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A computational model of NMDA receptor dependent and independent long-term potentiation in hippocampal pyramidal neurons
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A computational model of NMDA receptor dependent and independent long-term potentiation in hippocampal pyramidal neurons
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A Computational Model of NMDA Receptor Dependent and Independent Long-term Potentiation in Hippocampal Pyramidal Neurons by Sunil S. Dalai A Thesis Presented to the FACULTY OF THE SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN BIOMEDICAL ENGINEERING August 1995 Copyright 1995 Sunil S. Dalai This thesis, written by .Sun.il...S^...Dfl.la.l. . . . . . . . . . . . . . . . . . . . under the guidance ofhjs Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of Master of Science in .Biomedic^ Da/e July. 18, 1995................ Chairman Dedication This thesis is dedicated to my parents, Suresh and Bharti, and my brother Sarang. Acknowledgements I thank Dr. Charles E. Niesen and Dr. Michel Baudry for many thought provoking discussions and encouragement for pursuing this work. I also acknowledge the helpful critques provided by Dr. Theodore W. Berger and Dr. Vasilis Z. Marmarelis. Table of Contents Page 1.0 INTRODUCTION 1 1.1 Long-term Potentiation (LTP) 1 1.2 Spine Morphology 3 1.3 Dendritic Spine Cytology 4 1.4 Ontogeny of Spines and Development of LTP 6 1.5 Voltage-gated Calcium Channels 7 2.0 CALCIUM DYNAMICS MODEL 8 2.1 Spine Morphology 8 2.2 Calcium Flux Model 10 2.3 Spine Head Potential 17 2.4 Computational Strategy 17 3.0 INDUCTION OF NMDA-DEPENDENT LTP 19 3.1 NMDA Model 20 3.2.0 NMDA Simulation Results 23 3.2.1 Comparison of Geometries 24 3.2.2 Long-thin Geometry 26 3.2.3 Stubby Geometry 28 3.2.4 Mushroom Geometry 30 3.3 Discussion 33 4.0 THE ROLE OF AMPA RECEPTORS 35 4.1 AMPA Model ~ 36 4.2.0 AMPA Simulation Results 38 4.2.1 Comparison of Geometries 40 4.2.2 Long-thin Geometry 40 4.2.3 Stubby Geometry 42 IV 4.2.4 Mushroom Geometry 45 4.3 Discussion 47 5.0 INDUCTION OF NMDA RECEPTOR-INDEPENDENT LTP 48 5.1 Voltage-dependent Calcium Channel (VDCC) Model 50 5.2.0 Non-NMDA Mediated LTP Simulation Results 54 5.2.1 Comparison of Geometries 59 5.2.2 Long-thin Geometry 60 5.2.3 Stubby Geometry 63 5.2.4 Mushroom Geometry 66 5.2.5 L-type Calcium Channel 67 5.2.6 N-type Calcium Channel 68 5.2.7 T-type Calcium Channel 69 5.3 Discussion 71 6.0 COLOCALIZATION OF VGCCS AND GLURS 73 6.1.0 Colocalization Simulation Results 75 6.1.1 Comparison of Geometries 76 6.1.2 Long-thin Geometry 78 6.1.3 Mushroom Geometry 80 6.1.4 Stubby Geometry 83 6.2 Discussion 85 7.0 COMPARISONS WITH OTHER SPINE MODELS 87 7.1 Comparison with Imaging Studies 88 7.2 Future Studies 89 8.0 CONCLUSION 91 9.0 BIBLIOGRAPHY 93 10.0 APPENDIX 104 V List of Tables Table Page 2-1 Model #1 "Long-thin" Dimensions and [Bt] 11 2-2 Model #2 "Stubby" Dimensions and [Bt] 12 2-3 Model #3 "Mushroom" Dimensions and [Bt] 13 5-1 State Variables and Conductances 52 5-2 Calcium Channel Parameters 54 List of Figures Figure Page 2-1 Schematic of Dendritic Spine Function 9 2-2 Long-thin Spine 11 2-3 Stubby Geometry 12 2-4 Mushroom Geometry 13 2-5 Spine Head Potential 18 3-1 NMDA-dependent LTP Model 20 3-2 NMDA Calcium Current 24 3-3 Comparison of Compartment 1 Calcium Concentration 25 3-4 Comparison of Compartment 1 Buffer Concentration 25 3-5a Long-thin Spine: Regional Calcium Fluctuations 26 3-5b Long-thin Spine: Calcium Profile 27 3-6a Long-thin Spine: Regional Buffer Fluctuations 27 3-6b Long-thin Spine Buffer Profile 28 3-7a Long-thin Spine: Regional Calcium Fluctuations 29 3-7b Long-thin Spine: Calcium Profile 29 3-8a Stubby Spine: Regional Buffer Fluctuations 30 3-8b Stubby Spine: Buffer Profile 30 3-9a Mushroom Spine: Regional Calcium Fluctuations 31 3-9b Mushroom Spine: Calcium Profile 32 3-10a Mushroom Spine: Regional Buffer Fluctuations 32 3-10b Mushroom Spine: Buffer Profile 33 4-1 Isolated AMPA Model 36 4-2 AMPA Calcium Current 38 4-3 Comparison of Compartment 1 Calcium Concentration 39 vii 4-4 Comparison of Compartment 1 Buffer Concentration 39 4-5a Long-thin Spine: Regional Calcium Fluctuations 40 4-5b Long-thin Spine: Calcium Profile 41 4-6a Long-thin Spine Regional Buffer Fluctuations 41 4-6b Long-thin Spine: Buffer Profile 42 4-7a Long-thin Spine: Regional Calcium Fluctuations 43 4-7b Long-thin Spine: Calcium Profile 43 4-8a Stubby Spine: Regional Buffer Fluctuations 44 4-8b Stubby Spine: Buffer Profile 44 4-9a Mushroom Spine: Regional Calcium Fluctuations 45 4-9b Mushroom Spine: Calcium Profile 45 4-10a Mushroom Spine: Regional Buffer Fluctuations 46 4-10b Mushroom Spine: Buffer Profile 46 5-1 VGCC Dependent LTP Spine Model 50 5-2a Steady State Activation Variable 55 5-2b Activation Time Constant 55 5-3a Steady State Inactivation Variable 56 5-3b Inactivation Time Constant 56 5-4a Inactivation Variable 57 5-4b Square of Activation Variable 57 5-5a Calcium Currents 58 5-5b Calcium Currents (expanded view) 58 5-6 Comparison of Compartment 1 Calcium Concentration 59 5-7 Comparison of Compartment 1 Buffer Concentration 60 5-8a Long-thin Spine: Regional Calcium Fluctuations 61 5-8b Long-thin Spine: Calcium Profile 61 5-9a Long-thin Spine: Regional Buffer Fluctuations 62 viii 5-9b Long-thin Spine: Buffer Profile 62 5-10a Long-thin Spine: Regional Calcium Fluctuations 63 5-10b Long-thin Spine: Calcium Profile 63 5-1 la Stubby Spine: Regional Buffer Fluctuations 64 5-1 lb Stubby Spine: Buffer Profile 64 5-12a Mushroom Spine: Regional Calcium Fluctuations 65 5-12b Mushroom Spine: Calcium Profile 65 5-13a Mushroom Spine: Regional Buffer Fluctuations 66 5-13b Mushroom Spine: Buffer Profile 66 5-14 L-type Channel: Calcium Fluctuation Comparison 67 5-15 L-type Channel: Buffer Fluctuation Comparison 68 5-16 N-type Channel: Calcium Fluctuation Comparison 69 5-17 N-type Channel: Buffer Fluctuation Comparison 69 5-18 T-type Channel: Calcium Fluctuation Comparison 70 5-19 T-type Channel: Buffer Fluctuations Comparison 70 6-1 Colocalized VGCC and GluR Spine Model 74 6-2a Calcium Current Breakdown 75 6-2b Calcium Current Profile 76 6-3 Comparison of Compartment 1 Calcium Concentration 77 6-4 Comparison of Compartment 1 Buffer Concentration 77 6-5a Long-thin Spine: Regional Calcium Fluctuations 78 6-5b Long-thin Spine: Calcium Profile 79 6-6a Long-thin Spine Regional Buffer Fluctuations 79 6-6b Long-thin Spine: Buffer Profile 80 6-7a Mushroom Spine: Regional Calcium Fluctuations 81 6-7b Mushroom Spine: Calcium Profile 81 6-8a Mushroom Spine: Regional Buffer Fluctuations 82 ix Mushroom Spine: Buffer Profile Stubby Spine: Regional Calcium Fluctuations Stubby Spine: Calcium Profile Stubby Spine: Regional Buffer Fluctuations Stubby Spine: Buffer Profile Abstract A biophysical computer model was developed to simulate calcium dynamics in dendritic spines of rat hippocampal pyramidal neurons. This model accounts for calcium influx into dendritic spines from both glutamate receptors (GluRs) and voltage-gated calcium channels (VGCCs). Mechanisms simulating diffusion, pumping, and buffering of calcium were derived from the methods described by others (Holmes and Levy, 1990). Results obtained from the sim ulations are in good agreem ent with calcium imaging studies and electrophysiological data in the hippocampal slice, i.e., presynaptic stimulation elevates calcium levels in the spine head to tens of micromoles. In adult spines, both the N-type and T-type VGCC as well as the NMDA GluR are major contributors to this calcium boost which induces long-term potentiation (LTP). Neither the L-type VGCC nor the AMPA GluR alone can induce LTP. Both NM DA-independent LTP and NMDA-dependent LTP can be induced in the model. 1.0 INTRODUCTION Dendritic spines, the major postsynaptic targets for excitatory innervation in the central nervous system, play a significant role in synaptic transmission. They integrate excitatory action and play a great role in signal processing, learn ing and memory. They are located at 90% of excitatory synapses in the central nervous system (Harris and Kater, 1994). However, they elude study by conven tional techniques such as light microscopy and microelectrode recording due to their small size, which is on the order of a micron. With the emergence of confo- cal microscopy, and other advancements that allow us to see at least physical properties such as spine dimensions and molecular composition, computational models of spine function can now reasonably be generated. Analytical modeling based on experimental hypothesis will ultimately explain spine function. Computer simulations are implemented to vary structural dimensions and locations of active molecules. Ideal geometries such as cylinders are used to mimic spine morphology which can vary from fat, thin to long, short, to branched, unbranched (Harris et ai, 1992). These different categories of spine geometry may have direct implications on their function. The goals of this mod eling study are to: (i) determine what role these categories of spines have in de termining whether or not long-term potentiation (LTP) will be induced, and (ii) determine what contribution the various glutamate receptors (GluRs) and volt age-gated calcium channels (VGCCs) have, if any, on induction of the various forms of LTP. 1.1 Long-term potentiation (LTP) Long-term potentiation (LTP) has become the widely accepted model of memory by neuroscientists (Bliss and Collingridge, 1993). The amount of depo larization caused by a fixed amount of neurotransmitter in the postsynaptic cell 1 has been termed the "strength" of synaptic transmission. LTP refers to an in crease in synaptic strength lasting hours to weeks. It has been defined as "stable potentiation of excitatory synaptic responses induced by coactivation of both presynaptic and postsynaptic elements" (Kullman et ah, 1992). Two types of LTP have been identified in hippocampus: NMDA-dependent LTP and NMDA-in- dependent LTP (Grover and Teyler, 1990; Aniksztejn and Ben-Ari, 1990; Kullman et al., 1992). Both forms of LTP require a postsynaptic increase in intracellular calcium concentration. Dendritic spines have been hypothesized to isolate and amplify such a synaptically induced calcium increase (Koch et al., 1992). In the CA1 region of hippocampus, NMDA-dependent LTP is triggered at the Schaffer collateral input to pyramidal neurons. LTP is induced when the ex citatory neurotransmitter glutamate is released from the presynaptic terminal and then depolarizes the postsynaptic membrane, causing an increase in the con centration of intracellular calcium, [Ca]^ at the postsynaptic cell via activation of the N-m ethyl-D-aspartate (NMDA) glutamate receptor channel complex (Malenka and Nicoll, 1993). NMDA-independent LTP involves calcium influx through voltage-depen dent calcium channels (VDCCs) (Grover and Teyler, 1995; Huber et ah, 1995). In area CA1 of hippocampus, calcium-dependent NMDA receptor-independent LTP can be induced via 200 Hz tetanic stimulation of presynaptic fibers (Grover and Teyler, 1990; Grover and Teyler, 1992), activating VDCCs. Another form of NMDA-independent LTP, LTP(k), also exists. LTP(k) is induced by briefly ap plying the potassium channel blocker tetraethylammonium (TEA) and then stimulating the Schaffer collateral/commissural fibers (Aniksztejn and Ben-Ari, 1991). In both of these forms of NMDA-independent LTP, the subsequent rise in intracellular calcium concentration following stimulation is not mediated by 2 NMDA receptors because the NMDA receptor antagonist D,L-2-amino-5-phos- phonovalerate (APV) has no effect on this type of potentiation. Another pathway which is important for the induction of LTP is the metabotropic glutamate receptor (mGluR). This receptor is not linked to a chan nel, but instead has the effect of releasing calcium from intracellular storage sites such as mitochondria and endoplasmic reticulum. Through G-proteins, phos- pholipase C gets activated. Subsequently, an activated a-subunit hydrolyzes the membrane lipid phosphatidylinositol 4,5-biphosphate (PIP2) producing inositol triphosphate (IP3) and diacylglycerol (DAG). IP3 releases calcium from the en doplasmic reticulum, while DAG activates protein kinase C. The above m en tioned mGluR cascade, which is activated by glutamate, probably depends on the initial NMDA receptor-mediated calcium transient and thus produces an amplified calcium signal (Nicholls et al, 1992). Some have suggested that LTP is maintained in part by presynaptic mech anisms. They postulate that a "retrograde messenger" is released from the post synaptic site of induction to initiate increased transmitter release from the presy naptic terminal. Proteins, arachidonic acid, and nitric oxide have been proposed as possible messenger signals. However, the time course of their action seems to be too slow to play a part in a rapid process (Bliss and Collingridge, 1993). Other possible presynaptic mechanisms include regulation of transmitter release by means of alterations in the events that lead to calcium entry, exocytosis, mobiliza tion, docking, and fusion of vesicles at the release sites in the presynaptic termi nal. This regulation might have a role in the maintenance of LTP. 1.2 Spine Morphology Dendritic spines are protoplasmic protuberances covering the dendrites of many neurons (Koch and Zador, 1993). In general, spines terminate with a bul 3 bous "head" and have a "neck" that emerges from the dendritic shaft. Spine density has been quantified at three spines per micrometer of dendrite in CA1 hippocampal pyram idal cells. In these pyramidal neurons, spine dimensions vary immensely. Spine necks range in diameter from 0.04 to 0.46 pm, and in length from 0.08 to 1.58 pm (Harris and Stevens, 1989). Dendritic spines have been categorized according to their geometry. If the length of a spine is much greater than the neck diameter, they are characterized as being "long-thin." Spines are termed "mushroom" if the diameter of the head is much greater than the diameter of the neck. They are categorized as being "stubby" if the diameter of the neck and the total length of the spine are similar. On rare occasions, spines with more than one head are found. These are catego rized as being "branched" (Harris et al, 1992). There has been much speculation that spine shape is plastic in vivo. Spine shape has been reported to change in response to stimuli such as light or motor activity. Also, high frequency electrical stimulation of hippocampal pathways has caused dendritic spines to increase their number on dendritic shafts and alter their morphology, i.e., change in spine stem shape, frequent appearance of con cave spine heads, or appearance of larger spine heads (Koch et al, 1992). The spines of the dentate gyrus of hippocampus have been found to swell after in duction of LTP (Harris and Kater, 1994). 