Abstract
Shortterm synaptic plasticity (STP) can significantly alter the amplitudes of synaptic responses in ways that depend on presynaptic history. Thus, it is widely assumed that STP acts as a filter for specific patterns of presynaptic inputs, and as a result can play key roles in neuronal information processing. To evaluate this assumption and directly quantify the effects of STP on information transmission, we consider a population of independent synaptic inputs to a model neuron. We show using standard information theoretic approaches that the changes in synaptic response amplitude resulting from STP interact with the related effects on fluctuations in membrane conductance, such that information transmission is broadband (no frequencydependent filtering occurs), regardless of whether synaptic depression or facilitation dominates. Interestingly, this broadband transmission is preserved in the postsynaptic spike train as long as the postsynaptic neuron's baseline firing rate is relatively high; in contrast, low baseline firing rates lead to STPdependent effects. Thus, background inputs that control the firing state of a postsynaptic neuron can gate the effects of STP on information transmission.
 coherence
 facilitation
 information transmission
 neural dynamics
 shortterm plasticity
 synaptic depression
Introduction
Shortterm synaptic plasticity (STP) acts on a variety of timescales to change the amplitude of postsynaptic responses to presynaptic spike trains (Abbott and Regehr, 2004; Morrison et al., 2008). The functional roles that STP is thought to play depend on the nature of the presynaptic inputs. STP can enhance the detection of transient inputs, such as presynaptic bursts and abrupt changes in input rate (Abbott et al., 1997; Lisman, 1997; Puccini et al., 2007). In the case of stationary presynaptic inputs (i.e., inputs with constant mean rate), STP is generally thought to lead to highpass or lowpass filtering (Dittman et al., 2000; Fortune and Rose, 2001; Abbott and Regehr, 2004). This idea is based on the changes in synaptic response amplitude (synaptic gain) during stimulation of presynaptic inputs at different rates (Dittman et al., 2000; Lewis and Maler, 2002; Klyachko and Stevens, 2006): facilitation leads to increases in amplitude with increasing input rate, while shortterm depression leads to decreases, suggesting highpass and lowpass filtering, respectively.
A description of such changes in synaptic amplitude is not sufficient, however, to fully describe information transmission across a synapse. To do this, one must also consider changes in synaptic amplitude in the context of accompanying noise sources (Rieke et al., 1996; Gabbiani and Koch, 1998; Stein et al., 2005). Under natural conditions, synaptic inputs can involve populations of asynchronously firing presynaptic neurons, with the resulting synaptic activity providing a major source of membrane voltage fluctuations, or noise (Jacobson et al., 2005). In addition, one must consider how timedependent synaptic amplitudes and associated fluctuations are affected by timevarying inputs. Finally, the influence of these dynamic synaptic inputs on postsynaptic spike trains must be understood. Fluctuations in membrane voltage near spiking threshold can greatly influence postsynaptic spike trains (Chance et al., 2002; Kuhn et al., 2004; Stein et al., 2005). How these different features interact to influence information transmission across the synapse is not clear.
In this study, we investigate how STP shapes information transmission in a spiking model neuron receiving input from a population of independent synapses using a common formalism to model synaptic plasticity (Abbott et al., 1997; Tsodyks and Markram, 1997; Dittman et al., 2000; Lewis and Maler, 2002; Abbott and Regehr, 2004; Puccini et al., 2007; Mongillo et al., 2008; Morrison et al., 2008). We first characterize the synaptic activity in the context of spontaneous background inputs, and then quantify information transmission of a signal encoded by a ratemodulated input. We show that information transmission across a synapse can be independent of the frequency of this ratemodulated input regardless of whether facilitation or depression dominates. Information transmission is thus broadband, with no STPdependent filtering. Interestingly, the expression of this effect in a postsynaptic spike train depends on the state of the postsynaptic neuron: broadband coding is preserved when the neuron is firing at moderate to high rates, but STPdependent filtering occurs when the neuron is firing at low rates. These results suggest the intriguing possibility that the background synaptic inputs shown to be involved in controlling firing rate, response gain, stochastic resonance, and phase locking (French et al., 1972; Knight, 1972; Longtin, 1993; Doiron et al., 2001; Chance et al., 2002; Le Masson et al., 2002; Stein et al., 2005), can also play a more complex role in neural coding by controlling a switch between broadband information transmission and STPdependent filtering.
Materials and Methods
Our overall approach involves well established techniques to model and quantify shortterm synaptic plasticity, neuronal firing dynamics, and information transmission (Rieke et al., 1996; Gabbiani and Koch, 1998; Dayan and Abbott, 2001). We summarize the important aspects below and include additional technical details in the Appendix.
FDmodeling formalism and LIF neuron model.
In the first part of our study, we consider only spontaneous background synaptic inputs. In this case, a population of inputs (N = 100) are described by Poissonian presynaptic spike trains denoted by ξ_{j}(t) = Σ_{i} δ(t − t_{i,j}), where t_{i,j} is the ith spiking time of the jth presynaptic input and the mean firing rate for each input is r = 〉ξ_{j}(t)〈 (chosen to be equal for all synapses and thus independent of j). Here and in the following, 〉〈 denotes an ensemble average. The synaptic amplitude due to the ith input spike at the jth synapse is given by the product F_{i,j}D_{i,j}, where F_{j}(t) and D_{j}(t) are the facilitation and depression variables of the jth synapse following the dynamics given by the following equations: At the time of an input spike (t = t_{i,j}), D is first updated (decreased by an amount F_{i,j}D_{i,j}); then F is updated by an increment Δ; for the synaptic amplitudes, we take the values at a time t_{i}^{−} just before these updates, i.e., F_{i,j} = F_{j}(t_{i}^{−}), D_{i,j} = D_{j}(t_{i}^{−}). We have introduced an upper bound for F, i.e., F ≤ 1, to prevent negative values of the update factor of the depression variable. For very low input spike rates, F ≈ F_{0} and D ≈ 1. The FDmodulated synaptic input at the jth synapse is x_{j}(t) = Σ_{i} F_{i,j}D_{i,j}δ(t − t_{i,j}) and the total synaptic input (summed over all N = 100 synapses) is denoted by X(t) = Σ_{i,j} F_{i,j}D_{i,j}δ(t − t_{i,j}). The mean amplitude at low input rate can be approximated by the following (see Appendix): The formula fits the simulation results even at larger rates (compare Fig. 1B).
