Synaptic filtering of rate-coded information

PHYSICAL REVIEW E 81, 041921 共2010兲

Synaptic filtering of rate-coded information Matthias Merkel and Benjamin Lindner Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany 共Received 3 December 2009; revised manuscript received 26 February 2010; published 29 April 2010兲 In this paper, we analytically examine the influence of synaptic short-term plasticity 共STP兲 on the transfer of rate-coded information through synapses. STP endows each presynaptic input spike with an amplitude that depends on previous input spikes. We develop a method to calculate the spectral statistics of this amplitude modulated spike train 共postsynaptic input兲 for the case of an inhomogeneous Poisson process. We derive in particular analytical approximations for cross-spectra, power spectra, and for the coherence function between the postsynaptic input and the time-dependent rate modulation for a specific model. We give simple expressions for the coherence in the limiting cases of pure facilitation and pure depression. Using our analytical results and extensive numerical simulations, we study the spectral coherence function for postsynaptic input resulting from a single synapse or from a group of synapses. For a single synapse, we find that the synaptic coherence function is largely independent of frequency indicating broadband information transmission. This effect is even more pronounced for a large number of synapses. However, additional noise gives rise to frequency-dependent information filtering. DOI: 10.1103/PhysRevE.81.041921

PACS number共s兲: 87.19.lo, 87.19.lc, 87.18.Sn

I. INTRODUCTION

Synapses constitute the connections between nerve cells 共neurons兲 in the brain. However, they are not just bare conveyers of information from one neuron to another, but play a far more active role 关1兴. First of all, slow changes in their efficacy 共i.e., in their postsynaptic amplitude兲, called longterm plasticity, are believed to be the physiological basis for learning and memory in the brain. Furthermore, synapses also display changes in their efficacy on shorter time scales 共⬃100 ms to 1 s兲. The latter effect is referred to as shortterm plasticity 共STP兲 and is the subject of our work. Depending on the most recent spiking history, synapses modify the amplitudes of the action potentials they transmit. Repeated presynaptic spiking may increase the postsynaptic amplitude 共facilitation兲 or decrease it 共depression兲. Both kinds of STP can coexist in a single synapse because they rely on a variety of different physiological mechanisms 关2,3兴. A number of phenomenological models describing STP have been established in the past 关4–10兴. In this paper, we focus on facilitation-depression 共FD兲 models as proposed and studied in 关8,11–13兴. The fact that synapses are active elements in the neural dynamics suggests, that they may also play an important role in information processing in the brain. STP may be important for synaptic gain control, for the detection of transients and of bursts, and may be linked to working memory 关6,9,14–16兴. In addition, synaptic filtering properties have been examined in terms of the average postsynaptic amplitude depending on the presynaptic firing rate 关6,8,14,17兴. The amount information about the presynaptic spiking history carried by the synaptic strength has been studied in 关18兴. Recently, it was shown that information transmission about a rate-coded time-dependent stimulus across an ensemble of dynamic synapses is broadband, i.e., independent of frequency 关13兴. More precisely, it was shown by numerical simulations that neither facilitation nor depression introduces a frequency-dependent filtering of the spectral input1539-3755/2010/81共4兲/041921共19兲

output coherence. In this work, we study this problem analytically. We consider an ensemble of independent dynamic synapses driven by rate-modulated Poisson processes and derive expressions for the power spectra, cross-spectra, and coherence functions. We investigate these statistics in detail for a single FD synapse and discuss their basic features for the physiologically accessible part of the parameter space. For an ensemble of many synapses, we furthermore inspect the effect of an additional synaptic noise on information transfer. Our results contribute to a deeper understanding of the conditions for broadband coding by dynamic synapses. Our study can also be regarded as a contribution to the theory of point processes. The main problem in this paper is the calculation of the spectral statistics of a spike train with time-dependent amplitudes which are functionals of the underlying point process itself. We would like to point out that our results may also be of interest for other areas where pulse trains with variable amplitudes are encountered, as for instance, in the time series of earthquakes or in drop-outs of lasers. II. MODEL AND MEASURES

We consider a population of facilitating and depressing synapses, by which a neuron 共the “target neuron”兲 receives independent Poissonian spike train input from other neurons 共the “input neurons”兲; see Fig. 1. For a part of this input ensemble 共the “signal neurons”兲 a time-dependent stimulus is encoded in the instantaneous firing rate of the input spike trains. The transfer of this stimulus is affected by the nature 共facilitating or depressing兲 of the “signal synapses” connecting the signal neurons to the target neuron. The remaining “noise synapses” receive Poissonian input spike trains with constant rates. They may have a different STP character and they will also contribute to the total postsynaptic input to the target neuron. We would like to stress in this context, that we use the term “postsynaptic input” for the amplitude-modulated input

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©2010 The American Physical Society

PHYSICAL REVIEW E 81, 041921 共2010兲

MATTHIAS MERKEL AND BENJAMIN LINDNER

Poisson

external stimulus:

presynaptic spike trains: x0,N (t):


postsynaptic inputs:

signal synapse

t















x0,2 (t):

t

signal synapse

x0,1 (t):

t

t

x1 (t):

1st t

X(t):

x2 (t):

2nd

t

total postsynaptic input:

xN (t):

Nth t

R(t):

STP

signal synapse

t

η(t): additional input via noise synapses: t

FIG. 1. Scheme illustrating Eqs. 共1兲–共3兲 and 共9兲: The external stimulus R共t兲 modulates the rate of Poissonian spike trains x0,i共t兲. Each of these spike trains is modulated by a dynamic synapse according to the FD model turning it into xi共t兲. Finally, the total postsynaptic input X共t兲 to the target neuron is given by the sum of these modulated spike trains with an additional noise ␩共t兲. The additional noise accounts for input from synapses that are not involved in the transmission of the stimulus signal R.

that a synapse provides to the target neuron. Similarly, the “total postsynaptic input” is the sum over all the synaptic inputs that the target neuron receives. In this paper, we do not consider the output spike train of the target neuron, which is often termed “postsynaptic spike train.” However, in Sec. II E, we briefly show how to extend our results about the total postsynaptic input to the membrane conductance and the subthreshold membrane voltage of the target neuron. Note that in the following, we strictly distinguish between the notions of rate and frequency. Rate refers to a neuronal firing rate, whereas frequency is solely used for the argument of the Fourier transform.

correlations between cortical spike trains are common 关19兴. The time-dependent firing rate ␯共t兲 is

␯共t兲 =

冎

r关1 + ␧R共t兲兴 for 1 + ␧R共t兲 ⱖ 0 and 0

else.

共2兲

Here, r is the baseline rate, which is weakly modulated by the external stimulus R共t兲 with small amplitude ␧. We assume that R共t兲 is a band-limited Gaussian signal with 具R典 = 0, 具R2典 = 1, and a limiting frequency of 50 Hz. Generally, we neglect the possibility of negative rates, i.e., 1 + ␧R共t兲 ⬍ 0. Note that for the amplitudes used in this paper 共␧ ⱕ 0.2兲, the probability of 1 + ␧R共t兲 ⬍ 0 at an arbitrary instance t is less than 3 ⫻ 10−7.

A. Time-dependent signals as rate modulation

B. Short-term plasticity

We write the presynaptic spike train associated to the ith signal neuron as x0,i共t兲 = 兺 ␦ 共t − ti,j兲.

再

共1兲

j

We assume that the Poissonian spike trains from different signal neurons are statistically independent from each other except for a common rate modulation—the external stimulus R共t兲. This is clearly a simplification, since it is known, that

For each of the considered synapses in the population, we use a deterministic FD model in order to describe short-term plasticity. The corresponding dynamical system is similar to those used in 关8,11–13兴. Generally, facilitation and depression are modeled via different variables F and D, respectively. The product of these variables is the synaptic amplitude A. The postsynaptic input from the synapse corresponding to the ith input neuron is given by 共see also Fig. 1兲

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SYNAPTIC FILTERING OF RATE-CODED INFORMATION

⫾

Here and in the following, f共t 兲 refers to lim ␩⬎0 f共t ⫾ ␩兲. ␩→0 The synaptic amplitude Ai共t兲 is the product of a facilitation variable Fi共t兲 and a depression variable Di共t兲, Ai共t兲 = Fi共t兲Di共t兲.

共4兲

The dynamics of the facilitation and depression variables, Fi共t兲 and Di共t兲, depend on the respective presynaptic spiking history x0,i共t兲 of each synapse. For a static synapse we would have 共5兲

i.e., the postsynaptic input would, apart from a factor, coincide with the presynaptic spike train. 1. Facilitation

The facilitation dynamics models the increase of the release probability of synaptic vesicles due to an increase of residual calcium in the presynaptic terminal 关8,11兴. FC共t兲 corresponds to the concentration of a calcium bound molecule within the presynaptic terminal. Upon spiking, this concentration is increased by the constant ⌬. However, this concentration decays with a rate 1 / ␶F. These features are expressed by the following differential equation: 共6兲

The calcium bound molecule may bind to the release site altering the release probability of neurotransmitter into the synaptic cleft. This release probability is our facilitation variable F共t兲, given by F共t兲 = F0 +

1 共1 − F0兲−1 + FC−1共t兲

.

0.4 0.2

FC(t) F(t)=A(t)

0.8 0.6

∆

0.4 0.2 0.0 0.3 0.2 0.1 0.0

0

1

2

3

4

5

time [s]

FIG. 2. 共Color online兲 Illustration of facilitation-only dynamics as described by Eq. 共3兲 with A共t兲 = F共t兲, and Eqs. 共6兲 and 共7兲. Upper panel: presynaptic spike train x0共t兲. Middle panel: dynamics of FC共t兲 and facilitation dynamics F共t兲. Lower panel: postsynaptic input x共t兲. For illustration purposes, the presynaptic spike times were chosen manually. Parameters: F0 = 0.1, ⌬ = 0.3, and ␶F = 200 ms.

dD共t兲 1 − D共t兲 = − F共t−兲D共t−兲x0共t兲. dt ␶D

共8兲

The initial value for D is chosen such that D varies between 0 and 1. The depression dynamics is illustrated in Fig. 3. In contrast to the facilitation dynamics, a rapid succession of presynaptic spikes leads to a decrease in the synaptic amplitude, whereas for long pauses in the presynaptic input, the synaptic amplitude recovers. The combined effect of both, facilitation and depression, is illustrated in Fig. 4. C. Total postsynaptic input

The total postsynaptic input X共t兲 to the target neuron includes the postsynaptic input xi共t兲 from the N signal synapses

共7兲 1.0

The constant F0 is the baseline release probability, i.e., the release probability in the limit of low presynaptic spiking rates. If FC Ⰶ 1 − F0, we obtain F ⬇ F0 + FC. If, on the other hand, FC is large, F approaches 1. The facilitation dynamics is illustrated in Fig. 2. As can be seen in the amplitude dynamics 共lower panel兲, a rapid succession of presynaptic spikes evokes an increase of the synaptic amplitude, whereas during large pauses of the presynaptic spike train, the synaptic amplitude drops again. 2. Depression

As in 关11–13兴, we simplify the approach chosen in 关8兴. We assume that depression is due to a refractoriness of synaptic release sites. Release sites enter a refractory state upon neurotransmitter release. Our depression variable D共t兲 is the fraction of sites that are ready for immediate release. Sites recover from the refractory state into the release-ready state with the rate 1 / ␶D 共in 关8兴, ␶D is defined differently兲. This is expressed by

presynaptic spike train

dFC共t兲 FC共t兲 =− + ⌬ ⫻ x0共t兲. dt ␶F

0.8 0.6

0.0

0.8 0.6 0.4 0.2 0.0

postsynaptic input FD variables

Ai共t兲 = A0 = const,

1.0

presynaptic spike train

共3兲

postsynaptic input FD variables

xi共t兲 = Ai共t−兲x0,i共t兲.

