Effect of Channel Stochasticity on Spike Timing Dependent Plasticity

by

Harshit Sam Talasila

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Department of Electrical and Computer Engineering University of Toronto

© Copyright by Harshit Sam Talasila 2011

ii

Effect of Channel Stochasticity on Spike Timing Dependent

Plasticity

Harshit Sam Talasila

Master of Applied Science

Department of Electrical and Computer Engineering

University of Toronto

2011

Abstract

The variability of the postsynaptic response following a presynaptic action potential arises from:

i) the neurotransmitter release being probabilistic and ii) channels in the postsynaptic cell

involved in the response to neurotransmitter release, having stochastic properties. Spike timing

dependent plasticity (STDP) is a form of plasticity that exhibits LTP or LTD depending on the

precise order and timing of the firing of the synaptic cells. STDP plays a role in fundamental

tasks such as learning and memory, thus understanding and characterizing the effect variability

in synaptic transmission has on STDP is essential. To that end a model incorporating both forms

of variability was constructed. It was shown that ion channel stochasticity increased the

magnitude of maximal potentiation, increased the window of potentiation and severely reduced

the post-LTP associated LTD in the STDP curves. The variability due to short term plasticity

decreased the magnitude of maximal potentiation.

iii

Acknowledgments

It is with great gratitude that I acknowledge the people that have made this thesis possible. It has

been a long journey that has culminated in this work.

I would like to thank my family for their continued support throughout my career in academia

and especially this past year. Without their prayer and moral support I would not have made it

very far.

From the Cellular Bioelectricity Lab, I would to thank past and present members: Angela Lee,

Osbert Zalay, Marija Cotic, Eunji Kang, Mirna Guirgis, and Ryan McGinn. Special thanks to

David Stanley for sharing his model so I could continue the work and also for his helpful

suggestions during our many Skype sessions. I would also like to thank Sinisa Colic and Josh

Dian for making life more interesting in the lab with lively discussions. I shall truly miss these.

I would like to especially thank my thesis supervisor Berj Bardakjian for his guidance throughout

my three years with CBL. His passion for work and life are truly unique and working with him

has been rewarding.

Last but not least, thanks be to God for the divine help through all tests in the past years.

iv

Table of Contents

1. Introduction and Motivation ....................................................................................................... 1

1.1 Synaptic Plasticity ............................................................................................................... 2

1.1.1 Spike Timing Dependent Plasticity ........................................................................ 2

1.1.2 Short Term Plasticity .............................................................................................. 4

1.2 Variability in Synaptic Transmission .................................................................................. 4

1.3 Hypothesis ........................................................................................................................... 6

2. Methodology ............................................................................................................................... 7

2.1 Hippocampal Pyramidal Cell Model .................................................................................. 7

2.1.1 Traub Model ............................................................................................................ 7

2.1.2 Markovian Kinetic Model ..................................................................................... 10

2.2 Plasticity Models ............................................................................................................... 11

2.2.1 Spike Timing Dependent Plasticity ...................................................................... 11

2.2.2 Short Term Plasticity ............................................................................................ 14

2.3 Simulation ......................................................................................................................... 16

2.3.1 Simulation Environment ....................................................................................... 16

2.3.2 Network Setup and Simulation Protocol ............................................................... 16

3. Results ....................................................................................................................................... 18

3.1 Calcium Interspike Interval Charts ................................................................................... 18

3.2 Spike Timing Dependent Plasticity Curves ...................................................................... 21

v

4. Discussion and Future work ...................................................................................................... 24

4.1 Effect of Ion Channel Stochasticity on Spike Timing Dependent Plasticity .................... 24

4.1.1 Post Long Term Potentiation Associated Long Term Depression ........................ 26

4.2 Effect of Short Term Plasticity on Spike Timing Dependent Plasticity ........................... 26

4.3 Future Work ...................................................................................................................... 27

5. Conclusions ............................................................................................................................... 28

References ..................................................................................................................................... 29

Appendix ....................................................................................................................................... 34

vi

List of Figures

Figure 1.1 A schematic of Spike Timing Dependent Plasticity ..................................................... 3

Figure 2.1 A compartmental model of a hippocampal pyramidal cell .......................................... 8

Figure 2.2 Two of the functions in the Calcium-dependent plasticity model .............................. 12

Figure 2.3 A normalized spike timing dependent plasticity curve .............................................. 14

Figure 2.4 A schematic of a synapse ............................................................................................ 15

Figure 3.1 The peak Ca2+

concentration trace for the Default and Markovian case .................... 19

Figure 3.2 The peak Ca2+

influx one standard deviation above and below the mean .................. 19

Figure 3.3 The peak Ca2+

concentration for the Default case, with short term plasticity ............ 20

Figure 3.4 The peak Ca2+

concentration for the Markovian case, with short term plasticity ...... 21

Figure 3.5 The STDP curves for the Default case Markovian case ............................................ 22

Figure 3.6 The STDP curves of the Default case, with short term plasticity ............................... 23

Figure 3.7 The STDP curves for the Markovian case, with short term plasticity . ....................... 23

Figure 4.1 The current output of the calcium channel at peak ISI ............................................... 25

vii

List of Equations

Equation 2.1 Hodgkin-Huxley type equation for Na+................................................................... 9

Equation 2.2 Rate functions of gate activation and inactivation ................................................. 9

Equation 2.3 Channel kinetic parameters for NaPer ...................................................................... 10

Equation 2.4 Continuous time discrete state markov process for Na+

......................................... 10

Equation 2.5 Current output of Na+

.............................................................................................. 11

Equation 2.6 Calcium dependent plasticity model ...................................................................... 12

Equation 2.7 Conductance of a synapse ...................................................................................... 13

Equation 2.8 Kinetic parameters for a Ca2+

pool ......................................................................... 13

Equation 2.9 Facilitation of synapse during short term plasticity ................................................ 15

viii

List of Appendices

Appendix A: Computational Model Details

Table A1: Changes to Synaptic Parameters from Traub Model ................................................... 34

Table A2: Calcium Dependent Plasticity model parameters ........................................................ 35

Table A3: Short term plasticity parameter values ......................................................................... 35

ix

List of Abbreviations

AMPA α-amino-3-hydroxyl-5-methyl-4-isoxazolepropionate

[Ca] Intracellular Calcium Concentration

CaDP Calcium-Dependent Plasticity Model

CaMK II alcium-calmodulin dependent protein kinase II

GABA gamma-Aminobutyric acid

HFS High Frequency Stimulation

ISI Interspike Interval

LFS Low Frequency Stimulation

LTD Long Term Depression

LTP Long Term Potentiation

NMDA N-methyl-D-aspartic acid

NMDAR NMDA Receptor

STDP Spike Timing Dependent Plasticity

STP Short Term Plasticity

1

Chapter 1

Introduction and Motivation

Hebb, in 1949 [1] postulated that: When an axon of cell A is near enough to excite a cell B and

repeatedly or persistently takes part in firing it, some growth process or metabolic changes take

place in one or both cells such that A’s efficiency as one of the cells firing B, is increased. Since

that time, several decades have revealed much about the rules and mechanisms that underlie the

long-term changes in synaptic strength. Understanding the mechanisms that underlie these

changes are imperative, as such changes in synaptic strength (plasticity) are thought to be the

underling mechanisms of learning and memory [2-4].

