Short-Term Synaptic Depression Causes a Non-Monotonic Response to Correlated Stimuli

The synapse model

Because connected neurons rarely are connected by a single synapse, and synapses sometimes have more than one synaptic specialization (the particular area in which vesicle exocytosis takes place) (Walmsley et al., 1998), we will consider connections made of M “functional contacts,” the term used to refer to any synaptic specialization in which release takes place (Zador, 1998). Contacts sharing the same presynaptic cell will be referred to as “common” contacts. Figure 1, A and B, illustrates two examples in which each presynaptic cell establishes M = 3 and M = 2 common contacts, respectively. Cases with different values of M will be compared at a fixed total number of contacts. This is done by varying the number of presynaptic neurons as in the examples shown in Figure 1, A and B.

At each of these contacts, a simple stochastic model of vesicle turnover is implemented (Vere-Jones, 1966; Maass and Zador, 1999; Wang, 1999). The model assumes a vesicle “releasable pool” (RP) of size N₀, that is, it can hold up to N₀ vesicles. Vesicles within this pool are thought to be “docked” and will also be referred to as “readily releasable” vesicles. We assume the “univesicular release hypothesis,” which states that, independently of the number of docked vesicles, an action potential (AP) can at most trigger the fusion of one vesicle per functional contact (Edwards et al., 1976). The release of a docked vesicle is modeled stochastically, that is, on arrival of a spike, a release is triggered with a certain probability. This probability is a function of the current number of releasable vesicles, n, as follows: (1) as proposed in a study of the hippocampal CA3–CA1 synapse (Dobrunz and Stevens, 1997; Dobrunz, 2002). This dependence on n is illustrated in Figure 1 D, where one can observe that the parameter U represents the release probability when there is only one vesicle ready. The recovery of vesicles when the RP is not fully replenished is also modeled randomly: given that the number of vesicles ready to become docked is much larger than N₀ (De Camilli et al., 2001), the replenishment of a vacancy in the RP can be taken as the first event from a Poisson process with homogeneous mean recovery time τ_v (Fig. 1C). This modeling is compatible with the experimental observation that the recovery of the release probability from depletion can be fitted with an exponential (Dobrunz and Stevens, 1997) (data not shown). When comparing instances with RPs of different sizes, the recovery time τ_v will be normalized so that the ratio τ_v /N₀ is the same in all of the examples. This normalization is sketched in the two examples shown in Figure 1C, in which the case with N₀ = 4 has a recovery time twice as large as that with N₀ = 2. This scaling makes the comparison between synapses with different N₀ values easier, because (as will be shown later) the current statistics in the limit of large ν depends only on the ratio τ_v /N₀.

The average state of the synapses will be described by the probability that an afferent spike reaching a functional contact triggers the release of a vesicle. This “transmission probability” (P_t) will be obtained by computing over a long stimulus the fraction of functional contacts that, being hit by an AP, elicit a synaptic response.

Presynaptic activity: description of the stimuli

We will consider four different types of presynaptic activity: (1) “uncorrelated” stimuli, (2) “synchronous” stimuli, (3) “autocorrelated” stimuli, and (4) “phase-locked periodic” stimuli.

Figure 2 shows a spike rastergram for each stimulus type, which serves to explain graphically the reason of their choice. Because spikes are independent, the raster obtained with the uncorrelated stimulus looks very homogeneous in comparison with the rest, because it lacks any kind of spatial or temporal structure (Fig. 2 A). In contrast, the other three stimuli were chosen as simple examples with only spatial (Fig. 2 B, synchronous), only temporal (Fig. 2C, autocorrelated), and both spatial and temporal structure (Fig. 2 D, phase-locked). In each of these rasters, the corresponding structure is easily caught by the eye. Because of the presence of correlations, the global instantaneous activity of each stimulus (drawn below each raster), presents a different profile: despite its stochastic nature, the uncorrelated stimulus displays a much smaller temporal variability than the other three.

The stimuli are defined in terms of the “statistics” of the spike trains. Therefore, different trials do not reproduce the position of the spikes precisely, only the statistics of the presynaptic trains is the same. During a trial, each of the N presynaptic neurons elicits a spike train composed of a sequence of APs at the times , which we write as (i = 1,2,... N): (2) Its mean firing rate (ν) is defined as the mean number of spikes per unit time. Cross-correlations are quantified in terms of the correlation between two spikes of different spike trains (i ≠ j), (3) the brackets indicating average over trials, whereas a similar expression is used to define the autocorrelation, which measures the degree of correlation between two spikes of the same spike train, (4) Uncorrelated stimuli. This stimulus is built under the assumption of complete independence between any two spikes (regardless of whether they are fired by the same or different cells). In practice, the stimulus consists of a population of presynaptic neurons firing independent Poisson trains with the same constant rate, ν. We will use these stimuli to probe the synaptic model and the effects of varying different synaptic parameters; the elicited neuron response will serve as a control condition with which to compare the responses to correlated stimuli.

