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CHAPTER 7

The Synapse

This chapter covers a spectrum of models for both chemical and electricalsynapses. Different levels of detail are delineated in terms of model complex-ity and suitability for different situations. These range from empirical modelsof voltage waveforms to more detailed kinetic schemes, and to complex MonteCarlo models including vesicle recycling and release. Simple static modelsthat produce the same presynaptic response for every presynaptic actionpotential are compared with more realistic models incorporating short-termdynamics that produce facilitation and depression of the postsynaptic re-sponse. Different types of excitatory and inhibitory chemical synapses, in-cluding those with AMPA-receptor-mediated and NMDA-receptor-mediatedresponses, are considered.

7.1 Synaptic Input

So far we have considered neuronal inputs in the form of electrical stimula-tion via an electrode, as in an electrophysiological experiment. Many neu-ronal modelling endeavours start by trying to reproduce the electrical activ-ity seen in particular experiments. However, once a model is established onthe basis of such experimental data, it is often desired to explore the model insettings that are not reproducible in an experiment. For example, how doesthe complex model neuron respond to patterns of synaptic input? How doesa model network of neurons function? What sort of activity patterns can anetwork produce? These questions, and many others besides, require us tobe able to model synaptic input. We discuss chemical synapses in most detailas they are the principle mediators of targetted neuronal communication.Electrical synapses are discussed in Section 7.7.

The chemical synapse is a complex signal transduction device that pro-duces a postsynaptic response when an action potential arrives at the presy-naptic terminal. A schematic of the fundamental components of a chemicalsynapse is shown in Figure 7.1. We describe models of chemical synapsesbased on the conceptual view that a synapse consists of one or more activezones that contain a presynaptic readily-releasable pool of vesicles (RRVP)

1

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2 THE SYNAPSE

RRVP

Recycling

Reservevesicles

Postsynapticreceptors

Transmitter

Actionpotential

[Ca ]2+

PSCReleasemachinery

Fig 7.1 Schematic of a chemicalsynapse. In this example, thepresynaptic terminal consists ofa single active zone containing areadily-releasable vesicle pool(RRVP) which is replenishedfrom a single reserve pool. Apresynaptic action potential (AP)leads to calcium entry throughvoltage-gated calcium channelswhich may result in a vesicle inthe RRVP fusing with thepresynaptic membrane andreleasing neurotransmitter intothe synaptic cleft.Neurotransmitter diffuses in thecleft and binds with postsynapticreceptors which then open,inducing a postsynaptic current(PSC).

which on release may activate a corresponding pool of postsynaptic recep-tors (Walmsley et al., 1998). The RRVP is replenished from a large reservepool. The reality is likely to be more complex than this with vesicles in theRRVP possibly consisting of a number of subpools, each in different states ofreadiness (Thomson, 2000b). Recycling of vesicles may also involve a numberof distinguishable reserve pools (Rizzoli and Betz, 2005; Thomson, 2000b).

A model of such a synapse could itself be very complex. The first stepin creating a synapse model is identifying the scientific question we wish toaddress. This will affect the level of detail that needs to be included. Verydifferent models will be used if our aim is to investigate the dynamics of aneural network involving thousands of synapses compared to exploring theinfluence of transmitter diffusion on the time course of a miniature EPSC.In this chapter, we outline the wide range of mathematical descriptions thatcan be used to model both chemical and electrical synapses. We start withthe simplest models that capture the essence of the postsynaptic electricalresponse, before gradually including increasing levels of detail.

7.2 The Postsynaptic Response

The aim of a synapse model is to describe accurately the postsynaptic re-sponse generated by the arrival of an action potential at a presynaptic termi-nal. We assume that the response of interest is electrical, but it could equallybe chemical, such as an influx of calcium or the triggering of a second mes-senger cascade. For an electrical response, the fundamental quantity to bemodelled is the time course of the postsynaptic receptor conductance. Thiscan be captured by simple phenomenological waveforms, or by more com-plex kinetic schemes that are analogous to the models of membrane-boundion channels discussed in Chapter ??.

7.2.1 Simple conductance waveformsThe electrical current that results from the release of a unit amount of neu-rotransmitter at time ts is, for t ≥ ts:

Isyn(t ) = gsyn(t )(V (t )− Esyn) (7.1)

where the effect of transmitter binding to and opening postsynaptic recep-tors is a conductance change gsyn(t ) in the postsynaptic membrane. V is thevoltage across the postsynaptic membrane and Esyn is the reversal potential

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THE SYNAPSE 3

t (ms) 20 64 8

(a) (b) (c)

t (ms) 20 64 8

0

co

nd

ucta

nce

0.4

0.6

0.8

1.0

0.2

t (ms) 20 64 8

Fig 7.2 Three waveforms forsynaptic conductance: (a) singleexponential decay with τ = 3 ms,(b) alpha function with τ = 1 ms,and (c) dual exponential withτ1 = 3 ms and τ2 = 1 ms.Response to a single presynapticactional potential arriving attime=1 ms. All conductances arescaled to a maximum of 1(arbitrary units).

of the ion channels that mediate the synaptic current. Simple waveforms areused to describe the time course of the synaptic conductance gsyn(t ) for thetime after the arrival of a presynaptic spike, t ≥ ts. Three commonly usedwaveform equations are illustrated in Figure 7.2, and in the following orderare, (a) single exponential decay, (b) alpha function (Rall, 1967) and (c) dualexponential:

gsyn(t ) = g syn exp

�

−t − ts

τ

�

gsyn(t ) = g syn

t − ts

τexp

�

−t − ts

τ

�

gsyn(t ) = g syn

τ1τ2

τ1− τ2

�

exp

�

−t − ts

τ1

�

− exp

�

−t − ts

τ2

��

The alpha and dual exponential waveforms are a more realistic repre-sentation of the conductance change at a typical synapse and good fits ofEquation 7.1 using these functions for gsyn can often be obtained to recordedsynaptic currents. The dual exponential is needed when the rise and fall timesmust be set independently.

Response to a train of action potentials

If it is required to model the synaptic response to a series of transmitterreleases due to the arrival of a stream of action potentials at the presynapticterminal, then the synaptic conductance is given by the sum of the effectsof the individual waveforms resulting from each release. For example, if thealpha function is used then for the time following the arrival of the nth spike(t > tn ):

gsyn(t ) =n∑

i=1

g syn

t − ti

τexp

�

−t − ti

τ

�

(7.2)

where the time of arrival of each spike i is ti . An example of the response toa train of releases is shown in Figure 7.3.