The postsynaptic conductance due to the total synaptic input X(t) is given in our model by Ġ = −G/τ + (g_{0}/N)X(t), where τ = 5 ms and g_{0}/N = 4 nS. This conductance then drives a leaky integrateandfire (LIF) model neuron described by C_{m}V̇ = −g_{L}(V − V_{L}) − G(t)(V − V_{E}) + μ and a fireandreset rule. Here, V̇ represents the time derivative of V; V_{L} = −70 mV and V_{E} = 0 mV denote the leak and the excitatory reversal potentials, respectively; g_{L} = 100 nS is the leak conductance, C_{m} = 1 nF is the membrane capacitance, and μ is a constant bias current. The firing threshold is V_{thr} = −65 mV, and after firing the voltage is reset to V_{reset} = −70 mV. We denote the LIF output spike train by y(t) = Σ_{k} δ(t − t_{k}).
Spectral measures of membrane conductance fluctuations.
To quantify the membrane fluctuations at different frequencies, we use the power spectrum. Given a certain time series Z(t) (input or output spike train or conductance fluctuations) in a time window [0, T], we can determine its Fourier transform by Z̃ = ∫_{0}^{T}dtZ(t)e^{2πift}; its power spectrum is then defined as the scaled variance of the Fourier transform where the asterisk denotes complex conjugation. Since the conductance dynamics is just a linear filter acting on the total synaptic input, the power spectra of the conductance fluctuation S_{GG} and that of the total input S_{XX} are related by For the latter spectrum, we find for low input rate r the following approximation (see Appendix): where A_{i,j} = F_{i,j}D_{i,j} and τ̃ = (τ_{F}^{−1} + τ_{D}^{−1})^{−1}. An approximation for the second moment of the synaptic amplitude needed in this formula is sketched in Appendix.
Ratemodulated input and quantifying information transmission.
We modulate the rate of the Poisson process that drives the synaptic input in two different ways: first, by a sinusoidal signal R_{sin}(t) with frequency f_{s}, and then by a bandlimited Gaussian signal R_{ran}(t) (cutoff frequency 50 Hz). For sinusoidal signals, we use the signaltonoise ratio (SNR) to quantify the response (McNamara and Wiesenfeld, 1989; Gammaitoni et al., 1998). For weak periodic driving, the power spectrum contains peaks at the driving frequency (inversely proportional to the frequency bin) and the negative driving frequency, on top of a continuous background spectrum (noise floor S_{background}). For a finite simulation time T_{sim} = Δ_{f}^{−1}, the spectrogram of the conductance S_{GG} (f = i_{f} × Δ_{f}) reads where δ_{i,j} is the Kronecker function and g_{1} is the amplitude of the timedependent mean value of the conductance. The signaltonoise ratio is calculated by dividing the peak at the driving frequency by the continuous background spectrum which we obtain from an interpolation over the frequency bins surrounding i_{fs}.
For the Gaussian signals, we measured the crossspectra between the ratemodulated input and the outputs (conductance or output spike train of the LIF model) and calculated from these the coherence function (Rieke et al., 1996; Gabbiani and Koch, 1998): where S_{ZR}(f) is the crossspectrum and S_{ZZ}(f) and S_{RR}(f) are the respective power spectra of the output Z and the input R_{ran}(t).
For the coherence data we simulated 5000 realizations over a time window of ∼10^{6} time steps of Δt = 4 × 10^{−4} s; crossspectra and power spectra were averaged over 20 frequency bins (we ensured that this did not smooth out any true frequency dependencies) to obtain smooth estimates of the coherence function.
Designing static synapses for comparison to dynamic synapses.
We want to compare the statistics of a postsynaptic spike train generated by an LIF with FD dynamics (i.e., with dynamic synapses) to the same kind of statistics for static synapses, i.e., synapses without FD dynamics. A first approach might be to use a constant synaptic amplitude equal to the mean amplitude of the FD synapse in the conductance dynamics (i.e., g_{0} = 〉FD〈). It turns out that the resulting variability of the conductance in this case is much lower (because the amplitude of the presynaptic input does not vary); thus, the LIF output and consequently the LIF's coherence with the rate modulation is drastically lower as well. A better approach is to set up the synaptic conductance dynamics such that the mean and variance of the conductance are approximately the same for static and dynamic synapses. To this end we proceed as follows. In a first simulation, mean and variance of the conductance 〉G〈 and 〉ΔG^{2}〈 are measured for a population of dynamic synapses. For the staticsynapse setup, we then choose the following dynamics for g_{s}(t): The task solved in the following is to determine the parameters δ_{1} and δ_{2} such that The first condition together with a steadystate average of Equation 10 [the lefthand side (lhs) vanishes] yields (here we also assumed that any rectification of the rate modulation can be neglected and hence the mean firing rate over N synapses is just the Nfold baseline rate r). To obtain a second relation, we express the variance of g_{s} by the integral over the power spectrum We can easily determine the conductance spectrum S_{gsgs} from the spectrum of the summed constantamplitude input spike trains ξ(t) = Σ_{j}ξ_{j}(t), denoted by S_{ξξ}, via Fourier transforming Equation 10: The spectrum of the summed ratemodulated Poisson process reads [see related calculations by Gabbiani and Koch (1998)] Here the product of the Heaviside function θ(·) implements the bandpass limitation to frequencies between the cutoffs −F_{c} and F_{c}. Inserting the full formula for the spectrum S_{gsgs} into Equation 13, performing the integral, setting the resulting variance equal to that of the dynamic variance, and solving finally for δ_{2} yields Using Equations 16 and 12, δ_{1} and δ_{2} can be determined. We note that in all our simulations the static and the dynamic synapses showed very similar mean and variance for the conductance (<2% difference); higherorder statistics as well as the tail (e.g., the stationary probability density at high G values) differed slightly.
