation of basket cells. The other condition is that the CA3 network became disinhibited, i.e., synaptic strength of the basket cell inhibition of CA3 pyramidal cells is low, leading to higher associational activity that, in turn, over-activated basket cells. When either one or both of the conditions were satised, the basket cells were unable to generate action potentials due to the large amount and/or magnitude of their excitatory inputs and were then unable to provide any inhibition to the CA3 pyramidal cells. 70 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.18: Basket cell activity due to the interactions between the associational system and feedback inhibtion. Interactions between Feedforward and Feedback Inhibition Here, we explored the interactions between feedforward and feedback inhibition in a CA3 network that included the associational system (Fig. 3.19). These simula- tions represent the fully connected entorhinal-dentate-CA3 network and investigates how the synaptic weights of all inhibitory circuits aect the CA3 dynamics. In the gure above, the top row corresponds to a CA3 network with only feedback inhi- bition. They are the conditions for feedback inhibition previously shown in section 3.5.2 under which the CA3 network transitioned from a synchronized, oscillatory state to a sparsely ring, temporally synchronized but spatially discontinuous state. Under these conditions, both feedforward inhibition circuits were added. The second row of the gure corresponds to the CA3 pyramidal cell response to feedback inhibition and only the dentate feedforward inhibition circuit. The absence clusters seen in section 3.5.1.2 are not as prominent, but they are present. As the synaptic strength of the associational activation of basket cells is increased from 0.01 to 0.1, the absence clusters become more apparent. 71 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Figure 3.19: CA3 activity due to the interactions between the associational system, feedback inhibition, and both feedforward inhibition circuits. Starting from the third row of the gure, both entorhinal and dentate feedforward inhibition are present with feedback inhibition, and the synaptic strength of the entorhinal activation of basket cells is increased with the subsequent rows of the gure. Under the synchronized, oscillatory state due to feedback inhibition, i.e., the left-most column, entorhinal feedforward inhibition is shown to increase the presence of the dentate feedforward-induced absence clusters, decrease the density of the bands, and increase the spatial discontinuities within the bands. For the other feedback conditions, entorhinal feedforward inhibition provides a general inhibition until, at a synaptic strength of 0.15, the CA3 network is unable to generate activity. These results display how the individual eects of entorhinal feedforward inhibi- tion, dentate feedforward inhibition, and associational feedback inhibition, mediated by the pyramidal basket cell, are integrated and expressed in some non-linear com- bination by the CA3 activity. 72 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL 3.3.6 Summary of CA3 Work The present work explored the contributions of various projections on the spatio- temporal dynamics of CA3 pyramidal cells. The large-scale network used a topographically- organized connectivity scheme which used spatially-decaying connection probabili- ties that were constrained using anatomical axon terminal eld data specic to each neuron and projection type. Each projection was restricted to specic strata as re- ported by literature, and the density of each projection was dependent on the spine densities in the dendrites of the postsynaptic population, which were separately dened for each strata and neuron type. The synaptic weights for each presynaptic- postsynaptic neuron pair were constrained to match unitary postsynaptic potential characteristics. Based on the projection type, connectivity could be moderately sparse and weak (entorhinal), sparse and strong (dentate), or extremely dense and weak (associational). However, the connections were predominantly weak, resulting in a weakly-coupled network. The weak-coupling and spatially-decaying nature of the connectivity created soft clusters of neurons within the network in which cluster membership was a contin- uous value depending on the spatial distribution of the axon terminal elds. The cluster membership was quantied functionally at the spiking level using pair-wise correlation between neurons, i.e., noise correlation. The span of the axon terminal eld determined the spatial distance at which a neuron pair could be correlated. For cases when the network did not exhibit synchronized and oscillatory behavior, the peak correlations were between 0.005 and 0.02 which are values that have been re- ported in experimental works. If an asynchronous state is to be assumed for normal in vivo network activity, then the results demonstrate that the large-scale model is able to accurately generate a biologically plausible pairwise correlation. Each projection type generated a unique correlation structure based on their ax- onal anatomy. The entorhinal projection covered 10% of the longitudinal extent of the dentate gyrus and CA3. The mossy bers, characterized as long, single axonal processes with very few connections, covered less than 1% of the longitudinal extent of the CA3. The associational axons covered upwards of 70% of the longitudinal ex- tent of the CA3. Basket cells in the dentate gyrus and CA3 made local connections. Dentate basket cell axons spanned 1.3% of the longitudinal extent of the dentate gyrus, and CA3 basket cell axons spanned 0.5% of the longitudinal extent of the CA3. Axons were reported to span almost the entire distance along the transverse 73 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL and proximodistal axes, so clusters were not found along this axis. However, the action potential propagation delay caused travelling waves to move away from the crest in the dentate gyrus or move proximally to distally in the CA3. One major nding of the work was the robustness of cluster formation in hip- pocampus as a result of the characteristics of the aerent projections to the hip- pocampus. The entorhinal cortex employs a similar axonal organization in both dentate and CA3 regions. In both regions, random entorhinal activity resulted in the formation of clusters. The dierence in clusters arose from biophysics and num- ber of inputs. The CA3 pyramidal cells are much more excitable than dentate gyrus but also receive far fewer synaptic inputs from entorhinal cortex than granule cells. The excitability of the CA3 pyramidal cells contributed to more background activity and the fewer inputs reduced the number of shared inputs that neighboring CA3 pyramidal cells could have. These factors resulted in a lower correlation in CA3. In contrast to the entorhinal projection, the mossy ber system is organized in a lamellar manner and connects very sparsely with CA3 pyramidal cells. The sparse connectivity prevents CA3 pyramidal cells from sharing input from granule cells. Any spatial correlation observed in the granule cell activity is only propagated to the CA3 due to the lamellar organization of mossy bers, and specically only longitudinally-organized spatial correlation is preserved. Thus, the spatial correla- tions of the granule cells, due to the entorhinal projection, can be passed disynap- tically via the mossy bers to the CA3. Although mossy bers elicit very strong excitatory postsynaptic potentials, alone, they did not activate strong clusters in the CA3. The combination of the entorhinal and dentate activation resulted in a CA3 response with a spatio-temporal pattern that was very similar to the dentate response. A cross-correlation between the den- tate activity and the CA3 activity revealed that the CA3 activity occurred 16 ms after the dentate activity. This value is within the reported range of latencies for dentate vs CA3 activation after stimulation of the perforant path, a stimulation that would engage both the monosynaptic and disynaptic input pathways to CA3, and provides another form of validation of the large-scale model. The addition of the associational system required that the various forms of inhi- bition in the CA3 be included as the associational system forms an extremely dense and long-range connectivity. Each CA3 pyramidal cell receives tens of thousands of associational inputs and axons of CA3 pyramidal cells can span a majority of 74 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL the longitudinal and proximodistal extent of the CA3. Without inhibitory mecha- nisms to alleviate the strong excitation provided by the associational system, the CA3 network generated fast rhythmic activity and also lost all spatial and temporal correlations that were observed before the associational system was added. When inhibitory elements were added into the network, the spatially and tem- porally synchronous, rhythmic, and dense activity was transformed into temporally synchronous, spatially discontinuous, sparse activity. Feedforward inhibition elim- inated spatial synchrony. Feedback inhibition preserved temporal synchrony and contributed to spatial discontinuities. Feedforward inhibition also made the activity extremely sparse. The entorhinal feedforward inhibition was so strong that unless its synaptic strength was set to below 20% of its original value, CA3 pyramidal cell activity would be completely silenced. In contrast to excitatory inputs to CA3 pyra- midal cells which were reported to elicit postsynaptic potentials with amplitudes in the sub-millivolt range, excitatory inputs to the basket cells were found to generate postsynaptic potentials with amplitudes in the 2-4 mV range. With CA3 basket cells receiving many more entorhinal inputs than dentate inputs, the entorhinal ac- tivity was able to activate the basket cells much more strongly. The reduction of the input to below 20% of its original value reduced the amplitude to the sub-millivolt range. It is possible that the experimental data used to constrain the postsynaptic potential values were not from a unitary synaptic response. All of the data used to constrain the eect of entorhinal, dentate, and associational stimulation on basket cells came from the same study. 3.4 Discussion The large numbers of neurons in the neuronal network and the topographic con- nectivity revealed a novel emergent property. Given spatially and temporally un- correlated input, the network responded with weakly correlated activity that could be observed as spatio-temporal clusters of activity. Clusters of activity in the den- tate gyrus were then propagated to the CA3 which transformed the input clusters into dierent output clusters. The extent of the correlations were mediated by the topography, the synaptic dynamics, and the interactions between excitatory and inhibitory circuits. 75 CHAPTER 3. SPATIO-TEMPORAL PATTERNS OF THE LARGE-SCALE MODEL Autocorrelation maps were constructed primarily to identify the spatial and temporal extents to which activity within a hippocampal subeld was correlated and the magnitude of the correlations. Though useful in demonstrating the emergence of weak correlations given uncorrelated input, the autocorrelation maps could not quantify the transformations of spatio-temporal activity that were occurring. They did not represent the spatio-temporal ltering that was occurring as a consequence of dierent neural architectures. The next chapter describes the method used to extract the spatio-temporal lters of the dentate gyrus and CA3. 3.5 References Acs ady, L., Kamondi, A., S k, A., Freund, T., and Buzs aki, G. (1998). Gabaergic cells are the major postsynaptic targets of mossy bers in the rat hippocampus. Journal of Neuroscience, 18(9):3386{3403. De Wolf, T. and Holvoet, T. (2005). Emergence versus self-organisation: Dierent concepts but promising when combined. In Brueckner, S. A., Di Marzo Seru- gendo, G., Karageorgos, A., and Nagpal, R., editors, Engineering Self-Organising Systems, pages 1{15, Berlin, Heidelberg. Springer Berlin Heidelberg. Goldstein, J. (1999). Emergence as a construct: History and issues. Emergence, 1(1):49{72. Hinneburg, A. and Gabriel, H.