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A kinetic model of AMPA and NMDA receptors
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A kinetic model of AMPA and NMDA receptors
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A Kinetic Model of AMPA and NMDA Receptors by Deanna Elizabeth Najman A Thesis Presented to the FACULTY OF THE ENGINEERING SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Biomedical Engineering) August 1995 Copyright 1995 Deanna E. Najman This thesis, written by Deanna Elizabeth Najman under the guidance of 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 M aster— -at— Science..................................... Biomedical Engineering n „„ August 1995 Faculty Committee Acknowledgements I would like to thank Dr. David D'Argenio in his instruction in the mathematical modeling and manipulation o f ADAPT for use in a neurobiological model, as well as for his advice and guidance in the research for this project; Dr. Jim-Shih Liaw whose education in this field both from a biological and computational stand point was invaluable; and Mr. William Warne and my father, Mirko Najman, for the preparation and presentation o f this document. Dedication This work is dedicated to my family for their support both financially and emotionally they have given me for the past twenty-five years both in my education and in life. Tabl e of Contents Acknowledgements ......................................................................................................ii Dedication ................................................................................................................. iii List of Figures ............................................................................................................... v List of T a b le s ............................................................................................................. vi C hapter 1 ...................................................................................................................... 1 In tro d u c tio n ................................................................................................... ] 1.1 Basic Neuroscience...................................................................... 1 1.2 Background Information on Receptor Desensitization..............8 1.2.1 Non-NMD A Desensitization .................................. 10 1.2.2 NMD A Desensitization ........................................... II 1.3 Specific A im s............................................................................. 14 1.4 Outline o f T h e sis....................................................................... 15 C hapter 2 ................................................................................................................... 17 Model Development ................................................................................. 37 2.1 The AMPA Model ................................................................... 17 2.2 The NMD A M o d el...................................................................... 20 2.3 The Combined M o d el..................................................................22 2.4 Computational M ethod................................................................26 2.5 Simulations................................................................................. 29 C hapter 3 ......................................................................................................................30 R e su lts.............................................................................................................30 3.1 Probability of Channels O pening............................................... 30 3.2 Membrane Potential ................................................................. 35 3.3 High Frequency Stimulation Increases Calcium .......................41 3.4 Associative Synaptic Activation.................................................44 C hapter 4 ......................................................................................................................51 Conclusions ................................................................................................. 51 4.1 The Kinetic Model ................................................................... 51 4.2 Future M odeling........................................................................ 52 References ................................................................................................................. 54 Appendix A ............................................................................................................... 57 Appendix B ..................................................................................................................58 List of Figures Figure 1.1: Depolarization Necessary to Remove Magnesium Block ................. 4 Figure 1.2: View of the Hippocampal Formation: Hippocampus & Dentate Gyrus ........................................................................................................ 4 Figure 1.3: Voltage-Dependent Magnesium B lock................................................. 7 Figure 1.4: Calcium-Dependent Second Messengers .............................................7 Figure 2.1: Kinetic Model o f the AMPA Receptor Complex .......................... 18 Figure 2.2: Kinetic Equations of the AMPA M o d e l.......................................... 18 Figure 2.3: Kinetic Model of the NMDA Receptor C om plex.............................21 Figure 2.4: Kinetic Equations for the NMDA Model .......................................... 21 Figure 2.5: Circuit Describing Cell Membrane of N euron....................................25 Figure 2.6: Membrane Potentials for Non-NMDA, NMDA, and Combined . . 25 Figure 3.1: Probability AMPA & NMDA in Open State in 0 mM Magnesium . . 33 Figure 3.2: Probability AMPA & NMDA in Open State in 1.2 mM M agnesium .......................................................................................... 34 Figure 3.3: AMPA & NMDA Membrane Potentials.............................................36 Figure 3.4: AMPA & NMDA Membrane Potentials of Current Model without Magnesium B lo ck ..................................................................................36 Figure 3.5: AMPA & NMDA Membrane Potentials of Current Model with 1.2 mM Magnesium Block ............................................................................... 37 Figure 3.6: Pharmacological Block o f AMP A and NMDA Receptors .............39 Figure 3.7: Combined AMPA & NMDA epsps for 0 and 1.2 mM M agnesium .......................................................................................... 39 Figure 3.8: AMPA & NMDA Membrane Potentials in 1.2 mM Mg21 .............40 Figure 3.9: Combined AMPA & NMDA epsps in 0 Mg2+ at Frequencies of 100 & 200 H z ...................................................................................... 42 Figure 3.10: Combined AMPA & NMDA epsps in 1.2mM Mg2' at Frequencie of 100 & 200 Hz ...................................................................................... 42 Figure 3.11: AMPA and NMDA epsp and Conductance .....................................43 Figure 3.12: Coactivation of AMPA & NMDA Weak/Strong Inputs at 400 Hz ........................................................................................................................................ 45 Figure 3.13: Activation of AMPA & NMDA Weak/Strong Inputs at 400 Hz with I msec delay..................................................................................48 Figure 3.14: Activation o f AMPA & NMDA Weak/Strong Inputs at 400 Hz with 8 msec Delay ................................................................................49 Figure 3.15: Activation o f AMPA & NMDA Weak/Strong Inputs at 400 Hz with 20 msec Delay ........................................................................... 50 v List f Tables T able 1.1: Non-NMDA and NMDA receptor mediated conductance parameters .............................................................................................................................9 T able 2.1: Non-NMDA receptor mediated conductance parameters .............. 18 T able 2.2: NM DA mediated conductance p a ra m ete rs........................................ 21 Table 2.3: Model equation definitions ....................................................................28 vi Chapter 1 Introduction 1.1 Basic Neuroscience There exists a substantial amount o f physiological evidence that glutamate is the primary excitatory neurotransmitter in the central nervous system [1]. There are several types o f glutamate receptors which can be generalized into two main classes, NMDA and non-NMDA (kainate- or quisqualate-AMPA receptors) which are based on their sensitivity to the glutamate agonist, N-methyl-D-aspartate (NMDA). Excitatory postsynaptic potentials in the CNS are composed o f a fast, rapidly decaying non-NMDA component (AMPA) and a slowly decaying NMDA component. The a-amino-3-hydroxy-5-methyl-4-isoxazolenproprianate (AMPA) component has fast receptor kinetics with a time to peak o f approximately 5-15 msec and a duration of 50 msec. The NMDA component peaks at about 70 msec with a longer duration o f approximately 120 msec. The AMPA and NMDA receptors are primarily localized on the dendritic spine head. The AMPA receptor is directly linked to a channel which regulates the influx o f sodium and/or efflux of potassium ions down their electrochemical gradients. 