Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
A probabilistic model predicting retinal ganglion cell responses to natural stimuli
(USC Thesis Other)
A probabilistic model predicting retinal ganglion cell responses to natural stimuli
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
i UNIVERSITY OF SOUTHERN CALIFORNIA VITERBI SCHOOL OF ENGINEERING DEPARTMENT OF BIOMEDICAL ENGINEERING A PROBABILISTIC MODEL PREDICTING RETINAL GANGLION CELL RESPONSES TO NATURAL STIMULI NADAV IVZAN ADVISER: NORBERTO M. GRZYWACZ DISSERTATION MAY 2015 ii Acknowledgements First and foremost, I would like to thank my adviser Norberto M. Grzywacz for giving me the opportunity to pursue a doctoral degree in such a remarkable field. I would have never been able to do it without Norberto’s guidance, support & endless enthusiasm. I would also like to make special thanks to Dr. Judith A. Hirsch, Dr. Dong Song, Dr. Bosco S. Tjan, Dr. James D. Weiland, Dr. Bardia F. Behabadi, Dr. Raymond Sarkissian, Victor Barres, Dr. Arvind Iyer, Dr. Wan-Qing Yu, Dr. Eun- jin Lee, Dr. Junkwan Lee, Dr. Joaquin Rapella, Dr. Yerina Ji, Dr. Xiwu Cao, Dr. Justin Lui, Dr. Peter Yan, Rishabh Jain, Rachel Millin, Vandana Suresh, Ulas Ciftcioglu, Mischalgrace Diasanta, Dr. Viviane Ghaderi, Etti Israel, Dr. Reem Taha, Dr. Francois Cadieux, Vina Vo, Scott Ko, Tamay Kiper, Deniz Kahriman, Brendan parker, Sinan Bekdogan, April Adonis, Alex Kates, The Dor family, Dr. Zahava Israeli, Gai Edri, Gadi Feldman, Ronen Abraham, Revital raviv, The Itskovich family, My brother Ofer Ivzan, and last but not least my father, Israel Ivzan, who passed away shortly before this day, this work is dedicated to you, with love Nadav. iii Table of Contents Acknowledgements ................................................................................................................ ii List of Figures ...................................................................................................................... vi List of Acronyms .................................................................................................................. ix 1 Background .................................................................................................................. 1 1.1 Clinical Significance ................................................................................................ 1 1.2 Models of Retinal Responses .................................................................................. 2 1.3 Predicting Responses to Natural Images ................................................................ 3 1.4 Probabilistic Response ............................................................................................ 3 2 Specific Aims (SAs) ...................................................................................................... 4 3 SA1: A Probabilistic Model Predicting RGC Responses to Natural Images .................. 5 3.1 Introduction ............................................................................................................. 5 3.2 Methods .................................................................................................................. 5 3.2.1 Animal model ................................................................................................... 5 3.2.2 Preparation and recording ................................................................................ 6 3.2.3 Stimuli .............................................................................................................. 6 3.2.4 Receptive field analysis .................................................................................... 7 3.2.5 Image projections and binning .......................................................................... 8 3.2.6 Data partitioning and fit evaluation ................................................................... 8 3.2.7 Testing for Linearity ........................................................................................ 10 3.3 Results .................................................................................................................. 11 3.3.1 Difference between natural and artificial images. ........................................... 11 3.3.2 Model development ........................................................................................ 13 iv 3.3.3 Probability of Firing ......................................................................................... 26 3.3.4 3D probability-histogram distributions and model fits ...................................... 29 3.3.5 Analysis of resulting model parameters and curves ........................................ 33 3.3.6 σ*, K* and A parameters ................................................................................ 33 3.3.7 Linearity of PoF curves ................................................................................... 34 3.3.8 Non-linear subunit model ................................................................................ 36 3.4 Discussion ............................................................................................................. 38 3.4.1 Responses to natural and artificial images ..................................................... 38 3.4.2 STA filter ........................................................................................................ 38 3.4.3 Choice of probability distribution and parameters ........................................... 38 3.4.4 PoF curves ..................................................................................................... 39 3.4.5 Fit quality ........................................................................................................ 40 3.4.6 K and A parameters ....................................................................................... 40 3.4.7 Saturation sensitivity ...................................................................................... 41 4 Biophysical Model of the Probabilistic Responses ..................................................... 43 4.1 Introduction ........................................................................................................... 43 4.2 Methods ................................................................................................................ 43 4.3 Results .................................................................................................................. 47 4.3.1 Discussion ...................................................................................................... 56 5 Overall discussion ...................................................................................................... 57 5.1 Future work ........................................................................................................... 58 5.1.1 Natural movies ............................................................................................... 58 v 5.1.2 RF estimation ................................................................................................. 59 6 References ................................................................................................................. 60 6.1 Appendix A - SPDf derivation ................................................................................ 65 6.2 Appendix B1- Normal distribution fit ...................................................................... 68 6.3 Appendix B2 – SPDf fit .......................................................................................... 69 6.4 Appendix B3 – ON RGC stimulated with NI and AI SPDm fit ................................ 70 6.5 Appendix B4 – OFF RGC stimulated with NI and AI SPDm fit ............................... 72 6.6 Appendix B5 – RGC sensitive to spike saturation SPDm fit .................................. 74 6.7 Appendix C1- ON RGC STA ................................................................................. 76 6.8 Appendix C2 – OFF RGC STA .............................................................................. 77 6.9 Appendix C3 – STA of OFF RGC sensitive to spike saturation ............................. 78 vi List of Figures Figure 1-1 Retinal prosthesis and structure. ......................................................................... 1 Figure 1-2 Linear Non-linear (LN) model. ............................................................................. 2 Figure 3-1 Natural and artificial images .............................................................................. 11 Figure 3-2 difference in responses between natural and artificial images ........................... 12 Figure 3-3 maximum likelihood estimate of non-linearity. ................................................... 13 Figure 3-4 STA filter of a on type RGC ............................................................................... 14 Figure 3-5 Image projections and spiking event. ................................................................. 15 Figure 3-6 probability of Firing (PoF) .................................................................................. 16 Figure 3-7 3D probability distribution of spikes fired (F) vs. image projections (I) ............... 17 Figure 3-8 model probability assumption. ........................................................................... 18 Figure 3-9 Δ curve. ............................................................................................................. 19 Figure 3-10 Fit of normal distribution function to 3D probability distribution. ....................... 20 Figure 3-11 Example of an RGC sensitive to spike saturation ............................................ 21 Figure 3-12 spike limiting function....................................................................................... 22 Figure 3-13 Spiking probability distribution function (SPDf) ................................................ 23 Figure 3-14 SPDf fit to RGC sensitive to spike saturation ................................................... 24 Figure 3-15 SPDf and spike limiting function. ..................................................................... 25 Figure 3-16 PoF and Δ curves of ON and OFF type RGCs ................................................ 27 Figure 3-17 PoF and Δ curves of an off RGC with sensitivity to saturation ......................... 28 Figure 3-18 3D Spike probability dist. For ON and OFF RGCs ........................................... 30 vii Figure 3-19 3D probability dist. and model fit for an off RGC with sensitivity to saturation. . 31 Figure 3-20 mean of correlations (R 2 ) between data and model fits .................................... 32 Figure 3-21 Optimized parameters. .................................................................................... 34 Figure 3-22 Rainbow test results ........................................................................................ 35 Figure 3-23 Possible pixel combinations. ............................................................................ 36 Figure 3-24 Combinatorial model. ....................................................................................... 37 Figure 4-1 Retina level processes....................................................................................... 48 Figure 4-2 Bipolar Level processes..................................................................................... 49 Figure 4-3 Ganglion Cell Level Processing ......................................................................... 50 Figure 4-4 Spike Firing ....................................................................................................... 51 Figure 4-5 Spike Triggered Averages produced from NI and AI ......................................... 52 Figure 4-6 SPDm fed with biophysical model data .............................................................. 53 Figure 4-7 Varying all RF sizes by the same factor ............................................................. 54 Figure 4-8 varying the RF STD independently .................................................................... 55 Figure 4-9 σ* versus different RF sizes .............................................................................. 56 Figure 6-1 Cross section plots of figure 3.10 ....................................................................... 68 Figure 6-2 Cross section plots of figure 3.14. ...................................................................... 69 Figure 6-3 Cross section plots of ON-NI panel presented in figure 3.18. ............................ 70 Figure 6-4 Cross section plots of ON-AI panel presented in figure 3.18. ............................. 71 Figure 6-5 Cross section plots of OFF-NI panel presented in figure 3.18. ........................... 72 Figure 6-6 Cross section plots of OFF-AI panel presented in figure 3.18. ........................... 73 viii Figure 6-7 Cross section plots of NI panel presented in figure 3.19. ................................... 74 Figure 6-8 Cross section plots of AI panel presented in figure 3.19. ................................... 75 Figure 6-9 STA filter of ON RGC presented in figure 3.18 .................................................. 76 Figure 6-10 STA filter of OFF RGC presented in Figure 3.18 ............................................. 77 Figure 6-11 STA filter of OFF type RGC that is presented in figure 3.19 ............................ 