-H. (2007). Denclue 2.0: Fast clustering based on kernel density estimation. In R. Berthold, M., Shawe-Taylor, J., and Lavra c, N., editors, Advances in Intelligent Data Analysis VII, pages 70{80, Berlin, Heidel- berg. Springer Berlin Heidelberg. Yu, G. J., Hendrickson, P. J., Song, D., and Berger, T. W. (2015). Topography- dependent spatio-temporal correlations in the entorhinal-dentate-ca3 circuit in a large-scale computational model of the rat hippocampus. In 2015 37th Annual In- ternational Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pages 3965{3968. 76 Chapter 4 Spatio-Temporal Filters for Population Dynamics 77 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS 4.1 Introduction Under white noise input, the population dynamics of hippocampus was found to be organized into spatially and temporally correlated regions of activity (Marmarelis, 1980). Entorhinal input is converted into a spatio-temporal pattern in dentate gyrus which is then converted to dierent spatio-temporal patterns by the CA3. However, it is dicult to compare the dierences in the spatio-temporal patterns and dif- ferences in the transformations that are taking place. To quantify the functional dierences, in terms of population dynamics, it is necessary to identify the trans- formations taking place. There is no current method to extract the transformations occurring at the large-scale population level. However, there are two elds that are relevant to this problem: Input-output modeling and neural mass modeling. Input-output models are a class of model that describes the statistical relations between input and output signals that uses zero or minimal a priori knowledge of the mechanisms that underlie the system. Input-output models are a data-driven approach that are able to capture the functional properties of a system using a general mathematical form. One approach to input-output modeling is to choose a set of basis functions and estimate their coecients to represent the dynamics of the system. The combination of the basis functions and their coecients then form a lter bank that represents the transformation performed by the system. This approach has been applied to computational neuroscience to construct input-output models of synapses (Hu et al., 2015), neurons, and small neuronal networks (Song et al., 2009; Marmarelis et al., 2013; Geng and Marmarelis, 2017), but it has yet to be applied to large-scale population dynamics due to the lack of large-scale models and large-scale population recordings in the eld. A type of modeling called neural mass modeling had been developed to represent the output of a population of neurons (Freeman, 1978). A neural mass is dened as a population of neighboring neurons that lie in close proximity to each other. The input to the neural mass are all of the presynaptic spike times that are received by the population, and the output becomes the average membrane potential response of the population in response to the inputs. In general, a single set of equations and parameters are used to describe the dynamics of a single neuron. Then an average membrane potential is computed for the neural mass. From this general framework, neural mass modeling has been extended to estimate local eld potentials (David and Friston, 2003) and the hemodynamic response observed during functional 78 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS magnetic resonance imaging (Bojak et al., 2010). Though the structure of neural mass modeling is not applicable to this work, the neural mass nomenclature will be used to describe the population being modeled. This chapter will describe the computational framework which used a General- ized Volterra Model (Berger et al., 2011) structure to estimate the spatio-temporal lter underlying the transformations performed by a hippocampal subeld. The spatial and temporal aspects of the lter were assumed to be independent and were represented using dierent basis function sets. The spatial basis functions used a Chebyshev basis, and the temporal basis functions used a Laguerre basis. Each neural mass was represented as a multi-input single-output (MISO) model, and the concatenation of multiple neural masses represented the population dynamics of an entire subeld and resulted in a multi-input multi-output (MIMO) model. 4.2 Methods 4.2.1 Neural Mass and Population Spiking Rate Previously, a neural mass was dened to be a population of neighboring neu- rons that are spatially adjacent. The size of the population that the neural mass represents can then be specied based on the size of the spatial region within the neural system that can be considered a neural mass; this is the spatial resolution. A maximal spatial resolution would correspond to individual neurons, and decreas- ing spatial resolutions would correspond to a neural mass containing an increasing number of neurons. Before dening the input-output signals relevant for neural mass modeling, it is useful to rst consider an input-output model of a single neuron. Typically, input- output models of single neurons estimate the kernels for each input with the input representing a spike train. A temporal resolution would then be chosen such that the input would be binary. However, hippocampal neurons can receive upwards of tens of thousands of inputs which would make the estimation process unfeasible. Therefore, the inputs were grouped by cell type, e.g., all entorhinal inputs were combined into a single input signal, all granule cell inputs were combined into a single input signal, etc. However, the number of neurons that can comprise each input type would require an extremely ne temporal resolution to achieve a binary signal. Furthermore because a neural mass is comprised of many neurons, a binary 79 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.1: Mathematical framework for a MISO neural mass model using a Generalized Volterra Model structure. signal is not an ideal to represent the input and output. Because each bin in the signal can contain multiple spikes that originate from many dierent neurons, the relevant signals for a neural mass model are called population spiking rates. Thus, an input-output model of a neural mass would use the population spiking rates from dierent input neuron types as the input signals and the population spiking rate of the neurons contained in the neural mass as the output. 4.2.2 Computational Framework A single MISO neural mass model used a model structure initially based on the work of Song et al. (2009) which combined the Volterra series and basis expan- sion with a generalized linear model to implement the Generalized Volterra Model structure (Fig. 4.1. Each MISO model consisted of ve components: 1) A feedforward block for the input population spiking rates was used to iden- tify the average postsynaptic potential (PSP) elicited by the input type which in- corporates the synaptic dynamics and the propagation of the PSP from the synapse location to the soma. 2) A feedback block was included to capture the average afterhyperpolarization of the population that would be generated after a neuron would spike. It also captures the response due to neural circuits that may be embedded within the neural mass such as feedback inhibition due to basket cells or feedback excitation due to the CA3 associational system. 80 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS 3) A cross-linking block that uses the output of other neural masses was included due to the local topographic connectivity. The length of the axon collaterals can allow monosynaptic connections to be made between neurons of separate neural masses. 4) An adder is used to sum the ltered input signals. 5) The link function of the generalized linear model is used to ensure that the model output conforms to the proper support, i.e. the output must be non-negative because it represents a population spiking rate. The concatenation of all MISO neural mass models would result in a MIMO neu- ral region model representing the population dynamics of the hippocampal subeld. Generalized Volterra Model A Volterra functional series was used as the initial mathematical formulation for the Generalized Volterra Model. The Volterra functional series uses the following form: u(t) =k 0 + N X n=1 M k X =0 k (n) 1 ()x n (t) + N X n=1 M k X 1 =0 M k X 2 =0 k (n) 2s ( 1 ; 2 )x n (t 1 )x n (t 2 ) + N X n=1 M k X 1 =0 M k X 2 =0 M k X 3 =0 k (n) 3s ( 1 ; 2 ; 3 )x n (t 1 )x n (t 2 )x n (t 3 ) +::: (4.1) Using the Laguerre-Volterra expansion (Marmarelis, 1993), rather than estimat- ing the eect of each past input sample within the memory window on the current output, the shape of the resulting lter was assumed to follow the form of Laguerre basis functions (Fig. 4.2), which are a set of orthogonal functions that have been well suited for representing neural dynamics. The estimation then only needs opti- mize the coecients of each Laguerre basis function which dramatically decreases the amount of memory and data necessary to generate a predictive model. The temporal dynamics of the spatio-temporal lter, for both the feedforward kernels 81 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.2: Examples of basis functions. Laguerre basis functions are on the left, and Chebyshev basis functions are on the right. and feedback kernels, were represented with Laguerre basis functions. The Laguerre basis functions can be generated from the following equations: b j () = 8 < : (1) (j)=2 (1) 1=2 P k=0 (1) k k j k k (1) k ; 0 <j (1) j (j)=2 (1) 1=2 P j k=0 (1) k k j k jk (1) k ; jM (4.2) The Chebyshev basis functions were used to represent the spatial variation in the lter (Fig. 4.2). The Chebyshev basis functions are also a set of orthogonal functions and have had applications towards representing irregularities in surfaces such as lenses (Mason and Handscomb, 2002; Rayces, 1992). Because symmetry is not a necessary feature of the basis set, they can be used to t any arbitrary surface. The Chebyshev basis functions can also take advantage of the computation-reducing features of the Laguerre-Volterra expansion. The Chebyshev basis functions can be computed using the following equation: T n (x) = dn=2e X m=0 n 2m x n2m (x 2 1) m (4.3) For this work, it was assumed that the spatial and temporal aspects of the lter were independent. Therefore, the basis functions were combined by multiplying every pair combination between the Laguerre and Chebyshev basis functions (Fig. 4.3). Thus, Equation (4.1) can be rewritten as follows: 82 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.3: Spatio-temporal basis functions as a combination of Laguerre and Chebyshev basis functions. u(t) =c 0 + N X n=1 Lt X jt=1 Ls X js=1 c (n) 1 v (n) jt (t)w (n) js (x) + N X n=1 Lt X j t1 =1 j 1t X j t2 =1 Ls X j s1 =1 j s1 X j s2 =1 c (n) 2 v (n) j t1 (t)v (n) j t2 (t)w (n) j s1 (x)w (n) j s2 (x) +::: (4.4) A log link function was chosen as for the generalized linear model due to its upper and lower bounds. The lower bound of the log link function is zero which is ideal as the population spiking rate must be non-negative. An upper bound is necessary because the population spiking rate can saturate. However, the asymptotes of the log link function are zero and one. Therefore, the output population spiking rate was normalized by dividing by the maximum output population spiking rate in the data. 4.2.3 Optimization of Basis Function Parameters The Laguerre and Chebyshev basis functions have several metaparameters that must be optimized to result in the best estimation. Both share the following: the order of the basis and the order of the self-kernel. The order of the basis determines 83 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS the number of basis functions to use for a given basis set, e.g. a Laguerre basis of order three is comprised of three basis functions. The order of the self-kernel deter- mines whether non-linear interactions between samples of the same input signal will be included in the lter. The rst order self-kernel can be simply referred to as the rst order kernel and is entirely linear. The second order self-kernel incorporates non-linear pairwise interactions of the input, third order self-kernels incorporate the non-linear triple-wise interactions of the input, etc. The incorporation of non- linearities is important, but the result is that many more coecients need to be estimated which requires more data and computational resources and can lead to overtting. Because the spatial and temporal basis functions are multiplied to con- struct the spatio-temporal lters, the limitations for the order of the basis function and the order of the self-kernel become more severe. A metaparameter specic to the Laguerre basis is. This parameter determines the time-scale of the dynamics of the Laguerre basis functions and must be between 0 and 1. Lower values of result in faster dynamics, and higher values result in slower dynamics. A metaparameter specic to the Chebyshev basis is the radius r. The Chebyshev basis is dened within the interval [1, -1]. Therefore, the distance between neural masses must be normalized before the estimation procedure. The radius assumes that all the spatial variation lies within the radius, and any neural masses outside of the radius do not contribute to the predictive power of the model. For each MISO neural mass model all metaparameters are individually opti- mized for each input type. After the ideal metaparameters have been found, a nal estimation is performed in which all input types are included with their optimal metaparameters. Thus, a number of spatio-temporal lters equal to the number of MISO neural mass models that comprise the hippocampal subeld are computed. 