1 The N M D A receptor is also directly linked to a channel which causes a change in conductance primarily for calcium. But this channel conductance requires more than just the binding o f the glutamate ligand to the receptors, it also requires depolarization o f the postsynaptic cell. It is not the voltage-sensitivity o f the channel that requires depolarization, but rather that magnesium ions bind to another binding site within the channel as to block the conductance by any other ion. The magnesium ions block the channel at normal membrane resting potentials o f about -70 mV, but when the requirem ents o f glutamate binding to both the AM PA and NM DA receptors, the subsequent opening o f the AM PA channel, and influx o f sodium ions, the NM DA channel opens and the magnesium block is removed allowing the influx o f calcium into the cell. Figure 1.1 illustrates the depolarization necessary to remove the voltage-dependent magnesium blockage. W hen neurons are at their resting membrane potentials, NM DA channels contribute very little to the excitatory postsynaptic potentials (epsps) produced by the glutaminergic inputs, but when neurons are depolarized by a brief high frequency chain o f stimuli whereby the presynapse and postsynapse o f the cell are depolarized, NMDA channels significantly contribute to the epsp. Subsequently, and perhaps more importantly, calcium influx is substantiated within the cell, that triggers many secondary messenger systems. It is this associative synaptic mechanism which appears to underlie the cognitive processes o f learning and memory within the brain. 2 Within the temporal lobe of the brain, there are two entwined C-shaped structures o f the cortex known as the hippocampus and the dentate gyrus {Figure 1,2), together with the subiculum they compose the hippocampal formation. Insight into the role of the hippocampal formation in learning and memory came from patients whose temporal lobes were bilaterally ablated to alleviate symptoms o f temporal lobe epilepsy [2]. These patients exhibited losses in converting short-term memory into long-term memoiy. For example, if inquired about a name immediately after remembering it, they could recall it, but any minor distraction resulted in complete loss o f memory. The importance o f the hippocampus in memory consolidation was established by two cases in which memory impairment was linked to bilateral lesions in the hippocampus [3], These observations made it possible to assume that long-lasting changes in efficacy and signaling between neurons in the hippocampus might be involved in the consolidation o f memoiy and recall. The cellular mechanism underlying the long- lasting changes in synaptic efficacy was demonstrated by Bliss and Lomo in 1973, who demonstrated that high frequency stimulation of neurons in the dentate gyrus produced epsps that would last for hours, days, or even weeks in intact animals. The name given to this phenomena was long-term potentiation (LTP) [4] , LTP occurs in many different pathways in the hippocampus each having its own necessary pattern of stimuli, decay rate, and mechanism o f attainment. The synapses made by Schaffer collaterols/commissural tract onto pyramidal cells in the CA1 region o f the hippocampus display two forms o f LTP. 3 Figure 1.1: Depolarization Necessary to Remove Magnesium Block Magnesium blockade of the NM DA receptor is voltage-dependent. The following are single channel recordings from an outside-out patch of rat hippocampal cells bathed in an extracellular medium. On the left., the extracellular [Mg + ] is 0 mM, and the opening and closing of the cell is not voltage dependent. The channel is open at -60 mV, and reverses at 0 m V. On the right, at an [Mg2+] of 1.2 mM, the channel is closed due to the Mg2 + blockade at the resting potential of -60 mV, and requires a depolarization of +30 mV to open. H o ld in g potential (mV) Open C lo w a * 6 0 0 M g * * « * ~ 1 * —*H- 1.2 mM M g** I * ' — i— * 5 ff*— ---------------- w < j — r* mfirmr— r '- jr ^ r m - i r m ik% t m f b U r ' ] 2 pA 25 m n c Figure 1.2: View of the Hippocampal Formation: Hippocampus & Dentate Gyrus CAl CA3 4 Glutamate is released from these synapses and interacts on the postsynaptic cell which contains both AMPA and NMDA receptors (Figure J.3), Mechanisms o f LTP are based on calcium influx into the postsynaptic cell. Brown et al. were the first to demonstrate LTP in the CA1 region [5], They placed two extracellular stimulating electrodes in two different subpopulations o f cells in the dendrites o f Schaffer collaterols/commissural tract and recorded intracellularly from a pyramidal cell in the CA1 region. The stimulus intensities o f the intracellular recording electrodes were adjusted such that one responded with a higher postsynaptic potential than the other. When a brief tetanus o f 100 Hz for 1 second repeated once after 5 seconds was applied to the higher intensity stimulus, long-lasting post-synaptic potentials sufficient to remove the magnesium block of the NMDA receptors allowed the influx o f calcium into the postsynaptic cell initiating LTP. This was similar to the LTP demonstrated by Bliss and Lomo in the dentate gyrus. But LTP was not potentiated at the site of the weaker recording electrode. Similarly, when the weak recording electrode was stimulated independent o f the strong, LTP was not observed since subsequent stimulation at this input was too small stimulate depolarization o f the CAI region to unblock magnesium. When separate weak and strong inputs arrive at the same region of the dendrite o f the pyramidal cells simultaneously, the weak input was potentiated when co-activated with the strong input. This was termed associative LTP since depolarization at the strong input was sufficient to unblock the magnesium at the weaker allowing the influx o f calcium, and hence induced LTP at this site. 5 There is sufficient evidence that establishes the role of the NMDA receptors in potentiation of LTP. This evidence includes that application of antagonists (i.e. APV) to the NMDA receptors block the induction of LTP, but do not reduce LTP if applied after its occurrence [6], The importance of calcium of influx through the NMDA channel can be substantiated by two lines of evidence. The first was demonstrated when nitr-5 (a photable calcium chelator that releases calcium in response to ultraviolet light) was injected into CA1 pyramidal cells. When the neuron was illuminated with ultraviolet light, the increased intracellular calcium showed LTP synaptic enhancement [7] The second line of evidence illustrated that when intracellular calcium was maintained at a low level by injecting a calcium buffer, BAPTA, into the postsynaptic cell, LTP was blocked [8], How does calcium initiate LTP? The work of Nichol and Tsien indicates that 2 1 persistent enhancement of synaptic transmission by activating two Ca -dependent protein kinases; the Ca2'/calmodulin kinase and protein kinase C [9], These kinases are activated by the high frequency tetanus induce LTP via the relief of the magnesium blockage of the postsynaptic cell allowing calcium influx as described above. The mechanism of LTP exists through two lines of current thought, a postsynaptic versus a presynaptic mechanism as illustrated in Figure 1.4. The postsynaptic mechanism is based on the fact the release of these kinases within the postsynaptic cell causes alterations in the physical properties of the membrane such that its fluidity increases allowing new receptors to be inserted into the membrane. 