1.3 Dendritic Spine Cytology The postsynaptic density (PSD) occupies the first 50 nm of the spine head. It contains a num ber of proteins which have important neuronal functions. Neuroreceptor glycoproteins form calcium ion channels. Protein kinases such as calcium/calmodulin-dependent protein kinase type II (CaM-kinase II) and pro tein kinase C (PKC), which are immediate biochemical processes activated by 4 calcium that are responsible for LTP (Malenka et al, 1989), exist in the PSD. Antagonists applied to the regulatory protein calmodulin have been found to block LTP (Malenka et al, 1989). Fifty percent of the PSD contains the molecule CaM-kinase II (Harris and Kater, 1994). After exposure to calmodulin and calcium, this molecule undergoes a state change, going from a calmodulin/calcium-dependent state to a calmod ulin/calcium independent, autophosphorylating state. This state change might lead to phosphorylation of other proteins such as tubulin or MAP2, altering a spine's synaptic efficacy. Activation of calmodulin and kinase activity in the postsynaptic cell are required for the generation of LTP (Malenka et al, 1989). Structural proteins such as spectrin/fodrin, microtubule associated pro tein 2 (MAP2), actin, and myosin are also found in the PSD, making up the spine cytoskeleton. The transmitter glutamate has been found to degrade fodrin via the protease calpain I. Harris and Kater believe that this mechanisms might be responsible for spine morphology changes that have been observed following in duction of LTP (Harris and Kater, 1994). In addition, spines contain the organelle smooth endoplasmic reticulum (SER), which stores calcium and is involved in membrane synthesis. SER se questers and releases calcium intracellularly. IP3 receptors, which are activated by calcium in the cytoplasm, have been located in the SER of spines (Harris and Kater, 1994). Hence, a transient rise in intracellular calcium can release stored calcium. Dendritic spines isolate changes in spine calcium concentration from cal cium entering from other sources, i.e., voltage-gated calcium channels on the dendritic shaft (Gold and Bear, 1994). Hence, the metabolic machinery in the spine that induces LTP is isolated from dendritic shaft intracellular calcium. Also, dendritic spines augment the nonlinear relationship between magnitude of LTP induction and strength of synaptic stimulation (Jaffe et al., 1994b). This might occur because of the saturation of low capacity buffers or because of the nonlinear relation between spinous intracellular calcium and enzyme activation (Gold and Bear, 1994). These characteristics of spines implicate their role in forming the Hebbian nature of input specific LTP (Brown and Chattarji, 1995). 1.4 Ontogeny of Spines and Development of LTP Dynamic changes occur in spine structure and glutamate receptor distri bution during development (Harris and Kater, 1994). From birth through postnatal day (PND) 7, NMDA receptors are present at about 75% of adult values in area CA1 of rat hippocampus. Yet, at birth, no potentiation is elicited from tetanic stimulation in area CA1. By PND 3-4, post-tetanic potentiation (PTP) lasting less than a minute can be induced. By PND 5-7, dendritic spines appear, and potentiation lasts 45 minutes. By PND 10-11, potentiation lasts 2.5 hours. At day 15, animals show persistent LTP (Harris and Kater, 1994). The length and diameter of the spine neck have been shown to undergo changes during the course of neuronal development (Harris et al, 1992). Spines are first present at PND 5-7. They appear when a nonpersistent form of LTP is first induced. With maturation, more spines have constricted necks, and LTP has been found to be induced at lower stimulation intensities (Harris and Kater, 1994). At day 15, thin, stubby, and mushroom shaped spines are present at equal frequencies. When animals become young adults, the majority of synapses arises on small, thin spines. Hence, a sufficient number of spines with constricted necks are required for persistent LTP. In addition, decay times of the NMDA postsynaptic current change through the course of development. Its characterization can be well simulated with exponential decay functions. The time constants of the exponential function 6 vary from 204 ms for a 13-day-old animal to 67 ms for a 20-day-old animal (Hestrin et al., 1992). Because waveform summation from tetanic stimulation de termines calcium influx, the time course of the NMDA-mediated currents deter mines whether or not LTP will ultimately be induced. 1.5 Voltage-gated Calcium Channels Fluctuations of calcium levels in functioning dendritic spines have re cently been visualized with the aid of fluorescent dyes. Experiments have shown that micromolar levels of calcium are rapidly reached at the spines of distal den drites in response to presynaptic stimulation (Conner et al., 1994). NMDA recep tor-mediated calcium current has been shown to be a source of the influx that re sults in elevated spine calcium levels in the distal dendrites of CA3 neurons (Conner et al., 1994). It has long been assumed that activation of NMDA recep tors is solely responsible for this increase in spine calcium. However, recent con- focal microscopy imaging studies have revealed the existence of voltage-gated calcium channels within at least a subset of hippocampal spines (Jaffe et al., 1994 and Mills et al., 1994). In those studies, large fluorescent increases were observed in some cases in the spine but not in the dendritic shaft. Hence, a judiciously chosen distribution of voltage-gated calcium channels must be incorporated as a source of calcium influx into any model of calcium dynamics within dendritic spines (Brown and Chattarji, 1995). 7 2.0 CALCIUM DYNAMICS MODEL In this section, we'll develop a general hippocampal pyramidal neuron dendritic spine calcium dynamics compartmental model that will be applied in sim ulating the mechanisms underlying traditional NM DA-dependent LTP (Section 3), NMDA-independent LTP (Section 5), and a hybrid LTP model (Section 6). Figure 2-1 summarizes the essence of spine model function. High frequency presynaptic stimulation activates postsynaptic conductances localized to the spine head. NMDA receptors, AMPA receptors, an d /o r voltage-gated calcium channels (VGCCs) become activated, each emitting its own characteristic calcium current, modeled as a point current source, causing calcium influx into the spine head (Zador and Koch, 1994). This calcium then diffuses along the spine neck into the dendrite, binds to buffers, and gets extruded into the extracellular space via calcium pumps. Diffusional resistance to flow of calcium ions is provided by the spine neck (Holmes and Rail, 1995). Our analytical model is based on the experimental data from various studies of hippocam pal pyramidal neurons in rat hippocampus. Although we sometimes take data from region CA1 of hippocampus, and at other times from region CA3, the general features of pyramidal neurons are known to be similar in both regions (Jaffe et al, 1994b). 2.1 Spine Morphology Rat CA1 hippocampal pyramidal neuron spine models were based upon the three dimensional series sample reconstruction study performed by Harris et al. (Harris et al., 1992). The spine morphology was modeled as a fat cylinder (representing the spine head) overlaid on a skinny cylinder (representing the spine neck) concentrically. The attachment site of the spine to the dendritic shaft 8 High Frequency Stimulation Presynaptic Cell / o o o o d \ O O O CUD O Voltage-gated Calcium Channels Glutamate Receptors Ca Ca Calcium Buffers Calcium Pumps Diffusion Dendritic Spine Postsynaptic Cell Figure 2-1 Schematic of Dendritic Spine Function was in stratum radiatum, which is located between the stratum locunosum- moleculare and stratum pyramidale. Eleven or twelve compartments were devised. Compartments 1 and 2 represent the postsynaptic density. For the stubby and mushroom geometries, com partm ents 9-12 correspond to the dendritic shaft. For the long-thin geometry, compartments 8-11 correspond to the dendritic shaft. Compartments 3-8 have different connotations depending on the spine morphology chosen. For Model #1 (long-thin), compartm ents 1-4 represent the spine head, while compartments 5-7 represent the spine neck (see Fig. 2-2). For Model #3 (mushroom), compartments 1-5 represent the spine head, while compartments 6- 8 represent the spine neck (see Fig. 2-3). In Model #2 (stubby), there is no head/neck differentiation, so compartments 1-8 represent a long neck (see Fig. 2- 4). Tables 2-1,2-2, and 2-3 show the spine dimensions for each of the above three models. 2.2 Calcium Flux Model The major unknown parameters in modeling studies of dendritic spines are the density of calcium pumps, pum p rate, buffer rate, and buffer capacity. It is assumed that the spine has a high density of calcium pumps, even though there is no direct evidence to support this (Jaffe et al., 1994a). The ATP- dependent calcium pum p was taken to have a rate constant of 1.4 X 10 "'em /s (Holmes and Levy, 1990). Sequential and noncooperative buffering mechanisms were chosen to have a forward rate constant of 0.5 jiM 'ms'1 and a backward rate constant of 0.5 ms'1 , based on a Kd of calmodulin of 1 pM (Zador et al, 1990). The postsynaptic density of the spine, which contains calmodulin, calbindin, calcineurin, and other binding proteins, was chosen to have a calcium buffer concentration of 200 pM. This value was based on the assumption that there are 10 4 S. 3 _ 4 5 6 7 10 11 Figure 2-2 Long-thin Spine Table 2-1 Model #1 "Thin" Dimensions and Initial Buffer [Bt] Compartment Number diam (mm) thick (mm) Bt (HM) 1 0.55 0.05 200.0 2 0.55 0.05 200.0 3 0.55 0.25 100.0 4 0 . 55 0.20 100.0 5 0.10 0 .20 100.0 6 0.10 0.33 100.0 7 0.10 0.20 100.0 8 1.00 0 .50 100.0 9 1.00 0.50 100.0 10 1.00 0 .50 100.0 11 1.00 0.50 100.0 11 Figure 2-3 Stubby Geometry Table 2-2 M odel #2 "Stubby" Dimensions and Initial Buffer [Bt] Compartment Number diam (mm) thick (mm) Bt (HM) 1 0.76 0.05 200.0 2 0.76 0.05 200.0 3 0.76 0.20 100.0 4 0.76 0.19 100.0 5 0.76 0.20 100.0 6 0.76 0.10 100 .0 7 0.76 0.10 100.0 8 0.76 0.10 100.0 9 1.00 0.50 100.0 10 1.00 0.50 100.0 11 1.00 0.50 100.0 12 1.00 0.50 100.0 12 2 3 4 7 11 12 10 Figure 2-4 Mushroom Geometry Table 2-3 M odel #3 "Mushroom" Dimensions and Initial Buffer [Bt] Compartment Number diam (mm) thick (mm) Bt (MM) 1 0.77 0.10 200.0 2 0.77 0.10 200.0 3 0.77 0.20 100 .0 4 0.77 0.20 100 . 0 5 0.77 0.17 100 .0 6 0.20 - 0.15 100.0 7 0.20 0.23 100.0 8 0.20 0.15 100.0 9 1.00 0.50 100.0 10 1.00 0.50 100.0 11 1.00 0.50 100.0 12 1.00 0.50 100.0 13 four independent binding sites per calmodulin molecule, each molecule having a concentration of 40 jiM. The concentrations of the other molecules are unknown (Holmes and Levy, 1990). The Ca2+ current elicited from the NMDA receptor complex, the AMPA receptor complex, and/or voltage gated calcium channels (VGCCs) after strong presynaptic stim ulation under postsynaptic voltage clamp conditions is mimicked by equations 3-3, 4-3, and 5-1 (see their respective sections for details on their implementation): ^C a = P f^N M D A (3"3) Ara = ^//\MPA (4-3) 1 _ ^ a2 ^ £2 F v'R T I Av [C a2+]n 7 Ca^ C a -+ l - e 2F v,R T ( } These are the driving functions for the spine model. We'll consider them collectively as Eq. 2-1. This calcium then diffuses to the next compartment via one-dimensional calcium diffusion, becomes sequestered by endogenous calcium buffering, an d /o r is extruded via calcium pumps. The calcium influx due to these calcium currents is given by (Yamada et ah, 1989) d[Ca2 + ]n _ /C a (f + Af) + /C a (r) dt - 4FV, (2_ 2) where F corresponds to Faraday's constant and Vi corresponds to volume of the ith compartment. The changes in calcium concentration due to diffusion to adjacent compartments is given by the one-dimensional diffusion equation (Holmes and Levy, 1990): 14 = ~ t(j)u -,([ C < * l - i c a b ,) + (j ) u„([Ca], - [CV.],„)} (2-3) where (2-3b) D ~ diffusion constant for calcium = 0.6 |im 2 /m s V t = volume of the zth compartment [Ca], = concentration of calcium in the zth compartment = coupling coefficient between compartments z and j A■ = cross-sectional area of the zth compartment Aj = cross-sectional area of the ;th compartment I; = length (or thickness) of compartment z lj = length (or thickness) of compartment j ATP pum ps which exist on the postsynaptic element are sim ulated by the following term (Holmes and Levy, 1990): = -kPi{[Cal - [Ca]r) (2-4) at 15 kp,=kp^r where [Ca]r = resting calcium concentration = 20 nM (2-4b) kp = velocity of the ATP-dependent calcium pump = 1.4 X 10- 1 cm /s Aj = cross-sectional area of the zth compartment V t = volume of the zth compartment The change in calcium concentration due to sequestration by buffers is given by (Holmes and Levy, 1990): = -kbf [Ca],[Bl + kbb([Bt]j - [5],) (2-5) at where kbf = forward buffer rate constant (i.e., constant for association of calcium and buffer) = 0.5 jiM^ms'1 kbh = backward buffer rate constant (i.e., constant for disassociation of calcium and buffer) = 0.5 ms'1 [£], = concentration of free buffer in the zth compartment 16 [Bt\ = total buffer concentration in the z'th compartment = 200 pM near postsynaptic density (i.e., compartments 1 and 2) = 100 pM away from postsynaptic density (i.e., compartments 3-12) The change in amount of available buffer after application of calcium is given by = -kbf [Cal[Bl + kbb([Btl - [5],) (2-6) 2.3 Spine Head Potential Postsynaptic depolarization w ithout high frequency presynaptic stimulation does not induce LTP in CA1 pyram idal neurons (Bliss and Collingridge, 1992). Also, LTP can be induced in an associative manner; i.e., a weak input, itself not capable of inducing LTP, when paired with a strong input, induces LTP (Barrionuevo and Brown, 1983). The change in membrane potential at one spine head, Vm, due to 50 Hz presynaptic stimulation is depicted in Fig. 2- 5 (Holmes and Levy, 1990). It is based upon eight synaptic activations with 96 synapses coactivated at 50 Hz. Synapses were distributed uniformly along the same apical dendrite as the spine. Hence, the required postsynaptic depolarization was provided by strong coactive synaptic input. 2.4 Computational Strategy Equations 2-3, 2-4, and 2-5 were lumped together to create a single term j r y-t l for — This lumped parameter along with Eq. 2-6 for the change in buffer 17 concentration forms a simultaneous set of differential equations that was solved using the Fourth Order Runge-Kutta Method with a time step of 2 |is. A general-purpose computer program was developed with the vision that it should enable us to perform parametric studies involving sensitivity to model and channel parameters, variation of diffusion properties, location and value of influx currents or potentials, etc. A detailed description as well as a complete listing of the source code is provided in Section 10 (Appendix). The conceptual flow chart of the computer program is as follows: (i) Define the analytical model (ii) Provide channel information (i.e., AMP A, NMDA and/or VGCC) (iii) Provide influx information (iv) Specify time-history analysis parameters (v) Perform dynamic analysis (vi) Save and post-process results. - 10- -2 0 V -30 m (m V ) -4 0 -5 0 -6 0 -7 0 0 20 40 60 80 100 120 140 160 180 200 Time (ms) Figure 2-5 Spine Head Potential 18 3.0 INDUCTION OF NMDA-DEPENDENT LTP The first step in the induction of LTP is the depolarization of the post synaptic cell by high frequency presynaptic input (Bliss and Collingridge, 1992), resulting in the initial activation of AMPA receptors by glutamate, an amino acid which is released from presynaptic terminals. The resistive barrier of the spine neck constriction amplifies this depolarization (Harris and Kater, 1994). NMDA receptors, by contrast, are not initially activated by glutamate because the chan nel is still blocked by magnesium ions when the membrane potential is still at the resting level. With greater depolarizations of the postsynaptic membrane, mag nesium ions are removed from the channel, allowing the influx of calcium ions through the NMDA receptor channel as a result of glutamate binding to the re ceptor (Perkel et al., 1993). In addition to all of this, glycine must occupy its bind ing site on the NMDA receptor-channel complex for the NMDA channel to func tion at all (Liaw et al, 1995). The increased influx of calcium through the NMDA channels activates a number of intracellular biochemical pathways, including CAM kinase and possi bly protein kinase C, lipases and proteases. This activation causes (i) an increase in the sensitivity of AMPA receptors; (ii) an increase in the number of post synaptic receptors; and (iii) an alteration of channel kinetics. Other factors involved in the maintenance of LTP include "presynaptic modifications which result in an increase in the amount of glutamate released per impulse" or "an extrasynaptic change, such as a reduction in uptake of glutamate by glial cells leading to increased neurotransmitter availability at the receptors" (Bliss and Collingridge, 1993). Maintenance also involves changes in the physical properties of the postsynaptic side, i.e., widening of the neck to let more ions through. 