4.2.4 Parameter Estimation For training and validation, the rst 10 seconds of the simulated neural activity were discarded to avoid the transient eects of the initial input activation. Then, the rst 80% of the data was used for training, and the remaining 20% of the data was used for validation. To identify the optimal metaparameters to use for the basis functions, a range of values were explored for each input type, i.e. entorhinal, dentate, and CA3 84 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS input and feedback, and for every MISO neural mass model, and every combination of metaparameter values was tested, i.e. every combination of basis order, kernel order, and alpha/radius were tested for each input type. Then for each individual MISO neural mass model, the metaparameter combinations that resulted in the lowest normalized root-mean-square error (NRMSE) were then identied for each input type. The metaparameter values were then averaged to nd a global set of metaparameter values that would yield the best predictions for all MISO neural mass models. The equation for NRMSE is below. NRMSE = s P i (^ y i y i ) 2 P i y 2 i (4.5) Once the optimal set of metaparameters was identied, the coecients to the basis functions were estimated for each MISO neural mass model. The outputs of all of the MISO neural mass models would then become the nal output of the MIMO neural region model. 4.2.5 Simulation Data The estimation of the spatio-temporal lter was tested on simulation data gen- erated from the large-scale model of the entorhinal-dentate-CA3 network using the 3 Hz white noise input. A neural mass model of the CA3 was constructed due to its extensive associational system that would require the cross-linking block. A simula- tion representing 50 seconds of activity was performed to obtain a sucient amount of data to estimate the model coecients. 4.3 Results Neural masses were dened using bin sizes of 0.1 mm which were below the spatial Nyquist criterion for representing the clusters. This resulted in a 100 MISO neural mass models being generated. A temporal sampling rate of 200 Hz, i.e. a temporal bin size of 5 ms, was used to sample the population spiking rate. 85 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS 4.3.1 Metaparameter Optimization From the metaparameter optimizing procedure, a broad range of values were explored. For the Laguerre basis functions, basis orders of 4-6 were tested with rst-order through third-order self-kernels. The values incremented from 0.1 to 0.97 in increments of 0.05. Additionally, values of 0.025, 0.05, and 0.075 were tested in the low end with an additional value of 0.975 on the high end. For the Chebyshev basis functions, basis orders of 4 and 5 were tested with rst-order and second-order self-kernels. Radius values from 0.5 mm to 3.0 mm were explored in increments of 0.25 mm. The total number of metaparameter combinations that were tested per input were 6,336. With three dierent inputs and 100 MISO neural mass models, the total estimations that were performed was 1,900,800. After evaluating which combinations resulted in the lowest NRMSE for the in- dividual inputs, the best metaparameters for each MISO neural mass model were averaged to determine the best combination of metaparameters to use across all MISO models. Fifth-order Laguerre basis functions were chosen for all input types with a maximum self-order of two. The values were 0.25, 0.33, and 0.27 for the entorhinal, dentate, and CA3 inputs respectively. The entorhinal and dentate Chebyshev basis functions were fourth-order, and the CA3 Chebyshev basis functions were fth-order. The entorhinal kernels had a maximum self-order of one, and the dentate and CA3 kernels used a maximum self- order of two. The radius values were 2.5275, 1.625, and 1.15 mm for the entorhinal, dentate, and CA3 inputs respectively. The feedback kernel used fourth-order Laguerre basis functions with a maximum self-order of two. 4.3.2 Model Prediction Performance The NRMSE from the validation for each MISO neural mass model was calcu- lated with a mean NRMSE of 0.282 0.03. The NRMSE was also plotted as a function of the positions of the neural masses (Fig. 4.5). The NRMSE was higher at the boundaries which is likely due to the reduced number of inputs from which the output can be predicted. A scatter plot between the validation data and the model estimates was created, and a linear regression was performed (Fig. 4.5). The r 2 value of the regression was 0.79 which meant that the model captured 79% of the variance of the system. 86 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.4: Comparison between the simulation data and the model estimates. (Left) The output of all MISO neural mass models are shown. The simulation data is on top, and the estimates are below. (Left) The output of a single MISO neural mass model is plotted over the validation data. The neural mass model is the same as the signal from the left plot from the middle row at position 5.0 mm. Figure 4.5: (Left) Error is plotted as a function of septo-temporal position. (Right) A scatter plot with the validation data on the x-axis and the estimates on the y-data are plotted. A linear regression indicates how well the model captured the variance of the system. These evaluations of the model are important because they determine how well the computed spatio-temporal lters describe the input-output transformations. Focusing on the output of a single MISO neural mass model (Fig. 4.4), it can be seen that there are some nonlinearities that are unable to be captured by the model, particularly during parts of the signal were there are peaks. The large peaks can be underestimated and overestimated by the model. However, a higher-order self- kernel did necessary improve the performance. The metaparameter optimization procedure selected for a second-order self-kernel although third-order self-kernels were also explored. It is possible that the log link function, which introdues a nonlinearity to the model, may be insucient for capturing the nonlinearity of the system. 87 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.6: The spatio-temporal lters for the dierent input types from neural masses at dierent septo-temporal positions are shown. The left column plots the lters for a neural mass originating at a temporal location, and the right column corresponds to a septal location. The rst row corresponds to the spatio-temporal lter for entorhinal cortex. The second row is for dentate gyrus input. The third row is for CA3 input. The nal row plots the feedback kernels. 4.3.3 Spatio-Temporal Filters Because the spatio-temporal lters are two-dimensional, only the rst-order ker- nels were plotted and analyzed because the dimensional space for visualizing for higher-order kernels goes beyond what can easily be printed. For example, the second-order kernel would require a 4-dimensional visualization. A selection of spatio-temporal lters from neural masses at equally spaced posi- tions through the longitudinal axis of the CA3 are shown (Fig. 4.6). It can be seen that each neural mass forms a dierent spatio-temporal lter that is unique to their position. This is due to the non-uniform topography of the CA3 network. The blue areas correspond to negative values, and the red areas correspond to positive values. White space corresponds to a value of zero. 88 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Figure 4.7: The average spatio-temporal lters for each input for all neural mass models are plotted. An average lter was constructed by computing the mean of the spatio-temporal lters for all the MISO neural mass models (Fig. 4.7). 4.4 Discussion A mathematical framework was proposed to construct input-output models for large-scale population dynamics of neural systems. The major innovations were the application of input-output modeling to neural mass modeling and the addition of spatial basis functions to use the population ring rates of adjacent neural mass models in the prediction of the output of a dierent neural mass. The MIMO neural region model was able to capture 79% of the variance of the simulation data with a mean NRMSE of 0.282. This indicates that the spatio- temporal lters estimated by the model are able to capture the spatio-temporal transformations that are occurring between a presynaptic and postsynaptic region. In future work, the extracted spatio-temporal lters can be used to compare the transformations performed the dierent neural regions. Other directions in- clude further developing the model to reduce the NRMSE. For example, a more sophisticated nonlinearity can be introduced such as associated nonlinear functions. Another direction would be to perform studies to demonstrate which mechanisms 89 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS contribute to the shape of the spatial aspect of the lters. Potential mechanisms include the distribution of the axon terminal eld, either from aerent neurons or neurons within the neural region, and the dynamics of feedforward and feedback excitation/inhibition circuits. 4.5 References Berger, T. W., Hampson, R. E., Song, D., Goonawardena, A., Marmarelis, V. Z., and Deadwyler, S. A. (2011). A cortical neural prosthesis for restoring and enhancing memory. Journal of Neural Engineering, 8(4):046017. Bojak, I., Oostendorp, T. F., Reid, A. T., and K otter, R. (2010). Connecting mean eld models of neural activity to eeg and fmri data. Brain Topography, 23(2):139{ 149. David, O. and Friston, K. J. (2003). A neural mass model for meg/eeg:: coupling and neuronal dynamics. NeuroImage, 20(3):1743 { 1755. Freeman, W. (1978). Models of the dynamics of neural populations. Electroen- cephalography and clinical neurophysiology. Supplement, (34):9|18. Geng, K. and Marmarelis, V. Z. (2017). Methodology of recurrent laguerre{volterra network for modeling nonlinear dynamic systems. IEEE Transactions on Neural Networks and Learning Systems, 28(9):2196{2208. Hu, E., Bouteiller, J.-M., Song, D., Baudry, M., and Berger, T. (2015). Volterra representation enables modeling of complex synaptic nonlinear dynamics in large- scale simulations. Frontiers in Computational Neuroscience, 9:112. Marmarelis, V. Z. (1980). Identication of nonlinear systems by use of nonstationary white-noise inputs. Applied Mathematical Modelling, 4(2):117 { 124. Marmarelis, V. Z. (1993). Identication of nonlinear biological systems using la- guerre expansions of kernels. Annals of Biomedical Engineering, 21(6):573{589. Marmarelis, V. Z., Shin, D. C., Song, D., Hampson, R. E., Deadwyler, S. A., and Berger, T. W. (2013). Nonlinear modeling of dynamic interactions within neu- ronal ensembles using principal dynamic modes. Journal of Computational Neu- roscience, 34(1):73{87. 