Recently, with the advent of improved equipment and new experimental protocols the

probabilistic nature of synaptic transmission is being deeply studied [5, 6], and there is a growing

realization that the stochastic nature of underlying processes of plasticity do have a functional

effect. Although usually, fluctuations arising from stochastic elements decrease the sensitivity of

the temporal pattern of a stimulation, under particular conditions it can enhance the response, in

an effect called stochastic resonance [7, 8]. Additionally, it was recently proposed that

fluctuations which give rise to neural noise could play a role in dynamical diseases, such as

epilepsy, by providing the necessary impulse to push the brain from a normal state into a seizing

state [9].

How the intrinsic stochastic nature of cellular processes contribute to the variability in synaptic

transmission, its sensitivity and robustness to fluctuations has significant implications for

understanding the nature and mechanisms of synaptic plasticity.

The purpose of this thesis is to investigate the effect of variability in synaptic transmission on

plasticity. To that end we have constructed a biophysically inspired model of a hippocampal

pyramidal cell in order to investigate the effect ion channel stochasticity has on spike timing

2

dependent plasticity (STDP) and the effect of the variability introduced by short term plasticity

on STDP.

We have found that ion channel stochasticity does influence the response of the postsynaptic

neuron, and thus the STDP curve. Furthermore, the variability introduced by short term plasticity

also affects the STDP curve by decreasing the magnitude of maximal potentiation.

To summarize the contents of this document, first we present a review of the relevant literature

followed by the hypothesis guiding this work, all of which are in this chapter. In chapter 2, we

present an overview of the methodology utilized. Chapter 3 contains the results, which are

discussed in Chapter 4 along with their implications. Chapter 5 presents the conclusions of this

work.

1.1 Synaptic Plasticity

Synaptic plasticity is the ability of a synaptic connection between neurons to change its efficacy

of transmission. This activity dependent potentiation or depression in synaptic efficacy is thought

to be one of the underlying mechanisms of learning and memory [10].

In many parts of the brain, high frequency stimulation (HFS) of the presynaptic afferent neuron,

or by pairing presynaptic stimulation with a strong postsynaptic depolarization, results in an

increase in synaptic efficacy. This increase in efficacy lasts a long period of time, and is known

as long term potentiation (LTP) [8, 11]. Conversely, similar low frequency stimulation (LFS),

resulting in a decrease of synaptic efficacy, is called long term depression (LTD) [12]. Neurons

that express LTP have also been known to express LTD, and are known to be bi-directionally

plastic.

1.1.1 Spike Timing Dependent Plasticity

In addition to HFS and LFS induced plasticity, both long term -potentiation and -depression can

be induced by repeated and precisely timed pre- and postsynaptic spikes. The order and

interspike interval (ISI) in between these spikes has a significant effect on the magnitude and

type (LTP vs. LTD) of plasticity. A presynaptic spike followed closely by a postsynaptic spike

3

(positive ISI) results in LTP, whereas a postsynaptic spike followed by a presynaptic spike

(negative ISI) results in LTD [13-16]. A schematic presented in figure 1.1 [17], portrays the

nature of STDP, and highlights the requirement for precise temporal dynamics. A discontinuity

is present around zero ISI, where a difference of few milliseconds could result in LTP or LTD.

Figure 1.1 A schematic showing the order and timing specific nature of Spike Timing

Dependent Plasticity. Maximal plasticity is produced by small interspike intervals (ISI),

and there exists a sharp discontinuity at approximately zero ISI. Figure taken from

Karmarkar et al. [17].

Experimental studies have shown that slow and small increase in postsynaptic intracellular

concentration ([Ca]), leads to depression, while rapid and large increases in [Ca] results in

4

potentiation [18-21]. This calcium dependent model of synaptic plasticity can qualitatively

account for STDP, and one such model is presented in the following chapter. This calcium

dependent model also emphasizes the influence [Ca] dynamics could have over STDP.

1.1.2 Short Term Plasticity

In addition to LTP/LTD which account for changes in synaptic efficacy over long time scales,

there exist mechanisms that account for changes in synaptic efficacy that occur in the

millisecond time scale, known as short term plasticity. Short Term plasticity plays a role in both

the enhancement and depression of the synapse.

There are several different forms of enhancements such as facilitation, augmentation and post-

tetanic potentiation, which are usually connected with elevation in presynaptic [Ca], that trigger

certain molecular targets. Facilitation is visible with pairs of stimuli, where the second stimuli

results in a postsynaptic response that could be up to five times the size of the first. Facilitation

decays with a time course of approximately 100ms [22].

Depression is a form of short term plasticity, which is usually connected with the depletion of a

pool of readily releasable vesicles. These pools are re filled with a certain time course, which is

not instantaneous, thus resulting in a temporary stunting of synaptic transmission [22].

1.2 Variability in Synaptic Transmission

The transfer of information at a chemical synapse occurs due to an action potential at the pre-

synaptic cell, resulting in exocytosis of synaptic vesicles which produce a postsynaptic electrical

signal.

At synapses all across the nervous system, identical electrical stimulations results in non-

identical, probabilistic exocytosis [5, 6, 23-25]. This probabilistic response has additional effects

downstream, resulting in a variable postsynaptic response [26]. This variability in response to a

single action potential can be attributed to two general sources: The neurotransmitter release is

probabilistic, and the channels involved in the postsynaptic response to neurotransmitter release

have variable temporal and amplitude profiles [27].

