# On the Relation between Bursts and Dynamic Synapse Properties: A Modulation-Based Ansatz.

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**ABSTRACT:**Dual intracellular recordings from pairs of synaptically connected neurones have demonstrated that the frequency-dependent pattern of transmitter release varies dramatically between different classes of connections. Somewhat surprisingly, these patterns are not determined by the class of neurone supplying the axon alone, but to a large degree by the class of postsynaptic neurone. A wide range of presynaptic mechanisms, some that depress the release of transmitter and others that enhance release have been identified. It is the selective expression of these different mechanisms that determines the unique frequency- and pattern-dependent properties of each class of connection. Although the molecular interactions underlying these several mechanisms have yet to be fully identified, the wealth and complexity of the protein-protein and protein-lipid interactions that have been shown to control the release of transmitter suggest many ways in which the properties of a synapse may be tuned to respond to particular patterns and frequencies.Journal of Computational Neuroscience 01/2003; 15(2):159-202. · 2.44 Impact Factor -
##### Article: Bursting neurons signal input slope.

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**ABSTRACT:**Brief bursts of high-frequency action potentials represent a common firing mode of pyramidal neurons, and there are indications that they represent a special neural code. It is therefore of interest to determine whether there are particular spatial and temporal features of neuronal inputs that trigger bursts. Recent work on pyramidal cells indicates that bursts can be initiated by a specific spatial arrangement of inputs in which there is coincident proximal and distal dendritic excitation (Larkum et al., 1999). Here we have used a computational model of an important class of bursting neurons to investigate whether there are special temporal features of inputs that trigger bursts. We find that when a model pyramidal neuron receives sinusoidally or randomly varying inputs, bursts occur preferentially on the positive slope of the input signal. We further find that the number of spikes per burst can signal the magnitude of the slope in a graded manner. We show how these computations can be understood in terms of the biophysical mechanism of burst generation. There are several examples in the literature suggesting that bursts indeed occur preferentially on positive slopes (Guido et al., 1992; Gabbiani et al., 1996). Our results suggest that this selectivity could be a simple consequence of the biophysics of burst generation. Our observations also raise the possibility that neurons use a burst duration code useful for rapid information transmission. This possibility could be further examined experimentally by looking for correlations between burst duration and stimulus variables.Journal of Neuroscience 11/2002; 22(20):9053-62. · 6.91 Impact Factor - SourceAvailable from: Joaquín J Torres[Show abstract] [Hide abstract]

**ABSTRACT:**We study the effect of competition between short-term synaptic depression and facilitation on the dynamic properties of attractor neural networks, using Monte Carlo simulation and a mean-field analysis. Depending on the balance of depression, facilitation, and the underlying noise, the network displays different behaviors, including associative memory and switching of activity between different attractors. We conclude that synaptic facilitation enhances the attractor instability in a way that (1) intensifies the system adaptability to external stimuli, which is in agreement with experiments, and (2) favors the retrieval of information with less error during short time intervals.Neural Computation 11/2007; 19(10):2739-55. · 1.76 Impact Factor

Page 1

Hindawi Publishing Corporation

Computational Intelligence and Neuroscience

Volume 2009, Article ID 658474, 13 pages

doi:10.1155/2009/658474

Research Article

Onthe RelationbetweenBursts and DynamicSynapseProperties:

AModulation-Based Ansatz

ChristianMayr,Johannes Partzsch,and Rene Sch¨ uffny

Chair for Parallel VLSI Systems and Neural Circuits, Dresden University of Technology, 01062 Dresden, Germany

Correspondence should be addressed to Christian Mayr, mayr@iee.et.tu-dresden.de

Received 12 June 2008; Revised 31 January 2009; Accepted 11 March 2009

Recommended by Rodrigo Quiroga

When entering a synapse, presynaptic pulse trains are filtered according to the recent pulse history at the synapse and also with

respect to their own pulse time course. Various behavioral models have tried to reproduce these complex filtering properties. In

particular, the quantal model of neurotransmitter release has been shown to be highly selective for particular presynaptic pulse

patterns. However, since the original, pulse-iterative quantal model does not lend itself to mathematical analysis, investigations

have only been carried out via simulations. In contrast, we derive a comprehensive explicit expression for the quantal model. We

show the correlation between the parameters of this explicit expression and the preferred spike train pattern of the synapse. In

particular, our analysis of the transmission of modulated pulse trains across a dynamic synapse links the original parameters of the

quantal model to the transmission efficacy of two major spiking regimes, that is, bursting and constant-rate ones.

Copyright © 2009 Christian Mayr et al.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.Introduction

The main computational function of artificial neural net-

works has traditionally been modeled as an adjustment of

the coupling weight between neurons. In biological nets, this

coupling weight is provided by the synapse, where an incom-

ing (presynaptic) pulse causes a release of neurotransmitters,

which in turn generate a postsynaptic current (PSC) that

charges the postsynaptic (i.e., receiving) neuron membrane

[1]. The synaptic weight W (size of the PSC) can be modeled

as a function of three different variables [2]:

W = f?n, p,q?.

Mechanisms acting on the number of release sites n seem

to be targeted at long-term learning, while plasticity of the

neurotransmitter release probability p and release quantity q

bothactontimescalesof0.1–1secondsandarethereforewell

suited for extracting temporal fine structure of presynaptic

pulse trains [3, 4]. Even for long-term learning, this short-

term synaptic filtering may influence the type of learning

[5]. Thus, dynamic synapses carry out various crucial signal

transformations; for a review, see [3]. These transformations

are used for processing sensory information, for example, in

the auditory cortex [6].

(1)

Dynamic synapses also interact in a complex manner

with another important component of neural information

transmission, modulated pulse trains [7, 8], that is, spike

trains characterized by regular shifts between high and low

pulse rates [9]. In biology, these bursting spike trains have

been implicated in the rapid transmission of information,

encoding of stimuli, and population synchrony [7]. This

interactionhasbeenshowninsimulationsofmodelsdescrib-

ing the plasticity of the synaptic release probability p [4]

and also in models of the plasticity of the release quantity q

[8, 10]. With regard to this interaction, it is often postulated

as a general neural principle that a new stimulus is favorably

transmitted over a steady-state one. This would mean that

modulated spike trains, in which the stimulus continuously

changes, would be favored over regular-rate stimuli. We

will critically examine this assumption, extending the q-

plasticity-based calculations of Natschlaeger and Maass [10].