Synchronous stimuli. The synchronous stimulus represents the simplest example of instantaneous cooperative behavior between cells. To isolate the effect of spatial correlations from temporal correlations, we consider individual trains without autocorrelations (i.e., they all follow a Poisson statistics with the same rate). The different trains, however, are not generated independently, but they have a certain degree of synchrony, which means that each cell, beyond discharges occurring synchronously by chance, emits a fraction of its spikes at the same time as some other cells (Fig. 2 B, inset). This is quantified in terms of the cross-correlation between the spike trains of two presynaptic cells, which takes the following form: (5) The correlation coefficient (ρ) gives the probability above chance that, given that fiber i produced a spike at time t, fiber j produces another spike at the same time. It quantifies with values between 0 and 1 the strength of the cross-correlations. It is 0 when all of the presynaptic neurons fire independently, and it is 1 when they fire replicas of essentially the same spike train.

To numerically generate the synchronous stimulus, we first generated a Poisson “mother” train with rate ν/ρ. Next, each individual train was built as a thinned copy of the mother train from which some spikes, randomly picked, were eliminated with a probability of 1 – ρ per spike.

Autocorrelated stimulus. This stimulus is a simple example of cooperative behavior between spikes from the same cell. We chose a stimulus with positive temporal correlations between pairs of spikes from the same cell but not between spikes from different presynaptic trains, which has a simple numerical implementation. More specifically, we generated independent renewal trains with exponential autocorrelations [for details, see de la Rocha et al. (2002)]: (6) and without cross-correlations: C_ij(t,t′) = 0(i ≠ j). The magnitude of the autocorrelations is measured by the dimensionless parameter α, which intuitively sets the excess of probability of, given one spike, finding another one within a time range τ_c. Notice that α can be varied keeping the firing rate ν fixed. The inset in Figure 2C shows three examples of spike trains generated with the same rate ν = 10 Hz, the same correlation range τ_c = 2 ms, but different α values: 0 (which is a Poisson process), 0.6, and 1.5. It is clear that, for larger values of α, the APs cluster in bursts made of more spikes (although their exact number varies from burst to burst). The interval between consecutive spikes within a burst is of the order of τ_c.

Phase-locked periodic activity. As a last example, we combined a simple periodic temporal structure with a spatial organization in which the cells coordinate their firing by phase-locking their spikes to an external oscillation of frequency f. Individual spike trains are constructed in the following way: each cell emits one AP at the same temporal phase of the oscillation, but with a certain jitter. The value of the jitter of every spike is drawn from a Gaussian distribution of deviation Σ, truncated on the sides at a half-period distance (to prevent the tails of consecutive Gaussians from overlapping). The parameter Σ sets the precision of the phase-locking of each neuron, and its value is kept fixed as the oscillation frequency is varied. This pattern of spikes, which represents the output activity of a network that oscillates rhythmically (Buzsaki and Draguhn, 2004), is illustrated in Figure 2 D.

Synaptic current

The activity of the N neurons of the presynaptic population gives rise to the stimulus excitatory current I_stim(t). Each AP can produce at most the release of M vesicles. The release from a single vesicle is modeled as an instantaneous pulse of current of size J. Thus, the transformations of the train of APs into a sequence of instant current pulses is formulated as follows^a: (7) The sums on the right run over presynaptic neurons (i = 1... N), over common contacts for each presynaptic neuron (n = 1... M), and over the releases produced at each particular contact (k = 1... rel). We ignored the variability in the number of common contacts found in experiments (Gil et al., 1999; Silver et al., 2003), so that all connections are made of the same number of them. On the contrary, we captured the variability of quantum response across contacts by taking the efficacies J_i,n randomly distributed with a Gaussian of mean J and coefficient of variation Δ= 0.2–0.4 (Gil et al., 1999). In the simulations, we will specify the value of the ratio of the efficacy and the membrane capacitance, J/C_m, which measures the quantal amplitude (i.e., the mean amplitude of the EPSP produced by the release of a single vesicle) in voltage units.

The background current I_bg(t), which represents the afferent activity from other cells of the network or from different brain areas, is generated by Poisson events activating one excitatory and one inhibitory synapse at constant rates ν_E and ν_I, respectively. These rates are of the order of several spikes per millisecond, because they represent the superposition of thousands of presynaptic trains at low cortical spontaneous frequencies (e.g., ∼1–2 Hz).

The total instantaneous synaptic current reaching the target neuron is the sum I(t) = I_stim(t) + I_bg(t). We will normally be interested in its mean and SD. To evaluate them, we will divide the duration T of a long trial into small bins (of size dt), build the histogram of the current, and compute its mean μ_I and deviation σ_I as follows: (8) (9) where data are taken starting from a time when the response of the target neuron has reached its stationary state (depending on the input rate the stationary regimen has reached after 0.1–1 s).