78 ix List of Acronyms AI Artificial Image DoG Difference of Gaussians ePPR Enhanced Projection Pursuit Regression LN Linear Non-linear LNP Linear Non-linear Poisson MID Maximal Informative Dimensions NI Natural Image NICD Normal Inverse Cumulative Distribution function NM Natural Movie PoF Probability of Firing RF Receptive Field RGC Retinal Ganglion Cell SPDf Spiking Probability Distribution function SPDm Spiking Probability Distribution model STA Spike Triggered Average STC Spike Triggered Covariance VPL Visual Processing Laboratory 1 1 Background 1.1 Clinical Significance There is great interest in developing good mathematical models that explain how the retina encodes visual information. Such models will of course be important to understand neural operations, but also have practical applications. For instance, epi-retinal prosthetic devices 1–4 take images captured by a camera and transform them to the equivalent electrical stimulus that is applied to the retina of patients suffering from retinal degeneration, thus allowing the patients to regain some of their visual sense back (figure 1.1). To create such transformations we need accurate computational models to predict the visual neurons’ response for a given image. Retinal Ganglion cells (RGCs) are ideal for the development of models explaining neural operations because, although the retina is physically separate from the brain, it is by definition a brain tissue and it responds to light stimuli 5,6 . The RGCs output action potentials, like neurons in the brain, and those can be recorded using an electrode array system. Studying visual neurons elsewhere in the brain is much harder as it is difficult to isolate them or reach them with an electrode system. FIGURE 1-1 RETINAL PROSTHESIS AND STRUCTURE. AN EPI-RETINAL PROSTHETIC DEVICE 64 IS PRESENTED ON THE LEFT TRANSFORMING IMAGES CAPTURED BY A CAMERA TO THEIR EQUIVALENT ELECTRICAL STIMULATION PATTERN THAT IS APPLIED TO RGCS IN THE BACK OF THE EYE. THE CELLULAR STRUCTURE OF THE RETINA 65 IS PRESENTED ON THE RIGHT WITH THE RODS AND CONES THAT TRANSFORM THE LIGHT PHOTONS INTO A GRADED ELECTRICAL SIGNAL AND THE RGCS THAT GENERATE ACTION POTENTIALS THAT TRAVEL TO THE BRAIN. Retinal Ganglion Cells (RGC) Bipolar cells Photoreceptors (rods & cones) 2 1.2 Models of Retinal Responses Several groups have attempted to build computational models for connecting images with visual neuron responses. Generally those models are composed of a linear spatio- temporal filter followed by a static non-linearity (LN type models) 7 . The simplest of those models is the Linear-Nonlinear-Poisson (LNP) cascade model 8–10 , which estimates the linear spatio-temporal filter by weight averaging those images that caused a response (Spike Triggered Average – STA 11–13 ). The non-linearity is estimated by fitting a sigmoidal function to the maximum likelihood estimate of the data points relating the dot product of the spatio-temporal filter and the images to the number of spikes generated by them. Because the non-linear sigmoidal function has two free parameters, STA models require optimizing two parameters only, which makes it fast to optimize and implement in real time. Spike Triggered Covariance (STC) is a complementary analysis to the STA method, estimating higher order filters that capture additional features in the stimuli 9,10,14–16 . The advantages of the STA and STC methods are their simplicity and fast application, as they contain a small number of parameters that can be estimated quickly. Moreover, these models do a great job predicting the RGC response to artificial stimuli like Gaussian white noise, sinusoidal grating, moving bars and flashes of different intensities. However these models fail to predict the RGCs responses to natural images 17–22 , which are the stimuli visual neurons are evolutionarily designed to encode 23–26 . Spikes Image projection Average firing rate Non-linear function Spatio-temporal filter Image FIGURE 1-2 LINEAR NON-LINEAR (LN) MODEL. PREDICTING RGC SPIKING RESPONSES TO IMAGES BY CALCULATING THE DOT PRODUCT OF THE IMAGE WITH A LINEAR FILTER AND TRANSFORMING THE RESULTING IMAGE PROJECTION TO SPIKING RESPONSE BY A NON-LINEAR FUNCTION. 3 1.3 Predicting Responses to Natural Images LN models fail to predict responses to natural images because the statistics of those images have special properties 27–31 , which make the response to them unique 32–34 . One of these properties is sparseness, or high kurtosis 35–37 . Thus one should predict that visual neurons would respond in a sparse manner as well. Therefore, the most likely response is zero or small, making the determination of parameters of the STA and STC models difficult. Advanced models that accurately predict the responses of higher level visual cells (cortical neurons) to natural images exist 38–40 . However, for RGCs, those advanced models predict the same as simple STA and STC models. This is because at those higher level cells, the visual information is more refined after extensive processing, usually encoding specific features like direction and orientation. Because RGCs form the first “digital” visual processing stage, converting the visual information into action potentials, the signal they carry is still crude, containing much noise, and those advanced models fail to extract the significant information from it. 1.4 Probabilistic Response Lastly, all the aforementioned computational models predicting the response of visual neurons aim to predict the typical or average response of the neurons. However, neuronal responses are probabilistic, thus varying between trials 41–44 . Some models incorporate a Poisson process that randomizes the response; however, this is not strictly accurate 45–48 . The response variability is due to many reasons including Quantum catch and Synaptic noise. Quantum catch refers to the proportion of incident photons that are captured by the photoreceptor; the inability to capture all the photons causes variability 49,50 . Synaptic noise, is the result of many noisy bipolar cells synapsing on an RGC unit, thus often resulting in false alarms 51,52 . This variability in response applies for RGCs’ responses to both artificial and natural images. 4 2 Specific Aims (SAs) 1. For this aim I developed a probabilistic model that fitted the responses of rat RGCs that I recorded using a multi-electrode array. Briefly, the model projected images to the RF obtained using the STA method, and then added a specially designed noise to the projections. 2. The second aim was to investigate why the STA method using the first aim worked although NI have spatial correlations and to understand why parameters of the probabilistic model depended on the statistics of the image. I developed a biophysically realistic model of the retina and applied to it the analysis used in the first aim. 5 3 SA1: A Probabilistic Model Predicting RGC Responses to Natural Images 3.1 Introduction Because existing computational models fail in predicting the responses of RGCs to natural stimulation, there is a need to develop models that can. Regardless of the type of stimuli, the responses of RGCs are probabilistic. Therefore, aiming to develop a model that predicts the response probability distribution instead of the average or absolute response, like existing models do, will likely result in a more accurate prediction and will also account for the variability in the response. In this chapter of my work, I describe the development of a probabilistic model predicting RGC responses to a sequence of natural images and I present results of the model fit and evaluation of the model parameters. 3.2 Methods 3.2.1 Animal model I used 4-6 week-old Long Evans rats of either sex. All surgical and experimental procedures were in conformance with the guide for care and use of laboratory animals (National Institute of Health) and were approved by the Institutional Animal Care and Use Committee (IACUC) at the University of Southern California. The animal care facility at the university was fully accredited by the American Association for Accreditation of Laboratory Animal Care (AAALAC) during data collection. The animals were dark adapted for one hour before surgery, and all surgery was done in dim red light. The rats were anesthetized using a mixture of Ketamine, Xylazine and Phosphate Buffered Saline (PBS) solution (2:0.27:2 parts respectively). Anesthesia was checked by testing for corneal reflexes and reactions to a paw pinch. An eye was enucleated, and the animal subsequently euthanized with an I.V. overdose of Pentobarbital Sodium. 6 3.2.2 Preparation and recording For isolated retinal recording, the enucleated eye was hemisected and the retina was gently separated from the pigment epithelium and then placed, RGC-side down, over a hole punched in a Whatman Filter paper. The filter paper was then flipped and mounted in the recording chamber so that the RGCs would be accessible through the hole. The isolated retina was placed in the recording chamber within five minutes of interrupting the blood supply in vivo. Once in the chamber, the retina was continuously superfused with oxygenated bicarbonate-buffered Ames solution (Sigma, St. Louis, MO) at 37°C at a flow rate of 3-5 ml/min. The in vitro tissue remained healthy for about 6 hours post isolation. Extracellular recordings were made with a commercial multi-electrode array (UT-96, Blackrock Microsystems, Salt Lake City, UT) having a 10x10 grid with 500-μm inter- electrode spacing. An array was lowered onto the ganglion cell layer and fixed in position when responses were recorded on a maximum number of electrodes. Selected channels of data were digitized with a commercial data-acquisition system (Cerebus, Blackrock Microsystems, Salt Lake City, UT). A custom analysis program sampled the data at 10kHz and sorted the recordings for later analysis 53 . I performed spike sorting offline using the customized program, POWERNAP (Blackrock Microsystems, Salt Lake City, UT) and then used MATLAB (Mathworks, Natick, MA) to analyze the data set. In total I recorded 108 RGCs from 7 different retinas. 3.2.3 Stimuli Light stimuli were provided by an RGC monitor (Accusync LCD 51V, NEC, Itasca, IL) with a refresh rate of 60Hz that was gamma-corrected. The monitor image was reflected off a half-silvered mirror under the microscope stage, and focused on the photoreceptor layer of the retina by the condenser lens to produce an image of 1x1 cm on the retina. I wrote the code for stimulus control in MATLAB using the Psychophysics Toolbox extensions (PTB3). 7 The stimuli included constant illumination at the mean background intensity, full field steps, moving square-wave gratings and fast sequence of natural and artificial images. I displayed the full field step stimulus with 1-sec ON period, 1-sec OFF period for 60 trials. The light step was at 90% contrast above and below the mean background illumination of 9.1 cd/m 2 . Each cell’s RF size and position was next determined by analyzing responses to 25 presentations of a very low spatial-frequency square wave grating (0.042 cycles/mm) drifting upward, downward, leftwards and rightward. The gratings were such that a single dark or bright edge was present on the screen at a given time (temporal-frequency =0.5 cycles/sec). The natural images used in this paper were obtained from an online calibrated-image database 54 . I further calibrated them to be linear with respect to the luminance of the monitor (gamma-correction). The source images with 1536 x 1024 pixels were cropped down to patches with 300 x 300 pixels and stored in 8-bit unsigned format for easy extraction. The artificial images consisted of 100 X 100 squares with different intensities pulled from a normal distribution that was set to cover the full luminance range of the display. Each experiment consisted of a set of natural images followed by a set of artificial images and each set consisted of either 7020 or 10530 images. The images were presented at a fast frame rate (30Hz) and the presentation of the consecutive natural and artificial sets was repeated four times. 3.2.4 Receptive field analysis I used both the square wave grating and images stimuli to estimate the RF of the cells. The method I used to estimate the RF size and location with the square wave grating was described previously 55 . Using the images, presented to the retinas, I calculated the STA by weight averaging those images that caused a spiking response for 16 delays (500 ms duration). For most RGCs, the two methods coincided, yielding roughly the same RF size and location. For the rest of the cells, one of the methods yielded the location and size of 8 the RF and the other didn’t. To make the implementation of the model faster, I used the location of the RF center to cut 50 x 50 pixel patches from all the images presented to each cell around its center location and saved those size-reduced images for further analysis. 3.2.5 Image projections and binning To calculate the image projection values I used the 50 x 50 pixel image patches and their spiking responses to calculate the STA again for 16 time delays. This resulted in 16 consecutive spatio-temporal filters. I subtracted the mean background value from every filter and every image and calculated their dot product, yielding the image projection value for every sequence of 16 images. Binning those values into 15 bins included ordering the values of the projections yielding spiking response from high to low and putting them in 15 equal count bins. I put those projections with least value in the remainder and eliminated them from the data set, considering them to be noise. I used the values of the average of the highest and lowest projection in adjacent bins to determine the borders between the bins. Once bin borders were established, I looked back at all the image projection values, both those that yielded a spiking and those that didn’t, and binned them using the borders. This allowed me to calculate the Probability of Firing (PoF) curve by dividing the number of elements in the equal size bins by the number of total image projections falling into each bin. 3.2.6 Data partitioning and fit evaluation For each fit, the parameters of the model where optimized upon the first half of the data set (training portion) and the second half of the data was used to evaluate the quality of the fit (testing portion). The evaluation was done using qualitative, quantitative and statistical methods: 9 3.2.6.1 Qualitatively Plotting the testing portion of the data in three dimensional histogram bar plot (bar3, Matlab, Mathworks, Nattick, MA) and overlying the model fit by plotting it as a waterfall type plot (waterfall, Matlab, Mathworks, Nattick, MA) resulted in a good qualitative measure of how the model fits the data visually. Additionally, by generating cross section figures of the three dimensional plot and using the four trials done for each experiment to calculate the standard error of the testing portion, I created a more careful qualitative visualization of each three dimensional histogram. Such cross section plots are provided in the appendix figures. 