6 Figure 1.3: Voltage-Dependent Magnesium Block Magnesium blocks the NM D A channel such that depolarization of the AMPA channel is necessary for ion flow and conductance. nm da' Figure 1.4: Calcium-Dependent Second Messengers Activation o f the NM D A channel allows an influx of calcium; it is this increase of calcium within the post-synaptic cell which triggers second messengers O i l N , ' C i - ' ■ C u 0 S * V * V NMDA • f ir Ci5VOImoduiin n ia u Proton kfeiawC 7 Hence, upon activation at a repetitively high frequency stimulus, the neurotransmitter binds to more receptors upon subsequent stimulation inducing LTP. The second hypothesis involves a retrograde messenger such as nitric oxide or arachidonic acid which is released from the postsynapse and communicates back to the presynaptic side stimulating an increase in the quantal release of neurotransmitter with each subsequent presynaptic stimulation. A third, more recently proposed mechanism by Lynch el al. suggest that the alteration in channel kinetics accompanies LTP [10], Two recent observations support this hypothesis. The first is that the decay time constant of the synaptic response is reduced after induction of LTP [11], Secondly, aniracetam, a drug that selectively acts on the AMPA receptor exhibits different waveforms depending on potentiated versus unpotentiatied responses [12], Regardless of the exact mechanism, the levels o f intracellular calcium appear to underlie the induction of LTP and its maintenance. 1 .2, Background Information on Receptor Desensitization Currently Holmes and Levy have proposed a very simple kinetic model of the non-NMDA and NMDA receptor reaction. Table 1.1 summarizes the equations they used to predict whether calcium influx by itself was sufficient enough to maintain LTP throughout the non-NMDA and NMDA channels. Holmes and Levy fail to consider the desensitization that has been reported experimentally at the receptor sites. Table 1.1: Non-NMDA and NMDA receptor mediated conductance parameters taken from Holmes and Levy, 1993. Parameter Description Value A. Non-NMDA-receptor Reaction P A + R «- AR > 1 7 ? * k_i « a inverse of the mean channel open time 2.0 ms’1 P transition to the open channel state 2.0 ms*1 k-i receptor-transmitter dissociation constant 2.0 ms'1 AR(O) receptor-transmitter complexes after a pulse of transmitter 200 Y single channel conductance 5 pS B. NMDA-receptor Reaction P «2 A +R AR ^ AR* - * • A +R k-i «, AR*u n b lo ck cd /AR*to ta ! = (k'/k+ )/(k7k4 ) + [[ Mg2 1 ]] «i inverse of the mean channel open time 0.17 m s * 1 inverse of the mean channel open time 0.17 m s * 1 P transition to the open channel state 0.0006ms',k*1 receptor-transmitter dissociation constant 0.02 ms*1 AR(O) receptor-transmitter complexes after a pulse of transmitter 200 k'/k1 Km for Mg2 1 reaction 8.8 x 10‘3exp(V/12.5) extracellular Mg concentration 1.2 mM 10.6:1.0:1.0 9 1 .2 .1 N o n -N M D A D csen sili/aL ion The response to a neurotransmitter that decreases during repeated or prolonged application has been termed desensitization. Desensitization was first reported by Katz and Thesleffin 1957 at the end plate o f the neuromuscular junction when with repeated application o f acetylcholine the depolarizing response o f the muscle declined [13]. The direct mechanism o f desensitization is unknown, but it appears to be a mediated by the molecular mechanism o f the receptors themselves, yet the rate o f desensitization is modulated by receptor phosphorylation [14]. The activation o f non-NMDA receptors in the CNS and sensory ganglion neurons in response to the application o f AMPA, quisqualate, and other transmitters including L-glutamate produces a rapidly desensitizing response [15], Traditionally responses to kainate and AMPA were believed to occur at different receptor complexes, but more recently it has been suggested through electrophysiological recordings that responses to kainate and AMPA occur at the same receptor. The kainate receptor was isolated and was characterized by cDNAs coding for four glutam ate receptor subunits which respond to kainate, AMPA, quisqualate, and L- glutamate when expressed individually or in combination in oocytes and transfected cell lines [16]. The antagonistic response o f kainate from the application o f quisqualate, AM PA and L-glutamate has been demonstrated over the past 5 years [17]. There are several hypotheses which attempt to explain this antagonism; these include: 10 1) The effects of quisqualate, kainate, and glutamate appear to mediated by a common receptor complex and that quisqualate cross-desensitizes responses to kainate. This was demonstrated in experiments in horizontal retinal cells o f goldfish by Ishida and Neyton [18], 2) The rapid desensitization observed in mammalian neurons is produced upon application o f L-glutamate, quisqualate, and AMPA, but not quisqualate. 3) AMPA, quisqualate, and glutamate act as partial agonists at receptors completely activated by kainate and rapidly desensitized by quisqualate [19], 4) There exists two separate AMPA and kainate receptors where quisqualate act as an antagonist or weak partial agonist at the kainate receptor, yet acts as a complete agonist at the desensitizing AMPA receptor [20], The above hypotheses fail to account for the findings o f the competitive interactions caused by kainate and quisqualate and the relief o f a quisqualate caused block upon repeated application o f increasing concentrations of glutamate [21]. A cyclic model of desensitization as originally proposed by Katz and Thesleff for the ACh receptor has been proposed by Patneau and Mayer in attempt to account for the cross-desensitization responses of kainate with the application o f AMPA and quisqualate and the competitive interaction between these various agonists . 1 .2 .2 NM DA Desensitivmtion Even though the receptors for AMPA and NMDA have been colocalized on the same synapse, the decay times of the two synaptic components differ by more than 11 100-fold [22], This difference can be attributed to the variation in the receptor’s affinity for the agonist, L-glutamate. The AMPA receptor has a relatively low affinity and hence unbinds the ligand very rapidly. In contrast, the NMDA receptor has a higher affinity which sustains lengthened binding, giving the channel the ability to open repeatedly. Consequently, the rebinding of the transmitter is not necessary for the NMDA channel’s reopening . Like its non-NMDA counterpart, the NMDA receptor also desensitizes. This rapid desensitization o f synaptic receptors not only affects the time course of synaptic responses but also results in a refractory period during which a subpopulation o f previously activated receptors are no able to respond to transmitter released by subsequent stimuli [23], The decay of the NMDA current can be attributed to the receptor desensitization and disassociation of the agonist from the receptor. Prolonged high frequency applications o f L-glutamate cause glutamate to remain bound to the NMDA receptor. In addition, lengthened applications of L-glutamate results in receptor desensitization [24], Hence, it has been proposed by Lester ei al. (1992) that desensitization and unbinding of the agonist underlies the decay course o f the end- plate postsynaptic current (EPSC). In 1992, Lester et al. proved this by using out-side out patches to test for receptor desensitization. By increasing the duration o f the applied agonist and measuring the amplitude of the patch current, the maximum amount of glutamate necessary to saturate a pulse was found to be 3 mM for 0.8 msec. After determining 12 the maximum pulse for saturation, a second pulse was applied for the same duration and concentration. Since the peak amplitude o f the second response was smaller relative to the first, Lester concluded that it was not the unbinding of the agonist alone that caused decay of the response since in this case the amplitude o f the peak current upon a second pulse application would have been the same as the first. Thus considerable desensitization is measurable with even a brief application o f agonist. Other agonists such as L-cystate and L-aspartate when applied to the NMDA receptor also exhibit desensitization. If the time course o f the NMDA EPSC is determined by how long the transmitter remains bound to the receptor, then its decay should be dependent on the affinity of transmitter released, reflecting the disassociation rate o f the neurotransmitter from the receptor. When applications o f L-cystate and L-aspartate were applied whose affinities are approximately 10-fold and 100-fold lower than L-glutamate respectively, the rate o f decay o f these agonists increased with decreasing agonist affinity. Thus, Lester et a/, concluded that the disassociation rate of the agonist limited the time course o f the response. When 4 msec applications o f 20-30 mM o f L-cystate were applied to the NMDA receptor, the response did not saturate, and the L-cystate response increased with progressive application o f the neurotransmitter. From this finding it was concluded that for L-cystate the unbinding rate is fast enough for a number o f receptors to become unbound and rebind the neurotransmitter upon successive application, unlike L-glutamate. These findings conclude that the EPSC responses are dependent on the agonist affinity for the receptor. In order to describe the differences 13 in agonist affinities for the NMDA receptor, Lester et al. (1992) proposed a two- binding site kinetic model. 