The plasticity of q has been modeled in an influential

manuscript by Markram et al. [11]. They introduced a

formulation of quantal neurotransmitter release based on a

descriptive model of biological mechanisms and measure-

ments (in the following referred to as quantal model). Over

the intervening years, the quantal model has been exten-

sively studied with respect to its information transmission

Page 2

2 Computational Intelligence and Neuroscience

properties [3, 8, 10, 12]. It has also been combined with

other synaptic plasticity mechanisms to investigate possible

interrelations with long-term learning [3, 5] or probabilistic

release models [8]. Various state-of-the-art neuroscience

efforts still employ the original model, for example, in

studies of pain reception [5], the differing modes of memory

retrieval [13], or in the ongoing effort to fully characterize

the model itself and its various processing characteristics

[5, 13, 14]. Most of this work has been carried out via

simulations, probably caused by the iterative, pulse-based

nature of the model, making a closed solution, that is, some

kindoftransferfunction,intractable.However,especiallythe

causal dependency of the model’s behavior on its parameters

cannot be fully explained with simulations such as the ones

in [10]. Rather, some kind of analytical expression is needed.

This is especially interesting since biological synapses show

very complex interdependences between their state variables

and behavior [15, 16]; so an analytical expression of the

biophysical model in [11] could be employed to identify the

governing variables and mechanisms.

To derive this expression, we show that for regular pulse

rates,themodelbyMarkrametal.canbeexpressedexplicitly

as an exponential decay function. We use this function in

Section 2 to deduce the response of a dynamic synapse to

frequencymodulatedpulsetrains.Theveracityoftheexplicit

expression is shown by comparison to simulations of the

original quantal model in Section 3. Furthermore, we extend

theoptimalityanalysisof[10]toawiderparameterspectrum

and give an explanation for the favored transmission of

modulated spike trains in dynamic synapses.

2.SynapticTransmissionof Modulated

Pulse Trains

2.1. Model of Activity-Dependent Synapses. The model devel-

oped by Markram et al. [11] is governed by two parameters,

utilization of synaptic efficacy un and available synaptic

efficacy Rn. These are normalized as fractions of overall

efficacy at pulse n of the pulse train. The model is based

on a formulation of the refractoriness of neurotransmitter

release, where available synaptic efficacy is dependent on

the fraction used up in previous pulses. This increased

usage is counteracted by a facilitation mechanism, which

increasestheutilization ofsynaptic efficacy(i.e.,theavailable

neurotransmitter amount) with rising pulse rate. Thus,

utilization u is increased (facilitated) with each pulse and

recovers with a time constant τfacil, while synaptic efficacy R

recovers with τrec, dependent on the current utilization. The

iterative equations governing the evolution of unand Rnare

as follows [11] (For Rn, we use the index correction stated by

Natschlaeger and Maass [10].):

un+1= une−Δtn/τfacil+U ·

Rn+1= Rn(1 −un)e−Δtn/τrec+1 −e−Δtn/τrec,

where Δtndenotes the time elapsed between pulses (n − 1)

and n of the pulse train. The starting terms for (2) and (3)

are computed from the utilization U of a relaxed synapse as

?

1 −une−Δtn/τfacil?

(2)

(3)

0

5

10

15

20

EPSC (pA)

00.511.52

Time (s)

2.533.544.5

Input pulses and excitatory postsynaptic current (EPSC)

Figure 1: Behavior of the quantal synaptic short-term adaption,

protocol similar to [11], Figure 4, but with regular pulse rates

instead of Poisson, frequency step after 1.5 seconds and 3 seconds,

pulse rates 15s-1→ 30s-1→ 80s-1, the continuous curve denotes

the resulting PSC. The parameters are identical to [11], Figure 4,

that is, τf acil=530 milliseconds, τrec=130 milliseconds, A=1540

pA, U=0.03. To derive a continuous PSC from the pulse-PSC of

(4), pulses with a duration of 1.4 milliseconds are weighted with

the responses from (4), similar to the sum of PSCs as used in [10].

However, in contrast to [10], a moving average with a window of

100 milliseconds is computed to obtain a time curve rather than a

scalarfigureofmerit.Thepulsedurationisnotexplicitlymentioned

in [11], but biological evidence [1] and the similarity with [11]

support this value.

u1 = U or R1 = 1 − U, respectively [11]. The PSC caused

by a presynaptic pulse is defined as the product of unand Rn,

weightedwiththeabsolutesynaptic efficacyA(ratiobetween

release quantity and resultant PSC):

PSCn= A · Rn·un.

(4)

The effect of this adaption can best be described as

transmission of transients, that is, changes in the presynaptic

pulseratearetransmittedwiththeirfulldynamicrangetothe

postsynaptic neuron, but the response to steady-state input

pulse rates diminishes. This seems to be a universal feature of

biological neural nets, where novel stimuli receive increased

responses compared to static ones [1, 3].

For a steady-state signal, the above response can be

thought of as a signal compression, so that the high dynamic

range of, for example, sensory input is adapted to the limited

range of the pulse response of a neuron [3]. The steady-state

values that u and R settle to for a given pulse rate (Figure 1)

can be computed by equating unand un+1in (2) for a fixed

pulse rate λ [11]:

uc(λ) =

U

1 −(1 −U) ·e−1/(λ·τfacil).

(5)

Using this uc and a similar equalization approach, the

convergent Rcis derived as

Rc(λ) =

1 −e−1/(λ·τrec)

1 −(1 −uc(λ)) ·e−1/(λ·τrec).

(6)

As expected from the model, steady-state utilization u

increases with higher pulse rate, whereas available synaptic

efficacy R decreases. Due to the different time constants,

these changes do not cancel out completely but lead to a

maximum single PSC at around 20Hz with slight decay for

pulse rates below or above this value [11].

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Computational Intelligence and Neuroscience3

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Rel. deviation from converged value for PSC

020406080 100

Pulse rate (Hz)

30Hz

15Hz

1Hz

Figure 2: Relative PSC 5 Pulses after a step change in pulse rate.

The synapse was in a converged state for a pulse rate of 1Hz, 15Hz,

and 30Hz, respectively. Then, the pulse rate was changed to the one

denoted on the abscissa for 5 consecutive pulses. The mean PSC of

the time window corresponding to the 5 pulses was calculated. This

valuewasnormalizedtothemeanPSCofasynapsebeingconverged

to the pulse rate after the step.

However, this steady-state analysis does not do justice to

the complex transmission characteristics across a dynamic

synapse. Consequently, in the following we analyze the

response of a synapse to a single transient pulse rate

transition.