3.2.6.2 Quantitatively Calculating the two dimensional correlation between the histogram bars arising from the test data and the model fit (corr2, Matlab, Mathworks, Nattick, MA) resulted in a single value evaluating the quality of the fit. 3.2.6.3 Statistically Because I felt the quantitative test was too relaxed, I created a stringent statistical test to evaluate the fit quality. The statistics are based on a multinomial test with randomization. I used the multinomial test with randomization to know the probability of our fit to the data arising by chance. I first calculated the probability density associated with the test data and model fit using the multinomial test function: < 𝑒 𝑞𝑢𝑎𝑡𝑖𝑜𝑛 1 > Pr(𝑋 ) 0 = 𝑁 ! ∏ 𝜋 𝑖 𝑥 𝑖 𝑥 𝑖 ! 𝑘 𝑖 =1 Where π are the probabilities predicted by the model and x are the number of counts in each bin, and N is the total number of counts. Next I generated 1000 random distributions 10 with the same marginal probabilities as the original fit and repeated the calculation of the probability density for each randomization. By dividing the number of the randomization resulting probability density values below the fit’s by the total number of randomizations (n=1000), I was able to obtain a significance value that indicates the probability of the fit arising by chance. 3.2.7 Testing for Linearity 3.2.7.1 Rainbow test for lack of fit in regression The rainbow test for non-linearity is explained in full length in a previous publication 56 . In brief, to implement the rainbow test I fitted two linear fits to each Δ curve – one consisting of the full 15 data points of the curve and the second to the 7 middle data points of the curve. The error between the data and the fit for both cases was calculated and the F statistics were yielded using the following function: < 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2 > 𝐹 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 = 𝐸 𝑓𝑢𝑙𝑙 − 𝐸 ℎ𝑎𝑙𝑓 𝐷𝑂 𝐴 𝑓𝑢𝑙𝑙 − 𝐷𝑂 𝐴 ℎ𝑎𝑙𝑓 ∗ 𝐷𝑂 𝐴 𝑓𝑢𝑙𝑙 𝐸 𝑓𝑢𝑙𝑙 To calculate the significance level of the data using the F statistics I used the F- cumulative distribution function using the appropriate degrees of freedom (DOAfull=13, DOAhalf=5): < 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3 > 𝑃 = 1 − 𝑓𝑐𝑑𝑓 (𝐹 , 𝐷𝑂 𝐴 𝑓𝑢𝑙𝑙 − 𝐷𝑂 𝐴 ℎ𝑎𝑙𝑓 , 𝐷𝑂 𝐴 𝑓𝑢𝑙𝑙 ) To give some intuition, the lower the F statistics value is, the more linear the curve under examination is. Moreover, a low F statistics value results in a high P value, indicating that the half and full range fits result from the same statistical distribution. 11 3.3 Results 3.3.1 Difference between natural and artificial images. The STA type models that exist in the literature work well for artificial images 8,9,12 , but fail predicting the response to natural images 13 . I wanted to know why those models fail and see the difference in responses and image statistics between natural and artificial images. Figure 3.1 presents examples of the natural and artificial images I use (NI and AI respectively). The NI were obtained from the Van Hateran data base 54 and contain many large areas of similar intensity. For example, the sky presented in the third image (from the top left) takes a large portion of the image and is formed of pixels with nearly equal intensity. These large areas of equi-luminance result in the low spatial power governing natural images 29,35 . In contrast, the AI set shows much variability in the images. The intensity of the 100µm squares was pulled from a Gaussian distribution with a mean of gray intensity value and a standard deviation aimed to cover the full range of intensities, from pitch black to bright white. Therefore, most of the pixels are gray, but many squares with darker or brighter intensity lay between them as can be seen in the images. Now showing the difference in responses to the stimuli, figure 3.2 presents the responses to natural and artificial images from the same RGC for 30 consecutive images. Each set of images was presented four times at a fast rate (30Hz). In the figure, every row presents one of those four repetitions. For both types of images the response shows much variability with the spike magnitudes varying between trials. However, the responses to NI FIGURE 3-1 NATURAL AND ARTIFICIAL IMAGES. ON THE LEFT, 7 NATURAL IMAGES (NI) FROM THE VAN HATERAN DATABASE ARE PRESENTED. ON THE RIGHT, A SEQUENCE OF 7 ARTIFICIAL IMAGES CREATED WITH PIXEL INTENSITIES PULLED FROM A NORMAL DISTRIBUTION WITH A BACKGROUND MEAN AND A STANDARD DEVIATION VALUE AIMED TO COVER THE FULL LUMINANCE RANGE OF OUR PROJECTION SCREEN 12 show clusters of repeated and significant increase in the spike firing magnitude (0.1-0.3, ~0.5 and 0.7-0.9 seconds) and long periods of quiescent activity (0.5-0.7 seconds). In contrast, the AI elicits a constant activity of relatively smaller magnitude. What’s important to notice is that, for the AI, the response is very noisy, thus frequent, as the RGC is bombarded with pixels of different intensities, therefore the quiescent periods are rare to non-existent. Next, I wanted to know what in the STA type models causes the failure to estimate the RGC’s response to NI. In figure 3.3 I present the maximum likelihood estimate of the data points, called image projections, that are used to fit the non-linear function in the STA model 57 . I will explain how I calculate the image projection value in the next section discussing developing the model. For now, what’s important to mention is that each blue star in figure 3.3 relates a sequence of images with the average number of spikes generated by it. Therefore, as the image projection value increases, the potential of the image sequence to elicit a spiking event grows as well. The red lines are the maximum likelihood estimate of those data points, used to fit the non-linear function upon in the LN Time (Sec) Time (Sec) Spikes per Image Response to Natural Images (NI) 0 1 2 3 0 1 2 3 0 1 2 3 0 0.2 0.4 0.6 0.8 1 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 0.2 0.4 0.6 0.8 1 0 1 2 3 Spikes per Image Response to Artificial Images (AI) Figure 3-2 difference in responses between natural and artificial images. Spiking responses over four trials to sets of 30 natural and 30 artificial images presented at a 30Hz frame rate to a retinal ganglion cell. 13 model. For the AI case, the maximum likelihood estimate of the data points rises well, giving a good estimate of the data points for the non-linear function to fit on. However, for NI, the maximum likelihood estimate fails to rise for most of the image projection range (-0.8 to 0.4) and rises only minutely (0.5 spikes per image) for the remaining range. This is because for NI, the most likely response is zero arising from the low spatial power component dominating them. I believe that this is the reason why computational models that easily predict the response for artificial stimuli fail when tested with natural images. I therefore set out to develop a model that will be able to predict the responses to both natural and artificial images and account for the response variability as well. 3.3.2 Model development Because deterministic models with few free parameters do not work very well for natural images, I endeavor to develop a probabilistic model. I wanted the model to have components that are compatible with known retinal anatomy and physiology. I consider the receptive field (RF) of the RGC as the linear filter of an STA type model capturing the -0.8 0 0.8 0 0.5 1 -0.8 0 0.8 0 0.5 1 Image projections Image projections Natural Images Artificial Images Spikes per image Spikes per image FIGURE 3-3 MAXIMUM LIKELIHOOD ESTIMATE OF NON-LINEARITY. THE IMAGE PROJECTIONS FOR NI AND AI ARE PRESENTED BY THE BLUE STAR MARKERS VERSUS THEIR RESPECTIVE FIRING RATES, THE MAXIMUM LIKELIHOOD ESTIMATE OF THE PROJECTIONS IS MARKED BY THE RED CURVES. 14 important elements of an image 8 . The filter is calculated by weight averaging those images that caused a response. Because the spiking response can happen with delay, I needed to perform the weight averaging calculation for images previously presented to the RGC. So I repeated this image weight averaging for 15 previous images presented to the cell. This resulted in a spatio-temporal filter with 16 delays. Figure 3.4 shows such a filter for a ON type cell with 6 delays. The figure shows how the RF changes over time from the most current time (bottom-right) to previous times (leftward). The rightmost panel, which corresponds to the 0 delay filter, shows that the cell likes bright objects presented at the center of its RF. Looking at the previous delay (2 nd square from the left), the bright spot gets bigger, but from then on it shrinks, inverting to a darker intensity after the 4 th delay, and eventually becoming gray again. That means that more than the cell likes bright objects presented at the center of its RF, it likes objects that change from dark to bright intensity over time, presented at the center of its RF. Thus it is an ON type RGC. Every neuron has a pre-firing signal (R) that is measured by the voltage difference across its membrane. When this pre-firing signal crosses a threshold, spike firing occurs. I recorded the spike firing events using extracellular recording, however I still needed an estimate of the pre-firing signal to predict when spiking events are likely to occur. By FIGURE 3-4 STA FILTER OF A ON TYPE RGC. THE FILTER SEQUENCE SHOWS HOW THIS PARTICULAR RGC PREFERS TRANSITIONS FROM DARK (3 LEFT FILTERS) TO BRIGHT (3 RIGHT FILTERS) OBJECTS PRESENTED AT THE CENTER OF ITS RF. 15 calculating the dot product of the linear filter with a sequence of images that were presented to the cell, I obtained a single value for a given time that measures the potential of the sequence to generate a spiking response. This value is an estimate of the pre-firing signal and I’ll refer to it as the image projection (I). Figure 3.5 shows how the projection values (blue curve) relate to the spiking events I recorded (red vertical-dotted lines). As the image projection value increases, the likelihood of spike firing increases. For example, the large increase in the value at 2.5 seconds from -0.1 to 0.5 indicates the RGC pre-firing signal crossing the firing threshold, resulting in the two spiking events that follow it. However, because I is only an estimate, and not an empirical measure, sometimes spike firing occurs even though the projection value is low (1.7 seconds). This also happens because neurons are noisy, sometimes discharging without the optimal stimulus presented to them. But, in the model, I use the image projection value as an estimate of the pre-firing signal, because the estimate is accurate enough. Figure 3-5 Image projections and spiking events. The image projection values are presented by the blue curve fluctuating about zero and the recorded spiking events are shown by the dashed-vertical lines. 16 Next I developed an alternative solution to the fact that the maximum likelihood estimate of the data points fails to rise for natural images (figure 3.3). Instead of calculating the maximum likelihood estimate of the data points, I calculated the probability of generating a response for increasing projection values. I did this by binning those image projection values generating a response (Spikes>0) into 15 equal count bins. The size of the bin was determined by dividing the total number of projections generating a response by 15 and removing out of the data those projections with least value that fall into the remainder. I considered them to be noise. I divided the size of each bin by all the image projection values falling into it (Spikes >=0). This resulted in the conditional probability for generating a spiking response given a projection value. Figure 3.6 shows those conditional probability values for an RGC presented with images. Notice that the probability values rise monotonically with increasing values of I. I will refer to this curve as the Probability of Firing (PoF) curve because it succeeds in establishing a set of data points that captures the RGC’s response to NI. FIGURE 3-6 PROBABILITY OF FIRING (POF). PRESENTING THE RISE IN THE PROBABILITY OF SPIKE GENERATION BY THE RGC WITH INCREASING IMAGE PROJECTION VALUES 17 Because each probability data point in the PoF curve is the probability of generating any spiking response, it is also the sum of the probabilities of firing increasing number of spikes (F). For instance, the highest projection bin (I=0.50) presenting a spiking response probability of 0.37 is composed of the probabilities to generate 1 spike per image (P(F=1|I=0.50)=0.27) and 2 spikes per image (P(F=2|I=0.50)=0.08) and 3 spikes per image (P(F=3| I=0.58)=0.01). Repeating this breakdown of probabilities for all the image projection values, I generate a three dimensional probability distribution as presented in figure 3.7. The probabilities increase with rising image projection values and decay with increased firing rate. FIGURE 3-7 3D PROBABILITY DISTRIBUTION OF SPIKES FIRED (F) VS. IMAGE PROJECTIONS (I) 18 Because the PoF curve and the 3D histogram are composed of probabilities, I set out to find a probability distribution function that can characterize this data. Looking back at the structure of the retina, it is known that each RGC receives information from many bipolar cell units. The response of the bipolar units is noisy, being independent and random. This conforms well with the Central Limit Theorem (CLT) which states that the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed 58 . Since each RGC collects the responses of its bipolar units, it is safe to assume that the RGCs pre-firing response is normally distributed about a mean value (µ), with a characteristic standard deviation (σ) and there exists a threshold (θ), above which those signals generate spiking responses. This can be visualized in figure 3.8, where those pre-firing signals above θ are represented by the gray area in the distribution and cause a firing response. FIGURE 3-8 MODEL PROBABILITY ASSUMPTION. RGC PRE-FIRING SIGNALS ARE NORMALLY DISTRIBUTED ABOUT A MEAN (µ), WITH A CHARACTERISTIC STANDARD DEVIATION (σ) AND THERE EXISTS A THRESHOLD (Θ) ABOVE WHICH THOSE SIGNALS GENERATE A SPIKE FIRING RESPONSE. 