D(t) A(t)

0.8 -

0.6

F0D(t )

0.4 0.2 0.0 0.2 0.1 0.0

0

1

2

3

4

5

time [s]

FIG. 3. 共Color online兲 Illustration of depression-only dynamics as described by Eq. 共3兲 with A共t兲 = F0 · D共t兲 and Eq. 共8兲 with F共t兲 = F0. Upper panel: presynaptic spike train x0共t兲. Middle panel: depression dynamics D共t兲 and synaptic amplitude A共t兲. Lower panel: postsynaptic input x共t兲. Parameters: F0 = 0.4 and ␶D = 2000 ms; spike times as in Fig. 2.

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MATTHIAS MERKEL AND BENJAMIN LINDNER

presynaptic spike train

1.0

Correlations between two signals a and b can be quantified by the coherence function given by

0.8 0.6 0.4

Cab共f兲 =

0.2

postsynaptic input FD variables

0.0 F(t) D(t) A(t)

0.8 0.6

兩Sab共f兲兩2 . Saa共f兲Sbb共f兲

共14兲

This function attains values between zero and one and measures the normalized linear correlation in the frequency domain.

0.4 0.2 0.0 0.2

E. Relation to the conductance and membrane potential

0.1 0.0

0

1

2

3

4

5

time [s]

FIG. 4. 共Color online兲 Illustration of the full FD dynamics as described by Eq. 共3兲 with A共t兲 = F共t兲 · D共t兲 and Eqs. 共6兲–共8兲. Upper panel: presynaptic spike train x0共t兲. Middle panel: facilitation dynamics F共t兲, depression dynamics D共t兲, and synaptic amplitude A共t兲. Lower panel: postsynaptic input x共t兲. Parameters: F0 = 0.1, ⌬ = 0.3, ␶F = 500 ms, and ␶D = 2000 ms; spike times as in Fig. 2.

and the total synaptic input ␩共t兲 by the noise synapses 共see also Fig. 1兲: N

X共t兲 = ␩ 共t兲 + 兺 xi共t兲.

共9兲

i=1

The noise ␩ 共t兲 could likewise include intrinsic fluctuations of the target neuron, for instance, channel noise of the postsynaptic membrane. Below, we will however mainly consider the case, in which the noise statistics is shaped by either facilitation or depression, i.e., the case where the noise arises from presynaptic input. D. Spectral measures

We define the correlation function Kab共␶兲 between two stochastic processes a共t兲 and b共t兲 by Kab共␶兲 = 具a共t兲b共t + ␶兲典 − 具a共t兲典具b共t兲典.

共10兲

Kab does not depend on the absolute time t, because we assume stationarity 共i.e., we ignore any kind of transient兲. The spectrum Sab共f兲 is the Fourier transformation of the correlation function Kab共␶兲, Sab共f兲 =

冕

⬁

e2␲if ␶Kab共␶兲d␶ .

共11兲

−⬁

If a = b, then Kab共␶兲 is the autocorrelation and Sab共f兲 is the power spectrum of a共t兲. If a ⫽ b, then Kab共␶兲 is the crosscorrelation and Sab共f兲 is the cross-spectrum of a共t兲 and b共t兲. Another way to calculate the spectra is given by 关20兴 ˜ ⴱ共f ⬘兲典 , ␦共f − f ⬘兲Sab共f兲 = 具˜a共f兲b

冕

⬁

−⬁

e−2␲ifta共t兲dt.

III. THEORY

The main interest of this paper is the study of effects of STP on the information transmission across the synapse. To this end, we aim at the calculation of the coherence function between the external stimulus R and the total postsynaptic input X to the target neuron. In Appendix A, we find 1 1 N−1 1 1 = ⫻ + ⫻ CRX共f兲 N CRx共f兲 N CR具x典共f兲 +

共12兲

where ˜bⴱ denote the complex conjugate of ˜b, and ˜a denote the Fourier transform of a, ˜a共f兲 =

In this paper, we are mainly interested in the coherence CRX共f兲 between rate modulation R共t兲 and the total postsynaptic input X共t兲 and in the cross- and power spectra SRX共f兲 and SXX共f兲. However, we briefly want to point out relations to statistics which is more easily accessible in experiments than the postsynaptic input X, as for example the total membrane conductance G of the target neuron or its membrane voltage V. In voltage-clamp experiments, G共t兲 is proportional to the excitatory postsynaptic current 共EPSC兲. According to a common model for conductance-based synapses 关21兴, the conductance G is the low-pass filtered postsynaptic input X. Hence, the cross- and power spectra SRG共f兲 and SGG共f兲 are low-pass filtered versions of SRX共f兲 and SXX共f兲. We conclude that CRG共f兲 equals CRX共f兲. One may use a conductance-based leaky integrator model in order to characterize the subthreshold membrane voltage V. For a low variance of the conductance G, which is a realistic assumption for many neuronal systems, one can approximate the conductance-based leaky integrator by a current-based neuron model via the effective time constant approximation 关22,23兴 共for critical evaluations of this approximation and for improved approximation schemes, see 关24,25兴兲. With these assumptions, the cross- and power spectra SRV共f兲 and SVV共f兲 are twice low-pass filtered versions of SRX共f兲 and SXX共f兲; and furthermore, the coherence CRV共f兲 is well approximated by CRX共f兲. However, as soon as a spiking mechanism for the target neuron is taken into account, nontrivial consequences arise 关13兴.

共13兲

1 1 S␩␩共f兲 ⫻ ⫻ . N CRx共f兲 NSxx共f兲

共15兲

This is one of the central formulas used in this work, because it illustrates the influence of the number of synapses N, the single synapse coherence CRx共f兲, and noise S␩␩共f兲 on the coherence CRX共f兲. The coherence CR具x典共f兲 between R and 具x典x0兩R turns out to be approximately one 共see Appendix A 2兲.

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Note that additional noise enters Eq. 共15兲 only via its power spectrum S␩␩. In the following sections, we first present our results for a single synapse 共see Sec. IV兲 and then discuss the general case including multiple synapses and additional noise 共see Sec. V兲. In this section, we start with some general considerations about CRx共f兲 and CRX共f兲.

well suited to describe this linear information transmission is the coherence function C共f兲 as defined in Eq. 共14兲. It quantifies, how well a transmission can be expressed by a linear filter—and in addition, for Gaussian signals R, it gives a lower bound on the mean information rate Rinfo共R , x兲 关28–31兴, Rinfo共R,x兲 ⱖ Rinfo,LB共R,x兲 = −

In this section, we treat the single-synapse situation 共i.e., N = 1兲 from a more formal point of view. We consider the mean information rate Rinfo 共i.e., the mutual information that is transmitted per unit of time兲 over a single synapse in the absence of external noise ␩. First of all, we ask for Rinfo共R , x0兲 between R共t兲 and the presynaptic spike train x0共t兲. For a rate-modulated Poisson process, this information rate reads to the lowest significant order in ␧ 关using Eq. 共18兲 from 关26兴; compare also 关27兴兴 1 2

冕

⬁

log2关1 + ␧2rSRR共f兲兴df ⬇

−⬁

Note that without STP, the postsynaptic response x equals the presynaptic spike train x0. Hence, Rinfo共R , x0兲 is also the mean information rate between R共t兲 and the postsynaptic response in a situation without STP. In this paper, we are interested in the effect of STP on the information transmission of a synapse. In a strict sense, STP as modeled in Eqs. 共3兲–共8兲 and in the absence of external noise does not change the mean information rate: for ␩ ⬅ 0.

log2关1 − CRx共f兲兴df . 共18兲

Note that x does not need to be Gaussian for the lower bound condition in Eq. 共18兲 to hold; according to 关30兴, it suffices that the stimulus R is Gaussian—as we assume in this work. First, we want to consider the coherence function CRx0共f兲 between R and the presynaptic spike train x0. Following 关30,32兴, we can write to the lowest significant order in ␧: CRx0共f兲 ⬇ ␧2rSRR共f兲.

共19兲

Putting this into Eq. 共18兲 and comparing to Eq. 共16兲, one sees

␧ 2r . 2 ln 2 共16兲

Rinfo共R,x兲 = Rinfo共R,x0兲

⬁

0

A. Information rate and coherence function

Rinfo共R,x0兲 ⬇

冕

1 2

Rinfo,LB共R,x0兲 ⬇

⬁

log2关1 + ␧2rSRR共f兲兴df ⬇ Rinfo共R,x0兲.

−⬁

共20兲 Hence, for a Poissonian spike train with constant amplitude, the lower bound coincides with the information rate itself. Using Eq. 共18兲, we can conclude that—to the lowest significant order in ␧—the integral of the single-synapse coherence CRx共f兲 cannot become greater than the integral of the coherence of the presynaptic spike train CRx0共f兲:

冕

共17兲

⬁

0

This can be seen as follows. On the one hand, the information carried by the postsynaptic input x共t兲 cannot be more than that carried by the presynaptic spike train x0共t兲, because x共t兲 depends exclusively on x0 关according to Eqs. 共3兲–共8兲兴. Put differently, x cannot contain additional information on the rate modulation R共t兲 which is not yet present in x0共t兲. On the other hand, x contains not less information than x0, because one can reconstruct x0 from x by simply extracting only the spike times from x and ignoring the amplitudes 共this could be realized, for instance, by a spike detector with arbitrarily small threshold兲. Therefore, whatever information is carried by x0, it must also be carried by x. Hence, in the absence of additional noise and for the deterministic STP model treated here, synaptic plasticity does not change the basic information content of a single spike train, although the amplitudes of the spikes are modulated. However, the dynamics of the neuron that is driven by the postsynaptic input x may profit or suffer from this amplitude modulation. One might also regard a dynamic synapse as preprocessing instance for the neuron in the sense that it prepares the spike train for neuronal processing. In this work, we want to consider a case, where the neuron’s dynamics depend mainly linearly on the synaptic input it receives. Therefore, we are interested in the linear information transmission properties of a dynamic synapse. A quantity that is

冕

CRx共f兲df ⱕ

冕

⬁

0

CRx0共f兲df .