Without background component, the mean current μ_I has a simple analytical expression, because it is equivalent to the mean total charge entering the cell per unit time: because each of the NM contacts is hit by ν APs per unit time, and only a fraction P_t of them trigger the influx of the quantum charge J, we have the following: (10) The deviation σ_I measures the magnitude of the “fluctuations” produced by the stochasticity of the incoming spikes and of the synaptic transmission.^b

The model target neuron and the analysis of the response

We model the target neuron as a simple leaky integrate-and-fire (LIF) unit (Ricciardi, 1977). This model describes the dynamics of the membrane potential V(t) when it is below threshold by the following equation: (11) Here, C_m is the total membrane capacitance, g_L is the leak conductance, and E_L is the leak potential. The membrane time constant is τ_m = C_m/g_L. The terms I_stim(t) and I_bg(t) represent the stimulus and background current, respectively (see above). Eq. 11 is used until V(t) reaches the spike generation threshold θ. At that point, an AP is discharged, and the potential V(t) is reset to H, where it is held during a refractory time τ_ref. The “current threshold” is defined as the value of the current that depolarizes the neuron exactly at its firing threshold; for the integrate-and-fire neuron, it reads θ_I = θg_L.

In this work, we will deal only with the “stationary” regimen, that is, the response of the neuron will be computed after the transmission probability of its synapses has reached its steady state. We will characterize the response mainly by means of the output firing rate (ν_out), defined as the mean number of output APs per unit time. The output firing rate as a function of the afferent rate will be referred to as the “response function.”

We will also compute the coefficient of variation of the response interspike intervals, CV_out, defined as the ratio between the SD and the mean of the output interspike interval. This is a measure of the variability of the output trains, which is 0 for purely periodic trains, equals 1 for Poisson inputs, and is usually >1 for bursty spike trains. For the case of a phase-locked stimulus, we will also compute the response vector strength, VS_out, which measures the degree of phase-locking of spike responses varying between 0, for no temporal alignment, and 1, for perfect phase-locking (Goldberg and Brown, 1969).

Parameter values

Unless specified otherwise, we will use the following parameter values: integrate-and-fire model, E_L = 0 mV, C_m = 300 pF, τ_m = C_m/g_L = 10 ms, τ_ref = 2 ms, θ = 15–9 mV, H = 10–6 mV; synaptic model, τ_v /N₀ = 600 ms, U = 0.75, J/C_m = 0.25 mV, N₀ = 1–8; background, ν_E = 3.7 spikes/ms, ν_I = 1.2 spikes/ms, J_E/C_m = 0.25 mV, J_I/C_m =–0.35 mV. The total number of synaptic functional contacts N × M equals 2000.

Simulations

Because the synaptic current is made of instantaneous pulses of current formally modeled as Dirac δ functions (Eq. 7), the evolution of the potential (Eq. 11) can be solved in the simulations exactly [i.e., the numerical method used was not approximative (e.g., Runge-Kutta), but it was an exact integration]. The response of the neuron was simulated from three to five trials of 40–100 s, obtaining statistical errors for the output variables of the size of the symbols used in the plots. All simulations were performed on a PC running under SUSE Linux. All graphs were made using the plotting tools Grace and Xfig.

The fluctuation-driven regimen

One of the most outstanding effects of STD is that the mean stationary EPSP amplitude decreases with the afferent spike rate ν, behaving as 1/ν for large enough presynaptic rate (Abbott et al., 1997; Tsodyks and Markram, 1997) (Fig. 3A). The mean current μ_I is proportional to the product of the mean EPSP and ν (Eq. 10) and therefore saturates as ν increases (Fig. 3B). The average membrane potential, which can be approximated by 〈V〉≈ μ_I/g_L, is also upper bounded and its maximum value is given by the following: (12) where μ_max is simply obtained by substituting in Eq. 10 the rate of releases ν P_t by its maximum value, given by the rate at which vesicles recover, 1/τ_v. The question now is whether this upper bound can constrain the cell to be, regardless of the presynaptic rate, in the so-called “fluctuation-driven regimen” (Gerstein and Mandelbrot, 1964; Calvin and Stevens, 1968; Shadlen and Newsome, 1994). In the FDR, the mean depolarization must be lower than the threshold, a condition that is formulated as follows: (13) which after substituting 〈V〉_max by the expression above can be reformulated as follows: (14) Assigning some realistic values to the parameters on the right-hand side allows us to estimate the number of presynaptic neurons required to violate this inequality. Taking θ = 15 mV, MJ/C_m = 1 mV (which is a representative value of the nondepressed EPSP size), τ_v = 500 ms, and τ_m = 10 ms, we obtain that, if N < 750, the condition given in Eq. 13 is satisfied. We confirmed this result by going through the same analysis with a conductance-based LIF model obtaining a boundary of N < 600 (see Appendix). Although the total number of afferent connections received by a cortical neuron exceeds these figures (Braitenberg and Schüz, 1991), estimations of the size of what can be considered a population of neurons encoding the same information fall below this number [e.g., around a hundred (Shadlen and Newsome, 1998)].^c

Depression therefore implies that a neuron receiving an input signal from a few hundred cells will work in the FDR. This is true in the stationary state, that is, after the short transient interval required to depress the synapses. In the FDR, current fluctuations play a fundamental role in driving the response of the neuron, because the membrane potential reaches θ only on the arrival of positive fluctuations of I(t). That is why their correct modeling takes special importance in this regimen.