Results
Modeling shortterm facilitation and depression
The shortterm dynamics of a wide range of synapses are well described by a combination of facilitation and depression, the socalled FD formalism (Abbott et al., 1997; Tsodyks and Markram, 1997; Dittman et al., 2000; Dayan and Abbott, 2001; Lewis and Maler, 2002; Abbott and Regehr, 2004). In FD models, F describes a facilitation process that tends to increase synaptic strength, and D a depression process that tends to decrease synaptic strength, with the net synaptic strength given by the product FD. The dynamics of F and D arise from how they change after a presynaptic input spike (Dayan and Abbott, 2001). At the time of each presynaptic spike, F is incremented by a fixed amount Δ and then decays exponentially toward a baseline value of F_{0} with a time constant τ_{F}. Thus, F can be directly related to the transient increase in calcium concentration due to an action potential in the presynaptic terminal and the associated change in transmitter release probability (Zucker and Regehr, 2002); F_{0} is then the baseline probability of release. Similarly, at each presynaptic spike, D is multiplied by (1 − F), then decays back toward 1 with a time constant τ_{D}. The variable D reflects presynaptic Ca^{2+} channel inactivation (Mochida et al., 2008) and the amount of available transmitter, which is depleted by a fraction (1 − F) after each presynaptic spike (Zucker and Regehr, 2002). While the relation to biophysical quantities is not precise, it is important to realize that such models are able to capture a wide range of the shortterm synaptic dynamics observed experimentally (Abbott and Regehr, 2004; Morrison et al., 2008).
In general, as F_{0} increases from 0 to 1, a synapse changes from being facilitation dominated to depression dominated. Typically, synapses in different neural pathways are dominated by either facilitation or depression, but in some cases can transition between the two regimes over longer timescales (Markram and Tsodyks, 1996; Abbott and Regehr, 2004; Lewis and Maler, 2004). In principle, multiple F and D processes can be used to describe a particular synapse, but we consider here just one of each. As is the convention, we use FD to scale an excitatory synaptic conductance, G, having a time constant τ (see Materials and Methods). A population of these synaptic conductances (N = 100), each with independent FD dynamics and independent Poissonian input spike trains, provides the input to an LIF model neuron. To investigate the specific effects of synaptic dynamics without the additional nonlinearities associated with action potential generation, we first consider the condition when spiking is blocked, and then subsequently we describe the effects on output spike trains.
It is important to note that our approach (1) is directly related to many previous studies of synaptic dynamics (Abbott et al., 1997; Tsodyks and Markram, 1997; Dayan and Abbott, 2001; Abbott and Regehr, 2004, and references therein), (2) allows us to consider a range of relative strengths of facilitation and depression in a synaptic pathway, and specifically focuses on two distinct regimes: facilitationdominated or depressiondominated, and (3) does not address the issues of a single synaptic pathway dynamically switching between facilitation and depression, nor the stochastic nature of synaptic transmission.
Dynamics of synaptic amplitudes with facilitation and depression
Our analysis focuses first on the mean synaptic amplitude (synaptic gain) as a function of the mean rate r of a single presynaptic input (Fig. 1A,B). Previous studies have shown this relationship to take two qualitatively different forms (Abbott and Regehr, 2004), which we illustrate in Figure 1B with two different parameter sets. The first (red circles) is taken from a experimentally based model described previously (Lewis and Maler, 2004). Here, the amplitude grows with increasing input rate, reaches a maximum at ∼20 Hz, and then drops to 0 for high rates. Up to an input rate of 20 Hz, facilitation dominates in the sense that increases in input rate lead to a larger mean synaptic amplitude. We refer to the parameter range of (F_{0}, Δ), in which the mean amplitude grows with input rate r, as the facilitationdominated regime or FDR. Strictly speaking, this will of course depend on r, but here the definition applies for low to moderate rates. A second parameter set (green squares) results in a monotonic decrease in mean synaptic amplitude with input rate r. In this case, depression dominates for the entire range of input rates. We refer to this parameter range as the depressiondominated regime or DDR. Such dynamics in the synaptic amplitudes have led to the hypothesis that facilitation acts as a highpass filter, and depression acts as a lowpass filter (for review, see Abbott and Regehr, 2004). In this context, highpass and lowpass refer to the range of input rates for which the mean synaptic amplitude (or gain) is large relative to that for other rates. Later, we will also consider highpass and lowpass in the context of the frequency at which a baseline Poisson rate is modulated.