5

There are several different aspects of variability that influences synaptic transmission at the

presynaptic terminal. These include the variability of the calcium channel response to incoming

action potential, the Brownian motion of Ca2+

ions which have to bind to synaptotagmin,

enabling the release of synaptic vesicles, to facilitate the transfer of information. In particular,

the release or fusion of synaptic vesicles is probabilistic, and although up to 20 vesicles are

docked at a given active zone, the number of vesicles that fuse with the membrane in response to

an incoming action potential is often zero or one [28].At the hippocampal synapse the probability

of release in response to an action potential has been known to vary between 0.1 to 0.9 [27].

A Vesicle (or multiple vesicles) fusing into the synaptic cleft, results in the diffusion of

neurotransmitters across the cleft and activation of postsynaptic receptors. Activation of these

receptors could result in postsynaptic depolarization which in turn triggers signaling pathways

involved in synaptic plasticity. One such pathway is the calcium-calmodulin dependent protein

kinase II (CaMK II) pathway which is activated during induction of LTP [29]. In this pathway,

calcium ions flowing into the postsynaptic compartment result in the activation of CaMKII

subunits, through calmodulin. When sufficient subunits are activated, autophosphrylation takes

place between adjacent subunits, which then remain active for long periods of time [30]. A

stochastic model developed to study the effect of calcium fluctuation on the activation of CaMK

II subunits showed that a small increase in the number of open N-methyl-D-aspartic acid

receptors resulted in up to two fold increase in activated subunits [31]. This nonlinear increase

showcase the large affect, small fluctuations in Ca2+

could have on synaptic plasticity. This

places a particular emphasis on Ca2+ efflux channels (NMDAR and voltage gated Ca2+

channel)

as they pertain to STDP, where the intracellular calcium response of the postsynaptic cell

determines the magnitude and type of plasticity.

It is then, reasonable to surmise, this nature of the synapse could have implication in information

processing in the neuron, as it relies on the integration of various signals in a consistent manner

in order to generate specific temporal response to input signals [32]. It is then essential for

characterization of such processes and assessing how stochastic fluctuations effect synaptic

transmission, and ultimately how they influence phenomenon such as plasticity.

6

1.3 Hypothesis

We hypothesize that the stochastic properties of ion channels will have an effect on spike timing

dependent plasticity curves. As discussed above, spike timing dependent plasticity relies on

precise timing and response of the pre- and postsynaptic neurons. Thus adding the element of

ionic channel stochasticity will affect the calcium response of the postsynaptic neuron, resulting

in a change in the shape of the STDP curve.

We also hypothesize that the inclusion of variability due to short term plasticity will have a

depressive effect on the synapse, and decrease the magnitude of maximal LTP in the STDP

curves. As mentioned above, the synapse in the hippocampus has a very low vesicle release

probability, thus the introduction of such a synapse would decrease the chance of synaptic

transmission, resulting in less instances where spike timing plasticity could occur, during

induction.

To test this hypothesis, we have built a biophysically inspired computational model of a

hippocampal pyramidal cell with ion channels that can switch between stochastic and default

modes. Furthermore, we have implemented a model of a short term plasticity enabled synapse

that could switch between default (no short term plasticity) and short term plasticity modes. If

our hypothesis is correct, the resulting change in ionic channels kinetics from default to

stochastic mode as well as the change in synapses from default to one where short term plasticity

is enabled would result in changes in the shape of the STDP curves.

7

Chapter 2

Methodology

This chapter outlines the methodology undertaken during the course of this thesis. Details

regarding the cellular components of the simulation, both normal and stochastic, the plasticity

paradigms implements (STDP and short term), as well as the simulation environment, will be

outlined.

2.1 Hippocampal Pyramidal Cell Model

We used a hippocampal pyramidal cell model, initially developed by Traub et al. [33](Traub

model) containing 66 compartments: 63 comprising of the apical and basil dendritic structure, 1

representing a spherical soma compartment, one making the axon initial segment and one the

axon itself. The compartmental structure is illustrated below in figure 2.1.

2.1.1 Traub Model

A variant of the cellular model developed by Traub et al. [33] was used to represent a

hippocampal pyramidal cell under normal conditions (Default). The Traub model was chosen

because of its basis on experimental data and its wide use and acceptance.

This variant of the traub model was developed by K.M. Menne [34], the major difference being,

that the axon was modeled by two segments, while the Traub model used four segments. The

axon compartments notwithstanding, the Menne variant is identical to the Traub model in its, cell

geometry, channel type, channel kinetics and channel densities. As such it contains six types of

ion channels, namely Sodium (Na+), voltage gated calcium (Ca

2+), delayed rectifying potassium

channel (KDR), A-type transient potassium channel (KA), Calcium-activated potassium after-

hyperpolarization channel (KAHP) and Calcium dependent potassium channel (KC).

8

Figure 2.1 A compartmental model of a hippocampal pyramidal cell, showing the spherical

soma, dendritic branches, and the axon( originating from the soma and moving to the

right). Figure taken from Menne et al. [34].

The model also contains all the synaptic inputs included in the Traub model, namely α-amino-3-

hydroxyl-5-methyl-4-isoxazolepropionate (AMPA), N-methyl-D-aspartic acid (NMDA),

gamma-Aminobutyric acid (GABA). However this variant contains some changes to the synaptic

input location and density. These changes are outlined in Appendix A1.

The ionic channels are modeled using Hodgkin-Huxley-type equations (Default). An example of

the dynamics governing a voltage gated sodium channel is given by equation 2.1. All other

voltage gated ion channels are governed by similar principles.

9

(2.1)

In these set of equations, V is the membrane potential (mV) and ENa is the reversal potential for

sodium (mV). Furthermore, INa is the channel current (µA/cm2) and gmax is the maximal

conductance (mS/cm2). The variables m and h represent the channel activation and inactivation,

and their exponents are representing their respective number of activation gates (m), and

inactivation gates (n). The variables m and h are not only time dependent, as explicitly stated in

equation 2.1, they are also voltage dependent through their α(V) and β(v) rate functions (ms-1

).

These rate functions represent the rates of the opening and closing of their respective gates and

are governed by the equations described by equation 2.2.

(2.2)

Another ion channel was implemented as an addition to this model, in order to incorporate the

effects of the persistent sodium channels (Naper). The properties of Naper were identical to the

ones presented in work by Traub et al. [35], and are governed by the set of equations presented in

equation 2.3

10

for

for (2.3)

where,

The variables m∞ represents the steady state conductance, τm represents the time constant (ms) of

the activation gate, and v is the membrane potential (mv) from the cells resting membrane

potential.

2.1.2 Markovian Kinetic Model

There exist several viable models to represent the inherent stochastic nature of ion channels.