Current fluctuations in the deterministic and stochastic models of STD

To show the relevance of the stochastic nature of the synaptic transmission, we compare the current deviation and the neuron response produced by two models of STD, namely, the stochastic model of vesicle turnover described in Materials and Methods and the widely used deterministic model of “averaged” synaptic responses (Abbott et al., 1997; Tsodyks and Markram, 1997). When N₀ = 1, both models give, by construction, the same mean current (Fig. 3B), but they lead to current fluctuations of different magnitude and different behavior as a function of the rate (Fig. 3C). Because the deterministic model was proposed to fit synaptic responses averaged over trials (Tsodyks and Markram, 1997; Varela et al., 1997), it lacks the trial-to-trial variability observed in synaptic transmission (Stevens and Wang, 1994), and therefore the fluctuations it generates are smaller than those produced by the stochastic model, particularly at high input rates (Fig. 3C). Specifically, in the stochastic model, a smaller probability of transmission, P_t, increases the fraction of failure spikes. In contrast, P_t equals the normalized mean EPSC in the deterministic model, and its decrease produces just smaller PSC amplitudes (something that contradicts the quantal hypothesis) without filtering any of the incoming spikes. At very large rates, the current in the deterministic model is composed of a series of PSCs of negligible size, closely spaced in time. This synaptic current has almost no fluctuations and resembles, in the limit of large input rate, an injection of constant current. This subtle difference makes σ_I grow monotonically with ν toward its saturation value in the stochastic model, whereas in the deterministic model σ_I shows a non-monotonic behavior tending to 0 at high input rates (Fig. 3C).

Because the target cell is in the FDR, the amplitude of the afferent fluctuations mainly determines its response function. Thus, ν_out is much larger and saturates monotonically in the stochastic model, whereas in the deterministic model it inherits the non-monotonic behavior of σ_I and vanishes at high rates (Fig. 3D). Figure 3, E and F, illustrates traces generated by both models at ν = 60 Hz. Both produce the same mean depolarization, but the smaller magnitude of the current fluctuations in the deterministic model generates smaller fluctuations of the potential, making the neuron fire at a lower rate.

If the mean current saturates above threshold (i.e., the suprathreshold regimen), the two models give similar rates, because in this case the response is basically determined by the mean current drive and the fluctuations play a secondary role (Fig. 3D, inset).

The stochastic model seems, in summary, a more appropriate model to investigate the response in the FDR, given that the deterministic model leads to the artifact of vanishing fluctuations at high ν, an effect that would contaminate the analysis performed in what follows.

Effect of the synaptic parameters M and N₀ on the neuron response

We turn now to investigate the relevance of the size of the RP (N₀) and the number of functional contacts per presynaptic neuron (M) in the response to uncorrelated stimuli. Studying the response produced by synapses with several functional contacts will serve us to understand the impact of spike synchrony analyzed later. This happens because multiple functional contacts trivially give rise to the synchronous release of vesicles, which will produce the same effect qualitatively as a synchronous stimulus.

The size of the RP is a controversial issue. Different experiments in hippocampal slices have led to different values: whereas Stevens and collaborators report values between 2 and 20 with mean 〈N₀〉∼5 (Dobrunz and Stevens, 1997; Murthy et al., 2001), other groups have reported that the immediately releasable pool, or “primed” pool, has a size close to 1 (Hanse and Gustafsson, 2001). Theoretical analysis of paired-pulse depression in cortical neurons has reached the same conclusion, namely, that in some synapses there must be a last “bottleneck” pool that only holds around one vesicle (Matveev and Wang, 2000b).^d For a fair comparison of synapses with different N₀, the recovery time was adjusted to keep the ratio τ_v/N₀ fixed (see Materials and Methods) (Fig. 1C).

Regarding the number of common contacts, it is a widely variable quantity across brain areas. An average number from the somatosensory cortex lies around 5 for pyramidal cells (Markram et al., 1997; Silver et al., 2003) and 17 for GABAergic interneurons (Gupta et al., 2000). Among noncortical synapses, one can find examples in which M rises up to 15–20 in the spinal cord (Walmsley, 1991), 20 in the cerebellum (Pedroarena and Schwarz, 2003), and >1000 in the famous end-bulbs of Held (Held, 1893; Schneggenburger et al., 2002). For a fair comparison of cases with different M values in the following analysis, the total number of contacts, N × M, is kept fixed (Fig. 1A,B).