19 I therefore set out to explore this possibility. I started by assuming that, although the σ is characteristic, the mean (µ) of the distribution rises proportionally with increased likelihood of spike firing. Now, I do not have access to this µ value because I did not record the pre- firing signals directly (i.e. intracellular recording). I just have an estimate of the pre-firing signals through the image projection values. But, because I know the probability of generating a response for increasing image projection values (PoF curve), I can calculate the distance from the mean (µ) to the threshold ( ϴ) in units of standard deviation (σ) based on the probability values, using the Normal Inverse Cumulative Distribution function (NICD, norminv, Matlab, Mathworks, Nattick, MA). I will refer to the resulting values as Δ because they represent the difference between µ and ϴ in units of σ: < 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 4 > Δ = 𝜃 − 𝜇 𝜎 Calculating the Δ values based on the probabilities in the PoF curve (figure 3.6), results in a sequence of unitless data points (Figure 3.9) that rises with increasing image projection values and closely follows the PoF curve from which it emanates. FIGURE 3-9 Δ CURVE. DIFFERENCE BETWEEN THE MEAN (µ) AND THE THRESHOLD ( ϴ) IN UNITS OF STANDARD DEVIATION (σ) DECREASES FOR INCREASING IMAGE PROJECTION VALUES EMANATING FROM POF IN FIGURE 3.6. 20 Now that I have the Δs, all I have left to do to see how well a normal distribution will match the data is fit the firing probabilities at each projection (I) with a normal distribution function having Δ values taken from figure 3.9 with σ optimized through a unconstrained non-linear optimization method (fminsearch, Matlab, Mathworks, Nattick, MA). This results in an excellent qualitative fit as presented in figure 3.10. Appendix figure B1 presents cross section plots of this 3D figure. The fit curve presented by the black lines cut through the middle of the histogram bars, showing how the distribution mean shifts leftward for increasing projection values. This impressive qualitative fit is achieved with only a single free parameter optimized σ, which equals 0.88. FIGURE 3-10 FIT OF NORMAL DISTRIBUTION FUNCTION TO 3D PROBABILITY DISTRIBUTION. 21 Using normal distribution function worked well for most of the RGCs I recorded from (98 out of 108). However, for the rest of the cells, the spike firing probability did not decrease monotonically with increasing number of spikes fired, but peaked at an intermediate spike firing magnitude, like the RGC presented in figure 3.11. This could be because of the cells’ refractory period, which limits the number of spikes that the RGCs can generate in a given period of time and therefore induces the highest firing probability for an intermediate spike firing rate. I call this phenomena spike saturation and I developed a new probability distribution function by incorporating a spike limiting function into the normal distribution function to account for this data as well. FIGURE 3-11 EXAMPLE OF AN RGC SENSITIVE TO SPIKE SATURATION, WHERE THE PROBABILITY TO FIRE 2 SPIKES SURPASSES THE PROBABILITY TO FIRE OTHER SPIKE MAGNITUDES FOR HIGH PROJECTION VALUES. 22 The spike limiting function (Figure 3.12) produces a firing magnitude F in response to a pre-firing signal R and holds three variables A, K and θ: < Equation 5> 𝐹 (𝑅 ) = 𝐴 (𝑅 −𝜃 ) 𝐾 +(𝑅 −𝜃 ) To understand how this function works, I will explain its variables and their function. θ is the same pre-firing signal threshold I presented before in the normal distribution (Figure 3.8) above which, spike firing occurs. The A parameter represents the maximal number of spikes the RGC can fire per image. K, the semi-saturation constant, controls the sensitivity of the saturation mechanism. For example, if K is equal to zero, the value of F for any R above θ is A, meaning that the sensitivity to saturation is high. However, if K is very large (∞), F would equal to zero for any value of R, so it represents insensitivity to saturation. I incorporated the spike limiting function into the normal distribution function by using a change of variables operation (detailed derivation is provided in Appendix A). This resulted in a distribution function that is not necessarily symmetric about a mean value like the normal distribution, but it can be skewed (Figure 3.13). I will refer to this new function as FIGURE 3-12 SPIKE LIMITING FUNCTION PREDICTING THE SPIKE FIRING MAGNITUDE FOR INCREASING PRE-FIRING SIGNALS AND INCORPORATING THE PHYSIOLOGICAL REFRACTORY PERIOD OF EACH RGC THROUGH A MAXIMAL FIRING RATE A. 23 Spike Probability Distribution function (SPDf) and the model it constitutes as SPDm. The function holds two free parameters that need to be optimized K and A: <Equation 6> 𝑍 (𝑦 ) = 𝑦 𝐴 −𝑦 <Equation 7> 𝑓 𝑦 (𝑦 ) = | 𝐾 (1+𝑍 (𝑦 )) 2 𝐴 | 1 √2𝜋 𝑒 − (𝛥 +𝐾𝑍 (𝑦 )) 2 2 Those parameters hold the same purpose as in the spike limiting function. As figure 3.13 shows, the peak of the distribution function occurs within the gray area, which presents those pre-firing signals causing a spike firing response. This is important because I needed this distribution shape to model those RGCs showing spike saturation behavior. FIGURE 3-13 SPIKING PROBABILITY DISTRIBUTION FUNCTION (SPDF) SHOWING HOW THE FUNCTION CAN CAPTURE THE UN-SYMMETRIC PROBABILITY STRUCTURE OF RGCS SENSITIVE TO SPIKE SATURATION 24 Fitting the SPDf function was done using a non-linear constrained optimization method (fminunc, Matlab, Mathworks, Nattick, MA). This method was used because I needed to constrain the value of A to remain above the maximal recorded firing rate. An example of the fit to the same RGC presented in figure 3.11 is provided in figure 3.14 (Appendix B2 provides cross section plots of this figure with error bars). As you can see, the fit lines generally follow the probability data, and at those projection values the spiking saturation is becoming prominent (I>4), the SPDf transitions to capture the rise in the probability for intermediate firing rate value (F=2). Evaluating the fit quantitatively through a two dimensional correlation (corr2, Matlab, Mathworks, Natick, MA) results in an excellent fit (R 2 =0.99). And using a stringent multinomial statistical test with randomization yielded a significance level of 16%, indicating that the SPD model successfully captures the probabilistic data. FIGURE 3-14 SPDF FIT TO RGC SENSITIVE TO SPIKE SATURATION 25 To provide some intuition about the SPDf function and its parameters, figure 3.15 presents the SPD function (left panel) and the spike limiting function (right panel) with three different K and A parameter sets and a Δ equal to zero. In the simplest case when K and A are equal to each other but much greater than Δ (blue line: K and A equal to a 100), the function takes the form of a normal distribution with a standard deviation equal to 1. However, when the K and A are equal but relatively close to Δ (green line: K and A equal to 3), the function is no longer similar to a normal distribution, but presents spike saturation with an asymmetry about zero. Lastly, when K and A are not equal to each other (red line: K=1 and A=5), the function models a spiking saturation with the probability increasing to a peak and falling to zero as the pre-firing signal reaches A. In sum, the function is versatile and can model both RGCs that are sensitive and insensitive to spike saturation. FIGURE 3-15 SPDF AND SPIKE LIMITING FUNCTION (LEFT AND RIGHT PANELS RESPECTIVELY) PLOTTED WITH THREE DIFFERENT PARAMETER SETS AND Δ EQUAL TO ZERO. 26 3.3.3 Probability of Firing The transition from the Probability of Firing curve (PoF figure 3.6) to its corresponding Δ curve (figure 3.9) presented an interesting transformation, between the probability of the image projection to generate a spike firing response to the distance between the distribution mean (µ) and the firing threshold (θ) in units of σ, namely Δ. This was key in developing the model. I wanted to see if the monotonic rise in the Δ and probability values is universal and if there is a difference in this transformation between the responses to natural and artificial images, between different cell types (ON and OFF), and between those RGCs that are sensitive to spike saturation and the rest. Then I wanted to quantify this difference and see if it holds statistical significance. 3.3.3.1 ON and OFF RGCs The panels in Figure 3.16 present the Probability of Firing and Δ curves for a ON and OFF type RGCs, respectively (The STA filters of those RGCs are presented in appendices C1 and C2, respectively). In figure 3.16, the RGCs are stimulated with Natural (NI- left column) and artificial (AI – right column) images. For instance, the top left panel of the figure presents the PoF and Δ curves for a ON type RGC stimulated with NI. Now, because both curves share the same abscissa, namely the image projections, I was able to present them in the same panel. More specifically, the PoF is presented by the blue stars connected by a line and its ordinate is on the left side of the panel. The Δ curve is presented by the red squares and its ordinate is on the right side of the panel. Observing the curves of the ON cell more closely, in response to NI stimulation, the PoF and Δ curves both rise for increasing projection values. However, the rise is not monotonic holding minute reductions in it, like the decrease in the I=2 point. In contrast, for the AI stimulation (top-right panel), both curves rise monotonically, following each other closely. The non-monotonic rise of the curves for the NI stimulation might result in from the 27 special properties of those images compared to artificial ones, producing sparser spiking events with greater magnitude. Considering the PoF and Δ curves of the OFF RGC (bottom two panels), the results are similar to the ON type results where for both the natural and artificial image stimulation both the PoF and Δ curves rise for increasing image projection values. Also, for natural image stimulation, the rise is non-monotonic again with minute reductions, like the one at I=0.6. Because when I calculate the image projections I subtract the mean background from both the images and STA filters, the resulting image projections are fluctuating about zero and show affinity for spike firing by increasing value for both the ON and OFF type RGCs (Figure 3.5). This results in a standardization of the model response for both types of RGCs where, for increasing projection values, the potential for firing rises. FIGURE 3-16 POF AND Δ CURVES OF ON AND OFF TYPE RGCS STIMULATED WITH NATURAL AND ARTIFICIAL IMAGES -1 0 1 2 0 0.5 1 -1 0 1 2 -2 0 2 -1 0 1 2 0 0.2 0.4 0.6 -1 0 1 2 -1.5 -1 -0.5 0 -4 -2 0 2 4 6 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 6 -1 -0.5 0 0.5 1 -5 0 5 0 0.5 1 -5 0 5 -1 0 1 NI ON Prob ability of Firing (PoF) Image Projection (I) Image Projection (I) Δ Image Projection (I) Image Projection (I) OFF Probability of Firing Probability of Firing Probability of Firing Probability of Firing Δ Δ Δ AI 28 3.3.3.2 OFF RGC with sensitivity to saturation Because RGCs sensitive to spike saturation present a special category in my data, I wanted to see how their PoF and Δ curves differ from the rest of the RGCs. Figure 3.17 presents the PoF and Δ curves for a OFF type RGC that is sensitive to spike saturation. The RGC spatio-temporal filter is presented in appendix figure C3, showing the RGC prefers transitions from bright to dark intensities presented at the center of its RF. In figure 3.17, in response to NI stimulation, the PoF and Δ curves rise non-monotonically with increasing image projection values. This is similar to the result in RGCs that are not sensitive to spike saturation (Figure 3.16). However unlike the RGCs with low sensitivity to spike saturation for AI stimulation, the RGC in figure 3.17 presents Δ and PoF curves with a non-monotonic rise for increasing image projection values. Therefore, for the RGC sensitive to spike saturation, the curves for both natural and artificial images rise non-monotonically. FIGURE 3-17 POF AND Δ CURVES OF AN OFF RGC WITH SENSITIVITY TO SATURATION 29 3.3.4 3D probability-histogram distributions and model fits 3.3.4.1 ON and OFF RGCs I wanted to examine how the 3D probability histograms and model fits vary between natural and artificial image stimulation and between ON and OFF type RGCs. Figure 3.18 presents the distributions and model fits for the same ON and OFF RGCs whose curves were presented in figure 3.16. On each panel in the figure I present the K and A parameter values of the model fit and their ratio (K/A). To evaluate the quality of the fit I use both a quantitative and statistical approach. By using a two dimensional correlation (R 2 ) between the histogram bars and model fits I calculate a quantitative value and by using a statistical multinomial test with randomization I evaluate the fit statistically. Considering the top left panel in the figure (ON RGC stimulated with NI) both the histogram bars and model fit rise for increasing projection values and decay for increasing number of spikes, following each other closely and resulting in an excellent qualitative fit. Looking closely, you can observe the minute reduction in the probability at I=2 ,which arises from the minute reduction in figure 3.16. Quantitatively, the model fits the data well, resulting in a near perfect fit (R 2 =0.99). Evaluating the fit with a stringent statistical test, the multinomial test with randomizations, the model fails (P=0), indicating that the model fit and test data do not arise from the same distribution. The ON-AI result (top right panel) present similar quality fit (R 2 =0.99) with rising probability for increasing projection values and decaying probability for increasing magnitude of spikes fired. The P value again is unsuccessful in capturing the statistical significance of the fit. However compared to the NI stimulation the AI firing probabilities are slightly greater for all the projection bins. Comparing the K and A parameters value for the fits those values increase by more than a tenfold from NI to AI (K: 15.10 372.70, A: 13.05 213.52). However, the ratio of K/A presents an increase of lesser magnitude (K/A: 1.15 1.74). Cross section plots of this ON RGC 3D distribution and model fit are provided in appendix B3. 