Figure 2 shows the response of the quantal synapse to a

step change in pulse rate. The synapse starts out with one

of three initial converged values for pulse rates 1Hz, 15Hz,

and 30Hz, respectively. We then transition to a new pulse

rate as denoted on the abscissa for 5 consecutive pulses.

The mean PSC of these 5 pulses after the step in pulse rate

was normalised. The reference value for normalization is the

converged PSC that would result from the steady-state PSC

response for the pulse rate after the step, that is, the response

if the new pulse train was not stopped after 5 pulses.

For decreasing pulse rate, the PSC response will contin-

uously decrease, making the transient response bigger than

the converged value. At first glance, one would expect the

opposite for increasing pulse rate: if the PSC continuously

increased, the transient PSC should be smaller than the

converged value, and the quotient between both values

should diminish for higher pulse rate differences because of

τfacilbeingbiggerthanτrecaswellastheshortertimewindow.

Incontrasttothat,Figure 2showstransientPSCshigherthan

equilibrium for bigger step-ups in pulse rate; especially, for

an initial 30Hz rate, this is the case for all frequencies after

the step change. This effect is caused by two processes: first,

the time constant for utilization u considerably decreases

with higher pulse rate, making it roughly equal to the time

constant for efficacy R (see (A.7) in the appendix); second,

the value for a single PSC decreases above a pulse rate of

approximately 20Hz [11], so that the resulting mean PSC

sharply increases with the frequency step-up due to the

higher number of releases per time but then is regulated

downbythedecreasingamplitudeofasinglePSC.Thiseffect

is also visible from the PSC time course in Figure 1.

λ(t)

λ1

λ2

t

(a)

Pulse event

t

T = 1/fm

(b)

Figure 3: From top to bottom: bursty spike train, generated from

a sine-modulated Poisson process and regularized approximation

with rectangular modulation between high pulse rate λ1and low

pulse rate λ2with a modulation frequency of fm.

As shown in this section, the response of dynamic

synapses cannot be fully characterized by the transmission

characteristics for regular pulse rates. The response for most

cases of transients is amplified compared to the steady-state

response. This is as expected from biology, where changes in

pulse rate are a source of information, while static stimuli

should be attenuated in favor of these transients [1, 3].

2.2. Analytical Approach to Synaptic Transmission. In a

generalization of the analysis carried out in Figure 2, in this

section we derive the response of the quantal model to fully

transient stimuli. Thus, we do not start from a converged

valueofuandRasinFigure 2,butweuserepeatingtransient

stimuli that result in regular variations in pulse rate (i.e., a

modulated pulse signal; see Figure 3).

A modulated pulse rate can be thought of as a sequence

of bursts and as such represents a generic model for various

types of neural pulse signaling, where the information is

encoded in the temporal fine structure of the pulse signal

[8, 9] or where bursts represent mechanisms in memory

retrieval [13].

In the upper part of Figure 3, we generate a sine-

modulated stochastic pulse train using a Poisson process [1]

with time-variable pulse rate:

f(Δt) = λ(t) ·e−λ(t)·Δt,

Δt ≥ 0,(7)

where f(Δt) is the probability density function of the time

between two successive spikes. In contrast to [1], we do

not employ a fixed pulse rate λ, but one periodically sine-

modulated between high pulse rate λ1 and low pulse rate

λ2. We use this formulation for the simulations carried out

in Section 2.3. However, for the mathematical analysis, we

further simplify the stochastic bursting spike train in the

upper part of Figure 3 to one that switches with a period

of 1/fm between two fixed pulse rates (see lower part of

Figure 3). We additionally introduce a duty cycle b as the

fraction of high rate stimulation per period. This enables a

close approximation of different spiking modes (bursting,

Page 4

4Computational Intelligence and Neuroscience

λ(t)

λ1

λ2

t

t

T = 1/fm

1

3

2

uc,λ1

ux,λ1

ux,λ2

uc,λ2

u(t)

τu,λ1

τu,λ2

Figure 4: Time course of u(t) and its dependencies on modulation

frequency fmand convergence limits ucfor high and low pulse rates

λ1and λ2, respectively. b = 0.5 in this example.

stuttering, etc.). For the approximation of the sine-wave,

b = 0.5 is chosen.

Figure 4 qualitatively shows the time course of u(t) for

this switched modulated stimulus. Its value oscillates inside

a fixed amplitude interval [ux,λ2,ux,λ1] that depends on the

modulation frequency fm, the duty cycle b, the convergence

limits uc for low and high pulse rate, as well as the time

constants τu,λ1and τu,λ2defined by (A.7) in the appendix.

For the derivation of the PSC’s modulation dependency,

we start with the explicit expression of (2) as derived in the

appendix:

u(t) = (u0−uc)e−t/τu,λ+uc.

(8)

Dependentonthesignoftheterm(u0−uc),thisequation

describes one increasing or decreasing part of the time

course, respectively. For a complete formulation, the initial

values for each cycle must be calculated. These are generally

not the limits of convergence, but intermediate values, as can

be seen from Figure 4. Their calculation will be shown as an

example for ux,λ2in the following. Our approach is based

on the observation that the value of u(t) at points 1 and 3

in Figure 4 is the same in a steady-state. Following the time

course of u(t) beginning at point 1 (assuming t = 0 there)

gives

u(t) =?ux,λ2− uc,λ1

b

fm

?e−t/τu,λ1+uc,λ1

?e−b/(fmτu,λ1)+uc,λ1,

− → u

??

=?ux,λ2− uc,λ1

(9)

with the second equation determining the value of u(t) at

the end of the high rate interval. An analogous relation for

the low-rate interval, that is, the time course from point 2 to

3, results in:

u

?

1

fm

?

= ux,λ2=

?

u

?

b

fm

?

−uc,λ2

?

e−(b−1)/(fmτu,λ2)+uc,λ2.

(10)

Evaluating (9) and (10) leads to the following expression

for ux,λ2

ux,λ2

=uc,λ1e−(1−b)/(fmτu,λ2)?

1−e−b/(fmτu,λ1)?

1−e−b/(fm·τu,λ1)·e−(1−b)/(fm·τu,λ2)

+uc,λ2

?

1−e−(1−b)/(fmτu,λ2)?

.

(11)

Results for ux,λ1, Rx,λ1and Rx,λ2can be derived with similar

approaches.

Now, the mean synaptic release quantity uR = PSC/A

can be calculated. This is done by integrating the product

u(t) ·R(t), normalizing the result with the integration

interval. For the high rate interval, that is, the time course

between points 1 and 2, the following holds:

uR12=fm

b

??Rx,λ2−Rc,λ1

·

?b/fm

0

??ux,λ2− uc,λ1

?e−t/τR,λ1+Rc,λ1

?e−t/τu,λ1+uc,λ1

?