19 3.1 NMDA Model We simulate classic NMDA-dependent LTP in this section (see Fig. 3-1). The NMDA receptor complex was modeled after experimental data taken by Spruston et al. (Spruston et al, 1995). Sufficient depolarization is required to re lieve its magnesium block. The AMPA receptor contributes to this depolariza tion, playing an important role in mediating NMDA currents physiologically. Hence, the AMPA model is coupled to the NMDA model (see Section 4). NMDA GluR Calcium Buffers Calcium Pumps Diffusion Dendritic Spine Postsynaptic Cell Figure 3-1 NMDA-dependent LTP Model The NMDA receptor-mediated current in the presence of magnesium is given by J + [ M g 2 + ] 0 ^ - S z F V / R T K° (3-1) The current in magnesium-free solution is Io=8<y-Vm ) (3-lb) The outward rectification observed in nominally magnesium-free solution is where 8 = 8\ + T 2 —S r 1 + e (3-lc) I0 = current in the Mg2 + free solution g = conductance V = membrane potential Vr e v = reversal potential of the current = -0.7 mV [Mg2 + ]u = concentration of magnesium in the external solution = 1 mM K0 = IC50 at 0 mV = 1 mM 21 8 = electrical distance of the magnesium binding site from the outside of the membrane = 0.8 z = valence = +2 F = Faraday's constant R = gas constant T = temperature (in degrees Kelvin) = 298K g, = the lowest conductance (at very negative potentials) = 40 pS g2 = the highest conductance (at very positive potentials) = 247 pS a = steepness of the voltage dependent transition from gi to g2 = 0.01 The time course of the NMDA current is simulated by the following double ex ponential expression 22 - f - t I nmdaW ~ h ( e ' e 2) (3-2) where T j = 80 ms and r2 = 0.67 ms (Zador et al., 1990). The calcium current elicited from activation of the NMDA receptor complex is where Pf= the permeability of the NMDA receptor to calcium (Schneggen- burger et al., 1993) = 5%. 3.2.0 NMDA Simulation Results The change in spine head membrane potential Vm due to 50 Hz presynap tic stimulation is depicted in Fig. 2-5 (Holmes and Levy, 1990). The subsequent NMDA excitatory postsynaptic current (epsc) evoked by this stimulation is com puted by placing this Vm in Eq. 3-2. Figure 3-2 represents the calcium current lea/ which is believed to be 5% of the NMDA epsc. This current has a peak am plitude of 0.15 pA and is the driving stimulus of the spine model which was de veloped in Section 2. The rise in spinous intracellular calcium concentration and the fall in buffer concentration following presynaptic stimulation are traced in each com partment for each of the three models we are considering. We first show com parisons of the three models and then analyze each model individually. Model given by f NMDA (3-3) 23 #1 corresponds to the long-thin geometry, Model #2 corresponds to the stubby geometry, and Model #3 corresponds to the mushroom geometry. -.0 2 - .0 4 - 1 2 - 14- - . 16 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-2 NMDA Calcium Current 3.2.1 Comparison of Geometries Figure 3-3 compares the intracellular calcium concentration, [Ca] j, in com partment 1 for each of the three models. The long-thin geometry had a tremen dous rise in [Ca]j, reaching levels of 42 |oM, while the mushroom and stubby ge ometries had significantly less of an increase, approaching peak levels of only 5 jiM and 2 jiM respectively. Since rise in intracellular calcium is required for the induction of LTP, we conclude that the long-thin geometry is ideal. Figure 3-4 summarizes the change in spinous buffer concentration in com partment 1 following calcium influx into the spine head. The long-thin geometry experienced the steepest drop in buffer concentration, followed by the mushroom geometry, followed by stubby. Intuitively, this makes sense, because the long-thin geometry experienced the steepest rise in intracellular calcium concentration. Therefore, it should experience the greatest chelation by buffers. 24 45 4 0 - . 35 3 0 - 2 0- - 15- 1 0 - ■ 5 - . 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-3 Comparison of Compartment 1 Calcium Concentration 200 180 160 140- 8 0 - 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-4 Comparison of Compartment 1 Buffer Concentration 25 3.2.2 Long-thin Geometry Figure 3-5a traces the spinous calcium concentration over time in all 11 compartments for the long-thin geometry following activation of the neck, and 0.5 pM in the dendritic shaft. Figure 3-5b shows the three-dimensional (3-D) cal cium concentration profile for the entire spine. It peaks at 42 pM in the spine head and at 20 pM in the spine neck. It shows a saddle-shaped profile in the spine head and neck followed by a flat plane at the dendritic shaft. 45 40 Hedd Sfiine 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-5a Long-thin Spine: Regional Calcium Fluctuations Figure 3-6a traces the corresponding change in spinous buffer concentra tion over the 11 compartments. Practically all of the buffer was used to chelate the massive rise in concentration of calcium in the postsynaptic density (compartments 1-2). In compartments 4-6, the buffer level fell from its initial value of 100 pM to a steady state value of about 70 pM. Very little change in buffer concentration was noted in compartments 7-11. This resulted in a buffer "sink" in the 3-D profile of buffer concentration as shown in Fig. 3-6b. 26 [c a] (M M ) Time (ms) Figure 3-5b Long-thin Spine: Calcium Profile 200 180- 160 140 1 2 0 - [B] (|iM ) 10° - 80- 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-6a Long-thin Spine: Regional Buffer Fluctuations [B] O iM ) Time (ms) Figure 3-6b Long-thin Spine: Buffer Profile 3.2.3 Stubby Geometry Figure 3-7a traces the spinous calcium concentration over time in all 12 compartments for the stubby geometry following activation of the NMDA recep tor. It peaks at 1.7 pM in the spine head, 0.8 pM in the spine neck, and 0.6 pM in the dendritic shaft. Figure 3-7b shows the 3-D calcium concentration profile for the entire spine. It is relatively flat compared to the long-thin geometry. Figure 3-8a traces the corresponding change in spinous buffer concentra tion over the 12 compartments. From an initial value of 200 pM, the concentra tion of buffer in the postsynaptic density (compartments 1-2) dropped down to 70 pM. In compartments 3-12, the buffer level fell from its initial value of 100 pM to a steady state value of about 60 pM. The 3-D profile of buffer concentration was relatively flat, as is to be expected considering that the 3-D profile for cal cium profile was also flat (Fig. 3-8b). 28 1 .4 - .4 - Time (ms) Figure 3-7a Stubby Spine: Regional Calcium Fluctuations Time (ms) Figure 3-7b Stubby Spine: Calcium Profile 29 200 180 160 140 1 2 0 - [B] (|x M ) ,0°- 8 0 - 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-8a Stubby Spine: Regional Buffer Fluctuations Figure 3-8b Stubby Spine: Buffer Profile 30 3.2.4 Mushroom Geometry Figure 3-9a traces the spinous calcium concentration over time in all 12 compartments for the mushroom geometry following activation of the NMDA receptor. It peaks at 5 pM in the spine head, 2.25 pM in the spine neck, and 0.4 pM in the dendritic shaft. Figure 3-9b shows the 3-D calcium concentration pro file for the entire spine. Although it shows a saddle-shaped profile in the spine head followed by a flat plane at the dendritic shaft as in the long-thin geometry, the magnitude of the calcium amplification was nine-fold less. Figure 3-10a traces the corresponding change in spinous buffer concentra tion over the 12 compartments. From an initial value of 200 pM, the concentra tion of buffer in the postsynaptic density (compartments 1-2) dropped down to 20 pM. In compartments 4-7, the buffer level fell from its initial value of 100 pM to a steady-state value of about 45 pM. In the dendritic shaft, buffer concentra tion reached a steady-state value of 70 pM. This resulted in a buffer "sink" in the 3-D profile of buffer concentration, although the magnitude of the sink was not as great as that for the long-thin geometry (Fig. 3-10b). 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-9a Mushroom Spine: Regional Calcium Fluctuations 31 [Ca] GiM) Time (ms) Figure 3-9b Mushroom Spine: Calcium Profile 200 180- 160- 140- 120 - 8 0 - 6 0 - 4 0 - 20 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 3-10a Mushroom Spine: Regional Buffer Fluctuations 32 [B] (fiM) Time (ms) Figure 3-10b Mushroom Geometry: Buffer Profile 3.3 Discussion Flow of ions is impeded by the large diffusional resistance of the thin spine neck. This resistance isolates the spine head from the spine neck. Hence, diffusional resistance of the spine neck regulates calcium concentration in the spine head (Holmes, 1990). Since the diffusional resistance of the neck varies with geometry, i.e., long-thin>mushroom>stubby, the rate of calcium dependent reactions also varies with geometry. Peak spine head calcium concentration is an order of magnitude larger in long-thin spines than in stubby or mushroom shaped spines following presynaptic stimulation. This observation is well in line with experimental data showing that LTP induction (which is dependent upon postsynaptic increase in intracellular calcium) and magnitude are both hindered in rats younger than 15 days due to lack of predominance of long-thin spines (Harris et al, 1992). 33 Synaptic calcium currents are both long and strong enough to overwhelm transient calcium buffer and pump capacities, especially in the long-thin spine (Holmes, 1990). Once the buffering and pumping mechanisms are overwhelmed, the diffusional resistance of the spine neck amplifies intracellular calcium concentration in the spine head (Holmes, 1990). Because the long-thin spine has a smaller head compared to the mushroom and stubby geometries, it has fewer calcium binding sites which results in the greatest calcium amplification. At first glance, it would appear that elevation of intracellular calcium to levels of 40 fi.M in a part of a cell would be toxic. This is true in most parts of a cell, but not in dendritic spine heads. Localized to the small volume of a spine head, transient-elevated calcium levels are harmless, and as we have shown, per form an important function in the induction of LTP. Calcium buffers and endo plasmic reticulum bring down calcium concentration to negligible levels follow ing synaptic activation (Harris and Kater, 1994). Dendritic shaft calcium concen tration is only marginally affected by calcium which diffuses through the spine neck because of the large volume of the dendrite (Holmes and Rail, 1995). 34 4.0 THE ROLE OF AMPA RECEPTORS Presynaptic activity activates the fast, voltage independent a-amino-3-hy- droxy-5-methyl-4-isoxazole propionic acid (AMPA) conductance. The AMPA re ceptor depolarizes the cell to a level that facilitates activation of NMDA receptors (see Section 3). However, controversy exists as to whether only the AMPA com ponent of the synaptic current increases following induction of LTP ("postsynaptic hypothesis"), or whether both the AMPA and NMDA compo nents increase ("pre- or postsynaptic hypothesis") (Edwards, 1995). The AMPA receptor-mediated component of synaptic transmission is modified in LTP. It has been postulated that kinases directly phosphorylate ion channels that mediate synaptic transmission. As evidence of this, when the ki nase inhibitor K-252b was applied, the increase in AMPA sensitivity following the induction of LTP was prevented (Bliss and Collingridge, 1993). Phospholipase-A-2 (PLA2) increases the affinity of AMPA receptors by modify ing its binding properties. Baudry suggests that a mechanism exists whereby calpains, which are linked to the intracellular cytoskeleton, increase the number of AMPA receptors at the synapse (Baudry et al., 1992). The influx of calcium through NMDA recep tors is hypothesized to cause either insertion or activation of new AMPA recep tors on the postsynaptic membrane (Edwards, 1995). Furthermore, it is possible that LTP might be a reflection of a change in expression of the AMPA flip and flop receptor subunit (which have different conductance properties) or might involve a change in the expression of GluR 1-4 subtypes of the AMPA receptor or might even be the result of regulation of RNA editing (Bliss and Collingridge, 1993). 35 4.1 AMPA Model In this section, we consider the case of an AMPA receptor uncoupled to NMD A receptors (see Figure 4-1). AMPA GluR Calcium Pumps Calcium Buffers Diffusion Dendritic Spine Postsynaptic Cell Figure 4-1 Isolated AMPA Model The change in the voltage independent AMPA synaptic conductance is (Jaffe, et al, 1994a): A^a m p a = G 'a m p a 0^ (4-l) 36 where a = time to peak of the AMPA conductance change = 300 s'1 Ga m p a = maximum conductance of the AMPA receptor = 0.5 nS The synaptic AMPA current is derived by the following equation ^ A M P A = A ^ A M P A ( K i — -^ A M P a) where V m = membrane potential £a m p a = reversal potential of the AMPA current = 0 mV The calcium component of the AMPA current is A r a — P f l AMPA where P f = calcium permeability fraction of the AMPA current (4-2) (4-3) 37 The perm eability of the AMPA receptor to calcium was set to 1% (Schneggenburger et al., 1993). 4.2.0 AMPA Simulation Results The change in spine head membrane potential Vm due to 50 Hz presynap- tic stimulation is depicted in Fig. 2-5 (Holmes and Levy, 1990). The subsequent AMPA excitatory postsynaptic current (epsc) evoked by this stimulation is com puted by placing this Vm in Eq. 4-2. o - .0 5 - . 1 X Ca - .1 5 (pA) -.2 - .2 5 - . 3 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-2 AMPA Calcium Current Figure 4-2 represents the calcium current lea/ which is believed to be 1% of the AMPA epsc. This current has a peak amplitude of 0.25 pA and is the driving stimulus of the spine model which was developed in Section 2. The rise in spinous intracellular calcium concentration and the fall in buffer concentration following presynaptic stimulation are traced in each com partm ent for each of the three models we are considering. We first show comparisons of the three models and then analyze each model individually. 38 Model #1 corresponds to the long-thin geometry, Model #2 corresponds to the stubby geometry, and Model #3 corresponds to the mushroom geometry. 2 - ■ 1 .7 5 - 1 .2 5 - .#3. .25 1 * 2- 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-3 Comparison of Compartment 1 Calcium Concentration 200 180 * 2' 160- .*3' 140- 12 0 - m 100 i (HM) 8 0 - 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-4 Comparison of Compartment 1 Buffer Concentration 39 4.2.1 Comparison of Geometries Figure 4-3 compares the intracellular calcium concentration, [Ca]f, in compartment 1 for each of the three models. The long-thin geometry had the greatest rise in [Ca] j, reaching 2 pM, while the mushroom and stubby geometries had less of an increase, approaching peak levels of only 0.75 pM and 1 pM respectively. These values are 20-fold less than those attained with the NMDA receptor. Hence the AMPA receptor alone can not induce LTP. Figure 4-4 sum marizes the change in spinous buffer concentration in compartment 1 following calcium influx into the spine head. The long-thin geometry experienced the largest drop in buffer concentration, dropping to 70 pM from its initial value of 200 pM. Buffer concentration in the stubby geometry dropped to 100 pM, while it dropped to 110 pM for mushroom. 