共21兲

Indeed, we observe that even CRx共f兲 ⱕ CRx0共f兲 for all f ⬎ 0 共see below兲. In this work, we want to examine the coherence function CRx共f兲 between R共t兲 and the postsynaptic input x共t兲 in dependence on frequency. We use CRx共f兲 in order to judge, whether dynamic synapses can be considered as preprocessing units for those neurons which evaluate mainly the linear information content of their postsynaptic input. If, for example, at a certain frequency f 1, CRx共f 1兲 is considerably less than CRx0共f 1兲 while at another frequency f 2, CRx共f 2兲 is close to CRx0共f 2兲, frequency-dependent information filtering would come into play. We emphasize, that Eq. 共17兲 holds true for a single synapse neglecting stochasticity of transmitter release. For the total synaptic input, the coherence and the mutual information is expected to be reduced, for instance, by an external noise ␩共t兲. The extent of this reduction is addressed in Sec. V B. B. Main idea for the calculation of CRx

We present a way to calculate the coherence function CRx共f兲 according to Eq. 共14兲. In order to do this, we show how to derive the power spectrum Sxx共f兲 as well as the crossspectrum SRx共f兲.

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冋

Using Eqs. 共10兲 and 共11兲 and stationarity, we can calculate Sxx共f兲 for f ⫽ 0 by the Fourier transformation of 具x共t兲x共t + ␶兲典 = 具具x共t兲x共t + ␶兲典x0典R

共22兲

and SRx共f兲 for f ⫽ 0 by the Fourier transformation of 具R共t兲x共t + ␶兲典 = 具R共t兲具x共t + ␶兲典x0典R ,

共23兲

where 具 · 典x0 indicates averaging over the x0 ensemble for frozen R and 具 · 典R indicates averaging over the R ensemble. The main difficulty of the derivation is the calculation of the averages 具x共t兲x共t + ␶兲典x0

具x共t + ␶兲典x0

and

F共t兲 ⬇ Flin共t兲 = F0,lin + ⌬lin

冕

where F0,lin and ⌬lin are constants that depend on the model parameters and r. For linear facilitation models as used for example in 关13兴, F0,lin and ⌬lin correspond to F0 and ⌬, respectively. For the depression dynamics, we also use a linearized equation in which, however, the input enters in a multiplicative fashion 共cf. Appendix D兲. The solution is for this reason more complicated and reads D共t兲 ⬇ Dlin共t兲 =

冕

t

−⬁

e−共t−t⬘兲/␶De共t⬘+,t兲

冉

1 − x 共t⬘兲 ␶D 0

冊

共26兲

册

x0共t⬘兲dt⬘ ,

ta

ta ⱕ tb . 共27兲

共28兲

where the time instants t1 , t2 , . . . , t j , t j+1 , . . . may in general be completely arbitrary 共including, that any two of these time variables may be identical兲, except for: t j ⱕ t j+1,

t j+2 ⱕ t j+3,

... .

For the sake of illustration of how the averages Eq. 共28兲 appear in the calculation of spectral measures, let us consider the simple case of pure depression 共F共t兲 ⬅ F0兲. For the crossspectrum SRx共f兲, we obtain SRx共f兲 = F0

冕

⬁

−⬁

e2␲if ␶具R共t兲具Dlin共t + ␶−兲x0共t + ␶兲典x0典Rd␶ . 共29兲

Inserting the explicit solution for Dlin共t兲, we find for the average 具Dlin共t−兲x0共t兲典x0 共using that for pure depression Flin = F 0兲 具Dlin共t−兲x0共t兲典x0 =

1 ␶D

冕

t−

−⬁

e−共t−t⬘兲/␶D具x0共t兲e共t⬘+,t−兲典x0dt⬘

− 共具Dlin典F0 + 具FlinDlin典 − 2具Flin典具Dlin典兲

⫻

冕

t−

−⬁

=

1 ␶D

e−共t−t⬘兲/␶D具x0共t⬘兲x0共t兲e共t⬘+,t−兲典x0dt⬘

冕

t−

−⬁

e−共t−t⬘兲/␶D具x0共t兲e共t⬘+,t−兲典x0dt⬘ .

共30兲

This way, we express the average 具Dlin共t−兲x0共t兲典x0 in terms of 具x0共t兲e共t⬘+ , t−兲典x0, which is of the kind described by Eq. 共28兲. For the more involved case of full FD dynamics, the derivation of the power spectrum requires the calculation of averages as in Eq. 共28兲 with up to eight x0 factors. Given, that we can reduce the spectral measures to averages like in Eq. 共28兲, we now have to clarify how to actually calculate these averages. Our method of solution for this problem 共outlined in Appendix B兲 is to express averages like in Eq. 共28兲 in terms of averages of products of another inhomogeneous Poissonian ␦ spike train xˆ0 具xˆ0共t1兲xˆ0共t2兲¯典xˆ0 ,

⫻关具Dlin典Flin共t⬘−兲 + 具FlinDlin典 − 2具Flin典具Dlin典兴 dt⬘ ,

where we define the term e共ta , tb兲 as follows:

tb

具x0共t1兲x0共t2兲 ¯ e共t j,t j+1兲e共t j+2,t j+3兲¯典x0 ,

e−共t−t⬘兲/␶Fx0共t⬘兲dt⬘ , 共25兲

−⬁

冕

The constants F1 = 具Flin典, 具Dlin典, and 具FlinDlin典 depend merely on model parameters and r 共cf. Appendix D兲. Regarding the explicit dependencies of F and D on x0, we find, that the averages in Eq. 共24兲 take the form of sums of multiple convolution integrals over averages of the kind

共24兲

to the lowest significant order in ␧. Once expressions for these averages have been found, the subsequent averaging over the R ensemble and the Fourier transformation are elementary. For the calculation of the averages in Eq. 共24兲, we adopt the following strategy: First, we attempt to express x explicitly in terms of the unmodulated spike train x0. Second, we calculate these averages, now only involving x0. In the following, we describe both steps in more detail. Concerning the first step, we can write the averages in Eq. 共24兲 in terms of F, D, and x0, because, x共t兲 = F共t−兲D共t−兲x0共t兲 关cf. Eqs. 共3兲 and 共4兲兴. What remains is the derivation of explicit formulas expressing F and D in terms of x0共t兲. Unfortunately, because of the static nonlinearity in F共FC兲 in the facilitation dynamics and because of the term F ⫻ D in the depression dynamics, this appears to be disproportionally complicated. For this reason, we linearize the F and D dynamics as outlined below. For the facilitation dynamics, we linearize the dependency F共FC兲 as discussed in Appendix C. In our linear approximation, F is a constant plus a convolution of x0: t

e共ta,tb兲 = exp ln共1 − F1兲

共31兲

where the firing rate of xˆ0 is

␯ˆ 共t兲 = ␯共t兲共1 − F1兲k . These terms can be calculated following Stratonovich 关20兴. For the special cases of either pure facilitation or pure de-

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SYNAPTIC FILTERING OF RATE-CODED INFORMATION

具A典 ⬇ 具具Flin共t兲Dlin共t兲典x0典R .

~|cross 2 spectrum|

0.2

simulation theory 0.1

A 0.0

~power spectrum

pression, the required averages can be calculated by hand. In the general case of both facilitation and depression present, we used a Python script to compute the formulas for the averages. Note that the average of any product of F, D, and x0 can be derived in that way—for instance, the average synaptic amplitude

0.1

B

Remarkably, the resulting expression for 具A典 turns out to be the same as that derived in 关13兴, although different approximations have been made.

coherence

0.0 0.004 0.002

C 0.000

It is convenient to introduce the following dimensionless single synapse spectra 共approximating the spectra to the lowest significant order in ␧兲 sˆxx共f兲 ⬇

Sxx共f兲 , r

sˆRx共f兲 ⬇

SRx共f兲 . ␧rSRR共f兲

共32兲

Without short-term plasticity, they become exactly one for all frequencies: sˆx0x0共f兲 = sˆRx0共f兲 = 1 共this can be derived as explained in Sec. III B or as previously described in 关30,32兴兲. Short-term plasticity turns them into frequency-dependent functions. We express Eq. 共15兲 by means of these dimensionless spectra. Using CR具x典共f兲 ⬇ 1 共see Appendix A 2兲, we obtain to the most significant order in ␧ 1 1 N−1 1 sˆxx共f兲 ⬇ ⫻ 2 ⫻ + 2 CRX共f兲 N ␧ rSRR共f兲 兩sˆRx共f兲兩 N +

1 1 S␩␩共f兲 ⫻ ⫻ . N ␧2rSRR共f兲 Nr兩sˆRx共f兲兩2

0.1

1

10

frequency [Hz]

C. Contributions to CRX

共33兲

A number of interesting results are apparent in this equation. First, a small ␧ leads to a large first and third term in Eq. 共33兲. Hence, a weak influence of the external stimulus on the rate modulation diminishes linear information transmission. Similarly, a low firing rate r combined with a low power spectrum of the external stimulus SRR decreases the linear information transmission. Finally, a large number of input neurons N improves linear information transmission, in particular when N␧2rSRR ⲏ 1. Moreover, one can always bring the coherence arbitrarily close to 1 by increasing N sufficiently. In the following two sections, we investigate the effect of each of the three terms in Eqs. 共15兲 and 共33兲.

FIG. 5. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲 for the case of facilitation only. The parameter values are F0 = 0.1, ⌬ = 0.3, r = 10 Hz, ␶F = 80 ms, and ␧ = 0.2. Circles show the results of a simulation without any approximations made 共error bars within symbol size兲; while the solid lines illustrate the theoretical predictions expressed in Eqs. 共38兲, 共41兲, and 共34兲, respectively. The spectra show pronounced low-pass behavior while the coherence function is rather flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴.

are odd with respect to ␧. This is a direct consequence of the two facts that on the one hand, ␧ accompanies R as factor; and on the other hand, the distribution of R is symmetric under sign reversal. In addition, we always assume f ⫽ 0 共i.e., we will ignore the DC components of the spectra兲. For convenience, we will use the dimensionless versions sˆxx共f兲 and sˆRx共f兲 of the spectra as defined in Eq. 共32兲. The coherence without noise is then given by Eq. 共33兲 with N = 1 and S␩␩ = 0, 兩sˆRx共f兲兩2 . sˆxx共f兲

共34兲

CRx共f兲 = CRx0共f兲 ⬇ ␧2rSRR共f兲,

共35兲

CRx共f兲 ⬇ ␧2rSRR共f兲 Note that for a static synapse,

because in that case, sˆx0x0共f兲 = sˆRx0共f兲 = 1 共see above兲. We discuss the special cases of pure facilitation and pure depression, before we make some remarks about the general case. A. Pure facilitation

For the pure facilitation case, we use

IV. RESULTS FOR A SINGLE SYNAPSE

Without noise 关␩共t兲 ⬅ 0兴 and for N = 1, only the first term in Eq. 共15兲 remains. In this section, we present our results for CRx共f兲 and spectra Sxx共f兲 and SRx共f兲. We calculate everything to the lowest significant order in ␧ 共results that have been approximated in such a way are marked by “⬇”兲. Simulations show that for ␧ ⱗ 0.2, this is sufficient to characterize the spectra 共see Figs. 5, 7–10, and 13–16兲. This can be justified by the fact that all terms in Sxx and in 兩SRx兩2 vanish that

Ai共t兲 = Flin,i共t兲

共36兲

instead of Eq. 共4兲. In other words, we ignore the depression variable 共D共t兲 ⬅ 1兲 and use the linearized F-dynamics from Appendix C. The facilitation-only dynamics is exemplified in Fig. 2. The results are discussed in the following and illustrated in Figs. 5–7. For the cross-spectrum and its absolute square, we obtain

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MATTHIAS MERKEL AND BENJAMIN LINDNER

|S

R x

( 0 ) |2 / |S

R x

/ F lin

∆ -1

1 0

r τ R x

0

1 0

1 0

R x

(∞

F1 = F0,lin + ⌬linr␶F .