1.3 Specific Aims The current model by Holmes and Levy (1993) fails to account for the desensitization observed experimentally; the proposed model will combine the AMPA and NMDA kinetic models proposed by Lynch et a!, and Lester et al. respectively in order to create a working kinetic model that considers the desensitization, burst behavior, and differential binding o f agonist o f affinities o f the receptors involved in long term potentiation. This combined model will allow for application o f any frequency of stimuli o f glutamate input in order to predict calcium accumulation within the neuron since, as discussed previously, it is the internal level of calcium which underlies the induction and maintenance of LTP, The goals and direction of this thesis is the development and implementation o f the kinetic models o f the receptors involved in LTP includes the following: 1. Implementation of the AMPA model as proposed by Lynch et al. to predict the probability of the channel opening. 2. Implementation o f the NMDA model of Lester et al. to predict probability of channel opening. 3. Prediction o f epsp(s) in a synapse colocalized with both AMPA and NMDA receptors. 14 4. Using the AMPA and NMDA model: a. The AMPA/NMDA synapse will be coactivated with different frequencies of glutamate pulses in order to determine the optimal stimuli that results in calcium build up within the cell. b. The AMPA/NMDA synapse will be stimulated with weak and strong high frequency inputs to determine the pattern o f coactivation o f multiple synapses on a dendrite that results in optimal temporal summation o f stimuli to sustain calcium increase within the post-synaptic cell. 1.4 Outline of Thesis The thesis is organized as the following: 1. Chapter 2 will propose the model development o f the AMPA and NMDA kinetic reactions, the combined AMPA and NMDA model for a single synapse, and the computational methods o f the model approach. 2. Chapter 3 will analyze the results o f the models including the probability a given AM PA and NMDA channel will open, the resultant epsp upon coactivation o f the AMPA and NMDA receptor, the affects o f magnesium blockage o f the NMDA channel and its affects on the probability o f opening and channel kinetics. In addition, the response o f a single synapse to a high frequency input, as well as the temporal properties o f the response due to the stimulation o f multiple synapses in an associative manner will be analyzed. 15 3. Chapter 4 will address the conclusions of the model simulations as well as future progress and questions to be addressed in the regards to the model development and computational approach. 16 Chapter 2 Model Development 2.1 The AMPA Model The kinetic reaction used in this mode! is the cyclic model proposed by Lynch et a/. as illustrated in Figure 2.1 [25]. The values o f the kinetic parameters are defined in Table 2. /, and the kinetic equations of the model are specified in Figure 2.2. Lynch proposes a five state model where A represents the agonist and OPEN represents the conducting state o f the channel. The remaining non-conducting sates, R and Rih are bound desensitized states, respectively, and RA and RA^ are ligand bound open and desensitized states o f the complex. The neurotransmitter clearance rate xa(t') was modeled as time varying to allow for realistic analysis o f synaptic transmission. The time course o f transmitter concentration was modeled by the following differential equation: dxa(t)/dt = - xa tx a(i) for t&O (2.4) where xu r is the background concentration and was equal to 1 pM, and xa (0) is the peak agonist concentration which was 1000 pM. 17 Figure 2.1: Kinetic Model of the AMPA Receptor Complex [11 k,xa (t) [2] kQ [5] R +A ^ RA ^ Open k-i kc k j 1 - kr 1 I - kd k3 xa(t) Rd +A [41 k.3 [3] Table 2. T . Non-NMDA receptor mediated conductance parameters taken from Lynch et aL, 1993. Parameter Description Value k, association rate for sensitized state 1 p M 'V k., disassociation rate for desensitized state 1/1 m s1 k3 rate for desensitized state 10 g M 'V k-3 rate for desensitized state 1/9.97 ms'1 kd desensitized rate o f ligand-bound state 1/1.36 ms'1 kr resensitized rate o f ligand-bound state 1/61 ms'1 k4 rate for unbound state 1/1000 m s'1 k-4 rate for unbound state 1/450 ms'1 kQ opening rate o f channel 1/1.1 m s'1 kc closing rate channel 1/2 m s'1 Figure 2.2: Kinetic Equations of the AMPA Model d[R \A J/dl = - k,xa(t) - k4[R \ AJ + k J R A ] + k J R d+ A] d[RAJ/dt = k jX a ( i) [R i A] - ( k .,+ k d + k a )[RAJ + kJR jA J + kJOpen]M W A dfRjAJ/dt = kJR A J -(k„3 + kr )[Rc iA] + k3 xa(l)[Rd + A] d{Ra vA]/dt = kJR + A ] + k3 fR j4 ] - ( k3 xa(t) + k-4 )[Rd i - AJ d f Open]M II, A /dt = k J lU ] - kJO penjM IP A dxa(t)/di = - xatx a(i) 18 A " n,.is the transmitter clearance rate and is 1/1.25ms'1 . The number of receptors the transmitter is initially allowed to bind to is 1 . Although many o f the parameters o f the AMPA receptor have not been measured experimentally, estimates for these model parameters have been obtained by Lynch et al by satisfying various physical, biochemical, and physiological properties. The following properties were those that were considered by Lynch (1993) in the estimation o f the kinetic parameters: 1) Microreversibility where k,kdk.3k^ = k.]krk3k4. 2) The affinity Kd for L-glutamate was estimated between 30-100 pM. 3) Mean open time ( T = l/k c ) was estimated 1-3 msec. 4) Mean burst time ( T b ) was not much greater than T 0. 5) k, and k3 are estimated as rates of those measured for the NMD A receptor 1-13 pM"'s’t. 6) Desensitized state has a higher affinity than the sensitized state ( k.j/k, > k.t/k3 ). 7) State probabilities favor unbound sensitized state over the unbound desensitized state in the presence o f a low background agonist concentration and in response to a fast increase in transmitter concentration. 8) Shut times are characterized by short events ( 0.3-0.6 msec ) followed by two longer ones ( 30-94 and 70-400 msec ). 19 2.2 The NMDA Model Lester and Jahr observed that NMDA receptor exhibits different response time courses depending on the affinity o f the agonist applied [26], In order to account for the variability in the NMDA receptor's binding ability for different agonists, they implemented a two independent binding site model which was originally proposed by Patneau and Mayer in 1990. The evidence presented to date indicates that the decay of the NMDA response is not simply based on the closure o f the NMDA channel and unbinding of the agonist since this would be predicted by an exponential decay, but instead a significant proportion o f the number of receptors become desensitized. Figure 2.3 illustrates the simplest model proposed to fit Lester and Jahr's data. R represents the NMDA receptor with two available binding sites for the glutamate agonist, A. O is the open state o f the channel, and A *£) is the desensitized state of the channel. Table 2.2 summarizes the values of the rate parameters o f this model, and Figure 2.4 illustrates the equations that describe the kinetic reaction. Magnesium block o f the NMDA channel was simulated using the same equation as proposed by Holmes and Levy as the following: ^ \ n b i o C kc</AR*lohl, = (k"/k+ )/(k7k+ ) + [[ Mg2']] (2.5) k'/k' - 8.8 x 10'3 exp(V(t) /12.5) 20 Figure 2.3: Kinetic Model of the NMDA Receptor Complex [6] [7] 2ko n [6] [8] ko n [9] p [10] 2A + R ^ A + AR A2R Open ko fr 2ko fr kr 1 U c a A P [11] Table 2.2: NMDA mediated conductance parameters taken from Lester, 1992. Symbol Definition Dimension kon binding rate 5 p M 'V 1 koff unbinding rate 3.6 s'1 P opening rate 79.7 s 1 a closing rate 121 s 1 Figure 2.4: Kinetic Equations for the NMDA Model dfAJ/di = -2 k( J A ][R ] + ko M [A][AR] - kon[A][AR] dfRj/dt = -2kJA J[R ] - K gA][AR] d[AR]/dl = 2ko J /fA jlR J - K gAJfARJ - kJA J[A R J + K g A J i] + a[OJ - kd[A Ji] - a[A2R] d[Open]m tD A /dt = ^[A JiJ - a[Open]mDA d [ A p j/d t = - k J A p ] + k J A p j 21 2*3 The Comh ined Model In order to find the resultant epsps for a synaptic response with both AMPA and NMDA on the dendritic spine head, one can consider each synapse as a patch of membrane as a capacitor and resistor in parallel as illustrated in Figure 2.5. The membrane resistance Rm is defined as the resistance o f one square centimeter of membrane. Rin is independent o f geometry which allows for the comparison between neurons regardless o f their size and shape. Rm is primarily determined by the resting permeability to potassium and chloride. In the current model Rm = 10,000 fi-cm 2. In addition to allowing the passive flow o f ionic currents, charge accumulates on the membrane through the build up on the capacitive membrane element Cm. The charge develops as the difference between the inner and outer membrane o f the cell