4.2.2 Long-thin Geometry Figure 4-5a traces the spinous calcium concentration over time in all 11 compartments for the long-thin geometry following activation of the AMPA re- 1 .7 5 - 1 .2 5 - .7 5 - . 5 - .2 5 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-5a Long-thin Spine: Regional Calcium Fluctuations 40 ceptor. It peaks at 2 |iM in the spine head, 0.65 |iM in the spine neck, and 0.1 pM in the dendritic shaft. Figure 4-5b shows the three-dimensional (3-D) calcium concentration pro file for the entire spine. It clearly shows a transient spike followed by a decay as the profile in the spine head followed by a flat plane at the dendritic shaft. Figure 4-5b Long-thin Spine: Calcium Profile 200 180- 160 140 1 2 0 - [B] (HM) 100' i 8 0 - ~ 7; 60 40 20 “ t r r r p r r n - f ptt r f r r r r [ r t r t f r r r r j r ptt p r r r r j p r r rj-r r r r p - r r r f r r r r j r n r r j r r r r y r r r rj-r i-rr — 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-6a Long-thin Spine: Regional Buffer Fluctuations 41 Figure 4-6a traces the corresponding change in spinous buffer concentration over the 11 compartments. Buffer concentration dropped to 70 pM from its initial value of 200 pM in the postsynaptic density (compartments 1-2). In compartments 3-5, the buffer level fell from its initial value of 100 pM to 40 pM, while in compartment 6 it dropped to 60 pM. Very little change in buffer concentration was noted in compartments 7-11. This resulted in a buffer "sink" in the 3-D profile of buffer concentration. 4.2.3 Stubby Geometry Figure 4-7a traces the spinous calcium concentration over time in all 12 compartments for the stubby geometry following activation of the AMPA recep tor. It peaks at 1 pM in the spine head, 0.3 pM in the spine neck, and 0.15 pM in the dendritic shaft. Figure 4-7b shows the 3-D calcium concentration profile for the entire spine. It is relatively flat compared to the long-thin geometry. [B] ' (H M ) Time (ms) Figure 4-6b Long-thin Spine: Buffer Profile 42 [C a ] . (HM) 8 6 4 2 r * I O - 0 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-7a Stubby Spine: Regional Calcium Fluctuations J W l r Time (ms) Figure 4-7b Stubby Spine: Calcium Profile Figure 4-8a traces the corresponding change in spinous buffer concentra tion over the 12 compartments. From an initial value of 200 |iM, the concentra tion of buffer in the postsynaptic density (compartments 1-2) dropped down to 43 100 (J.M . In compartments 3-11, the buffer level fell from its initial value of 100 pM to a steady state value of about 90 (iM. The 3-D profile of buffer concentration was relatively flat, as is to be expected considering that the 3-D profile for calcium profile was also flat (Fig. 4-8b). 200 180 160 140 120 [B] (H M ) 100 80 60 40 20 0 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-8a Stubby Spine: Regional Buffer Fluctuations 0 j T ------------------------------------------------------------------------------------------------- j Time (ms) Figure 4-8b Stubby Spine: Buffer Profile 44 4.2.4 Mushroom Geometry Figure 4-9a traces the spinous calcium concentration over time in all 12 compartments for the mushroom geometry following activation of the AMPA receptor. It peaks at 0.78 pM in the spine head, 0.38 |iM in the spine neck, and /s: -1-4; 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-9a Mushroom Spine: Regional Calcium Fluctuations J m i w \ Time (ms) Figure 4-9b Mushroom Spine Calcium Profile 45 0.05 (J.M in the dendritic shaft. Figure 4-9b shows the 3-D calcium concentration profile for the entire spine. The profile was relatively flat. Figure 4-10a traces the corresponding change in spinous buffer concentra tion over the 12 compartments. From an initial value of 200 |i.M, the concentration of buffer in the postsynaptic density (compartments 1-2) dropped 200 160- 140- 1 2 0 - 10 0 - 80 60 40 20 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 4-10a Mushroom Spine: Regional Buffer Fluctuations [B] OiM) Time (ms) Figure 4-10b Mushroom Spine: Buffer Profile 46 down to 110 pM. In compartments 4-7, the buffer level fell from its initial value of 100 pM to a steady state value of about 80 pM. In the dendritic shaft, buffer concentration reached a steady-state value of 95 pM. The 3-D profile of buffer concentration thus turns out to be flat (Fig. 4-10b). 4.3 Discussion We considered here the case of a spine possessing an AMPA receptor without NMDA receptors. Although the AMPA receptor plays a role in provid ing sufficient depolarization to relieve the magnesium block of NMDA receptors, the AMPA receptor alone can not induce LTP. Our simulations show that for even the ideal long-thin geometry case, peak spinous [Ca]i reached only 2 pM, whereas for the NMDA case, it reached levels of 42 pM. Large increases in cal cium concentration is a requirement for induction of LTP. The distribution of kainate/AMPA type glutamate receptors has been shown to vary during neuronal development (Pellegrini-Giampiertro et al., 1992). Pellegrini-Giampiertro's group has shown through in situ hybridization that in area CA1 and CA3 of hippocampus, a larger number of calcium permeable kainate/AMPA channels exist in early neonatal life. They showed an increase from postnatal week 1 to week 3, followed by a decline. The model presented here considered only an "adult" AMPA conductance. Experimental data regard ing conductance changes of the AMPA receptor over time has not yet been col lected. 47 5.0 INDUCTION OF NMDA RECEPTOR- INDEPENDENT LTP Non-NMDA receptor mechanisms of calcium entry involving voltage-de pendent calcium channels (VDCCs) have been shown at normal and potentiated synapses in CA1 pyramidal neurons (Grover and Teyler, 1990; Huber et al., 1995). The model presented in this section explores the consequences of localizing VDCCs to dendritic spines. Several studies pointing to the existence of calcium channels on distal den drites and possibly on spines provide justification for their placement on spine heads in this model, which created active calcium conductances on these structures. Confocal microscopy studies utilizing the fluorescein conjugate of the N-type VDCC blocker co-conotoxin (Fl-co-CgTx) have revealed the possible localization of N-type channels on a subpopulation of dendritic spines in CA1 pyramidal neurons (Mills et al, 1994). Immunohistological experiments utilizing antibody labeling show that N-type VDCCs are distributed in distal dendrites (Westenbroek et al., 1992). L-type calcium channels have also been identified with antibody labeling at the base of dendrites in hippocampal pyram idal neurons (Westenbroek et al., 1990). Aniksztejn and Ben-Ari were the first to show that by blocking the IC / Im, and delayed rectifier (Ik) potassium currents transiently with tetraethlyammo- nium (TEA), a NMDA-independent form of LTP, termed LTP(k), occurs in CA1 region of hippocampus (Aniksztejn and Ben-Ari, 1991). The NMDA receptor an tagonist APV did not prevent LTP(k). However, the AMPA receptor antagonist 6-cyano-7-nitroquinoxaline-3,3-dione (CNQX)*did block LTP(k), indicating that LTP(k) is induced by an enhanced release of glutamate acting on non-NMDA re ceptors. Also, the calcium channel blocker flunarizine prevented LTP(k). The authors of the study speculate that potentiation elicited by TEA is due to a tran sient enhanced release of glutamate which generates a depolarization by way of 48 quisqualate receptors sufficient to cause activation of voltage-dependent flunar- izine-sensitive calcium channels. Huang and Malenka also showed that synaptic enhancement induced by TEA does not require NMDA receptor activation (Huang and Malenka, 1993). They blocked LTP(k) by either applying the L-type VGCC blocker nifedipine or injecting the calcium chelator BAPTA into the postsynaptic cell. Nifedipine had no effect on the magnitude and duration of tetanus induced NMDA-receptor in dependent LTP. Hence, it would seem from this study that L-type VGCCs are required for induction of LTP(k) but not NMDA receptor-dependent LTP. However, an occlusion experiment was done in which tetanus induced NMDA- dependent LTP reduced the magnitude of TEA-induced synaptic enhancement and vice versa, indicating that a common expression mechanism is shared and NMDA receptor-dependent LTP can be generated via calcium flux through VGCCs. Huber's group facilitated induction of VGCC-dependent TEA LTP in CA1 hippocampal neurons by either alvear 25 Hz stimulation or by Schaffer collateral stimulation in isolated CA1 slices (Huber et al., 1995). This observation implies that "high frequency stimulation conditions may be required to increase postsy naptic depolarization to sufficiently activate VDCCs and engage potentiation mechanisms." Corroborating evidence to this claim is provided by Kullman et al., who showed that compared to sustained depolarization, repeated depolariz ing pulses were more effective in producing VGCC potentiation (Kullman et al., 1992). In Kullman's study, in presence of the*NMDA antagonist APV, LTP was induced by synaptic stimulation in conjunction with postsynaptic depolarization. The LTP model created in this section addresses these issues (see Fig. 5-1). 49 N-type T-type VGCC VGCC 2 + 2+ Calcium Buffers Calcium Pumps Diffusion Dendritic Spine Postsynaptic Cell Figure 5-1 VGCC Dependent LTP Spine Model 5.1 Voltage-dependent Calcium Channel (VDCC) Model The VDCC model includes the three main types of calcium channels: T- type, N-type, and L-type. Most excitable membranes described to date contain VDCCs. Three types of neuronal calcium channels have been described by whole-cell patch clamp recordings in dorsal root ganglion neurons: the L-type current, which slowly decays over hundreds of milliseconds; the T-type current which is activated by weak depolarizations from a hyperpolarized holding po 50 tential and then rapidly inactivates; and the N-type current, a larger transient current that is also activated from a hyperpolarized holding potential with a stronger depolarization than that is required for the T-type current (Fox et al, 1987). Each type has its own structural, pharmacological, kinetic, and voltage dependent properties. The equation characterizing calcium current elicited by activation of volt age-dependent calcium channels is (Hagiwara and Byerly, 1981 and Jaffe et al, 1994a): ^ c 2Fv/RT I 2 += s Av— [Ca~ J; ■ ■ =----- (5-1) C a 2+ <SC a 2 + \ - e 2FvlR T V ' where g = maximum channel conductance for each C a2+ particular VDCC (L-type, N-type, or T-type) A = Hodgkin-Huxley state variable v = membrane potential [Ca2 + ], = intracellular calcium concentration [Ca2+]0 = extracellular calcium concentration F = Faraday's constant R = universal gas constant T = temperature (in degrees K) The state variables and conductances for each"VDCC type are listed in Table 5-1, where m is the Hodgkin-Huxley channel activation variable and h is the inactiva tion variable. Conductance values were taken as in Jaffe (Jaffe et al., 1994b). It is assumed that there are 5 to 20 channels in a 1 |im2 patch of membrane and that the T:N:L channel ratio is 1:10:10 throughout the cell. 51 Table 5-1 State Variables and Conductances VDCC Type State Variable, A Conductance L-type m 2 0.0025 S/cm 2 N-type m2h 0.0025 S/cm 2 T-type m2h 0.00025 S/cm 2 These voltage-dependent state variables are computed by the following formal ism. For channel activation dm _ mx - m dt T m where the steady-state variable mjis defined as (5-2) m = ------------ (5-3) a . m + p . m and time constant zm is * ,.= ------------ (5-4) «.(V)tA(V) Taking a Laplace transform of Eq. 5-2, we get m(t) = m„( l-e ~ K') (5-5) 52 The Hodgkin-Huxley equations describing a m and are « L ( V ) - 2 ^ e 1 0 - 1 (3 m(V) = cS-vld) where variables a, b, and c are channel dependent and given in Table 5-2. In an analogous fashion, for channel inactivation dh _ h ^ -h dt ~ r, where the steady-state variable hx is defined as a*(V) + A(V) and time constant r,( is 1 “.w+AOO Taking a Laplace transform of Eq. 5-8, we get h(t) = h„( l - e 'f t ) (5-6) (5-7) (5-8) (5-9) (5-10) (5-11) 53 The Hodgkin-Huxley equations describing a h and (3 h are: a h(V) = eei-v/f) (5-12) — (5-13) e 10 +1 where variables d, e, f, and g are channel dependent and given in Table 5-2 (Jaffe et al., 1994a). Table 5-2 Calcium Channel Parameters imeter T-Channel N-Channel L-Channel a 0.2 0.19 15.69 b 19.26 19.88 81.5 c 0.009 0.046 0.29 d 22.03 20.73 10.86 e 10-6 1.6X10-4 — f 16.26 48.4 — g 29.79 39 — 5.2.0 Non-NMDA Mediated LTP Simulation Results The change in spine head membrane potential Vm due to 50 Hz presynap- tic stimulation is depicted in Fig. 2-5 (Holmes and Levy, 1990). The subsequent calcium currents (lea) evoked by this stimulation is computed by placing this Vm in Eq. 5-1 and following through with the algorithm described in Section 2. 54 .6 - m oo -1 0 0 -9 0 -8 0 -7 0 -6 0 -5 0 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 50 Membrane Voltage (mV) Figure 5-2a Steady-State Activation Variable 1 0 - 8 - 6 - (ms) 4 - 2 - 0 i r r t T ' n i 1 , Vi'Tpi ■ ■ ■ ■ j ■ i ■ i j i . n -p r 1 1 1 111 i i i i 11 -1 0 0 -9 0 -8 0 -7 0 -6 0 -5 0 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 50 Membrane Voltage (mV) Figure 5-2b Activation Time Constant Figure 5-2a depicts graphically the steady-state value of the activation variables m for the T-type, N-type, and L-type VDCC computed from Eq. 5-2. while Fig. 5-2b shows the corresponding activation time constants computed by Eq. 5-3. In an analogous fashion, Fig. 5-3a depicts graphically the steady-state 55 value of the inactivation variable h for the T-type and N-type VDCCs computed from Eq. 5-9, while Fig. 5-3b shows the corresponding inactivation time constants computed by Eq. 5-10. Since the L-type calcium channel does not inactivate, h was set to unity. . 8 -- . 7 - - . 6 - • h . 4 - - . 3 - - -1 0 0 -9 0 -8 0 -7 0 -SO -5 0 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 50 Membrane Voltage (mV) Figure 5-3a Steady-State Inactivation Variable 9000 8000 7000 6000 5 0 0 0 - Th 4000 - (ms) 3 0 0 0 - 2000 1 0 0 0 - • -1 0 0 -9 0 -8 0 -7 0 -6 0 -5 0 -4 0 -3 0 -2 0 -1 0 0 10 20 30 40 50 Membrane Voltage (mV) Figure 5-3b Inactivation Time Constant 56 The activation parameter A in Eq. 5-1 required computation of both the inactivation variable h and the square of the activation variable m2. Plots of these variables versus time are shown in Fig. 5-4a and Fig. 5-4b respectively. i ! i i i i i i I i i i z : i i j 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-4a Inactivation Variable .02 .0 1 7 5 - .0 1 5 - .0 1 2 5 - .0 1 - .0025 - - .0 0 2 5 - - .0 0 5 0 20 40 60 80 100 120 140 160 180 200 Time (ms) Figure 5-4b Square of Activation Variable 57 The resulting calcium currents are shown in Fig. 5-5. The L-type current was nearly negligible, the T-type current had a peak amplitude of 0.08 pA, while the N-type current had a peak amplitude of -0.39 pA. Hence, the aggregate calcium current lea was -0.47 pA. This current is the driving stimulus of the spine model which was developed in Section 2. - . 2 - su n - . 3 - - . 4 - - . 5 - - , 6 _. 20 40 60 80 100 120 <60 180 200 0 140 Time (ms) Figure 5-5a Calcium Currents - . 2 - xCa (pA ) - . 4 - - . 5 - ■ - . 6 -. 