The comparison of the transmission probability (Fig. 4A) and the mean number of readily releasable vesicles (Fig. 4B) for two different RP sizes (N₀ = 1 and N₀ = 4) reveals that these two synaptic types differ only at low frequencies (ν < 15 Hz). In this range, synapses with larger N₀ have a larger P_t and, on average, have more vesicles ready for release. However, as the input rate increases, more spikes reach the synapses per unit time and the RP becomes more depleted. Both the transmission probability and the mean number of releasable vesicles tend to 0 as 1/ν (dashed lines in Fig. 4A,B represent the fit 1/ν). At these high rates, synapses with different N₀ are indistinguishable. In other words, a synapse with N₀ > 1 behaves as a synapse with a single vesicle RP, but with a recovery time constant τ_v/N₀ (Matveev and Wang, 2000a; de la Rocha, 2003). This can be understood by noting that, at high input rates, the RP is normally empty and sometimes has at most one docked vesicle. Thus, if for instance N₀ = 2, after complete depletion of the RP, the replenishment of one vesicle can come from the recovery of any of the two vacancies, so that it can be viewed as a single vesicle process with a twice as large recovery rate.

The mean current produced by synapses with N₀ = 1 and N₀ = 4 is plotted in Figure 4C. In both cases, vesicle depletion produces the saturation of μ_I at high input rates. Because, as it was just shown, at high rates these synaptic types are indistinguishable, it is expected that they produce the same current statistics. Before the mean saturates, synapses with larger N₀ generate a larger μ_I because the transmission probability is larger. With the parameters chosen here, μ_I saturates below current threshold (Fig. 4C, dotted line), which implies that the target cell works in the FDR at all input rates.

Synchronous stimulus

We will now discuss a correlated stimulus in which the presynaptic neurons fire synchronous Poisson trains with the same constant rate ν and a correlation coefficient ρ (see Materials and Methods). We will show that the impact of synchrony across afferent APs resembles very much that of synchronous releases produced by common synaptic contacts, examined above.

As a general rule, modifying the correlations between presynaptic trains, while keeping their rate fixed, does not alter the “mean” synaptic current but may change the current deviation substantially. As a consequence, any information conveyed by the spatiotemporal correlations of the input is transmitted by the fluctuations of the current but not by the mean drive. For reliable static synapses, synchronizing the afferent spikes increases the probability of synchronous releases, which give rise to PSPs composed of several unitary events. This increases the magnitude of the fluctuations (Shadlen and Newsome, 1998; Salinas and Sejnowski, 2000; Moreno et al., 2002b), something that generally increases the output rate, particularly in the FDR (Salinas and Sejnowski, 2000; Moreno et al., 2002b).^e For synapses showing STD, synchrony increases σ_I too, although in a nonuniform way with ν (Fig. 7A): for low rates, the increase is larger than for high rates generating a non-monotonic behavior of σ_I as a function of ν (Moreno et al., 2002a). This occurs because the impact of synchrony is weighted by the transmission probability P_t, which decreases with ν (Fig. 3A) (J. de la Rocha, R. Moreno-Bote, and N. Parga, unpublished observations). In other words, at high rates, the input synchrony does not affect the response, because it becomes rather unlikely that the simultaneous arrival of APs triggers the corresponding synchronous releases.

The non-monotonic behavior of σ_I is again inherited by ν_out, which displays a tuned dependence on ν with a “preferred” presynaptic rate, ν_p. The preferred rate depends on the synaptic parameters τ_v, U, and N₀ (data not shown). However, it remains almost unaltered to variations of ρ within a wide range of values of the correlation coefficient (ρ ∼ 0.01–0.1 for N₀ = 1). Furthermore, ρ sets the amplitude of the tuning curve acting as a “gain control parameter,” for low and moderate input rates, within the range ρ ∼ 0.01–0.06 (for N₀ = 1, see Fig. 7B). The maximum response initially follows an approximately linear relationship with ρ and later saturates (Fig. 7C).

In Figure 7D, we lowered the threshold while keeping the rest of the parameters fixed, so that the cell sits in the suprathreshold regimen. As expected, the response no longer has a resonant shape but increases monotonically for all correlation values. Increasing the degree of synchrony increases the rate, especially at low ν at which μ_I < θ_I (i.e., the cell is still in the FDR), and depression is not very prominent. This change, however, does not represent an increase of the gain, as it approximately does in the subthreshold regimen (Fig. 7B). Hereafter, we will therefore restrict the analysis to the FDR and will come back to the general case in Discussion.

The gain modulation observed in the FDR and attributable to the afferent spike correlations has important consequences in the transmission of information carried by the rate ν to the postsynaptic cell. In the absence of synchrony, σ_I is barely modulated by the input rate (Fig. 7A, full circles), so that the information can be transmitted by μ_I only in the range of low and moderate ν values, because at high presynaptic rates, μ_I undergoes very little modulation (Fig. 7A, solid line). Synchronizing the afferent APs acts as a gain for σ_I (Fig. 7A), which because of depression may acquire a prominent non-monotonic behavior. Thus, when the mean current is saturated, the current fluctuations become the main carrier of information, enabling the modulation of the response (1) within a larger dynamical range of the response and (2) for a wider range of ν (Fig. 7A,B). For instance, for ρ = 0.05, the response undergoes a ∼10 Hz variation in the range ν = 20–100 Hz (Fig. 7B), whereas μ_I barely varies in this range (Fig. 7A, solid line).