90 CHAPTER 4. SPATIO-TEMPORAL FILTERS FOR POPULATION DYNAMICS Mason, J. and Handscomb, D. (2002). Chebyshev Polynomials. CRC Press. Rayces, J. L. (1992). Least-squares tting of orthogonal polynomials to the wave- aberration function. Appl. Opt., 31(13):2223{2228. Song, D., Chan, R. H., Marmarelis, V. Z., Hampson, R. E., Deadwyler, S. A., and Berger, T. W. (2009). Nonlinear modeling of neural population dynamics for hippocampal prostheses. Neural Networks, 22(9):1340 { 1351. Brain-Machine Interface. 91 Chapter 5 Connectivity-Dependent Information Processing 92 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING 5.1 Introduction The function of the hippocampus is strongly implicated in the formation of episodic memory (Fortin et al., 2002; SCOVILLE and MILNER, 1957; Squire, 1992). The basis of such a function must arise from the collective properties of the neural components within the hippocampal system and their interactions. Such properties include the network/circuit architecture, the neuron morphology, their biophysical properties, synaptic dynamics, and synaptic plasticity. Therefore, by using the vast amounts of quantitative data available on the hippocampus to constrain and represent these properties in a computational model, a computational approximation of the hippocampus and its functions can be simulated. We are developing a platform from which a full-scale, biologically realistic, spiking neuronal network model of the entorhinal-dentate system of the rat hippocampus was constructed. The model represents one complete dentate gyrus including one million compartmental models of granule cells with realistic dendritic morphologies and over 3 billion perforant path synapses (Hendrickson et al., 2016). Our previous work with this model used random synaptic inputs to characterize its baseline activity. We found a strong role for topography in determining hip- pocampal system dynamics, as has been observed in theoretical models of other brain regions (Ambros-Ingerson et al., 1990; Coultrip et al., 1992; Granger et al., 1996; Rodriguez et al., 2004). Topography refers to the ordered anatomical arrange- ment of axonal projections, originating from a presynaptic population and synapsing onto a postsynaptic population, that results in a system-level connectivity which may impose an organization to the information being transmitted. Such an organization has been characterized for the entorhinal-dentate projection by Dolorfo and Amaral (Dolorfo and Amaral, 1998), and this data was quantied for use in the network con- nectivity for our model (Hendrickson et al., 2016). The anatomical distribution of information then may represent a foundational "bias" that the neural system must then incorporate into its response. However, the random nature of the input in the previous work, though useful in characterizing the dynamical properties, precluded an analysis relating topography, or other model parameters, to higher level functions of the system. Therefore, physiological inputs were sought to allow the activity of the model to be interpreted with respect to higher level function. Grid cells of the medial entorhinal cortex (MEC) encode information about posi- tion, expressing spatial receptive elds in a grid-like pattern that span the environ- 93 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING ment that an animal explores (Hafting et al., 2005). They represent a large portion of the inputs to the hippocampus and are necessary for the formation of place cells within the hippocampus(de Almeida et al., 2009; Miller and Best, 1980; O'Keefe and Dostrovsky, 1971; Solstad et al., 2006; Van Cauter et al., 2008). Thus, grid cells and place cells are important as they provide the spatial context for memory events (McKenzie et al., 2014). Measurement of various properties of the grid cells in rat have revealed that the size of the spatial receptive elds, i.e., the grid elds, varies among grid cells following an anatomical gradient (Hafting et al., 2005; Brun et al., 2008; Stensola et al., 2012). Using the spatially-correlated activity provided by grid cells to drive a large- scale, mechanistic, spiking neuronal network model of the rat entorhinal-dentate system which included 120,000 granule cells and 5,600 basket cells, the in uence of topography on a network's ability to encode spatial information was investigated. The size of the axon terminal eld of the entorhinal-dentate projection was varied to explore how dierent feedforward architectures aected network dynamics, spatial representation, i.e., place eld properties, and neuronal spatial information. A point process lter based on the work of Brown et al. (1998a) was used to assess the ability of the dierent feedforward architectures to encode spatial information and quantify it at the network level. Using these methods, we explore the role of network architecture on the neural encoding of spatial information. 5.2 Methods The large-scale model simulations described here were designed to be analogous to typical in vivo experiments for determining place elds. In such experiments, a rat explores an environment during which neuronal spiking activity is recorded. The distribution of spikes is plotted against the location of the rat to identify regions in space where the neuron prefers to re. Ideally, a rat will randomly and uniformly sample its entire environment for a sucient time to allow the conditional ring probability as a function of x- and y-position to be calculated for each neuron. A random sampling is ideal to eliminate correlations that would arise based on patterned movements, and a uniform sampling is ideal to ensure that sucient numbers of samples are obtained equally across all possible locations within the environment. 94 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING For computational simulations, a random path model was used to mimic the movement of a rat in a xed environment. Experimentally, recordings are often performed during sessions that range from 20 minutes to one hour (Van Cauter et al., 2008; Jung et al., 1994; Neunuebel and Knierim, 2012). However, the com- putational complexity of the large-scale model had previously limited simulation times to 10 seconds. In order to complete 20 minute simulations in a reasonable amount of computational time, the complex neuron morphologies were simplied using an equivalent circuit algorithm developed by Marasco et al., 2012 (Marasco et al., 2012). The methods section will rst describe the large-scale model and the equivalent circuit algorithm. It will then describe the methods for generating realistic grid cell activity, the procedure for extracting and characterizing place elds, and the statistical methods used to decode position and quantify information at the network level. 5.2.1 Large-Scale Model Neuron models were constructed and simulated using the NEURON simulation environment (Carnevale and Hines, 2006). Granule cells were represented using a multi-compartment model with the electrophysical properties of each compartment being modeled by an electrical circuit. The reduced granule cell models consisted of four compartments corresponding to the soma and the dendrites in the inner, middle, and outer thirds of the molecular layer. The parameters of the reduced model were obtained by creating an equivalent circuit model based on the complex morphology (Marasco et al., 2012). Compartments in series and parallel could be combined using standard circuit approaches to create a reduced model that pre- served the electrophysiological properties characteristic of granule cells. Basket cells were represented using a single compartment model with properties obtained from (Santhakumar et al., 2005). Gaarskjaer (1978) measured the dimensions of an un- folded rat dentate gyrus and reported a septo-temporal length of 10 mm. Neurons were instantiated in a dentate gyrus map following those dimensions. Synaptic connections between neurons were simulated using a distance-based rule that followed anatomically derived constraints. Axons were functionally repre- sented as time delays that were computed based on the axonal path length distance between the soma at which the action potential was initiated and the receiving 95 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING postsynaptic neuron using an action potential propagation velocity of 0.32 mm/ms (Tielen et al., 1981). The entorhinal-dentate projection was mapped based on a comprehensive study that used a series of retrograde tracers injected into the den- tate gyrus (Dolorfo and Amaral, 1998). Quantication of the data is described in (Hendrickson et al., 2016). Connectivity was stochastically generated by converting spatial axon density data into distance-based Gaussian probabilities for entorhinal cortical, granule, and basket cells. Convergence values were estimated using den- dritic lengths, spine density counts, and presynaptic-postsynaptic population ratios (Hendrickson et al., 2016). Basket cells provided both feedforward and feedback inhibition to granule cells (Hendrickson et al., 2015). Entorhinal input to basket cells provided a basis for feedforward inhibition, and granule cell input to the basket cells activated feedback inhibition. Suprapyramidal granule cells received 2117 grid cell inputs and 2417 LEC inputs. Infrapyramidal granule cells received 1253 grid cell inputs and 1479 LEC inputs. These numbers were derived from morphological and spine density data (Hendrick- son et al., 2016) which report that suprapyramidal granule cells had a larger total dendritic length (Claiborne et al., 1990) and higher spine densities than infrapyra- midal granule cells (Hama et al., 1989; Desmond and Levy, 1985). Suprapyramidal and infrapyramidal granule cells received 108 and 68 basket cell inputs, respectively. Each basket cell received 871 MEC inputs, 1161 LEC inputs, and 915 granule cells. The conductance time course of synapses upon activation was modeled using two exponentials (Santhakumar et al., 2005; H ausser and Roth, 1997) with parameters optimized to match excitatory and inhibitory postsynaptic potential data for the re- spective synapses. The excitatory conductances represented AMPA receptors, and the inhibitory conductances represented GABAA receptors. 