C

0 ,lin

R x

0

The absolute square of the cross-spectrum 兩sˆRx共f兲兩 can be understood as the sum of a constant plus a Lorentzian lowpass spectrum, i.e., pure facilitation has a low-pass effect. The low- and high-frequency limits of the squared crossspectrum are given by 1

1 0

F

(0 )/ C

R x

(∞

∆

lin

∆

1 .1

1 0

)

-1

兩sˆRx共f兲兩2 ⬇

-2

1 .0

-1

1 0

r τ

0

1 0

1

1 0

r τ

0

1 0

1

FIG. 6. 共Color online兲 Theoretical results for 兩SRx共0兲兩2 / 兩SRx共⬁兲兩2 and for CRx共0兲 / CRx共⬁兲. 共A兲 兩SRx共0兲兩2 / 兩SRx共⬁兲兩2 depending on r␶F and the facilitation intensity expressed by the parameters for linear facilitation dynamics ⌬lin / F0,lin. 共B兲 兩SRx共0兲兩2 / 兩SRx共⬁兲兩2 for F0 = 0.04, depending on r␶F and ⌬. 共C兲 CRx共0兲 / CRx共⬁兲 depending on r␶F and ⌬lin / F0,lin. 共D兲 CRx共0兲 / CRx共⬁兲 for F0 = 0.04, depending on r␶F and ⌬. The ⌬ axes in 共B兲 and 共D兲 are scaled such that ⌬ / F0 ranges from 0.1 to 10 like ⌬lin / F0,lin in 共A兲 and 共C兲.

sˆRx共f兲 ⬇ F1 +

⌬linr␶F , 1 – 2␲if ␶F

共37兲

and

~|cross 2 spectrum|

simulation theory

-3

A

~power spectrum

0×10

-3

4×10

-3

2×10

-3

B

coherence

0×10

-4

4×10

-4

2×10

-4

0×10

共F0,lin + ⌬linr␶F兲2

for f → ⬁.

共40兲

共41兲

Both cross- and power spectra predicted by these formulas show good agreement with the results of numerical simulations 共see, for example, Figs. 5 and 7, 共A兲 and 共B兲, respectively兲. There is a slight deviation in the power spectrum, which is overestimated by Eq. 共41兲. We have verified that this deviation is due to the linearization approximation for the F dynamics outlined in Appendix C. The coherence CRx共f兲 as calculated from cross- and power spectra reads

-3

1×10

冎

共F0,lin + 2⌬linr␶F兲2 for f → 0 and

1 2 sˆxx共f兲 ⬇ 兩sˆRx共f兲兩2 + ⌬lin r␶F . 2

-3

-3

再

Hence, the low-pass behavior is most pronounced for a high ratio ⌬linr␶F / F0,lin 共i.e., for high rates and for high ⌬lin compared to F0,lin兲. The ratio between the two limits is shown in Fig. 6共A兲 as a function of r␶F and ⌬lin / F0,lin and illustrates the parameter region of low-pass behavior of the cross spectrum. In terms of the original parameters ⌬ and r␶F 关compare Fig. 6共B兲兴, we see a similar effect: For high input rate and a large increment ⌬, the low-pass behavior is most pronounced. An exception to this is observed in the limit of very large ⌬ and r␶F, which is due to the saturation effect in the nonlinearity F共FC兲 in Eq. 共7兲: for a very high-input rate and high ⌬, the effective increment ⌬lin decreases with growing ⌬ and, thus, leads to a weaker facilitation effect and a less pronounced low-pass behavior of the cross spectrum. Remarkably, the expression for the power spectrum is similar to that of the absolute square of the cross-spectrum. To the lowest significant order in ␧, sˆxx共f兲 equals 兩sˆRx共f兲兩2 up 2 to a shift of 21 ⌬lin r ␶ F:

3×10 2×10

共39兲 2

1 .3 0

1 关共F1 + ⌬linr␶F兲2 + 共2␲ f ␶F兲2F21兴 , 1 + 共2␲ f ␶F兲2

-2

D 1 0

兩sˆRx共f兲兩2 ⬇

where

1 0

)

) |2

(∞

共38兲

F

(0 )/ C

R x

-1

r τ

1 .2

1 0

( 0 ) |2 / |S

1

1 0

1 .4 1 0

R x

∆

0 ,lin 0

C

/ F

|S

3 .6 3 .2 2 .8 2 .4 2 .0 1 .6 1 .2

1 0

1 0

B

) |2

(∞

C 0.1

1

冋

CRx共f兲 ⬇ ␧2rSRR共f兲 1 +

10

frequency [Hz]

FIG. 7. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲 for the case of facilitation only. The parameter values are F0 = 0.01, ⌬ = 0.3, r = 1 Hz, ␶F = 80 ms, and ␧ = 0.2. Circles show the results of a simulation without any approximations made 共error bars within symbol size兲; while the solid lines illustrate the theoretical predictions expressed in Eqs. 共38兲, 共41兲, and 共34兲, respectively. The spectra and the coherence function show pronounced low-pass behavior. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴.

冋

= CRx0共f兲 1 +

2 r␶F/2 关1 + 共2␲ f ␶F兲2兴⌬lin

共F1 + ⌬linr␶F兲2 + 共2␲ f ␶F兲2F21 2 r␶F/2 关1 + 共2␲ f ␶F兲2兴⌬lin

共F1 + ⌬linr␶F兲2 + 共2␲ f ␶F兲2F21

册

册

−1

−1

. 共42兲

The latter relation reflects the fact that the coherence of dynamic synapse CRx共f兲 is always smaller than that of the static synapse CRx0共f兲, because the inverted square bracket is always smaller than one.

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共43兲

Note that this ratio only depends on two dimensionless parameters: ⌬lin / F0,lin and r␶F. The first characterizes the strength of facilitation approaching zero in the absence of facilitation 共i.e., for ⌬lin = 0兲. The second parameter, r␶F, is the ratio of two time scales: the decay time constant of facilitation and the mean interspike interval of the input spike train. The coherence is broadband if one or both of the following conditions are met: 1 r␶F Ⰶ 共F0,lin/⌬lin + r␶F兲2 2

and

r␶F Ⰶ F0,lin/⌬lin . 共44兲

Closer inspection of these inequalities reveals that indeed for most parameter values, the synapse performs broadband coding. The ratio in Eq. 共43兲 shown in Fig. 6共C兲 as a function of ⌬lin / F0,lin and r␶F is close to one unless for strong facilitation and for intermediate values of the time scale ratio r␶F, where the coherence differs by a moderate factor of 1.5 between the low and high-frequency limits. The effect is less pronounced in terms of the original parameters ⌬ and r␶F as depicted in Fig. 6共D兲 for F0 = 0.04. An example for a low-pass coherence is shown in Fig. 7, where the coherence suffers a considerable overall reduction compared to the case of a static synapse 共dotted line兲. Note that in this case, the ratio between ⌬ and F0 is extremely large—typical values are rather ⌬ / F0 ⱗ 5 关2,8,12兴.

~|cross 2 spectrum|

0.03

A

0.02

simulation theory

0.01 0.00

~power spectrum

2 + r␶F/共F0,lin/⌬lin + r␶F兲2 CRx共0兲 = . CRx共⬁兲 2 + r␶F/共F0,lin/⌬lin + 2r␶F兲2

0.04

0.03

B

0.02 0.01 0.00

coherence

In order to find conditions for broadband coding, we inspect the ratio between low and high-frequency limits of the coherence. This ratio is given by

0.004 0.002 0.000

C 0.1

1

FIG. 8. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲 for the case of depression only. The parameter values are F0 = 0.4, r = 10 Hz, ␶D = 300 ms, and ␧ = 0.2. Circles show the results of a simulation without any approximations made 共error bars within symbol size兲; while the solid lines illustrate the theoretical predictions expressed in Eqs. 共47兲, 共50兲, and 共34兲. The spectra show pronounced high-pass behavior while the coherence function is perfectly flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴.

to a high-pass effect of the transfer function for pure depression. The low- and high-frequency limits are given by

再

兩sˆRx共f兲兩2 ⬇

F20/␤2 for f → ⬁.

冋

sˆxx共f兲 ⬇ 兩sˆRx共f兲兩2 1 −

instead of Eq. 共4兲 关the facilitation variable is set to F共t兲 = F0兴. Note that the parameter F0 not only scales the amplitude in Eq. 共45兲, but also the multiplicative decrease in the D variable 关compare Fig. 3 and Eq. 共8兲兴. The results calculated to the lowest significant order in ␧, are illustrated in Fig. 8 and discussed in the following. We obtain for the cross-spectrum and its absolute square: sˆRx共f兲 ⬇

冉

F 0r ␶ D/ ␤ F0 1− ␤ 1 − 2␲if ␶D/␤

冊

共46兲

and F2 1 + 共2␲ f ␶D/␤兲2␤2 , 兩sˆRx共f兲兩2 ⬇ 40 ⫻ ␤ 1 + 共2␲ f ␶D/␤兲2

共49兲

Hence, the high-pass behavior is most pronounced for large F0r␶D 共i.e., for strong depression and high rates兲. The power spectrum sˆxx共f兲 shows 共up to a constant factor兲 exactly the same behavior,

For the pure depression case, we use 共45兲

冎

F20/␤4 for f → 0 and

B. Pure depression

Ai共t兲 = F0Dlin,i共t兲,

10

frequency [Hz]

F20r␶D 2␤

册

−1

.