just as a capacitor, and the fluid on either side o f the membrane is the conductor. The membrane is only about 7 nm thick and hence has the ability of storing a significant amount o f charge. The capacitance is defined as how much charge (q) can accumulate per volt ( V) applied to it (C = q/V). The capacitance is defined in terms of Farads (coulombs / Volt), and is typically on the order o f 1 pF/cm2. For a steady state resting membrane potential to exist in neurons there are three major requirements which include: the intracellular and extracellular solutions must be electrically neutral, the cell must be osmotically balanced, and there is no net movement of ions inside or outside o f the cell [27]. The steady state resting membrane potential of the cell used in this simulation is -70 raV. 22 The current flowing into and out o f the capacitor o f the membrane can be deduced from the relationship between charge and voltage. The current / is expressed in units o f amperes ( ampere = coulomb/sec). Hence from the derivative o f C = q/V, the current can be expressed as the following: / = dq/dt = C( d V /dt) (2.6) Equation 2.6 can be rearranged as to solve for the change in voltage across the membrane: dV/dt = i/C (2.7) where the total current i is equal to the input currents minus the leakage current through the membrane. The input currents are composed o f the flow o f ions through both the AMPA and NMDA channels. Hence, i = Iam p a + In m d a - ( 2 .8 ) To measure the amount o f current / flowing through a particular membrane at various potentials o f Vm, the following equation applies for each ionic current whether I(W 1 ,A , In m d ai a n (^ I a m pa = [ P o p e n (V )]a m i’a N y am pa ( - E AM !>A ) (2.9) In this equation, Popi-nO) > s the probability varying over time predicted by the kinetic equations in Appendix A that the channel will open for either the AMPA or NMDA channel. N is the number o f channels activated in the synapse and was chosen to fit the amplitude o f the change in membrane potential over time ( d v/d t) o f the Holmes and Levy model (Figure 2.6). 23 The number of conducting channels for AMPA was estimated to be 2500, and for N M D A was 400 x 10s, y is the single channel conductance which is 5 pS for AMPA, 50 pS for the NMDA channel, and 1 x 10'7 pS for IL. E is the equilibrium or reversal potential where there is no net passive movement o f permeant ion species into or out o f the cell. EL was selected to be 30 mV; EM [P A and EN M D A both are 0. Thus, the total current (/r) can be calculated via the following equation: I'r= [PqpenCO a m ivJ^ yC Vm )]A+ [Poi’E nO nmimN y (V|h)]n " (Vm -E, )]L (2.10) Substituting l r o f Equation 2.10 for i o f Equation 2.7, one can integrate to find the change in membrane potential over time Vm(t) which is the resultant epsp o f the AMPA, NMDA, and leakage current. 24 Figure 2.5: Circuit Describing Cell Membrane of Neuron (Cm= membrane capacitance, = membrane resistance) C m 1 Outside R m I Inside Figure 2.6: Membrane Potentials for Non-NMDA, NMDA, and Combined taken from Lynch et **/., 1993. A 1.0 0.8 0.6 0.4 0.2 0.0 0 20 40 60 ao 100 120 > B E, 0.4 o ’ 0.2 a > 2 0 _____ S 0 20 40 60 SO 100 120 > c 1.0 0.8 0.6 0.4 0.2 0.0 0 20 40 60 80 100 120 Time (ms) non-NMOA 25 25 -4 Computationa 1 Metli od Since the AMPA model is a non-homogeneous Poisson process, Lynch’s approach was to numerically solve the system of differential equations by defining a probability matrix equation where P'(f)=R(0P(0 where each entry pf (/) of the column vector P(/) represents the probability the receptor is in state /' at time /, and R(t) [28], The fifth-order Runge-Kutta method with adaptive size control was then implemented to solve the differential equations. The solution to the differential equations in the present model used ADAPT II (a pharmacokinetic/pharmacodynamic program) where the process under study is is assumed to be described as the following differential/state equations and output equations respectively: jc = f( x, a ,/• ,/), x(o) = c (2.11) y = h (x,a ,r,() (2.12) In order to simulate pulse applications of neurotransmitter concentrations what is equivalent to a bolus input in drug applications was used. Pulse type inputs are simulated using instantaneous changes in model states via the following. x(dtj1 ) = jc( dtj') + b( d(t), nd (2.13) T where A(t) is the pulse (bolus) input vector (b(l) = [/?,(/)......^n (0] ) where - and + indicate times immediately preceding and following the transmitter pulse application. Longer pulse durations >lm s can be simulated using a rate vector r rand instantaneous changes in state are represented by the following equation: 26 ij ( t )= T j_ ,, dti < t ^ dt;, i=2, nd+1 (2.14) The dose times di, represents any time the input vectors r(t) or b(t) change value as upon administration of a new transmitter pulse. Each transmitter dose event "nd" is entered into ADAPT with the time and value o f the rate and bolus inputs. Table 2.3 summarizes the model equation symbols. Linear homogeneous systems can be simplified to the following equation and initial condition: x = A(a)x, *(0) = c (2.15) The Simulation (SIM) program in ADAPT was used to solve the differential equations o f the kinetic reactions (Appendix A) of both the AMPA and NMDA receptors. Solutions to the model differential equations are solved using the LSODA (Livermore Solver for Ordinary Differential equations with Automatic method switching for stiff and nonstiff problems). LSODA uses variable order, variable step size formulations o f Adam's method and Gear's method to solve stiff and nonstiff problems. 27 Table 2.3: Model equation definitions as taken from ADAPT □ User's Guide D'Argenio, 1992. Symbol Definition Dimenaan X T state vector, x(t) = [^ (t)...^ ^ )] n a system parameter vector P r input vector, /’ (t) = [/ ,(t)...rk (t)] k c initial condition vector n y output vector, .y(t) = [y,(t).../(t)]T e t time scalar X dxtdl ^ n b pulse (bolus) vector, b(t) = [b^ty.-.b^t)] n 28 2-5 Simulations The following simulations w ere demonstrated by implementation o f the individual kinetic models and/or combined AM PA/NMDA model: 1. The kinetic equations as described in Appendix A w ere implemented to predict the probability that the AMPA and NM DA channels were open given a 1000 pM 1 msec pulse or pulses o f glutamate both in the presence o f a 1.2 mM magnesium block and 0 mM magnesium. 2. These probabilities o f the opening were incorporated into a single synaptic model o f a resistance o f capacitance in parallel in which the current, and ultimately m em brane potential (voltage) were predicted for a frequency o f glutamate input at 100 and 200 Hz in order to predict the optimal stimuli to maintain calcium within the post-synaptic cell. 3. Temporal summation was demonstrated by activating a weak and strong synaptic input in order to analyze what pattern and frequency o f coactivation resulted in increased calcium levels within the post-synaptic cell. The weak input was modeled as an activation o f 5 synapses, whereas a strong synaptic input was modeled as 50 synapses. 29 Chapter 3 Results 3.1 Probability of Ch annels Opening The kinetic equations as described in Appendix A predicted the probability of the AMPA and NMDA channels opening. For the AMPA receptor, 1000 pM of glutam ate was applied for I msec. It was assumed that the receptor bound instantaneously in a 1:1 to ratio with the glutamate applied. Hence, it assumed that the transition from R T A RA is extremely fast, and the transmitter-receptor ratio is 1000. Thus state 1 in the model file considered R + A to be 1000 as the initial condition (XI(0) = 1000 ) in the ADAPT model file Appendix B. In order to find the probability of the interaction of one receptor, the probability was normalized by 1000 in the output equation o f the model file (Y(l) = X(5) / 1000). For the NMDA receptor, 1000 pM o f glutamate was concurrently applied with the AMPA receptor in state 6 (X6(0) = 1000) o f the ADAPT model file (Appendix B), but in this case the 30 initial value of the receptor was 1, (X8(0) = 7). Hence, 1000 pM was allowed to bind with one receptor. Figure 3.J predicts the probability of the AMPA and NMDA channels opening over time for the above conditions. The probability of the AMPA channel opening had a time to peak of about 5 msec with a 0.25 or a 25 % maximum probability of opening. The probability of the NMDA channel opening was predicted in Figure 3.1 in the absence of the magnesium block. The time to peak of the channel was approximately 15 msec with a maximum amplitude of approximately 0.1, or in other words, 10 % of the channels were open at the peak of the NMDA response. The time course of the probability of opening was greater than 140 msec. The next simulation took into consideration a 1.2 mM magnesium block. Equation 2.5 as proposed by Holmes and Levy to describe the number of channels that remain unblocked was used to predict the probability a single channel remained open in the current model. The equation was modified to the following to predict the probability of a single channel opening rather than the number of channels that remained unblocked: [f> (O unblocked]N M lM = [P(t)oPENJNMIM(k /k+)/(k /k+) + [[ Mg2+J] (3.1) k"/k+ = 8.8 x 10'3 exp(V(t) /12.5) This new probability of the number of NMDA channels that remains unblocked in the presence of magnesium replaces the probability of Equation 2.9 ( [ P opbnW J n m im ) to become: In M D A [ U nbluckcdlN M D A N Y nmda ( Vm - Enmda ) (3.2) When this equation was integrated into the model file, the probability that the NMDA channel was open was significantly reduced. Figure 3.2 illustrates that the time course and peak amplitude o f the AMPA channel remained unchanged as well as the time course of the NMDA channel, but the peak amplitude was significantly reduced to approximately 7.5 x 10 A, a 0.075% probability of being open. Hence, magnesium is a significant factor in blocking the conductance of calcium through the NMDA channel. 