Facilitation and depression differentially influence fluctuations in synaptic conductance
We now consider the effects of a population of synaptic inputs on fluctuations in membrane conductance (Fig. 1C); this is analogous to recording synaptic current in a voltageclamp experiment. (Note: the context here involves spontaneous activity due to Poisson inputs with constant mean rate; encoding a ratemodulated signal is considered later.) While the synaptic amplitude is a conventional measure of synaptic efficacy and can indicate whether facilitation or depression dominates at a synapse (as illustrated in the previous section), it does not reveal anything about the temporal fluctuations introduced by STP (FD dynamics). Likewise, it does not reveal how these fluctuations shape the noise background at different frequencies, which is relevant for signal detection and transmission. Instead, to investigate these features we use the power spectrum. The power spectrum describes any timevarying signal (membrane current, spike train, etc.) in terms of the relative strength of different frequency components; note that we will reserve the use of the term frequency for this context, and will use rate exclusively to refer to the number of input or output spike events per unit time.
From the power spectrum of the total membrane conductance G(t), we see that the main effect of STP at low and moderate input rates (e.g., mean rate, r = 10 Hz) is different for the FDR and DDR (Fig. 1D, middle). In the FDR, conductance fluctuations (spectral power) in the lowfrequency ranges are large relative to those in the DDR. Further, in the FDR, spectral power decreases with increasing frequency, whereas in the DDR, spectral power first increases then decreases, with a spectral peak near 7–10 Hz. Note that for low to moderate frequencies, the effect of facilitation and depression is the opposite of that observed for the mean synaptic amplitude (Fig. 1B): here, facilitation leads to lowpass filtering, while depression implements a highpass filter. At very high input rates (r = 100 Hz) however, depression dominates for both parameter sets, resulting in low power in the lowfrequency ranges for both conditions (Fig. 1D, bottom). In all cases, the loss of power in the highfrequency range (> 30 Hz) is due to depression in combination with the conductance dynamics, which effectively lowpass filter the input spike train (see Materials and Methods).
Sinusoidal modulation of input rate and the postsynaptic conductance
We now consider a simple example in which a sinusoidal signal is transmitted across the synapse. We assume the signal is initially encoded by a sinusoidal modulation of the presynaptic spike rate; this rate modulation is referred to as R_{sin}(t), which denotes a pure sine wave of amplitude 1 and frequency f_{s}. In other words, the Poisson firing rate of the population of 100 synaptic inputs is modulated by R_{sin}(t), as r[1 + εR_{sin}(t)], where r is the mean input rate and ε is the amplitude of the rate modulation. This scenario is outlined schematically in Figure 2A. Figure 2B shows the FD dynamics (FDR) for a single input (synaptic amplitudes) and the total postsynaptic conductance G(t) for two different sinusoidal rate modulations (f_{s} = 0.5 and 50 Hz; mean rate r = 10 Hz; ε = 0.1). Because of the large fluctuations involved, it is difficult to visually assess the transmission of the sinusoidal signal. In this context, signal transmission can be quantified by the conductance power spectra and the signaltonoise ratio, SNR (Fig. 2C, left) (see also Materials and Methods). Note that the SNR varies very little for the three sinusoidal frequencies tested (SNR = 156, 153, and 148 for f_{s} = 0.5, 10, and 50 Hz, respectively; FDR). This is also true in the DDR (Fig. 2C, right), although comparatively, the SNR is slightly greater (SNR = 170, 169, 171 for f_{s} = 0.5, 10, and 50 Hz, respectively; DDR). As we will show in the following sections, the fact that these different ratemodulated signals result in a relatively constant SNR (for both FDR and DDR) has important implications for synaptic information transmission.
Information transmission of a ratemodulated signal to the postsynaptic conductance
While sinusoidal rate modulations can be instructive, to quantify information transmission in a general sense, we use a more complex signal to drive the rate modulation of the presynaptic input spike trains. Specifically, this signal is taken as a bandlimited [0, 50 Hz] Gaussian random signal R_{ran}(t). The Poisson firing rate of the population of 100 synaptic inputs is then modulated by R_{ran}(t), as r[1 + εR_{ran}(t)]. To quantify information transmission between the input signal R_{ran}(t) and a particular output Z(t), we use the spectral coherence, C_{ZR} (Eq. 9) (Rieke et al., 1996; Gabbiani and Koch, 1998). The coherence C_{ZR}(f) is a measure that ranges, at each frequency, between 0 (in which case the output is not linearly related to the input at that frequency, due to noise or nonlinearity) and 1 (in which case there is perfect linear transmission of the input stimulus at that frequency). For weak stimuli, a lower bound on the mutual information rate is directly related to the spectral coherence (Rieke et al., 1996). As in the previous sections, we consider the total synaptic conductance G(t) as our output measure (Fig. 3A), and thus the crossspectra S_{GR}(f) of the rate modulation with the synaptic conductance. Note that the crossspectrum is directly proportional to the frequencydependent gain of the system (also commonly referred to as the transfer function). Surprisingly, for both FDR and DDR, the shape of the squared crossspectra (Fig. 3B) is very similar to that of the power spectra S_{GG} seen for the unmodulated condition (i.e., Poisson inputs with fixed mean rate) (Fig. 1D). In the FDR, the crosspower decreases with frequency for low and moderate input rates (r = 1 and 10 Hz); in this case, the synaptic dynamics act as a lowpass filter on the rate modulation. In the DDR, power is relatively suppressed in the lowfrequency range for moderate to high input rates (r = 10 and 100 Hz); the synaptic dynamics, in this case, act as a highpass filter from the crossspectral point of view. For very low input rate (r = 1 Hz), lowpass behavior is also seen for DDR. We note for reference that a static synapse (constant synaptic amplitude with Poisson input spike train; see Materials and Methods) leads to a flat crossspectrum which, for a weak rate modulation, can also be derived analytically (Gabbiani and Koch, 1998).