?

·

dt.

(12)

Evaluating this integral results in

uR12=fm

b

??ux,λ2−uc,λ1

×

+?ux,λ2−uc,λ1

+?Rx,λ2−Rc,λ1

+b ·uc,λ1Rc,λ1

??Rx,λ2−Rc,λ1

1 −e−b(τu,λ1+τR,λ1)/(fm·τu,λ1·τR,λ1)?

?τu,λ1Rc,λ1

?τR,λ1uc,λ1

?

? τu,λ1τR,λ1

τu,λ1+τR,λ1

?

?

?

1 −e−b/(fmτu,λ1)?

1 −e−b/(fmτR,λ1)?

fm

.

(13)

Integrating over the low-rate interval, that is, the time

course between points 2 and 3, in the same way yields the

corresponding value uR23.

As mentioned together with Figure 1, these mean values

must be weighted by the number of pulses that occurred in

the corresponding time interval. This can be done by using

the ratio between the total time any pulse was active and the

time interval:

PSCxy= A ·TpulseNpulse,x

Tnorm

·uRxy

= A ·Tpulse(λxTnorm)

Tnorm

·uRxy.

(14)

For the high rate interval, Tnorm= b/fm, whereas for the low-

rate interval, Tnorm = (1 − b)/fm. Using the corresponding

constant pulse rate, Npulsecan be calculated for each interval

as Npulse,x= λx· Tnorm. For calculations, we will set Tpulse=

1.4 milliseconds, which is in agreement with the parameters

used in [11].

Page 5

Computational Intelligence and Neuroscience5

Jittered optimized

00.20.40.60.81

Time (s)

Mod.

Reg.

Opt.

(a)

0

5

10

15

20

25

Number of occurrences

3.13.23.33.43.53.6

ΣuR

Reg. spiketrain

Mod. spiketrain

Jit. opt.

Spiketrain

(b)

Figure 5: Left: spike trains as applied to the original quantal

model,frombottomtotop:high(100Hz)andlow(2Hz)frequency

modulated spike train (fm = 4.2Hz, b = 0.12), regular spike rate

(20Hz), spike train as extracted from Figure 5(a) in [10] (U = 0.15,

τrec = 144 milliseconds, τfacil = 62 milliseconds), and five jittered

versions of this spike train (σ = 5 milliseconds). Right: histogram

of the sum over the product uR, with the data points for modulated

and regular spike train indicated.

When calculating an overall mean PSC, the duty cycle

(i.e., the fraction each PSCxywas active) has to be taken into

account. This results in a weighted average formula:

PSC = b · PSC12+(1 −b) ·PSC23

?

2.3. Results. The explicit expressions derived in Section 2.2

describe the behavior of PSC transmission dependent on

the modulation frequency. To evaluate these equations, we

compare our model to numerical simulations of the original

= ATpulse·

bλ1uR12+(1 −b)λ2uR23

?

.

(15)

iterative equations (2) and (3). In particular, Natschlaeger

et al. [10] treat the quantal model to a rigorous numerical

analysis;soweapplyourmodeltotheirframework.Sincethe

optimal spike trains of [10] differ from our modulated pulse

rate assumption, we have to validate that the sum over the

product uR, that is, the PSC efficacy criterion, has the same

quantitative and qualitative behavior for the modulated rate

as for the optimized spike train. An initial validation can be

donebyextractingasamplespiketrainforasingleparameter

set from [10], applying a jitter to account for extraction

errors, and comparing it to a modulated spike train which is

parameterized to exhibit a similar burstiness. This is shown

in Figure 5.

The parameters were chosen to resemble the experiment

ofFigure 5in[10],with20pulsesdistributedinaone-second

interval. All spike trains were processed with the original

quantal model. As can be seen, the original optimized

spike train shows a strong burstiness, so as expected, the

regular spike train has a much lower synaptic efficacy. Also,

the modulated spike rate is well within bandwidth of the

statistical variations of the optimized spike train and also

shows significantly larger synaptic efficacy than the regular

rate. From this limited example (and others below), the

initial assumption for our derivation seems valid, that is,

a modulated pulse rate exhibits the same behavior with

respect to the Markram model as a more precisely optimized

one.

In the following, we will thus apply the derivation

of Section 2.2, in particular the new non-iterative time

constants,toextendtheanalysisof[10]andespeciallytestthe

predictive and explanatory power of our analytical expres-

sions. Two major activity regimes can be discerned from

Figure 5 of [10]: one where the grouping of pulses into short

activity bursts results in a large synaptic efficacy, and one

whereincontrastaregulardistributionofall20pulsesacross

the time interval is advantageous. If we relate this back to our

model, the pulse regime is determined by the modulation

frequency. Thus, with the explicit expression for the mean

PSC (15), we can state an alternative optimality approach

to [10]. Maximizing the mean PSC over the modulation

frequency corresponds to finding the optimum pulse regime

for a synapse. The optimum modulation frequency fm,optin

that sense can be derived using the necessary condition:

0 =∂PSC

∂fm

= ATpulse·

?

bλ1∂uR12

∂fm

+(1 −b)λ2∂uR23

∂fm

?

.

(16)

In general, this equation cannot be brought in an explicit

form. Approximate explicit expressions could be derived, for

example, assuming fmτu,λ ? 1, but the respective approxi-

mationsarenotvalidovertheentiresynapseparameterspace

and optimization range. Thus, we will solve the optimization

equation numerically. Modulated (i.e., bursty) spike trains

are only generated in the fm-range (0,1/2λ1); other values

result in a regular spike train. Thus, if the partial derivative

∂PSC/∂fmdoes not change sign inside this interval, that is,

Page 6

6Computational Intelligence and Neuroscience

1.8

2

2.2

2.4

uR

0102030 4050

Modulation frequency (Hz)

U = 0.09

τfacil= 50ms

τrec= 250ms

(a)

1.8

2

2.2

2.4

uR

0 1020 30 4050

Modulation frequency (Hz)

U = 0.2

τfacil= 200ms

τrec= 200ms

(b)

Figure 6: Mean synaptic efficacy per spike uR with respect to

modulation frequency fm for two parameter sets. Dashed line:

optimum modulation frequency for the plotted range. λ1= 100Hz,

λ2= 5Hz in both cases.

no local maximum exists therein, a regular spike train will

result in a maximum mean PSC.