30 Looking at the OFF type RGC (bottom two panels), the model fits the test data well, resulting in R 2 value of 0.99 for both the natural and artificial image stimulations. For both fits the probability rises for increasing image projection values and decays for increasing magnitude of spikes fired, resulting in a good qualitative fit. However, both fits fail statistically, resulting in a P value of 0. Like in the case for the ON RGC, the response probabilities for the AI stimulation are slightly greater, but for both the NI and AI stimulation, the probabilities of the OFF RGC are lower compared to the ON RGC. The K and A parameters are lower compared with the ON RGC. However, like the case of the ON cell, they rise from NI to AI stimulation and the ratio of K/A rises as well (NI AI, K: 2.94 3.64, A: 3.61 3.95, K/A: 0.81 0.92). For a better evaluation of the fits, cross section plots of the three dimensional figure for this OFF RGC are provided in Appendix B4. Figure 3-18 3D SPIKE FIRING PROBABILITY DISTRIBUTIONS AND MODEL FITS FOR ON AND OFF TYPE RGCS STIMULATED WITH NATURAL AND ARTIFICIAL IMAGES 31 3.3.4.2 OFF RGC with sensitivity to saturation Figure 3.19 presents the 3D probability distributions for the same saturation sensitive OFF type RGC whose curves are presented in figure 3.17. For the natural and artificial image result, the probability of response increases with increasing projection values, but the probability does not decay with an increasing number of spikes fired for the highest image projection values (I= 8). It peaks at an intermediate value of F= 2. Quantitatively, the model fits both the artificial and natural image data well (R 2 =0.99), and qualitatively, the model fits the data well. However it is not as accurate as the fits for the RGCs insensitive to spike saturation (Figure 3.18) and holds minute “glitches” like the one at I=5 and F=2 for NI, and I=6 and F=2 for the AI. Testing the fit statistically for the NI stimulation, the resulting P=0.003 value fails to indicate statistical significance. However, for the AI case, the P=0.09 value presents a statistical significance of above 5%. Comparing the K and A parameters value between the two types of stimulation, the A value decreases minutely and the K value increases by a slight amount from NI to AI (K: 1.79 2.06, A: 3.84 3.73). The K over A ratio slightly increases from NI to AI (K/A: 0.46 0.55) but its value is much lower compared to the insensitive ON and OFF RGCs. Lastly, cross section plots of this figure are presented in appendix figure B5. FIGURE 3-19 3D PROBABILITY- HISTOGRAM DISTRIBUTIONS AND MODEL FITS FOR AN OFF RGC WITH SENSITIVITY TO SATURATION THAT WAS PRESENTED WITH NATURAL AND ARTIFICIAL IMAGES. 32 3.3.4.3 Quantitative and Statistical fit analysis of all RGCs Considering all the 108 RGCs, figure 3.20 presents the mean and standard error of the correlations between data and model fits for NI and AI. The mean correlation for natural images (R 2 =0.971±0.006) is greater than the mean correlation for artificial image stimulation (R 2 =0.90±0.02). Testing all the RGCs statistically with the randomized multinomial test, 31 of the RGCs stimulated with NI and 35 of the RGCs stimulated with AI resulted in statistical significance of above 5% out of the total 108 RGCs. FIGURE 3-20 MEAN OF CORRELATIONS (R 2 ) BETWEEN DATA AND MODEL FITS FOR ALL RGCS STIMULATED WITH NI AND AI. THE ERROR BARS CORRESPOND TO THE STANDARD ERROR OF THOSE CORRELATIONS 33 3.3.5 Analysis of resulting model parameters and curves Because I noticed differences in the model parameters between NI and AI stimulation and between those RGCs sensitive to saturation and the rest, I wanted to know if those differences applied to all of the RGCs. To start this analysis I reworked the parameters of the model into a non-dimensional form. Mathematical details of the work appear in the appendices (section 6.1). Of particular importance the appendix defines K* as the ratio of K/σ, a unitless representation of the 50% saturation point. Moreover the appendix introduces σ* as K* divided by A. 3.3.6 σ*, K* and A parameters Throughout the process of fitting the SPDm to the RGC data I noticed consistent differences in those fits to the different stimuli types and between those RGCs sensitive to saturation and the rest. The optimized free parameters for all the RGCs’ I fitted are presented in panels 3-21a and 3-21b. Panel 3-21a presents the σ* values of all the RGCs arising from NI versus AI, with the parameters of those RGCs sensitive to saturation marked by the full circles. This panel shows two significant results. The first is that AI stimulation results in a greater σ* value compared with NI, that is because most circles lie above the diagonal line, and the second is that those RGCs sensitive to saturation have a lower σ* value compared with most other RGCs. Now, because for saturation-sensitive RGCs K* and A are not strictly dependent panel 3-21b presents those, with each RGC’s NI (hollow circle) and AI (full circle) data points, connected by a line. The horizontal and vertical lines crossing the panel present the medians of the A and K* data points (A=4 and K*=1.1). The interesting thing about the values in panel 3-21b is that K* is greater for AI compared with NI except for one RGC. 34 FIGURE 3-21 (A) OPTIMIZED σ* OF ALL RGCS FOR NI VS. AI WITH THOSE SATURATION SENSITIVE RGCS PRESENTED BY THE FULL-BLACK CIRCLES AND THE REST BY THE HOLLOW CIRCLES. (B) K* VS. A OPTIMIZED PARAMETERS FOR THOSE SATURATIONS SENSITIVE RGCS FOR NI (HOLLOW) AND AI (BLACK-FULL) STIMULATION, WITH EACH RGC DATA CIRCLES CONNECTED BY A LINE. 3.3.7 Linearity of PoF curves Now I wanted to go back and look at the PoF curves since they are the first step in my models’ ability to successfully predict the RGCs’ responses to NI. For this, I implemented the rainbow test for lack of fit in regression 56 on the PoF curves to check if they are linear or not. Panel 3-22a presents the test results in log scale for all the RGCs for NI versus AI. The panel shows that for the majority of the RGCs I cannot reject the linearity of both the NI and AI PoF curves at a significance level of 1% (n=50, circles within the upper right quadrant of the panel). However, I also noticed that a significant portion of the RGCs have AI-PoF curves that are statistically non-linear and NI-PoF curves that are linear (n=35, bottom-right quadrant). Looking more closely at those curves I found that the AI-PoF curves hold a particular shape, examples of which are presented in panel 3-22b. In this panel I present five exemplary AI-PoF curves, which happen to overlay on each other, and whose origin RGCs are marked by the star symbols in panel 3-22a. As you can see for those 35 RGCs the AI-PoF, initially rises for increasing image projection values, then plateaus, and finally rises again. I noticed this type of curve for roughly forty of the RGCs and I was curious about its origin. The NI-PoF curves of the exemplary RGCs are presented in panel 3-22c. In general, those present an increase in the probability for greater image projection values. FIGURE 3-22 (A) RESULTS OF THE RAINBOW TEST FOR LACK OF FIT IN REGRESSION FOR THE RGCS POF CURVES FOR NI VERSUS AI PLOTTED ON A LOG SCALE. (B) FIVE EXEMPLARY AI-POF CURVES WHOSE RAINBOW TEST RESULTS ARE PRESENTED BY THE STAR DATA POINTS IN PANEL 3-22A. (C) NI-POF CURVES OF THE SAME RGCS WHOSE AI-POF CURVES ARE PRESENTED IN PANEL 3-22B. 36 3.3.8 Non-linear subunit model To explain the peculiar shape of the curves presented in panel 3-22b I developed a simplistic computational model that uses combinatorial effects and our knowledge of the retinal physiology to reproduce those shapes. For this model I assume that an RGC is innervated by two bipolar units. Each of those bipolar units is stimulated by one pixel that can hold one of three intensities: white (1), gray (0) or black (-1). The summation of the intensities collected by the two bipolar units results in the RGC pre-firing signal, which is the equivalent of the image projection in the SPD model. Now, because I know that retinal bipolar units hold non-linearities 59 I assume that a single white pixel is enough to cause the RGC to cross threshold and fire an action potential. Next to account for the difference between NI and AI I assume that for NI the two pixels stimulating the bipolar units hold the same intensity, accounting for those images low spatial power, and for AI the pixels can take any combination of intensities. The possible combinations for each stimuli type and their resulting image projection and probability to fire are presented in figure 3-23. As you can see AI holds three times more combinations than NI (nine versus three). FIGURE 3-23 ALL POSSIBLE PIXEL COMBINATIONS WITH THEIR RESULTING IMAGE PROJECTION AND PROBABILITY OF FIRING VALUES FOR NATURAL AND ARTIFICIAL STIMULI. 37 Panel 3-24a presents plots of those probabilities versus their respective image projection values and for the AI plot the simulation results in a slope decrease in the middle portion of the curve, which is similar to the plateau in panel 3-22b. For the NI case the plot rises for increasing image projection values like the curves presented in panel 3-22c. Lastly, I used this simplistic model to generate a simulated train of spikes for both the NI and AI cases, using a random signal generator. Those trains are presented in panel 3- 24b, for NI the spikes are sparser but hold greater magnitude compared with AI which are more frequent. In sum, this is similar to what I observe in the physiological data presented in panel 3-2 which suggests that our simplistic model might hold the right basics. FIGURE 3-24 (A) RESULTING PROBABILITY OF FIRING CURVES (POF) FOR NI AND AI. (B) SIMULATED SPIKE TRAINS FOR NI AND AI CASES BASED ON MODEL ASSUMPTION. 38 3.4 Discussion 3.4.1 Responses to natural and artificial images The characteristics of the responses between natural and artificial images vary markedly with the responses to natural images being sparser, but of higher magnitude (Figure 3.2). The sparseness of the response to NI makes sense because the images producing them are also sparse showing large areas of equi-luminance (Figure 3.1). Thus a sparse input pattern is likely to produce a sparse output response. Classic LN type models have difficulty in predicting a sparse response pattern because they aim to estimate the average response, but, if the response is sparse, the most likely prediction is zero (Figure 3.3). However, one common feature for both responses to NI and AI is that they are probabilistic, reproducing between trials with a certain probability. This motivated my effort to develop a model that will successfully predict responses to NI using a probabilistic approach. 3.4.2 STA filter The first stage in applying the model I developed is calculating the STA of the RGCs (Figure 3.4). I chose to incorporate STA into the model because of its simple and fast calculation. Although the STA overestimates the cells’ RFs 13 , it still results in excellent fits to the model. One interpretation of why the STA works is that it is because the correlation structure is relatively small compared to the RF size, yielding negligible enlargement given that the cell is noisy. Also, it is known that the RF size of rats is especially large 60 , so if using a different animal model, it might be necessary to use a more complex RF estimation method. In chapter 4 I will provide support to these hypotheses. 3.4.3 Choice of probability distribution and parameters My initial choice of a normal distribution (Figure 3.8) for the modeling arose from the functional structure of the retina conforming to the central limit theorem. The distribution proved correct for most RGCs, as for the firing signals I recorded, their relation with the pre- 39 firing signal estimates a thresholded normal distribution (figure 3.10), allowing to model their response with a single free parameter (σ). For those RGCs sensitive to saturation (Figure 3.11), the incorporation of the spike limiting function resulted in two free parameters, K and A. 3.4.4 PoF curves I wanted to concentrate on the Probability of Firing data I generated because they are the first real step in our model’s ability to predict the responses to natural stimuli. Although most of those curves did not reveal significant differences across stimuli type (panel 3-22a), I did find some RGCs to produce peculiar shaped PoF curves in response to artificial stimuli (panel 3-22b). This shape of curves is not common in biological systems, which usually try to emulate a switch like behavior using a sigmoidal curve (panel 3-22c for instance). Therefore, it is interesting to ask how is this shape advantageous for the sensory system. I found some answers to this in the non-linear subunit model that was able to replicate the shape of the PoF curves for each stimuli type (panel 3-24a). Based on the simulation result I can suggest that this shape is designed to better encode the increased pixel combinations that are presented by AI. That is in contrast to NI where the RGC aims to encode strictly the combination that it’s most tuned for. This result may suggest that there might exist a trade-off between a large number of RGCs each tuned for a specific feature and a few RGCs, each encoding multiple features by plasticity of their PoF curve. Therefore, it will be interesting to investigate the cellular mechanisms behind the PoF curve shapes using intracellular methods and see how those change between the stimuli and RGC types. Lastly, by producing a simulated train of spikes, for each stimuli type, using the non- linear subunit model (panel 3-24b). I was able to replicate the patterns seen in the 40 physiological data (Figure 3-2). Therefore, I got some evidence that I am on the right track to understand how the retina encodes the visual information. 