32 Probability Open Figure 3.1: Probability AMPA & NMDA in Open State in 0 mM Magnesium Probability AMPA in Open State vs. Time 0.3 0.1 CL 40 20 80 Time (msec) 120 60 100 140 Probability NMDA in Open State vs. Time {0 Mg2+) 0.12 0.1 0.08 0.06 0.04 -i 0.02 40 20 60 8 < Time (msec) 100 120 140 33 Probability Open Figure 3.2: Probability AMPA & NMDA in Open State in 1.2 mM Magnesium Probability AMPA in Open State vs. Time 0.3 0.2 0.1 140 40 100 120 20 Time (msec) Probability NMDA in Open State vs. Time (1.2 Mg2+) E O . 140 40 100 120 20 Time (msec) 34 3.2 Me mb rane Potential In order to analyze whether the time course and change in membrane potential of the current model was similar to what occurs experimentally and the model of Holmes and Levy, Figure 3.3 illustrates the results of the change in membrane potential (dVm/di ) modeled by Holmes and Levy while Figure 3.4 illustrates the change in membrane potential over time o f the present model [28], The AMPA channels appear to be synonymous in it's time course and kinetics both having a time to peak of about 5 ms, and a duration of about 40 ms. On the other hand, the NMDA membrane potential in the Holmes and Levy model has a duration of about 120 ms and a time to peak of about 15 ms, but the current model has a much longer duration, greater than 140 ms. Perhaps, this could be due to the desensitization of the receptor A2D which can resensitize to a conducting state after a period of time. Figure 3.5 illustrates the time course and kinetics o f the change in membrane potential for a 1.2 mM magnesium block. 35 dVm/dt {mV/msec) dVm/dt (mV/msec) Figure 3.3: AMPA & NMDA Membrane Potentials, taken from Holmes and Levy, 1993. o.a 0.6 0 .4 0.2 0.0 0 20 SO 1 0 0 1 2 0 4 0 6 0 > c 1.0 T non-N M O A + NMDA 0 2 0 4 0 GO 8 0 1 0 0 1 2 0 Time (ms) Figure 3.4: AMPA & NMDA Membrane Potentials of Current Model without Magnesium Block Membrane Potential Chanpeffime AMPA (1 00(}*jmol ot Glu) 0.6 0.4 0.2 40 100 1 20 140 20 60 8 < Time (msec) Membrane Potential Change/Time NMDA (OmM Mg+) 0.4 0.3 0.2 200 300 150 Time (msec) 250 100 36 d V m /d t (m V /m sec) Figure 3.S: AIVEPA & NMDA Membrane Potentials of Current Model with 1.2 mM Magnesium Block M em brane Potential C hange/Tim e AMPA {1000pnal of Glu) 0.6 0.4 0.2 100 120 140 20 60 8( Time (m sec) 40 M em brane Potential C hange/Tim e NMDA (1.2mM Mg2+) x 1 0 ' E2 120 60 3 < Time (msec) 100 40 37 The time to peak and duration remain unchanged, but the membrane potential is significantly decreased as expected due to the magnesium blockage of the channel. In any case, these results are very similar to those achieved experimentally in which the epsp of one component was pharmacologically blocked [30] as seen in Figure 3.6. Figure 3.7 illustrates the resulting combined epsp of coactivation of AMPA and NMDA receptors for both a 0 mM and 1.2 mM magnesium block. Figure 3.7 illustrates that although the epsp is reduced in amplitude, it is a not a significant difference since both cells are depolarized to approximately -30 mV. These results are similar to those of Holmes and Levy in 1993 as illustrated in Figure 3.3, and can be attributed to the fact the AMPA channel is the major conducting channel that carries most of the current. It is more important to note the change in duration, rise times, and repolarization times of the two simulations. The cell in 1.2 mM magnesium, has a faster rise time and time to peak than the cell in 0 magnesium. This is due to the fact that 1.2 mM of magnesium almost completely blocks the conduction through the NMDA channel as shown in Figure 3.8', hence, most of the current is carried by the AMPA channel that's kinetics are much faster. Even more pronounced is the duration of repolarization of the cell. The time course of the epsp changes from 60 ms in 0 mM magnesium to over twice that in 1.2 mM. This is obviously due to the necessary removal o f the magnesium block which takes the time and energy of the cell. 38 Figure 3.6: Pharmacological Block of AMPA and NMDA Receptors AP V blocks the N M D A com ponent o f the ep sp . N o te th e tim e cou rse o f th e N M D A and no'rt- N M D A com ponents reflect that o f the current m odel. U « » -to Figure 3.7: Combined AMPA & NMDA epsps for " 0 and 1.2 mM Mg2+ AMPA & NMDA ep sp vs. Tim s 100Qum Glu(0mM Mg2+ & 1.2 mM) - 2 5 -3 0 - 3 5 -SO -5 5 -6 0 -7 0 1 2 0 140 40 60 (H Time (m sec) 100 20 0 mM M g2* 1.2 mM Mg2+ 39 Voltage(mV) Figure 3.8: AMPA & NMDA Membrane Potentials in 1.2 mM Mg2+ NMDA & AMPA ep sp vs. Time (1000 micromol pulse of Glu, 1.2mM Mg2+) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 - 0.1 20 60 Time (m sec) 100 120 AMPA NMDA 40 3.3 High Frequency Stimulation Increases Calcium In order to examine the synaptic conductance of repetitive high frequency stimulation of the cell as would be characterize LTP, the cell was stimulated in Figure 3,9 with 100 Hz and 200 Hz 1000 pM glutamate inputs. Each pulse had a I msec duration . Figure 3.9 illustrates the effects in the absence of magnesium. It is obvious from the plot that a high frequency stimulation (i.e. 200 Hz) depolarizes the cell sufficiently more than lower frequencies (i.e. 100 Hz). This can be attributed to the fact calcium is released in the cell upon each subsequent stimulation and then resequestered by the sarcoplasmic reticulum. At a high frequency input of 200 Hz, calcium is being released from internal stores at a greater rate than its being resequestered. Hence, calcium accumulates within the cell and potentiates a higher level of depolarization and summation of epsps. Figure 3.10 illustrates high frequency stimulation in the presence of 1.2 mM magnesium. At 100 Hz, the potentials are weakly fused due to the separation in time between the synaptic inputs being too great to sufficiently relieve the magnesium block. Hence, the resulting conductance through the NMDA channel was low, and calcium influx was small. At a 200 Hz, activation produced temporal summation of the spine head potential; thus, relieving the magnesium block, and increasing the NMDA conductance. These results were similar to those found by Holmes and Levy as illustrated in Figure 3.11. 41 Voltage (mV) Figure 3.9: Combined A MPA & NMDA epsps in 0 M g2+ at Frequencies of 100 & 200 Hz AMPA & NMOA epsp in 0 M02+(100 Hz & 200 Hz) -10 -20 -50 -60 - 7 0 60 10 200 Hz 100 Hz Figure 3.10: /IMPA and NMDA epsp and Conductance, taken from Holmes and Levy, 1993. 