However, there often is a trade-off between accuracy and ease of implementation. Markov

models have shown to address some of the limitations of Hodgkin-Huxley models [36], while

general enough to represent a range of channels. Thus, to model the stochastic nature of ion

channels, a custom markovian model developed by Stanley et al. [37] was utilized. A summary

of the relevant details are presented here.

Using the previously presented equations 2.1 and 2.2, we can represent each channel as a

continuous-time discrete state markov process. The six state markov kinetic model, representing

a voltage gated sodium channel is presented below in equation 2.4:

(2.4)

m0h0

m0h1

m1h0

m1h1

m2h0

m2h1

2αm

αm

αh

αm

2αm

αh α

h β

h

βm

2βm

βm

2βm

βh β

h

11

There are six states, taking into account all possible gate state involving two of the activation

gates and one inactivation gate. Each channel is represented by such a markov jumping process,

and in this case, state m2h1 represents the channel open case. The current output of such a

channel can be expressed as:

(2.5)

In this case the variables Gmax, m and h are replaced by Nm2h1 and γNa. The variable Nm2h1

represents the number of channels in state m2h1 (open state), while γNa is the single channel

conductance of the voltage gate sodium channel. A custom algorithm used by Mino et al. [38],

and Stanley et al. [37] known as Channel Number Tracking Algorithm, is used to keep track of

the number of channels in the open state.

Equivalent stochastic models of ion channels present in the original Traub model, namely

voltage gated Sodium (Na+)

, voltage gated calcium (Ca2+

), KDR, KA, KAHP and KC, were

implemented. All parameters are left at default (Traub) settings, which are tuned to reproduce

biologically realistic bursting [37].

2.2 Plasticity Models

2.2.1 Spike Timing Dependent Plasticity

The spike timing dependent plasticity model used here was described previously by Shouval et

al. [39]. This mathematical model describes bidirectional synaptic plasticity induced by several

different induction protocols with a fixed set of parameters. This is a calcium dependent

plasticity model (CaDP) and is guided by the observations that small increase in postsynaptic

intracellular concentration [Ca] leads to depression, while rapid and large increases in [Ca]

results in potentiation[18-21]. A summary of the relevant details are presented here.

12

The concentration of intracellular calcium in the postsynaptic compartment determines the sign

and magnitude of synaptic plasticity as is presented in equation 2.6 below:

(2.6)

Here, wi represents the synaptic weight of particular synapse i. The change in this weight is

directed by the Ω function and the learning rate function ε. These two functions are intracellular

calcium ([Ca]I) dependent and are depicted in figure 2.2. Details regarding the parameters are

outlined in Appendix A2. The Ω function, translates a calcium concentration to a target weight,

while the ε function determines the rate at which the change in synaptic weight will occur.

Figure 2.2 Two of the functions in the Calcium-dependent plasticity model. A. The Ω

function determines the sign and magnitude of synaptic plasticity. B. The η function is the

calcium-dependent learning rate.

13

The influx of calcium comes from two sources with independent kinetic properties [17]. The first

source is the NMDA receptor (NMDAR), which is activated by application of neurotransmitter,

and whose conductance (Gk, mS/cm2) changes with a damped second-order characteristics with

a time course given by τ1 and τ2 (ms) , as shown in equation 2.7. The second source is the voltage

gated calcium channels (Ca2+

), governed by equations outlined earlier in the chapter.

(2.7)

The calcium entering the cell through NMDAR and Ca2+

channels is aggregated in an

intracellular Ca2+

pool immediately inside the cell membrane, initially developed by Traub et al.

[33]. This single pool model for Ca2+

concentration is governed by equation 2.8 shown below:

(2.8)

Here Ca is the Ca2+

concentration (arbitrary units), Cabase is the baseline calcium concentration, Ik

is the input current (A) and B is a parameter to be fitted based on the compartment volume. The

time constant τ (15 ms) dictates the time course of the decay. The calcium concentration has

arbitrary units, following the convention used by Traub et al. [33]. Since [ca] was described in

arbitrary units, the [Ca] output of the model had to be fitted to the calcium dependent function

describing CaDP. To that end, the calcium output of the model to the STDP induction protocol

was obtained, and scaled to fit the CaDP function, in order to produce STDP curves similar to

Shouval et al. [40] as presented in figure 2.3.

14

Figure 2.3 A normalized spike timing dependent plasticity curve, produced using Default

ion channel kinetics, without short term plasticity. Positive weight suggests LTP, while

negative weight points to LTD.

2.2.2 Short Term Plasticity

A model of the synapse capable of short time plasticity used here was described previously by

Cai et al. [41]. A summary of the relevant details are presented here.

The basic schematic of the synapse is presented in figure 2.4. This schematic showcases the

general structure of the synapse and outlines the various components of the synapse involved in

short term plasticity.

-40

-20

0

20

40

60

80

100

120

-150 -100 -50 0 50 100 150

No

rmal

ize

d w

eig

ht

(%)

ISI (ms)

15

Figure 2.4 A schematic of a synapse showcasing the general structure of the synapse and

the various components of the synapse involved in short term plasticity. An action potential

may trigger a fusion of a vesicle with the membrane, from the readily releasable vesicle

dock, releasing neurotransmitters. The dock is then replenished from the available vesicle

pool. The released neurotransmitters diffuse across the synaptic cleft and may activate the

receptors on the postsynaptic cell.

This implemented model of the synapse can undergo both short term synaptic depression and

facilitation. The probability of presynaptic transmitter release undergoes depression, after a

readily releasable vesicle is released, and returns to its default value with a certain time constant.

Facilitation, increases the probability of transmitter release (pr) by a fixed percentage (υ)

immediately following a presynaptic spike, and recovers to its default values (pdr0) with a certain

time constant (τf), as described in equation 2.9 below:

(2.9)

Pre synaptic cell

Post synaptic cell

Receptor

Readily releasable vesicle

docks

Available vesicle pool

16

The model parameters values of this short term plasticity enabled synapse are presented in

Appendix A3.

The major difference between a short term plasticity enabled synapse and the synapse used in the

Traub model [33], is the fact that the Traub model transmits every action potential across a

synapse, and while the short term enabled synapse is variable and transmits only with a certain

probability, which could be facilitated or depressed.

2.3 Simulation

2.3.1 Simulation Environment

An open source simulator called General Neural Simulation System (GENESIS) version 2.3 [15]

was chosen as the platform to house and run the simulation on Intel Pentium computer running

Ubuntu 7.10 Linux operating system.