3.4.5 Fit quality The quantitative fit between the model and the RGCs data presented near prefect correlation for both NI and AI stimulation (Figure 3.21). Because I felt the 2-D correlation coefficient might be too lenient, I developed a strict multinomial statistical test with randomization to determine which of the fits hold statistical significance, meaning that the fit and testing data arise from the same distribution statistically. At 5% significance level, roughly a third of all the RGCs proved statistical significance (section 3.3.4.3), which is impressive, given the strictness of the test. 3.4.6 K and A parameters Throughout the process of laying out the results I noticed consistent and quantifiable differences between the responses and the parameters resulting from each stimuli type. To explain those differences I will start by suggesting that an optimal model would yield the same parameter set regardless of the stimuli it is tested by. Thus my model is not perfect. However and in spite of the difference in parameter sets the model still predicts very well the responses to each stimuli type. An answer to why that is so might lie in the response patterns each stimulus generates and its relation to the role that each parameter represents in the model. Now, considering the difference in σ* (panel 3-22a), I noticed that for most RGCs its value is greater for AI compared with NI. This is interesting, because in the model implementation, σ* represents the relation of the firing signals (F) to the deviation (σ) of the pre-firing signals (R) about their mean (µ). Therefore, for a given µ, a greater σ* value means that more of the pre-firing signals are above threshold (θ), thus generating a spiking response. This relates well with our knowledge about the difference in response pattern each stimuli generates, since a higher portion of the pre-firing signal distribution above θ 41 means that the likelihood of a spiking event is greater, just like the responses AI generate compared with NI (figure 3-2). I can also relate the individual K* and A parameters with the change in σ*, but I need to present my reasoning carefully here. The maximal firing rate of each RGC is a physiological parameter that is invariant to the stimulus type. Ideally, the A parameter which estimates the maximal firing rate should predict this number, regardless of the stimuli, leaving K* as the only parameter to vary. In that case a lower K* indicates a higher sensitivity to saturation (Equation 4) and results in a lower σ* as well. Now NI present a lower σ* and generates responses of greater magnitude (panels 3-24 and 3-2) compared with AI. Thus, the greater response magnitude NI generates renders the RGC to be increasingly sensitive to saturation, yielding the lower K* and σ* parameters in turn. The opposite explanation applies to AI. Some evidence for this can be found in panel 3-21a where except for one saturation sensitive RGC the rest present a higher K* value for AI compared with NI. However, the A parameter does not remain constant across stimuli in this panel, which weakens my argument. To conclude, in the above I suggested explanations as to how the model parameters vary with each stimuli type. Other explanations might be true as well, however because my model is novel it will take time to investigate it on a deeper level. In chapter 4 I will provide further analysis of why σ* is different in AI and NI. 3.4.7 Saturation sensitivity The model’s independent parameters not only varied between stimulus types but also varied between those RGCs that are sensitive to saturation and the rest as well, with those RGC’s sensitive to saturation yielding a lower σ* compared with most other cells (panel 3-21a). This lower σ* might result from the same reasons I mentioned above. Specifically, since A estimates the RGCs maximal firing its value should be relatively similar among RGCs. Now σ* is the ratio of K* over A and a lower K* characterizes the saturation 42 sensitive behavior those RGCs experience. So the relatively similar A and lower K* parameters, compared with insensitive RGCs, results in the lower σ*. Now in our model I assume that the saturation behavior arises from the RGCs’ refractory period limiting their ability to fire spikes at infinite rate. However this behavior might result from other sources as well. For example, neuronal adaptation or deterioration in the health of the RGCs post-separation from the pigment epithelium might be the reason. Thus, I am not certain about the origin of the saturation behavior and will try to design experiments and analysis techniques to elucidate it. 43 4 Biophysical Model of the Probabilistic Responses 4.1 Introduction So far I used a simple STA method to estimate the linear filter portion of my model. This method worked well, unveiling well known characteristics in the RGCs’ RFs, like the liking of ON type RGCs for stimuli that transitions from dark to bright intensity presented at the center of its RF and the opposite liking of OFF type cells. However, it is well known that the STA method tends to overestimate the RF size of visual cells 13 (Section 3.4.2). Moreover, the σ* parameter was consistently larger for AI stimulus compared with NI (Panel 3-21a). But since σ* is a parameter of retina computation it should not depend on the stimulus (section 3.4.6). I try to answer the question of why the STA worked well and why σ* was greater for AI by building a realistic biophysical model of the retina. This biophysical model includes photoreceptor, bipolar and ganglion cells with noise, center-surround implementation and divisive inhibition. 4.2 Methods To evaluate how well the STA method estimates the visual RF I need to know what the true RF is first, so I can compare it with the reconstructed estimation. Using a real RGC, this is impossible, because I do not have access to its true RF structure. However, using a biophysical model, I can simulate a retinal structure holding antagonistic center-surround mechanisms. 44 Fortunately we have enough information about retinal circuitries and function to be able to build a biophysically realistic model that includes relevant noise processes, non-linearities and center-surround interactions 5 . The input to the biophysical model are the same images I used in the physiology experiment described in the first part of my thesis. I will now explain every stage of the model in detail. At the photoreceptor level, the response is equal to the image plus noise. Modeling the basic response as proportional to the image is a consequence of assuming low contrasts (as in both natural images 29 and my artificial images) and thus not needing to model photoreceptor non-linearities. The noise is dependent on every pixel intensity (Equation 8), increasing proportionally with the square root of the signal because of photon-catch variability. The noise (εph) added to each pixel intensity (I) is drawn from a normal distribution with zero mean and σ proportional to the square root of intensity. 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 8: 𝜀 𝑃 ℎ = 𝐺 (𝜇 𝑝 ℎ = 0, 𝜎 𝑝 ℎ = 𝛼 𝑝 ℎ √𝐼 ) where G is a Gaussian probability distribution (Equation 9) and αph>0 is the constant of proportionality of the equation, 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 9: 𝐺 = 1 𝜎 √2𝜋 𝑒 − (𝑥 −𝜇 ) 2 2 𝜎 2 At the bipolar level I convolve the output of the photoreceptor with center and surround Gaussian filters, representing horizontal cell interactions (Equations 10a & 10b), 𝐸 𝑞𝑢𝑎𝑡𝑖𝑜𝑛 10𝑎 : 𝐵 𝑃 𝑐𝑒𝑛𝑡𝑒𝑟 = (𝐼 + 𝜀 𝑃 ℎ ) ∗ 𝐺 (𝜇 = 0, 𝜎 𝐵𝑃 𝐶𝑒𝑛𝑡𝑒𝑟 ) 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 10𝑏 : 𝐵 𝑃 𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑 = (𝐼 + 𝜀 𝑃 ℎ ) ∗ 𝐺 (𝜇 = 0, 𝜎 𝐵𝑃 𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑 ) 45 where * stands for convolution and σBP center and σBP surround are the standard deviation parameters of the equation. Next, I use divisive inhibition to model the horizontal-cell center- surround mechanism 61 The result is the bipolar-RGC presynaptic signal: 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 11: 𝐵 𝑃 𝑃𝑟𝑒 = 𝐵 𝑃 𝑐𝑒𝑛𝑡𝑒𝑟 𝐵 𝑃 𝑆𝑢𝑟𝑟𝑜𝑢𝑛𝑑 + 𝐾 where K is the divisive inhibition constant. I added Gaussian noise (εBP) to the output of the presynaptic signal to simulate probabilistic vesicle release 62 . Then I fed the pre-synaptic output to a sigmoidal non-linearity (Equation 12) to simulate a post-synaptic response. 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 12: 𝐵 𝑃 𝑜𝑢𝑡𝑝𝑢𝑡 = 1 + erf(𝐵 𝑃 𝑔𝑎𝑖𝑛 (𝐵 𝑃 𝑝𝑟𝑒 + 𝜀 𝐵𝑃 − 𝐵 𝑃 𝑜𝑓𝑓𝑠𝑒𝑡 )) 2 where erf is the error function and BPgain>0 and BPoffset are the parameters of the equation, The output of BP was fed into the Ganglion cell level where it is convolved with its center and surround Gaussian filters and the difference of those is calculated (Equation 13), 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 13: 𝑅𝐺 𝐶 𝑙𝑖𝑛𝑒𝑎𝑟 = 𝐵 𝑃 𝑜𝑢𝑡𝑝𝑢𝑡 ∗ 𝐺 (𝜇 = 0, 𝜎 𝑅𝐺𝐶 𝑐𝑒𝑛𝑡𝑒𝑟 ) − 𝐵 𝑃 𝑜𝑢𝑡𝑝𝑢𝑡 ∗ 𝐺 (𝜇 = 0, 𝜎 𝑅𝐺𝐶 𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑 ) where, σRGC center and σRGC surround are the standard deviation parameters of the RGC filters. Because each bipolar cell has a non-linear output, and these outputs sum through the convolution operation, they implement the non-linear subunits of the RGC receptive field 59 . The size of the subunit are the filters of the center and surround of the bipolar cell. This synaptic RGC signal is fed to a non-linear function to simulate the spiking threshold and then randomized using a Poisson process. 46 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 14: 𝑅𝐺𝐶 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑃 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 (𝐹 : 𝜆 = 𝛼 𝑔𝑎𝑖𝑛 ( 1+erf(𝑅𝐺𝐶 𝑔𝑎𝑖𝑛 (𝑅𝐺𝐶 𝐿𝑖𝑛𝑒𝑎𝑟 −𝑅𝐺𝐶 𝑜𝑓𝑓𝑠𝑒𝑡 )) 2 )) where F is the number of fired spikes and RGCgain>0, RGCoffset and αgain>0 are the parameters of the equation, and PPoisson is the Poisson function, 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 15: 𝑃 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 (𝐹 : 𝜆 ) = 𝜆 𝐹 𝐹 ! 𝑒 −𝜆 I optimized the parameters of the biophysical model to generate realistic spike firing. The values of the optimized parameters are presented in table 1 below. Level Parameter Value Photoreceptor cell αph 1 Bipolar cell σ BP center 1.5 σ BP surround 3 K 25 εBP 0.1 BP offset 0.7 BP gain 5 Ganglion cell σ RGC center 3 47 σ RGC surround 6 RGC offset 1.1 RGC gain 1 α gain 1.2 TABLE 1 BIOPHYSICAL MODEL PARAMETERS I used the images and spikes from the biophysical model as input to the Spiking Probability Distribution model (Chapter 3). Every biophysical simulation that was fed to the SPDm used 6000 AI and NI images with four repetitions. In order to evaluate the effect of the receptive field size on the STA estimation I varied the BP and RGC receptive fields in two ways. Firstly, I changed the size of all those receptive fields by the same factor. Second, I varied the size of each receptive field independently leaving the rest constant. Next, I estimated the standard deviation of the resulting STAs using a Gaussian fit to find the difference between the original standard deviation values and the ones resulting from the STA estimation. Last I varied all the RFs by the same factor to evaluate their effect on the model σ* parameter for natural and artificial images. 4.3 Results Now that I have a working biophysical model I will show what every stage of the model output results in. Panel 4-1a presents an exemplary natural image fed into the biophysical model. Panel b presents the output of the processing that takes place at the photoreceptor 48 level where Gaussian noise with proportionality to the square root of pixel intensity is added to the image (Equations 8 & 9). FIGURE 4-1 INITIAL PROCESSING (A) ORIGINAL IMAGE FED INTO BIOPHYSICAL MODEL (B) GAUSSIAN NOISE PROPORTIONAL TO PIXEL INTENSITY ADDED TO THE IMAGE AT THE PHOTORECEPTOR LEVEL. Now at the bipolar cell level multiple processes are taking place. Figures 4-2 presents those processes. Panels 4-2 a and b present the center and surround filtered outputs with the surround (b) being more smeared due to the larger standard deviation of the surround Gaussian filter (Equations 10a & 10b). Panel 4-2 c presents the output of the center- surround divisive inhibition process where the contour outlines are being amplified (Equation 11). Now, in panel 4-2d I add Gaussian noise to the output of the divisive inhibition which contributes a grainier texture to the image (Equation 12). Panels e and f present the output of the bipolar processing following the non-linear sigmoidal function (Equation 12). For panel e the brighter pixels are amplified (ON bipolar) and for panel f the darker pixels are amplified (OFF bipolar). In this work ON and OFF RGCs gave similar results and thus I only show the results and equations for ON type cells. (a) Original image (b) Photoreceptor level 49 FIGURE 4-2 BIPOLAR LEVEL PROCESSES (a) Bipolar center (b) Bipolar surround (c) Divisive inhibition (d) Noise addition (e) Bipolar ON (f) Bipolar OFF 50 At the ganglion cell level the output of the bipolar is filtered through center and surround Gaussian filters (panels 4-3 a&b, Equation 13) of which the difference is calculated resulting in a center-surround mechanism (panel 4-3 c, Equation 13). Next the difference is put through a non-linear sigmoidal function that enhances the brightest pixels and suppresses the darker ones (Fig 4-3 d, Equation 14) FIGURE 4-3 GANGLION CELL LEVEL PROCESSING (a) RGC center (b) RGC surround (c) RGC Difference of Gaussians (d) RGC nonlinearity 51 Lastly every pixel generates a firing response by