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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.
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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.
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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
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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
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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
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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
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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