The important consequences of the similarities between Figures 1D and 3B become evident when we consider the coherence between the rate modulation and the conductance, C_{GR}. First, overall coherence increases with growing mean input rate r (Fig. 3C). More importantly however, for both FDR and DDR, the coherence is flat with respect to the rate modulation of the input [frequency content of the signal R_{ran}(t)]. In other words, the FD dynamics do not result in a frequencydependent transfer of information. Such a flat, or broadband, coherence coding property is usually expected for a linear system with additive white noise and white input signal. In particular, the static synapse case shown in Figure 3C is a close approximation of this linear case; the coherence is flat because the crossspectrum and power spectrum are similarly lowpass (data not shown). However, in the present context (a bandlimited signal encoded in presynaptic spike trains driving nonlinear synaptic dynamics), this broadband property is unexpected. Indeed, it arises because the power spectra of the membrane conductance fluctuations G(t), with and without the signal R_{ran}(t), are both similar in shape to the crossspectra between R_{ran}(t) and G(t); thus, their ratio in the coherence function cancels out any frequency dependence. In other words, in the FDR/DDR, STP not only results in increases/decreases in mean synaptic amplitude, but it also acts to lowpass/highpass filter the ratemodulated signal. The combination of these two directly related effects results in a frequencyindependent transfer of information to the postsynaptic conductance.
Further, a peak in the power spectra of the conductance fluctuations at frequencies near 10–20 Hz is seen for the DDR (Fig. 1D, middle and bottom), as well as for the FDR (at high rates when depression dominates) (Fig. 1D, bottom). Such a peak is also present in the crossspectra (Fig. 3B) for the same mean rates, r. These peaks, being associated with dominantdepression, suggest an increased sensitivity to the highfrequency content of a ratemodulated signal. This is in contrast to the synaptic amplitudeversusinput rate plots (Fig. 1B), which show a peak for the FDR and lowpass behavior for the DDR. These contrasting behaviors (Figs. 1D, 2B) are at the root of the previously observed enhanced response of depressing synapses to abrupt changes in the input rate, when firing rates are sufficiently high (Abbott et al., 1997; Abbott and Regehr, 2004).
Information transmission of a ratemodulated signal to the output spike train
Thus far, our analyses have been limited to information encoded in the synaptic conductance. In the context of information transfer, it is of course critical to consider the membrane potential fluctuations and the spiking dynamics of the postsynaptic neuron. Under conditions in which spiking is blocked (subthreshold membrane potential), the membrane potential behaves similarly to the synaptic conductance (data not shown).
We now consider a spiking LIF neuron being driven by dynamic synaptic inputs (Fig. 4A). This situation is identical to that described for the synaptic conductance (Fig. 3), but with the conductance now driving the membrane potential and an LIF spiking mechanism (see Materials and Methods). In addition, we consider both the DDR (Fig. 4B) and the FDR (Fig. 4C) compared with static synapses having no STP but causing conductance fluctuations with the same mean and variance as the respective dynamic synapses (see Materials and Methods). To separate out the effects of the LIF itself from the synaptic dynamics arising from STP, we also apply different levels of a constant bias current (μ). In doing so, we find that the coherence between the input rate modulation and the spike output depends on the firing regime of the LIF. When the bias current is large, causing the neuron to fire at a high rate, the broadband coding observed for the synaptic conductance (Fig. 3) is preserved for both DDR and FDR (and is essentially the same as that for the static synapses). However, for decreasing levels of bias current, such that the mean firing rate of the LIF neuron decreases, the STPdependent filtering becomes apparent in the lowfrequency range (compare with static synapse, μ = 0 pA) (Fig. 4B,C). In addition, the increased nonlinearity of the LIF in this regime (μ = 0 pA) results in a drop in coherence in the highfrequency range for both DDR and FDR. Figure 5 illustrates that the LIF firing rates for the FDR and DDR, both in the unmodulated and ratemodulated cases, are essentially identical to their respective matched static synapses, for a given bias current μ, and thus cannot be the cause of any differences between static and dynamic synapses. However in general, the lower coherence for the FDR is due to a slightly lower firing rate of the LIF in the FDR (Fig. 5).
Thus, we find that information transmission via the output spike train depends on the state of the LIF neuron: broadband coding is preserved when the neuron is well above threshold, but filtering due to both STP and LIF dynamics is apparent when the neuron is near threshold (μ = 0 pA), in the socalled fluctuationdriven regime (where spike threshold is crossed only for sufficiently large fluctuations). In this latter case (μ = 0 pA), the LIF spike train is dominated by fluctuationdriven firing such that its power spectrum is whitened (i.e., it becomes flatter indicating more Poissonlike firing). We illustrate the significance of this for the FDR in Figure 6 (DDR is similar). Example LIF spike trains are shown for μ = 150 pA (lowfiring state, fluctuationdriven firing) (Fig. 6A) and μ = 500 pA (highfiring state) (Fig. 6B). Note that we use intermediate values of μ to make it easier to compare the two spike regimes on the same timescale. The LIF spike train power spectra for low and highfiring states are shown in Figure 6, C and D (top); it is clear that the spectra are more flat for μ = 0 pA. Figure 6, C and D (bottom), shows the crossspectra between the input, R_{ran}(t), and the LIF spike trains. Now, recall that coherence is related to the ratio of the crossspectrum and power spectrum (Eq. 9). The consequence of the flat power spectra (Fig. 6C, top) is that the coherence now reflects the filtering properties exhibited by the crossspectra (Fig. 6C, bottom). On the other hand, when the power spectra reflect the same frequency dependencies as the crossspectra, the coherence is flat. In a more natural condition, this bias current could arise from populations of independent background, or feedback, inputs (mediated through glutamatergic synapses), and thus would not be constant but would include additional fluctuations and noise. If these inputs had zero mean (similar to the μ = 0 pA case), the output spike train power spectra would be even more flat, such that the coherence would reveal the same filtering effects shown by the crossspectra (data not shown). For large mean input levels (similar to μ = 750 pA), the main effect of adding noise would be to drastically decrease the overall coherence; for the remaining coherence, a modest effect of the synaptic filter, in terms of suppression or enhancement of coherence at low frequencies, can be observed (data not shown).