10 15 25 5 20 30 45 0 35 50 40 Time (ms) Figure 5-5b Calcium Currents (expanded view) 58 The rise in spinous intracellular calcium concentration and the fall in buffer concentration following presynaptic stimulation are traced in each com partm ent for each of the three models we are considering. We first show comparisons of the three models and then analyze each model individually. Model #1 corresponds to the long-thin geometry, Model #2 corresponds to the stubby geometry, and Model #3 corresponds to the mushroom geometry. 5.2.1 Comparison of Geometries Figure 5-6 compares the intracellular calcium concentration, [Ca]j, in com partm ent 1 for each of the three models. The long-thin geometry had a hu- mongous rise in [Cajj, reaching levels of 85 pM, while the mushroom and stubby geometries had significantly less of an increase, approaching peak levels of only 20 (iM and 5 fiM respectively. Since rise in intracellular calcium is required for the induction of LTP, we conclude that the long-thin geometry is ideal. 90 80 70 60 2 0 - 1 0 - '»3. * 2 - - 1 0 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-6 Comparison of Compartment 1 Calcium Concentration 59 Figure 5-7 summarizes the change in spinous buffer concentration in com partment 1 following calcium influx into the spine head. The long-thin geometry experienced the steepest drop in buffer concentration, depleting all 200 pM of initial buffer. The buffer concentration fell to 10 pM in the mushroom case, while falling to 40 pM in the stubby case. Intuitively, this makes sense, because the long-thin geometry experienced the steepest rise in intracellular calcium concen tration. Therefore, it should experience the greatest chelation by buffers. 200 180- 160- 140- » 2- 6 0 - L»3‘ 40 20 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-7 Comparison of Compartment 1 Buffer Concentration 5.2.2 Long-thin Geometry Figure 5-8a traces the spinous calcium concentration over time in all 11 compartments for the long-thin geometry following activation of the VDCCs. It peaks at 85 pM in the spine head, 40 pM in the spine neck, and less than 1 pM in the dendritic shaft. Figure 5-8b shows the three-dimensional (3-D) calcium con centration profile for the entire spine. It clearly shows a saddle shaped profile in the spine head followed by a flat plane at the dendritic shaft. 60 90 80 70 60 4 0 - - 30 20 - • 10 - • 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-8a Long-thin Spine: Regional Calcium Fluctuations Time (ms) Figure 5-8b Long-thin Spine: Calcium Profile Figure 5-9a traces the corresponding change in spinous buffer concentra tion over the 11 compartments. Practically all of the buffer was used up to chelate the massive rise in concentration of calcium in the postsynaptic density (compartments 1-2). In compartments 3-6, the buffer level fell from its initial 61 value of 100 pM to a steady-state value of about 30 |iM. Buffer concentration in compartments 7-11 settled to about 50 fiM. This resulted in a buffer "sink" in the 3-D profile of buffer concentration. 200 180- 160 140 120 [B] (HM) 100 80 6 0 - 40- 2 0 - -frvr, ■ . I i i . i I i , i i j , i . i I i i i , ; i . i , , j.,.,.,.. - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-9a Long-thin Spine: Regional Buffer Fluctuations [B] (H M ) Time (ms) Figure 5-9b Long-thin Spine: Buffer Profile 62 5.2.3 Stubby Geometry Figure 5-10a traces the spinous calcium concentration over time in all 11 compartments for the stubby geometry following activation of the NMDA recep tor. It peaks at 5.5 (iM in the spine head, 2.5 (iM in the spine neck, and 1.5 |iM in the dendritic shaft. Figure 5-10b shows the 3-D calcium concentration profile for 3 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-10a Stubby Spine: Regional Calcium Fluctuations Time (ms) Figure 5-10b Stubby Spine: Calcium Profile 63 the entire spine. It is relatively flat compared to the long-thin geometry. Figure 5-lla traces the corresponding change in spinous buffer concentra tion over the 11 compartments. From an initial value of 200 (iM, the concentra tion of buffer in the postsynaptic density (compartments 1-2) dropped down to 200 180 160 140 120 [B] (HM) 100 80 60 40 20 0 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-lla Stubby Spine: Regional Buffer Fluctuations [B] OiM) Figure 5-llb Stubby Spine: Buffer Profile Time (ms) 64 20 |iM. In compartments 3-11, the buffer level fell from its initial value of 100 pM to a steady-state value of 40 |iM. The 3-D profile of buffer concentration was rel atively flat, as is to be expected considering that the 3-D profile for calcium profile was also flat (Fig. 5-1 lb). 25 2 2 .5 20 17.5 7 . 5 - 2 . 5 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-12a Mushroom Spine: Regional Calcium Fluctuations Time (ms) Figure 5-12b Mushroom Spine: Calcium Profile 65 5.2.4 Mushroom Geometry Figure 5-12a traces the spinous calcium concentration over time in all 11 compartments for the mushroom geometry following activation of the NMDA receptor. It peaks at 20 pM in the spine head, 8 pM in the spine neck and 1.25 p.M in the dendritic shaft. Figure 5-12b shows the three-dimensional (3-D) calcium 200 180 160 140- 1 2 0 - [B] (HM ) ,0° - 8 0 - 6 0 - 40 20 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-13a Mushroom Geometry: Regional Buffer Fluctuations Time (ms) Figure 5-13b Mushroom Spine: Buffer Profile 66 concentration profile for the entire spine. Although it shows a saddle shaped profile in the spine head followed by a flat plane at the dendritic shaft as in the long-thin geometry, the magnitude of the calcium amplification was four-fold less. Figure 5-13a traces the corresponding change in spinous buffer concentra tion over the 11 compartments. From an initial value of 200 |iM, the concentra tion of buffer in the postsynaptic density (compartments 1-2) dropped down to 5 (iM. In compartments 3-7, the buffer level fell from its initial value of 100 |iM to a steady-state value of about 40 (J.M . In the dendritic shaft, buffer concentration reached a steady-state value of 50 (iM. This resulted in a buffer "sink" in the 3-D profile of buffer concentration (Fig. 5-13b). 5.2.5 L-type Calcium Channel In this section, we consider whether or not the L-type calcium channel alone is sufficient to elicit LTP following presynaptic stimulation. The postsy naptic L-type calcium current following presynaptic 50 Hz stimulation is shown in Fig. 5-5. Figure 5-14 compares the intracellular calcium concentration, [Ca]f, in .0 3 - .029- .028- .027-; .026- .025-; [C a]. 024i 1 .023-1 (HM) Q22 j . 0 2 1 - . 0 2 - .019- .018-1 .017-; .016-; .015-^ » 3 - "2 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-14 L-type Channel: Calcium Fluctuation Comparison 67 compartment 1 for each of the three models. Figure 5-15 compares the corre sponding change in buffer concentration. Insignificant rise in [Ca]j occurred in all three geometries considered. Hence, LTP can not be mediated by the L-type calcium channel alone. 19? 1 9 6 .8 - 1 9 6 .5 - 1 9 6 .4 - 1 9 6 .2 - [B] (HM) 195‘ 1 9 5 .8 - #3 1 9 5 .6 - 1 9 5 .4 - 1 9 5 .2 - 195 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-15 L-type Channel: Buffer Fluctuation Comparison 5.2.6 N-type Calcium Channel Next, we considered whether or not stimulation of the N-type calcium channel alone was sufficient to induce LTP. The postsynaptic N-type calcium current following presynaptic 50 Hz stimulation is shown in Fig. 5-5. Figure 5-16 compares the intracellular calcium concentration, [CaJj, in compartment 1 for each of the three models. Figure 5-15 compares the corresponding change in buffer concentration. Significant rise in [Ca]j occurred in all three geometries considered, reaching levels of 88 |iM for the long-thin geometry, 20 (iM for the stubby geometry, and 6 (J.M for the mushroom geometry. Hence, LTP can be mediated by the N-type calcium channel alone. 6 8 9 0 80 70 3 0 - 2 0-- 1 0- - #3 »2 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-16 N-type Channel: Buffer Fluctuation Comparison 200 180 160 140- 120 #2 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-17 N-type Channel: Buffer Fluctuation Comparison 5.2.7 T-type Calcium Channel Finally, we determined if excitation of the T-type channel alone could in duce LTP. The postsynaptic T-type calcium current following presynaptic 50 Hz stimulation is shown in Fig. 5-5. Figure 5-18 compares the intracellular calcium 69 concentration, [Ca]j, in compartment 1 for each of the three models. Figure 5-19 compares the corresponding change in buffer concentration. 50 4 5 - 4 0 - 3 5 - 15- 1 0 - 5 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-18 T-type Channel: Calcium Fluctuation Comparison 200 180- 160- 140- 1 2 0 - [B] (|lM ) 100 7 8 0 - 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 5-19 T-type Channel: Buffer Fluctuation Comparison 70 Significant rise in [Ca]j occurred in all three geometries considered, reaching levels of 48 |iM for the long-thin geometry, 10 |J.M for the stubby geometry, and 3 (iM for the mushroom geometry. Hence, LTP can be mediated by the T-type cal cium channel alone. 5.3 Discussion The simulations clearly show that LTP can be elicited solely by voltage- gated calcium channels (VGCCs). Activation of VGCCs coupled voltage changes to local calcium influx, amplifying free calcium in discrete subcellular compart ments, mediating biochemical changes associated with local calcium dependent events, in particular LTP. As was the case with NMDA-mediated LTP, peak spine head calcium concentration was an order of magnitude larger in long-thin spines than in stubby or mushroom shaped spines following presynaptic stimu lation. Activation of low-threshold T-type calcium channels in postsynaptic cells has been shown by Komatsu and Iwakiri to mediate induction of NMDA-inde- pendent long-term potentiation in developing kitten visual cortex (Komatsu and Iwakiri, 1992). LTP was induced by 2 Hz stimulation applied for 15 minutes. The NMDA receptor antagonist APV did not affect induction of LTP in their study, confirming the notion that a non-NMDA receptor was responsible for the calcium influx required to induce LTP. Application of the T-type VGCC blocker nickel, however, did block induction of LTP. Also, injection of the calcium chela tor BAPTA into the postsynaptic cell blocked'LTP induction, confirming the T- channels were postsynaptic. The high-threshold L-type VGCC blocker nifedip ine, however, did not effect induction of LTP. These observations are consistent with our model in that: (i) T-type channels are capable on their own of inducing LTP; and (ii) L-type channels did not provide the rise intracellular calcium re- 71 quired to produce LTP, indicating that any rise in postsynaptic calcium is buffered and does not significantly affect calcium levels within spines (Huang and Malenka, 1993). 72 6.0 COLOCALIZATION OF VGCCS AND GLURS The model presented here colocalizes VGCCs with glutamate receptors. Pharmacological studies confirm the involvement of both NMD A receptors and VGCC's in the induction of LTP. The NMDA receptor antagonist D,L-2-amino-5- phosphonovalerate (APV) has been found to diminish the rise in intracellular cal cium concentration following synaptic stimulation, thus suppressing LTP (Perkel et al., 1993). Yet, the T-channel blocker Ni2+ suppresses 60% of 0-burst induced LTP in guinea-pig hippocampal CA1 neurons while the P-channel blocker ca- AgalVA supresses 78% of the potentiation (Ito et al, 1995). However, the L-chan- nel blocker nifedipine has no effect on LTP induced by 100 Hz tetanic stimulation (Kullman et al., 1992). Thus, VGCCs and glutamate receptors might both be re quired to be activated to generate tetanus induced LTP. It is now believed, in contrast to the literature findings presented in Sec tion 5, that TEA elicits two distinct forms of potentiation in area CA1 of hippo campus, each based on a different type of synaptic modification: (i) a NMDA receptor-dependent form; and (ii) another form distinct from NMDA receptor- dependent LTP, dependent on VGCCs (Hanse and Gustafsson, 1994). In con trast to the studies by Aniksztejn and Ben-Ari and Huang and Malenka, Hanse and Gustafsson show a potentiation induced by TEA in the presence of an tagonists to VGCCs. This potentiation was confirmed to be NMDA-dependent by occluding it prior to induction of tetanus induced LTP. The discrepency between the studies may lie in the fact that Hanse and Gustafson applied picro- toxin to their slices, facilitating activation of NMDA receptor channels through GABAa mediated inhibition. The recent study by Huber and colleagues attempted to systematically analyze the NMDA-dependent and NMDA-independent induction mechanisms (Huber et al., 1995). Using hippocampal slices with intact CA3-CA1 connections, 73 they found that TEA induced LTP had two components: one dependent on NMDA receptor activity, and the other dependent on VGCC activation. They were able to induce LTP(k) in the presence of APV or nifidepine, but not when APV and nifedipine were applied concomitantly. Hence they concluded that "calcium influx through VDCCs is additive and/or synergistic during induction of TEA LTP." NMDA AMPA Voltage-gated Calcium Channels 2 + 2+ Calcium Buffers Calcium Pumps Diffusion Dendritic Spine Postsynaptic Cell Figure 6-1 Colocalized VGCC and GluR Spine Model 74 Huber and colleagues also contend that NMDA-receptor dependent TEA LTP and tetanus induced LTP are mediated by common cellular mechanisms (Huber et al., 1995). Both require presynaptic stimulation and pathway speci ficity. Both require protein kinase activity. These observations are accounted for in the LTP model presented in this section (see Figure 6-1). 6.1.0 Colocalized Simulation Results The change in spine head membrane potential Vm due to 50 Hz pre synaptic stimulation is depicted in Fig. 2-5 (Holmes and Levy, 1990). The sub sequent excitatory postsynaptic currents (epsc) evoked by this stimulation is computed by placing this Vm in the appropriate equations derived for the NMDA, AMPA, and VDCC models. Figure 6-2a breaks down the calcium cur rent lea elicited by all three models and also depicts the aggregate current. This current has a peak amplitude of 0.8 pA and is the driving stimulus of the spine UGCC '.Total - . 4 - . 5 - - . 6 - - . 7 - - . 8 - - . 9 - S O 0 30 4 0 6 0 10 20 7 0 80 90 100 Time (ms) Figure 6-2a Calcium Current Breakdown 75 model which was developed in Section 2. Figure 6-2b shows the various currents in profile form. T otal C pfo Figure 6-2b Calcium Current Profile The rise in spinous intracellular calcium concentration and the fall in buf fer concentration following presynaptic stimulation are traced in each compart ment for each of the three models we are considering. We first show compari sons of the three models and then analyze each model individually. Model #1 corresponds to the long-thin geometry, Model #2 corresponds to the stubby geo metry, and Model #3 corresponds to the mushroom geometry. 6.1.1 Comparison of Geometries Figure 6-3 compares the intracellular calcium concentration, [Cajf, in com partment one for each of the three models. The long-thin geometry had a hu- mongous rise in [Ca]j, reaching levels of 140 pM, while the mushroom and stub by geometries had significantly less of an increase, approaching peak levels of only 30 pM and 15 pM respectively. Since rise in intracellular calcium is required 76 for the induction of LTP, we conclude that all three geometries could induce LTP under these conditions, with the long-thin geometry being the ideal one. 130 140-j 130 H 120 -j 110-1 100 -j [Ca]. “ 1 1 80 H (HM) 70 J 60 H 50 H 40 -j 30 -j • 20-j. 10 -j 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-3 Comparison of Compartment 1 Calcium Concentration 200 180- 160 140 80- 6 0 - 40- 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-4 Comparison of Compartment 1 Buffer Concentration 77 Figure 6-4 summarizes the change in spinous buffer concentration in compartment 1 following calcium influx into the spine head. The long-thin geometry experienced the steepest drop in buffer concentration, followed by the mushroom geometry, followed by stubby. Intuitively, this makes sense, because the long-thin geometry experiences the steepest rise in intracellular calcium concentration. Therefore, it should experience the greatest chelation by buffers. 