GENESIS is a script language based on C which provides a wide range of pre-defined libraries.

Several custom objects were added to this library dealing with short term and spike timing

dependent plasticity.

2.3.2 Network Setup and Simulation Protocol

A presynaptic and a postsynaptic pyramidal cell were simulated. The presynaptic cell was

connected to the postsynaptic cell at an apical dendritic compartment by an excitatory synapse

with both AMPA and NMDA receptors.

The presynaptic and postsynaptic spike paring was simulated by eliciting an action potential in

each cell at ISIs ranging from -100 ms to +130ms. ISI were defined by the start time of the first

spike to the start time of the second spike. The simulation ran for 100s, with a stimulus just

sufficient to elicit an action potential delivered at the rate of 1Hz [41]. The simulation time step

was 0.01ms.

To examine the effects of stochastic channels, the ion channels as defined by the Traub model

(Default) were converted to their markovian counterpart (Markovian). Furthermore, to examine

17

the effect of short term plasticity, the normal synapse was converted to its short term plasticity

(STP) enabled counterpart.

The raw calcium concentration trace was analyzed, and the peak calcium concentration was

obtained and averaged across the 100 times the stimulus was given, for each ISI. The calcium

concentrations were then normalized with respect to the value at -80 ISI, as it consistently

provided the lowest average value in the negative ISI realm. The synaptic weight at the end of

the simulation was obtained, normalized against the initial weight of 0.25 and graphed with

respect to ISI.

18

Chapter 3

Results

This chapter presents the results examining the effect of ion channel stochasticity and short term

plasticity, as per the methods detailed in Chapter 2. First were present the intracellular Calcium

response of the model at each interspike interval, and later present the resulting spike dependent

plasticity (STDP) curves.

3.1 Calcium Interspike Interval Charts

This section displays the charts of calcium influx into the intracellular calcium compartment in

the postsynaptic cell at each interspike interval (ISI). All the four cases, Default, Markovian,

Default with STP, and Markovian with STP are presented.

Figure 3.1 displays the peak calcium influx for the Default case, as well as for the Markovian

case. Introducing markovian kinetics resulted in a higher maximum (130%) peak Ca2+

change,

when compared with default kinetics (101%). Furthermore, under markovian conditions the

curve reaches its maximum value slower (+20ms) than its Default counterpart (+10ms). It is also

apparent that although at negative ISIs both trace are very similar, the Markovian trace is higher

between +10 to +70 ISI. The default peak Ca2+

change trace is similar to such traces found in

literature [17], although, this trace does not reach the baseline Ca2+

concentration as quickly, for

positive ISIs.

The addition of markovian kinetics has an additional effect of increasing the variability of the

peak Ca2+

response, as highlighted in Figure 3.2. It appears that this variability, as shown in the

form of curves one standard deviation above and below the mean, is higher at large ISIs.

19

Figure 3.1 The two traces display the peak Ca2+

influx for the Default case (red) and the

Markovian case (blue) at each ISI from -100 to + 130ms.

Figure 3.2 The peak Ca2+

influx for the A) Default case (red) and B) the Markovian case

(blue) at each ISI from -100 to + 130ms. The dashed lines represent the peak Ca2+

influx

one standard deviation above and below the mean.

0

20

40

60

80

100

120

140

160

180

-150 -100 -50 0 50 100 150

Pe

ak C

a2+

(arb

itra

ry u

nit

s)

ISI (ms)

Default

Markovian

A B

20

Figures 3.3 and 3.4 display the peak calcium influx for the Default, and Markovian case after the

synapse with short term plasticity have been implemented. In both the figures, it appears that the

variability introduced by the short term plasticity enabled synapse does not have an effect on the

magnitude of highest peak Ca2+

. However the inclusion of short term plasticity reduces the peak

Ca2+

output at large negative and positive ISIs.

Figure 3.3 The two traces display the peak Ca2+

change for the Default case (red), and

where short term plasticity is enabled (green) at each ISI from -100 to + 130ms.

0

20

40

60

80

100

120

140

160

180

-150 -100 -50 0 50 100 150

Pe

ak C

a2+

(arb

itra

ry u

nit

s)

ISI (ms)

Default

Default_STP

21

Figure 3.4 The two traces display the peak Ca2+

change for the Markovian case, where

short term plasticity is enabled (purple) and where short term plasticity is not implemented

(blue), at each ISI from -100 to + 130ms.

3.2 Spike Timing Dependent Plasticity Curves

This section displays the STDP curves for the following four cases, Default, Markovian, and

Default with STP and Markovian with STP

Figure 3.5 displays the STDP curve for the Default and the Markovian case. Introduction of

markovian ion channel kinetics resulted in a greater magnitude of maximal potentiation (118%),

when compared with the Default curve (96%). Furthermore, under Markovian conditions the

point of maximal potential is reached slower (+20ms) than its Default counterpart (+10ms), a

characteristic carried over from the peak Ca2+

change traces as presented in figure 3.1. It is also

apparent that under markovian conditions the curve does not experience the region of post

maximum LTP, associated LTD (post-LTP associated LTD) that is evident in the Default curve

(from +30 to +50 ISI).

0

20

40

60

80

100

120

140

160

180

-150 -100 -50 0 50 100 150

Pe

ak C

a2+

(arb

itra

ry u

nit

s)

ISI (ms)

Markovian

Markovian + STP

22

Figure 3.5 The two STDP curves are presented for the Default case (blue) and the

Markovian case (red) for ISIs ranging from -100 to + 130ms.

Figures 3.6 and 3.7 displays the STDP curves for the Default case, and the Markovian case after

the short term plasticity synapse has been implemented. In both the figures the introduction of

variability through the implementation of short term plasticity enabled synapse, resulted in a

lower magnitude of maximal potentiation. Furthermore, in Figure 3.6, the introduction of short

term plasticity, resulted in a longer period of post-LTP associated LTD, while with Markovian

kinetics, the presence of post-LTP associated LTD continues to be missing.

-40

-20

0

20

40

60

80

100

120

140

-150 -100 -50 0 50 100 150

No

rmal

ize

d w

eig

ht

(%)

ISI (ms)

Default

Markovian

23

Figure 3.6 The STDP curves for the Default case (red), and where short term plasticity is

enabled (green) at each ISI from -100 to + 130ms.

Figure 3.7 The STDP curves for the Markovian case (blue), and where short term plasticity

is enabled (purple) in addition to markovian dynamics, at ISIs from -100 to + 130ms.