A single neuron may receive thousands of inputs. Efficient numericalcalculation of synaptic conductance is often crucial. In a large scale networkmodel, calculation of synaptic input may be the limiting factor in the speedof simulation. The three conductance waveforms considered are all solutionsof the impulse response of a damped oscillator, which is given by the second-order ODE for the synaptic conductance:

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4 THE SYNAPSE

t (ms) 200 6040 80

0

co

nd

ucta

nce

0.4

0.6

0.8

1.0

0.2

100

Fig 7.3 Alpha functionconductance with τ = 10 msresponding to action potentialsoccurring at 20, 40, 60 and 80 ms.Conductance is scaled to amaximum of 1 (arbitrary units).

τ1τ2

d 2 g

d t 2+ (τ1+ τ2)

d g

d t+ g = g syn x(t ) (7.3)

The function x(t ) represents the contribution from the stream of trans-mitter releases. It take the value 1 if a release occurs at time t and is zerootherwise. The single exponential form occurs when τ1 = 0 and the alphafunction when τ1 = τ2 = τ.

This ODE can be integrated using a suitable numerical integration rou-tine to give the synaptic conductance over time (Protopapas et al., 1998)in a way that does not require storing spike times or the impulse responsewaveform (both of which are required for solving equation 7.2; a methodfor handling equation 7.2 directly that does not require storing spike timesand is potentially faster and more accurate than numerically integrating theimpulse response is proposed in Srinivasan and Chiel (1993)).

Voltage dependence of response

These simple waveforms describe a synaptic conductance that is independentof the state of the postsynaptic cell. Certain receptor types are influenced bymembrane voltage and molecular concentrations. For example, NMDA re-ceptors are both voltage sensitive and are affected by the level of extracellularmagnesium. The basic waveforms can be extended to capture these sort ofdependencies. For example (Mel, 1993):

gNMDA(t ) = g syn

exp(−(t − ts)/τ1)− exp(−(t − ts)/τ2)

(1+µ[Mg2+] exp(−γV ))(7.4)

where µ and γ set the magnesium and voltage dependencies, respectively.In this model the magnesium concentration, [Mg2+] is usually set at a pre-determined, constant level (e.g. 1 mM). The voltage V , is the postsynapticmembrane potential, which will vary with time.

7.2.2 Kinetic schemesA significant limitation of the simple waveform description of synaptic con-ductance is that it does not capture the actual behaviour seen at manysynapses when trains of action potentials arrive. A new release of neurotrans-mitter soon after a previous release should not be expected to contribute asmuch to the postsynaptic conductance due to saturation of postsynaptic re-ceptors by previously released transmitter and the fact that some receptorswill be already open. Certain receptor types also exhibit desensitization that

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THE SYNAPSE 5

co

nd

ucta

nce

t (ms)

62 840

t (ms)

co

nce

ntr

atio

n

0

0.4

0.6

0.8

1.0

0.2

(a) (b)

62 840

0

0.4

0.6

0.8

1.0

0.2

Fig 7.4 Response of the simple2-gate kinetic receptor model toa single pulse of neurotransmitterof amplitude 1 mM and duration1 ms. Rates are α = mM−1ms−1

and β = ms−1. Conductancewaveform scaled to an amplitudeof 1 and compared with an alphafunction with τ = 1 ms (dottedline).

prevents them (re)opening for a period after transmitter-binding, in the sameway that the sodium channels underlying the action potential inactivate. Tocapture these phenomena successfully, kinetic (Markov) models can be used.Here we outline this approach. More detailed treatments can be found in thework of Destexhe et al. (1994b, 1998).

Basic model

The simplest kinetic model is a two state scheme in which receptors can beeither closed, C, or open, O, and the transition between states depends ontransmitter concentration, [T], in the synaptic cleft:

Cα[T]−+)−β

O

where α and β are voltage-independent forward and backward rate con-stants. For a pool of receptors, states C and O can range from 0 to 1 anddescribe the fraction of receptors in the closed and open states, respectively.The synaptic conductance is:

gsyn(t ) = g synO(t )

A complication of this model compared to the simple conductance wave-forms discussed above is the need to describe the time course of transmitterconcentration in the synaptic cleft. One approach is to assume that each re-lease results in an impulse of transmitter of a given amplitude, Tmax, andfixed duration. This enables easy calculation of synaptic conductance withthe two state model (see Box 7.1 for details). An example response to such apulse of transmitter is shown in Figure 7.4. The response of this scheme to atrain of pulses at 100 Hz is shown in Figure 7.6(a). However, more complextransmitter pulses may be needed, as discussed below.

The neurotransmitter transientThe neurotransmitter concentration transient in the synaptic cleft follow-ing release of a vesicle is characterised typically by a fast rise time followedby a decay that may exhibit one or two time constants, due to transmitteruptake and diffusion of transmitter out of the cleft (Clements et al., 1992;Destexhe et al., 1998; Walmsley et al., 1998). This can be described by thesame sort of mathematical waveforms (e.g. alpha function) used to model

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6 THE SYNAPSE

Box 7.1 Solving the two state modelThe basic 2-state kinetic scheme (equation 7.2.2), is equivalent to an ordi-nary differential equation (ODE), in which the rate of change of O is equalto the fraction converted from state C minus the fraction converted from O

to C :dO

dt= α[T ](1 − O) − βO (a)

where O + C = 1.If the neurotransmitter transient is modelled as a square-wave pulse with

amplitude, Tmax, numerical integration of this ODE yields (Destexhe et al.,1994a, 1998):

Ot+1 = O∞ + (Ot − O∞) exp(

−∆t

τO

)if [T ] > 0 (b)

Ot+1 = Ot exp(−β∆t) if [T ] = 0 (c)

for time step ∆t. In the presence of neurotransmitter the fraction of openreceptors approaches O∞ = αTmax/(αTmax + β) with time constant τO =1/(αTmax + β).

the postsynaptic conductance itself (see Section 7.2.1). However, a simplesquare-wave pulse for the neurotransmitter transient is often a reasonableapproximation for use with a kinetic model of the postsynaptic conductance(Destexhe et al., 1994a, 1998), as illustrated above.