6.1.2 Long-thin Geometry Figure 6-5a traces the spinous calcium concentration over time in all 11 compartments for the long-thin geometry following activation of the NMDA receptor. It peaks at 140 pM in the spine head, 68 pM in the spine neck, and 8 pM in the dendritic shaft. Figure 6-5b shows the three-dimensional (3-D) calcium concentration profile for the entire spine. It clearly shows saddle shaped calcium spikes in the spine head followed by a flat plane at the dendritic shaft. 150- 140- 130- 1 2 0 -j 110 -j 100 H [ca], 9 0 i l 8 0 H ( M M ) 7 0 _; 60 -i 50- 4 0 i 30 -j 20 -i 1 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-5a Long-thin Spine: Regional Calcium Fluctuations 78 Time (jus) Figure 6-5b Long-thin Spine: Calcium Profile Figure 6-6a traces the corresponding change in spinous buffer concen tration over the 11 compartments. Practically all of the buffer was used to chelate the massive rise in concentration of calcium in the postsynaptic density (compartments 1-2). In compartments 3-11, the buffer level fell from its initial 200 180 160 140 120 [B] <H M ) ,0° 80 6 0 - 40 2 0 - H fi* i 'H ■ ■ . j......i ,, ■ . i............ j ■ ■ }...... f 0 25 50 75- 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-6a Long-thin Spine: Regional Buffer Fluctuations 79 value of 100 |iM to a steady state value of about 20 (iM. The 3-D buffer profile is shown in Fig. 6-6b. 6.1.3 Mushroom Geometry Figure 6-7a traces the spinous calcium concentration over time in all 12 compartments for the mushroom geometry following activation of the NMDA receptor. It peaks at 30 fiM in the spine head, 13 pM in the spine neck, and 5 |J.M in the dendritic shaft. Figure 6-7b shows the 3-D calcium concentration profile for the entire spine. Although it clearly shows saddle shaped calcium spikes in the spine head followed by a flat plane at the dendritic shaft as in the long-thin geometry, its magnitude was five fold less. 7 [B] CiM) Time (ms) Figure 6-6b Long-thin Spine: Buffer Profile 80 14- 12 - 1 0 - 8 - 6 - 4 - 2 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-7a Mushroom Spine: Regional Calcium Fluctuations Figure 6-7b Mushroom Spine: Calcium Profile Figure 6-8a traces the corresponding change in spinous buffer concen tration over the 12 compartments. From an initial value of 200 pM, the concentration of buffer in the postsynaptic density (compartments 1-2) dropped 81 down to practically 0 JJ.M . In compartments 3-12, the buffer level fell from its initial value of 100 |iM to a steady state value of about 20 |iM. The 3-D profile of buffer concentration is shown in Fig. 6-8b. 200 180- ■ 160- 140- 1 2 0 - 1 0 0 - 8 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-8a Mushroom Spine: Regional Buffer Fluctuation [B] ( J ± M ) - Figure 6-8b Mushroom Spine: Buffer Profile 82 6.1.4 Stubby Geometry Figure 6-9a traces the spinous calcium concentration over time in all 12 compartments for the stubby geometry following activation of the NMDA receptor. It peaks at 15 pM in the spine head, 13 |iM in the spine neck, and 10 |oM 3 0 - ■ 2 5 - 2 0 - - [ C a ] t ; (|lM ) 1 5 -- 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-9a Stubby Spine: Regional Calcium Fluctuations Time (ms) Figure 6-9b Stubby Spine: Calcium Profile 83 in the dendritic shaft. Figure 6-9b shows the 3-D calcium concentration profile for the entire spine. Figure 6-10a traces the corresponding change in spinous buffer con centration over the 11 compartments. From an initial value of 200 |iM, the concentration of buffer in the postsynaptic density (compartments 1-2) 200 180 160 140 120 - [B] (|iM ) ,0°- 8 0 - 6 0 - 4 0 - 2 0 - 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 Time (ms) Figure 6-10a Stubby Spine: Regional Buffer Variation Figure 6-10b Stubby Spine: Buffer Profile 84 dropped down to 30 pM. In compartments 3-12, the buffer level fell from its initial value of 100 pM to a steady state value of about 20 pM. The 3-D profile of buffer concentration is shown in Fig. 6-10b. 6.2 Discussion When NMDA receptors, AMPA receptors, and voltage gated calcium channels are colocalized on the spine, phenomenonal increases in intracellular calcium concentration occured in both the long-thin and stubby geometries. Experimental data are not yet available to validate these results. This enormous amplification, to levels upwards of 140 pM for the long-thin geometry, occurred due to a number of reasons. First, calcium buffering did not have sufficient capacity and pumping was not fast enough to prevent transient amplification of intracellular calcium. Second, the colocalization of voltage-dependent calcium channels on dendritic spines with glutamate receptors resulted in a potent and long duration aggregate calcium current. Organelles, long-term buffering mechanisms, if present in dendritic spines, would have limited the enormous in crease in intracelhilar calcium (Holmes, 1990). However, such structures have not been shown to exist in spines, and hence were not included in the model. In addition, basic constants such as pump rate, buffer rate, and postsynaptic density buffer capacity are unknown and were estimated. Calcium entry through both NMDA receptor channels and VGCCs might be necessary conditions for the induction of LTP. Ito and coworkers have shown that the T-type and P-type voltage-gated calcium channels play a role in theta- burst induction of AP5-sensitive NMDA-dependent LTP in guinea-pig hippo campal CA1 neurons (Ito et al., 1995). When they applied the T-channel blocker nickel, LTP magnitude was suppressed by 60%. The P-channel blocker omega- AgalVA suppressed LTP by 78%. They speculate that calcium entry through 85 VGCCs occurred because the NMDA-mediated excitatory postsynaptic potential (epsp) during theta-burst provided enough depolarization to drive calcium entry through VGCCs. Consistent with the results of our model, the L-type VGCC blocker nifedipine was ineffective in suppressing LTP. However, Ito's findings are in contradiction with the studies conducted by Grover and Teyler which found that nifedipine attenuated NMDA-independent LTP induced by 200 Hz stimulation (Grover and Teyler, 1990; Grover and Teyler 1992). The cause of this apparent contradiction is probably that robust tetanization activated enough L- type channels to cause sufficient calcium entry postsynaptically, overcoming endogeneous buffering mechanisms and activating processes within the spine that are responsible for LTP (Huang and Malenka, 1993). The classic NMDA theory of LTP featured: (i) input specificity, (ii) cooperativity, and (iii) associativity (Bliss and Collingridge, 1993). The principle of input specificity might not be obeyed by the calcium influx through VGCCs provided for in this model because its entry might not be localized to sites of synaptic input (Jaffe et al., 1994b). Cooperativity and associativity are obeyed in that synaptic inputs supply enough depolarization to activate VGCCs and cause an intracellular calcium rise (Ito et al., 1995). 8 6 7.0 COMPARISONS WITH OTHER SPINE MODELS Many of the mechanisms incorporated in this model were taken from the methods described by Holmes and Levy (Holmes and Levy, 1990). They mod eled associative LTP in the dentate gyrus granule cells, whereas we have modeled LTP in CA1 hippocampal pyramidal neurons. They did not include VDCCs in their model because induction of LTP in the dentate gyrus is solely induced by activation of NMDA receptor channels and the burst discharges generated by voltage-dependent calcium channels in CA1 pyramidal cells have not been observed to occur in granule cells. Nevertheless, Holmes has shown that for the long-thin, m ushroom, and stubby geometries, peak calcium concentration in the spine head rises to 28.4 pM, 1.59 pM, and 1.57 pM respectively following NMDA receptor-activation (Holmes, 1990). Our pyramidal cell model produced similar results predicting 42 pM, 5 pM, and 2 pM, respectively. Gold and Bear have also modeled the implications of NMDA receptors on long-thin and stubby spines, predicting intracelluar calcium concentrations of 29.8 pM for long-thin and 1.9 pM for stubby spines (Gold and Bear, 1994). It has previously been shown theoretically that if spines were endowed with VDCCs, even small synaptic inputs could trigger large EPSPs in a passive dendrite (Shepherd et al., 1985). At that time, however, evidence mapping such VDCCs was lacking and the phenomena of non-NMDA mediated LTP was not shown. Now that these have been shown experimentally to be true, the model presented here takes this data into account, ft also shows that VDCCs amplify synaptic input. Jaffe et al. have proposed a "second-generation" dendritic spine model of calcium dynamics (Jaffe et al., 1994a). They account for observations that contra dict the assumptions made in the previous spine models proposed by both 87 Holmes and Levy and Zador and Koch (Holmes and Levy, 1990, and Zador et al., 1990), namely: (i) induction of LTP does not require NMDA receptors; (ii) influx of calcium through voltage gated calcium channels is much greater than the in flux mediated by NMDA receptor-gated channels in CA1 pyramidal neurons; and (iii) activation of metabotropic glutamate receptors in addition to NMDA re ceptor-mediated currents are necessary for LTP induction (Jaffe et al., 1994a). Nevertheless, they still assume that calcium entry through NMDA receptors is still the primary source of calcium that triggers LTP and that entry of calcium through AMPA receptors or voltage-gated calcium channels are subthreshold for inducing LTP. 7.1 Comparison with Imaging Studies Imaging studies have been performed in CA3 hippocampal pyramidal cells and cerebellar Purkinje cells that measure rise in postsynaptic calcium in dendritic spines following afferent stimulation. Muller et al. have shown using the dye fura-2 that in CA3 pyramidal cells, after stimulation of associational- commissural fibers, NMDA receptor-mediated calcium influx in the micromolar range was localized to spines, with hardly any influx occurring in the dendritic shaft (Muller et al., 1994). When they applied the NMDA antagonist AP5, calcium influx into spines was eliminated. They found that even with the loss of calcium influx in the spine, calcium changes in the dendritic shaft remained the same. They hypothesize that "calcium in the spines has more or less free access to the dendritic shaft, but that when only a Small number of spines along the shaft are activated, the parent dendrite can very effectively buffer this input." The simulations presented here are in agreement with this trend. Eilers et al. have shown in cerebellar Purkinje neurons subthreshold cal cium signaling restricted to dendritic compartments consisting of fine dendrites 88 and spines upon activation of parallel fiber synapses (Eilers et al, 1995). They concluded that their observed postsynaptic calcium signals were due to "AMPA receptors locally depolarizing the postsynaptic membrane potential to open VGCCs in the region of dendritic membrane in the immediate vicinity of activated parallel fibers." Guthrie and collaborators visualized calcium gradiants in region CA1 pyramidal cells, also in the micromolar range (Guthrie et al., 1991). They also found that no diffusion barrier exists between the spine and dendritic shaft. This finding validates the assumption made by the current model that calcium buffer uptake systems and pumps isolate calcium in the spine head (Koch et al., 1992). It also supports the observation that when calcium concentration is artificially elevated in the dendrite without first applying tetanic stimuli to the spine, LTP induction fails (Koch et al., 1992). 7.2 Future Studies We have considered the case of one synapse for the present study. Its extension to neural network proportions (e.g., the effect of five spines on a dendritic shaft) would be quite enlightening. We have shown that not all spine morphologies restrict diffusion. Our model assumes that the distribution of calcium stores and pumps are the same in all spines. However, it has been shown that long-thin spines have only a thin tube of smooth endoplasmic reticulum while mushroom shaped spines have laminated spinus apparatuses (Harris and Kater, 1994). The distribution of endo- geneous mechanisms may also change over development. The link between the interplay of dendritic spine morphology with development of glutamate receptors and voltage-gated calcium channels was only briefly touched upon here. We have shown that the long-thin spine, which 89 is found on the majority of adult synapses, is optimal for inducing the various forms of LTP, compared to the mushroom or stubby spines. However, we did not vary the distribution of the various glutamate receptors and voltage-gated calcium channels on the spine over the course of development. Consistent experimental data is not yet available to perform such a study. Once these values are determined experimentally, the corresponding parameters can be adjusted in our parametric model. 90 8.0 CONCLUSION Calcium dynamics in dendritic spines was modeled as a nonlinear function of diffusion, pumping, and buffering. The rate of calcium flux along the longitudinal axis of the spine is determined by the diffusion constant. The pumping action causes calcium ions to leak through the membrane; hence it can be thought of as a membrane conductance (Zador and Koch, 1994). The buffer is a storage device, acting as a capacitor, which affects only the transient behavior. Synaptic inputs applied on dendritic spines amplify the increase in intra cellular calcium concentration as compared to the inputs applied directly to the dendritic shaft (Koch et al, 1992). This amplification is caused by restricted dif fusion out of the spine head/neck junction as well as by the spine's large surface to volume ratio (Jaffe et al., 1994b). The rush of incoming calcium ions into the head cannot be pumped out nor buffered instantaneously. The steep rise in cal cium concentration in the spine head is responsible for the nonlinear induction of LTP (Gold and Bear, 1994). Spines act as biochemical compartments (Koch and Zador, 1993). Den dritic spines also compartmentalize calcium transients (Jaffe et al, 1994b). Cal cium concentration has been found to be negligible in the dendritic shaft while steep increases in calcium concentration are restricted to the spine head follow ing presynaptic stimulation (Gold and Bear, 1994). Thus, spines are a micro environment that localizes changes in the concentration of the second messenger calcium (Zador and Koch, 1994). Results obtained from the analytical simulations are in good agreement with calcium imaging studies and electrophysiological data in the hippocampal slice. Presynaptic stimulation elevates calcium levels in the spine head to tens of micromoles. In adult spines, both the N-type and T-type VGCC as well as the NMDA GluR are the major contributors to this calcium boost which induces 91 long-term potentiation (LTP). Neither the L-type VGCC nor the AMPA GluR alone can induce LTP. Both NMDA-independent LTP and NMDA-dependent LTP were induced in the model. The results of our model also address speculation that spine shape is plastic in vivo. Some indirect evidence shows a shortening and widening of the neck after high frequency stimulation sufficient to induce NMDA receptor- dependent LTP in CA1 neurons. This simulation shows that such plasticity would result in increased diffusion of calcium out of the subsynaptic region. Such efflux of calcium probably explains the phenomenon of saturation of LTP, i.e., why, after transient stimulation of NMDA receptors, LTP is difficult to in duce. The sharp rise in calcium concentration required to trigger LTP no longer occurs because of the change in spine neck geometry (Gold and Bear, 1994). 