-40

-20

0

20

40

60

80

100

120

140

-150 -100 -50 0 50 100 150

No

rmal

ize

d w

eig

ht

(%)

ISI (ms)

Default

Default_STP

-40

-20

0

20

40

60

80

100

120

140

-150 -100 -50 0 50 100 150

No

rmal

ize

d w

eig

ht

(%)

ISI (ms)

Markovian

Markovian + STP

24

Chapter 4

Discussion and Future work

The results from chapter 3 have shown that ion channel stochasticity does have an effect on the

calcium response of the postsynaptic cell, which translates to a change in the spike timing

dependent plasticity (STDP) curve. Furthermore, it was seen that with the implementation of

short term plasticity enabled synapse the magnitude of maximal potentiation is decreased in the

STDP curves.

This chapter further summarizes the results and discusses significant finding. It ends with a

section outlining future work that could be conducted in light of these results.

4.1 Effect of Ion Channel Stochasticity on Spike Timing Dependent Plasticity

Examining the effect of ion channel stochasticity on STDP, by comparing the Default peak Ca2+

traces and STDP curves with their stochastic counter parts reveal that stochasticity, implemented

through markovian schemes of ion channels does have an effect on the shape of the STDP curve.

The first affect readily visible, is the higher magnitude of peak potentiation in the Markovian

curves when compared with their Default counterparts (figure 3.5). This increase in peak LTP

lets the cell experience greater potentiation. This observation is consistent with the peak Ca2+

traces (figure 3.1). Furthermore the attribute observed in the Markovian Ca2+

traces, such as ISI

at peak potentiation being +20 ISI is also translated to the STDP curves.

Another consequence of ion channel stochasticity is the longer window during which LTP takes

place, stretching the positive ISI window limit from approximately +30 ISI to +50 ISI (figure

3.5). This observation is in agreement with the peak Ca2+

traces (figure 3.1), where the

25

Markovian trace exhibits higher peak Ca2+

response change when compared to the Default case

at the same ISI.

In order to investigate this phenomenon, the current output of the Default voltage gated calcium

channel was compared with its Markovian counterpart, as presented in figure 4.1. Here the

current output of the calcium channel at ISI where maximum potentiation was experienced (+10

for Default, +20 for Markovian), are normalized to their respective outputs at -80 ISI. This

reveals that, the Markovian model of the voltage gated Ca2+

channel generates a greater output of

calcium current with respect to its baseline levels, when compared with its Default counterpart.

This could explain the higher maximum peak Ca2+

change and subsequently, a higher maximum

potentiation on the STDP curve for the Markovian model. It should be noted that the Default

and Markovian channels have the same parameters, as set by Traub et al. [33] and on average

produce similar conductance in response to a single current injection.

Figure 4.1 The current output of the calcium channel at peak ISI ( +10ms for default,

+20ms for Markovian), normalized to their respective current outputs at -80 ISI, recorded

during the STDP induction protocol.

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

ISI Injection

Cal

ciu

m C

urr

en

t O

utp

ut

(%

Ch

ange

)

Default

Markovian

26

4.1.1 Post Long Term Potentiation associated, Long Term Depression

In all the Markovian STDP curves, the post-LTP associated LTD is, either entirely abolished or

its effects severely diminished (figure 3.5, 3.7). The experimental support for the feature of post-

LTP associated LTD was not strong when the Shouval et al. [40]. model was initially proposed

in 2002. However some physiological experiments in subsequent years have showed that this

post-LTP associated LTD phenomenon does exist [42-45].

Here we have shown that in the Default case, post-LTP associated LTD exists, and that it

disappears for the Markovian case. This leads to the belief that this phenomenon of post-LTP

associated LTD could be associated with the stochastic nature of the underlying system.

Locations with small number of channels, where the stochastic nature of channels has its greatest

influence, could result in STDP curves without post-LTP associated LTD, as seen in the work by

Shouval et al. [46].

4.2 Effect of short term plasticity on Spike Timing Dependent Plasticity

The short term plasticity enabled synapse had no impact on the magnitude of maximal peak Ca2+

change, however in the STDP curves, the magnitude of maximal potentiation was decreased for

both, channels with default and markovian kinetics.

The lower magnitude of maximal potentiation for STDP curves with a short term plasticity

enabled synapse can be attributed to the variable nature of the synapse. This variability, with

depression as well as facilitation, ensures that every presynaptic action potential is not

transmitted across to the postsynaptic neuron. Although this probability of transmission is aided

by short term facilitation, on average there are lower numbers of action potentials that are

transmitted and thus can aid in the potentiation of the synapse, when compared with the default

synapse that is able to transmit every single presynaptic action potential. It is interesting that, this

reduction of potentiation has its greatest effect at small (negative and positive) ISIs. This is

attributed to the fact that, at ISIs where peak potentiation takes place, the system is already very

close to right side of the Ω function and the ε function. This means that any pre-synaptic action

potential that gets transmitted, not only sets a high target weight, but the rate of weight change is

27

also accelerated, resulting in a larger weight change, when compared with other ISIs. Thus with

the variability of short term plasticity, as the number of transmitted action potentials decrease,

their effect on the STDP curve is reduced.

In the case of peak Ca2+

influx, short term plasticity results in no change in the higher peak Ca2+

,

but reduced the Ca2+

influx at large negative and positive ISIs. The probabilistic nature of short

term plasticity enabled synapse ensures that a vesicle is not released every time there is a

presynaptic spike. Thus, during the course of the induction process, the cumulative effect of

calcium influx due to a vesicle release as well as the post synaptic spike occurs, only a few times.

On average, this results in Ca2+

influx, only due to the postsynaptic cell firing, which produces a

lower Ca2+

at large ISIs.

4.3 Future Work

The model constructed for this work could serve as a launching point for further analysis of the

effect of synaptic transmission variability on plasticity.

One of the immediate extension of this work, would be to incorporate more realistic mechanisms

of variability in the pre-synaptic cell. Depression and facilitation aspect of short term plasticity

are not purely random events, but are heavily influenced by intracellular calcium [22].

Implementing calcium dependent depression and facilitation mechanisms will provide us with a

more realistic model to investigate very fundamental phenomenons.

Another step towards constructing a more realistic model would be the implementation of a

stochastic synaptic input channel. Work by Serletis et al. [47], has shown that the presence of

higher complexity in membrane potential could be attributed to synaptic noise, thus

implementation of a realistic synapse, would enable the study of how variability in postsynaptic

responses could influence plasticity.