5.2.2 Grid Cell Activity Blair et al. (2007) developed a mathematical description to model grid maps involving the summation of three cosines that are rotated in increments of 60deg to form the characteristic triangular lattice pattern of grid maps: G(r;;;c) =g( 3 X k=1 cos( 4 p 3 u( k +) (rc))) (5.1) 96 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.1: Distribution of grid eld parameters and generation of grid cell activity. (A) Data from Hafting et al., 2005 (left and middle) and Stensola et al., 2012 (right) were used to constrain the grid eld properties. (B) The grid eld properties were normalized along the dorso-ventral axis of the medial entorhinal cortex using a generalized logisitic function such that the grid eld properties were represented approximately equally. (C) The gradient of grid eld parameters in the medial entorhinal cortex (left) and the mapping between medial entorhinal cortex and dentate gyrus (center) determine where the grid eld information is communicated to within the dentate gyrus. The nal distribution of grid eld parameters results in a gradient in the dentate gyrus (right). (D) An example grid eld is shown with notation describing the eld area, distance, and orientation properties. (E) Left: The movement of a virtual rat in white is overlaid on a grid eld with a triangle and circle denoting the start and end points of the movement, respectively. Right: The ring rate (red) is determined using a grid eld and the movement of the rat through the eld. A non-homogeneous Poisson process is used to generate spiking activity (black) using the ring rate. 97 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING where the function u( k +)=(cos( k +),sin( k +) ) determines the relative and global rotation of the cosines. The vector r corresponds to the position (x,y) with c=(x 0 ,y 0 ) determining the spatial oset of the grid map, and the variable sets the distance between the grid elds. The function g(x)=e (a(xb)) -1 was chosen to act as a monotonically increasing gain function where the parameter a determines the width of the place elds, and b= 3 2 is used to set the minimum value of the function to zero. The grid maps then were normalized such that the peak ring rate was 50 Hz. Figure 5.1D summarizes the major properties of grid maps. The position of a virtual rat as it explored an 80 cm by 80 cm square environment was used as an input to the above function to obtain a position-dependent, time- varying ring rate. Movement was modeled by sampling from uniform random distributions that dened the speed of the rat (0-30 cm/s), the direction of movement (0deg-360deg), and the period of time during which the rat would move with the sampled velocity vector (0-500 ms). If the trajectory of the rat would take it beyond the boundaries of the environment, the rat's movement would be re ected by o the boundary. The resulting ring rate was used as input to an inhomogeneous Poisson renewal process to generate spike times (Fig. 5.1E). When a spike was elicited by a Poisson process, the ring rate after the spike time would be modied by a refractory period with an exponential time course having a time constant of 35 ms, preventing inter- spike intervals from becoming too short. The time constant was estimated from data published by Alonso and Klink (1993). The parameters of the grid maps were constrained based on the supplemen- tary data published by Hafting et al. (2005) (Fig. 5.1A). They reported a linear relationship between both the grid spacing and the grid eld area and the entorhi- nal distance from the postrhinal border which corresponds approximately with the dorso-ventral position within the MEC. Stensola et al. (2012) further reported a linear relationship between grid spacing and grid orientation. Linear regressions between the grid eld properties and MEC dorso-ventral position were performed to quantify the anatomical organization of grid eld properties. A uniform distribution for grid eld sizes and grid spacing was assumed for the model such that each grid eld size was equally represented. Uniform distributions were constructed by transforming the grid map gradient with a sigmoidal function which was based on the distribution of grid cells within the MEC (Fig. 5.1B): 98 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING y(x) =A + KA (1 +e B(xM) ) 1 v (5.2) where y(x) represents the grid eld size at a given dorso-ventral position, A = 0.3256, K = 2.4239, B = 2.15, M = -1.7858, and v = 0.01. 5.2.3 Place Field Calculation An initial rate map was created by discretizing the environment explored by the virtual rat into bins of size 2 cm by 2 cm. The number of spikes that were elicited by a granule cell in each bin was counted and divided by the amount of time spent in each bin. The rate maps then were smoothed using a modied version of the adaptive smoothing procedure described by Skaggs et al. (1996). The original procedure expanded a circle around each bin and used a criterion based on a meta parameter and the number of samples available within the circle to determine the size of the circle, i.e., the extent of the smoothing. The circle weighted all bins equally. The modied procedure weighted each bin based on the distance from the center using a Gaussian function. The standard deviation of the Gaussian function was increased until the weighted number of samples satised the criterion set by the same meta parameter using the following equation: N spikes > N 2 occ 2 (5.3) where N spikes corresponds to the number of spikes transformed by the Gaussian function, N occ is the number of samples transformed by the Gaussian function, is the standard deviation of the Gaussian, and is the meta parameter that controls the amount of smoothing, set at 1:0 10 13 . A grid-based version of the density clustering algorithm, DENCLUE, was used to quantify the number of place elds and areas of the place elds of the smoothed rate maps. The original DENCLUE algorithm smooths the data using a density kernel and uses a local hill climbing procedure to identify data points that share the same local maxima as clusters (Hinneburg and Gabriel, 2007). A grid-based version was developed to reduce the number of data points that needed to be clustered and increase the speed of the algorithm. The density kernel was a Gaussian function that used the standard deviations that satised the sampling criterion during the 99 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING adaptive smoothing procedure. Clusters were subjected to a threshold such that bins less than 40% of the maximum value of the cluster were removed. Processed clusters with an area less than 200 cm 2 , as described by Muller and Kubie (1989), were not considered to be clusters. 5.2.4 Spatial Information Score The spatial information score was calculated using a histogram-based method proposed by Skaggs et al. (1996) with SI = N X i=1 p i R i R log 2 R i R (5.4) where the environment was divided into non-overlapping bins indexed by i = 1;:::;N,p i is the probability that the rat is in bin i,R i is the mean ring rate for bin i, and R is the overall mean ring rate. The metric is called spatial information score in this work to dierentiate it from information as it is dened by information theory. The spatial information score resembles the calculation of the mutual information but does not follow the exact form in terms of joint and marginal probabilities. 5.2.5 Recursive Point Process Filter for Decoding Brown et al. (1998b) developed a statistically-based, recursive point process l- tering technique that could use the receptive eld and spiking activity of one or more neurons to estimate the quantity that was being encoded. The same technique was applied to estimate the position of the rat using the following equations: One-Step Prediction x kjk1 =x k1jk1 +Fx k1jk1 (5.5) One-Step Prediction Variance W kjk1 =FW k1jk1 F T +RW " (5.6) 100 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Posterior Variance W 1 kjk =W 1 kjk1 + C X c1 [[rlog( c k )[ c k ] T ]r 2 log( c k )[n c k c k ]] (5.7) Posterior Mode x kjk =x kjk1 +W kjk1 C X c=1 rlog( c k )[n c k c k ] (5.8) The temporal spiking activity for each neuron was discretized into bins such that a one signies that an action potential was generated and a zero signies no action potential. The bins were indexed by the variable k which denotes the bin number, and denes the bin size which was 1 ms. The variable c k indicates the ring rate for the neuron indexed by the variable c and at time k. The generation of an action potential at time k is denoted by n c k . Thus, the probability of ring an action potential is determined by multiplying the ring rate with the bin size, c k . Ther andr 2 variables represent functions for the rst and second derivatives with respect to position. The state evolution matrix F and covariance matrix W " were calculated by cal- culating a rst-order autoregressive model of the path of the rat. During One-Step Prediction, the position x kjk1 , is estimated using the autoregressive model param- eters and the previous position estimate x k1jk1 , which incorporates the spiking history within the interval (0; (k 1)). Similarly, One-Step Prediction Variance estimates the covariance matrix of the One-Step Prediction position based on the autoregressive model parameters and the previous covariance matrix estimate. The Posterior Mode and Variance use the neuron ring probabilities and the spiking activity at timek to update the one-step estimates and obtain the estimate of the current position,x kjk , and the covariance matrix of the posterior,W kjk , under the assumption that the neurons are conditionally independent. These estimates incorporate the spiking history within the interval (0;k). The derivatives of the log of the ring rates are weighted by the ring probabilities and spike counts to compute these estimates. The derivation of equations (5.5-5.8) are detailed by Barbieri et al. (2004). 101 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.2: Smoothed rate maps from simulated dentate granule cells at dierent locations along the dorso-ventral axis. A dorso-ventral gradient for the size of the place elds was observed. Ventrally-located granule cells exhibited larger place elds, and dorsally-located granule cells exhibited smaller place elds. The color scale denotes high ring rates in red and low ring rates in blue. To facilitate the computation of the rst and second derivatives of the rate maps during the decoding process, the rate maps were represented by a set of Legen- dre polynomials which are dierentiable and form an orthonormal basis within the interval [-1,1] (Lebedev and Silverman, 1972). 5.2.6 Lower Bound of Mutual Information Encoded by Net- work Pillow et al. (2011) derived a generalized estimate for the lower bound of the mutual information and was used in the present study as an information-theoretic measure to characterize the ability of the network to encode position. I[x;r] 1 2 [log 2 [(2e) 2 jW " j] (log 2 jE[rr T ]j +log 2 (2e))] (5.9) Here,W " refers to the covariance matrix in (6), and E[rr T ] corresponds to the covariance of the residuals where r =x ^ x or the error of the estimate. 5.3 Results 5.3.1 Emergence of Place Fields The network was driven by input from both the medial and lateral entorhinal cortices. The MEC provided spatially-correlated grid cell input, and the LEC pro- vided spatially-uncorrelated input as random, Poisson activity, contributing noise 102 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING to the grid cell input. Entorhinal-dentate system simulations were performed using experimentally-derived constraints for all aspects of the model (e.g., cellular bio- physics, convergence, divergence, EPSP/IPSP parameters, grid cell receptive eld properties, etc.). Each simulation represented 2,000 seconds which took 98 hours per simulation using 1,000 processors. Simulation results revealed that under these con- ditions granule cell activity exhibited multiple, irregularly spaced place elds which matched experimental descriptions of granule cell place elds (Fig. 5.2) (Jung and McNaughton, 1993; Leutgeb et al., 2007). Experimental studies reported an average of 2.2 place elds with an area of 667.3cm 2 (de Almeida et al., 2009; Leutgeb et al., 2007). The average number of place elds per granule cell from simulation was 4.10 1.66, and the average area was 719.5 185.6 cm 2 . Neunuebel and Knierim (2012) had reported an average spatial information score of 1.1 0.56 bits/spike for experimentally recorded granule cells, and the simulations reported here resulted in an average spatial information of 0.83 0.03 bits/spike. The quantitative sim- ilarities between the simulated and experimental place elds demonstrate that the large-scale model, in addition to accurately representing cellular dynamics, can ad- equately recreate phenomena at a higher level involving network dynamics essential for spatial cognition. 5.3.2 Evaluating Gradients in Dentate Gyrus The gradient of grid map properties within the MEC and the gradient of entorhinal- dentate projections align such that grid cells associated with smaller place elds are transmitted along projections terminating within the dorsal dentate gyrus, while grid cells associated with larger place elds are transmitted along projections ter- minating within the ventral dentate gyrus (Fig. 5.1C). This relation suggested that the dentate gyrus would exhibit place eld properties such that granule cells within the dorsal dentate gyrus would express smaller place elds and granule cells within the ventral dentate gyrus would express larger place elds. Therefore, an analysis was performed to quantify the emergence of any possible gradient in the place eld properties of granule cells with respect to dorso-ventral position. 