Overall, these results suggest that synaptic filtering depends critically on the state of the postsynaptic neuron and thus may be actively controlled by background synaptic inputs. More specifically, when a neuron is tonically firing at a high rate, neural coding will not be influenced by STP, whereas in the lowfiringrate regimes, more likely associated with spiketiming codes, STP could play an important role in shaping coherence, especially at low frequencies.
Synaptic properties related to the FDR and DDR
Given that the broadband effect we describe depends on the interaction between synaptic gain (amplitude) (Fig. 1B) and fluctuations (conductance spectrum) (Fig. 1D), it is important to consider the different parameter regimes more carefully. To do this, we again consider the effects of spontaneous background inputs (as in Fig. 1). Figure 7 shows the (F_{0}, Δ) parameter space for both the FDR and DDR (for the related theoretical analyses, see Appendix). By moving along a horizontal line in this plot (at some fixed value of Δ), for instance, it is easy to see how changes in F_{0} alone can change a synapse from being facilitationdominated to depressiondominated; in other words, this illustrates how increasing the baseline probability of transmitter release increases the tendency for a synapse to show depression. Most importantly, the diagram illustrates that the distinct behaviors, observed in FDR and DDR, for both the mean synaptic amplitude versus input rate r (existence of a maximum or monotonic decrease) (Fig. 1B) and the conductance fluctuations (relative increase or decrease in power at low frequencies) are directly related. Indeed, we find the same simple analytical condition on the synaptic parameters for each (gray line in Fig. 7) (see Appendix). The condition for dominating facilitation reads as follows: The presence of finite timescales τ_{D} and τ_{F} is essential for a nontrivial separation of the parameter space: taking one of them to zero we obtain either the trivial condition Δ > 0 (for τ_{D}→0) or the nonsatisfiable condition Δ > ∞ (for τ_{F}→0).
We also observed the broadband coherence effect with various combinations of τ_{D} and τ_{F}. When these timeconstants are >500 ms, however, a highpass effect is found (data not shown). Together, these results show that the relationship between synaptic gain (amplitude) and conductance fluctuations is not due to a delicate balance of parameter values, but rather is expected for a large range of parameters and is thus likely to play a role at many different types of synapses. It should also be noted that a synapse can exhibit facilitation and depression having multiple timescales through distinct overlapping dynamic mechanisms (Zucker and Regehr, 2002; Abbott and Regehr, 2004). The impact of these more complicated scenarios is beyond the scope of the present study and will be addressed in future work.
Discussion
STP is thought to play various computational roles in neural processing (Abbott and Regehr, 2004). In particular, the notion of synaptic filtering is often used generically to describe how dynamic synapses shape synaptic inputs (Dittman et al., 2000; Lewis and Maler, 2002; Zucker and Regehr, 2002; Abbott and Regehr, 2004; Destexhe and Marder, 2004). The results we present here suggest that this notion, at least under conditions involving the responses to timevarying inputs, should be considered carefully. Indeed, our results suggest that the filtering effects of synaptic dynamics can be controlled by independent inputs via their influence on postsynaptic neuron firing rate.
Given that STP results in synaptic amplitudes that vary with input rates over different timescales, it is reasonable to assume that information transfer through synapses will vary accordingly, with suppression of either high or low frequencies. However, this is not true with respect to the spectral coherence between a ratemodulated input and the STPmodulated postsynaptic membrane conductance. The coherence, which provides a signaltonoise measure, is flat and independent of frequency. This is because signal transfer (synaptic amplitude) and noise transfer (conductance fluctuations) are increased or decreased in equal proportions at each frequency. Thus, the postsynaptic conductance resulting from a ratemodulated input reflects broadband coding regardless of whether facilitation or depression dominates (Fig. 3).
These results also extend to the dynamics of the subthreshold membrane potential, even though the synaptic conductance is multiplied by the driving force voltage in the Hodgkin–Huxley formalism (see Materials and Methods). However, the threshold dynamics resulting from spiking provide the interesting possibility for the postsynaptic control of synaptic filtering (e.g., through parameters that govern the mean output firing rate). Our results indicate that when the postsynaptic neuron is above threshold and firing autonomously, the broadband coding observed in the membrane conductance and voltage is preserved. However, when the neuron is below threshold and firing is dominated by membrane fluctuations (fluctuationdriven firing, resulting in a flat power spectrum), the synaptic filtering expected from the transfer functions (crossspectra) in Figure 3B becomes evident.
Previous studies have investigated the effects of STP on input spike trains (Goldman et al., 2002) and information transmission (Fuhrmann et al., 2002). In theory, STP adds a historydependent amplitude to each spike, so it is expected that in certain cases the correlations of the spike train with the rate modulation will increase. Here, we have shown that this is not equivalent to an increase in information about a timevarying rate signal. Information transmission relies on measuring the signal with respect to a noise background, and thus both signal and noise amplitude are important. From a signaltonoise perspective, we found that information transfer can be broadband (independent of frequency) in terms of the coherence, even though STPdependent filtering is present in the transfer function (crossspectra) (Fig. 3B). Interestingly, our results in Figures 1D and 3B reveal that STP in combination with membrane conductance, results in a resonance for both DDR and FDR (r = 100 Hz). In addition, while the coherence exhibits minimal STPdependent effects, the different features evident in the power spectra and crossspectra (r = 10 Hz) could provide a method to assess the relative levels of facilitation and depression affecting the synapses onto a neuron of interest.