共50兲

This results in a coherence function, that is does not depend on frequency at all

冋

CRx共f兲 ⬇ ␧2rSRR共f兲 1 −

册

冋

册

F20r␶D F 2r ␶ D = CRx0共f兲 1 − 0 . 2␤ 2␤ 共51兲

The broadband coherence for pure depression is a nontrivial result. As in the case of pure facilitation, one can also verify for the case of pure depression that the coherence CRx共f兲 is lower than that of a static synapse CRx0共f兲. The reduction compared to the static case can be neglected CRx共f兲 ⬇ CRx0共f兲,

共47兲

F20r␶D Ⰶ 1. 2␤

共52兲

Hence, a substantial reduction of the coherence is only expected for high F0 and moderate to large r␶D.

with

␤ = 1 + F 0r ␶ D .

共48兲

The absolute square of the cross-spectrum 兩sˆRx共f兲兩 can be regarded as a constant minus a Lorentzian spectrum leading 2

C. General case

For the general case, we use the full coupled FD dynamics according to Eq. 共4兲, which is illustrated in Fig. 4. For

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MATTHIAS MERKEL AND BENJAMIN LINDNER

k=1

sˆc,k . 1 + 共2␲ f ␶k兲2

共53兲

Here, sˆc,⬁ is a constant indicating the high-frequency limit of 兩sˆRx共f兲兩2. The constants sˆc,k 共with k = 1 , . . . , 4兲 are factors for Lorentzian terms while the ␶k are the corresponding time constants. The expressions for these four time constants are simple

~|cross 2 spectrum|

0.1

0.0

~power spectrum

4

兩sˆRx共f兲兩2 ⬇ sˆc,⬁ + 兺

0.2

simulation theory

A

0.1

0.0

coherence

both—facilitation and depression—present, our results for power and cross-spectra become very involved. To the lowest significant order in ␧, we obtain the following general structure for the absolute square of the crossspectrum,

B

0.004 0.002

C 0.000

0.1

␶1 = ␶F , ␶D , 1 + F 1r ␶ D

␶3 = ␶F/2, 1 + F1r˜␶

,

共54兲

−1 −1 where F1 = F0,lin + ⌬linr␶F and ˜␶ = 关␶F−1 + ␶D 兴 . The expresˆ ˆ sions for the constants sc,⬁ and 兵sc,k其 are too cumbersome to be presented here. The general expression of the power spectrum is of the same kind as 兩sˆRx共f兲兩2, 4

sˆxx共f兲 ⬇ sˆ p,⬁ + 兺

k=1

sˆ p,k . 1 + 共2␲ f ␶k兲2

lations are in general in a good accordance with the theory— except for the aforementioned deviation, due to the Dlin approximation, which comes into play for high r␶D.

共55兲

The time constants ␶k of the Lorentzians are the same as for 兩sˆRx共f兲兩2. However, the coefficients in Eq. 共55兲 differ from those in Eq. 共53兲, are rather lengthy 共the complete formula amounts to ⬃100 kB of Python code兲, and will not be stated here explicitly. As discussed previously 关13兴, one can find parameter ranges where facilitation or depression will dominate and where the spectral statistics are similar to the cases of pure facilitation and depression discussed above. However, we can also find parameter sets, where new spectral features are observed, in particular if the time scales of facilitation and depression differ significantly. Examples are displayed in Figs. 9 and 10, they show a minimum or a maximum in the power and cross-spectra, which are predicted by theory and confirmed by simulations. Substantial deviations are seen in Fig. 10共A兲. Closer inspection reveals that these deviations are mainly due to the Dlin approximation 共cf. discussion in Appendix D兲. In order to characterize the changes induced by STP over a large part of the parameter space, we consider again the ratio between low and high frequency limits of power and cross-spectra 共see Fig. 11兲. We find that the low-pass property of both, 兩SRx共f兲兩2 and Sxx共f兲, behave similar over the whole domain shown. For moderate r␶F and low r␶D, we find a strong low-pass behavior. In contrast to that, for high r␶D, we find a strong high-pass behavior. The results of the simu-

0.02

~|cross 2 spectrum|

˜␶

A 0.01

simulation theory

0.00

~power spectrum

␶4 =

10

FIG. 9. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲 for the general case 共facilitation and depression兲. The parameter values are F0 = 0.1, ⌬ = 0.3, r = 10 Hz, ␶F = 300 ms, ␶D = 100 ms, and ␧ = 0.2. Circles show the results of a simulation without any approximations made 共error bars within symbol size兲; while the solid lines illustrate the theoretical predictions. The spectra show a pronounced minimum while the coherence function is rather flat. There is only a slight deviation of the spectra between theory and simulations for low frequencies that cancel out for the coherence. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴.

B 0.01

0.00

coherence

␶2 =

1

frequency [Hz]

0.004 0.002

C 0.000

0.1

1

10

frequency [Hz]

FIG. 10. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲 for the general case 共facilitation and depression兲. The parameter values are F0 = 0.1, ⌬ = 0.3, r = 10 Hz, ␶F = 100 ms, ␶D = 500 ms, and ␧ = 0.2. The circles show the results of a simulation without any approximations made 共error bars within symbol size兲; while the solid lines illustrate the theoretical predictions. The spectra show a pronounced maximum, while the simulated coherence is flat again. For the power spectrum, theory and simulation are in good accordance. However, the cross-spectrum shows a deviations for low frequencies, which leads to deviations in the coherence function. However, the coherence function as determined from simulations is rather flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴.

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A

B

c r o s s -s p e c tr u m 1

1 0

0 .5 7 ( 0 .5 3 )

0 .4 0 ( 0 .3 1 )

0 .1 6 ( 0 .1 5 )

p o w e r s p e c tr u m 1

1 0

2 .0 0

0 .1 0 ( 0 .1 1 )

0 .5 0 ( 0 .5 2 )

1 .7 5

0 .4 0 ( 0 .4 6 )

0 .1 6 ( 0 .3 2 )

0 .1 0 ( 0 .1 3 )

1 .5 0

D

1 0

1 .3 5 ( 1 .2 8 )

0 .9 2 ( 0 .8 0 )

1 .0 0

r τ

r τ

D

1 .2 5 0

1 0

0

1 .2 2 ( 1 .2 2 )

0 .9 3 ( 0 .8 1 )

0 .7 5

1 .6 2 ( 1 .6 1 ) 1 0

0 .5 0

1 .0 6 ( 1 .1 6 )

1 .4 6 ( 1 .5 0 )

0 .2 5

-1

1 0

-1

1 0

r τ

0

1 0

1 0 1

1 .0 2 ( 1 .1 4 )

-1

1 0

-1

1 0

r τ

0

1 0

1

FIG. 11. 共Color online兲 Theoretical results for 共A兲 兩SRx共0兲兩2 / 兩SRx共⬁兲兩2 and 共B兲 Sxx共0兲 / Sxx共⬁兲. The low-pass property is shown in dependence of r␶F and r␶D for a large part of the physiological parameter space. The dots with the annotations show simulation results for the ratios between the values at two different frequencies 共simulation uncertainties on the left side: less than 3%; on the right side: less than 0.1%兲, while the values in parentheses are the corresponding theoretical results. The values of the spectra have been calculated at the frequencies 1.5 mHz and 25 Hz. They have been chosen so that according to Eqs. 共53兲–共55兲, the spectra do not vary much outside of this interval. The dots with the red 共gray兲 annotations correspond to the simulations shown in Figs. 9 and 10. Here, the frequency values 15 mHz and 27 Hz have been used. We simulated at constant r = 1 Hz varying ␶F and ␶D. For theory and simulations, we used the parameter values F0 = 0.1 and ⌬ = 0.3.

Turning to the coherence function, the theory predicts for a large part of the physiological parameter space broadband coding, i.e., the coherence between low and high frequencies deviate not more than 10% 共see Fig. 12兲. For low r␶D, facilitation seems to dominate and consequently, we find a mild low-pass behavior which is also confirmed by simulations 共compare the numbers in Fig. 12兲. At large r␶D and moderate r␶F, the theory predicts a strong high-pass behavior of the coherence, which is not confirmed by simulations. An example for such a parameter set is shown in Fig. 10共C兲, which illustrates how the deviations in cross- and power spectra between theory and simulations lead to a strong error in their ratio, i.e., in the coherence function. We find from simulations, that in this region of parameter space, the coherence is rather flat. The above simulations and analytical calculations were carried out for fixed F0 = 0.1 and ⌬ = 0.3. We have observed, that for smaller values of ⌬, differences between low- and high-frequency values are even smaller as well as the general discrepancies between theory and simulations. Put differently, the parameter set considered in Fig. 10 represents an extreme example. In conclusion, while for parameter regimes of dominating facilitation or depression, we find low-pass and high-pass behavior, respectively, for the cross-spectrum, the coherence function for a single synapse is largely independent of frequency.

V. RESULTS FOR MANY SYNAPSES AND ADDITIONAL NOISE

Let us state again Eq. 共15兲, 1 1 1 N−1 1 = ⫻ + ⫻ CRX共f兲 N CRx共f兲 N CR具x典共f兲 +

1 1 S␩␩共f兲 ⫻ ⫻ . N CRx共f兲 NSxx共f兲

So far we have seen that the first term, which is essentially determined by the coherence for a single synapse, is largely independent of frequency. In this section, we want to examine the influence of the second and the third term in the above equation, reflecting the effects of a multitude of synapses and of an additional noise. In particular, we want to know whether the broadband property of CRX共f兲 is maintained or even enhanced for many synapses and/or in the presence of noise. A. Many synapses without additional noise

In general, averaging over many independent spike trains with common rate modulation improves the coherence with this rate modulation and thus, the coherence increases with N. Here we ask how the coherence increases with N and how its frequency dependence is affected by N. To this end, we

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1 0

1

1 .0 1 ( 0 .6 8 )

1 .0 5 ( 0 .8 5 )

0 .9 6 0 .8 8

r τ

D

0 .8 0 1 0

N=10000 N=1000

0.1

coherence CRX

1 .1 3 ( 1 .0 0 )

1 .0 0 ( 0 .4 7 )

1 .0 4

N=100

0.01

N=10

0.001

0

1 .1 0 ( 1 .0 5 )

1 .0 0 ( 0 .9 9 )

N=1

0 .7 2 0.0001

simulation theory

0 .6 4 0.01

1 .1 2 ( 1 .0 8 ) 1 0

-1

1 0

-1

1 .0 5 ( 1 .0 2 ) 1 0

r τ

0

0 .4 8 1 0

1

FIG. 12. 共Color online兲 Theoretical results for the low-pass property of the coherence function CRx共f兲 expressed by the ratio of the low- and the high-frequency limit: CRx共0兲 / CRx共⬁兲. The lowpass property is shown in dependence of r␶F and r␶D for a large part of the physiological parameter space. The dots with the annotations show simulation results for the ratios between the values at two different frequencies 共simulation uncertainties: less than 3%兲, while the values in parentheses are the corresponding theoretical results. The dots with the red 共gray兲 annotations correspond to the simulations shown in Figs. 9 and 10. Theory and simulation parameters are the same as in Fig. 11.

consider Eq. 共15兲 without additional noise 关S␩␩共f兲 = 0兴, 1 1 1 N−1 1 = ⫻ + ⫻ . CRX共f兲 N CRx共f兲 N CR具x典共f兲

NCRx Ⰶ 1.