inputting the RGC non-linear output value into a Poisson function resulting in an integer number of spikes (Figure 4-4, Equations 14&15) FIGURE 4-4 SPIKE FIRING Spike Firing 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 SPIKES 52 As a first evaluation of the biophysical model l calculated the Spike Triggered Average (STA) from 6000 50x50 image patches and the firing responses they produce. The left image is produced from natural Images and the right is produced from artificial ones. As you can observe the STA outcomes results in a clear receptive field for both stimuli cases. FIGURE 4-5 SPIKE TRIGGERED AVERAGES PRODUCED FROM NATURAL (LEFT) AND ARTIFICIAL (RIGHT) IMAGES APPLIED TO THE BIOPHYSICAL MODEL Now, that I know that the biophysical model results in a valid STA data I used the input- output data from the biophysical model to see if it would work with the Spiking Probability Distribution model (SPDm) which is described in the first part of my thesis (Figure 4-6). The results for both NI and AI are good showing the increase in probability for increased image projection value and the decrease in probability for increased firing rate. Natural image STA Artificial image STA 53 FIGURE 4-6 SPDM FED WITH BIOPHYSICAL MODEL DATA Now that I proved that the model is working I varied the biophysical model parameters to understand why although STA is known to be inaccurate it worked so well for the SPDm model. First I varied the standard deviation of the bipolar and ganglion, center and surround by the same factor (Figure 4-7) and measured the resulting STA standard deviation using a Gaussian fit. For large standard deviation parameters (size of largest σ > 66.6µm) there is a rise in both the simulated and measured RF size. However for the low RF size (size of largest σ < 66.6µm) the measured σ remains constant at a 100µm (Figure 4-7b). From this I conclude that NI have a correlation of σ around 100µm. Consistently with this conclusion, the σ estimated from the STA starts rising as a function of the σ of the biophysical-model filters when their size start approaching 100µm (Figure 4-7). Thus, at that point the filters start dominating the STA. That the NI have spatial correlations of around 100µm is important because in the physiological experiments I conducted I measured the RF to have a standard deviation of around 200µm (Figure 3-4 and dashed lines at Figure 4-7). Therefore the NI correlation is smaller than the physiological RF standard deviation which explain why STA might work so well. 54 FIGURE 4-7 VARYING ALL RF SIZES BY THE SAME FACTOR Next I varied the bipolar and ganglion, center-surround filter sizes independently and measured the resulting STA RF standard deviation using a Gaussian fit (4-8 a through d). I did this to see which of the parameters matters most for the RF estimation. The results show that besides the bipolar surround (4-8 b) all the filter factors affect the measured STA RF standard deviation. 55 FIGURE 4-8 VARYING THE RF STD INDEPENDENTLY Lastly I used the SPDm to measure the σ* of the resulting biophysical model data for natural and artificial images for RF sizes varied by the same factor (Figure 4-9). From this figure it is visible that σ* is greater for AI compared with NI, like in the experimental data (Figure 3-21a). Moreover when I used white noise with finer 33.3µm squares resulted in σ* that is greater than the one generated by 100µm squares I previously used. 0 100 200 300 400 0 100 200 300 (a) BP center Size of largest filter ( m) STA ( m) 0 100 200 300 400 0 100 200 300 (b) BP surround Size of largest filter ( m) STA ( m) 0 100 200 300 400 0 100 200 300 400 (c) RGC center Size of largest filter ( m) STA ( m) 0 100 200 300 400 0 50 100 150 200 250 (d) RGC surround Size of largest filter ( m) STA ( m) 56 FIGURE 4-9 σ* VERSUS DIFFERENT RF SIZES 4.3.1 Discussion The probabilistic model is consistent with our understanding of the retina. The STA method works well for us because the receptive field size of rats is bigger by a factor of two than the NI correlation structure so that the model filters are larger than the NI correlation structure. Also σ* is greater for AI compared with NI because of the spatial correlation structure being larger for the latter. 0 50 100 150 200 250 300 350 400 450 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 of largest filter ( m) * NI AI 100 m AI 33.3 m 57 5 Overall discussion This thesis has shown that it is possible to build models for the probabilistic responses of the retina and that such models predict better than deterministic ones. In such models the goal is not to fit the response but to fit the probability distribution of it. This probabilistic responses emerge from various noise stages in retinal processing. First, photoreceptors probabilistic nature of photon capture. Second, intrinsic noise in the biophysical mechanism of synaptic release (e.g. probabilistic vesicle release) and finally there is the probabilistic nature of firing due to the noisy opening and closing of gates. What is important is that because we have a population of interneurons (bipolars) the noise eventually becomes Gaussian making probabilistic modeling easy to analyze. The bipolar outputs are non-linear (Equation 12). This non-linearity transforms the probability distribution of each output to a non-Gaussian form. However, because multiple outputs are summed by the RGC (Equation 13) its noise is well approximated by a Gaussian according to the central limit theorem. These outputs are the so called non-linear outputs of the RGC receptive field 32 . Although the number of non-linear subunits in the real retina is finite I implemented an infinite number of them through the convolution operation for mathematical simplicity. This mathematical approximation was good enough for the purposes of this thesis, but future studies would be necessary to understand the deviations from Gaussianity due to the finite number of subunits. Modeling retinal responses as a probability is important from the point of view of visual function. The brain does not work with deterministic retinal output. It must build probabilistic models of arriving spikes. My work contributed to understanding what these models may be. In sum, although the end result of the neuronal computations taking place in our sensory systems is deterministic (we can most of the time quickly determine what it is that 58 we see or feel), the function of the neurons underlying this process is probabilistic. This has been shown in numerous psychophysical experiments where the ability of our sensory system to make a decision about a stimulus is pushed to the limit 49,63 . In my work, I have taken a first step into characterizing this probabilistic response at the level of the neuron. Thus, I aim to predict the RGCs response probability distribution instead of its absolute response. This may form the first building block to look at neuronal system responses as a result of many probabilistic signals, and may help us understand how our nervous system comes to make such accurate decisions. 5.1 Future work 5.1.1 Natural movies In the first part of my project I used natural images to stimulate RGCs and I developed a model (SPDm) that allows predicting the responses to those, probabilistically. I chose natural images because they hold special statistics that our visual system was evolutionary designed to encode, thus generating a “natural” response. However, the visual scene we experience daily is not only composed of image frames that are correlated spatially, like NI, but of images that are temporally correlated as well. For instance, if I observe an object moving at a distance (e.g. sailing boat), I would notice that although the object is moving over time, the rest of the scene remains constant, thus holding temporal correlations in addition to the spatial ones that are well characterized. In future work I would like to investigate those temporal correlations and the RGCs’ responses to them to see if the probabilistic approach used to develop the SPDm would work for those as well. 59 5.1.2 RF estimation So far I used a simple STA method to estimate the linear filter portion of my model. This method worked well, unveiling well known characteristics in the RGCs’ RFs, like the liking of ON type RGCs for stimuli that transitions from dark to bright intensity presented at the center of its RF and the opposite liking of OFF type cells. However, it is well known that the STA method tends to overestimate the RF size of visual cells 13 . In addition, the estimation of the RFs varies with the stimuli presented to the RGC, which might be the reason for the discrepancy in the optimized parameters (K and A) between NI and AI stimuli. Advanced methods to estimate the RFs of visual cells exist. For example, Spike Triggered Covariance (STC) 9 is a complementary method to STA, estimating higher order filters, and Maximally Informative Dimensions (MID) 40 is a technique that estimates the RF by maximizing the mutual information between the image stimuli and spike train output. In the future I would like to investigate how the different RF estimation methods affect the prediction of the SPDm model and its resulting parameter set. 60 6 References 1. Weiland, J. D., Liu, W. & Humayun, M. S. Retinal prosthesis. Annu. Rev. Biomed. Eng. 7, 361–401 (2005). 2. Margalit, E. et al. Retinal prosthesis for the blind. Surv. Ophthalmol. 47, 335–56 (2002). 3. Freeman, D. K., Rizzo, J. F. & Fried, S. I. Encoding visual information in retinal ganglion cells with prosthetic stimulation. J. Neural Eng. 8, 035005 (2011). 4. Humayun, M. S. et al. Visual perception in a blind subject with a chronic microelectronic retinal prosthesis. Vision Res. 43, 2573–81 (2003). 5. Purves, D. Neuroscience. Sinauer Associates (2004). 6. Kandel, E. R., Schwartz, J. H., Jessell, T. M., Siegelbaum, S. & Hudspeth, A. J. Principles of Neural Science. (McGraw-Hill Medical, 2000). 7. Hunter, I. W. & Korenberg, M. J. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55, 135–144 (1986). 8. Chichilnisky, E. J. A simple white noise analysis of neuronal light responses. Network- Computation Neural Syst. 12, 199–213 (2001). 9. Schwartz, O., Pillow, J. W., Rust, N. C. & Simoncelli, E. P. Spike-triggered neural characterization. J. Vis. 6, 484–507 (2006). 10. Simoncelli, E. P., Paninski, L., Pillow, J. W. & Schwartz, O. in The New Cognitive Neurosciences (ed. Gazzaniga, M.) (MIT Press, 2004). 11. De Boer, R. & Kuyper, P. Triggered correlation. IEEE Trans. Biomed. Eng. 15, 169– 79 (1968). 12. Marmarelis, P. Z. & Naka, K. White-noise analysis of a neuron chain: an application of the Wiener theory. Science 175, 1276–8 (1972). 13. Willmore, B. & Smyth, D. Methods for first-order kernel estimation: simple-cell receptive fields from responses to natural scenes. Network 14, 553–77 (2003). 61 14. Steveninck, R. D. R. V. & Bialek, W. Real-Time Performance of a Movement- Sensitive Neuron in the Blowfly Visual System: Coding and Information Transfer in Short Spike Sequences. Proc. R. Soc. B Biol. Sci. 234, 379–414 (1988). 15. Bryant, H. L. & Segundo, J. P. Spike initiation by transmembrane current: a white- noise analysis. J. Physiol. 260, 279–314 (1976). 16. Bialek, W. & van Steveninck, R. R. de R. Features and dimensions: Motion estimation in fly vision. 18 (2005). at <http://arxiv.org/abs/q-bio/0505003> 17. Laughlin, S. A simple coding procedure enhances a neuron’s information capacity. Z. Naturforsch. C. 36, 910–2 (1981). 18. Theunissen, F. E., Sen, K. & Doupe, A. J. Spectral-temporal receptive fields of nonlinear auditory neurons obtained using natural sounds. J. Neurosci. 20, 2315– 2331 (2000). 19. Theunissen, F. E. et al. Estimating spatio-temporal receptive fields of auditory and visual neurons from their responses to natural stimuli. Network 12, 289–316 (2001). 20. Dan, Y., Atick, J. J. & Reid, R. C. Efficient coding of natural scenes in the lateral geniculate nucleus: experimental test of a computational theory. J. Neurosci. 16, 3351–3362 (1996). 21. Touryan, J., Felsen, G. & Dan, Y. Spatial structure of complex cell receptive fields measured with natural images. Neuron 45, 781–91 (2005). 22. Baddeley, R. et al. Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proc. Biol. Sci. 264, 1775–1783 (1997). 23. Barlow, H. B. H. in Sensory Communication (ed. Rosenblith, W.) 217–234 (M.I.T. Press, 1961). doi:10.1080/15459620490885644 24. Simoncelli, E. P. & Olshausen, B. A. Natural Image Statistics and Neural Representation. Annu. Rev. Neurosci. 24, 1193–1216 (2001). 25. ATTNEAVE, F. Some informational aspects of visual perception. Psychol. Rev. 61, 183–193 (1954). 26. Olshausen, B. A. & Field, D. J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–9 (1996). 62 27. Dong, D. & Atick, J. Statistics of natural time-varying images. Network: Computation in Neural Systems 6, 345–358 (1995). 28. Burton, G. J. & Moorhead, I. R. Color and spatial structure in natural scenes. Appl. Opt. 26, 157–70 (1987). 29. Balboa, R. M. & Grzywacz, N. M. Power spectra and distribution of contrasts of natural images from different habitats. Vision Res. 43, 2527–2537 (2003). 30. Van der Schaaf, A. & Van Hateren, J. H. H. Modelling the power spectra of natural images: Statistics and information. Vision Res. 36, 2759–2770 (1996). 31. Tolhurst, D. J., Tadmor, Y. & Chao, T. Amplitude spectra of natural images. Ophthalmic Physiol. Opt. 12, 229–32 (1992). 32. Cao, X., Merwine, D. K. & Grzywacz, N. M. Dependence of the retinal Ganglion cell’s responses on local textures of natural scenes. J. Vis. 11, (2011). 33. David, S. V, Vinje, W. E. & Gallant, J. L. Natural stimulus statistics alter the receptive field structure of v1 neurons. J. Neurosci. 24, 6991–7006 (2004). 34. Dong, D. & Atick, J. Temporal decorrelation: a theory of lagged and nonlagged responses in the lateral geniculate nucleus. Network: Computation in Neural Systems 6, 159–178 (1995). 35. Field, D. J. Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. A. 4, 2379–2394 (1987). 36. Ruderman, D. L. & Bialek, W. Statistics of natural images: Scaling in the woods. Phys. Rev. Lett. 73, 814–817 (1994). 