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Being a general purpose program, it can be used for parametric studies involving sensitivity to model parameters such as the number of channels to be included and control parameters defining channel conductances, inhibition rates, etc. The size of the model (i.e., the number of compartments), the configuration of the model (i.e., diameters, thicknesses, and connectivity of compartments), variation of diffusion properties (e.g., buffer values and pum p rates, etc.) as well as location and value of the influx current or potentials can also be varied. The basic steps involved are summarized below: (i) Provide information for the analytical model: Input compartment diameter, thickness, connectivity, and parameters such as total buffer value, Bt and pum p rate parameter, kp. Compute the compartment areas and volumes. (ii) Provide calcium channel information in the case of the VGCC model. Input the number of channels and the corresponding parameters a, b, c, d, e, f, g, and the conductance values. Skip this step if influx input consists of calcium current (e.g., NMDA and AMP A analyses). (iii) Provide the influx information. Input the time-history of the influx as follows: (a) Spine head potential in the case of combined VGCC and diffusion models; or 104 (b) Influx current in the case of diffusion m odel for other channels. (iv) Provide the time-history analysis parameters. Specify values for length of time the analysis is to be carried out, the integration time step, the number of steps to be saved for post-processing, etc. (v) Perform the dynamic analysis using the procedure described in Section 2. Starting with the initial at-rest values, compute for each successive time step: (a) channel parameters m2 and h and the resulting i^a values for each channel in the case of VGCC model; (b) the compartment Ca and B values for the diffusion model; and (c) the derivative values dm /dt, d h /d t for VGCC model and d C a/dt and dB /dt for diffusion model using the analytical formulation described in Section 6. (vi) Save the results at the input specified time intervals for post processing such as plotting: (a) the tim e-histories of channel param eters m2 and h, individual and total calcium channel currents for VGCC models; and (b) the compartment calcium and buffer values for the diffusion model. A complete source program listing follows. 105 PROGRAM L T P SUNIL S. D A L A L Department o f Biomedical Engineering U niversity o f Southern California Copyright 1995 C --------- Spine D iffusion Model C C Solution o f s e t o f simultaneous f i r s t order linear C Ordinary D ifferen tia l Equations: C dVi/dt = f i ( t ,V i ) C with sp ecified i n i t ia l conditions Vi(0) C --------- by 4th Order Runge-Kutta Method. C C H A R A C TER Flx*7 REAL*8 FROMt.THRUt PA R A M ETER (FROMt=0.0D0,NVChan=2, NVComp=2) C O M M O N /MPARA1/ Neqnl,Neqn2 C O M M O N /CONST/ FARAD,R,TEMP REAL*8 FARAD,R,TEMP C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4,Car,DD,kBf,kBb,kp C O M M O N /VPARA1/ Ca2i,Ca20 REAL*8 Ca2i,Ca20 C O M M O N /VPARA2/ CRatio.FbyRT REAL*8 CRatio.FbyRT C O M M O N /FLX1/ Ibegin.Iend C O M M O N /FLX2/ T itle CH A R A C TER Title*80 C O M M O N /FLX3/ didt REAL*8 didt C O M M O N /STP/ h,h2,h6 REAL*8 h,h2,h6 C C O M M O N /FILES/ FILENM C H A R A C TER FILENM*20 C O M M O N /FILEUN/ Iin p ,Io u t,Ip lo t C INTEGER M T O T PA R A M ETER (MTOT=50000) REAL*8 AC M TO T) INTEGER NA(M TOT) EQUIVALENCE (A(1),NA(1)) C 3001 FO R M A T (1018) 3002 FO R M A T (5D15.6) 4001 FO R M A T ('***•* Blank Common Storage Exceeded:'/ * 7X,'Rerun with M T O T = ',1 6 ) C F4=4.0D0*FARAD*1 .0D+12 CRatio=Ca2i/Ca20 FbyRT=FARAD/(R*TEMP)*l.0D-3 NILAST=0 NChan=0 NComp=0 C 106 n n n CLOSE(Iout) O PEN (UNIT=Iout,FILE='vgdiff-out',STATUS=’U N K N O W N ’) -- Model Data C A LL M O D EL0 (NComp,NCmax) — Nij NI201 = NILAST -i- 1 — ij NI202 = NI201 + NCom p — Diam NI203 = NI202 + NComp*NCmax — Thick NI204 = NI203 + NCom p — Bti NI205 = NI204 + NCom p — Area NI206 = NI205 + NCom p — Vol NI207 = NI206 + NCom p — kpi NI208 = NI207 + NCom p NILAST = NI208 + NCom p - 1 IF CNILAST .GT. M TO T) T H E N 50 W RITE (*,4001) NILAST PAUSE G O T O 1000 EN D IF C IF (NComp .GT. 0) T H E N C Nij i j Diam Thick C A LL M O D E L (NComp,NA(NI201),NA(NI202),A(NI203),A(NI204) C Bti Area Vol kpi & ,A(NI205),A(NI206),A(NI207), A(NI208)) E N D IF Channels Data 60 C A LL CH ANL0 (NChan) IF (NChan .GT. 1) T H E N NChanl=NChan+l Flx='Voltage' ELSEIF (NChan .EQ. 1) T H E N NChanl=NChan Flx='Voltage' ELSE NChan=0 NChanl=NChan Flx='Current' C E N D IF — a NI101 = NILAST + 1 C — b NI102 = NI101 + NChan C — c NI103 = NI102 + NChan n n r i ri r> o o n o C — d NI104 = NI103 + NChan C — e NI105 = NI104 + NChan C — f NI106 = NI105 + NChan C — g NI107 = NI106 + NChan C — gmax NI108 = NI107 + NChan C — area NI109 = NI108 + NChan NILAST = NI109 + NChan - 1 IF (NILAST .GT. M TO T) G O T O 50 C IF (NChan .GT. 0) T H E N C a b e d C A LL C H A N L (NChan,A(NI101),A(NI102),A(NI103),A(NI104) C e f g gmax garea & ,A(NI105),A(NI106),ACNI107), A(NI108),A(NI109)) E N D IF C 70 Neqnl = NVChan*NChan Neqn2 = NVComp*NComp Neqn = Neqnl + Neqn2 IF (Neqn .LT. 1) G O T O 1000 Influx Data C A L L FLUX0 (NFlxin,Flx) — tin NI301 = NILAST + 1 — Flxin NI302 = NI301 + NFlxin — FlxFr NI303 = NI302 + NFlxin NILAST = NI303 + NCom p - 1 IF (NILAST .GT. M TO T) G O T O 50 IF (NFlxin .GT. 0) T H E N tin Flxin C A LL FLU X (NFlxin,A(NI301),A(NI302)) FlxFr IF (NComp .GT. 0) TH EN C A LL FLUX2 (NComp,A(NI303)) E N D IF E N D IF Dynamic Analysis --var NI401 = NILAST + 1 —deriv NI402 = NI401 + Neqn - y T NI403 = NI402 + Neqn --dyTdt NI404 = * NI403 + Neqn 108 C — dyMdt NI405 - NI404 + Neqn NILAST = NI405 + Neqn - 1 IF (NILAST .GT. M TO T) G O T O 50 C A LL DYN0 (FROMt,THRUt,Nstp,Nprnt,NprFr) — t NI451 = NILAST + 1 — V NI452 = NI451 + Nprnt NILAST = NI452 + Nprnt - 1 NW Chan = NChan*Nprnt — m m NI501 = NILAST + 1 — hh NI502 = NI501 + NChan — iiC al NI503 = NI502 + NChan — iiCa2 NI504 = NI503 + NChanl — m NI505 = NI504 + NChanl — h NI506 = NI505 + N W Chan — iCa NI507 = NI506 + N W Chan NILAST = NI507 + NWChan+Nprnt - 1 N W C om p = NComp*Nprnt — C a NI601 = NILAST + 1 —-B NI602 = NI601 + N W C om p — dCadt NI603 = NI602 + N W C om p — dBdt NI604 = NI603 + N W C om p NILAST - NI604 + N W C om p - 1 IF (NILAST .GT. M TO T) G O T O 50 C A L L D Y N (NChan,NChanl,NComp,Neqn,NFlxin Nij i j Diam Thick Bti Area & , NA(NI201), NA(NI202),A(NI203),A(NI204),A(NI205),A(NI206) Vol kpi & ,A(NI207),A(NI208) a b c d e f & ,A(NI101),A(NI102),A(NI103),A(NI104),A(NI105),A(NI106) g gmax garea & ,A(NI107),A(NI108),A(NI109) tin Flxin FlxFr t V & ,A(NI301),A(NI302),A(NI303),A(NI451),A(NI452) var deriv yT dyTdt dyMdt & ,A(NI401),A(NI402),A(NI403),A(NI404),A(NI405) m m hh iiC al iiCa2 & ,A(NI501),A(NI502),A(NI503),A(NI504) m h iCa & ,A(NI505),A(NI506),A(NI507) non on o non C Ca B dCadt dBdt & ,A(NI601), A(NI602),A(NI603),A(NI604) & , FROMt,THRUt, Nstp,NprFr,Nprnt) C 1000 E N D C-------------------------------------------------------------------------------------- SUBROUTINE CH ANL0 (NChan) C C O M M O N /FILES/ FILENM C H A R A C TE R FILENM*20 C O M M O N /FILEUN/ Iin p ,Io u t,Ip lo t REAL*8 a ,b ,c ,d ,e ,f ,g C 1001 FO R M A T ('* Type n or N i f no channel data inp u t') 1002 FO R M A T ('Channel Data F ile Name? ',$ ) C NChan=0 C W RITE (*,1001) W RITE (*,1002) FILENM='n' R EA D ( * , '(A )') FILENM IF (FILENM .NE. ' ' .AND. FILENM .NE. 'N' & .AND. FILENM .NE. 'n ') T H E N W R ITE ( * , ' (A )') FILENM W R ITE (lo u t ,'( A ) ') FILENM CLOSE(Iinp) O PEN (UNIT=Iinp, FILE=FILENM,STATUS='OLD') Count Channels 10 R E A D (Iinp,*,END=50) a ,b ,c ,d ,e ,f ,g NChan=NChan+l G O T O 10 E N D IF 50 R ETU R N E N D SUBROUTINE C H A N L (NChan,a,b,c,d,e,f,g,gm ax,garea) REAL*8 a ( l ) , b ( l ) , c ( l ) ,d ( l ) , e ( l ) , f ( 1 ) , g ( l ) , gmax(l),garea(l) C O M M O N /FILEUN/ Iin p ,Io u t,Ip lo t Input Channel Parameters R EW IN D Iinp D O 100 1=1,NChan R EA D (Iinp,*,END=100) & a ( I ) , b ( I ) , c ( I ) ,d ( I ) ,e ( I ) , f ( I ) , g ( I ) , gmax(I), garea(I) write (lo u t,* ) I , a ( I ) , b ( I ) ,c ( I ) , d ( I ) ,e ( I ) , f ( I ) ,g ( I ) ,gmax(I) 8i ,garea(I) 100 CONTINUE c pause '?' R ETU R N E N D 110 ooo rioo fioo r~ > o o r~ > < ~ i n SUBROUTINE M O D EL0 (NComp.NCmax) C C O M M O N /FILES/ FILENM C H A R A C TE R FILENM*20 C O M M O N /FILEUN/ Iin p ,Io u t,Ip lo t REAL*8 D,t,B C 1001 FO R M A T ('* Type n or N i f no model data inp u t') 1002 FO R M A T ('Model Data F ile Name? ’ ,$) C NComp=0 NCmax=0 C W RITE (*,1001) W RITE (*,1002) FILENM='n' R E A D ( * , '(A )') FILENM IF (FILENM .NE. ' ' .AND. FILENM .NE. 'N' & .AND. FILENM .NE. 'n ') TH E N W RITE ( * , '(A )') FILENM c W RITE ( lo u t ,'( A ) ') FILENM CLOSE(Iinp) O PEN (UNIT=Iinp, FILE=FILENM,STATUS='OLD') Count Compartments and Connectivity 10 R E A D (Iinp,*,END=50) D ,t,B ,N ij,( I J , J= l,N ij) NComp=NComp+l IF (N ij .GT. NCmax) NCmax=Nij G O T O 10 E N D IF 50 R E TU R N E N D SUBROUTINE M O D E L (NComp.Nij,IJ,Diam,Thick,Bti.Area,Vol,kpi) INTEGER N i j ( l ) , IJ(NComp,1) REAL*8 D iam (l),T h ick(l), A rea (l), V o l ( l ) , B t i ( l ) , k p i(l) Input Compartment Dimensions, Connectivity and Bt Values C A L L M 0DEL2 (NComp.Nij,IJ,Diam,Thick,Bti) — Compute Compartment Volumes C A LL G ETA V (NComp.Diam,Thick,Area,Vol) - - Compute Compartment Pump Rates C A L L GETkpi (NComp,Area,Vol,kpi) c pause '?' R ETU R N E N D SUBROUTINE M 0DEL2 (NComp.Nij,IJ,Diam,Thick,Bti) C 111 C O M M O N /FILEUN/ Iin p ,Io u t,Ip lo t INTEGER Ni j ( 1 ) , IJ(NComp,1) REAL*8 D ia m (l),T h ick (l),B ti(l) REAL*8 D,t,B C R EW IN D Iinp D O 100 IC-1,NComp R E A D CIinp,*,END=100) D, t , B, Ni j (IC ), (IJ (IC , J ) , J=1, Ni j (IC)) Diam(IC)=D*1.0D-6 T hick(IC )=t*l.0D-6 Bti(IC)=B*1.0D-6 w rite Clout,*) IC,Diam(IC),Thick(IC),BtiCIC) & ,N ij(IC ),(IJ (IC ,J ),J = l,N ij(IC )) 100 CONTINUE C c pause '?' R ETU R N E N D C------------------------------------------------------------------------------------------------------- SUBROUTINE FLUX0 (NFlxin.Flx) C C H A R A C TER Flx*7 C O M M O N /FILES/ FILENM C H A R A C TER FILENM*20 C O M M O N /FILEUN/ I in p ,lo u t,Ip lo t C O M M O N /FLX2/ T itle C H A R A C TER T itle*80 REAL*8 t.iC C 1001 F O R M A T C'InFlux ' ,A7 ,' Data F ile Name? ',$ ) C W RITE (*,1001) Fix FILENM=' ' R E A D C*, ' (A )') FILENM IF (FILENM .EQ." .OR. FILENM .EQ. ' ') FILENM='flux-in' W RITE ( * , ' (A )') FILENM W R ITE ( lo u t ,'( A ) ') FILENM CLOSE(Iinp) O PEN (UNIT=Iinp, FILE=FILENM,STATUS='OLD') Count Flux Data points NFlxin=0 T itle=' ' R EA D (I in p , ' (A )' , END=50) T itle W RITE ( * , '(A )') T itle W R ITE ( l o u t ,'( A ) ') T itle C 10 R E A D (Iinp,*,END=50) t,iC NFlxin=NFlxin+l G O T O 10 C 50 R ETU R N E N D C - ------------------------------------------------------------------------------------------------ SUBROUTINE FLUX (N F lxin,tin,F in) C REAL*8 t i n ( l ) , F i n ( l ) non C O M M O N /FILEUN/ I in p ,lo u t,Ip lo t C O M M O N /FLX1/ Ibegin.Iend C O M M O N /FLX2/ TITLE C H A R A C TER TITLE*80 C O M M O N /FLX3/ didt REAL*8 didt REAL*8 SLOPE C R EW IND Iinp R EA D (I in p , ' (A )1, END=100) TITLE D O 100 I=l,NFlxin R EA D (Iinp,*,END=100) tin (I ),F in ( I ) c Note: Input f i l e should have ms as units o f time c tin (I)= tin (I)* 1 .0 D -3 c Note: Input f i l e should have m V as units o f V m c Fin(I)=Fin(I)*1.0D+3 c Note: Input f i l e should have pA as units o f iCa c Fin(I)=Fin(I)*1.0D+12 c w rite (* ,* ) I ,t in ( I ) ,F in ( I ) c w rite (lo u t,* ) I ,t in ( I ) ,F in ( I ) 100 CONTINUE C IF (NFlxin .LT. 2) T H E N NFlxin=2 tin (l)= -1 .0 D 6 tin(2)=1.0D6 Fin(2)=Fin(l) E N D IF Ibegin=l Iend=2 didt=SLOPE(Ibegin, lend,t i n , Fin) C R ETU R N E N D C------------------------------------------------------------------------------------------------------- SUBROUTINE FLUX2 (NComp,FlxFr) - - InFlux Data REAL*8 FlxFr(l) C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C 1001 FO R M A T ( / ' * Provide Information on Location and Amount o f ', & 'InF lu x:'/ & 'Enter a compartment numberd or > ',1 3 ,' to e x i t ' ) 1002 FO R M A T ('Compartment number? ',$ ) 1003 FO R M A T ('Fraction o f InFlux Applied? ',$ ) 2001 FO R M A T ( / ' * InFlux Data:’) 2002 FO R M A T (I5.F 12.5) C D O 50 IC-1,NComp FlxFr(IC)=0.0D0 50 CONTINUE C W RITE (*,1001) NCom p 100 IC=0 W RITE (*,1002) R EA D (* ,* ) IC r > o IF (IC .LT. 1 .OR. IC .GT. NComp) G O T O 150 W RITE (*,1003) R E A D (* ,* ) FlxFr(IC) G O T O 100 C 150 W RITE (*,2001) W RITE ( l o u t ,2001) D O 200 IC=1,NComp W RITE (*,2002) IC,FlxFr(IC) W RITE ( l o u t ,2002) IC,FlxFr(IC) 200 CONTINUE C R E TU R N E N D C------------------------------------------------------------------------------------------------------- SUBROUTINE DYN0 (FROMt,THRUt, Nstp,Nprnt, NprFr) — Dynamic Analysis Data C O M M O N /FILEUN/ I in p ,lo u t,Ip lo t C O M M O N /STP/ h,h2,h6 REAL*8 h,h2,h6 REAL*8 FROMt,THRUt 1001 F O R M A T ('Carry Analysis from 0 ms through how many ms? ',$ ) 1002 FO R M A T ('Time step s iz e (m icrosec.)