103 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.3: Raster plots of the spiking activity due to dierent axon terminal eld extents. (A) Spiking activity is represented using black dots which represent the time and position along the longitudinal extent at which an action potential was generated. Clustered activity is apparent with smaller axon terminal eld sizes and organizes into vertical bands at larger axon terminal eld sizes. (B) A conceptual representation of the consequences of changing the axon terminal eld have on connectivity is depicted. As the axon terminal eld grows larger, a larger area of neurons can be contacted, and the input is more dispersed spatially. At smaller axon terminal elds, the area in which neurons can be contacted becomes restricted. A second gradient was hypothesized to emerge along the transverse axis due to the dierence in number of inputs the suprapyramidal and infrapyramidal blades received, so the data was further divided into suprapyramidal and infrapyrami- dal populations. Therefore, the subsequent analyses investigated the presence and magnitude of the dorso-ventral gradients that occurred in the suprapyramidal and infrapyramidal blades. 5.3.3 Place Field Area Gradient Depends on Axonal Anatomy A linear regression was performed between the mean place eld area and dorso- ventral position, and a gradient was discovered in which smaller place elds were generated dorsally and larger place elds were generated ventrally. To test the in uence of topography on the emergence of place eld area gradients, the dorso- ventral extent of the axon terminal elds of the entorhinal projection were varied in a log-linear manner, i.e., 0.1 mm, 0.5 mm, 1.0 mm, 2.0 mm, and 10.0 mm, due to the computational resources necessary to complete a single simulation. The extent of the axon terminal eld constrains the area in the dentate gyrus to which a single entorhinal neuron could form synaptic connections and was used as a continuous 104 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.4: The eect of axon eld terminal extent on granule cell place eld properties and suprapyramidal- infrapyramidal dierences. (A) The mean place eld area within 0.5 mm bins along the dorso-ventral axis were plotted as a scatter plot with the corresponding linear ts. The mean values within a bin were plotted for visual clarity, but the regressions were performed using the raw data. (B) The magnitude of the slope between place eld area and dorso-ventral position decreased log-linearly with the axon terminal eld extent. The error bars denote the standard error of the estimates of slope. (C) The mean place eld areas of the populations were weakly correlated with the axon terminal eld extent. The error bars denote standard error. Nsupra = 65,000 and Ninfra = 55,000 for each eld extent. quantity through which the entorhinal-dentate connectivity could be transitioned between a lamellar connectivity and a uniform random connectivity (Fig. 5.3B). The spatio-temporal activity due to dierent axon terminal eld extents and a conceptual representation of the axon terminal eld are depicted in Fig. 5.3A. As previously reported in Hendrickson et al. (2016), the axon terminal eld controls the spatial extent of the clusters. Shorter axon terminal eld extents resulted in a linear decrease in place eld area along the dorso-ventral axis (Fig. 5.4A). The slope was used as a measure to quantify the range of place eld sizes, i.e., spatial resolutions, that the network represented. The slope decreased following a power law, i.e., log-linearly, as the axon terminal eld extent increased (Fig. 5.4B). The mean place eld area for the population was weakly correlated with the axon terminal eld extent (Fig. 5.4C). The place eld area gradient was present in both the infrapyramidal and suprapyra- midal populations with both populations exhibiting larger ventral place elds and smaller dorsal place elds. However, the suprapyramidal blade exhibited a steeper gradient. When comparing the mean place eld areas between the infrapyramidal and suprapyramidal populations, the suprapyramidal granule cells had larger place elds regardless of the axon terminal eld extent (p<<0.001 for all axon eld extents). There was no dorso-ventral gradient for the number of place elds generated per granule cell (data not shown). Suprapyramidal granule cells exhibited a slightly lower number of place elds (0.14 0.03 place elds) though it was statistically signicant (p 0.001 for all axon eld extents). 105 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.5: The eect of axon terminal eld extent on spatial information score and the eect of multi-resolution input on spatial information score. (A) The mean spatial information score within 0.1 mm bins along the dorso- ventral axis were plotted with the corresponding linear ts. The mean values within a bin were plotted for visual clarity, but the regressions were performed using the raw data. (B) The magnitude of the slope between the spatial information score and dorso-ventral position decreased exponentially with the axon terminal eld extent. The error bars denote the standard error for the estimates for slope. (C) The mean spatial information increases exponentially with axon eld extent. The error bars denote standard deviation. (D) The standard deviation of the grid eld areas that a granule cell received were plotted against the corresponding spatial information score. (E) The standard deviation of grid eld areas in the input is correlated with the axon terminal eld extent. A larger axon terminal eld results in a granule cell receiving a total input with a larger variety of grid eld sizes. 5.3.4 Spatial Information Score Depends on Axonal Anatomy A linear regression was performed between the spatial information score and dorso-ventral position which revealed that a spatial information gradient was present in the data for the dentate gyrus. The spatial information score was highest for the activity emerging from the dorsal dentate and was lowest for activity from the ventral dentate. Like the place eld area gradient, the spatial information gradient approached zero as the topography became random (Fig. 5.5A and 5B). Contrary to the mean place eld area which had a very weak correlation with axon eld extent, the mean spatial information score for the population increased as the axon eld extent increased (Fig. 5.5C). 106 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING The 10 mm axon terminal eld network had the highest spatial information score, whereas the 0.1 mm axon terminal eld network had the lowest spatial information score. The spatial information gradient was present in both the infrapyramidal and suprapyramidal blades, but the suprapyramidal blade displayed both a steeper spatial information gradient and a larger mean spatial information score than the infrapyramidal blade (p<<0.001 for all axon eld extents). 5.3.5 Multi-Resolution Inputs Aect Spatial Information Score We next investigated the mechanism by which axon terminal eld contributed to the spatial information score. Theoretical studies have concluded that a multi- resolution representation of space in the input is important to generate singular place elds (Solstad et al., 2006; O'Keefe and Burgess, 2005). We hypothesized that wider axon terminal elds would result in granule cells receiving inputs composed of a larger variety of grid eld sizes which would cause the granule cells to exhibit a higher spatial information score. To test this hypothesis, a linear regression was performed between the spatial information score and the amount of dierent spatial resolutions, i.e., grid eld sizes, represented by the total input to a granule cell which demonstrated that the spatial information score is improved by a multi-resolution input (Fig. 5.5D). The amount of dierent resolutions encoded by the input was quantied by calculating the standard deviation of the grid eld areas that comprised the inputs to a granule cell. Additionally, the amount of dierent spatial resolutions received by granule cells was correlated with the extent of the axon terminal elds as the axon elds constrained the divergence of the input within the postsynaptic region (Fig. 5.5E). 5.3.6 Spatial Information Score and Multi-Resolution Rep- resentations of Space Predict Decoding Performance The next investigation evaluated position decoding performance using the ac- tivity generated by the neuronal network under the dierent axon terminal eld conditions. A subset of 2000 neurons from the suprapyramidal blade and 2000 neu- rons from the infrapyramidal blade were chosen using a uniform random sampling along the longitudinal axis of the dentate gyrus. For each blade, decoding was performed using the same 2000 neurons for each axon terminal eld extent. The 107 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Figure 5.6: Decoding performance as a function of axon terminal eld extent. (A) The position decoding estimates under dierent axon terminal eld conditions are plotted over the actual positions. (B) The average error, represented by the Euclidean distance between the predicted and actual position, is plotted against axon terminal eld extent (black). A linear regression was used to predict decoding performance using two variables: place eld area slope and spatial information score (red). The p-values of the regressions were << 0.001 for both blades. The coecients to Eq. 17 and the eect sizes are listed within each gure. (C) The estimated lower bound of the mutual information between position and the spiking of the neural population is plotted as a function of axon terminal eld extent. 108 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING position estimates and decoding performance are plotted in Fig. 5.6. Rather than a monotonic relation, decoding error varied non-linearly as a function of axon termi- nal eld extent. The smallest axon terminal eld extent at 0.1 mm yielded the best estimate for both blades, followed by a large increase in error at 0.5 mm, a local minimum at 2.0 mm, and another increase in error at 10.0 mm (Fig. 5.6B). The lower bound of spatial information was found to vary with the decoding error (Fig. 5.6C). To explore the relation between axon terminal eld size and decoding error, we performed an analysis using the results obtained in the previous sections. We had demonstrated two opposing gradients: the range of place eld areas that are represented by the population decreased as the axon eld increased, but the spatial information score increased as the axon eld increased. We hypothesized that, due to their opposing relations, decoding performance may be related to a combination of the two properties. The variables and their interaction were linearly combined using an equation of the form ^ E(x) =Af SI (x) +Bf G (x) +Cf SI (x)f G (x) +D (5.10) where ^ E(x) refers to the predicted decoding error, x refers to the axon terminal eld extent, f G (x) refers to power law equations for place eld size gradient from Fig. 5.4B, and f SI (x) corresponds to the power law equations for spatial informa- tion score from Fig. 5.5C. The power law equations were also used to interpolate additional points along the predicted decoding performance curve. The eect size of each variable was quantied using ! 2 which indicated that the interaction between spatial information score and slope was a signicant predictor of decoding error (Fig. 5.6B). These results suggest that a combination of a multi-resolution input and the spatial information score contributes to optimal spatial encoding. An interesting feature is the local minimum in the actual decoding error that appears at 2.0 mm for both blades and in the predicted decoding error at 2.43 mm and 1.32 mm for the suprapyramidal and infrapyramidal blades, respectively. These values are com- parable to the anatomical axon eld extents of 1-1.5 mm found in vivo (Tamamaki and Nojyo, 1993). 