To resemble the optimization regime of [10], we adjust

the duty cycle b(λ1,λ2) such that the mean frequency

f = 20Hz. The results of [10] show that a modulated regime

is optimal for low values of U and τfacil, whereas for higher

values, a regular spike train is favorable. Figure 6 confirms

this result with our analysis for an illustrative example: for

the low-value case (left), a maximum at approximately 4Hz

is present, whereas for the high-value case, the synaptic

efficacy monotonically increases with modulation frequency,

whichultimatelyleadstoaregularspiketrainasanoptimum.

Of course, as we have shown in the previous section and

the appendix, the preference of the quantal model depends

not only on U, τfacil, and τrec but also on the spike train

characteristics, that is, duty cycle b, high rate λ1, and low-

rate λ2. To show these dependencies, we extend the analysis

of Figure 6 to a full sweep across b, λ1and λ2, employing the

synapse parameter set of Figure 6, left.

Figure 7 shows the optimal modulation frequency fm,opt,

derived similar to Figure 6, in grey-scale. Data points are

only depicted if a distinct optimal fmis found, that is, if the

maximum as shown in Figure 6, left, is at least 1% above the

value of uR for the high modulation frequency (right side of

both graphs in Figure 6). Thus, nonsignificant maxima and

cases where a regular spike train is preferred (Figure 6, right)

are omitted. A good correspondence between the simulation

of the original quantal model and the mean uR as derived

from the analysis in the previous section can be observed,

showing the validity of our derivations.

There is almost no dependence of the optimal modula-

tion frequency and the burst preference on the low spike rate

λ2.Thismaybeduetothefactthatthereisonlyacertainlevel

of relaxation that can be obtained by the synapse during the

low-rate intervals. This means that, while the relaxation is

important to obtain a high uR, as will be explained together

with Figure 9, the exact low rate during this relaxation is not

important, only the fact that there is such a relaxation phase.

However, there is a clear dependence between the duty cycle

b and fm, where fmrises linearly with b at a certain λ1,λ2(see

columns in the plots of Figure 7). In other words, this can be

thought of as

b

fm

= const. = b ·T = Thigh,(17)

with T being the duration of a period and Thighbeing the

duration of the high rate interval therein, that is, the length

of a burst. Thus, if Thighis constant, the number of pulses

during a burst for a given high rate λ1is also constant. An

explanation for this could be that there exists an optimal

burst profile which maximizes uR for a given parameter set

U, τfacil, τrec, and a given λ1. Accordingly, if b is subjected

to a sweep, fmmust rise with it to keep this optimal profile.

At the same time, bursts are shifted closer, so that the mean

number of pulses in a fixed time interval rises linearly with b.

Equation (16) thus searches not so much for an optimal fm

but rather for an optimal burst profile.

Another interesting characteristic of the above plot is

the decrease of the maximum b at which a significant fm,opt

can be found with increase in λ1. This inverse relationship

between maximum b for a bursty spike train and λ1may hint

at an optimal profile or number of pulses in a burst almost

independent of λ1. According to (17), the number of spikes

inaburstcanbecomputedasb/fm· λ1,resultingina(mean)

number of pulses per burst of 4.3 for λ1= 150Hz and 3.6 for

λ1 = 50Hz. These similar values could be explained by the

fact that the optimum uR is governed by the evolution of u

andRduringtheburst.Theseinturndependontheabsolute

time constants derived in the appendix, which scale with λ;

thus, the scaling of the time constants and λ cancel each

other at least partially, resulting in very similar optimal burst

Page 7

Computational Intelligence and Neuroscience7

0

0.2

0.4

0.6

b

10

5

0

λ2(Hz)

150

100

λ1(Hz)

50

0.148

fm(Hz)

(a)

0

0.2

0.4

0.6

b

10

5

0

λ2(Hz)

150

100

50

λ1(Hz)

0.148

fm(Hz)

(b)

Figure7:Optimalmodulationfrequency fm,opt(greyscalecoded)withrespecttotheparametersofthemodulatedspiketrain.(a)simulation,

and (b) our analytical calculation. Only those cases are shown where a modulated spike train is favored. Parameters as in Figure 6, left.

0

0.2

0.4

τrec(s)

0.4

0.2

0

τfacil(s)

0

0.2

U

0.4

00.070.14

(uRmod−uRreg)/uRreg

(a)

0

0.2

0.4

τrec(s)

0.4

0.2

0

τfacil(s)

0

0.2

U

0.4

00.20.4

(uRmod−uRreg)/uRreg

(b)

Figure 8: Relative difference of a modulated spike train from a regular spike train, grey scale coded, with respect to the parameters of the

quantal model. (a) simulation, and (b) our analytical calculation. Only positive (i.e., modulation-favored) part is shown.

profiles despite the change in λ1. So the absolute value of uR

may vary with λ1, b and fm, but the qualitative behavior, that

is, the burst profile for which uR is maximum, seems to be

constant for a given synapse type. Interestingly, there exists

no optimal modulation frequency above 8Hz, that is, in this

range a regular spike train is always better than a modulated

one. This is probably due to the fact that at this fm, there is a

natural transition between bursty and regular spike train in

any case. That means, the burst phases are too short to allow

a real grouping of spikes, while the low-rate phases are too

short to obtain a significant recovery of u and R, so that the

same number of pulses achieves a higher uR if it is spaced

regularly across the given time span.

As already stated, one of the main questions behind

such analyses is, for which synapse types (i.e., parameters

U, τfacil and τrec) a modulated spike train is favored over

Page 8

8Computational Intelligence and Neuroscience

0

0.2

0.4

0.6

0.8

1

uc,Rc

0 50100 150

Pulse rate (Hz)

U = 0.09

τfacil= 50ms

τrec= 250ms −400ms

τrec

R

u

(a)

0

0.2

0.4

0.6

0.8

1

uc,Rc

050100150

Pulse rate (Hz)

U = 0.2

τfacil= 200ms

τrec= 200ms

R

u

(b)

0

50

100

150

200

250

Time constant (ms)

050100 150

Pulse rate (Hz)

U = 0.09

τfacil= 50ms

τrec= 250ms −400ms

τrec

τR

τu

(c)

0

50

100

150

200

250

Time constant (ms)

050100150

Pulse rate (Hz)

U = 0.2

τfacil= 200ms

τrec= 200ms

τR

τu

(d)

Figure 9: (a) Steady-state values of u and R over spike rate for a bursty parameter set, with additional indication of R dependency on τrec(b)

same as (a) for a regular parameter set (c) convergence time constants for the parameter set(s) of (a) (d) convergence time constants for the

parameter set of (b).

a regular spike train in terms of transmission. This question

was tackled in [10] only exemplarily for single-value sweeps.