共56兲

共57兲

Hence, for small N, the multiple synapse coherence grows proportionally with the number of synapses. Therefore, any low-pass or high-pass behavior in CRx共f兲 with respect to frequency 共i.e., deviations from broadband coding兲 is preserved in this case. In the opposite limit of large N, we obtain, CRX共f兲 ⬇ 1 −

1 NCRx共f兲

,

NCRx Ⰷ 1.

1

10

frequency [Hz]

FIG. 13. 共Color online兲 Influence of the number N of synapses on the coherence function CRX共f兲. The synapses are purely facilitating with parameters as in Fig. 7. We assume zero noise ␩ = 0. Simulations 共circles兲 agree well with the theoretical curves 共black solid lines兲 corresponding to Eq. 共56兲. For N ⱕ 103, the coherence function is proportional to N preserving the low-pass property 关compare Eq. 共57兲兴. However, for N = 104, we see that CRX共f兲 approaches 1 and starts to flatten out.

frequency limits of the coherence decreases for growing N. This indicates an enhancement of the broadband property for many synapses. In particular, the absolute and relative differences ␦C = 兩C共f 1兲 − C共f 2兲兩 and ␦C / C共f 1兲 decrease with N 共for arbitrary f 1 and f 2兲:

␦CRX ⬇

␦CRx 2 NCRx 共f 1兲

,

NCRx Ⰷ 1

共59兲

and

We would like to recall that CR具x典共f兲 ⬇ 1 关see Eq. 共A7兲 in Sec. A 2兴. It is instructive to consider the two limiting cases of small and large N. If N is small, we find that we can neglect the second term in Eq. 共56兲, CRX共f兲 ⬇ NCRx共f兲,

0.1

0 .5 6

␦CRX CRX共f 1兲

1 ␦CRx ␦CRx ⫻ Ⰶ , NCRx共f 1兲 − 1 CRx共f 1兲 CRx共f 1兲

NCRx Ⰷ 1. 共60兲

In order to further illustrate the N dependence of CRX共f兲, we consider a situation of N facilitating input synapses. The model parameters are chosen as in Fig. 7, so that the single synapse coherence CRx共f兲 exhibits a mild low-pass behavior. Simulations and theory for different N are shown in Fig. 13. With the parameters given, we have CRx ⬃ 10−4. For N Ⰶ 1 / CRx ⬃ 104, the coherence function CRX共f兲 is proportional to N. In a logarithmic plot, this manifests itself in the fact that the curves for N = 1 to 1000 are shifted versions of each other. However, if N ⲏ 1 / CRx ⬃ 104, the coherence function CRX共f兲 approaches one and becomes more flat. B. Many synapses and additional noise

共58兲

Thus, the coherence approaches 1 by a N−1 correction term. This describes the saturation in the limit of an infinite number of synapses in which a perfect noiseless signal transfer is achieved. The very same saturation effect is responsible for a suppression of a possible frequency dependence. In other words, the relative difference between low- and high-

⬇

Additional noise may arise from spike train input 共not modulated by the signal兲 stemming from other neurons. These inputs will enter via synapses that we refer to as noise synapses; equivalently, the rate-modulated spike trains enter via the signal synapses. Noise synapses and signal synapses may share the same STP character, i.e., both may be facilitation- or depression-dominated. They may however

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20

spectra [Hz]

spectra [Hz]

A 15 10 5

Sηη NSxx

30

Sηη NSxx

20 10

A 0

0

B

0.01

0.00 0.01

coherence

coherence

0.02

CRX (simulation) CRX (theory) 1

0.00 0.01

10

frequency [Hz]

also differ, e.g., signal synapses may be depressing while noise synapses may be facilitating. For the coherence, we can again distinguish between the limiting cases of small N and large N. For small N, we obtain, CRX共f兲 ⬇ NCRx共f兲 1 +

S␩␩共f兲 NSxx共f兲

冋

for NCRx 1 +

册

−1

S␩␩ NSxx

册

−1

while for large N, the coherence reads, 1 NCRx共f兲

CRX (simulation) CRX (theory)

冋 冋

1+

S␩␩共f兲 NSxx共f兲

for NCRx 1 +

0.1

1

10

frequency [Hz]

FIG. 14. 共Color online兲 Influence of noise ␩ on the coherence function CRX共f兲. We simulate N = 10 facilitating input synapses with F0 = 0.1, ⌬ = 0.3, r = 10 Hz, ␶F = 80 ms, and ␧ = 0.2. The noise ␩ originates in 10 static synapses with amplitude ANoise = 0.4 each of which is driven by a Poissonian process with constant rate rNoise = 10 Hz. The resulting coherence is well predicted by Eq. 共15兲 共black line兲. In this situation, a low-pass power spectrum of the input synapses together with a flat power spectrum of the additional noise leads to a low-pass coherence function.

CRX共f兲 ⬇ 1 −

0.01

B

0.1

冋

0.02

S␩␩ NSxx

册

Ⰶ 1,

共61兲

Ⰷ 1.

共62兲

册

−1

In both expressions, it becomes obvious that additional noise always reduces the coherence between the total postsynaptic input and the rate modulation. Furthermore, let us assume that S␩␩ results from Poissonian spike train input without rate modulation, but with similar baseline rate r and via similar FD dynamics for the involved synapses. Then, the additional noise terms in Eqs. 共61兲 and 共62兲 do not introduce any additional frequency dependence of the coherence. In these cases, the broadband coherence of synaptic filtering is maintained. We may, however, also consider, that noise arrives via static synapses or via dynamic synapses which are different from the synapses that transmit the signal R共t兲. In either case, the ratio S␩␩共f兲 / 关NSxx共f兲兴, if equal or larger than one, introduces a frequency dependence into the coherence function. This is illustrated for static noise synapses in Fig. 14, which

FIG. 15. 共Color online兲 Influence of noise ␩ on the coherence function CRX共f兲. We simulate N = 10 facilitating input synapses with parameters as in Fig. 14. The noise ␩ originates in 100 depressing synapses with F0,Noise = 0.4 and ␶D,Noise = 300 ms each of which is driven by a Poissonian process with constant rate rNoise = 10 Hz. The resulting coherence is well predicted by Eq. 共15兲. Comparing to Fig. 14, a high-pass noise power spectrum leads to an even stronger low-pass behavior of the coherence function.

lead in conjunction with facilitating signal synapses to a lowpass coherence function. The reduction of coherence at high frequencies results essentially from the frequencyindependent noise. Similarly, for static noise synapses and depressing signal synapses, one observes a high-pass coherence function 共not shown兲. The above filtering effects are even more enhanced, if noise and signal synapses have opposite STP character. For instance, facilitating signal synapses and depressing noise synapses lead to a strong lowpass filtering of information as quantified by the coherence function 共see Fig. 15兲. The opposite case is illustrated in Fig. 16: depressing signal synapses and facilitating noise synapses will lead to a pronounced high-pass filtering of the coherence. We note that this filtering effect is largest for a single signal synapse. As discussed in the previous section, also in the presence of additional noise, the coherence increases but also flattens out 共with respect to frequency兲 by increasing the number of signal synapses. To see a significant filter effect on the coherence without reducing its magnitude too much, the number of signal synapses should be intermediate, i.e., it should be roughly about N ⬃ 关1 + S␩␩ / 共NSxx兲兴 / CRx. VI. SUMMARY AND DISCUSSION

In this paper, we developed a method to calculate spectral measures of spike trains transmitted by dynamic synapses showing facilitation and depression. Mathematically, dynamic synapses assign a history-dependent amplitude to each spike. Besides this amplitude modulation, we also considered a weak modulation of the firing rate by a time-dependent signal. We described, how to compute time-dependent averages of these postsynaptic responses. These measures can then be used in order to obtain an expression for the coher-

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spectra [Hz]

40

A 30

Sηη NSxx

20 10 0

coherence

0.25

B

0.2 0.15

CRX (simulation) CRX (theory)

0.1 0.05 0 0.01

0.1

1

10

frequency [Hz] FIG. 16. 共Color online兲 Influence of noise ␩ on the coherence function CRX共f兲. We simulate N = 100 depressing input synapses with r = 10 Hz, F0 = 0.4, and ␶D = 300 ms. The noise ␩ originates in 10 facilitating synapses with parameters as in Fig. 14; each of which is driven by a Poissonian process with constant rate rNoise = 10 Hz.

ence function between the amplitude-modulated spike train and the rate modulation. By means of the coherence function, we estimated the effect of STP on information transfer through the synapse. Although we treated the rate modulation in linear response theory, assuming a small relative amplitude ␧ Ⰶ 1, the obtained expressions were also shown to be valid for moderate amplitudes 共␧ = 0.2兲. We systematically examined spectral information measures and found that if the presynaptic population is modulated with a time-dependent signal, STP does not filter the information content about such a rate modulation 共broadband coding兲. We thus confirmed the numerical results from 关13兴. In examining this effect, we observed that for a single synapse, the coherence function is flat for various parameter sets chosen from the physiological parameter space. Furthermore, analytical arguments revealed that a growing number of synapses does not only increase the information transmission, but also decreases any frequency-dependent information filtering effect. However, for an extended scheme with additional noise 共i.e., background synaptic activity兲, we showed that substantial information filtering becomes possible. This kind of filtering is particularly pronounced if the short-term plasticity character of signal and noise synapses differ 共e.g., one is facilitating and the other one is depressing or vice versa兲. The synaptic background input can switch the information transfer from broadband coding to frequency-dependent filtering. Our results make predictions about spectral measures that are accessible in experiments. For example, according to Sec. II E, the coherence CRV of the subthreshold membrane potential in the absence of postsynaptic spiking is to a good approximation given by the coherence of the total synaptic input CRX, which we studied in this paper. Broadband coding for the subthreshold membrane voltage under Poissonian rate modulation can be verified or falsified experimentally. Furthermore, according to Sec. II E, power and cross-spectra

would be simple low-pass-filtered versions of the synaptic input spectra which we discussed here. Fitting, for instance, power spectra to experimental data obtained for Poissonian stimulation may allow for an estimate of the parameters characterizing STP. Note, that experiments in vitro are closer to the single-synapse scenario, which we focused on in Sec. IV. This is so, because upon stimulation, a number of presynaptic input fibers receive one and the same spike train 关11,12兴. In our study, we made a number of simplifying assumptions. First of all, we considered Poisson statistics for the presynaptic spike trains. If the firing statistics is not exactly but close to Poissonian statistics, we do not expect any drastic changes of the main conclusions of the present study. Our results do not apply, however, to pacemakerlike or bursting presynaptic inputs 共for a study on the latter in the context of STP, see 关33兴兲. Second, we assumed statistical independence of the presynaptic spike trains 共apart from the common rate modulation兲. Weak correlations between input spike trains 共as observed in vitro 关34兴兲 will presumably diminish the beneficial averaging across synapses. In this case, the effective number Neff of statistically independent synapses is smaller than their actual number. For a study on the interplay between input cross-correlations and STP, see 关35兴. A third limitation of our results is due to neglecting the stochasticity of transmitter release. In a more detailed model, the synaptic amplitude A would be drawn from a probability distribution. We expect that the difference between stochastic synaptic response and the deterministic response can qualitatively be treated as noise. Therefore, the coherence CRX will be reduced when considering stochastic synapses. However, for many release sites with similar parameters contributing to transmission, this additional transmission noise is weak. The analytical methods developed in this paper can in principle be used to treat a broad range of models, for instance the model by Markram and Tsodyks 关7兴, models using multiple facilitation 关9兴 or multiple depression components 关5兴. These may be interesting subjects of future investigations. APPENDIX A: EXPRESSIONS FOR SPECIFIC COHERENCE FUNCTIONS