109 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING 5.4 Discussion 5.4.1 Large-Scale Modeling and the Incorporation of Bio- logical Constraints Perhaps the most signicant nding of the present study is the successful pre- diction of place eld properties of dentate granule cells in response to grid cell ring of entorhinal cortical neurons as inputs to the dentate. There was no information or constraint relative to place cell ring incorporated to any degree into the model. In this sense, then granule cell place elds truly represent a property that is emer- gent from the network. In addition to the simple emergence of place elds, the simulated entorhinal-dentate network also displayed multiple high-level properties of place elds displayed by granule cells in vivo (de Almeida et al., 2009). For ex- ample, simulated granule cells exhibited an average of 4.10 1.66 place elds with place eld areas of 719.5 185.6cm 2 which were comparable to the 2.2 place elds with areas of 667.3cm 2 that were measured in vivo (de Almeida et al., 2009; Leutgeb et al., 2007). Furthermore, the average spatial information of the simulated gran- ule cells was 0.83 0.03 bits/spike which were within the experimentally reported values of 1.1 0.56 bits/spike (Neunuebel and Knierim, 2012). The only substantial constraints included in the model were morphological (e.g. the division of the apical dendrites of the granule cell models into the granule cell layer and the inner, middle, and outer thirds of the molecular layer and the num- bers and densities of synapses), anatomical (e.g. the topographical projection of entorhinal axons to dentate gyrus and the numbers and distributions of neurons throughout the dentate gyrus), biophysical (e.g. the types and densities of the ion channels represented), and electrophysiological (e.g. the threshold, input resistance, spike frequency adaptation ratio, etc., see Table I), and the synaptic dynamics. Despite the number of components and interdependencies present in the model, the large-scale neuronal network was able to exhibit place eld properties and spatial information scores that were within the ranges reported experimentally. Whereas synaptic and cellular dynamics represent a bottom-up or lower level validation and are the means by which neuronal network models are typically validated, the place elds and spatial information, which rely on a combination of multiple elements outside of the individual cell models, represent a top-down validation or validation at a higher level. 110 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Validation at both levels, given the number of components in the model, suggest that at least an appropriate number of signicant components were included in the model to generate place eld behavior and that these components were adequately constrained with respect to the biology. 5.4.2 Dorso-Ventral Gradient in Dentate Gyrus A possible dorso-ventral organization in dentate gyrus due to the entorhinal- dentate topography and the organization of the grid cell receptive eld properties had been proposed by Solstad et al. (2006). However, the quantication of data regarding the topography of the entorhinal-dentate projection and grid cell receptive eld properties and the integration of both datasets into a comprehensive model had not been performed until this work. The large-scale model presented here predicts that the dentate gyrus has a functional, dorso-ventral organization in which smaller place elds are exhibited dorsally and larger place elds are exhibited ventrally. A similar gradient for place eld area has been reported in vivo for the CA3/4 subeld (Jung et al., 1994) but has not been investigated for dentate gyrus. However, given that the entorhinal-CA3 projection is organized in the same manner as the entorhinal-dentate projection and that the dentate-CA3 projection has a strongly lamellar organization (Amaral and Witter, 1989; Acs ady et al., 1998), it is plausible that the prediction of the large-scale model is accurate. 5.4.3 Transverse Gradient in Dentate Gyrus A transverse gradient was discovered between the suprapyramidal and infrapyra- midal blades. Suprapyramidal granule cells exhibited larger place eld areas, a greater place eld area gradient, and higher spatial information scores than in- frapyramidal granule cells. The key dierence between the suprapyramidal and infrapyramidal granule cell models was the number of inputs that each received with 2117 and 1253, respectively. The ratio of grid cell to LEC inputs was similar with 43% and 46% of the inputs being composed of grid cells for suprapyramidal and infrapyramidal granule cells, respectively. The mean ring rates between the blades were similar at 0.70 Hz (suprapyramidal) and 0.68 Hz (infrapyramidal). Al- though the number of inputs were dissimilar, the feedback inhibition provided by basket cells caused the ring rates to be approximately equal. The range of spatial resolutions available for either the suprapyramidal or infrapyramidal granule cells 111 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING in their input was also similar (Fig. 5.5E). This leaves the number of inputs as the main factor dierentiating the two populations in the present model. A study in- vestigating the dierences between suprapyramidal and infrapyramidal granule cells may reveal additional insights as to the functional dierences between the groups. 5.4.4 High-Level Constraints Predict Function of Lower- Levels Controlling certain properties in neural systems experimentally (e.g. the size of the axon terminal eld) can be impossible given current technology. Computational models, with sucient detail to represent phenomena within and across dierent physical scales, can predict the role of lower level properties in a larger context. The axon terminal eld was found to mediate a trade-o between encoding multiple spatial resolutions or achieving a high spatial information score. Beyond a highly multi-resolution representation of space as achieved by the 0.1 mm axon terminal eld extent which resulted in the greatest decoding performance, the optimal axon terminal eld extents to maximize spatial encoding was predicted to be 2.43 mm and 1.32 mm for the suprapyramidal and infrapyramidal blades which suggests that both properties are necessary for good encoding. Furthermore, these values lie within the range of the reported axon eld extent for entorhinal cortical axonal elds within the dentate gyrus (roughly 1-1.5 mm by Tamamaki and Nojyo (1993)). These results suggest that spatial encoding eciency may be one constraint that is used to determine the in vivo size of the entorhinal axon terminal eld and demonstrate how the large-scale model can be used to generate hypotheses that connect lower level biological properties to a system level function. In the context of quantifying high-level properties of neural system activity, in- formation theory has been essential by providing a framework through which the non-linear relations between external, e.g., neural, biological, and behavioral, cor- relates and single neuron or ensemble neural spiking activity can be reduced into single number metrics. Statistical approaches including decoding algorithms and neural functional connectivity estimation methods can be used to extract various information-theoretic quantities directly from the spiking activity as opposed to histogram-based methods (Barbieri et al., 2004; Pillow et al., 2011; Song et al., 2013). 112 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING The lower bound of mutual information used in this study directly uses the error of the estimate to quantify mutual information, but using and developing more sophisticated techniques are necessary to provide accurate metrics for higher-level function. 5.4.5 Additional Contributions to Spatial Encoding Though grid cells are an important physiological input to the hippocampus, there exist many other functional cell types in the entorhinal cortex that have not yet been incorporated which can add to the dimensions by which the large-scale model can be interpreted. Additional cell types from layer II of the MEC include boundary cells (Solstad et al., 2008) and speed cells (Krop et al., 2015). Deeper layers of the MEC have head direction cells and conjunctive cells (Sargolini et al., 2006). Less is known about specic, functional cell types in the LEC and their topographic organization. Studies in rat have shown that the LEC responds to multi-modal sensory stimuli such as olfaction and vision (Eichenbaum et al., 1996, 2007). Additionally, there is evidence that LEC cells can encode locations of objects within an environment (Deshmukh and Knierim, 2011). Basket cells in the model participated in both feedforward inhibition driven by the entorhinal cortex and feedback inhibition driven by the granule cells but a thor- ough investigation into their contributions on the processing of spatial information was outside the scope of this work. Feedback inhibition has been included in most computational studies of place eld formation to provide competitive inhibition and promote sparse activity (de Almeida et al., 2009; Solstad et al., 2006; Rolls et al., 2006), but the in uence of feedforward inhibition on place eld formation has not been investigated in the literature. Future works will study the contributions of feed- back and feedforward inhibition on spatial encoding. Another signicant interneuron population, mossy cells, comprises the associational system of the dentate gyrus and contributes long-ranging projections along the dorso-ventral axis. We have previ- ously explored the eects of mossy cells on network dynamics (Hendrickson et al., 2015) and will also explore their role in spatial encoding. Another aspect of place eld generation and decoding that has yet to be ad- dressed involves network oscillation which has been shown to encode spatial infor- mation (Tsodyks et al., 1996). As the large-scale model is able to generate spiking data for entire neural populations from which local eld potentials can be predicted 113 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING (Bingham et al., 2018), it is uniquely situated to investigate the functional con- sequences of network oscillations. Computational studies have demonstrated the importance of oscillations for communicating information (Salinas and Sejnowski, 2001; Sejnowski and Paulsen, 2006), and further work will be performed to integrate the analysis of oscillations with typical spiking based methods. 5.5 Conclusion The accessibility and interpretability of parameters allows parametric models of neural systems to simulate unique experiments and provide mechanistic explana- tions of the observed phenomena. The large-scale model presented here is novel in the scope and depth of detail included to describe biologically plausible population level activity approaching the full scale of a single hemisphere of rat hippocampus. With the introduction of grid cells as input, the output of the large-scale neuronal network model and the contributions of its model parameters can now be analyzed in the context of spatial cognition. In this work, the specic role of the axonal anatomy in the rat entorhinal-dentate system was investigated using simulations that would otherwise be impossible to recreate experimentally using present technology, and as such, this work oers one of the few investigations that could explain how network architecture within and between subelds, e.g., spatial processing between dentate gyrus and CA3/4 (Kilborn et al., 1998), aect neural encoding and neural system function. The successful inclusion of grid cells at this stage allows behaviorally rel- evant computational studies to be performed using the large-scale model as it is further expanded to include the CA3/4 and CA1/2 subelds, and the dierences in feedforward connectivity that exist among the subelds will oer additional oppor- tunities in studying how the dierences in architecture, and other model elements, aect the processing of spatial information as it is successively transformed by the trisynaptic pathway. 5.6 References Acs ady, L., Kamondi, A., S k, A., Freund, T., and Buzs aki, G. (1998). Gabaergic cells are the major postsynaptic targets of mossy bers in the rat hippocampus. Journal of Neuroscience, 18(9):3386{3403. 114 CHAPTER 5. CONNECTIVITY-DEPENDENT INFORMATION PROCESSING Alonso, A. and Klink, R. (1993). Dierential electroresponsiveness of stellate and pyramidal-like cells of medial entorhinal cortex layer ii. Journal of Neurophysiol- ogy, 70(1):128{143. PMID: 8395571. 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Abstract (if available)