Figure 7E shows traces of the current and the membrane potential at three presynaptic rates, namely, at 1 Hz, at the preferred frequency ν_p = 9 Hz, and at 80 Hz, for ρ = 0.04. Apart from the synchronized stimulus, the target cell receives a constant background activity (see Materials and Methods). As the rate increases, the positive fluctuations in the current (produced by synchronous releases) occur more often but are smaller, and the mean current increases, taking the mean depolarization closer to the threshold. Therefore, the neuron responds maximally at ν_p = 9 Hz, because the current reaches a compromise between large fluctuations and large mean (note that the preferred frequency in Fig. 7B does not produce the maximal fluctuations in Fig. 7A). The histograms of the current at these three rate values are shown in Figure 7, F–H, respectively (thick lines), along with the asynchronous case, ρ = 0 (thin lines), drawn for comparison. At each frequency, both histograms have the same mean (dotted lines), which always falls below θ_I (arrowheads in the figure), but at ν = 1 and 9 Hz they differ substantially (Fig. 7F,G). In these two cases, synchronous releases skew the distribution and increase its deviation, something that increases the output rate significantly. At higher rate (ν = 80 Hz), the skewness, which is the signature of synchronous releases, disappears and the two distributions become similar.

To better understand the consequences that synchrony and STD may have in the transmission of information, we went beyond the analysis of the output rate and studied higher-order statistics of the response. In this direction, we performed a simulation of two target neurons receiving APs from nonoverlapping subpopulations, which are part of the same population of synchronously firing cells. We then computed the output correlation coefficient of the target pair, ρ_out, which because of the afferent correlations is different from 0. The observed general trend is that, as synapses depress with an increasing afferent rate, the correlation coefficient ρ_out drops off to 0 (data not shown). This is completely consistent with what we just saw, namely, that as synapses become more unreliable the impact of input synchrony vanishes, affecting the output firing rate but also weakening the ability of the presynaptic population to generate additional synchrony in the output ensemble of cells. We will go over the implications of this result in Discussion.

To summarize, the impact of synchronized incoming spikes is very similar to that of synaptic connections with several common contacts, but the functional implications are very different, because ρ can be a dynamical variable, whereas M was a fixed parameter. Synchronized stimuli increase the current fluctuation gain at low and moderate input rates, something that enhances the gain of the non-monotonic response function.

Autocorrelated stimulus

Neurons in cortical and subcortical areas often display positive autocorrelations in their spike times (Bair et al., 1994; Dan et al., 1996; Baddeley et al., 1997; Goldman et al., 2002), meaning that the emission of a spike by a presynaptic neuron increases the probability of observing a new discharge of the same neuron over a short time interval (e.g., ∼10–20 ms). In other words, spikes are not homogeneously spread in time, but they tend to come in clusters. A typical case of short-range positive autocorrelations is the emission of bursts observed in many different areas (Lisman, 1997; Sherman, 2001; Krahe and Gabbiani, 2004). We used a particular instance of input statistics that produces spike trains with short-range positive exponential autocorrelations (see Materials and Methods). The magnitude of the autocorrelations is set up by a parameter α, which yields a Poisson process for α = 0 and trains with a bursty temporal structure for α ≥ 1.5 (“bursts” composed of a variable number of spikes with mean ∼4–5). Examples of these spike trains for several values of α can be seen in Figure 2C, inset.

For nondepressing synapses, positive autocorrelations have the same qualitative effect as spatial cross-correlations (Moreno et al., 2002b): the impact produced by incoming clusters of APs does not depend on whether the spikes within the cluster come from the same or from different presynaptic cells.^f This simple principle seems not to be applicable when synaptic transmission depends on the presynaptic activity. With facilitating synapses, for instance, a cluster of spikes coming from the same or from different cells, would not produce the same depolarization: only if they are emitted by the same presynaptic neuron, the synapse facilitates and gives rise to a larger depolarization. However, we will see that, under the choice of certain parameter values of the stochastic model, bursts of spikes may produce qualitatively the same effect as the synchronous activity considered before, although with a quantitatively smaller overall impact.

Clustering the incoming spikes can increase the fluctuations of the synaptic current, because it may induce the temporal summation of consecutive PSCs elicited by the spikes within the burst. This increase of the fluctuations is analyzed at the level of the release statistics in Figure 8, where we compare two inputs with the same rate but different correlation magnitudes: α = 0 (Poisson) and α = 1.5 (bursts). Figure 8, A and B, shows the number of spikes falling in bins of size τ_m versus time.^g As expected, the correlated case produces large positive fluctuations because of the existence of bursts and the uncorrelated case looks very homogeneous, producing one or no spikes per bin and only occasionally two. Figure 8, C and D, depicts the number of “releases” produced by those spike trains when the RP size is N₀ = 4 and M = 5. The bursty input still produces larger fluctuations than the Poisson stimulus. Notice that, because every AP can produce from zero up to five releases, the variability of the releases seems amplified with respect to that of the spike trains.