Shortterm plasticity is ubiquitously expressed in neuronal networks. And because our approach is generic and the results do not depend on specific parameter choices, the broadband coding we describe is likely to be operating in a variety of contexts. This has widespread implications for neuronal information processing. First, the fact that the firing state of the postsynaptic neuron can control the influence of STP also presents interesting possibilities for feedback control and the evolution of network organization. Synaptic inputs with identical presynaptic properties could confer differential plasticity solely due to differences in baseline firing rates of postsynaptic neurons. Thus, networks could evolve to extract different information from presynaptic inputs in parallel through postsynaptic neurons with different mean firing rates alone (making distinct control over the presynaptic neurons themselves unnecessary). Second, our results show that any steadystate changes in synaptic gain, due to depression for example (Abbott et al., 1997; Tsodyks and Markram, 1997), will not affect information transmission when the postsynaptic neuron is in the highfiring state. Thus, the prediction follows that pathways where information transmission is important, for example at early stages of sensory processing, will have neurons in a relatively highfiring state, whereas those involved in feature extraction (Gabbiani and Koch, 1998) will comprise neurons in a lowfiring state.
One particular example where these issues are relevant is in the electrosensory system of weakly electric fish, the study of which provided the initial motivation for our current analyses (Lewis and Maler, 2002, 2004). The first central nucleus in the electrosensory pathway, the ELL (electrosensory lateral line lobe), receives primary sensory input as well as extensive feedback input. Two classes of ELL projection neurons, the deep (DP) and superficial (SP) pyramidal neurons, are differentiated by their baseline firing rates and the extent to which longterm synaptic plasticity plays a role in their response to feedback inputs (Bastian et al., 2004): DP neurons have high firing rates and are considered nonplastic, whereas SP neurons have low firing rates and very plastic feedback inputs. In light of our present results, DP neurons could reliably encode the ratemodulated feedback inputs across all frequencies (i.e., in a broadband manner), whereas SP neurons could extract particular features of the feedback inputs (Bastian et al., 2004; Oswald et al., 2004).
It is important to realize that our results pertain to timevarying, but statistically stationary modulations of the firing rate of the presynaptic population. Other coding schemes (e.g., a signal encoded in temporal or spatial correlations in firing) or the detection of transient inputs (i.e., abrupt changes in firing rate) may be enhanced by the FD dynamics (Abbott et al., 1997; Lisman, 1997; Abbott and Regehr, 2004). Specifically, STP in excitatory and feedforward inhibitory inputs has been shown to regulate the postsynaptic response to transient inputs (Lewis and Maler, 2002; Klyachko and Stevens, 2006). In such cases, we may see a frequency dependence in information transmission, but appropriate quantification methods are lacking at present. It is also possible that signals much stronger than those we consider here may be able to profit from the information stored in the synaptic amplitudes. Regardless, our results illustrate the potential complexities brought about by synaptic dynamics, and emphasize that appropriate quantitative methods will be essential for evaluating neural coding and transmission of both stationary and transient, nonstationary inputs. More intriguing though, is the possibility that, under some stimulus conditions, the filtering effects of STP can be switched on and off by external inputs simply through their influence on postsynaptic firing rate. The latency of this switch would be short, limited only by the intrinsic neural dynamics. Such a switch is yet another dynamic feature that could be coopted by the neural code.
Appendix
Here, we derive the statistics of the FDmodulated Poissonian spike train: mean and variance of the amplitude of the synaptic input, as well as its power spectrum. Also derived is the condition Equation 17 for dominating facilitation (i.e., that defines the facilitationdominated regime, FDR, and the depressiondominated regime, DDR).
Mean synaptic amplitude
For the dynamics of a single FD model (a single synaptic input j), we can relate the values of F_{j}(t) and D_{j}(t) right before the ith and (i + 1)th input firing times (t_{i,j}, t_{i}_{+1,j}), denoted by F_{i,j}, D_{i,j} and F_{i}_{+1,j}, D_{i}_{+1,j}, respectively. Neglecting the constraint that F(t) < 1 and omitting the second index j (which indicated the synapse), we obtain the following maps: where T_{i} = (t_{i}_{+1} − t_{i}) denotes the ith interspike interval (ISI). These maps will be a good approximation as long as 〉F_{i}〈 + Δ < 1 (see also below).