Here, we perform a sweep over the full three-dimensional

parameter space of the quantal model, as shown in Figure 8.

Thereby we use the relative difference of a modulated spike

train and a regular spike train as a measure for the favored

spike mode. Parameters were again λ1= 100Hz, λ2= 5Hz,

and f = 20Hz, together with a modulation frequency fm=

8Hz, comparing to results of Figure 5 in [10].

We also used this parameter sweep to compare our

analytical calculation with the original iterative formula.

This is a hard test case, because already small deviations

in the calculations, for example, caused by the continuous-

time idealization and the approximations made with the

derivation of τR,λ, can lead to marked changes in the

relative difference value calculated for comparison. Taking

this sensitivity into account, our derivation is in good

agreementwiththesimulation.Especiallythediscrimination

between favored modulated or regular spike train is well

replicated.

The principal dependencies of the favored spike mode

on the synapse parameters as suggested by Figure 5 of [10]

are also present in the whole parameter space exploration:

A modulated spike train is only favored if U or τfacil are

low. Also, for U

= 0.32 and τfacil

a transition from regular-favored to modulation-favored

transmission with increasing τrec is present in the plot,

which is in agreement with [10]. This again shows that even

with the assumption of a fixed modulated, and potentially

nonoptimal, spike train, essentially the same predictions

= 62 milliseconds,

Page 9

Computational Intelligence and Neuroscience9

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

τR,λ1−τu,λ1

−0.2

−0.15

−0.1

−0.0500.050.10.15

(uRmod−uRreg)/uRreg

(a)

−3

−0.2

−2

−1

0

1

2

3

4

5

6

ln(τR,λ2−τu,λ1)

−0.15

−0.1

−0.0500.050.10.15

(uRmod−uRreg)/uRreg

τrec= 0.47

τrec= 0.05

(b)

Figure 10: Correlation between criteria based on the convergence time constants of (A.12) and (A.7) (y-axes) and the relative difference

between the synaptic efficacy uR for a regular and a modulated/grouped spike train (x-axes). Each of the dots corresponds to one data point

from Figure 8, that is, one parameter set (U,τfacil,τrec). In addition, the right figure has a decrease in τrecdenoted by increasing gray levels.

Also, the smallest (resp. largest) value for τrecis denoted in crosses resp. circles.

can be derived as with a single-spike optimization, but

with much less computational complexity. Also, further

dependencies can be extracted from the parameter sweep: if

bothU andτfacildecreasetolowvalues,therelativedifference

of the response to modulated and regular spike trains gets

moreandmoreindependentof τrec.Additionally,toacertain

extent, higher values for U can be compensated with lower

values for τfacil, and vice versa.

In the following, we try to analyze the above parameter

dependencies, based on our modeling of u and R as

exponential decays. In particular, based on our noniterative

time constants for u and R and on the converged values

uc and Rc that scale the exponential decay functions, we

postulatethefollowingmechanismsforapreferenceofeither

regular or grouped (bursty) spike trains by the synapse.

There is a dependency of this preference on the relation

between the time constants τR,λ1and τu,λ1, that is, for the

convergence during the high rates/bursts (Figure 9(c)). A

bursty spike train benefits if the time constant τu,λ1is

relativelylow,sothaturisesfasttoitsconvergedvalue,which

is a factor of five above its relaxed value (i.e., at the end of

the low-rate interval, Figure 9(a)), as this increases markedly

the total value of uR. On the other hand, R diminishes

to a small value for the high rate, so its convergence time

constant τR,λ1should be large relative to τu,λ1, so that most

of the spikes during the burst still “see” the high relaxed

value (Figure 9(a)). Compared to a parameter regime which

preferentially transmits a regular rate (Figure 9b and 9d), a

low time constant τR,λ1diminishes the value of R during a

burst, and a corresponding high time constant τu,λ1would

prevent u from rising to compensate this decrease in R,

especially if at the same time u has a smaller dynamic

range (Figure 9b). Thus, for this parameter regime and its

resulting time constants,a bursty regime would result in a

lower synaptic efficacy compared with a regular rate.

A second criterion based on which it can be predicted if a

burstyorregularregimeispreferredbythesynapse,wouldbe

the relation between the convergence time constant for R for

low and high rate τR,λ2and τR,λ1. This relation expresses the

basicintuitionthatτR,λ2shouldberelativelylowcomparedto

τR,λ1, so that R can relax very fast to a high value during the

low-rateintervals.Incontrast,τR,λ1shouldbehighcompared

to the burst duration so that R does not decrease too much

during the high rate intervals. So a parameter regime that

results in a low τR,λ2relative to τR,λ1should preferentially

transmit bursts.

From the above postulates, two criteria can be derived

where the time constants derived in this paper allow to

predict if a grouped/bursty or a regular regime is preferred

by the synapse. The first one would be the difference between

the convergence time constants for the high rates, that is,

τR,λ1−τu,λ1, see Figure 10(a).

As can be seen, there is a definite correlation in the

way suggested above, that τR,λ1should be larger than τu,λ1

in order for the synapse to transmit a bursty spike train

better than a regular one. In the figure, this is expressed

on the x-axis by the normalized difference between ΣuR for

a regular, respectively, a bursty spike train (for the same

quantal parameter set). The second criterion would be the

quotient between the convergence time constant for R for

low and high rate, as expressed by τR,λ2/τR,λ1. Figure 10(b)

shows a plot of this criterion, against the same synaptic

Page 10

10Computational Intelligence and Neuroscience

efficacy criterion as in Figure 10(a). For clarity reasons, the

natural logarithm of the above quotient is plotted rather

than the quotient itself. Again, as postulated above, there

is a clear correlation between a measure based on the

convergence time constants and the amount a bursty spike

train evokes more or less synaptic efficacy compared to

a regular spike train. Interestingly, there also seems to be

some parameter which causes a change in slope as well as

a shift of the correlation curve. When plotting the data

points based on their parameter values, it becomes evident

that this parameter is τrec, that is, for larger τrec, the spike

trains enter the bursty regime earlier. This trend towards

burstiness with increasing τrec can be explained based on

Figures 9(a) and 9(c). As can be seen, for larger τrec the

slope of the R curve increases, so that the (converged) value

of uR for a regular rate diminishes, while for the short

high rate episodes characteristic of a burst, the relaxed R

is still close to one. Due to the fact that τu,λ for the high

rate does not decrease, the burst benefits from this high

relaxed value in the same way as it did for lower τrec. At

the same time, τR,λ increases with increasing τrec, so that

there is a more pronounced trend towards long periods

of little activity in the spike train, so that R can reach its

relaxed value even if its convergence time constant becomes

larger. That the synapse exhibits a mechanism which prefers

modulated spike trains for certain parameter sets (as shown

above) might also provide an alternative, synapse-based

way for bursting behavior to emerge. This could comple-

ment the conductance-based bursting behavior shown in

[7].