These proposed future endeavors would result a more realistic computational model that will

provide us with a greater understanding of the underlying mechanisms of phenomenons such as

plasticity, which is imperative in fundamental human activities of learning and memory.

28

Chapter 5

Conclusions

In the quest to discover the effect, variable nature of synaptic transmission has on spike timing

dependent plasticity a model incorporating this variability in the form of short term plasticity and

stochastic ion channels was built. Using this model we saw that ion channel stochasticity results

in a higher magnitude of maximum LTP in the STDP curve, which lets the cell experience even

greater potentiation. It was also observed that ion channel stochasticity resulted in a larger

window of LTP, and a severe reduction of the post-LTP associated LTD. Furthermore, we also

saw that when variability in the form of short term plasticity was introduced, the magnitude of

maximal potentiation was decreased in the STDP curves. These results shed light on the

influence of synaptic transmission variability on fundamental phenomenon such as plasticity,

which has functional implications on essential tasks such as learning and memory.

29

References

[1] D. O. Hebb, Neurology, 4th Edition. 1949.

[2] T. Brown, E. Kairiss and C. Keenan, "Hebbian Synapses - Biophysical Mechanisms and

Algorithms," Annu. Rev. Neurosci., vol. 13, pp. 475-511, 1990.

[3] Y. Dan and M. Poo, "Spike timing-dependent plasticity of neural circuits," Neuron, vol. 44,

pp. 23-30, SEP 30, 2004.

[4] R. Malenka and M. Bear, "LTP and LTD: An embarrassment of riches," Neuron, vol. 44, pp.

5-21, SEP 30, 2004.

[5] N. Hessler, A. Shirke and R. Malinow, "The Probability of Transmitter Release at a

Mammalian Central Synapse Rid A-7328-2010," Nature, vol. 366, pp. 569-572, DEC 9, 1993.

[6] V. Murthy, T. Sejnowski and C. Stevens, "Heterogeneous release properties of visualized

individual hippocampal synapses," Neuron, vol. 18, pp. 599-612, APR, 1997.

[7] P. Hanggi, "Stochastic resonance in biology - How noise can enhance detection of weak

signals and help improve biological information processing RID B-4457-2008,"

ChemPhysChem, vol. 3, pp. 285-290, MAR 12, 2002.

[8] K. Wiesenfeld and F. Moss, "Stochastic Resonance and the Benefits of Noise - from Ice Ages

to Crayfish and Squids," Nature, vol. 373, pp. 33-36, JAN 5, 1995.

[9] F. da Silva, W. Blanes, S. Kalitzin, J. Parra and P. Suffczynski, "Dynamical diseases of brain

systems: Different routes to epileptic seizures," IEEE Trans. Biomed. Eng., vol. 50, pp. 540-548,

2003.

[10] M. Bear and C. Rittenhouse, "Molecular basis for induction of ocular dominance plasticity,"

J. Neurobiol., vol. 41, pp. 83-91, 1999.

30

[11] T. Bliss and T. Lomo, "Long-Lasting Potentiation of Synaptic Transmission in Dentate Area

of Anesthetized Rabbit Following Stimulation of Perforant Path," J. Physiol. -London, vol. 232,

pp. 331-356, 1973.

[12] S. Dudek and M. Bear, "Homosynaptic Long-Term Depression in Area Ca1 of

Hippocampus and Effects of N-Methyl-D-Aspartate Receptor Blockade," Proc. Natl. Acad. Sci.

U. S. A., vol. 89, pp. 4363-4367, MAY 15, 1992.

[13] W. Levy and O. Steward, "Temporal Contiguity Requirements for Long-Term Associative

Potentiation Depression in the Hippocampus," Neuroscience, vol. 8, pp. 791-797, 1983.

[14] D. Debanne, B. Gahwiler and S. Thompson, "Asynchronous Presynaptic and Postsynaptic

Activity Induces Associative Long-Term Depression in Area Ca1 of the Rat Hippocampus In-

Vitro," Proc. Natl. Acad. Sci. U. S. A., vol. 91, pp. 1148-1152, FEB 1, 1994.

[15] H. Markram and M. Tsodyks, "Redistribution of synaptic efficacy between neocortical

pyramidal neurons," Nature, vol. 382, pp. 807-810, AUG 29, 1996.

[16] G. Bi and M. Poo, "Synaptic modifications in cultured hippocampal neurons: Dependence

on spike timing, synaptic strength, and postsynaptic cell type," J. Neurosci., vol. 18, pp. 10464-

10472, DEC 15, 1998.

[17] U. Karmarkar and D. Buonomano, "A model of spike-timing dependent plasticity: One or

two coincidence detectors?" J. Neurophysiol., vol. 88, pp. 507-513, JUL, 2002.

[18] K. Cho, J. Aggleton, M. Brown and Z. Bashir, "An experimental test of the role of

postsynaptic calcium levels in determining synaptic strength using perirhinal cortex of rat," J.

Physiol. -London, vol. 532, pp. 459-466, APR 15, 2001.

[19] R. Cormier, A. Greenwood and J. Connor, "Bidirectional synaptic plasticity correlated with

the magnitude of dendritic calcium transients above a threshold," J. Neurophysiol., vol. 85, pp.

399-406, JAN, 2001.

[20] J. Cummings, R. Mulkey, R. Nicoll and R. Malenka, "Ca2+

signaling requirements for long-

term depression in the hippocampus," Neuron, vol. 16, pp. 825-833, APR, 1996.

31

[21] S. Yang, Y. Tang and R. Zucker, "Selective induction of LTP and LTD by postsynaptic

[Ca2+

], elevation," J. Neurophysiol., vol. 81, pp. 781-787, FEB, 1999.

[22] R. Zucker and W. Regehr, "Short-term synaptic plasticity," Annu. Rev. Physiol., vol. 64, pp.

355-405, 2002.

[23] J. Delcastillo and B. Katz, "Quantal Components of the End-Plate Potential," J. Physiol. -

London, vol. 124, pp. 560-573, 1954.

[24] C. Auger and A. Marty, "Quantal currents at single-site central synapses," J. Physiol. -

London, vol. 526, pp. 3-11, JUL 1, 2000.

[25] T. Branco and K. Staras, "PERSPECTIVES The probability of neurotransmitter release:

variability and feedback control at single synapses RID C-1503-2008," Nat. Rev. Neurosci., vol.

10, pp. 373-383, MAY, 2009.