We can use the maps to estimate the mean and variance of the synaptic amplitude A_{i} = F_{i}D_{i} as follows. We can assume that for a renewal spike train (with constant rate) (1) mean values of the form F_{i+1}^{n}D_{i+1}^{m} (with n, m being integer values) will not depend on the index i and thus 〉F_{i+1}^{n}D_{i+1}^{m}〈 = 〉F_{i}^{n}D_{i}^{n}〈 and (2) the value of a synaptic variable F_{i} (or D_{i}) is statistically independent of the subsequent interval T_{i}. By multiplying powers of the two maps with each other, we obtain a hierarchy of equations for the moment 〉F_{i}^{n}D_{i}^{n}〈 containing also averages over the exponentials of the ISI. The latter can be readily calculated for a Poisson process, e.g., Using this relation and averaging the stochastic maps, we obtain the relations From Equation 21 we can conclude that the map Equation 18 is a good approximation as long as Multiplying the lhs and righthand side (rhs) of both maps in Equations 18 and 19, followed by averaging, yields the equation This equation (and all higherorder equations) contain a higher power on the rhs (the term 〉F_{i}^{2}D_{i}〈). In other words, the hierarchy of equations for the moments is not closed, and therefore we have to make an approximation. Here we choose a cumulant expansion (Stratonovich, 1967; Gardiner, 1985). Our strategy is to neglect all cumulants higher than of second order, i.e., we make a Gaussian approximation for the joint probability density of D_{i} and F_{i}: where the double brackets indicate the cumulant of the respective quantity. To express the cumulants by moments, one can use a procedure due to van Kampen [see Gardiner (1985), sec. 2.7.1]. Specifically for the product F_{i}^{2}D_{i}, the cumulant reads Using Equation 25, we thus approximate the moment 〉F_{i}^{2}D_{i}〈 as follows: Inserting this expression into Equation 24, solving it for the mean 〉F_{i}^{2}D_{i}〈, using Equations 21 and 22, and finally, calculating the different exponentials of the ISI T_{i} by means of Equation 20, we arrive at the approximate formula for the mean value (Eq. 4).
The second moment of the amplitude
To calculate the variance of the spike train's amplitude, we need to calculate the second moment of F_{i}D_{i}, i.e., 〉F_{i}^{2}D_{i}^{2}〈. We use again Equation 25 to approximate 〉F_{i}^{2}D_{i}^{2}〈 and 〉F_{i}D_{i}^{2}〈 by lowerorder moments, yielding and Furthermore, from the stochastic maps, we can obtain equations for the second moments of F_{i} and D_{i}, reading The resulting five linear equations (Eqs. 27–31) uniquely determine the unknown moments 〉F_{i}^{2}〈, 〉D_{i}^{2}〈, 〉F_{i}D_{i}^{2}〈, 〉F_{i}^{2}D_{i}〈, and 〉F_{i}^{2}D_{i}^{2}〈. The resulting expression for the second moment of 〉F_{i}^{2}D_{i}^{2}〈 is too cumbersome to be presented here. It can, however, be readily obtained by solving the above equations using computer algebra software. Comparisons to simulation data show excellent agreement as long as the condition in Equation 23 is not violated (data not shown).
Power spectrum of the amplitudemodulated spike train
Here we calculate an approximation for the power spectrum S_{xx} at a single synapse x_{j}(t) (the index j is omitted again in the following). If the set of input spike times {t_{i}} corresponds to a Poissonian point process, we can write the power spectrum of a single synaptic input as follows (using the index k in place of i to avoid confusion with the imaginary i): where we used the abbreviation for the synaptic amplitude A_{k} = F_{k}D_{k} (k refers to the spike time, not to the synapse). In deriving Equation 32, we assume that the number of spikes divided by the time window gives the spike rate r in the limit. The triple correlations in the last line are hard to calculate. For low input rate, though, only correlations between A_{i}_{+1} and the previous ISI T_{i} dominate. This assumption leads to the following expression for the spectrum: Using the maps in Equations 18 and 19, the synaptic amplitude at the k + 1 spike can be expressed by the values of the synaptic variables at the kth spike: After multiplying the last equation with e^{2πifTk}, we can separately average all terms consisting of products of the synaptic variables and all exponentials containing the subsequent ISI (which is independent of F_{k} and D_{k} and their products). With S_{XX} = NS_{xx}, for statistically independent inputs [see also Lindner (2006) for a recent discussion], we obtain a lengthy expression for the power spectrum with Lorentzian correction terms, the prefactors of which contain the averages 〈D_{k}〉, 〈F_{k}〉, 〈F_{k}D_{k}〉, and 〈F^{2}_{k}D_{k}〉 (see previous sections): This expression and Equation 6 have been used in Figure 1D to calculate the theoretical curves of the conductance spectrum. If we set r = 0 in the square bracket in Equation 35, we obtain the more instructive expression given in the main text in Equation 7 in which the Lorentzian correction terms are given in a simple way by the snaptic parameters. At low rate (r = 1 Hz in Fig. 1D), Equations 35 and 7 yield very similar spectra. Quantitative deviations between these approximations have been observed for the FDR at r = 10 Hz.
Separating FDR and DDR
Here, we sketch out the derivation of Equation 17. We first investigate whether the mean synaptic amplitude increases or decreases by increasing the input's firing rate. This will depend on the rate itself (the amplitude can go through a maximum as a function of rate) (see Fig. 1B), and to be definite, we study the sign of the mean's derivative at zero rate. Taking the derivative of Equation 4 with respect to r at r = 0 yields and the condition that this is positive (the synapse is dominated by facilitation) leads to Equation 17.
Second, we may ask under which condition the modification of the power spectrum by the three Lorentzian terms is positive at zero frequency and for low input rate. Subtracting from the power spectrum (Eq. 7) its highfrequency limit rN 〈A^{2}_{i}〉, we find that the resulting difference is positive at zero frequency if which indeed corresponds to Equation 17. Hence, dominating facilitation is necessarily associated with increase of power at low frequencies.
Footnotes

This work was supported by Natural Sciences and Engineering Research Council of Canada Discovery grants (A.L., J.E.L.) and an Ontario Premier's Research Excellence Award (A.L.). We thank Matthias Merkel for helpful comments on this manuscript.
 Correspondence should be addressed to John E. Lewis, Department of Biology, University of Ottawa, 30 MarieCurie, Ottawa, Ontario, Canada K1N 6N5. jlewis{at}uottawa.ca