6 0 - r 2CO H i C O C 40 S O H i 5 20 ■■ 10 -- Time (ms) 42 -1 0 - -20 - - -30 -50 -■ •60 - - -70 Voltage (mV) Figure 3.11: Combined AMPA & NMDA epsps in 1.2mM Mg2+ at Frequencies o f 100 & 200 Hz AMPA & NMDA ep sp vs. Time 1.2mM Mg2+ (100 & 200 Hz) -10 -20 -3 0 -4 0 -5 0 -6 0 -7 0 30 4( Time (msec) 50 70 20 200 Hz 100 Hz 43 3.4t Associative Synaptic Activation As discussed earlier, associative LTP occurs with coactivation of weak and strong high frequency inputs which are temporally summated. All the previous simulations were for a single synapse, the following results illustrate the stimulation of a weak input of 5 synapses activated synchronously and asynchronously with a strong input o f 50 synapses. The modification o f whether 5 or 50 synapses were activated was considered as a multiplication factor of the total input current (Equation 2.16). For example, for the simulation o f 5 synapses the total current, Ir, was expressed as the following: It =5 [PopEN(t)A M i’ANY(Vm )]A +[P0pEN(t)N M nA N Y (Vm )]N - [Ny (Vm - EL )]L (3.3) The simulations used the same stimulation pattern as the experimental study by Levy and Steward in 1983 in which they found the weak contralateral projection had to be stimulated before or simultaneously with the strong ipsilateral projection to potentiate a response [31], Each projection was stimulated at 400 Hz and activated eight times. The influx of calcium was predicted via the increase in relative size o f the epsps upon subsequent stimulations. The first simulation involved coactivation o f the weak and strong input synchronously. The coactivation depolarized the cell to +60 mV. The epsp of this simulation is displayed in Figure 3.12. This simulation produced the largest calcium increase within the cell. 44 Voltage (mV) Figure 3.12: Coactivation of AM PA & NMDA W eak/Strong Inputs at 400 Hz C oactivation of Weak and Strong Inputs at 400 Hz 60 40 20 -2 0 -40 -60 -80 20 40 Time (m sec) 30 60 70 60 (m sec) 80 90 45 In contrast, Holmes and Levy found that the largest calcium influx occurred when the weak contralateral input was preceded by the strong input by 1 msec. The results presented in Figure 3.13 predict the peak depolarization of the cell to be just under +60 mV with a 1 ms delay between weak and strong inputs. With subsequent delays S ms and 20 ms, the as seen in Figure 3.14 and Figure 3.15 respectively predict reduced calcium influx. The varying levels of calcium influx as predicted by the change in the size of the epsps with different temporal activations can be explained by the varying conductance of the NMDA channels. When the weak input is activated alone, the conductance in the NMDA channel is small, when it is proceeded by the activation of the strong input, the conductance through the NMDA channel increases considerably. When the weak input was activated, the majority of NMDA channels were bound to the receptor (A but few channels were open due to the slow kinetics as well as the blockage of magnesium, Figure 3.14 indicates that once the strong input was activated, the voltage-dependent magnesium block was removed and the epsp increased significantly. The conclusion from these findings illustrates that even though no new transmitter was released, the NMDA conductance through the weak channels increases with removal of the magnesium block upon stimulation of the strong ipsilateral input. With increasing delays in time from 1 ms to 20 ms, the resulting temporal summation of the epsp decreases; hence, conductance through the NMDA channels also decreases (,Figure 3.15). This could be explained by the fact 46 that with longer delays more o f the NMDA channels begin to close and enter the desensitized state such that upon stimulation with the strong input, the magnesium is removed allowing more channels to conduct calcium, but even more channels have progressed to the desensitized and closed states with increasing time delays; subsequently, the number of conducting channels is reduced, decreasing the resulting epsp. 47 Voltage (mV) Figure 3.13: Activation of AMPA & NMDA Weak/Strong Inputs at 400 Hz with 1 msec Delay W eak and Strong Inputs with 1 m s Delay {400 Hz) 60 -2 0 -4 0 -6 0 -8 0 40 50 Time {msec) 60 48 Voltage (mV) Figure 3.14: Activation of AMPA & NMDA Weak/Strong Inputs at 400 Hz with 8 msec Delay Activation of W eak and Strong Inputs with 8 m s Delay (400 Hz) 60 20 -20 -4 0 -6 0 -8 0 90 100 40 Time (msec) 49 Voltage (mV) Figure 3.15: Activation of AMPA & NMDA Weak/Strong Inputs at 400 Hz with 20 msec Delay Activation of W eak and Strong Inputs with 20 m s Delay (400 Hz) -10 -20 -4 0 -5 0 -6 0 -7 0 90 100 20 Time (m sec) 50 Chapter 4t Conclusions 4.1 The Kinetic Model AMPA and NMDA receptors are the key to understanding the phenomena of long-term potentiation. Understanding the kinetics and desensitization of the receptors involved in LTP is important in developing kinetic models that describe the response of these receptors to applications of agonists. Computational models such as the one presented here are beneficial since they predict the response of a neuron or population o f neurons comparative to experimental electrophysiological studies. The kinetic model discussed here predicts the probability of the AMPA and NMDA channels opening, and the membrane potentials of the neuron with different frequency of stimuli. In addition, it is representative of the time course, kinetics, and response of what has occurred experimentally. Although the accumulation of calcium is not directly calculated nor predicted by the model, it is apparent that upon high frequency stimulation o f the presynaptic cell there is an increase in the membrane potential, and hence an increase in the current (I^mda) through the NMDA channel 51 with each subsequent stimulation. It can be assumed the current flow through the NM DA channel is equivalent to the influx o f calcium since calcium is the main ion carrying the current through the NM DA channel (IN M D A « [Ca2']). High frequency stimuli results in a build up o f calcium within the cell since the frequency o f re-uptake o f the internal levels o f calcium is less than the stimulation frequency. The result is temporal summation o f subsequent epsps and sustained levels o f calcium within the cell. It is this internal level o f calcium that triggers the possible mechanisms responsible for LTP such as the release o f protein kinases, calmodulin, and retrograde messengers. 4.2 Future Modeling? Although the current model considers temporal summation o f epsps, future consideration o f the model needs to incorporate spatial summation as well. Specifically, during associative LTP where different synapses along the same dendrite or different dendrites are stimulated and the epsps are summed at the soma o f the cell. The modeling o f the dendrite is described by cable theory in which the dendrite o f the cell is assumed to have a cylindrical conducting core which is insulated and has a finite electrical capacity and resistance. An electrical stimulus at one point o f the 'cable' spreads passively to neighboring regions by flow o f current in a local circuit down the 52 cylindrical axis. 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Neurosci. 12(2):63 5-643 (1992), [24] Sather, W., Johnson, J., Henderson, G., Ascher, P. Glycine-insensitive desensitization of the NMDA responses in cultured mouse embryonic neurons. Neuron. 4:725-731 (1990). [25] Ambros-Ingerson, J., Lynch, G. Channel gating kinetics and synaptic efficacy: a hypothesis for expression o f long-term potentiation, Proc. Natl. Acad. Sci. 90:7903 (1993). [26] Lester, R., Clements, J., Westbrook, G., Jahr, C. Channel kinetics determine the time course of NMDA receptor-mediated synaptic currents. Nature. 346:565-567 (1990). [27] Nicholls, J., Martin, A., Wallace, B., From Neuron to Brain (Sunderland: Sinauer Associates, Inc., 1992), p. 68. [28] D'Argenio, D., Schumitzky, A. ADAPT II: Mathematical Software for Pharmacokinetic/Pharmacodynamic System Analysis (Los Angeles: Biomedical Simulations Resource University of Southern California., 1992), p.p. 28-31. [29] Holmes, W., Levy, W. Insights into associative long-term potentiation from computational models of NMDA receptor-mediated calcium influx and intracellular calcium concentration changes. [30] Collingridge, G., Lester, R. Excitatory amino acid receptors in the vertebrate central nervous system. Pharmacol. Rev. 40:143-210 (1989). [31] Levy, W., Steward, O. Temporal contiguity requirements for long-term associative potentiation/depression in the hippocampus. Neuroscience. 8:791-797 (1983). 56 Appendix A Kinetic Equations of the AMPA Model dfR+A ]/dt = - k jxa(t) - k4(R-\-A ] + k2 [RA ] + k_4 /R d+ A j d[RA]/dt = k jx jt) [R + A] - ( k2 + kd+ kQ )[RAJ + k,.[Rc iAJ + kJOpenJ dfRjAj/dt = kJRAJ - ( k ^ + k ^fR jA J + k3 xa(t)fRd + A] d[Rd+AJ/dt = kJR+A] + kjfRjAJ - ( k3 xa(t) + k-4 )[Rd + A] dfP o p e n ) 'amp,/dt = kJR A ] - kJP O P F N ]M fP A dxa(l)/dt = - xm x jt) [R+A(0)]=1000 Kinetic Equations for the NMDA Model dfAJ/dt = -2 k J A ][R ] + koff[A][AR] - kon[A][AR] d[R]/dt = -2kJA ][R ] - ko (ffA][AR] dfARj/dt = 2kogfA][R] - kofj[A][AR] - kJA }[A R ] + ko J j[AJi]+*[(P0P F H )mDA] - k J A 2 RJ - a[A2RJ d f ( P o pe n) mm./iVdt = P [A 2 R] - a[OJ d[A2 D]/di = - k/A dJJ + k J A 2 RI IAj(0) 1000, [Rj=J Equation for Membrane Potential of AMPA & NMDA Receptor dVm /dt=(P0p,;N (t)A M p A N Y(Vn ,))A +(P0p E N (t)N M D A N v(Vin ))N -(N yfV^-E, ))L Ca 57 A p p e n d ix B h o l j n * 3 2 . f T h u J u n a 1 1 : 4 6 :4 6 1 9 9 5 1 C C C C ' c c c c c c c c c c c c c c c c* T h i s f i l e c o n t a i n s t h e F o r t r a n s u b r o u t i n e s l i s t e d b e l o w i n w h i c h t h e u s e r m u s t e n t e r t h e r e l e v a n t e q u a t i o n s a n d c o n s t a n t s . C o n s u l t t h e u s e r ' s m a n u a l f o r d e t a i l s c o n c e r n i n g t h e f o r m a t f o r e n t e r e d e q u a t i o n s a n d d e f i n i t i o n o f s y m b o l s . 1 . D i f f E q - S y s t e m d i f f e r e n t i a l e q u a t i o n s . 2 . A m a t - S y s t e m s t a t e m a t r i x . 