[26] E. Hanse and B. Gustafsson, "Quantal variability at glutamatergic synapses in area CA1 of

the rat neonatal hippocampus," J. Physiol. -London, vol. 531, pp. 467-480, MAR 1, 2001.

[27] C. Ribrault, K. Sekimoto and A. Triller, "From the stochasticity of molecular processes to

the variability of synaptic transmission," Nature Reviews Neuroscience, vol. 12, pp. 375-387,

JUL, 2011.

[28] H. Korn and D. Faber, "Quantal Analysis and Synaptic Efficacy in the Cns," Trends

Neurosci., vol. 14, pp. 439-445, OCT, 1991.

[29] J. Lisman, H. Schulman and H. Cline, "The molecular basis of CaMKII function in synaptic

and behavioural memory," Nat. Rev. Neurosci., vol. 3, pp. 175-190, MAR, 2002.

[30] P. Miller, A. Zhabotinsky, J. Lisman and X. Wang, "The stability of a stochastic CaMKII

switch: Dependence on the number of enzyme molecules and protein turnover RID D-2722-

2009," PLoS. Biol., vol. 3, pp. 705-717, APR, 2005.

[31] S. Zeng and W. R. Holmes, "The Effect of Noise on CaMKII Activation in a Dendritic

Spine During LTP Induction," J. Neurophysiol., vol. 103, pp. 1798-1808, APR, 2010.

32

[32] A. A. Faisal, L. P. J. Selen and D. M. Wolpert, "Noise in the nervous system RID A-2367-

2009 RID A-6066-2009," Nat. Rev. Neurosci., vol. 9, pp. 292-303, APR, 2008.

[33] R. Traub, J. Jefferys, R. Miles, M. Whittington and K. Toth, "A Branching Dendritic Model

of a Rodent CA3 Pyramidal Neuron," J. Physiol. (Lond. ), vol. 481, pp. 79-95, 1994.

[34] K. Menne, A. Folkers, T. Malina, R. Maex and U. Hofmann, "Test of spike-sorting

algorithms on the basis of simulated network data," Neurocomputing, vol. 44, pp. 1119-1126,

JUN, 2002.

[35] R. Traub, E. Buhl, T. Gloveli and M. Whittington, "Fast rhythmic bursting can be induced

in layer 2/3 cortical neurons by enhancing persistent Na+ conductance or by blocking BK

channels," J. Neurophysiol., vol. 89, pp. 909-921, FEB, 2003.

[36] A. Strassberg and L. Defelice, "Limitations of the Hodgkin-Huxley Formalism - Effects of

Single-Channel Kinetics on Transmembrane Voltage Dynamics," Neural Comput., vol. 5, pp.

843-855, NOV, 1993.

[37] D. Stanley, B. Bardakjian, M. Spano and W. Ditto, "Calcium-dependent subthreshold

fluctuations in membrane voltage; a modeling study," BMC Neuroscience, vol. 11, pp. P122,

2010.

[38] H. Mino, J. Rubinstein and J. White, "Comparison of algorithms for the simulation of action

potentials with stochastic sodium channels," Ann. Biomed. Eng., vol. 30, pp. 578-587, APR,

2002.

[39] H. Shouval, G. Castellani, B. Blais, L. Yeung and L. Cooper, "Converging evidence for a

simplified biophysical model of synaptic plasticity," Biol. Cybern., vol. 87, pp. 383-391, DEC,

2002.

[40] H. Shouval, M. Bear and L. Cooper, "A unified model of NMDA receptor-dependent

bidirectional synaptic plasticity," Proc. Natl. Acad. Sci. U. S. A., vol. 99, pp. 10831-10836, AUG

6, 2002.

33

[41] Y. Cai, J. P. Gavornik, L. N. Cooper, L. C. Yeung and H. Z. Shouval, "Effect of stochastic

synaptic and dendritic dynamics on synaptic plasticity in visual cortex and hippocampus," J.

Neurophysiol., vol. 97, pp. 375-386, JAN, 2007.

[42] H. Markram, J. Lubke, M. Frotscher and B. Sakmann, "Regulation of synaptic efficacy by

coincidence of postsynaptic APs and EPSPs," Science, vol. 275, pp. 213-215, JAN 10, 1997.

[43] G. Bi and M. Poo, "Synaptic modifications in cultured hippocampal neurons: Dependence

on spike timing, synaptic strength, and postsynaptic cell type," J. Neurosci., vol. 18, pp. 10464-

10472, DEC 15, 1998.

[44] D. Feldman, R. Nicoll, R. Malenka and J. Isaac, "Long-term depression at thalamocortical

synapses in developing rat somatosensory cortex," Neuron, vol. 21, pp. 347-357, AUG, 1998.

[45] M. Nishiyama, K. Hong, K. Mikoshiba, M. Poo and K. Kato, "Calcium stores regulate the

polarity and input specificity of synaptic modfication," Nature, vol. 408, pp. 584-588, NOV 30,

2000.

[46] H. Shouval and G. Kalantzis, "Stochastic properties of synaptic transmission affect the

shape of spike time-dependent plasticity curves," J. Neurophysiol., vol. 93, pp. 1069-1073, FEB,

2005.

[47] D. Serletis, O. C. Zalay, T. A. Valiante, B. L. Bardakjian and P. L. Carlen, "Complexity in

Neuronal Noise Depends on Network Interconnectivity," Ann. Biomed. Eng., vol. 39, pp. 1768-

1778, JUN, 2011.

34

Appendix

Appendix A: Computational Model Details

Table A1: Changes to Synaptic Parameters from Traub Model

Parameter Traub Model Model Implemented in

this thesis

Location Excitatory synapses Located 175μm or more

from the soma in long

dendrites 75μm or more

in short dendrites,

NMDA and AMPA

always occur together

Located in compartment

levels 1, 8, 9, 10. NMDA

and AMPA always occur

together

GABA_A Located in most proximal

dendrites

No difference

GABA_B Located in compartments

at least 240μm away

from soma

Located in compartments

9, 10, and 11

Number Excitatory synapses 33 AMPA and NMDA 29 AMPA and NMDA

GABA_A 96 43

GABA_B 10 23

35

Table A2: Calcium Dependent Plasticity model parameters

Variable Value

ɛ0 0.33333

ɛ1 0.22

ɛ2 0.39

δ1 80

δ2 40

p1 1.0

p2 0.28

p3 3.0

p4 1e-5

Table A3: Short term plasticity parameter values

Variable Value

Pdr0 0.19

τr 1000(ms)

τf 100(ms)

υ 0.8

n 2