Figure 8, E and F, also shows a sequence of releases but for synapses with N₀ = 1. In contrast to the case N₀ = 4, the large fluctuations of the bursty input have been severely filtered out so that now both bursty and Poisson inputs produce fluctuations of similar magnitude. This difference can be easily understood: bursts produced after a long silent interval might find the RP fully replenished. Then, if N₀ = 1, the burst can release at most one vesicle per contact, whereas if N₀ > 1, the same burst can release up to N₀ vesicles. This explains that, only in the case N₀ > 1 and when preceded by a long enough silent interval, a cluster of APs can be faithfully translated into a significant fluctuation in the number of releases. At high rates, the required long silent periods never occur. The model with N₀ > 1 becomes effectively a model with N₀ = 1 (as explained previously) (Fig. 4B), and the fluctuations coming from the bursts are again filtered.

In addition, clustering the incoming spikes decreases the mean release count, because a larger fraction of APs find the synapses depleted. The monotonically increasing lines in Figure 8 represent the cumulative number of events (spikes or releases) produced by Poisson and bursty inputs (scale on the right vertical axes). Because both stimuli have the same rate (ν = 5 Hz), the total number of spikes is about the same (∼200 spikes) (Fig. 8A,B). However, the total release count is smaller for the bursty stimulus regardless of the value of N₀ (Fig. 8C–F).

Positive input autocorrelations, therefore, may alter the response of the cell in opposite directions depending on the relative impact of these two effects, namely (1) the decrease of the mean release count which implies a decrease of μ_I and occurs for any size N₀ (Fig. 9A,B); (2) the increase of the fluctuation of the release count, which implies an increase of σ_I and crucially depends on the value of N₀: when N₀ = 1 the bursty stimulus generates fluctuations with a similar overall magnitude as the Poisson stimulus (Fig. 9C), whereas if N₀ > 1, σ_I increases substantially at low and intermediate input rates (Fig. 9D). As a consequence, only when N₀ > 1 (and M ≥ 5)^h, the increase in σ_I yields to an increase of ν_out with α, in that way enhancing the resonant behavior obtained previously with Poisson input (Fig. 9F). If N₀ = 1, it is the decrease in μ_I that dictates the decrease of the response (Fig. 9E). The behavior at high input rate is expected: synapses saturate and, for any α, both μ_I and σ_I (and in turn ν_out) converge to the value given by the uncorrelated stimulus.

Figure 9, G and H, shows the response CV_out for the two cases N₀ = 1 and N₀ = 4. As explained, when N₀ = 1, autocorrelations are severely filtered so that the CV_out remains unaffected when α is increased. In contrast, when N₀ = 4, bursty trains produce slightly larger CV_out values if the synapses are not very depressed, which in this example means ν < 30 Hz. Figure 9H, inset, shows this sensitivity of the CV_out to α at two values of the input rate, ν = 10 and 40 Hz. Although, for ν <30 Hz, bursts of spikes increase CV_out, the impact is rather small if one considers, for instance, that autocorrelated trains with α = 1.5 have themselves an input CV = 2 and produce a maximum CV_out ≈ 1.2.

Spike and release rasters together with current and potential traces neatly illustrate this effect (Fig. 10). Setting M = 5 for both types of synapse, a Poisson and a bursty stimulus were compared at a moderate input rate, ν = 5 Hz. When N₀ = 1, the bursts of spikes are filtered by depletion of the synapses, a fact evident in the nonbursty structure of the raster of releases compared with the corresponding raster of autocorrelated spikes (Fig. 10B, bottom and top rasters). Because of this efficient filtering of bursts, fewer releases are triggered in comparison with the Poisson case (Fig. 10A,B, compare bottom rasters) and the current trace shows a reduction of its mean with barely any variation in the fluctuations (Fig. 10A,B, compare top traces). As a result, the response firing rate is smaller for the bursty stimulus than for the Poisson input (Fig. 10A,B, bottom traces). In contrast, the example with N₀ = 4 shows that an afferent burst does usually elicit several consecutive releases (a thinned version of the original spike burst) (Fig. 10D, bottom raster). This leads to an increase of the current fluctuations that, despite the decrease of μ_I, yields an increase of the response firing rate (Fig. 10C,D, traces).

Phase-locked spike trains

We saw that, in the saturation regimen, synaptic depression may filter out the input synchrony between presynaptic trains and the positive autocorrelations of individual trains. We now show that this effect also occurs when cross-correlations are generated by a common drive. With nondepressing synapses, “signal-correlations” increase the output rate (Salinas and Sejnowski, 2000) (but see also Grande et al., 2004), which happens just because a coordinated population of neurons can produce larger depolarizations than a population of independent neurons. Let us then consider a simple configuration in which each presynaptic neuron fires, locked to a periodic signal of frequency f, one spike per cycle. The precision of the spike timing is limited by a fixed Gaussian jitter of SD Σ (for details, see Materials and Methods).