For many synapses, or at least individual active zones, it is highly likelythat at most a single vesicle is released per presynaptic action potential (Red-man, 1990; Thomson, 2000b). This makes the use of simple phenomenolog-ical waveforms for the transmitter transient both easy and sensible. How-ever, some synapses can exhibit multivesicular release at a single active zone(Wadiche and Jahr, 2001). The transmitter transients due to each releasedvesicle must then be summed to obtain the complete transient seen by thepostsynaptic receptor pool. This is perhaps most easily done when a continu-ous function (such as the alpha function) is used for the individual transient,with the resulting calculations being the same as those required for gener-ating the postsynaptic conductance due to a train of action potentials (seeSection 7.2.1).

A further complication is spillover of transmitter from neighbouring ac-tive zones. To compensate for this requires consideration of the spatial ar-rangement of active zones and the likely contribution of spillover due todiffusion (Barbour and Häusser, 1997; Destexhe and Sejnowski, 1995). Thiscan be described by a delayed, and smaller amplitude transmitter transientthat also must be summed together with all other transients at a particularactive zone. A synapse containing multiple active zones with spillover ofneurotransmitter between zones is illustrated in Figure 7.5.

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THE SYNAPSE 7

Actionpotential

PSC

RRVP

Activezones

Spillover

Fig 7.5 Schematic of achemical synapse with multipleactive zones. Single vesicles arereleasing at the lower two activezones, resulting in spillover ofneurotransmitter between zones.

More detailed modelsPostsynaptic conductances often exhibit more complex dynamics than canbe captured by the simple scheme used above. EPSCs may contain fast andslow components, and may decline in amplitude on successive presynapticpulses due to desensitization of the postsynaptic receptors. These factors canbe captured in kinetic schemes by adding further closed and open states, aswell as desensitized states (Destexhe et al., 1994b, 1998).

A basic 5-gate kinetic scheme that includes receptor desensitization is:

C0

Rb[T]−−+)−−

Ru1

C1

Rb[T]−−+)−−

Ru2

C2

Ro−+)−Rc

O (7.5)

Rd

−+ )−

Rr

D

where the binding of two transmitter molecules is required for opening, andfully bound receptors can desensitize (state D) before opening. An exam-ple of how such a scheme may be translated into an equivalent set of ordi-nary differential equations (ODEs) is given in Section ??. The response ofthis scheme to 100 Hz stimulation, obtained by numerically integrating theequivalent ODEs, is shown in Figure 7.6(b). The EPSC amplitude declineson successive stimuli due to receptor desensitization.

Variations on this scheme have been used to describe AMPA, NMDAand GABAA receptor responses (Destexhe et al., 1994b, 1998). More com-plex schemes include more closed, open and desensitized states to match the

t (ms)

6020 80400

t (ms)

EP

SC

(p

A)

(a) (b)

−60

−40

−20

0

100 6020 80400 100

Fig 7.6 Postsynaptic current inresponse to 100 Hz stimulationfrom (a) 2-gate kinetic receptormodel (α = 4 mM−1ms−1,β = 1 ms−1); (b) 5-gatedesensitizing model(Rb = 13 mM−1ms−1,Ru1 = 0.3 ms−1, Ru2 = 200 ms−1,Rd = 10 ms−1, Rr = 0.02 ms−1,Ro = 100 ms−1, Rc = 1 ms−1).Each presynaptic actionpotential is assumed to result inthe release of a vesicle ofneurotransmitter, giving asquare-wave transmitter pulseamplitude of 1 mM and durationof 1 ms. The current is calculatedas Isyn(t) = gsyn(t)(V (t) − Esyn).Esyn = 0 mV and cell is clampedat −65 mV. A conductanceapproaching 0.8 nS is reached onthe first pulse.

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8 THE SYNAPSE

experimental time course of postsynaptic current responses to applied andevoked neurotransmitter pulses. Transition rates may be constant, sensitiveto ligands such as neurotransmitter or neuromodulators, or voltage sensitive.

Models of metabotropic receptorsThe kinetic schemes discussed above are suitable for modelling the responseof ionotropic receptors in which the current-carrying ion channels are di-rectly gated by the neurotransmitter. Other receptors, such as GABAB, aremetabotropic receptors that gate remote ion channels through second mes-senger pathways. Kinetic schemes can also be used to describe this type ofbiochemical system. For example, the GABAB response has been modelledas (Destexhe and Sejnowski, 1995; Destexhe et al., 1998):

R0+T−+)−R−+)− D

R+G0−+)−RG−→R+G

G−→G0

C+nG−+)−O

where the receptors enter activated, R, and desensitized, D, states whenbound by transmitter, T. G-protein enters an activated state, G, catalyzedby R, which then proceeds to open the ion channels.

7.3 Presynaptic Neurotransmitter Release

Any postsynaptic response is dependant upon neurotransmitter being re-leased from the presynaptic terminal. In turn, this depends on the availabil-ity of releasable vesicles of neurotransmitter and the likelihood of such avesicle releasing due to a presynaptic action potential. A complete model ofsynaptic transmission needs to include terms describing the release of neuro-transmitter.

The simplest such model, which is commonly used when simulating neu-ral networks, is to assume that a single vesicle releases its quantum of neu-rotransmitter for each action potential that arrives at a presynaptic terminal(for example, see Figure 7.3). In practice, this is rarely a good model forsynaptic transmission at any chemical synapse. A better description is thatthe average release is given by n p where n is the number of releasable vesi-cles and p is the probability that any one vesicle will release. Both n and pmay vary with time, resulting in either facilitation or depression of release,and hence the postsynaptic response. Such short-term synaptic plasticity op-erates on the time scale of milliseconds to seconds and comes in a variety offorms with distinct molecular mechanisms, largely controlled by presynap-tic calcium levels (Magleby, 1987; Thomson, 2000a,b; Zucker, 1989, 1999;Zucker and Regehr, 2002)

We consider how to model n and p. We present relatively simple mod-els that describe a single active zone, with a readily-releasable vesicle pool(RRVP) that is replenished from a single reserve pool. Two alternatives thatspecify that an active zone contains either an unlimited or a limited numberof release sites are described, as illustrated in Figure 7.7.