? ',$ ) 1003 F O R M A T ('How many time steps to be saved? ',$ ) 2001 F O R M A T ('* Carry Analysis thru time: ',F 1 0 .6 ,' ms') 2002 FO R M A T ('* time step size: \F 1 0 .8 ,' ms') 2003 FO R M A T ('* No. o f time steps: ',110) 2004 FO R M A T ('* No. o f time steps to be saved: ',110) 2005 F O R M A T ('* Save Interval: ’ ,110) C W RITE (* ,* ) W R ITE (*,1001) THRUt=0.0D0 R E A D (* ,* ) THRUt C Nstp=0 Nprnt=0 NprFr=l IF (THRUt .GT. FROM t) TH E N W RITE (*,1002) h=1.0D0 R EA D (* ,* ) h IF (h .GT. 0.0D0) TH E N c h=h*1.0D-6 h=h*1.0D-3 Nstp=ABS(THRUt-FROMt)/h W RITE (*,2003) Nstp W RITE (*,1003) R EA D (* ,* ) Nprnt E N D IF C W RITE (* ,* ) W RITE (*,2001) THRUt W RITE ( lo u t ,2001) THRUt W RITE (*,2002) h o r \ o W RITE C lout,2002) h C IF (Nprnt .LT. 1) Nprnt=Nstp NprFr=Nstp/Nprnt Nstp=Nstp+l Nprnt=Nprnt+l E N D IF h2=h/2.0D0 h6=h/6.0D0 C W RITE (*,2003) Nstp W RITE ( l o u t ,2003) Nstp W RITE (*,2004) Nprnt W RITE ( l o u t ,2004) Nprnt W RITE (*,2005) NprFr W RITE ( l o u t ,2005) NprFr C R ETU R N E N D C- ------------------------------------------------------------------------- SUBROUTINE D Y N (NChan,NChanl,NComp,Neqn,NFlxin & ,N ij,IJ,D iam ,T hick,B ti,Area,Vol,kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, garea & ,tin ,F lx in ,F lx F r ,t,V & ,var,deriv,yT,dyTdt,dyMdt & ,mm,hh,iiCal,iiCa2 & ,cm,ch,iCa & ,Ca,B,dCadt,dBdt & , FROMt,THRUt, Nstp, NprFr,Nprnt) C INTEGER Ni j ( 1 ) , IJ(NComp,1) REAL*8 D iam (l),T h ick (l),A rea(l), V o l ( l ) ,B t i ( l ) , k p i(l) REAl*8 c h n a (l), ch n b (l), c h n c (l), ch n d (l), chne(l) & , chnf( 1 ) , c h n g (l),gmax(l), garea(l) REAL*8 t i n ( l ) , F lx in ( l) , FlxFr(l) REAL*8 t ( l ) , V ( l ) REAL*8 v a r ( l ) , d e r iv ( l) ,y T (l), dyT dt(l), dyMdt(l) REAL*8 m m (l),hh(l), i i C a l ( l ) , iiC a2(l) REAL*8 cm(NChan, 1 ) , ch(NChan, 1 ) , iCa(NChanl,1) REAL*8 Ca(NComp, 1 ) , B(NComp, 1 ) , dCadt(NComp,1 ) , dBdt(NComp,1) REAL*8 FROMt,THRUt C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /STP/ h,h2,h6 REAL*8 h,h2,h6 C O M M O N /FLX1/ Ibegin.Iend REAL*8 tt,Fnow,Fnext,GETF C 2002 FO R M A T (5X,1P5D14.5) 3001 FO R M A T (D15.6 ,2 1 6 ,3D15.6) C t(l)=FROMt D O 5 JT = 2 ,Nprnt t(JT)=t(JT-l)+h*NprFr 5 CONTINUE Set I n it ia l Conditions 115 n o n D O 10 I=l,NChanl iiCa2(I)=0.0D0 10 CONTINUE Fnow=GETF(Fromt,NFlxin,tin,Flxin) C A LL INIT (Fnow.NChan,NComp & , chna, chnb, chnc, chnd, chne, chnf, chng,B ti,var) C A LL DERIV1 (Fnow,NChan,NChanl & , chna, chnb, chnc, chnd, chne,chnf, chng, gmax, garea & ,var,deriv,m m ,hh,iiC al) C Fnext=GETF(Fromt+h2,NFlxin,tin,Flxin) C A LL DERIVS (Fnow.Fnext,NChan.NChanl,NComp & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, ga rea & ,var,deriv,m m ,hh,iiC al,iiC a2,FlxFr) C NPR-1 C A LL 0UT1 (Fromt,Npr,NChan,NChanl,NComp & ,var,deriv,V,Fnow & ,mm,hh,iiCa2,cm,ch,iCa & ,Ca,B,dCadt,dBdt) — RK 4 SOLUTION tt=FR0Mt JPRNT=l+NprFr D O 800 Jstp=2,Nstp C C A L L RK 4 (NChan,NChanl,NComp,Neqn,NFlxin & ,N ij,IJ,T hick,A rea,V ol,B t i, kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, ga rea & , tt,var,deriv,yT,dyTdt,dyM dt & ,mm,hh,iiCal,iiCa2 & ,tin,Flxin,FlxFr,Fnow,Fnext) C C A LL DERIVS (Fnow,Fnext,NChan,NChanl,NComp & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , chna, chnb, chnc,chnd, chne, chnf, chng,gmax, garea & ,var,deriv,m m ,hh,iiC al,iiC a2,FlxFr) C IF (JSTP .EQ. JPRNT) T H E N NPR=NPR+1 JPRNT=JPRNT+NprFr C A LL 0UT1 Ctt.Npr,NChan,NChanl,NComp & ,var,deriv,V,Fnow & ,mm,hh,iiCa2,cm,ch,iCa & ,Ca,B,dCadt,dBdt) E N D IF 800 CONTINUE C CLOSE(Iout) C A LL O U T (NChan.NChanl,NComp,Npr & ,t,V,cm ,ch,iCa,Ca,B,dCadt,dBdt,Iplot) C R ETU R N E N D C-------------------------------------------- ---------------------- ------- ----------- SUBROUTINE RK4 (NChan.NChanl,NComp,Neqn,NFlxin & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , chna, chnb, chnc,chnd, chne, chnf, chng,gmax, garea & , t , Y, dYdt,yT, dyTdt, dyMdt & ,mm,hh,iiCal,iiCa2 & ,tin,Flxin,FlxFr,Fnow.Fnext) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /STP/ h,h2,h6 REAL*8 h,h2,h6 C REAL*8 t,Fnow.Fnext INTEGER N i j ( l ) , IJ(NComp,1) REAL*8 T h ic k (l),A r ea (l),V o l ( l ) ,B t i ( l ) , kpiCl) REAL*8 c h n a (l), ch n b (l), ch n c (l), ch n d (l), chne(l) & , chnf( 1 ) , ch n g (l), gmax(l), garea(l) REAL*8 Y ( l ) , d Y d t(l),yTCl), dyT dt(l), dyMdt(l) REAL*8 mm(l), h h ( l) , i i C a l ( l ) , iiC a2(l) REAL*8 t i n ( l ) , F lxinC l), FlxFr(l) C REAL*8 th2,th,GETF C c h2=h/2.0D0 c h6=h/6.0D0 th2=t+h2 th=t+h c C A LL COPYVi (Fnow.Fnext,iiCal,iiCa2,NChanl) Fnext=GETF(th,NFlxin,tin,Flxin) C D O 110 Jeqn = l.Neqn yT(Jeqn) = YCJeqn) + h2*dYdt(Jeqn) 110 CONTINUE C A L L DERIVS (Fnow.Fnext,NChan.NChanl,NComp 8i , Ni j ,IJ , Thick, Area, V ol, B t i, kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, garea & ,yT,dyTdt,mm,hh,iiCal,iiCa2,FlxFr) C D O 120 Jeqn = l.Neqn yT(Jeqn) = Y(Jeqn) + h2*dyTdt(Jeqn) 120 CONTINUE C A L L DERIVS (Fnow.Fnext,NChan.NChanl,NComp & ,Ni j , I J ,Thick,Area,Vol,Bti,kpi & , chna, chnb, chnc,chnd, chne, chnf, chng, gmax, garea & ,yT,dyMdt,mm,hh,iiCal,iiCa2,FlxFr) C D O 130 Jeqn = l.Neqn yT(Jeqn) = Y(Jeqn) + h*dyMdt(Jeqn) dyMdt(Jeqn) = dyTdt(Jeqn) + dyMdt(Jeqn) 130 CONTINUE C t=th C A LL COPYVi (Fnow.Fnext,iiCal,iiCa2,NChanl) Fnext=GETF(t+h2,NFlxin,tin,Flxin) c C A LL DERIVS (Fnow.Fnext,NChan.NChanl,NComp 8i ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, garea & ,yT,dyTdt,mm,hh,iiCal,iiCa2,FlxFr) 117 nnn c D O 140 Jeqn = l.Neqn Y(Jeqn) = Y(Jeqn) + h6*(dYdt(Jeqn) & + dyTdt(Jeqn) + 2.0D0*dyMdt(Jeqn)) 140 CONTINUE C RETU R N E N D C------------------------------------------------------------------------------------------------------ SUBROUTINE INIT (V.NChan,NComp & , chna,chnb,chnc,chnd,chne,chnf, chng,Bti,var) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C REAL*8 ch n a (l), ch n b (l), c h n c(l), ch n d(l), chne(l) & ,ch n f(l),ch n g (r) REAL*8 v a r(l) REAL*8 B t i( l) C C O M M O N /MPARA1/ Neqnl,Neqn2 C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4,Car,DD,kBf,kBb,kp REAL*8 taum.tauh SET INITIAL CONDITION D O 100 I=l,NChan IV2=NChan+I C A L L G E T m CV,chna(I),chnb(I),chncCI),chndCI),taum,varCI)) C A LL GETh CV,chneCI),chnfCI),chngCI),tauh,varCIV2)) 100 CONTINUE C D O 200 1=1,NCom p IVl=Neqnl+I IV2=NComp+IVl var(IVl)=Car varCIV2)=kBb*BtiCI)/(kBf*var(IVl)+kBb) 200 CONTINUE C R ETU R N E N D C-------------------------- -------------------------------------------- ----------- ------------------------ SUBROUTINE COPYVi (Fnow.Fnext,iiCal,iiCa2,NChanl) C REAL*8 Fnow.Fnext REAL*8 iiC a l(l),iiC a 2 (l> C Fnow=Fnext D O 10 I=l,NChanl iiC a l(I)= iiC a 2 (I) 10 CONTINUE C RETU R N E N D C------------------------------------ --------- -------------------- ---------- ------------------------- SUBROUTINE DERIVS (Fnow,Fnext,NChan.NChanl,NComp & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax,garea 118 & ,var,deriv,m m ,hh,iiCal,iiCa2,FlxFr) C C O M M O N /FILEUN/ Iin p ,lo u t,I p lo t C REAL*8 Fnow.Fnext INTEGER Nij(l),IJ(NComp,1) REAL*8 T h ic k (l) ,A r e a (l),V o l(l),B t i( l) ,k p i( l) REAL*8 ch n a (l), ch n b (l), ch n c (l), ch n d (l), chne(l) & , chnf( 1 ) , ch n g (l),gm ax(l), garea(l) REAL*8 v a r ( l) ,d e r iv ( l) REAL*8 mm(l), h h ( l) , i i C a l ( l ) , iiC a 2 (l) REAL*8 FlxFr(l) C REAL*8 iiCa C IF (NChan .GT. 0) TH E N C A LL DERIV1 (Fnext,NChan.NChanl & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, garea & ,var,deriv,mm,hh,iiCa2) E N D IF C IF (NComp .GT. 0) TH E N IF (NChan .GT. 0) TH EN iiCa=iiCal(NChanl)+iiCa2(NChanl) ELSE iiCa=Fnow+Fnext E N D IF C A LL DERIV2 (NComp & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & ,var,d eriv,iiC a,F lxF r) E N D IF C R E TU R N E N D C--------------------------------------------------------------------------------------------------- SUBROUTINE DERIV1 (V,NChan.NChanl & , chna, chnb, chnc, chnd, chne, chnf, chng, gmax, garea & , var,deriv,mm,hh.iCa) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C REAL*8 V REAL*8 c h n a (l), ch n b (l), c h n c (l), ch n d (l), chne(l) & , chnf( 1 ) , ch n g (l), gmax(l), garea(l) REAL*8 v a r ( l) ,d e r iv ( l) REAL*8 m m (l),h h (l),iC a(l) C REAL*8 taum.minf,tauh,hinf REAL*8 FACTR,VF,SUM C VF=FACTR(V) SUM=0.0D0 D O 100 1=1,NChan IV2=NChan+I C A LL G E T m (V ,chna(I),chnb(I),chnc(I),chnd(I),taum .m inf) deriv(I)=(m inf-var(I))/taum mm(I)=var(I)*var(I) c 119 C A LL G ETh 0 /,c h n e(I),ch n f(I),ch n g (I),ta u h ,h in f) hh(I)=var(IV2) deriv(IV 2)=(hinf-var(IV2))/tauh c iCa(I)=gmax(I)*garea(I)*mm(I)*hh(I)*VF SUM=SUM+iCa(I) 100 CONTINUE iCa(NChanl)=SUM C R ETU R N E N D C------------------------------------------------------------ ------- ---------------------------------- SUBROUTINE DERIV2 (NComp & ,N ij,IJ,T hick,A rea,V ol,B ti,kpi & , var, deriv, iiCa.FlxFr) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /MPARA1/ Neqnl.NeqnZ C INTEGER N i j ( l ) , IJ(NComp,1) REAL*8 T h ic k (l) ,A r e a (l),V o l(l),B t i(l),k p i(l) REAL*8 v a r ( l) ,d e r iv ( l) REAL*8 FlxFr(l) REAL*8 iiCa C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4,Car,DD,kBf,kBb,kp C REAL*8 ABYD, SU M C D O 200 1=1,NCom p IVl=Neqnl+I IV2=NComp+IVl deriv(IV2>-kBf*var(IVl)*var(IV2)+kBb*(Bti(I)-var(IV2)) SUM=0.0D0 D O 150 J = l,N ij(I) C A LL COPLIJ(Thick,A r e a ,I J (I ,J ),I ,ABYD) SU M = SU M + ABYD*(var(IVl)-var(Neqnl+IJ(I,J))) 150 CO NTINUE C deriv(IVl)=-DD/Vol(I)*SUM-kpi(I)*(var(IVl)-Car)+deriv(IV2) & -iiC a*FlxFr(I)/F4/V ol(I) deriv(IVl)=deriv(IVl)*1.0D-3 deriv(IV2)=deriv(IV2)*1.0D-3 200 CO NTINUE C R E T U R N E N D C ------------------------------------------------- — SUBROUTINE OUT1 (t.Jstp,NChan.NChanl,NComp & ,var,deriv,V ,V t & ,mm,hh,iica,cm,ch,iCa & ,Ca,B,dCadt,dBdt) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /MPARA1/ Neqnl,Neqn2 C REAL*8 t,V t REAL*8 v a r ( l) ,d e r iv ( l) ,V ( l) REAL*8 m m (l),h h (l),iiC a (l) REAL*8 cm(NChan, 1 ) , ch(NChan, 1 ) , iCa(NChanl,1) REAL*8 Ca(NComp,l),B(NComp,l) REAL*8 dCadtCNComp, 1 ) , dBdt(NComp,1) C 2002 FO R M A T (10X,1P5D14.5) C V(Jstp)=Vt C D O 40 1=1,NChan cm(I,JSTP)=mm(I) ch(I,JSTP)=hh(I) iCa(I,JSTP)=iiCa(I) 40 CONTINUE IF (NChanl .GT. NChan) & iCa(NChanl,JSTP)=iiCa(NChanl) C D O 80 1=1,NCom p IVl=Neqnl+I IV2=NComp+IVl Ca(I, Jstp )= var(IV l)*l. 0D6 B (I,Jstp)=var(IV 2)*l.0D6 dCadtCI, Jstp )= d eriv(IV l)*l.0DB dBdtCI, Jstp)=deriv(IV 2)*l.0D3 80 CONTINUE C R ETU R N E N D C---------------------------------------------------------------------------------------------------- SUBROUTINE 0UT2 (VAL,NComp,Npr.t,Iplot) C REAL*8 t C l) ,VAL(NComp,1) C O M M O N /FLX2/ T itle C H A R A C TER T itle*80 C 2001 FO R M A T (1PD14.5) 2002 FO R M A T (10X,1P5D14.5) 2012 FO R M A T (216,' 1 ') C W RITE ( I p l o t ,2012) NComp,Npr W RITE (I p lo t,* ) T itle C D O 20 Jpr=l,Npr w rite ( I p l o t ,2001) t(Jpr) w rite ( I p lo t ,2002) (VAL(Ic,Jpr),Ic=l,NComp) 20 CONTINUE C R E TU R N E N D C ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ — SUBROUTINE O U T (NChan,NChanl,NComp,Npr & ,t,V,m ,h,iCa,Ca,B,dCadt,dBdt,Iplot) C REAL*8 t ( l ) , V ( l ) REAL*8 m(NChan, 1 ) , h(NChan, 1 ) , iCa(NChanl,1) REAL*8 Ca(NComp, 1 ) , B(NComp,1 ) , dCadt(NComp, 1 ) , dBdt(NComp,1) C C H A R A C TER FILENM*12 c IF (NChan .GT. 0) T H E N FILENM='vgdiff-vin' ELSE FILENM='vgdiff-icain' E N D IF CLOSE(Iplot) O PEN (UNIT=Iplot, FILE=FILENM,STATUS='U N K N O W N ’) C A L L OUT2 (V ,1 ,N p r,t,Ip lo t) C IF (NChan .GT. 0) TH E N CLOSE(Iplot) O PEN (UNIT=Iplot, FILE-'vgdiff-m 2' , STATUS='U N K N O W N ') C A LL OUT2(m,NChan,NPR,t,Iplot) C CLOSE(Iplot) O PEN (UNIT=Iplot,FILE='vgdiff-h', STATUS='U N K N O W N ’) C A LL 0UT2(h,NChan,NPR,t,Iplot) C CLOSE(Iplot) O PEN (UNIT=Iplot,FILE=’v g d if f - ic a ', STATUS='U N K N O W N ’) C A L L 0UT2(iCa,NChanl,NPR,t,Iplot) E N D IF C IF (NComp .GT. 0) T H E N CLOSE(Iplot) O PEN (UNIT=Iplot, FILE='vgdiff-C a' , STATUS-'UNKNOWN') C A L L 0UT2 (Ca,NComp,Npr,t,Iplot) C CLOSE(Iplot) O PEN (UNIT=Iplot,FILE='vgdiff-B', STATUS='UNKNOW N') C A LL 0UT2 (B,NComp,Npr,t,Iplot) C c CLOSE(Iplot) c O PEN (UNIT=Iplot, FILE='vgdiff-dB dt' , STATUS='U N K N O W N 1) c C A LL 0UT2 (dBdt,NComp,Npr,t,Iplot) C c CLOSE(Iplot) c O PEN (UNIT=Iplot, FILE='vgdiff-dCadt' , STATUS='U N K N O W N ') c C A L L 0UT2 (dCadt,NComp,Npr,t,Iplot) E N D IF C R E TU R N E N D C-------------------------------------------- ------------------------------------------------------- SUBROUTINE G ET A V (Ncomp,Diam,Thick,Area,Vol) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t REAL*8 D iam (l),T h ick(l),A rea (l), V ol(l) - C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4, Car,DD, kBf, kBb, kp C write ( lo u t ,* ) ’G ETAV' PIBY4=DAC0S(- 1 .0D0)/4. 0D0 D O 20 Ic=l,Ncomp Area(Ic)=PIBY4*Diam(Ic)**2 Vol(Ic)=Area(Ic)*Thick(Ic) w rite (lo u t ,* ) Ic,D iam (Ic),T hick(Ic),A rea(Ic),V ol(Ic) 122 20 CONTINUE C R ETU R N E N D C- --------------------------------------- ---------------- --------------- SUBROUTINE GETkpi (Ncomp,Area,Vol,kpi) C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4, Car, DD, kBf, kBb, kp REAL*8 A r e a (l),V o l(l),k p i(l) write (lo u t,* ) 'GETkpi' D O 20 Ic=l,Ncomp kpi(Ic)=kp*Area(Ic)/Vol(Ic) write (lo u t,* ) Ic,A rea (Ic),V o l(Ic),k p i(Ic) 20 CONTINUE R ETU R N E N D SUBROUTINE COPLIJ (Thick,Area,I,J.ABYDIJ) C O M M O N /FILEUN/ I in p ,lo u t,Ip lo t REAL*8 T hick(l),A rea(l) REAL*8 ABYDIJ ABYDIJ=2.0D0*(Area(I)*Thick(I)+Area(J)*Thick(J))/ & (Thick(I)+Thick(J))**2 R E T U R N E N D FUNCTION SLOPE(Ibegin, lend, t i n , Fin) REAL*8 t i n ( l ) ,F in ( l ) REAL*8 SLO PE SLO PE=(Fin(Iend)-Fin(Ibegin))/(tin(Iend)-tin(Ibegin)) R ETU R N E N D FUNCTION G ETF (t,N F lxin,tin,F IN ) REAL*8 tin (l),F I N (l) C O M M O N /FLX1/ Ibegin.Iend C O M M O N /FLX3/ didt REAL*8 didt REAL*8 GETF,t,SLOPE IF ( t .LE. tin (Ib eg in )) T H E N GETF=FIN(Ibegin) ELSEIF ( t .GE. tin(N Flxin)) TH E N GETF=FIN(NFlxin) ELSE D O 100 W H ILE (Ibegin .LT. NFlxin .AND. t .GE. tin (Ien d )) Ibegin=Iend Iend=Ibegin+l didt=SLOPECIbegin, lend, t i n , FIN) 100 CONTINUE GETF=FIN(Ibegin)+didt*(t-tin(Ibegin)) E N D IF C R ETU R N E N D C— --------------------------------------------------------------------------- FUNCTION FA C TR (V) C C O M M O N /FILEUN/ I in p ,lo u t,Ip lo t C REAL*8 FACTR,V C O M M O N /VPARA2/ CRatio.FbyRT REAL*8 CRatio.FbyRT REAL*8 FVRT.VF C VF=DABS(V) IF (VF .LT. 1.0D-3) VF=1.0D-3 c FVRT=DEXP(-2.0D0*DABS(VF)*FbyRT) FVRT=DEXP(- 2 .0D0*(VF)* FbyRT) FACTR=-DABS((1.0D0-CRatio*FVRT)/(1.0D0-FVRT)*VF) C R ETU R N E N D C ------------------------------------------------------------------------ ------------------- ---------- ---------- FUNCTION A L PH A m (V ,a,b) C REAL*8 V,a,b,Vb REAL*8 A L PH A m C Vb=-V+b i f (DABS(Vb) .I t . 1.0D-6) then ALPHAm=10.0D0*a e lse ALPHAm=a*Vb/(DEXP(Vb/10.0D0)-1.0D0) end i f C R ETU R N E N D C----------------------------------------------- ----------------------- ------------- FUNCTION BE TA m (V ,c,d) C REAL*8 V,c,d REAL*8 BE TA m C BETAm=c*(DEXP(-V/d)) C R ETU R N E N D C------------------------------------------------------------------------------------- FUNCTION ALPHAh (V ,e ,f) C REAL*8 V ,e ,f REAL*8 ALPHAh C 124 ALPHAh=e*(DEXP(-V/f)) C R E T U R N E N D C-------------------------------------------------------------------------- FUNCTION BETAh (V,g) C REAL*8 V,g REAL*8 BETAh C BETAh=l.0D0/CDEXPCC-V+g)/10.0D0)+1. 0D0) C R E TU R N E N D C— ------------------------------------------------------------- SUBROUTINE G E T m CV,a(b,c,d,taum.minf) C REAL*8 V ,a,b ,c,d REAL*8 ALPH Am , BETAm , taum,minf, alpha, beta C alpha=ALPHAmCV, a , b) beta=BETAm(V, c , d) taum=l. 0D0/(alpha+beta) minf=alpha*taum C R ETU R N E N D C------------------------------------------------------------------------ SUBROUTINE GETh (V ,e ,f,g ,ta u h ,h in f) C REAL*8 V ,e,f,g ,ta u h ,h in f REAL*8 ALPHAh,BETAh,alpha,beta C alpha=ALPHAh(V,e,f) beta=BETAhCV,g) tauh=1.0D0/(alpha+beta) hinf=alpha*tauh C R ETU R N E N D C ---------------------------------------------------------------------------------------------------------------------- - BLO C K D A T A C C O M M O N /FILEUN/ Iin p ,lo u t,Ip lo t C O M M O N /CONST/ FARAD,R,TEM P REAL*8 FARAD,R,TEM P C O M M O N /DPARA1/ F4,Car,DD,kBf,kBb,kp REAL*8 F4,Car,DD,kBf,kBb,kp C O M M O N /VPARA1/ Ca2i,Ca20 REAL*8 Ca2i,Ca20 C D A TA Iin p ,lo u t,Ip lo t & / 1, 2, 3/ C D A TA FARAD, R, T E M P & / 9 . 6485D+7,8 .3 15D+3,298.0D0/ C D A TA Car, DD, kBf, kBb, kp 125 & /2 0 .0D-9,0 .6D-9, 0 . 5D+9,0 .5D+3,1 .4D-6/ C D A TA Ca2i, Ca20 & /50.0D -9.1.8D -3/ C E N D C—---------------------------------- ------------------------------------- 126 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. 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Creator
Dalal, Sunil S.
(author)
Core Title
A computational model of NMDA receptor dependent and independent long-term potentiation in hippocampal pyramidal neurons
School
School of Engineering
Degree
Master of Science
Degree Program
Biomedical Engineering
Degree Conferral Date
1995-08
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
biology, neuroscience,biophysics, medical,engineering, biomedical,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
Berger, Theodore W. (
committee chair
), Niesen, Charles E. (
committee member
), Savage, Jay Mathers (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c18-7730
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UC11357611
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7730
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Dalal, Sunil S.
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texts
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University of Southern California Dissertations and Theses
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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 au...
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Tags
biology, neuroscience
biophysics, medical
engineering, biomedical