37. Thomson, M. G. Beats, kurtosis and visual coding. Network 12, 271–87 (2001). 38. Rapela, J., Mendel, J. M. & Grzywacz, N. M. Estimating nonlinear receptive fields from natural images. J. Vis. 6, 441–474 (2006). 39. Rapela, J., Felsen, G., Touryan, J., Mendel, J. M. & Grzywacz, N. M. ePPR: a new strategy for the characterization of sensory cells from input/output data. Network 21, 35–90 (2010). 40. Sharpee, T., Rust, N. C. & Bialek, W. Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). 63 41. Croner, L. J., Purpura, K. & Kaplan, E. Response variability in retinal ganglion cells of primates. Proc Natl Acad Sci U S A 90, 8128–8130 (1993). 42. Stein, R. B., Gossen, E. R. & Jones, K. E. Neuronal variability: noise or part of the signal? Nat. Rev. Neurosci. 6, 389–397 (2005). 43. Meister, M. & Berry, M. J. The Neural Code of the Retina. Neuron 22, 435–450 (1999). 44. Keat, J., Reinagel, P., Reid, R. C. & Meister, M. Predicting every spike: a model for the responses of visual neurons. Neuron 30, 803–17 (2001). 45. Berry, M. J., Warland, D. K. & Meister, M. The structure and precision of retinal spike trains. Proc. Natl. Acad. Sci. 94, 5411–5416 (1997). 46. Tolhurst, D. J., Movshon, J. A. & Dean, A. F. The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res. 23, 775–85 (1983). 47. Reich, D. S., Victor, J. D. & Knight, B. W. The power ratio and the interval map: spiking models and extracellular recordings. J. Neurosci. 18, 10090–104 (1998). 48. Rieke, F., Warland, D., De Ruyter Van Steveninck, R. & Bialek, W. Spikes: Exploring the Neural Code. Computational Neuroscience 20, (1997). 49. Hecht, S., Shlaer, S. & Pirenne, M. H. ENERGY, QUANTA, AND VISION. J. Gen. Physiol. 25, 819–40 (1942). 50. De Vries, H. L. The quantum character of light and its bearing upon threshold of vision, the differential sensitivity and visual acuity of the eye. Physica 10, 553–564 (1943). 51. Calvin, W. H. & Stevens, C. F. Synaptic noise as a source of variability in the interval between action potentials. Science 155, 842–844 (1967). 52. Allen, C. & Stevens, C. F. An evaluation of causes for unreliability of synaptic transmission. Proc. Natl. Acad. Sci. U. S. A. 91, 10380–10383 (1994). 53. Guillory, K. S. & Normann, R. A. A 100-channel system for real time detection and storage of extracellular spike waveforms. J. Neurosci. Methods 91, 21–29 (1999). 64 54. Van Hateren, J. H. & van der Schaaf, A. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. Biol. Sci. 265, 359– 66 (1998). 55. Chatterjee, S., Merwine, D. K., Amthor, F. R. & Grzywacz, N. M. Properties of stimulus-dependent synchrony in retinal ganglion cells. Vis. Neurosci. 24, 827–43 56. Utts, J. M. The rainbow test for lack of fit in regression. Commun. Stat. - Theory Methods 11, 2801–2815 (1982). 57. Nykamp, D. Q. & Ringach, D. L. Full identification of a linear-nonlinear system via cross-correlation analysis. J. Vis. 2, 1–11 (2002). 58. Glantz, S. Primer of Biostatistics, Seventh Edition. 2011, (McGraw-Hill Education, 2011). 59. Field, G. D. & Rieke, F. Nonlinear signal transfer from mouse rods to bipolar cells and implications for visual sensitivity. Neuron 34, 773–85 (2002). 60. Heine, W. F. & Passaglia, C. L. Spatial receptive field properties of rat retinal ganglion cells. Vis. Neurosci. 28, 403–17 (2011). 61. Merwine, D. K., Amthor, F. R. & Grzywacz, N. M. Interaction between center and surround in rabbit retinal ganglion cells. J. Neurophysiol. 73, 1547–1567 (1995). 62. Hessler, N. A., Shirke, A. M. & Malinow, R. The probability of transmitter release at a mammalian central synapse. Nature 366, 569–72 (1993). 63. Campbell, F. W. & Robson, J. G. Application of Fourier analysis to the visibility of gratings. J. Physiol. 197, 551–566 (1968). 64. USC, ERC, BMES. Epi-retinal prosthesis. (2013). at <http://bmes- erc.usc.edu/research/testbeds.htm> 65. Kolb, H. Simple organization of the retina. Webvision (2013). at <http://webvision.med.utah.edu/book/part-i-foundations/simple-anatomy-of-the- retina/> 65 6.1 Appendix A - SPDf derivation I assume that the distribution of the pre-firing signal x is Gaussian with mean μ, i.e., (1) 𝑓 𝑥 (𝑥 ) = 1 √2𝜋 𝜎 𝑒 − (𝑥 −𝜇 ) 2 2𝜎 2 where σ>0, the standard deviation of the Gaussian, is a constant. I also assume that the firing is a saturation function y of how much x is above a threshold θ, i.e., (2) 𝑦 = 𝑔 (𝑥 ) = 𝐴 (𝑥 −𝜃 ) 𝐾 +(𝑥 −𝜃 ) where K>0 and A>0 are constants representing the semi-saturation constant and maximal firing respectively. The distribution of firing is then calculated as follows: (3) 𝑓 𝑦 (𝑦 ) = | 1 𝑔 ′ (𝑔 −1 (𝑦 )) | 𝑓 𝑥 (𝑔 −1 (𝑦 )) 𝑔 ′ (𝑥 ) = 𝐴𝐾 (𝐾 + (𝑥 − 𝜃 )) 2 𝑔 −1 (𝑦 ) = 𝑥 = 𝜃 + 𝑦𝐾 𝐴 − 𝑦 𝑓 𝑦 (𝑦 ) = | (𝐾 + (𝑔 −1 (𝑦 ) − 𝜃 )) 2 𝐴𝐾 | 𝑓 𝑥 (𝑔 −1 (𝑦 )) (4) 𝑓 𝑦 (𝑦 ) = | (𝐾 + 𝑦𝐾 𝐴 −𝑦 ) 2 𝐴𝐾 | 1 √2𝜋 𝜎 𝑒 − (𝜃 −𝜇 + 𝑦𝐾 𝐴 −𝑦 ) 2 2𝜎 2 with this function being defined for 0≤y≤A. Because I assume that the pre-firing signal is Gaussian and that it must cross the threshold to cause firing, from the probability of firing, I can calculate (θ -μ)/σ. Hence, I can define 66 (5) 𝜃 ∗ = (𝜃 −𝜇 ) 𝜎 (6) 𝐾 ∗ = 𝐾 𝜎 (7) 𝑍 (𝑦 ) = 𝑦 𝐴 −𝑦 To rewrite Eq. 4 as (8) 𝑓 𝑦 (𝑦 ) = 𝐾 ∗ (1+𝑍 (𝑦 )) 2 √2𝜋 𝐴 𝑒 − (𝜃 ∗ +𝐾 ∗ 𝑍 (𝑦 )) 2 2 with this function being defined for 0≤y≤A. The case without saturation has K ≫x- θ, and thus, 𝑔 ′ (𝑥 ) = 𝐴 𝐾 𝑔 −1 (𝑦 ) = 𝑥 = 𝜃 + 𝑦𝐾 𝐴 𝑓 𝑦 (𝑦 ) = 𝐾 𝐴 𝑓 𝑥 (𝑔 −1 (𝑦 )) 𝑓 𝑦 (𝑦 ) = 𝐾 𝐴 1 √2𝜋 𝜎 𝑒 − (𝜃 −𝜇 + 𝑦𝐾 𝐴 ) 2 2𝜎 2 which, by defining (9) 𝜎 ∗ = 𝐾 ∗ 𝐴 I get 67 (10) 𝑓 𝑦 (𝑦 ) = 1 √2𝜋 𝜎 ∗ 𝑒 − (𝜃 ∗ 𝜎 ∗ +𝑦 ) 2 2𝜎 ∗ 2 68 6.2 Appendix B1- Normal distribution fit -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = -0.06 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.02 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.06 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.09 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.11 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.14 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.17 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.19 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.21 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.24 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.27 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.29 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.33 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.38 -10 0 10 0 0.1 0.2 0.3 0.4 0.5 I = 0.51 Spikes Fired (F) Probability P(F|I) FIGURE 6-1 CROSS SECTION PLOTS OF FIGURE 3.10 PRESENTING THE NORMAL DISTRIBUTION FIT ALONG WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS 69 6.3 Appendix B2 – SPDf fit 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 1 2 3 0 0.5 1 I=9.41 I=6.82 I=5.37 I=4.39 I=3.64 I=2.86 I=2.16 I=1.40 I=0.71 I=0.06 I=-0.59 I=-1.36 I=-2.16 I=-3.31 I=-5.45 Spikes Fired (F) Probability P(F|I) FIGURE 6-2 CROSS SECTION PLOTS OF FIGURE 3.14 PRESENTING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 70 6.4 Appendix B3 – ON RGC stimulated with NI and AI SPDm fit 2 4 0 0.5 1 I= -4.77 2 4 0 0.5 1 I= -2.69 2 4 0 0.5 1 I= -1.80 2 4 0 0.5 1 I= -1.16 2 4 0 0.5 1 I= -0.60 2 4 0 0.5 1 I= -0.14 2 4 0 0.5 1 I= 0.31 2 4 0 0.5 1 I= 0.76 2 4 0 0.5 1 I= 1.19 2 4 0 0.5 1 I= 1.68 2 4 0 0.5 1 I= 2.27 2 4 0 0.5 1 I= 2.89 2 4 0 0.5 1 I= 3.69 2 4 0 0.5 1 I= 4.62 2 4 0 0.5 1 I= 7.45 NI Spikes Fired (F) Probability P(F|I) FIGURE 6-3 CROSS SECTION PLOTS OF ON-NI PANEL PRESENTED IN FIGURE 3.18 PRESENTING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 71 1 2 3 0 0.5 1 I= -3.01 1 2 3 0 0.5 1 I= -1.84 1 2 3 0 0.5 1 I= -1.21 1 2 3 0 0.5 1 I= -0.72 1 2 3 0 0.5 1 I= -0.26 1 2 3 0 0.5 1 I= 0.14 1 2 3 0 0.5 1 I= 0.55 1 2 3 0 0.5 1 I= 0.86 1 2 3 0 0.5 1 I= 1.20 1 2 3 0 0.5 1 I= 1.56 1 2 3 0 0.5 1 I= 1.94 1 2 3 0 0.5 1 I= 2.37 1 2 3 0 0.5 1 I= 2.83 1 2 3 0 0.5 1 I= 3.50 1 2 3 0 0.5 1 I= 5.01 AI Spikes Fired (F) Probability P(F|I) FIGURE 6-4 CROSS SECTION PLOTS OF ON-AI PANEL PRESENTED IN FIGURE 3.18 PRESENTING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 72 6.5 Appendix B4 – OFF RGC stimulated with NI and AI SPDm fit FIGURE 6-5 CROSS SECTION PLOTS OF OFF-NI PANEL PRESENTED IN FIGURE 3.18 SHOWING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 1 2 3 0 0.5 1 I= -1.37 1 2 3 0 0.5 1 I= -0.78 1 2 3 0 0.5 1 I= -0.54 1 2 3 0 0.5 1 I= -0.33 1 2 3 0 0.5 1 I= -0.18 1 2 3 0 0.5 1 I= -0.05 1 2 3 0 0.5 1 I= 0.07 1 2 3 0 0.5 1 I= 0.20 1 2 3 0 0.5 1 I= 0.33 1 2 3 0 0.5 1 I= 0.46 1 2 3 0 0.5 1 I= 0.59 1 2 3 0 0.5 1 I= 0.76 1 2 3 0 0.5 1 I= 0.96 1 2 3 0 0.5 1 I= 1.27 1 2 3 0 0.5 1 I= 2.07 NI Spikes Fired (F) Probability P(F|I) 73 1 2 3 0 0.5 1 I= -1.03 1 2 3 0 0.5 1 I= -0.55 1 2 3 0 0.5 1 I= -0.31 1 2 3 0 0.5 1 I= -0.10 1 2 3 0 0.5 1 I= 0.08 1 2 3 0 0.5 1 I= 0.24 1 2 3 0 0.5 1 I= 0.39 1 2 3 0 0.5 1 I= 0.54 1 2 3 0 0.5 1 I= 0.69 1 2 3 0 0.5 1 I= 0.85 1 2 3 0 0.5 1 I= 1.00 1 2 3 0 0.5 1 I= 1.16 1 2 3 0 0.5 1 I= 1.38 1 2 3 0 0.5 1 I= 1.73 1 2 3 0 0.5 1 I= 2.36 AI Spikes Fired (F) Probability P(F|I) FIGURE 6-6 CROSS SECTION PLOTS OF OFF-AI PANEL PRESENTED IN FIGURE 3.18 SHOWING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 74 6.6 Appendix B5 – RGC sensitive to spike saturation SPDm fit 1 2 3 0 0.5 1 I= -3.66 1 2 3 0 0.5 1 I= -1.87 1 2 3 0 0.5 1 I= -1.11 1 2 3 0 0.5 1 I= -0.61 1 2 3 0 0.5 1 I= -0.11 1 2 3 0 0.5 1 I= 0.38 1 2 3 0 0.5 1 I= 0.87 1 2 3 0 0.5 1 I= 1.29 1 2 3 0 0.5 1 I= 1.72 1 2 3 0 0.5 1 I= 2.20 1 2 3 0 0.5 1 I= 2.70 1 2 3 0 0.5 1 I= 3.27 1 2 3 0 0.5 1 I= 4.07 1 2 3 0 0.5 1 I= 5.17 1 2 3 0 0.5 1 I= 7.67 NI Spikes Fired (F) Probability P(F|I) FIGURE 6-7 CROSS SECTION PLOTS OF NI PANEL PRESENTED IN FIGURE 3.19 SHOWING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 75 1 2 3 0 0.5 1 I= -2.61 1 2 3 0 0.5 1 I= -1.06 1 2 3 0 0.5 1 I= -0.30 1 2 3 0 0.5 1 I= 0.27 1 2 3 0 0.5 1 I= 0.82 1 2 3 0 0.5 1 I= 1.28 1 2 3 0 0.5 1 I= 1.75 1 2 3 0 0.5 1 I= 2.25 1 2 3 0 0.5 1 I= 2.74 1 2 3 0 0.5 1 I= 3.20 1 2 3 0 0.5 1 I= 3.70 1 2 3 0 0.5 1 I= 4.30 1 2 3 0 0.5 1 I= 4.96 1 2 3 0 0.5 1 I= 5.86 1 2 3 0 0.5 1 I= 7.89 AI Spikes Fired (F) Probability P(F|I) FIGURE 6-8 CROSS SECTION PLOTS OF AI PANEL PRESENTED IN FIGURE 3.19 SHOWING THE SPDF FIT (BLUE LINES) WITH THE HISTOGRAM DATA POINTS (RED STARS) FOR ALL THE IMAGE PROJECTIONS. THE RED ERROR BARS ARE THE STANDARD ERRORS OF THE TESTING DATA CALCULATED FROM THE FOUR REPETITIONS OF THE EXPERIMENT. 76 6.7 Appendix C1- ON RGC STA FIGURE 6-9 STA FILTER OF ON RGC PRESENTED IN FIGURE 3.18 77 6.8 Appendix C2 – OFF RGC STA FIGURE 6-10 STA FILTER OF OFF RGC PRESENTED IN FIGURE 3.18 78 6.9 Appendix C3 – STA of OFF RGC sensitive to spike saturation FIGURE 6-11 STA FILTER OF OFF TYPE RGC WITH SENSITIVITY TO SATURATION THAT IS PRESENTED IN FIGURE 3.19
Abstract (if available)
Abstract
Developing computational models that predict the spiking responses of neurons is important to understand how the brain functions, and cure neuronal diseases. In the first part of my thesis I have developed a probabilistic computational model that accurately predicts the spiking responses of retinal ganglion cells to natural and artificial image stimuli. This model holds a biophysical meaning with its parameters arising from the internal structure of the retina. In the second part of my thesis I developed a computational model of the retina and used it to proof the probabilistic model I developed first. This work advances the field of neural engineering by showing that probabilistic models can predict the noisy responses of neurons.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Encoding of natural images by retinal ganglion cells
PDF
Dependence of rabbit retinal synchrony on visual stimulation parameters
PDF
Characterization of visual cells using generic models and natural stimuli
PDF
Manipulation of RGCs response using different stimulation strategies for retinal prosthesis
PDF
Neural spiketrain decoder formulation and performance analysis
PDF
Experimental and computational models for seizure prediction
PDF
Excitatory-inhibitory interactions in pyramidal neurons
PDF
Parametric and non‐parametric modeling of autonomous physiologic systems: applications and multi‐scale modeling of sepsis
PDF
Computational investigation of glutamatergic synaptic dynamics: role of ionotropic receptor distribution and astrocytic modulation of neuronal spike timing
PDF
Effect of continuous electrical stimulation on retinal structure and function
PDF
Probabilistic framework for mining knowledge from georeferenced social annotation
PDF
Adaptive event-driven simulation strategies for accurate and high performance retinal simulation
PDF
Nonlinear modeling of causal interrelationships in neuronal ensembles: an application to the rat hippocampus
PDF
Scalable exact inference in probabilistic graphical models on multi-core platforms
PDF
Real-world evaluation and deployment of wildlife crime prediction models
PDF
Neuromorphic motion sensing circuits in a silicon retina
PDF
Mechanistic model of chimeric antigen receptor T cell activation
PDF
Functional consequences of network architecture in rat hippocampus: a computational study
PDF
Learning contour statistics from natural images
PDF
Multiscale spike-field network causality identification
Asset Metadata
Creator
Ivzan, Nadav
(author)
Core Title
A probabilistic model predicting retinal ganglion cell responses to natural stimuli
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Biomedical Engineering
Publication Date
04/15/2015
Defense Date
12/15/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
computational modeling,electophysiology,natural images,OAI-PMH Harvest,retina,vision
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Grzywacz, Norberto M. (
committee chair
), Hirsch, Judith A. (
committee member
), Song, Dong (
committee member
)
Creator Email
ivzan@usc.edu,kingos82@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-548002
Unique identifier
UC11297896
Identifier
etd-IvzanNadav-3303.pdf (filename),usctheses-c3-548002 (legacy record id)
Legacy Identifier
etd-IvzanNadav-3303.pdf
Dmrecord
548002
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Ivzan, Nadav
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
Tags
computational modeling
electophysiology
natural images
retina