Figure 11 shows the response of the target neuron as a function of the oscillation frequency for four values of Σ along with a case in which each cell fires phase-locked to the same signal but at a different phase (incoherent mode). As expected, coordinating the firing across afferent trains in a coherent manner gives a larger response, but only for low and intermediate input frequencies. As with the two previous stimuli, in the coherent condition, ν_out shows a resonant behavior, with a preferred frequency f_p (Fig. 11A). Beyond f_p, responses decrease sharply until a critical value f_c is reached (e.g., f_c ∼ 23 Hz for Σ= 10 ms), at which the response coincides with that of the incoherent case. Beyond f_c, the output rate in the two modes are indistinguishable.

Figure 11B illustrates the input (dashed lines, VS_in) and output (symbols, VS_out) vector strengths, a measure that quantifies between 0 and 1 the precision of the phase-locking of the spikes. Because the parameter Σ is held constant as f varies, the jitter relative to the cycle length increases with the frequency, except for the case Σ= 0 ms. This produces the monotonic decrease of VS_in if Σ > 0. Remarkably, in these three cases, VS_out is always larger than VS_in, although eventually it also drops to 0 as the frequency increases.

The non-monotonic behavior of the response function occurs because, for depressing synapses, P_t decreases with f so that releases in a given synaptic contact occur for a smaller fraction of cycles, a fact that decreases the correlations between the releases across contacts. In addition, if Σ > 0, correlations become weaker, because the relative jitter increases. For f > f_c, the correlations are so small that they do not affect the response rate of the cell. However, the output spikes still carry information about the input signal, as can be observed, in that VS_out becomes 0 at a frequency significantly higher than f_c (Fig. 11B). This is particularly true when Σ= 0, at which the input vector strength is always 1, and as a consequence VS_out ∼ 1in a very large range of f (up to 1 kHz).

Because for non-zero jitter, VS_in decreases with the frequency, one might be tempted to attribute the non-monotonic shape of ν_out to this characteristic of the stimulus or to its combined effect with depression. The nonphysiological case Σ= 0 allows us to isolate the effect of depression from that of the VS_in decrease. Because the non-monotonicity occurs here without decrease in VS_in, it becomes clear that the resonant effect is solely attributable to depression and that the decrease of VS_in only affects the position of the maximum.

Finally, it is worth mentioning that the resonant effect is still present in the more general case in which cells fire at every cycle with a certain probability p < 1 (data not shown).

Generalization of the results

In the present work, we constrained the analysis to “temporally homogeneous” stimuli. However, it is clear that, in the general case, the variability of the synaptic current registered from a cell would have a large contribution from temporal variations of the afferent signal. In a recent work, we explored the effect of depression over time-varying signals defined by their instantaneous firing rate, finding that, if the synapses work in saturation, the time-varying structure of the input is also severely weakened as it is converted into a synaptic current (J. de la Rocha and N. Parga, unpublished observations). This damping of the afferent structure, which implies a loss of spatiotemporal information, seems therefore to be a general property on synapses whose transmission reliability is substantially decreased as the presynaptic activity increases. This is not an exclusive feature of vesicle depletion; a number of presynaptic mechanisms also yield a decrease of P_t: inactivation of release machinery (Hsu et al., 1996), presynaptic inhibition via metabotropic receptors (Davies and Collingridge, 1990; Scanziani et al., 1997), and others (for review, see Zucker and Regehr, 2002). The non-monotonic behavior in ν_out is achieved if P_t decreases at least as 1/ν, something observed in the cases cited above (Matveev and Wang, 2000a) but violated by synapses showing activity-dependent recovery, that is, when τ_v decreases with ν (Dittman and Regehr, 1998; Stevens and Wesseling, 1998; Wang and Kaczmarek, 1998; Fuhrmann et al., 2004).

If synapses exhibit both depression and facilitation (Thomson, 1997; Markram et al., 1998), the resonance is still present because facilitation does not prevent the synapse from saturating at high ν because of vesicle depletion (de la Rocha et al., 2004). With facilitation alone, the result would however be very different, namely, that only at high input rates, when the synapses are facilitated, correlations would have a significant impact. The response function would no longer be non-monotonic but would produce superlinear behavior.

We used a somehow detailed model of the presynaptic terminal, but an oversimplified leaky integrate-and-fire neuron model without neither synaptic conductances nor synaptic filters. However, preliminary simulations using an LIF model with synaptic conductances confirm our main results (data not shown). This occurs because the kind of correlations used here change the variance of the total current but do not alter the mean total conductance substantially. Furthermore, depression constrains the increase in the mean total conductance obtained from an increase in the presynaptic rate. This is attributable to the saturation of the release rate, which imposes an upper bound on the mean conductance that an ensemble of cells can generate. In other words, a finite presynaptic population cannot make the effective membrane time constant of a cell increasingly smaller by simply bombarding the cell at a higher rate, because synaptic depression prevents it.