3 . O u t p u t - S y s t e m o u t p u t e q u a t i o n s . 4 . S y m b o l - P a r a m e t e r s y m b o l s a n d m o d e l c o n s t a n t s . 5 . V a r m o d - E r r o r v a r i a n c e m o d e l e q u a t i o n s . 6 . P r i o r - P a r a m e t e r m e a n a n d c o v a r i a n c e v a l u e s M O D E L A D A P T I I R e l e a s e 3 S u b r o u t i n e D I F F E Q ( T , X , X P ) I m p l i c i t N o n e I n c l u d e ' g l o b a l s . i n c ' I n c l u d e ' m o d e l . i n c ' R e a l * 8 T , X ( M a x N D E ) , X P ( M a x N D E ) C 1 . E n t e r D i f f e r e n t i a l E q u a t i o n s B e l o w l e . g . X P f l ) - - P ( 1 ) * X ( 1 ) ) C X P ( 1 J — < — P ( 1 ) * X ( 1 3 ) - P ( 7 ) ) * X ( 1 ) + P ( 2 ) * X ( 2 ) + P ( 8 ) * X { 4 ) X P ( 2 > - P < 1 ) - J C ( 1 3 ) * X ( l ) + ( - P ( 2 ) - P ( 5 ) - P ( 9 ) ) * X ( 2 ) + F ( 6 ) * X ( 3 ) + P ( 1 0 ) * X ( 5 ) X P ( 3 ) - P ( 5 ) * X ( 2 ) + ( - P ( 6 ) - P ( 4 ) ) * X ( 3 ) + P ( 3 ) * X ( 1 3 ) * X ( 4 ) X P ( 4 ) - P ( 7 ) * X ( 1 ) + P ( 3 ) * X ( 3 ) + ( { — P ( 3 ) * X ( 1 3 ) - P ( 8 ) ) ) * X ( 4 ) X P ( 5 ) - P ( 9 ) * X ( 2 ) ~ P ( 1 0 ) * X ( 5 ) X P ( 6 ) — 2 * P ( 1 3 ) * X { 6 ) * X ( 7 ) + P ( 1 4 ) * X ( 6 ) * X ( 8 ) - P < 1 3 ) * X ( 6 ) * X < 8 ) + 2 * P ( 1 4 ) * X { 9 ) X P < 7 ) — 2 * P ( 1 3 ) * X ( 6 ) * X ( 7 ) - P ( 1 4 ) * X ( 6 ) * X ( 8 ) X P ( 8 ) » 2 * P ( 1 4 ) " X ( 6 ) * X ( 7 ) - P ( 1 4 ) * X ( 6 ) * X ( 8 ) - P ( 1 3 ) * X ( 6 ) * X ( 8 ) + 2 * P ( 1 4 ) * X ( 9 ) X ? ( 9 ) - P ( 1 3 ) * X ( 6 ) * X ( 8 ) - 2 ’ P ( 1 4 1 » X ( 9 ) - P ( 1 5 ) * X < 9 ) + P ( 1 6 ) * X ( 1 0 ) - P ( 1 8 ) * X ( 9 ) + P ( 1 7 ) * X ( 1 1 ) X P ( 1 0 ) - P ( 1 5 ) * X ( 9 ) - P ( 1 6 ) * X ( 1 0 ) X P ( 1 1 ) — ? ( 1 7 ) * X ( U J + P ( 1 8 ) * X ( 9 ) X ? ( 1 2 ) - 5 0 * ( { — 2 5 0 0 * 5 0 e - 6 * X ( 1 0 ) * ( X < 1 2 ) - . 1 ) + - 4 0 0 0 0 0 0 0 * 5 e - 6 * ( X ( 5 ) / 1 0 0 0 ) * ( X ( 1 2 ) - . 1 } - . 1 * ( X ( 1 2 ) - . 0 3 ) ) ) X P ( 1 3 ) — . 8 * X ( 1 3 ) c -------------------------------------------------------------------------------------------------------------------- c c cc c --- - c c ■ c c R e tu r n E nd holiM32.fi T hu J u n 8 1 1 : 4 6 :4 6 1995 2 S u b r o u t i n e A M A T ( A ) I m p l i c i t N o n e I n c l u d e ' g l o b a l s . i n c ' I n c l u d e ' m o d e l . i n c ' I n t e g e r I , J R e a l * 3 A < M a x N D E , M a x N D E ) D O I - l , N d e q s D o J - l , N d e q s A { I , J ) - O . O D O E n d D o E n d D o C C c-------------------------------------------------------------------------------------------------------------------c C 2 . E n t e r n o n z e r o e l e m e n t s o f s t a t e m a t r i x ( e . g . A ( l , l ) - - ? ( 1 ) ] C c -------------------------------------------------------------------------------------------------------------------c c -------------------------------------------------------------------------------------------------------------------c c R e t u r n E n d S u b r o u t i n e O U T P U T ( Y , t , X ) I m p l i c i t N o n e I n c l u d e ' g l o b a l s . i n c ' I n c l u d e ' m o d e l . i n c ' R e a l * 8 Y ( M a x N O E ) , T , X ( M a x N D E ) C C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C C 3 . E n t e r O u t p u t E q u a t i o n s B e l o w ( e . g . Y ( l ) “ X ( l ) / P [ 2 ) ( C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C Y ( l > - X ( 5 ) / 1 0 0 0 Y < 2 ) - X U O I Y ( 3 ) - ( X ( 1 2 ) - . 1 ) * 1 0 0 0 C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ■ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C ------------------------------------------------------------------------------------------------------------------ c * R e t u r n E n d C # # # # ♦ ♦ # # # # * # t * * * * * * * * * S u b r o u t i n e S Y M B O L r m p l i c i t N o n e I n c l u d e ' g l o b a l s . i n c ' I n c l u d e ' m o d e l . i n c ' I n t e g e r l e q s o l 59 h o lm a s 2 . £ T hu J u n 3 1 1 : 4 6 : 4 6 199S 3 c h a r a c t e r * 6 0 d e s c r c o m m a r / e q s o l / l e q s o l c o m m o n / d e s c r / D a i c r C C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C C 4 . E n t e r a a I n d i c a t e d C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C N D E q s — 1 3 ! E n t e r I o f D i f f . E q a . N S P a r a m - 1 8 ! E n t e r f o f S y s t e m P a r a m e t e r s . N V p a r a m - 0 ! E n t e r t o f V a r i a n c e P a r a m e t e r s . l e q s o l - 1 ! M o d e l t y p e : 1 - D I F F E Q , 2 - A M A T , 3 - O U T P U T o n l y . D e s c r - ' P R O B A B I L I T Y A M P A a n d N M D A I N O P E N S T A T E ' C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C c C C c --------------------------------------------------------------------------------------------------------------------- c C 4 . E n t e r S y m b o l f o r E a c h S y s t e m P a r a m e t e r ( e g . P s y m ( l ) - ' K e l ' ) C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C P s y m ( l ) - ' K l ' P s y m ( 2 ) K - l ' P s y m { 3 ) - ' K 3 ' P s y m ( 4 ) - ' K - 3 ' P s y m ( 5 ) - ' K d ' P s y m ( S ) — ' K r ' P s y m ( 7 ) K 4 ' P s y m ( 8 ) K - 4 ' P s y m ( 9 ) - ' K o ' P s y m ( 1 0 ) - ' K C ' P s y m d U - ' X a b ' P s y m ( 1 2 ) - ' X a o ' P s y m ( 1 3 ) - ' K o n ' P s y m ( 1 4 ) K o f f ' P s y m ( l S ) " “ ' B e t a ' P s y m ( 1 6 ) - ' A l p h a ' P s y m ( 1 7 ) - ' K r ' P s y m d 8 ) - ' K d ' c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C c --------------------------------------------------------------------------------------------------------------------- c C C c --------------------------------------------------------------------------------------------------------------------- c C 4 . E n t e r S y m b o l f o r E a c h V a r i a n c e P a r a m e t e r ( e g : P V s y m ( l ) - ' S i g m a ' I C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C R e t u r n E n d S u b r o u t i n e V A R M O D ( V , T , X , Y ) I m p l i c i t N o n e 6 0 h o lm e a 2 . f «&u J u a a 1 1 : 4 6 :4 6 199S 4 I n c l u d e ' g l o b a l a . i n c ' I n c l u d e ' m o d a l . i n c ' R e a l » 8 V ( M a j c N O E ) , T , X ( M a x N D E ) ( M a x N O E ) C C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C C S . E n t e r V a r i a n c e M o d e l E q u a t i o n s B e l o w c C ( e . g . V ( l ) - P V ( 1 ) » » 2 * Y ( 1 ) * * P V ( 2 > ( C C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C c -------------------------------------------------------------------------------------------------------------------c c R e t u r n E n d S u b r o u t i n e P r i o r ( P m e a n , P c o v ) I m p l i c i t H o n e I n c l u d e ' g l o b a l s . i n c ' I n c l u d e ' m o d e l . i n c ' I n t e g e r I , J R e a l * S P m e a n ( M a x N S P + M a x N D E ) R e a l * 8 P c o v ( M a x N S P + M a x N D E , M a x N S P + M a x N D E ) D o I “ l , N S p a r a m P m e a n ( I ) - O . O D O D o J - l , N S p a r a m P c o v { J , I ) - O . O D O E n d D o E n d D o C C c------------------------------------------------------------------------------------------------------ c C S . E n t e r N o n z e r o E l e m e n t s o f P r i o r M e a n V e c t o r C C ( e . g . P m e a n < 2 ) - 1 0 . 0 ) C c------------------------------------------------------------------------------------------------------ c c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c C C c ------------------------------------------------------------------------------------------------------------------ c C S . E n t e r N o n z e r o E l e m e n t s o f C o v a r i a n c e M a t r i x ( L o w e r T r i a n g . ) C C ( e . g . P c o v < 2 , 1 1 - 0 . 2 5 ) C C------------------------------------------------------------------------------------------------------ c C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - C - C R e tu r n E nd 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 of 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 bleed through, substandard margins, and improper alignment can adversely affect reproduction. 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Najman, Deanna Elizabeth
(author)
Core Title
A kinetic model of AMPA and NMDA receptors
School
School of Engineering
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Master of Science
Degree Program
Biomedical Engineering
Degree Conferral Date
1995-08
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University of Southern California
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Tag
biology, neuroscience,biophysics, general,OAI-PMH Harvest
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English
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D'Argenio, David B. (
committee chair
), Berger, Theodore W. (
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committee member
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