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THE SYNAPSE 9

RRVPReservevesicles

p

n

p

p

kr

nk

(a) Vesicle−state model

RRVPReservevesicles

p

n<=n

p

p

kr

nk

(b) Release−site model

(n −n)T

T

Fig 7.7 Two models of vesiclerecycling and release at anactive zone: (a) vesicle-statemodel in which the number ofvesicles in the RRVP is limitedonly by vesicle recycling andrelease rates; (b) release-sitemodel in which there is a limitednumber of release sites. n is thenumber of vesicles available forrelease; p is the releaseprobability of a vesicle; kn is thearrival rate from the reserve pool;kr is the return rate to thereserve pool.7.3.1 Vesicle release

Firstly we consider models for p, the probability that a vesicle will be re-leased. There is a direct relationship between p and presynaptic calcium lev-els. Release probability is determined by the calcium influx through one,or very few, voltage-gated calcium channels close to a release site, with pvarying according to a power (usually 3 to 4) of the calcium concentra-tion (Redman, 1990; Thomson, 2000b; Zucker and Fogelson, 1986). Mostrelease occurs in synchrony with the arrival of a presynaptic action poten-tial, which results in a rapid influx of calcium. Residual calcium levels be-tween action potentials can result in spontaneous asynchronous release andmay enhance release (facilitation and augmentation) on subsequent actionpotentials (Thomson, 2000a,b; Zucker, 1999).

Complex models can be conceived that include explicitly the volt-age waveform of the action potential, the accompanying calcium currentthrough voltage-gated calcium channels, the associated change in intracellu-lar calcium concentration, and then calcium-activated signalling pathwaysthat eventually result in transmitter release. Unless it is the release processitself that is the subject of study, this level of detail is unnecessary. Here weuse simpler approaches which describe the vesicle release probability as adirect function of a stereotypical action potential or the resultant calciumtransient.

Phenomenological model of facilitation

At many synapses, trains of presynaptic action potentials cause the releaseprobability to increase in response to the arrival of successive spikes, dueto residual calcium priming the release mechanism (Del Castillo and Katz,1954; Katz and Miledi, 1968; Thomson, 2000a,b; Zucker, 1974). Releaseprobability returns to baseline with a time constant of around 50-300 ms(Thomson, 2000a).

A simple phenomenological model which can capture the amplitude andtime course of experimentally-observed facilitation (Tsodyks et al., 1998) in-crements the probability of release, p, by an amount ∆p at each action po-tential, with p decaying back to baseline p0 with a fixed time constant τf:

d p

d t=−(p − p0)

τf

+∑

s

∆p.(1− p)δ(t − ts) (7.6)

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10 THE SYNAPSE

t (ms) 1000 300200 400

0

pro

ba

bili

ty

0.2

0.3

0.4

0.5

0.1

500

Fig 7.8 Facilitation of releaseprobability in the basicphenomenological model.Stimulation at 50 Hz withp0 = 0.1, ∆p = 0.1 andτf = 100 ms.

where the Kronecker delta function δ(t − ts) = 1 at spike time t = ts and iszero otherwise. Only the times of presynaptic action potentials, ts, appearexplicitly in the model. The resultant calcium influx implicitly determinesp, but is not explicitly modelled.

For an interspike interval of ∆t , p will return towards p0 by the frac-tional amount, 1− exp(−∆t/τf), from its value immediately following theprevious action potential. This leads to an expression for the probability ofrelease at the nth spike in terms of the probability at the (n− 1)th spike:

pn = p+n−1+ (1− exp(−∆t/τf))(p0− p+

n−1)

where

p+n−1= pn−1+∆p.(1− pn−1)

is the release probability attained just following the previous action poten-tial due to facilitation. For a stream of spikes at a constant rate, the releaseprobability reaches a steady-state value per spike of:

p∞ =p0(1− exp(−∆t/τf))+∆p. exp(−∆t/τf)

1− (1−∆p)exp(−∆t/τf)

An example of facilitation in release is shown in Figure 7.8. Though p iscalculated as a continuous function of time, the model is strictly only usedto determine the release probability of a vesicle at the time of arrival of eachpresynaptic action potential.

In this model there is no attempt made to provide a causal mechanismfor the facilitation of release. Other slightly more complex models have at-tempted to relate the magnitude and time course of facilitation explicitly tocalcium entry (Dittman et al., 2000; Trommershäuser et al., 2003).

Kinetic gating scheme

A different approach to modelling p attempts to represent the interactionbetween calcium and the release machinery by using a kinetic scheme drivenby an explicit calcium concentration to capture the time course of release, inthe same way that such schemes driven by the neurotransmitter transient canbe used to describe the postsynaptic conductance change (see Section 7.2.2).A basic scheme (Bertram et al., 1996) specifies p as the product of the pro-portions of open states of a number N (typically 2 to 4) of different gates:

p(t ) =O1(t )×O2(t ) . . .×ON (t )

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THE SYNAPSE 11

3000 100

t (ms)

Pro

ba

bili

ty

(c)

(e)

(b)

0

1.0

0.5

0

1.0

0.5

198

t (ms)

0

1.0

0.5

(d)

(f)0

1.0

0.5

0

1.0

0.5

0

1.0

0.5

(a)

200

Ga

te 1

Op

en

Ga

te 2

Op

en

204200 202

Fig 7.9 Facilitation oftransmitter release in a kinetic2-gate model. Left-hand columnshows stimulation at 50 Hz.Right-hand column showsrelease and gating transients atthe tenth pulse. Fast gate:k+

1 = 200 mM−1ms−1,k−

1 = 3 ms−1; Slow gate:k+

2 = 0.25 mM−1ms−1,k−

2 = 0.01 ms−1; square-wave[Ca2+]r pulse: 1 mM amplitude,1 ms duration.

where O j (t ) is the proportion of open states of gate j , which is determinedby the local calcium concentration at the release site:

C j

k+j[Ca2+]r−−−−+)−−−−

k−j

O j

This requires a model of the calcium transient, [Ca2+]r, seen by the releasesite in response to the arrival of a presynaptic action potential. For manypurposes a simple stereotypical waveform, such as a brief square-wave pulse,is sufficient. This model is mathematically identical to the simple kineticscheme for postsynaptic conductance considered earlier (Section 7.2.2.) Morecomplex models of the effect of the local calcium concentration, that includethe accumulation of residual calcium, may also be used.

An example of the time course of p(t ) =O1O2 resulting from a two-gate model, with one fast gate and one slow gate, is shown in Figure 7.9.Fast gates determine the total release probability attained for each action po-tential. They will close completely between action potentials. Slower gatesmay remain partially open between action potentials and gradually summateto achieve higher open states on successive action potentials. This results infacilitation of the probability of release obtained on successive action poten-tials.

This model is actually specifying the average rate of vesicle release as afunction of time, rather than the probability that a vesicle will release. Thisrate can be turned into a probability that a vesicle will release by determiningthe area under the curve, p, in the interval [t , t +∆t], for some small timeinterval ∆t . This gives the probability, p∗, that a vesicle will release duringthat interval. A crude, but useful approximation is:

p∗ = p(t )∆t

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12 THE SYNAPSE

In principle this form of model can describe both release due to presy-naptic action potentials and spontaneous release between action potentials.However, with this gating scheme it is usual that at least one gate will closecompletely between action potentials, reducing r (and hence p) to zero, asillustrated in Figure 7.9. Note that release due to an action potential is spreadrealistically over a short time interval (right-hand colum of Figure 7.9.)

Other factors

These basic models of release can be extended to include other factors thataffect the release probability, such as changes in the calcium transient due tocalcium channel inactivation (Forsythe et al., 1998) or activation of presy-naptic metabotropic glutamate receptors(mGluRs). For example, mGluRactivation has been modelled as a decrease by a small amount ∆pb in thebaseline release probability pb following each action potential (Billups et al.,2005):

d pb

d t=

p0− pb

τb

−∑

s

∆pb.pbδ(t − ts)

This baseline recovers slowly with time constant τb (on the order of sev-eral seconds) to the initial release probability p0. Baseline release pb is usedin place of p0 in Equation 7.6 when determining the spike-related releaseprobability p.

7.3.2 Vesicle availabilityThe second part of the presynaptic model concerns the number n of vesi-cles available for release at any particular time i.e. the size of the readily-releasable vesicle pool (RRVP). The average size of this pool is determinedby the average rate of vesicle release and the rate at which the pool is re-plenished from reserve pools and vesicle recycling. Depletion of this poolhas long been identified as a major component of synaptic depression (Betz,1970; Del Castillo and Katz, 1954; Elmqvist and Quastel, 1965; Kusano andLandau, 1975; Liley and North, 1953; Thomson, 2000b).

Continuous modelsA basic model considers the RRVP, Vp, to be replenished at rate kn from asingle reserve pool, Vr, containing nr vesicles. The n vesicles in the RRVPmay spontaneously return to the reserve pool at rate kr. Exocytosis (vesiclerelease) occurs at rate p, which typically reaches a brief maximum followingarrival of a presynaptic action potential. This is described by the kineticscheme:

Vr

kn−+)−kr

Vp

p−→T (7.7)

This is an example of a vesicle-state model (Gingrich and Byrne, 1985;Heinemann et al., 1993; Weis et al., 1999), as vesicles may be either in re-serve, release-ready, or released, and the size of the RRVP is not limited(Figure 7.7a).

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THE SYNAPSE 13

For simplicity, we assume that the number of vesicles in the RRVP andreserve pool is sufficiently large that we can treat n and nr as real num-bers, representing the average number of vesicles. This is reasonable for largesynapses, such as the neuromuscular junction and giant calyceal synapses inthe auditory system. It could also represent the average over many trials at asingle central synapse, or the average over many synapses onto a single cell.The kinetic scheme (Equation 7.7) is then equivalent to the ODE:

d n

d t= knnr− krn− n p

A further simplification is to assume that the reserve pool is so largethat nr is effectively constant over the stimulation period. The model thenreduces to:

d n

d t= k∗

n− krn− n p (7.8)

with constant replenishment rate k∗n= knnr.

If no action potentials arrive at the synapse (p(t ) = 0 for all t ) the size ofthe RRVP reaches a steady-state value of:

n∞ = k∗n/kr (7.9)

If a vesicle release takes place with constant probability ps on the arrivalof each presynaptic action potential, the rate of vesicle release is then:

p(t ) = psδ(t − ts)

for a spike arriving at time ts, with p(t ) = 0 for t 6= ts. For action potentialsarriving with a constant frequency f , the release rate averaged over time is:

p(t ) = f ps

Substituting this value into Equation 7.8 and setting the rate of change in nto be zero, we see that the synaptic stimulation rate f results in steady-stateRRVP size being decreased to:

n∞ = k∗n/(kr + f ps)

compared to the resting size (Equation 7.9). Increasing stimulation frequencyf depletes the RRVP further (n∞ decreases as f increases), resulting in adepression of the steady-state postsynaptic response, which is proportionalto n∞ p.

A typical example from this model is illustrated, in combination withthe phenomenological facilitation model for release probability p (Equa-tion 7.6), in Figure 7.10 (left-hand column; solid lines).

An alternative formulation of the n p active zone model assumes that thenumber of release sites in an active zone is physically limited to a maximumsize, nmax (Figure 7.7b). This release-site model (Dittman and Regehr, 1998;Matveev and Wang, 2000; Weis et al., 1999; Vere-Jones, 1966) has differentsteady-state and dynamic characteristics from the vesicle-state model. Eachrelease site can either contain a vesicle (state Vp) or be empty (state Ve).A site may lose its vesicle through transmitter release, or via spontaneous

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14 THE SYNAPSE

Vesicle−state Release−site

3000 100

t (ms)

p

(b)

0

1.0

0.5

0

0.2

0.1

198

t (ms)

0

0.4

0.2

0

1.0

0.5

0

0.4

0.2

0

0.4

0.2

(a)

200

nT

=n

p

204200 202400

Fig 7.10 Facilitationdepression in deterministicmodels of short-termdynamics. Phenomenologicalmodel of facilition:∆p = 0.05; τf = 100(a) Vesicle-state model:k∗

n = kr = 0.2 s−1.(b) Release-site model:kn = kr = 0.2 s−1. Solidns = 0; dotted lines:Synapse stimulated

undocking and removal to the reserve pool. An empty site can be refilledfrom the reserve pool. The state of a release site is described by the kineticscheme:

Ve

kn−+)−k∗

r

Vr

where k∗r= p+ kr is the sum of the rate of vesicle release and the rate of

spontaneous removal of vesicles to the reserve pool. Empty sites becomefilled with release-ready vesicles from an effectively infinite reserve pool atrate kn. Given a total of nmax release sites, the size of the RRVP over time isdescribed by the ODE:

d n

d t= kn(nmax− n)− k∗

rn

In the absence of presynaptic activity (p(t ) = 0), the size of the RRVPreaches a steady-state of:

n∞ = knnmax/(kn+ kr)

which is simply nmax if vesicles do not spontaneously undock (kr = 0).For stimulation with constant frequency f , the steady-state decreases to:

n∞ = knnmax/(kn+ kr+ f ps)

An example for this model is shown in Figure 7.10 (right-hand column;solid lines). Note that if n is significantly less than nmax then the term:

kn(nmax− n)≈ knnmax = k∗n

is constant and this model is now equivalent to the vesicle-state model.So far we have assumed that the refilling rate, kn, of the RRVP is constant.

The reality is likely to be rather more complex, with kn being a direct, or

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THE SYNAPSE 15

indirect function of the presynaptic calcium concentration (Gingrich andByrne, 1985; Dittman et al., 2000; Trommershäuser et al., 2003; Weis et al.,1999). In particular, the rate of filling of the RRVP may be dependent onpresynaptic activity. At the neuromuscular junction (Worden et al., 1997)and the calyx of Held (Wong et al., 2003), recorded postsynaptic responsesare well fit if it is assumed that a fraction (or number) of new vesicles, ns, areadded to the RRVP following each presynaptic action potential.

For a regular stimulation frequency f , the release-site model is now:

d n

d t= kn(nmax− n)− krn+ (ns(nmax− n)− n ps) f

The vesicle-state model, in which there is no hard limit to the size of theRRVP, now becomes:

d n

d t= k∗

n− krn+ (ns− n ps) f

The effect of ns on the model responses is shown in Figure 7.10 (dottedlines). The activity-dependent recovery of the RRVP increases the steady-state size of the RRVP for both models. For high frequency stimulation ofthe vesicle-state model the size of the RRVP approaches the steady value:

n∞ = (k∗n+ f ns)/(kr+ f ps)≈ ns/ps

This steady state may be less than (depletion) or greater than (facilitation)the steady state RRVP size in the absence of activity (k∗

n/kr), depending on

the level of activity-dependent replenishment (ns).Many variations and extensions to these model components are possi-

ble, including one or more finite-sized reserve pools which may deplete, andmore complex models of vesicle recycling and refilling of the RRVP.

Stochastic models

In the models discussed so far, we have been treating the number of presy-naptic vesicles as a continuous variable representing an average over severalspatially-distributed synapses that are synchronously active, or over a num-ber of trials at a single synapse (Dittman et al., 2000; Markram et al., 1998;Tsodyks and Markram, 1997). It is also possible to treat n as a count of thenumber of vesicles in the RRVP, with release of each vesicle being stochasticwith probability p (Fuhrmann et al., 2002; Matveev and Wang, 2000; Vere-Jones, 1966). This latter case is computationally more expensive but repre-sents more realistically the result of a single trial at typical central synapsesfor which the number of vesicles available for release at any time is limited,and failures of release can be common. This treatment also allows determi-nation of the variation in synaptic response on a trial-by-trial basis.

Instead of treating kinetic models for the size of the RRVP, n, as beingequivalent to continuous ODEs, they can be turned into discrete stochasticmodels by treating n as an integer representing the exact number of vesiclescurrently in the RRVP. The rates of vesicle recycling (kn and kr) and release(p) determine the probability that a new vesicle will enter (or leave) theRRVP in some small time interval, ∆t .

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16 THE SYNAPSE

Box 7.2 Simulating a stochastic synapse modelMonte Carlo simulations of a model can be carried out using a per-time-stepmethod, as outlined previously (Section ??) for general kinetic schemes. Thealgorithm is as follows.

1. Time is divided into sufficiently small bins of duration ∆t, so that theprobability of a vesicle entering or leaving the RRVP during ∆t is sig-nificantly less than 1.

2. During each small time interval ∆t, the arrival of a new vesicle from thereserve pool has probability pa:

Vesicle-state model: pa = 1 − exp(−k∗n ∆t) ≈ k∗

n ∆t

Release-site model: pa ≈ kn(nmax − n)∆t

Arrivals are generated by testing whether a uniform random number inthe interval [0,1] is less than or equal to pa. If it is, n is incremented by1.

3. Similarly, spontaneous undocking of a vesicle from the RRVP is testedby generating n uniform random numbers and checking to see if eachnumber is less than or equal to kr∆t and decrementing n by one if so.

4. If a presynaptic action potential occurs in a particular time interval ∆t,then each of the remaining n vesicles is tested for release against releaseprobability ps, with a release occurring if a uniform random number inthe interval [0,1] is less than or equal to ps.

Computer simulation of this type of model is carried out using MonteCarlo techniques. An algorithm is detailed in Box 7.2. A single simulation isequivalent to a single experimental trial at a real synapse. Results from multi-ple simulations can be averaged to give equivalent results to the deterministicODE models. Single and average responses from a stochastic version of thevesicle-state model are shown in Figure 7.11. The average response matchesthe output of the deterministic model.

7.4 Complete Synaptic Models

A complete model of a synapse relates the arrival of the presynaptic ac-tion potential with a postsynaptic response. We have described a numberof approaches to modelling the pre- and postsynaptic mechanisms involved.These different approaches may be mixed-and-matched to produce a com-plete model which contains sufficient detail for the purpose. If the synapticmodel is used in a large neural network involving thousands of synapses,the simplest synaptic model will be sufficient. More complex models will bedesirable and necessary if we wish to examine the short-term dynamics of aparticular input pathway onto a single cell.

A distinguishing feature of complete models is whether the postsynapticmechanism is driven by (a) the time of arrival of a presynaptic spike, (b)

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THE SYNAPSE 17

3000 100

t (ms)

p(b)

0

2.0

1.0

0

0.4

0.2

t (ms)

0

0.4

0.2

0

2.0

1.0

0

1.0

0.5

0

0.4

0.2

(a)

200

nT

=n

p

400 3000 100 200 400

Single trial Average response7.11 Stochasticvesicle-state model of short-termdynamics at a single synaptic

zone. (a) A singlesimulation run, showing the

probability p, the actualof releasable vesicles, n

(initially 1), and neurotransmitterrelease, T . (b) Average values of

variables, taken over 10000Phenomenological model

facilitation: p0 = 0.2;.05; τf = 100 ms.

esicle-state model:= 0.2 s−1; ns = 0.1.

Synapse stimulated at 50Hz.

the average amount of transmitter released due to a presynaptic spike, or (c)the transmitter transient due to the precise number of vesicles released by aspike.

The kinetic schemes for the postsynaptic conductance are driven by anexplicit neurotransmitter transient. These postsynaptic models can be cou-pled with the presynaptic models of the RRVP, n, and release probability, p,by producing a transmitter transient due to release, T= n p. If releases due toa single presynaptic spike are assumed to be synchronous, then the parameterT can be used as a scaling factor for the amplitude of a predefined transmit-ter waveform (assuming interspike intervals are sufficiently large that succes-sive waveforms will not overlap.) If release is asynchronous, or transmitterwaveforms from successive spikes may overlap in time, then the transmit-ter transients due to individual releases will need to be summed to give thetransmitter concentration over time.

Figure 7.12 shows the output of an example model based on the giantcalyx of Held (Wong et al., 2003) that combines the stochastic version of thevesicle state model for vesicle recycling and release (Fig. 7.11) with a sim-ple 2-gate kinetic scheme for the AMPA receptor response (Fig. 7.6). Thisshows the summed EPSCs due to 500 independent active zones which con-tain on average a single releasable vesicle. Note the trial-to-trial variation dueto stochastic release and the interplay between facilitation of release and de-pletion of available vesicles.

If the trial-to-trial variation in the postsynaptic response is of interest, inaddition to using a stochastic model for vesicle recycling and release, vari-ations in quantal amplitude (the variance in postsynaptic conductance onrelease of a single vesicle of neurotransmitter) can be included. This is doneby introducing variation in the amplitude of the neurotransmitter transientdue to a single vesicle, and/or variation in the maximum conductance thatmay result (Fuhrmann et al., 2002).

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18 THE SYNAPSE

(a) (b) (c)

t (ms) 0 50 100

EP

SC

(n

A)

−6

−4

−2

0

−8

t (ms) 0 50 100 0 50 100

t (ms)

Fig 7.12 Three distincta complete synapse500 independent activeEach active zone usesstochastic vesicle-stateshort-term synapticcombined with themodel of AMPA receptors.Phenomenologicalfacilitation: p0 = 0.

τf = 100 ms. Vesicle-statek∗

n = kr = 0.2 s−1; n

kinetic receptor model:α = 4 mM−1ms−1, β

Esyn = 0 mV, with postsynapticcell clamped at −65stimulated at 100 Hz

7.5 Long Lasting Synaptic Plasticity

The models detailed above incorporate aspects of short-term plasticity atsynapses, such as variability in the availability of releasable vesicles and theprobability of their release. Learning and memory in the brain is hypoth-esised to at least in part be mediated by longer lasting changes in synapticstrength, known as long-term potentiation (LTP) and long-term depression(LTD). While short-term plasticity largely involves presynaptic mechanisms,longer lasting changes are mediated by increases or decreases in the magni-tude of the postsynaptic response to released neurotransmitter.

Models of LTP/LTD seek to map a functional relationship betweenpre- and postsynaptic activity and changes in the maximum conductanceproduced by the postsynaptic receptor pool, typically AMPARs and NM-DARs. The biology is revealing complex processes that lead to changes inthe state and number of receptor molecules (Ajay and Bhalla, 2005). As out-lined in the previous chapter (see Section ??) these processes involve calcium-mediated intracellular signalling pathways. Detailed models of such path-ways are being developed. However, for use in network models of learn-ing and memory, it is necessary to create computationally simple models ofLTP/LTD that capture the essence of these processes, while leaving out thedetail. In particular, a current challenge is to find the simplest model that ac-counts for experimental data on spike-timing-dependent plasticity (STDP).This data indicates that the precise timing of pre- and postsynaptic signalsdetermines the magnitude and direction of change of synaptic conductance.The presynaptic signal is the arrival time of an action potential, and thepostsynaptic signal is an action potential or a calcium transient.

There have been a large number of attempts at creating STDP modelsof varying levels of complexity. For example, the models of Abarbanel et al.(2003); Badoual et al. (2006); Castellani et al. (2005); Rubin et al. (2005). Toillustrate the general approach, we examine the phenomenological modelsof Badoual et al. (2006). Their simplest model relates pre- and postsynapticspike times directly to changes in synaptic strength, or “weight”, via theequation:

d w j i

d t=∑

k

P (t − t̃ j )δ(t − ti ,k )−∑

l

Q(t − t̃i )δ(t − t j ,l )

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THE SYNAPSE 19

where w j i is the strength (used as a scaling factor for the maximum con-ductance) of a synapse from presynaptic neuron j to postsynaptic neuron i .The sums are over all postsynaptic (ti ,k ) and presynaptic (t j ,l ) spike times.P and Q are exponentially decaying functions of the immediately previouspresynaptic ( t̃ j ) and postsynaptic ( t̃i ) spikes, respectively:

P (t ) = exp(−t/τP ), Q(t ) = exp(−t/τQ)

If a presynaptic spike precedes a postsynaptic spike by a short interval (de-termined by time constant τP ), then function P will result in an increase inthe synaptic weight, or LTP. Conversely, a postsynaptic spike preceding apresynaptic spike will result in a decrease in the weight (LTD) via functionQ . Time constants τP and τQ can be altered to match experimentally deter-mined STDP curves. Other components can be added to incorporate weightsaturation and spike triplet interactions (Badoual et al., 2006).