How could these results be applied in the wider neu-

roscience context? One important topic of current interest

is the interaction of the different forms of plasticity on

the same synapse, especially with regard to the different

temporal timescales of expression [5, 15–17]. Some studies

which employ both kinds of plasticity act on an abstract

idea of weight, but with basically unchanging parame-

ters of the dynamic synapse [5]. On the other hand,

Spike-Timing-Dependent Plasticity (STDP) is postulated

to depend on the modulation of neurotransmitter release

probability ([16, 17]), which in the model discussed herein

is expressed as the initial release quantity U [11]. As

evidenced by our analysis, this influences directly the spike

pattern preferences through the mechanisms postulated

in Figure 9. So these are not just plasticity mechanisms

overlaid on the same synaptic weight [17], instead STDP

might govern the operating regime of the short term

dynamics. Thus, STDP might not only provide a basis

for static, weight-based memory formation [18], but also

serve as a substrate for memory and computation in

dynamical models. Examples for this could be attractor

neural networks [13], or liquid computing [19], which

rely heavily on the short term dynamics of synapses. In

this respect, our analysis indicates several ways in which

a U modulated by some other plasticity mechanisms

might in turn govern the absolute temporal dynamics of

a synapse, namely through uc, τu,λ, and τR,λ. Pushing this

speculation further, there might also be a feedback path

back towards STDP, in which the absolute synaptic time

constants τu,λand τR,λof our derivation influence the time

course of the STDP learning window. Of course, classical

STDP relies on coincidence between pre- and postsynaptic

spikes, so the quantal release mechanism which only acts

on presynaptic spikes would not work in this context.

However, several newer forms of STDP rely on dendritic

spikes [20], which depend on coincident heterosynaptically

expressed presynaptic spike transmission rather than on

postsynaptic spikes. Thus, this form of STDP could, through

its influence on U, change τu,λ and τR,λ and these in

turn would impact on the temporal learning window. This

could form some kind of metaplasticity or homeostasis

[21], in which STDP influences its own expression at the

synapse.

3.Conclusion

We have derived an explicit expression for the iterative

quantal model [11] describing short term plasticity of

dynamic synapses. A wide range of naturally occurring pulse

trains could be subjected to detailed mathematical analysis

using this model. For example, our analysis is also valid if

the pulse rate during a burst is not constant (see Figure 5).

Thus, the selective treatment of bursts by dynamic synapses

as derived in Section 2 could also be extended to cases were

the information is contained in the fine structure of the

bursts [4, 8, 12]. Also, the modulation does not have to be

constant, that is, pauses between bursts could vary, so that

pulse trains derived in [8, 14] could also be treated with a

more rigorous, global approach, rather than an analysis via

simulations.

We have shown how the filtering characteristics might

be determined from the synaptic parameters. Specifically, we

have provided an explanation how the filtering characteristic

of a dynamic synapse depends on the effective time constants

τu,λand τR,λand their interaction with the converged values

for u and R (see Figures 5 and 10). Also, in extension of [10],

we have provided a more complete picture how the filtering

characteristic relates back to the original parameters (see

Figure 8). The mechanisms/correlations shown in Figure 10

could be applied to characterize the transfer/decoding

function of synaptic networks, such as the ones used in

[4]. Also, the closed expression for the transfer function

developed in this manuscript could be employed to deriving

thesynapticalparametersetfortheoptimalcodingofstimuli

in, for example, the auditory cortex [6]. We have shown a

limitedexampleofthisin Figure 6,whereweuseourtransfer

expression to derive the optimal modulation frequency for

a parameter set, which is in good agreement with the

numerical simulations of [10].

Our noniterative expression for the behavior of the

dynamic synapses of [11] could also have consequences

on plasticity mechanisms. Synapses exhibit very diverse

modulatory and plastic behaviors, where the interdepen-

dences and governing variables often cannot be clearly

Page 11

Computational Intelligence and Neuroscience 11

determined [15, 16]. Since the time constants of neural

actions are not very amenable to change [1], it might be

assumed that the temporal dynamics and preferences of

a synapse are relatively fixed. In contrast, our derivations

in this paper predict mechanisms by which a synapse

could change its effective time constants and spike pattern

preference based on U even though the basic temporal

parameters τrecand τfacil of the dynamic synapse might be

constant. In this context, we have speculated on possible

repercussions of this modulation of τu,λand τR,λon models

of long-term plasticity (STDP), especially with regard to

extending STDP to dynamical models of computation and

learning/memory.

Appendix

A.TransientAnalyticalDescription of

QuantalPlasticity

The convergence of iterative equations like those of the

quantal model [11] can only be expressed explicitly for

some special cases [22]. Whereas the convergence limits

for a constant presynaptic pulse rate λ can be derived

with relatively little effort [11], the time course and speed

of convergence is difficult to define, especially due to the

constant parts of the iterative equations for un, (2), and Rn,

(3).

However, as Figure 11 shows, in case of regular spike

trains with a defined pulse rate, the iterative descriptions

of u and R in (2) and (3) can be interpreted as settling of

transient responses to a steady-state value, comparable to

the exponential convergence of, for example, an RC voltage

settling curve.

In this case, an absolute time constant for this settling

maybederived,whichislikelytodependonthefundamental

time constants of the quantal model.

In the following, an explicit expression for the con-

vergence of u toward a steady-state value will be derived.

Equation (2), recursivelydescribing the valueofuafterinter-

spike interval (ISI) Δtn, can be rewritten as

un= un−1·e−Δtn−1/τfacil·(1 −U)+U.

(A.1)

Note that all variables are shifted by one ISI compared to the

original formulation. For deriving an explicit expression, we

restrict ourselves to pulse trains having a constant rate λ, so

that Δtn= 1/λ for all n. Recursively extending (A.1) by one

ISI yields

un= un−2·e−2/(λ·τfacil)·(1 −U)2

+U · e−1/(λ·τfacil)· (1 − U)+U.