In this section, we derive Eq. 共15兲 for CRX and show that CR具x典 equals one to leading order in ␧. 1. The coherence CRX

Using Eq. 共12兲, we write for the cross spectrum SR具x典共f兲 between R and 具x共t兲典x0

具


ⴱ 共f ⬘兲 ␦共f − f ⬘兲SR具x典共f兲 = ˜R共f兲具x典 x0

典 = 具˜R共f兲x˜ R

= ␦共f − f ⬘兲SRx共f兲. Hence,

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共f ⬘兲典x0,R 共A1兲

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SR具x典共f兲 = SRx共f兲,

␯共t兲 ⬇ r关1 + ␧R共t兲兴

共A2兲

i.e., the cross spectrum between stimulus R and response equals the cross spectrum between stimulus R and average response 具x典x0. Similarly, we put the definition of X 关cf. Eq. 共9兲兴 into Eq. 共12兲 with a = b = X using statistical independence among the x0,i and statistical independence between x0,i and ␩, SXX共f兲 = N共N − 1兲S具x典具x典共f兲 + NSxx共f兲 + S␩␩共f兲.

In the simple cases where Eq. 共B1兲 does not contain any e共· , ·兲-term, we use Eqs. 共6.28兲 and 共6.32兲 from 关20兴 in order to obtain the following scheme, 具x0共t1兲典x0 = ␯共t1兲, 具x0共t1兲x0共t2兲典x0 = ␯共t1兲␯共t2兲 + ␦共t1 − t2兲␯共t1兲,

共A3兲

共B5兲

¯= ¯,

Analogously, we find SRX共f兲 = NSRx共f兲.

共A4兲

The formulas 共A3兲 and 共A4兲 relate the many-synapses spectra to the single-synapse versions. We combine Eqs. 共A2兲–共A4兲 with the definition of the coherence 关cf. Eq. 共14兲兴 and obtain 1 1 1 1 S␩␩共f兲 N−1 1 1 + + = . CRX共f兲 N CR具x典共f兲 N CRx共f兲 N CRx共f兲 NSxx共f兲

which directly yields an expression for Eq. 共B1兲 in terms of r, ␧, and R. For the general case, where e共· , ·兲 terms may be present, there are two principal steps required. First, we express Eq. 共B1兲 in the following subsection by a product of averages of the kind 具x0共t1兲x0共t2兲 ¯ ek共ts,te兲典,

共A5兲

ts ⬍ t1,t2, . . . ⬍ te


˜ 共f兲␹共f兲, 具x典x0共f兲 ⬇ ␧R

共A6兲

where ␹共f兲 is the transfer function characterizing this linear transformation. There is no absolute term, because for ␧ = 0, stationarity has to be fulfilled. Hence, to the lowest order in ␧, we obtain a deterministic linear relation between R共t兲 and 具x共t兲典x0, for which the coherence is one CR具x典共f兲 ⬇ 1,

共A7兲

for all frequencies f.

共B6兲

where the instants t1 , t2 are of arbitrary order, but all of them are between ts and te

2. The coherence CRŠx‹

We want to evaluate the coherence CR具x典共f兲. To this end, we consider the dependence of the average modulated spike train 具x共t兲典x0 on the external stimulus R. As R enters this dependency only via the rate modulation, it is always accompanied by the factor ␧ 关cf. Eq. 共2兲兴. To the lowest significant order in ␧, we obtain a linear relationship in Fourier space:

共B4兲

共B7兲

and the exponent k ⱕ 0. Second, in the subsequent subsection, we show how to calculate the average in Eq. 共B6兲. 1. Reduction of Eq. (B1) to terms as in Eq. (B6)

For a case distinction between different orderings of the times appearing in Eq. 共B1兲, we use the Heaviside-function. For instance, for two different times, we can use the identity 1 = ⌰共t1 − t2兲 + ⌰共t2 − t1兲

for t1 ⫽ t2 ,

共B8兲

which, as a prefactor of the average Eq. 共B1兲, formalizes the distinction between the cases t1 ⬎ t2 and t2 ⬎ t1 corresponding to the two different terms on the right-hand side 共rhs兲 in Eq. 共B8兲. 共The case t1 = t2 may demand additional considerations 关36兴, not presented here.兲 In general, the products of Heaviside functions cycle through all possibilities of ordering the times. Formally, this can be written as follows: n−1

APPENDIX B: CALCULATION OF POISSONIAN AVERAGES

1=

We show how to calculate averages as in Eq. 共28兲, 具x0共t1兲x0共t2兲 ¯ e共t j,t j+1兲e共t j+2,t j+3兲¯典,

共B1兲

where the times t1 , t2 , . . . , t j , t j+1 , . . . are completely arbitrary 共including, that any two of these time variables may be identical兲, except for 关cf. Eq. 共27兲兴 t j ⱕ t j+1,

t j+2 ⱕ t j+3,

We define the term e共ta , tb兲 as

冋

e共ta,tb兲 = exp ln共1 − F1兲

冕

tb

册

x0共t⬘兲dt⬘ .

ta

共B9兲

where Pn is the set of all permutations of the sequence of integers from 1 to n. As an example, let us consider the computation of 具e共t j , t j+1兲e共t j+2 , t j+3兲典,

共B2兲

...

兺 兿 ⌰共t␴共i兲 − t␴共i+1兲兲,

␴苸Pn i=1

具e共t j,t j+1兲e共t j+2,t j+3兲典 = ⌰共t j+2 − t j+1兲具e共t j,t j+1兲e共t j+2,t j+3兲典 + ⌰共t j − t j+3兲具e共t j,t j+1兲e共t j+2,t j+3兲典 + ⌰共t j − t j+2兲⌰共t j+3 − t j+1兲具e共t j,t j+1兲e共t j+2,t j+3兲典

共B3兲

+ ⌰共t j+2 − t j兲⌰共t j+1 − t j+3兲具e共t j,t j+1兲e共t j+2,t j+3兲典 + ⌰共t j − t j+2兲⌰共t j+3 − t j兲⌰共t j+1 − t j+3兲 ⫻具e共t j,t j+1兲e共t j+2,t j+3兲典

We recall that the time-dependent rate of x0 is 041921-15

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MATTHIAS MERKEL AND BENJAMIN LINDNER

+ ⌰共t j+2 − t j兲⌰共t j+1 − t j+2兲⌰共t j+3 − t j+1兲

2. Calculation of terms as in Eq. (B6)

⫻具e共t j,t j+1兲e共t j+2,t j+3兲典,

共B10兲

where we have already used Eq. 共B2兲 reducing the number of terms on the rhs from 24 to six. For each term, the respective prefactor determines uniquely the time ordering, which is essential to calculate the average properly. For instance, for the term on the first line, we can assume the order t j ⱕ t j+1 ⱕ t j+2 ⱕ t j+3 and obtain for the average 共omitting the Heaviside function兲 具e共t j,t j+1兲e共t j+2,t j+3兲典 = 具e共t j,t j+1兲典 ⫻ 具e共t j+2,t j+3兲典, 共B11兲 because the averages are taken in nonoverlapping intervals and thus, the two e共· , ·兲 terms are statistically independent. The last term in Eq. 共B10兲 is only nonvanishing for t j ⱕ t j+2 ⱕ t j+1 ⱕ t j+3, which yields for this specific case

Here, we outline the proof for 具x0共t1兲x0共t2兲 ¯ ek共ts,te兲典 = Ck具xˆ0共t1兲xˆ0共t2兲¯典, 共B14兲 where xˆ0 is a Poissonian ␦-spike train with the rate ␯ˆ 共t兲 defined by

␯ˆ 共t兲 = ␯共t兲共1 − F1兲k ⬇ r关1 + ␧R共t兲兴共1 − F1兲k and

冋

冉

Ck = exp − r关1 − 共1 − F1兲k兴 ⫻ 共te − ts兲 + ␧

冕

te

共B15兲

R共t⬘兲dt⬘

ts

册冊

.

共B16兲 The terms 具xˆ0共t1兲xˆ0共t2兲¯典 are moment functions of the Poisson process xˆ0共t兲. These can be calculated according to the scheme 关cf. 共6.28兲 and 共6.32兲 in 关20兴兴

具e共t j,t j+1兲e共t j+2,t j+3兲典

具xˆ0共t1兲典xˆ0 = ␯ˆ 共t1兲,

= 具e共t j,t j+2兲e 共t j+2,t j+1兲e共t j+1,t j+3兲典 2

具xˆ0共t1兲xˆ0共t2兲典xˆ0 = ␯ˆ 共t1兲␯ˆ 共t2兲 + ␦共t1 − t2兲␯ˆ 共t1兲,

= 具e共t j,t j+2兲典 ⫻ 具e2共t j+2,t j+1兲典 ⫻ 具e共t j+1,t j+3兲典. 共B12兲 This example illustrates, that powers of the e共· , ·兲 term arise due to overlapping intervals. If there are additional x0共ti兲 prefactors in the average in Eq. 共B1兲, then we have to do more case distinctions for the respective time instances, which can be carried out by means of Heaviside functions. For example 具x0共t1兲e共t2,t3兲典 = ⌰共t2 − t1兲 ⫻ 具x0共t1兲典具e共t2,t3兲典 + ⌰共t3 − t1兲⌰共t1 − t2兲 ⫻ 具x0共t1兲e共t2,t3兲典 + ⌰共t1 − t3兲 ⫻ 具x0共t1兲典具e共t2,t3兲典,

具x0共t1兲x0共t2兲 ¯ e 共ts,te兲典x0 = lim k

Nd→⬁

= lim

Nd→⬁

= lim

Nd→⬁

冓冉 冋 冋

using the time-dependent rate ␯ˆ given by Eq. 共B15兲. Equation 共B14兲 together with the scheme in Eq. 共B17兲 allows for the analytical calculation of the averages in Eq. 共B6兲. In the following, we prove Eq. 共B14兲. The spike train within the interval 关ts , te兴 can be regarded as the limit case of a temporally discretized function. More precisely, in terms of the equally-sized intervals I j of width ⌬T = 共te − ts兲 / Nd, the spike counts n j = 兰I jx0共t⬘兲dt⬘, and the indicator functions ␹ j for the intervals I j, we write,

共B13兲

where we recall, that t2 ⱕ t3 by virtue of Eq. 共B3兲. The techniques presented here can be generalized in order to split averages of the type of Eq. 共B1兲 into products of the type of Eq. 共B6兲. For the necessary case distinctions, one can also use computer algebra software.