Abstract In the past decades, the combination of both the capability to simultaneously record multiple neurons and computational methods for studying the activity of neuronal populations, i.e. multiple neurons, have given weight to a hypothesis that may be added to the neuron doctrine: Though neurons are undisputedly the fundamental, discrete units that comprise a neural region, the function of a neural region emerges as a consequence to the activity of groups or ensembles of neurons rather than single neurons. The overall theme of this work is to understand how a large population of discrete elements, i.e. neurons, and the connectivity of these elements, influence the dynamical response of a neural system and achieve a collective function. ❧ The hippocampus is an ideal neural region to study the relation between connectivity and function because its molecular, cellular, and anatomical properties have been particularly well studied. Furthermore, the hippocampus expresses a distinct and topographically organized connectivity between each of its subfields. The function of the hippocampus also gathers significant interest as the hippocampus is crucial to memory formation. However, the exact functions of the hippocampus and the purpose of the hippocampal connectivity, and the other properties that comprise a hippocampus, are not known. ❧ To investigate the relation between connectivity and hippocampal function, I constructed a computational neuronal network model, designed with a three-dimensional geometry and a numerical scale approaching the full number of neurons and synapses of a single hemisphere of a rat hippocampus, which include the dentate gyrus, CA3, and CA1 subfields. Each subfield includes the principal neuron and at least one interneuron, the basket cell. This model was used to investigate the role of topographic connectivity on the emergent spatio-temporal dynamics that were generated by the hippocampal neurons and its influence on the encoding of spatial information. ❧ The thesis will be divided into four major sections: a description of the large-scale model and the method of estimating anatomical connectivity, investigations of the large-scale model using white noise input, the development of a MIMO model framework to quantify the spatio-temporal patterns that the hippocampal subfields perform, and investigations on the role of connectivity in processing spatial information using a physiologically realistic and behaviorally relevant input, i.e. grid cell activity.

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Asset Metadata

Creator Yu, Gene Jong (author)

Core Title Functional consequences of network architecture in rat hippocampus: a computational study

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Publication Date 12/04/2019

Defense Date 12/04/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag CA3,connectivity,dentate gyrus,entorhinal cortex,grid cells,hippocampus,large-scale,neuronal network model,OAI-PMH Harvest,spatial decoding,spatio-temporal correlation,spatio-temporal filter,topography

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Berger, Theodore William (committee chair), Lazzi, Gianluca (committee member), Marmarelis, Vasilis (committee member), Mel, Bartlett (committee member), Song, Dong (committee member)

Creator Email gene.jong.yu@gmail.com,geneyu@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-243573

Unique identifier UC11673964

Identifier etd-YuGeneJong-7989.pdf (filename),usctheses-c89-243573 (legacy record id)

Legacy Identifier etd-YuGeneJong-7989.pdf

Dmrecord 243573

Document Type Dissertation

Rights Yu, Gene Jong

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA