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. 2012 Jun 21;8(6):e1002557. doi: 10.1371/journal.pcbi.1002557

Short Term Synaptic Depression Imposes a Frequency Dependent Filter on Synaptic Information Transfer

Robert Rosenbaum 1,2,*, Jonathan Rubin 1,2, Brent Doiron 1,2
Editor: Olaf Sporns3
PMCID: PMC3380957  PMID: 22737062

Abstract

Depletion of synaptic neurotransmitter vesicles induces a form of short term depression in synapses throughout the nervous system. This plasticity affects how synapses filter presynaptic spike trains. The filtering properties of short term depression are often studied using a deterministic synapse model that predicts the mean synaptic response to a presynaptic spike train, but ignores variability introduced by the probabilistic nature of vesicle release and stochasticity in synaptic recovery time. We show that this additional variability has important consequences for the synaptic filtering of presynaptic information. In particular, a synapse model with stochastic vesicle dynamics suppresses information encoded at lower frequencies more than information encoded at higher frequencies, while a model that ignores this stochasticity transfers information encoded at any frequency equally well. This distinction between the two models persists even when large numbers of synaptic contacts are considered. Our study provides strong evidence that the stochastic nature neurotransmitter vesicle dynamics must be considered when analyzing the information flow across a synapse.

Author Summary

Neurons communicate through electro-chemical connections called synapses. Action potentials in a presynaptic neuron cause neurotransmitter vesicles to release their contents which then bind to nearby receptors on a postsynaptic neuron's membrane, transiently altering its conductance. After it is released, the replacement of a neurotransmitter vesicle takes time and the depletion of vesicles can prevent subsequent action potentials from eliciting a postsynaptic response, an effect that represents a form of short term synaptic depression. When a vesicle is available for release, an action potential elicits its release probabilistically and depleted vesicles are replenished randomly in time, making the transmission of presynaptic signals inherently unreliable. We analyze a mathematical model of vesicle release and recovery to understand how signals encoded in sequences of presynaptic action potentials are reflected in the fluctuations of a postsynaptic neuron's conductance. We find that slow modulations in the rate of presynaptic action potentials are more difficult for a postsynaptic neuron to detect than faster modulations. This phenomenon is only observed when randomness in vesicle release and replacement is taken into account. Thus, by including stochasticity in the workings of synaptic dynamics we give new qualitative understanding to how information is transferred in the nervous system.

Introduction

Synapses act as information gates in neuronal networks. Presynaptic action potentials are communicated to postsynaptic neurons by causing synaptic neurotransmitter vesicles to release their contents, which then bind to receptors on a postsynaptic neuron's membrane, evoking a transient change in membrane conductance. After a vesicle is released, it typically takes several hundred milliseconds for it to be replaced at a synaptic contact (see Fig. 1 for a schematic of synaptic release and recovery). This refractoriness induces a form of short term synaptic depression that alters the filtering properties of synapses [1]. An accurate description of synaptic vesicle dynamics and their impact of on information transfer is necessary for a thorough understanding of coding in neuronal networks.

Figure 1. Synaptic vesicle dynamics.

Figure 1

(A) The axon of a presynaptic neuron (orange) makes Inline graphic synaptic contacts onto a postsynaptic neuron (green). (B) Synaptic vesicles in the synaptic terminal of the presynaptic neuron contain neurotransmitter molecules. A presynaptic action potential releases these neurotransmitter molecules with some probability, Inline graphic. Once released, these molecules bind to the postsynaptic neuron's membrane and cause a transient change in membrane conductance. (C,D) After a vesicle is released, the synapse enters a refractory state where it is unavailable to release additional neurotransmitter until it recovers by replacing the released vesicle.

A widely used model of synaptic depression treats vesicle release and recovery as deterministic processes [2][6]. While this deterministic model accurately describes the trial-averaged synaptic response to a presynaptic spike train presented repeatedly to a cell [7][11], it fails to capture the variability introduced at each trial by the probabilistic nature of vesicle release and recovery [12]. Regardless, the model has been used in studies for which neural variability and information transfer are central themes [13][18]. The aim of our paper is to determine the impact (if any) of stochastic vesicle dynamics on the filtering properties of depressing synapses.

Past studies have begun to address this aim by considering how variability from stochastic vesicle release and recovery affects the amount of information transmitted through a synapse as well as the firing rate of a postsynaptic cell [12], [19], [20], but a thorough investigation of the impact of stochastic vesicle dynamics on synaptic filtering has not been performed. We derive a compact description of the filters imposed by short term synaptic depression when stochastic vesicle dynamics are taken into account and when they are ignored. We find that variability introduced by stochastic vesicle dynamics plays a fundamental role in shaping the way in which depressing synapses filter presynaptic information. In particular, a model that ignores this variability transmits presynaptic information encoded at any frequency with the same fidelity [16], [17]. In contrast, a model that captures this variability reduces overall information transmission, and transmits quickly varying signals with higher fidelity than slowly varying signals. Differences between the two models persist over a broad range of physiologically motivated parameter values, even when a large number of synaptic contacts is considered and even at the population level. Our results suggest important implications for how signals encoded at different timescales are propagated through the nervous system and show that synaptic variability must be taken into account to accurately address such questions.

Results

We study the synaptic filter induced by short term depression with both a stochastic model and a deterministic model of synaptic vesicle dynamics (see Fig. 2A–D for an illustration and Methods for a detailed discussion). For both models, we consider a presynaptic spike train, Inline graphic, with rate Inline graphic that induces a postsynaptic conductance,

graphic file with name pcbi.1002557.e005.jpg

Here, Inline graphic is the time of the Inline graphicth presynaptic spike, Inline graphic is the number of vesicles released by the Inline graphicth presynaptic spike, and Inline graphic represents the time course of conductance induced by the release of a single synaptic vesicle. The presynaptic cell makes Inline graphic contacts with the postsynaptic cell. We make a simplifying assumption that each contact contains only one release site, so that a single presynaptic action potential can release at most one vesicle per contact [21], hence Inline graphic. Alternately, to model biological settings where this single vesicle hypothesis is violated [22], [23], Inline graphic can be interpreted as the total number of release sites across all contacts (see Discussion). We rescale conductance units so that Inline graphic. This rescaling causes Inline graphic to have dimension Inline graphic but simplifies the exposition.

Figure 2. Stochastic versus deterministic models of short term depression.

Figure 2

(A) An example presynaptic spike train, Inline graphic. Each vertical bar represents an action potential. (B) The number of synaptic vesicles, Inline graphic, available for release and the conductance, Inline graphic, induced in the postsynaptic cell for one realization of the stochastic model. Filled circles in (B) represent vesicle recovery events. (C) A second realization of the stochastic model with the same input. Observe in (B) and (C) that the number of vesicles released by the stochastic model during one second is primarily determined by the number of recovery events during that second and does not reflect the number of presynaptic spikes. (D) The number of synaptic vesicles and the conductance induced by the deterministic model with the input from (A). Parameters in (A–D) were chosen for illustrative purposes as Inline graphic, Inline graphic, Inline graphic, and Inline graphic. (E) The steady state mean conductance, Inline graphic, as a function of the presynaptic firing rate, Inline graphic. The inset shows the gain, Inline graphic. (F) The steady state variance of Inline graphic as a function of Inline graphic for the deterministic (solid blue) and stochastic (dashed red) models of vesicle dynamics with Poisson inputs. Variability in the deterministic model is introduced only by variability in the input, Inline graphic. Synaptic parameters for (E–F) and for all subsequent figures are given in Table 1.

In the stochastic model of vesicle dynamics [12], [19], [24], [25], a presynaptic spike releases each available vesicle at each contact independently with probability Inline graphic. After a contact releases its vesicle, it is unavailable to release again until the vesicle is replaced, a process known as recovery. The waiting time until the vesicle is replaced follows an exponential distribution with mean Inline graphic (Fig. 2B,C). For the deterministic model of vesicle dynamics [2], the number of available vesicles is treated as a continuous variable where a proportion Inline graphic of the total available vesicles are released by each presynaptic spike and the number of available vesicles increases exponentially towards Inline graphic with timescale Inline graphic between releases (Fig. 2D). Stochasticity in the conductance, Inline graphic, produced by the deterministic model is introduced solely by the stochasticity in the input, Inline graphic. Several presentations of the same realization of Inline graphic produce the same Inline graphic for the deterministic model, but not for the stochastic model (Fig. 2A–D).

The conductance produced by the deterministic model represents the quantity that would be obtained by presenting the same realization of Inline graphic to the stochastic model over several trials, then computing the trial-averaged conductance. Despite the agreement of their trial-averages, though, individual realizations of the two models differ substantially. The deterministic model responds to every presynaptic input, but releases a fractional number of vesicles at each response (Fig. 2D). In contrast, the stochastic model responds to only a few inputs, but releases a larger, quantal number of vesicles at each response (Fig. 2B,C).

The steady state mean conductance induced by a presynaptic spike train Inline graphic with rate Inline graphic is given by Inline graphic for both the stochastic and deterministic models of vesicle dynamics (Fig. 2E and Eq. (25)). The degree to which a small shift of the presynaptic rate is reflected in a shift of the steady state mean conductance is measured by the gain,

graphic file with name pcbi.1002557.e043.jpg (1)

which is a decreasing function that decays to zero as Inline graphic increases, a well-known effect that is due to the saturation of the mean conductance for large presynaptic firing rates (see Fig. 2E, inset and [2], [3], [26]). However, the gain only measures changes in the steady state mean of Inline graphic after a sustained shift in the mean of Inline graphic, whereas the signal processing properties of a synapse also depend on the temporal response of Inline graphic to transient fluctuations in Inline graphic [3], [10], [27], [28]. Below, we use a cross-spectral measure to quantify the temporal response properties of Inline graphic.

The information processing capabilities of a synapse depend not only on the response of Inline graphic to temporal fluctuations in Inline graphic, but also on the temporal and trial-to-trial variability of Inline graphic. Noise introduced by stochastic vesicle release and recovery leads to larger variability in Inline graphic, as measured by its variance (Fig. 2F). However, the variance alone does not capture the timescale over which this variability occurs. Below, we use a power-spectral measure to describe the variability of Inline graphic over different timescales.

Synaptic filtering of a Poisson presynaptic spike train

To gain an intuition for the signal processing properties of depressing synapses, we first study the case of a single Poisson presynaptic spike train, Inline graphic, with constant rate Inline graphic. Since a homogeneous Poisson process has equal power at every frequency, this approach allows us to investigate synaptic filtering at all frequencies simultaneously. Later, we will consider the response to an inhomogeneous Poisson process whose rate encodes a signal.

The magnitude of the response of the conductance, Inline graphic, at frequency Inline graphic to fluctuations in the input, Inline graphic, is quantified by the cross-spectrum, Inline graphic, between these quantities (see Methods). For both the deterministic and stochastic models of vesicle dynamics, the cross-spectrum is given by (see Eq. (25) in Methods)

graphic file with name pcbi.1002557.e061.jpg (2)

where Inline graphic denotes the Fourier transform and Inline graphic is a kernel that captures the filtering properties of synaptic depression (see Eq. (20) in Methods and Fig. 3A). The fact that Inline graphic is identical for the stochastic and deterministic models can be understood intuitively by noting that stochasticity in vesicle dynamics is uncorrelated from Inline graphic and therefore does not contribute to the covariability of Inline graphic and Inline graphic. It should be noted that, while Eq. (2) is exact for the deterministic model, it is an approximation for the stochastic model (see Methods), which is validated by simulations (Fig. 3B).

Figure 3. Synaptic filtering of a single Poisson presynaptic spike train.

Figure 3

(A)(B) The low-pass filter, Inline graphic, and the high-pass filter, Inline graphic, are multiplied with the presynaptic rate (cf. Eq. (2)) to determine the band-pass cross-spectrum, Inline graphic, between a Poisson presynaptic spike train, Inline graphic, and postsynaptic conductance, Inline graphic. The cross-spectrum is identical for the stochastic (solid blue) and deterministic (dashed red) models. (C)(D) The power spectrum, Inline graphic, of the conductance is larger for the stochastic model than the deterministic model due to the additive terms, Inline graphic and Inline graphic, that quantify the increase in variability due to stochastic vesicle release and recovery (see Eq. (3)). For this and all subsequent figures, solid blue lines and dashed red lines show plots obtained from closed form expressions for the stochastic and deterministic models, respectively. Light blue and light red lines indicate simulations of the stochastic and deterministic models, respectively.

The shape of Inline graphic can be understood by its components in Eq. (2). The low-pass filter, Inline graphic, which captures postsynaptic channel dynamics, suppresses power at frequencies higher than Inline graphic (see Fig. 3A and [29]). The high-pass filter Inline graphic, which captures the deterministic dynamics of short term depression, suppresses power at frequencies lower than Inline graphic (see Fig. 3A, Methods and [17]). Their product, which determines Inline graphic through Eq. (2), is then band-pass with most of its power at frequencies between Inline graphic and Inline graphic (Fig. 3B). Thus, only fluctuations in the presynaptic input within this frequency band are reflected faithfully by fluctuations in the postsynaptic conductance.

The low-frequency limit of Inline graphic is nearly zero for the parameter values chosen in Table 1 (Fig. 3B). This can be explained by noting that the zero-frequency cross-spectrum is related to the gain by [30]

graphic file with name pcbi.1002557.e085.jpg

For large Inline graphic, the mean conductance saturates and the gain decays to zero like Inline graphic (see Eq. (1) and Fig. 2E). Thus, Inline graphic which decays to zero for large Inline graphic (Fig. 4Ai). More specifically, Inline graphic when vesicles become depleted, which occurs when release is faster than recovery, i.e., Inline graphic. Note, though, that Inline graphic is larger for higher frequencies, meaning that faster fluctuations in Inline graphic cause larger transient fluctuations in Inline graphic when compared to changes in the steady state mean conductance, Inline graphic, caused by static changes in Inline graphic [3], [10], [27], [28].

Table 1. Table of synaptic parameters.

Name Definition Default value
Inline graphic timescale of vesicle recovery Inline graphic
Inline graphic number of contacts between a pre- and postsynaptic cell Inline graphic
Inline graphic probability of release when vesicle is available Inline graphic
Inline graphic presynaptic rate Inline graphic Hz
Inline graphic synaptic activation kernel Inline graphic
Inline graphic time constant of postsynaptic channels Inline graphic
Inline graphic bandwidth of rate-coded signal 0.1 Hz
Inline graphic peak power of rate-coded signal 20 Hz
Inline graphic noise correlation between presynaptic spike trains 0.1

Parameters for synapses and presynaptic spike trains. These parameter values are used in all figures unless otherwise indicated. Here, Inline graphic represents the Heaviside step function.

Figure 4. Low frequency signal transfer in a variety of parameter regimes.

Figure 4

Low frequency cross-spectrum (Inline graphic), auto-spectrum (Inline graphic), and coherence (Inline graphic) between a Poisson presynaptic spike train, Inline graphic, and postsynaptic conductance, Inline graphic, plotted as a function of the presynaptic rate, Inline graphic (Ai–iii), the vesicle recovery timescale, Inline graphic (Bi–iii), the number of synaptic contacts, Inline graphic (Ci–iii), and presynaptic population size, Inline graphic (Di–iii). Columns A–C are for a single presynaptic spike train (Inline graphic). The zero-frequency coherence in Diii is shown for three values of the presynaptic correlation coefficient: Inline graphic, Inline graphic, and Inline graphic. The power spectrum and coherence predicted by the stochastic model (solid blue) and the deterministic model (dashed red) disagree by orders of magnitude unless Inline graphic is small, Inline graphic is large, Inline graphic is small, or Inline graphic is large with Inline graphic.

The trial-to-trial and temporal variability of the conductance at frequency Inline graphic is quantified by its power spectrum, Inline graphic, which is given by (see Eq. (25) in Methods)

graphic file with name pcbi.1002557.e133.jpg (3)

Here Inline graphic is a constant that represents variability introduced by the interaction of Poisson input with deterministic vesicle dynamics, Inline graphic captures variability introduced by stochastic recovery, and Inline graphic captures variability introduced by probabilistic vesicle release. For the deterministic model, Inline graphic, but Inline graphic and Inline graphic are positive for the stochastic model (see Methods and Fig. 3C). As a result, the stochastic model predicts a larger power spectrum than the deterministic model (Fig. 3D). The decay of Inline graphic at high frequencies is due to the low-pass nature of the synaptic conductance kernel, Inline graphic (see Fig. 3A and [29]).

The power spectrum predicted by the two models differs most significantly at low frequencies, where it is nearly zero for the deterministic model but much larger for the stochastic model (Fig. 3D). This can be understood by noting that [30]

graphic file with name pcbi.1002557.e142.jpg

where Inline graphic is the number of vesicles released in a window of length Inline graphic. For the parameter values in Table 1, Inline graphic so that vesicles are mostly depleted and therefore the number of vesicles released in a large time window is determined largely by the number of recovery events during that window (Fig. 2A–D). For the stochastic model, recovery events at each contact occur as a Poisson process with rate Inline graphic. Since there are Inline graphic contacts and a Poisson process has power equal to its rate, Inline graphic when Inline graphic is large. This intuition is confirmed by noting that Inline graphic for the stochastic model. In contrast, for the deterministic model, recovery is deterministic and therefore the amount of neurotransmitter taken up, and hence released, over a large time window has a small variance. This is confirmed by noting that Inline graphic for the deterministic model and therefore approaches zero for large Inline graphic. For the synaptic parameters in Table 1, the power spectra produced by the stochastic and deterministic models disagree for Inline graphic larger than a few Hz (Fig. 4Aii).

The fidelity with which fluctuations in the postsynaptic conductance, Inline graphic, reflect fluctuations of the input, Inline graphic, at frequency Inline graphic is quantified by their coherence

graphic file with name pcbi.1002557.e157.jpg

where Inline graphic is the power spectrum of the Poisson input. Since Inline graphic is identical for the two models, but Inline graphic is larger for the stochastic model (Fig. 3B,D), it follows that Inline graphic is smaller for the stochastic model (Fig. 5). We now investigate the differences between the coherences produced by the two models in more depth.

Figure 5. Coherence between a single presynaptic spike train and the postsynaptic conductance it induces.

Figure 5

The coherence, Inline graphic, between a Poisson presynaptic spike train, Inline graphic, and the resulting postsynaptic conductance, Inline graphic. The stochastic model (solid blue) yields a high pass coherence that is dramatically smaller than the flat coherence predicted by the deterministic model (dashed red).

Since Inline graphic for the deterministic model, the cross-spectrum, Inline graphic, and power spectrum, Inline graphic, are proportional to one another (see Eqs. (2) and (3)) so that dividing them gives a flat coherence (i.e., a coherence that does not depend on Inline graphic, Fig. 5 and [16], [17]),

graphic file with name pcbi.1002557.e169.jpg

Here and in subsequent expressions, a Inline graphic (Inline graphic) superscript indicates identities for the deterministic (stochastic) model. Synaptic variability in the stochastic model increases the power spectrum, giving a frequency-dependent coherence

graphic file with name pcbi.1002557.e172.jpg

which is high-pass (Fig. 5). Thus, stochastic vesicle dynamics introduce high-pass frequency dependence into the fidelity of a synaptic filter.

In addition to introducing frequency dependence, stochastic vesicle dynamics also decrease the coherence substantially, especially at lower frequencies where the coherence is nearly zero for the stochastic model (Fig. 5). The fact that coherence is small at low frequencies for the stochastic model can be understood intuitively through the following relation [30],

graphic file with name pcbi.1002557.e173.jpg

where Inline graphic is the Pearson correlation coefficient between the number of presynaptic spikes, Inline graphic, and the number of vesicles released, Inline graphic, in a window of length Inline graphic. When Inline graphic, synapses are mostly depleted in the steady state. As a result, the number of vesicles released during a long time interval is determined primarily by the number of recovery events in that time window and hence mostly independent of the number of presynaptic spikes (Fig. 2A–C and [31]). Therefore, for the stochastic model, the number of vesicles released over a long time window is uncorrelated from the number of presynaptic spikes and, as a result, Inline graphic is small.

These intuitions are confirmed by appealing to the asymptotic expressions derived for the cross-spectrum and power spectrum above. For the stochastic model, Inline graphic and Inline graphic when Inline graphic. Since Inline graphic for Poisson input, it is then clear that

graphic file with name pcbi.1002557.e184.jpg

for the stochastic model when Inline graphic. For the deterministic model, however, Inline graphic, Inline graphic, and Inline graphic so that Inline graphic approaches a positive constant for Inline graphic sufficiently larger than Inline graphic. For the parameter values in Table 1, the coherences for the stochastic and deterministic models disagree substantially when Inline graphic is more than a few Hz (Fig. 4Aiii).

The disagreement between the stochastic and deterministic models is most dramatic when Inline graphic since the postsynaptic response is determined primarily by vesicle recovery dynamics in this regime, as discussed above. In the figures considered so far, we have used Inline graphic, motivated by measurements of pyramidal–to–pyramidal synapses in rodent neocortex [2], [19]. However, both shorter and longer time constants have also been reported in cortex [5], [7], [8], [32], [33]. When other parameters are set to the values from Table 1, the two models disagree substantially when Inline graphic (see Fig. 4Bi–iii).

A proposed justification for using a deterministic model of vesicle dynamics is that stochasticity introduced at each contact averages out when a presynaptic cell makes several contacts [17]. The number, Inline graphic, of contacts a presynaptic cell makes with a single postsynaptic cell varies greatly across cell subtypes and brain regions. Rodent and cat pyramidal cells in the hippocampus and neocortex typically make Inline graphic–12 contacts onto other pyramidal cells or onto interneurons. Interneurons in the same regions make Inline graphic–17 contacts onto pyramidal cells. On the other hand, the Calyx of Held synapse can make more than Inline graphic contacts onto a single postsynaptic target in the rodent auditory brainstem and Purkinje cells can receive over Inline graphic contacts from single presynaptic cells in the rodent cerebellum (see [34] for values of Inline graphic measured in various animals and synapses). When other parameters are set to the values from Table 1, the stochastic and deterministic models disagree substantially for Inline graphic (see Fig. 4Ci–iii).

In summary, over a broad range of synaptic parameters, stochastic vesicle dynamics both attenuate and impart a high-pass nature to the coherence between a pre-synaptic spike train and the post-synaptic conductance response. We next explore the implications of these effects on the transfer of rate-coded information.

Synaptic filtering of a rate-coded signal

Time-varying stimuli are often encoded in fluctuations of the firing rate of neuronal populations [35]. To address the question of how information about a rate-coded signal is filtered by vesicle dynamics, we use a model from [16] and [17] in which a time-varying signal is encoded in the firing rate of a presynaptic spike train to yield a doubly stochastic Poisson process, Inline graphic (see Methods).

In this model, the instantaneous presynaptic rate conditioned on a signal, Inline graphic, is given by Inline graphic and, without conditioning on Inline graphic, is given by Inline graphic. The power spectrum of the presynaptic spike train is given by

graphic file with name pcbi.1002557.e208.jpg (4)

where Inline graphic is the power spectrum of Inline graphic. Eq. (4) can be interpreted as follows: Inline graphic represents the power of Poisson noise and Inline graphic represents the power of the signal. Unless Inline graphic is identically zero, Inline graphic inherits non-Poisson statistics from Inline graphic, which violates the Poisson assumptions used to derive the spectral properties given above. In the Methods, we derive a linear approximation (valid when Inline graphic) to the synaptic filter induced by the deterministic and stochastic models of vesicle dynamics and use it to obtain approximations to the cross-spectrum, Inline graphic, between the signal and conductance as well as the power spectrum, Inline graphic, of the conductance for this model (see Eqs. (27) and (28) in the Methods). These approximations allow an investigation of the information transfer of the signal across the synapse in various frequency bands.

We model Inline graphic as a Gaussian process with Gaussian-shaped power spectrum (Fig. 6A,B),

graphic file with name pcbi.1002557.e220.jpg (5)

where Inline graphic is the bandwidth, Inline graphic the central frequency, and Inline graphic the peak power of the signal. We use a narrow-band signal (Inline graphic small) to more clearly illustrate the dependence of synaptic fidelity on signal frequency. Since Inline graphic is Gaussian, there is a positive probability that Inline graphic so that the instantaneous firing rate of the presynaptic cells becomes negative. However, when Inline graphic, this occurs rarely and can be disregarded by considering negative rates as zero [17]. The coherence, Inline graphic, between the signal and the conductance quantifies the fidelity with which the signal, Inline graphic, is represented in the postsynaptic conductance, Inline graphic. For the deterministic model of vesicle dynamics, the coherence is given by (from Eqs. (27))

graphic file with name pcbi.1002557.e231.jpg

so that changing Inline graphic merely shifts Inline graphic, but does not change its amplitude (Fig. 6C,D dashed red line). Thus, a signal coded within any frequency band is transmitted with the same fidelity, consistent with the conclusions reached above using the Poisson model and also consistent with previous studies [16], [17]. For the stochastic model, however,

graphic file with name pcbi.1002557.e234.jpg

Since Inline graphic is high pass (Fig. 3A) and Inline graphic is mostly flat (Fig. 3B), Inline graphic is larger when Inline graphic concentrates its power in higher frequencies. For example, the amplitude of the coherence is larger when Inline graphic than when Inline graphic for the stochastic model, but independent of Inline graphic for the deterministic model (Fig. 6C,D).

Figure 6. Signal transfer at high and low frequencies.

Figure 6

The firing rate of a single presynaptic spike train (Inline graphic) is modulated by the signal, Inline graphic, producing a postsynaptic conductance, Inline graphic. The coherence between the signal and conductance for (A) a slowly varying signal with peak frequency Inline graphic and (B) a quickly varying signal with Inline graphic. The stochastic model (solid blue) transmits the higher frequency signal more reliably than the lower frequency signal. The deterministic model (dashed red) transmits the signal with equal fidelity in both cases.

The rate of linear information transferred from the signal to the conductance is given by [36], [37]

graphic file with name pcbi.1002557.e247.jpg

In particular, Inline graphic represents the total information per unit time that a linear decoder can obtain about the signal, Inline graphic, by observing the conductance, Inline graphic, and also represents a lower bound on the Shannon information [36], [37]. The stochastic model predicts a dramatically lower linear information rate than the deterministic model (Fig. 7A). Since, for the deterministic model, the amplitude of Inline graphic is independent of the central signal frequency, Inline graphic, the linear information rate is also independent of the central frequency (Fig. 7A). The stochastic model, however, transmits quickly varying signals with more fidelity than slowly varying signals (Fig. 7A). Hence, stochastic vesicle dynamics introduce frequency dependence into the transfer of linear information across a synapse.

Figure 7. Linear information transfer rate as a function of signal frequency.

Figure 7

The linear mutual information rate, Inline graphic, between a rate-coded signal, Inline graphic, and the total conductance, Inline graphic, produced by (A) Inline graphic, (B) n = 100, and (C) Inline graphic presynaptic spike trains, each encoding Inline graphic. The information rate is plotted as a function of the central frequency, Inline graphic, at which Inline graphic is encoded. The stochastic model (solid blue) transmits quickly varying signals more reliable than slowly varying signals. The deterministic model (dashed red) transmits information encoded at any frequency equally well.

In summary, our results show that the high pass nature of synaptic depression combined with low frequency synaptic noise limits the transfer of low frequency information through a synapse, while higher frequency information is transmitted more reliably. We next investigate these conclusions in a population setting.

Synaptic filtering at the population level

So far, we have studied the conductance induced by a single presynaptic spike train that makes several contacts onto a postsynaptic cell. However, information about a stimulus is often encoded by populations of several presynaptic cells. We now consider a population model in which a collection, Inline graphic, of Inline graphic presynaptic spike trains all encode the same signal, Inline graphic, as described for the single-cell model above. These inputs induce individual synaptic conductances, Inline graphic, in a single postsynaptic cell. Define the total presynaptic input, Inline graphic, and the conductance induced by this input, Inline graphic. For simplicity, we assume that all synapses have the same synaptic parameters Inline graphic, Inline graphic, Inline graphic, and Inline graphic.

The signal, Inline graphic, introduces variability that is shared between the presynaptic spike trains. Such shared variability is commonly referred to as signal correlation since it is informative of the signal. Populations of presynaptic neurons that code for the same stimulus also share non-informative variability, known as noise correlation [38], [39]. As a simple model of presynaptic noise correlation, we assume that each pair of spike trains, Inline graphic and Inline graphic with Inline graphic, share a proportion Inline graphic of their spike times. The pairwise cross-spectra are then given by

graphic file with name pcbi.1002557.e276.jpg

where Inline graphic represents the contribution of noise correlations and Inline graphic represents the contribution of signal correlations.

As we have done for the single input model above, we gain an intuition for the population-level filter imposed by short term depression by first considering purely Poisson spike trains, which is achieved by setting Inline graphic so that Inline graphic. Even though the cross-spectrum, Inline graphic, is identical for the stochastic and deterministic models, the power spectrum, Inline graphic, is larger for the stochastic model due to noise introduced by synaptic variability (see Fig. 8A,B and Eq. (29) in Methods). Therefore the coherence, Inline graphic, between the total presynaptic signal and the total conductance is smaller for the stochastic model. Moreover, the deterministic model predicts a flat coherence, while the stochastic model predicts a high-pass coherence (Fig. 8C). These conclusions are identical to those reached for a single input above, but the disparity between the two models is reduced at the population level (compare Figs. 3 and 5 with Fig. 8).

Figure 8. Synaptic filtering at the population level.

Figure 8

A population, Inline graphic, of Inline graphic Poisson presynaptic spike trains with pairwise correlation Inline graphic drive a postsynaptic neuron to produce postsynaptic conductances, Inline graphic. (A) The cross-spectrum between the total presynaptic input and the total conductance. (B) The power spectrum of the total conductance has maximal power within the beta frequency band for both the deterministic (dashed red) and stochastic (solid blue) models. (C) The coherence between the total presynaptic input and the total conductance. Stochastic vesicle dynamics increase the power spectrum and therefore decrease the coherence, especially at low frequencies. All three plots are obtained in the absence of a rate-coded signal (Inline graphic).

Notice also that the power spectrum, Inline graphic, is peaked within the beta frequency band even though the inputs are Poisson and therefore have a flat power spectrum. This effect could exaggerate beta frequencies in recorded data. We return to this topic in the Discussion.

A potential justification for using a deterministic model of vesicle dynamics is that, since stochastic release and recovery events are uncorrelated across all synapses, the extra variability introduced by synaptic noise averages out at the population level. So far, we have compared the two models for a population size of Inline graphic. For the parameter values in Table 1, the low frequency cross-spectrum is identical for the two models, but the coherence and power spectrum disagree considerably until Inline graphic (Fig. 4Di–iii). The value of Inline graphic at which the models begin to agree depends on the pairwise correlation, Inline graphic, between the presynaptic inputs. Notably, in the absence of correlations (Inline graphic and Inline graphic), the population-level coherence is identical to the individual coherences, Inline graphic, so that the coherence predicted by the stochastic and deterministic models disagree by the same amount for any value of Inline graphic (Fig. 4Diii, lightest lines). As Inline graphic increases, the two models agree at smaller population sizes (Fig. 4Diii, darker lines). Hence, presynaptic correlations must be present and Inline graphic must be large if the deterministic model is to be used in place of the stochastic model for large populations.

We now study the transfer of rate coded information at the population level by allowing Inline graphic. In particular, we are interested in how information about a rate-coded signal, Inline graphic, is transferred to the population conductance, Inline graphic. As above, we use a signal with Gaussian shaped power spectrum given by Eq. (5). A linear approximation to the cross-spectrum, Inline graphic, for this model is calculated in the Methods (see Eq. (29)), which allows us to calculate the coherence, Inline graphic, between the signal and the postsynaptic response and the linear information rate, Inline graphic, which depends on the central frequency at which the signal is coded in a qualitatively similar manner as for a single presynaptic spike train (compare Figs. 7A to 7B,C). In particular, low frequency information transfer is reduced for the stochastic model of synaptic depression. Moreover, the stochastic model transfers information in a frequency dependent manner and the deterministic model transfers information at all frequencies equally (Fig. 7). The disparity between the models is substantial when Inline graphic, but reduced considerably when Inline graphic (compare panels B and C in Fig. 7). We remind the reader that Inline graphic represents the number of presynaptic neurons that encode the shared signal, Inline graphic, which could be much smaller than the total number of presynaptic inputs a cell receives. This suggests that, due to the stochastic nature of vesicle release and recovery, large presynaptic populations must be used to encode slowly varying signals.

Discussion

We derived a concise mathematical description of the synaptic filter induced by short term depression arising from neurotransmitter vesicle depletion. We found that stochasticity in vesicle release and recovery plays an important role in shaping this filter and determining the information processing capabilities of depressing synapses. For example, ignoring the stochasticity introduced by stochastic vesicle dynamics gives rise to a filter that transmits rate-coded signals encoded at all frequencies equally well [16], [17], but taking this stochasticity into account reduces information transfer and causes slowly varying signals to be transferred with higher fidelity than slowly varying signals.

The deterministic model of short term depression provides a usable approximation to the stochastic model when considering large populations of correlated presynaptic spike trains (Figs. 4Di–iii and 7C). While a postsynaptic neuron typically receives thousands of inputs, only a fraction of these inputs might be devoted to encoding a single stimulus. Our results show that a slowly varying stimulus must be encoded by large presynaptic populations, but quickly varying stimuli can be encoded by smaller populations. This conclusion is not true the deterministic model of synaptic depression, which ignores the inherent randomness of vesicle dynamics.

Since the two models predict the same mean conductance, the deterministic model is valid for studies that focus on mean postsynaptic activity and for which noise is not a concern. For example, the deterministic model has been used to describe the effects of depression on gain and temporal changes in postsynaptic firing rate [3], [10], [26], [27]. Using the deterministic model in these cases is justified only if changes in postsynaptic firing rate result primarily from changes in the mean conductance and the variability of the conductance is inconsequential. When spiking is fluctuation driven, the postsynaptic firing rate is underestimated by the deterministic model [12].

A number of experimental studies have successfully fit parameters for the deterministic model to recorded neural data. This is achieved by first repeating the same presynaptic stimulus to a cell, then averaging the cell's response and fitting the averaged response to the response predicted by the deterministic model [2], [5], [7], [8], [18], [32], [33]. Since the stochastic model discussed here uses the same parameters as the deterministic model, the parameters obtained through this procedure can also be used to constrain the stochastic model.

Spectral analysis of synaptic depression

There is an extensive experimental and theoretical literature addressing how synapses that exhibit short term depression transmit different patterns of presynaptic spikes [3], [26], [27], [40], [41]. One recurring observation in these studies is that the steady state mean conductance (equivalently, the mean rate of vesicle release) saturates with the presynaptic firing rate, which causes the gain, Inline graphic, to approach zero for large presynaptic rates (Fig. 2E). However, the gain only captures the sensitivity of the steady-state mean, Inline graphic, to static changes in Inline graphic. Previous studies show that temporal changes in Inline graphic are reflected more reliably in the transient mean of Inline graphic than static changes of Inline graphic are reflected in the steady-state mean of Inline graphic [3], [10], [27], [28]. This observation can be understood through our analysis by noting that higher frequency components of Inline graphic are larger than the low-frequency components (Fig. 3B). Note that the decay of Inline graphic at very high frequencies is due to the low-pass properties of the post-synaptic conductance kernel, Inline graphic, (Fig. 3A and [29]) and not to synaptic depression. The filtering effects of depression are captured by the kernel Inline graphic, which is high-pass (Fig. 3A).

A second shortcoming of the gain as a descriptive quantity is that it does not capture the trial-to-trial variability in the conductance, which is a vital component of information transfer. We quantify this trial-to-trial variability as a function of frequency using the power spectrum, Inline graphic. We show that the frequency-independence of information transfer through a deterministic synapse model depends on the precise shape of Inline graphic [16], [17], and the high-pass frequency-dependence of information transfer through a stochastic synapse model likewise depends on the shape of Inline graphic. Furthermore, we show that stochastic vesicle dynamics cause an overall decrease in information transfer by increasing Inline graphic. Thus, trial-to-trial variability in Inline graphic must be considered to obtain an accurate description of information transfer through a synapse.

While other studies of synaptic depression have investigated the transfer of rate-coded signals at various frequencies, we are not aware of a study that derives an explicit approximation to the filter induced by a depressing synapse. Such an approximation is derived in the Methods, giving

graphic file with name pcbi.1002557.e326.jpg

where Inline graphic and Inline graphic are the Fourier transforms of the presynaptic spike train and postsynaptic conductance respectively (see Methods for definitions of other terms). This expression can be used to predict the spectral properties of the postsynaptic response to a presynaptic input with a given power spectrum. A generalization of this expression that can be used in the case of a population of correlated presynaptic spike trains is given by Eq. (26).

Synaptic depression and neural rhythms

For the parameters in Table 1, the power spectrum is peaked within the beta frequency band (Inline graphic) for both the stochastic and deterministic models (Fig. 8B). We emphasize that the presynaptic spike trains in this case are Poisson processes with flat power spectra and cross-spectra. Thus, the peaked power spectrum of the conductance is due completely to synaptic filtering: Frequencies below Inline graphic are suppressed by synaptic depression and frequencies above Inline graphic are suppressed by post-synaptic channel dynamics. The conductance power spectrum is peaked between these two frequencies. This effect could potentially cause an exaggeration of beta or other frequencies in recordings such as local field potentials that reflect large pools of synaptic currents. Parameters can be chosen within a physiologically realistic range to produce a more exaggerated peak than that shown in Fig. 8B or to produce a peak within another frequency band (not shown). Further work is needed to determine the role that synaptic filtering plays in generating or exaggerating rhythms within beta or other frequency bands in functioning neural circuits.

Possible extensions

We used a simplified model of neurotransmitter release and recovery. In particular, we assumed that each contact contains only one release site. However, individual contacts can have multiple release sites and recent results show that multiple vesicles can be released by a single contact in response to a single presynaptic action potential [22], [23]. Such situations can be modeled in our framework by interpreting Inline graphic as the total number of release sites at all contacts. However, this interpretation is only valid if the release of vesicles is statistically independent between release sites that share a contact. If the probability of release at one site depends on release at another site – for instance if a contact has several release sites but can only release one vesicle per presynaptic spike [12], [42] – then our model would need to be adjusted to account for this dependency. To the authors' knowledge, the precise structure of such dependencies are a subject of current research and not presently understood. In the depleted state (Inline graphic), a contact with several release sites will rarely have more than one vesicle available for release at any point in time and our single-vesicle model should provide an accurate approximation regardless of dependencies between release sites, as long as the recovery time constant is properly adjusted [12].

We modeled stochasticity introduced by probabilistic vesicle release and random recovery times, but did not model stochasticity introduced by randomness in the amount of neurotransmitter contained in each vesicle [43], [44]. In addition we did not model variability at the postsynaptic site (e.g., randomness in the number of bound receptors, the number of open channels, or the availability of messenger molecules), which could introduce variability in the amplitude of the postsynaptic conductance elicited by each vesicle released. Assuming statistical independence of these sources of variability between release events, they can be captured by multiplying each response amplitude, Inline graphic, by a random number. This would simply scale the power spectrum of the conductance linearly and would not alter our central conclusions.

The cross-spectrum between presynaptic input and postsynaptic conductance decays to zero at high frequencies, but the coherence between the two does not (Figs. 3A and 5). This is due to the fact that the power spectrum also decays at high frequencies and cancels perfectly with the cross-spectrum. However, any additional high frequency noise would destroy this balance. For example, if one were to instead compute the coherence between the presynaptic input and the current across the postsynaptic membrane, high frequency channel noise [45] could increase the power spectrum without increasing the cross-spectrum and therefore cause the coherence to decay at high frequencies. Thus, information transfer from presynaptic input to postsynaptic current is effectively bandpass. Similar observations were discussed in [17] for the deterministic model of vesicle dynamics with additive noise.

We used a linear approximation to predict the spectral properties of the postsynaptic conductance induced by non-Poisson presynaptic spike trains. However, the approximation is only assured to be accurate when inputs are approximately Poisson, i.e., have a nearly flat power spectrum. This restriction is implicit in our assumption that Inline graphic (see Eq. (4) and the surrounding discussion). Presynaptic spike trains that exhibit highly non-Poisson properties, such as bursts or a high degree of regularity, can interact with synaptic depression in a fundamentally different manner than Poisson spike trains [12], [46]. Further work is needed to extend our results to highly non-Poisson presynaptic spiking statistics.

We focused on short term depression caused by the depletion of synaptic neurotransmitter vesicles. However, other sources of short term depression as well as several forms of short term facilitation affect the filtering properties of synapses [1], [40]. Our mathematical methods could be extended to take these additional forms of plasticity into account.

Synaptic transmission of Shannon information

To quantify information transfer through a synapse, we used an information metric that only captures the amount of information available to a linear decoder observing the conductance. The Shannon information measures the maximum amount of information available to any decoder [47]. Interestingly, for our choice of Inline graphic, the deterministic model of vesicle dynamics transmits Shannon information perfectly because every presynaptic spike elicits a postsynaptic response (Fig. 2D) and hence each spike time can be resolved by detecting jumps in Inline graphic [17], [19]. In contrast, the stochastic model of vesicle dynamics exhibits failures due both to probabilistic release and to vesicle depletion (Fig. 2C,E). Due to the presence of synaptic failure, the stochastic model reduces Shannon information since some presynaptic spikes have no effect on the postsynaptic conductance.

A few studies have investigated the reduction of Shannon information through synapses with synaptic failure [20], [46], [48] but focus on the impact of probabilistic release and ignore stochasticity in vesicle recovery dynamics. In contrast, we studied the reduction of linear information induced by both probabilistic release and stochastic recovery. The qualitative differences we observed between stochastic and deterministic models depend on the stochasticity of vesicle recovery since it introduces low frequency variability into the conductance (Fig. 3C,D). To our knowledge, only one study [19] has investigated information transmission in a model with both probabilistic release and stochastic recovery. Using simulations, they found that stochastic vesicle dynamics reduce Shannon information by orders of magnitude, consistent with our results for linear information. These previous studies of information transmission do not quantify the dependence of information transfer on the frequency band in which presynaptic information is encoded. Furthermore, care must be taken when drawing conclusions about neural coding from studies of Shannon information. Shannon information quantifies the maximal information that can be extracted by a decoder, but it is not always clear whether a neural decoder can achieve optimal or even near-optimal decoding.

Methods

Definition of the models and derivation of first moments

Consider a single presynaptic neuron that fires action potentials at times Inline graphic and define the presynaptic spike train as a point process,

graphic file with name pcbi.1002557.e339.jpg

where Inline graphic is the Dirac delta function. The number of presynaptic spikes in Inline graphic is then given by Inline graphic. Define Inline graphic to be the number of functional contacts that the presynaptic neuron makes onto a postsynaptic cell [48] and, for simplicity, assume that each contact can have at most one vesicle available for release at any point in time. Let Inline graphic be the total number of vesicles available for release at time Inline graphic. Let Inline graphic be the number of vesicles released by the Inline graphicth presynaptic spike, with Inline graphic. The total number of vesicles released up to time Inline graphic is given by Inline graphic and the effective synaptic input is a marked point process defined by

graphic file with name pcbi.1002557.e351.jpg (6)

We first consider a model of synaptic vesicle dynamics that treats vesicle release and recovery stochastically [12], [19], [24], [25]. At each presynaptic spike time, Inline graphic, each contact at which a vesicle is available releases this vesicle independently with probability Inline graphic. After a synaptic contact releases its vesicle, vesicle recovery occurs as a Poisson process with rate Inline graphic. That is, the waiting time from vesicle release until recovery at a single contact is exponentially distributed with mean Inline graphic and independent from the state of other contacts, so that the probability of a recovery event during the interval Inline graphic is Inline graphic. This model can be described by the equation

graphic file with name pcbi.1002557.e358.jpg (7)

where Inline graphic is the increment of an inhomogeneous Poisson process with instantaneous rate that depends on Inline graphic through Inline graphic (here, Inline graphic denotes conditional expectation) and Inline graphic is given by Eq. (6) where each Inline graphic is a binomial random variable with mean Inline graphic. Since each trial with a fixed input, Inline graphic, yields a different, random realization of the response, Inline graphic, we hereafter refer to this model as the “stochastic model” of vesicle dynamics.

A popular simplification of the stochastic model replaces the random increments, Inline graphic and Inline graphic, in Eq. (7) with their expected values conditioned on Inline graphic and Inline graphic [2], [3], [5], [6]. Since Inline graphic and Inline graphic, this gives

graphic file with name pcbi.1002557.e374.jpg (8)

This model treats Inline graphic as a continuous variable where a proportion Inline graphic of the available vesicles are released at each input and recovery occurs exponentially with time constant Inline graphic. We hereafter refer to the model described by Eq. (8) as the “deterministic model” of vesicle dynamics since the response, Inline graphic, is determined completely by the presynaptic input, Inline graphic. Stochasticity in this model is only introduced by randomness in Inline graphic.

When Inline graphic is a homogeneous Poisson process, the deterministic model is analytically tractable: the first two moments of Inline graphic and Inline graphic can be derived exactly, as we show below. We also show that the first moments agree for two models. The second moments for the stochastic model are difficult to derive analytically, but we derive a more tractable diffusion approximation below. Furthermore, when Inline graphic is not a homogeneous Poisson processes, closed form approximations can be obtained for both the deterministic and stochastic models.

Assume that Inline graphic is a homogeneous Poisson process with rate Inline graphic. Then the increment, Inline graphic, is independent from the current value of Inline graphic so that, by taking expectations in Eq. (8), Inline graphic for the deterministic model. Similarly, Inline graphic. Combining these gives

graphic file with name pcbi.1002557.e391.jpg (9)

Eq. (9) is also obtained by taking expectations in Eq. (7), which implies that the deterministic model and the stochastic model yield the same means when Inline graphic is a homogeneous Poisson process. The following equation for Inline graphic can be obtained using Eq. (9) and the fact that Inline graphic,

graphic file with name pcbi.1002557.e395.jpg (10)

The stationary mean of Inline graphic is given by the unique steady state solution to Eq. (9) [4],

graphic file with name pcbi.1002557.e397.jpg (11)

Furthermore, after a perturbation of Inline graphic or starting from an initial condition Inline graphic, Inline graphic decays exponentially back to Inline graphic with time constant

graphic file with name pcbi.1002557.e402.jpg

The stationary mean number of vesicles released by each presynaptic spike is given by Inline graphic and the stationary mean of the postsynaptic signal is Inline graphic, which represents the steady state rate of vesicle release. Furthermore, Inline graphic approaches its steady state exponentially with the same time constant, Inline graphic, as Inline graphic.

The calculations of first moments above depend on the fact that Inline graphic and Inline graphic are independent for any Inline graphic. This can only be assumed to hold when Eq. (8) is interpreted in the ItInline graphic sense (so that Inline graphic is updated directly after a spike) and Inline graphic is a homogeneous Poisson process. If Inline graphic is not a homogeneous Poisson process, then the equations for the first moments are not valid and the first moments may not agree for the two models.

A diffusion approximation of the stochastic model

Second moments for the stochastic model are difficult to derive analytically, so we obtain approximations by considering a diffusion approximation

graphic file with name pcbi.1002557.e415.jpg (12)

where Inline graphic is a standard Wiener process that models stochasticity in vesicle recovery. Stochasticity in vesicle release is captured by the stationary process, Inline graphic, with moments given by Inline graphic, Inline graphic, and Inline graphic for Inline graphic. We assume that Inline graphic, Inline graphic, and Inline graphic are mutually independent. These equations should be interpreted in the ItInline graphic sense, so that the increments Inline graphic and Inline graphic are independent from the history of the noise terms, Inline graphic, for any time Inline graphic [49]. Since Inline graphic, it is clear that the diffusion approximation defined by Eq. (12) has first moments that satisfy Eq. (9).

The noise coefficients, Inline graphic and Inline graphic, quantify the degree of randomness introduced by stochastic release and recovery respectively. To find appropriate values for these coefficients, we compute the infinitesimal variance of Inline graphic and Inline graphic conditioned on the drift terms that appear in their respective equations in Eq. (12) [50]. Since vesicle recovery events are Poissonian, the variance of its increment is equal to its rate, giving the conditional variance

graphic file with name pcbi.1002557.e435.jpg

Note that the Inline graphic term that appears on the right hand side of Eq. (12) does not contribute to this conditional variance since Inline graphic. Conditioned on Inline graphic and the occurrence of a presynaptic spike, the number of vesicles released has a binomial distribution with mean Inline graphic and therefore has conditional variance given by

graphic file with name pcbi.1002557.e440.jpg

Optimally, we would set Inline graphic and Inline graphic, but doing so would give rise to nonlinear multiplicative noise in Eq. (12), which is difficult to treat mathematically. Instead, we obtain an approximation by replacing Inline graphic with its stationary mean, Inline graphic, to obtain

graphic file with name pcbi.1002557.e445.jpg (13)

All calculations for the stochastic model are carried out using the diffusion approximation from Eq. (12) with the noise coefficients from Eq. (13), and therefore expressions obtained are approximations to the full stochastic model described above. However, in all figures, simulations are performed using the full stochastic model from Eq. (7) (light blue lines) and show excellent agreement with the closed form approximations (dark blue lines).

Note that the deterministic model can be recovered by taking Inline graphic in Eq. (12). Thus, we can proceed in our analysis by considering Eq. (12) without instantiating Inline graphic or Inline graphic to obtain results that apply to both the deterministic and stochastic models.

Derivation of the auto-covariance and power spectrum of Inline graphic

We quantify temporal and trial-to-trial variability between two stationary processes, Inline graphic and Inline graphic, using the cross-covariance function,

graphic file with name pcbi.1002557.e452.jpg

and its Fourier transform, the cross-spectrum,

graphic file with name pcbi.1002557.e453.jpg

The cross-covariance (cross-spectrum) between a process and itself is called an auto-covariance (power spectrum). To quantify the variability of the postsynaptic response, we now derive the auto-covariance, Inline graphic, and the power spectrum, Inline graphic, for the synapse model in Eq. (12).

From Eqs. (9) and (10) it is apparent that, for Inline graphic, the expectations Inline graphic and Inline graphic decay exponentially to their steady state, given any initial distribution, Inline graphic, imposed on Inline graphic and Inline graphic. From this fact, it is apparent that Inline graphic should inherit this exponential shape and therefore that Inline graphic should have an exponential shape with time constant Inline graphic.

We now make this argument more precise using a regression theorem from [49]. Define the bivariate Markov process,

graphic file with name pcbi.1002557.e465.jpg

Then Eqs. (9) and (10) show that

graphic file with name pcbi.1002557.e466.jpg

for Inline graphic where

graphic file with name pcbi.1002557.e468.jpg

In Sec. 3.7.4 of [49], it is shown that this implies

graphic file with name pcbi.1002557.e469.jpg

for Inline graphic Solving this linear differential equation gives

graphic file with name pcbi.1002557.e471.jpg

for Inline graphic. Thus, due to stationarity,

graphic file with name pcbi.1002557.e473.jpg

for Inline graphic and where Inline graphic is a constant. By symmetry, we have Inline graphic. Note also that, since Inline graphic is a marked point process, there is a Dirac delta function that contributes to Inline graphic at Inline graphic [51]. Finally, we may conclude that the auto-covariance of Inline graphic has the form

graphic file with name pcbi.1002557.e481.jpg (14)

for some constants Inline graphic and Inline graphic.

To calculate the coefficients Inline graphic and Inline graphic in Eq. (14), we must first calculate a few infinitesimal moments using stochastic calculus techniques [52]. In our calculations, we ignore terms of order Inline graphic and higher, but must include terms of the form order Inline graphic and Inline graphic because their expectation is of the order Inline graphic [50].

The second moment of Inline graphic conditioned on Inline graphic is given by

graphic file with name pcbi.1002557.e492.jpg (15)
graphic file with name pcbi.1002557.e493.jpg (16)

where (15) follows from the fact that Inline graphic and Inline graphic are independent from each other and from Inline graphic, that Inline graphic, and that Inline graphic; and (16) follows from the fact that Inline graphic. The calculation of the conditional mixed moment, Inline graphic, is similar and gives

graphic file with name pcbi.1002557.e501.jpg

To calculate the stationary second moment, Inline graphic, we modify a strategy from Sec. 4.4.7c of [49] to derive a linear differential equation for the time dependent second moment and find its steady state. First note that

graphic file with name pcbi.1002557.e503.jpg

The first term in this sum is given by

graphic file with name pcbi.1002557.e504.jpg

where we used the fact that Inline graphic and Inline graphic are independent (see above) and the last line follows from the equation for Inline graphic derived above. Now calculate

graphic file with name pcbi.1002557.e508.jpg

where we have eliminated terms of order Inline graphic and used the fact that Inline graphic is independent from all other terms; and the last line follows from the equation for Inline graphic above. Combining these expressions gives a differential equation for the time course of the second moment of Inline graphic,

graphic file with name pcbi.1002557.e513.jpg

where Inline graphic is given by the solution of Eq. (9) above. The stable fixed point of this linear differential equation is the stationary second moment of Inline graphic,

graphic file with name pcbi.1002557.e516.jpg (17)

where Inline graphic is the stationary mean of Inline graphic, given in Eq. (11). The delta function in Inline graphic has area given by

graphic file with name pcbi.1002557.e520.jpg (18)

where we used Eq. (16) above and where Inline graphic is given by Eq. (17).

To calculate the one-sided limit, Inline graphic, first calculate

graphic file with name pcbi.1002557.e523.jpg

where we have used the fact that Inline graphic and Inline graphic are independent of all of the other terms when Inline graphic. Each of the terms in the sum above can be calculated by conditioning on a spike at time Inline graphic and on the value of Inline graphic,

graphic file with name pcbi.1002557.e529.jpg

where Inline graphic is expectation over the variable Inline graphic. Similarly,

graphic file with name pcbi.1002557.e532.jpg

where Inline graphic is expectation over Inline graphic. Combining the expressions above gives

graphic file with name pcbi.1002557.e535.jpg

Finally, since Inline graphic from above, we have

graphic file with name pcbi.1002557.e537.jpg (19)

where Inline graphic and Inline graphic are the stationary first and second moments of Inline graphic, given in Eqs. (11) and (17). The auto-covariance of Inline graphic is then given by Eq. (14) with Inline graphic and Inline graphic given by Eqs. (18) and (19).

The power spectrum is obtained from the auto-covariance through a Fourier transform,

graphic file with name pcbi.1002557.e544.jpg

where

graphic file with name pcbi.1002557.e545.jpg (20)

is a deterministic linear kernel,

graphic file with name pcbi.1002557.e546.jpg

is the noise intensity introduced by the interaction between the stochastic input and deterministic vesicle dynamics,

graphic file with name pcbi.1002557.e547.jpg

is the noise introduced by stochasticity in vesicle recovery, and

graphic file with name pcbi.1002557.e548.jpg

is the noise introduced by stochasticity in vesicle release. Note that Inline graphic for the deterministic model since Inline graphic.

Derivation of the cross-covariance and cross-spectrum between Inline graphic and Inline graphic

To measure the covariability between the presynaptic spike trains and the postsynaptic response, we now derive the cross-covariance between the input, Inline graphic, and the response Inline graphic. By a similar argument to the one made above for Inline graphic, we may conclude that Inline graphic is the sum of a delta function and an exponential, except that the exponential is one-sided since Inline graphic for Inline graphic. For Inline graphic, we can find the peak of the exponential by first conditioning on a spike at time Inline graphic, then conditioning on a spike at time Inline graphic,

graphic file with name pcbi.1002557.e562.jpg

since Inline graphic. Thus,

graphic file with name pcbi.1002557.e564.jpg

The area of the delta function in Inline graphic is given by

graphic file with name pcbi.1002557.e566.jpg

since Inline graphic. Thus, we have

graphic file with name pcbi.1002557.e568.jpg

where Inline graphic is the Heaviside step function. Taking the Fourier transform gives the cross-spectrum

graphic file with name pcbi.1002557.e570.jpg

where Inline graphic is defined in Eq. (20) above.

Postsynaptic response to several correlated presynaptic spike trains

The statistics of the postsynaptic response to a population, Inline graphic, of uncorrelated presynaptic spike trains can be easily calculated from the statistics of individual responses, which are calculated above. However, neurons that contact a shared postsynaptic cell often exhibit correlations between their spiking activity [39], [53]. To determine the postsynaptic response to a population of correlated presynaptic spike trains, we must first calculate the pairwise cross-spectra of the conductances induced by these inputs. Assume that each spike train, Inline graphic, in the presynaptic population is a Poisson process with rate Inline graphic. Introduce correlations by assuming that each pair, Inline graphic and Inline graphic, of spike trains share a proportion Inline graphic of their spike times so that Inline graphic [54]. We use subscripts to denote quantities associated with each spike train and double subscripts as necessary. For simplicity, assume that the synaptic parameters Inline graphic, Inline graphic, and Inline graphic are identical for all synapses. The asymmetric case can be treated identically, but the expressions obtained are more cumbersome. The power spectrum, Inline graphic, and the cross-spectrum, Inline graphic, are given above (where they are written as Inline graphic and Inline graphic). Below, we derive expressions for Inline graphic and Inline graphic for Inline graphic.

First, following the same arguments used above to derive the moments of Inline graphic and Inline graphic in the case of a single presynaptic spike train, we obtain the bivariate moments

graphic file with name pcbi.1002557.e591.jpg

Similarly,

graphic file with name pcbi.1002557.e592.jpg

and, equivalently,

graphic file with name pcbi.1002557.e593.jpg

We now derive a differential equation for Inline graphic to get the stationary second moment. First note that Inline graphic so that

graphic file with name pcbi.1002557.e596.jpg (21)

By symmetry, the first and second terms in Eq. (21) are the same and they can be derived from Eq. (12) as

graphic file with name pcbi.1002557.e597.jpg

The last term in Eq. (21) is given by

graphic file with name pcbi.1002557.e598.jpg

Combining these gives

graphic file with name pcbi.1002557.e599.jpg

which has a fixed point at

graphic file with name pcbi.1002557.e600.jpg (22)

We now calculate the cross-covariance between Inline graphic and Inline graphic. By a similar argument to that used to derive Eq. (14) above, the cross-covariance between Inline graphic and Inline graphic has the form

graphic file with name pcbi.1002557.e605.jpg (23)

where we have used the symmetry of Inline graphic and Inline graphic, inherited from the symmetry in parameters, to conclude that Inline graphic. The area of the delta function is given by

graphic file with name pcbi.1002557.e609.jpg

where Inline graphic is given in Eq. (22). To find Inline graphic, we first calculate

graphic file with name pcbi.1002557.e612.jpg

so that

graphic file with name pcbi.1002557.e613.jpg

which gives Inline graphic through Eqs. (22) and (23).

Finally, we will derive Inline graphic and Inline graphic. Once again, by linearity, each of these is the sum of a delta function and an exponential. The area of the delta function is given by

graphic file with name pcbi.1002557.e617.jpg

We also have

graphic file with name pcbi.1002557.e618.jpg

Thus,

graphic file with name pcbi.1002557.e619.jpg

and therefore

graphic file with name pcbi.1002557.e620.jpg

By symmetry, Inline graphic

Finally, the cross-spectra can now be found through a Fourier transform to obtain

graphic file with name pcbi.1002557.e622.jpg

where

graphic file with name pcbi.1002557.e623.jpg (24)

Statistics of the postsynaptic conductance

So far we have described the statistics of the processes, Inline graphic, which quantify the release of vesicles released over time. The postsynaptic conductance induced by vesicle release is then defined as Inline graphic where Inline graphic denotes convolution and Inline graphic represents the time course of conductance induced by the release of a single vesicle (with Inline graphic for Inline graphic). The statistics of Inline graphic can easily be derived from those of Inline graphic using standard signal processing identities [29] to give

graphic file with name pcbi.1002557.e632.jpg (25)

for Inline graphic and the steady state variance of Inline graphic is given by Inline graphic.

Synaptic filtering of presynaptic spike trains with rate coded signals

So far, we have discussed statistics of the conductance induced by a population of homogeneous Poisson presynaptic spike trains, but spike trains measured in vivo do not always exhibit homogeneous Poisson statistics [55]. For example, time-varying stimuli can induce fluctuations in the firing rate of presynaptic neurons. As a simple model of rate-coded signals, we assume that a shared, time-varying signal, Inline graphic, is encoded in the firing rates of a presynaptic population, Inline graphic.

In this model, each presynaptic spike train is a doubly stochastic Poisson process [51]. The instantaneous firing rate of each presynaptic neuron, conditioned on Inline graphic, is given by Inline graphic. Without loss of generality, we assume that the signal has zero bias, Inline graphic, so that the unconditioned firing rates are Inline graphic. Signal correlations are introduced in this model by the shared signal, Inline graphic. We include noise correlations, i.e., correlations that are not due to shared signal [38], [39], by assuming each pair of presynaptic spike trains share a proportion Inline graphic of their spike times.

To compute the auto- and cross-covariance functions we first note that, for Inline graphic,

graphic file with name pcbi.1002557.e645.jpg

where Inline graphic is distribution of Inline graphic in the steady state (Inline graphic). In addition, Inline graphic has a Dirac delta function at Inline graphic with mass equal to the rate of synchronous spikes, Inline graphic. Thus, Inline graphic for Inline graphic. The auto-covariance (Inline graphic) can be obtained by taking Inline graphic. The cross-covariance function between Inline graphic and Inline graphic is be computed similarly to obtain Inline graphic. Taking Fourier transforms gives the spectra,

graphic file with name pcbi.1002557.e659.jpg

where Inline graphic is the power spectrum of the signal.

Exact expressions for the statistics of the postsynaptic conductance are difficult to obtain for this inhomogeneous Poisson model because Inline graphic is correlated with Inline graphic and with Inline graphic, which invalidates the methods used in the derivations for the homogeneous Poisson model above. However, when Inline graphic, the firing rate inhomogeneities are weak compared to the background firing rate and temporal correlations are weak as a result (analogously, Inline graphic). In this case, a linear approximation to the synaptic response can be obtained. To obtain this approximation, we find a linear filter that maps presynaptic spike trains to conductances and that is consistent with Eqs. (25) when inputs are Poisson. The following filter satisfies this requirement

graphic file with name pcbi.1002557.e666.jpg (26)

Here, Inline graphic is standard Gaussian white noise, Inline graphic is unbiased stationary noise with power spectrum Inline graphic that accounts for stochasticity in vesicle recovery, and similarly for Inline graphic, which accounts for stochastic vesicle release. The noise terms Inline graphic and Inline graphic are zero for the deterministic model. All noise terms here are independent except that Inline graphic and Inline graphic are correlated with cross-spectrum

graphic file with name pcbi.1002557.e675.jpg

where Inline graphic is given by Eq. (24).

The spectra predicted by Eq. (26) can be easily calculated using the fact that Inline graphic for stationary processes, Inline graphic and Inline graphic, where Inline graphic and Inline graphic denotes complex conjugation [56]. Thus,

graphic file with name pcbi.1002557.e682.jpg

where we used the independence of the noise sources to eliminate several terms. Other spectra can be derived in a similar manner to obtain the following generalizations of Eqs. (25)

graphic file with name pcbi.1002557.e683.jpg (27)

for Inline graphic. These expressions agree with Eqs. (25) when inputs are Poisson, i.e., when Inline graphic, because Inline graphic and Inline graphic in this case. When Inline graphic and Inline graphic, these expressions give a linear approximation which is verified using simulations in several figures below. The fidelity with which the signal, Inline graphic, is represented in the conductances, Inline graphic, depends on the cross-spectrum which can be calculated in analogous manner to Inline graphic above to obtain

graphic file with name pcbi.1002557.e693.jpg (28)

We are especially interested in the population spectra, Inline graphic, Inline graphic and Inline graphic, where Inline graphic is the total presynaptic input and Inline graphic is the total conductance induced by Inline graphic. These are given by using the bilinearity of covariances to obtain

graphic file with name pcbi.1002557.e700.jpg (29)

A similar inhomogeneous Poisson input model was used in [17] to investigate the transfer of rate-coded signals for the deterministic model of synaptic depression. Their model is analogous to our deterministic model with Inline graphic (since their response amplitudes are normalized) and Inline graphic (since they consider the postsynaptic response, Inline graphic, before convolution with a conductance kernel). Under these substitutions, our expression for Inline graphic agrees with their expression for Inline graphic exactly (where we use an “Inline graphic” superscript to indicate expressions from [17]). However, our expression for Inline graphic for the deterministic model only agrees with their expression for Inline graphic when Inline graphic (i.e., when the input is a homogeneous Poisson process). Our expression has an additional term that accounts for power introduced by the signal Inline graphic. In particular, Inline graphic for the deterministic model when Inline graphic and Inline graphic.

Parameters used for figures

Theoretical results are obtained for arbitrary parameter values, but for all figures we use the parameters from Table 1, which are chosen to represent values from experimental studies. The values used for Inline graphic and Inline graphic have been deemed “typical” for pyramidal-to-pyramidal synapses in the rodent neocortex [2], [19] and the value of Inline graphic is typical for several cortical areas [34]. The form of Inline graphic is chosen to model AMPA dynamics and its units are rescaled so that Inline graphic. This rescaling simplifies the exposition in the Results.

Acknowledgments

We would like to thank Anne-Marie Oswald for comments on an early draft of this manuscript. We would also like to thank the reviewers for several helpful suggestions that improved the quality and clarity of this manuscript.

Footnotes

The authors have declared that no competing interests exist.

This work was supported by NIH-1R01NS070865-01A1, NSF-DMS-1021701, and NSF-DMS-1121784; and B Doiron is a Sloan Research Fellow. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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