(A.2)

The further recursion back to u0 is obvious from (A.2),

resulting in

un= u0·

?

(1−U)e−1/(λ·τfacil)?n+U ·

n−1

?

i=0

?

(1−U)e−1/(λ·τfacil)?i.

(A.3)

Because the term (1−U)e−1/(λ·τfacil)never exceeds the interval

[0,1), the geometric series of the second term converges, and

its sum can be calculated to yield [22]:

un=u0·

?

(1−U)e−1/(λ·τfacil)?n+U

?

(1−U)e−1/(λ·τfacil)?n

(1−U)e−1/(λ·τfacil)−1

−1

.

(A.4)

The limit for n → ∞ is the same as the value for uc(λ)

calculated in (5). In fact, equation (A.4) can be rewritten in

the following form to make this limit obvious:

un=

?

u0−

U

1 −(1 −U)e−1/(λ·τfacil)

?

·

?

(1 −U)e−1/(λ·τfacil)?n

+

U

1 − (1 − U)e−1/(λ·τfacil).

(A.5)

The speed of convergence is determined by the term

dependent on n. For introducing the notion of a time

constant,weextendunin(A.5)toacontinuous-timevariable

u(t) that is equal to unat the time of pulse n, which means

u(n·Δt) = unfor a constant pulse rate. At this point in time,

equalityn = λ·t holds, whichweuseto reformulatetheterm

dependent on n:

?

(1 −U)e−1/(λ·τfacil)?λ·t

= et·[λ·ln(1−U)−1/τfacil]= e−t/τu,λ,

(A.6)

with the time constant describing the speed of convergence

following:

τu,λ=

1

λ ·ln(1/(1 −U)) +1/τfacil.

(A.7)

The time constant thus is dependent on both the time

constant of the iteration, τfacil, and the pulse rate, λ.

Therefore, (A.5) can be modeled as follows:

u(t) = (u0−uc)e−t/τu,λ+uc.

(A.8)

Page 12

12 Computational Intelligence and Neuroscience

0

0.2

0.4

0.6

un, u(t)

024

Time (s)

Iterative

Explicit

(a)

0

0.5

1

Rn, R(t)

024

Time (s)

Iterative

Explicit

(b)

Figure 11: Comparison of simulated and analytically derived time

course of u (A) and R (B). Parameters are the same as used in

Figure 1.

An explicit expression for Rncan be derived in a similar

way, starting with an equation analogous to (A.2):

Rn= Rn−2·(1 −un−2)(1 −un−1)e−2/(λ·τrec)

+

?

1 −e−1/(λ·τrec)?

·

?

(1 −un−1)e−1/(λ·τrec)+1

?

.

(A.9)

This again makes the further recursion back to R0clear:

Rn= R0·e−n/(λ·τrec)

⎧

⎩

n−1

?

i=0

(1 −ui)+

?

1 −e−1/(λ·τrec)?

⎤

×

⎨

n−1

?

j=0

⎡

⎣e−(n−j)/(λ·τrec)·

n−1

?

i=j

(1 −ui)

⎦+1

⎫

⎬

⎭.

(A.10)

Because of the varying ui, the terms in the product are not

constant like in the derivation for u, so that the resulting

series is not geometric, making a straightforward derivation

impossible. Nevertheless, an exponential decay with a fixed

time constant like for u is the dominant behavior also for

R, as can be seen from Figure 11. We will therefore use a

heuristic approximation for deriving an explicit expression

for the time constant of R. We first insert the explicit

expression of un(A.8) and the starting value for R0to derive

the following expression:

Rn=

?

1 −e−1/(λτrec)?

⎧

⎩

·

⎨

n−1

?

j=0

??

−

M

1 −M·e−1/(λτrec)

1 − e−1/(λτfacil)−U · Mi?⎤

?j

×

n−1

?

i=n−j

?

⎦+1

⎫

⎬

⎭,

with

M = (1 −U) ·e−1/(λτfacil).

k

(A.11)

The constant factor outside the sum does not have an

influence on the convergence speed of R. When neglecting

the product inside the sum, the expression reduces to the

sum of a geometric series, whose time constant may be

determined analogously to (A.4) and (A.5). The deviations

from the original formulation then have to be corrected

by additional or modified terms in the time constant. The

exponential functionandthetermforM inthenumeratorof

the simplified expression result in terms 1/τrecand 1/τfacilin

the time constant, respectively. By using the approximation

(1 − ε) = eε, the term (1 − M) in the denominator can

be transformed to yield an inverse proportional relation

between U and the time constant for R, τR,λ.

Now, the product term in (A.11) has to be accounted

for. The term Miresults in a quadratic dependency between

the time constant for R and the pulse rate λ due to the

additional potentiation of M. At the same time, the term

(1 − U) of Miis again transformed into ln1/(1 − U),

resultinginλ2τfacil ln1/(1−U).TheconstantfactorU results

in an inverse linear dependency of τR,λ on λ, while U is

again transformed into ln 1/U. The term (1 − e−1/(λτfacil))

can be neglected for high frequencies λ, but has to be

taken into account for low frequencies. This can be done by

combining the constant terms 1/τrecand 1/τfacilwhile adding

a frequency-dependent factor U ·λ.

With all principal dependencies being identified, the

resulting time constant of R reads:

τR,λ=

1

U · [λ2· τfacil· ln(1/(1 −U))+2/3λ · ln(1/U)+2/(3τrec· Uτfacil· λ)],

(A.12)

Page 13

Computational Intelligence and Neuroscience13

Analogously to the derivation for u, an explicit formula-

tion for R can be stated using the time constant defined by

equation (A.12):

R(t) = (R0− Rc)e−t/τR,λ+Rc .

Equations (A.8) for u and (A.13) for R were verified

against simulations of the iteration formulae (2), (3) over

a wide range of parameters for τfacil, τrec and U, see

results. Despite the continuous-time generalization and the

approximations made in the derivation of τR,λ, analytical and

simulation results are in good agreement. Figure 11 shows

an example of the resulting time courses. The differences

between simulated and analytically derived u(t) are due

to the discrete nature of the original iterative equations

that were generalized to continuous time for the analytical

formulation.Slightlybigger,butstillnegligibledeviationsare

visible for R(t), which are due to the approximations made.

(A.13)

Acknowledgment

The authors would like to gratefully acknowledge financial

support by the European Union in the framework of

the Information Society Technologies program, Biologically

Inspired Information Systems branch, project FACETS (No.

15879).

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[3] L. F. Abbott and W. G. Regehr, “Synaptic computation,”

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