N −1

1 Nd→⬁ ⌬T

x0共t兲 = lim

冊冉

Nd−1

N −1

冊

N −1

␹ j共t兲n j . 兺 j=0

Nd−1

1 d 兺 ␹i 共t2兲ni2 ¯ ⌬T i2=0 2

1 1 兺 ␹i 共t1兲 兺 ␹i 共t2兲 ¯ ⌬T i1=0 1 ⌬T i2=0 2 N −1

Nd−1

共B18兲

For the averages of the spike count in the interval I j, we write ␭ j = 具n j典. Using this, we transform the average Eq. 共B6兲 as follows:

1 d 兺 ␹i 共t1兲ni1 ⌬T i1=0 1 Nd−1

共B17兲

¯= ¯,

1 d 1 d ␹ 共t 兲 兺 i 1 兺 ␹i 共t2兲 ¯ ⌬T i1=0 1 ⌬T i2=0 2

冓

j

Nd−1

共1 − F1兲n k 兿 j=0

n i1n i2 ¯

Nd−1

␦ 具ni 兿 j=0

i1 j

1

冔 冔册

共1 − F1兲n k 兿 j=0 j

␦

x0

册

x0

ni i2 j ¯ 共1 − F1兲n jk典x0 . 2

共B19兲

For the last step, we used, that the n j are statistically independent for different intervals I j. The average appearing on the last line can be rewritten as 041921-16

PHYSICAL REVIEW E 81, 041921 共2010兲

SYNAPTIC FILTERING OF RATE-CODED INFORMATION

⬁ e−␭ j␭sj m e−␭ j关␭ j共1 − F1兲k兴s m s 共1 − F1兲s·k = 兺 s s! s=0 s=0 s!

⬁

⬁

具nmj 共1 − F1兲n j·k典x0 = 兺 P共n j = s兲sm共1 − F1兲s·k = 兺 s=0

=e

−␭ j·关1−共1 − F1兲k兴

ˆ

k

e−␭ j共1 − F1兲 关␭ j共1 − F1兲k兴s m −␭ 关1−共1 − F 兲k兴 ⬁ e−␭ j␭ˆ sj m −␭ 关1−共1 − F 兲k兴 ⬁ 1 1 s =e j s =e j P共nˆ j = s兲sm 兺 兺 兺 s! s=0 s! s=0 s=0 ⬁

k

= e−␭ j关1−共1 − F1兲 兴具nˆmj 典xˆ0 ,

共B20兲

where xˆ0 is the aforementioned Poissonian ␦-spike train with the rate as defined in Eq. 共B15兲 and nˆ j and ␭ˆ j = 具nˆ j典 are the corresponding spike count and its average, respectively. Inserting Eq. 共B20兲 into Eq. 共B19兲 finally yields Eq. 共B14兲.

FC共t兲 = ⌬

冕

Q1 = 0,

共C2兲

where we used Eq. 共6兲 and ⌬lin = m⌬. The integration of this linear equation yields Eq. 共25兲. If one demands, that the mean square difference between the true and the approximated dynamics becomes minimal, i.e., f共m,F0,lin兲 = 具关F共FC兲 − Flin共FC兲兴2典 → min,

冋

册

⌬共1 − F0兲2 2⌬ ⌬2共1 + 3r␶F兲 1 − + , ␥2 3␥ 2␥2

冋

册

⌬共1 + 9r␶F兲 , ␥

⬁

⳵ kF k fC, k=0 k!

where we define f C = FC − 具FC典

f共m,F0,lin兲 =

具FC典 = ⌬r␶F .

d kF

dFCk

. 具FC典

⳵0F − F0,lin − m具FC典 + 共⳵1F − m兲f C

⳵ kF k fC k=2 k!

册冔 2

共C10兲

.

Standard calculus results in the following expressions for the minimum: mmin = ⳵1F +

1 ⬁ ⳵ kF 兺 Qk+1 , Q2 k=2 k!

共C11兲

and ⬁

冉

冊

⳵ kF Qk+1 Qk − 具FC典 . Q2 k=2 k!

We sketch briefly how to derive these expressions. First, we need the first four central moments of FC defined by Qk = 具关FC − 具FC典兴k典,

冓冋

⬁

共C6兲

冏 冏

⳵ kF =

and

+兺

共C5兲

共C9兲

Insertion of Eq. 共C9兲 into Eq. 共C3兲 yields

and

␥ = 1 − F0 + ⌬r␶F .

1 Q4 = ⌬4r␶F共3r␶F + 1兲. 4

F共FC兲 = 兺

共C4兲

共⌬r␶F兲2共1 − F0兲 ⌬2r␶F共1 − F0兲2 + F0,lin = F0 + ␥2 6␥3 ⫻ 1−

1 Q 2 = ⌬ 2r ␶ F , 2

The exact minimization of Eq. 共C3兲 is not feasible. Instead, we use a Taylor expansion of F共FC兲 around the mean value of FC,

共C3兲

one arrives 共considering, for simplicity, a constant rate兲 at the following expressions for the effective parameters of the linear dynamics: ⌬lin =

1 Q 3 = ⌬ 3r ␶ F, 3

共C1兲

dFlin共t兲 F0,lin − Flin共t兲 = + ⌬linx0共t兲, dt ␶F

共C8兲

and the Stratonovich scheme Eq. 共B5兲 with constant rate ␯共t兲 = r. The central moments read

We want to approximate the function F共FC兲 given in Eq. 共7兲 by a linear relationship,

which obeys the linear facilitation dynamics,

e−共t−t⬘兲/␶Fx0共t⬘兲dt⬘ ,

−⬁

APPENDIX C: LINEARIZATION OF F DYNAMICS

Flin共FC兲 = F0,lin + mFC ,

t

F0,lin,min = ⳵0F − ⳵1F具FC典 + 兺

共C7兲

共C12兲

These can be obtained using the explicit solution of the linear FC dynamics 关cf. Eq. 共6兲兴:

Using terms up to k = 3 results in the expressions in Eqs. 共C4兲 and 共C5兲.

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MATTHIAS MERKEL AND BENJAMIN LINDNER

Flin共t兲D共t兲 = 具Flin典具D典 + 具Flin典␦D + 具D典␦F + ␦F␦D ⬇ 具FlinD典 + 具Flin典␦D + 具D典␦F

共D1兲

coherence

where in the last line, we approximated ␦F␦D ⬇ 具␦F␦D典. This leads to the following approximate Dlin dynamics with Dlin ⬇ D: dDlin共t兲 1 − Dlin共t兲 = − 关具Dlin典Flin共t−兲 + 具Flin典Dlin共t−兲 dt ␶D + 具FlinDlin典 − 2具Flin典具Dlin典兴x0共t兲.

共D2兲

共D3兲

1 − r␶D关具FlinDlin典 − 具Flin典具Dlin典兴 , 1 + 具Flin典r␶D

共D4兲

具FlinDlin典 = 具Flin典具Dlin典 −

⌬linr˜␶ 1 + 具Flin典r˜␶

册

冋

where 共D6兲

For 具Dlin典 and 具FlinDlin典, these are self-consistent equations. The solutions are 具Dlin典 =

1 + r˜␶共具Flin典 + ⌬lin兲 , denom.

共D7兲

and 具FlinDlin典 =

2 具Flin典共1 + 具Flin典r˜␶兲 − 21 ⌬lin r˜␶r␶F

denom.

,

0.002

0.1

1

10

denom. = 共1 + 具Flin典r˜␶兲共1 + 具Flin典r␶D兲 + ⌬linr˜␶

冉

共D8兲

where the denominator of both expressions is given by

关1兴 L. F. Abbott and W. G. Regehr, Nature 共London兲 431, 796 共2004兲. 关2兴 S. A. Fisher, T. M. Fischer, and T. J. Carew, Trends Neurosci. 20, 170 共1997兲. 关3兴 R. S. Zucker and W. G. Regehr, Annu. Rev. Physiol. 64, 355 共2002兲. 关4兴 A. Mallart and A. R. Martin, J. Physiol. 共London兲 193, 679 共1967兲.

冊

1 ⫻ 1 − ⌬linr␶Fr␶D . 2

共D5兲

−1 −1 ˜␶ = 关␶F−1 + ␶D 兴 .

0.00 0.004

FIG. 17. 共Color online兲 兩sˆRx共f兲兩2, sˆxx共f兲, and CRx共f兲. Comparison between the results of approximated and original dynamics. “simulation FD”: simulation without any approximations made; “simulation FlinD”: simulation, where the approximated facilitation dynamics has been used; “simulation FlinDlin”: simulation, where approximated facilitation and depression dynamics has been used; “theory FlinDlin”: theoretical results for the approximated dynamics. The dotted coherence curve shows the coherence for the case without any synaptic plasticity 关i.e., F共t兲 = D共t兲 = 1兴. The parameters are the same as in Fig. 10.

具FlinDlin典

1 + 具Dlin典⌬linr␶F , 2

0.01

frequency [Hz]

具Flin典 = F0,lin + ⌬linr␶F ,

and

simulation FD simulation FlinD simulation FlinDlin theory FlinDlin

0.02

0.000

The explicit solution of this differential equation is Eq. 共26兲. We derive the averages 具Flin典, 具Dlin典, and 具FlinDlin典 for a constant spiking rate ␯共t兲 = r as explained in Sec. III B:

具Dlin典 =

0.01

0.00

~power spectrum

Assuming linear facilitation dynamics, we approximate the depression dynamics given by Eq. 共8兲 linearizing the product Flin共t兲D共t兲. With the deviations ␦F = Flin − 具Flin典 and ␦D = D − 具D典, we write

~|cross 2 spectrum|

0.02

APPENDIX D: APPROXIMATION OF D DYNAMICS

共D9兲

In the main text, we use the abbreviation F1 = 具Flin典. Using the approximation described in this appendix, we derived the single synapse spectra and the coherence function. In most situations, this leads to good agreements with simulations of the original dynamics 共cf. Figs. 9, 11, and 12兲, however, for large r␶D and moderate r␶F, we find substantial deviations. For the example shown in Fig. 10, these deviations are clearly due to the Dlin approximation as can be seen in Fig. 17: simulations where only the facilitation dynamics is approximated coincide fairly well with the original dynamics. However, as soon as the depression dynamics is approximated, substantial deviations are observed, in particular in the coherence function for low frequencies.

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