A more complex model attempts to capture the relationship betweenpostsynaptic calcium transients and changes in synaptic strength (Badoualet al., 2006). Rather than pre- and postsynaptic spike times, the major de-terminant of long-term synaptic changes is the postsynaptic calcium lev-els resulting from pre- and postsynaptic activity via calcium entry throughNMDA- and voltage-gated calcium channels (Ajay and Bhalla, 2005). Boththe magnitude and time course of calcium transients determine the sign andmagnitude of synaptic weight changes. Badoual et al. (2006) present a sim-ple, phenomenological intracellular signalling scheme to model the mappingfrom calcium to weight changes. An enzyme, K, mediates LTP in its acti-vated form, K∗, which results from binding with calcium:

K+ 4Ca2+ k+−+)−k−

K∗

The LTD enzyme, P, is activated by two other enzymes, M and H, that areactivated by calcium and neurotransmitter (T), respectively:

M+Ca2+ m+−+)−m−

M∗

H+Th+−+)−h−

H∗

M∗+H∗+Pp+

−→ P∗

The percentage increase in synaptic weight (maximum conductance) is pro-portional to K∗ and the percentage decrease is proportional to P∗, with thetotal weight change being the difference between these changes. This modelspecifies that the sign and magnitude of a weight change is determined by thepeak amplitude of a calcium transient. It captures the basic phenomena thatlower levels of calcium result in LTD, whereas higher levels result in LTP.

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20 THE SYNAPSE

7.6 Detailed Modelling of Synaptic Components

We have concentrated on models that describe synaptic input for use ineither detailed compartmental models of single neurons, or in neural net-works. Modelling can also be used to gain greater understanding of the com-ponents of synaptic transmission, such as the relationship between presynap-tic calcium concentrations and vesicle recycling and release (Bennett et al.,2000a,b; Yamada and Zucker, 1992; Zucker and Fogelson, 1986), or thespatial and temporal profile of neurotransmitter in the synaptic cleft (Bar-bour and Häusser, 1997; Coggan et al., 2005; Destexhe and Sejnowski, 1995;Franks et al., 2002; Rao-Mirotznik et al., 1998; Smart and McCammon, 1998;Sosinsky et al., 2005), as illustrated in Figure 7.13.

Presynaptic calcium(a)

calcium channels

vesicle

Neurotransmitter transient(b)

vesicle

fusionpore

cleft

Fig 7.13 (a) Spatialdistribution of voltage-gatedcalcium channels with respect tolocation of a docked vesicle.(b) Spatial and temporal profileof neurotransmitter in thesynaptic cleft on release of avesicle.

Such studies typically require modelling molecular diffusion in eithertwo or three dimensions. This can be done using the deterministic andstochastic approaches outlined in Chapter ?? in the context of modellingintracellular signalling pathways. Deterministic models calculate molecu-lar concentrations in spatial compartments and the average diffusion be-tween compartments (Barbour and Häusser, 1997; Destexhe and Sejnowski,1995; Rao-Mirotznik et al., 1998; Smart and McCammon, 1998; Yamadaand Zucker, 1992; Zucker and Fogelson, 1986). Stochastic models track themovement and reaction state of individual molecules (Bennett et al., 2000a,b;Franks et al., 2002). Increasingly, spatial finite element schemes based on 3-dimensional reconstructions of synaptic morphology are being employed(Sosinsky et al., 2005; Coggan et al., 2005). The simulation package MCELL(Stiles and Bartol, 2001) is specifically designed for implementing stochasticmodels based on realistic morphologies (see Appendix ??).

7.7 Gap Junctions

In certain areas of the nervous system, particularly in sensory and motorsystems, neurons may be connected by purely electrical synapses known asgap junctions (for a good description, see Koch (1999)). These electrical con-nections are typically dendrite-to-dendrite or axon-to-axon and are formedby channel proteins that span the membranes of both connected cells (Fig-

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THE SYNAPSE 21

Gap junction(a)

cell 1 cell 2

Junctionchannels

Equivalent circuit(b)

gcV V1 2

cell 1 cell 2

7.14 (a) Schematic of a gapconnection between two

apposed neurites (dendrites orwith (b) the equivalent

electrical circuit.

ure 7.14a). These channels are bidirectionally permeable to ions and othersmall molecules. Their permeability is not much modulated by the local en-vironment, such as membrane potential or molecular concentrations. Thusthey form a much simpler connection between two neurons than do chemi-cal synapses.

Given the fixed permeability of the gap junction channels, the electricalcurrent through a gap junction is modelled as being strictly ohmic, witha fixed, symmetric coupling conductance, gc . Simply, the current flowinginto each neuron is proportional to the voltage difference between the twoneurons at the point of connection (Figure 7.14b):

I1 = gc (V2−V1) (7.10)

I2 = gc (V1−V2) (7.11)

An example of the effect of a gap junction between two axons is shown inFigure 7.15. The gap junction is half-way along the two axons, and an actionpotential is initiated at the start of one axon. If the gap junction is sufficientlystrong then the action potential in the first axon can initiate an action poten-tial in the second axon, which then propagates in both directions along thisaxon.

7.8 Summary

The chemical synapse is a very complex device. Consequently a wide rangeof mathematical models of it can be developed. When embarking on devel-oping such a model it is essential that the questions for which answers willbe sought by the use of the model are clearly delineated.

In this chapter we have presented various models for chemical synapses,ranging from the purely phenomenological to models more closely tied tothe biophysics of vesicle recycling and release and neurotransmitter gating ofpostsynaptic receptors. Along the way we have highlighted when differenttypes of model may be appropriate and useful.

Many of the modelling techniques are the same as those seen in the previ-ous two chapters. In particular, the use of kinetic reaction schemes and theirequivalent ordinary differential equations. Stochastic algorithms have beenused when the ODE approach is not reasonable, such as for the recyclingand release of small numbers of vesicles at an active zone.

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22 THE SYNAPSE

0 5 10−100

−50

0

50(a)

V (

mV

)

0 5 10−66

−64

−62(c)

t (ms)

V (

mV

)

0 5 10−100

−50

0

50(b)

0 5 10−100

−50

0

50(d)

t (ms)

Fig 7.15 Action potentialtravelling along twoare joined by a gaphalf-way along theirMembrane potentialsrecorded at the start,end of the axons. (a,b)

which an action potentialinitiated by a currentinto one end. (c) Otherwith a 1 nS gap junction.(d) Other axon, withjunction. Axons are2 µm in diameter withHodgkin-Huxley sodium,potassium and leak

Finally we considered how subcomponents of the chemical synapse canalso be illuminated by mathematical modelling. Reaction-diffusion systemsat the ODE and stochastic levels are of importance here to investigate presy-naptic calcium transients and the neurotransmitter transient in the synapticcleft.

Mention was also made of modelling electrical gap junctions, which arerather simpler in operation than chemical synapses.

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THE SYNAPSE 23

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