Cerebral Cortex

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Cerebral Cortex, Vol. 10, No. 11, 1143-1153, November 2000
© 2000 Oxford University Press

Differential Short-term Synaptic Plasticity and Transmission of Complex Spike Trains: to Depress or to Facilitate?

Victor Matveev^1,2 and Xiao-Jing Wang¹

¹ Volen Center for Complex Systems, Brandeis University, Waltham, MA 02454 and, ² Mathematical Research Branch, NIDDK, National Institutes of Health, Bethesda, MD 20892, USA

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Notes References

 
Experimental studies have revealed conspicuous short-term facilitation and depression that are expressed differentially at distinct classes of cortical synapses. To explore computational implications of synaptic dynamics, we investigated transmission of complex spike trains through a stochastic model of cortical synapse endowed with short-term facilitation and vesicle depletion. Inputs to the synapse model were either real spike train data recorded from the visual and prefrontal cortices of behaving monkeys, or were generated numerically with prescribed temporal statistics. We tested the hypothesis that short-term facilitation could enable synapses to filter out single spikes and favor bursts of action potentials. We found that the ratio between release probabilities for a burst spike and an isolated spike grows monotonically with increasing number of spikes per burst, and with increasing interval between isolated spikes. Burst detection is optimal when the facilitation time constant matches the average burst duration. Using fractal-like spike patterns characterized by long-term power-law temporal correlations and similar to those seen in sensory neurons, we found that facilitation increases correlation at short time scales. In contrast, depression leads to a dramatic reduction in temporal correlations at all time scales, and to a flat (‘whitened’) power spectrum, thereby decorrelating natural input signals.

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Notes References

 
Transient activity-dependent synaptic plasticity is a prevalent feature of both vertebrate and invertebrate neural systems, regulating synaptic efficacy at a variety of time scales, from milliseconds to minutes (Magleby, 1987; Zucker, 1989, 1996; Fisher et al., 1997). Recently there has been a resurgence of interest in this topic, and a growing number of experimental results on cortical synapses (Stevens and Wang, 1995; Debanne et al., 1996; Markram and Tsodyks, 1996; Thomson et al., 1996; Abbott et al., 1997; Buhl et al., 1997; Dobrunz and Stevens, 1997; Castro-Alamancos and Connors, 1997; Thomson and Deuchars, 1997; Varela et al., 1997, 1999; Ali et al., 1998; Galarreta and Hestrin, 1998; Dittman et al., 2000; Kreitzer and Regehr, 2000). Theoretical studies have shown that short-term depression can have dramatic effects on synaptic response to inputs of changing firing rates (Abbott et al., 1997; Tsodyks and Markram, 1997), and it was also suggested that depression may be involved in generation of rhythmic activity in certain neural systems (Senn et al., 1996; O'Donovan and Chub, 1997; O'Donovan and Rinzel, 1997). It was also proposed that short-term plasticity may underlie the selectivity of neurons for temporal patterns in afferent signals (Buonomano and Merzenich, 1995; Buonomano, 2000). These results highlighted the notion that synapses are readily modifiable and play an active role in the computation carried out by a neural circuit.

Short-term plasticity is differentially expressed at synapses, in a target-cell-specific manner. For example, intracellular recordings from neocortical slices revealed that afferents of a pyramidal cell innervating another pyramidal cell and an interneuron display frequency-dependent depression and facilitation respectively (Thomson, 1997; Markram et al., 1998; Varela et al., 1999). In another study, synaptic responses evoked by a pyramidal cell in a bitufted interneuron showed facilitation, while the responses evoked by the same pyramidal cell in a multipolar interneuron exhibited short-term depression (Reyes et al., 1998). Target-cell specific short-term plasticity of synapses of pyramidal cells was also observed in the hippocampus (Ali and Thomson, 1998). All these data raise the questions of why short-term plasticity should be synapse-specific, and when facilitation or depression is desirable from a computational point of view. To address these questions, we need to study the interplay between presynaptic firing patterns and synaptic dynamics, for example by using natural spike trains from a behaving animal as stimulation patterns in studies of synaptic transmission (Dobrunz and Stevens, 1999). In the same spirit, we used a computational approach to investigate how a depressing or facilitating synapse would process complex spike trains similar to those occurring in the intact brain.

Although a fair number of modeling studies of short-term synaptic dynamics can be found in the literature, many of the existing biophysical models are concerned with a particular feature of synaptic response (Neher and Zucker, 1993; Tank et al., 1995; Bertram et al., 1996; Bennett et al., 1997; Dobrunz and Stevens, 1997; Canepari and Cerubini, 1998; Wu and Betz, 1998), while models used in investigating the functional roles of short-term plasticity tend to be phenomenological (Tsodyks and Markram, 1997; Varela et al., 1997, 1999). Furthermore, even the more detailed studies (Dittman and Regehr, 1998; Dittman et al., 2000) do not take into account the stochastic nature of synaptic response, and only consider simple (periodic or Poisson) input patterns. In the present work, we investigated a model of synaptic dynamics that incorporates both the stochastic vesicle recycling process and activity-dependent facilitation. Unlike most existing models, our model takes into account the fundamental assumption, believed to hold for central synapses, that at most one vesicle can be released per action potential (Redman, 1990; Arancio et al., 1994; Korn et al., 1994; Stevens and Wang, 1995; Somogyi et al., 1998; Walmsley et al., 1998). This condition provides an important constraint on response properties of a cortical synapse. We explored the response of this model synapse to ‘naturalistic’ inputs similar to neuronal spike trains recorded in vivo from the cortex, and focused on two common kinds of complex neuronal firing patterns: spike trains containing fast bursts as well as isolated spikes, and spike trains of fractal temporal structure with long-term correlations.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Notes References

 
Our synapse model is, in essence, a simple model of vesicle turnover (Fig. 1) (Wang, 1999; Matveev and Wang, 2000). It consists of a single vesicle pool of size N with an upper limit of N₀, which can lose one vesicle in response to a presynaptic action potential. Depletion of the vesicle pool leads to short-term depression, which recovers with a time constant _D equal to the inverse of the vacancy refill rate. Processes of vesicle release and recovery are treated stochastically (Vere-Jones, 1966, Melkonian and Kostopoulos, 1996; Quastel, 1997; Maass and Zador, 1999). Release of a single vesicle during an incoming pulse is governed by a Poisson process with some time-dependent rate _V(t), which we assume is significant only for the duration of the pulse. The integral _V = _V dt is the fusion rate for a single vesicle integrated over the duration of the presynaptic pulse. The single-vesicle release probability is then p_v = 1 – exp(–_V), and the single-vesicle failure probability is 1 – p_v = exp(–_V) (Dobrunz and Stevens, 1997). Vesicle recovery is assumed to be governed by a Poisson process with rate 1/_D. Therefore, the probability for a vacancy in the vesicle pool to be refilled during a time interval t is given by p_refill = 1 – exp(–t/_D).

View larger version (18K): [in this window] [in a new window]   Figure 1. Model of vesicle turnover. Release probability is described by a Poisson process with lateral inhibition between individual vesicles during exocytosis, and is given by one minus the failure rate, which is equal to exp(–VN), where V = V,0F is the fusion rate for a single vesicle, and F is the facilitation factor. Vacancy in the vesicle pool is refilled with a time constant of D, which determines the depression recovery dynamics.

 
We impose a constraint that no more than one vesicle can be released per single action potential, assuming that a vesicle release event transiently prevents other vesicles from being exocytosed, as suggested by Triller and Korn (Triller and Korn, 1982). Then, the release probability per stimulus is one minus the failure probability, given by the Nth power of single-vesicle release failure probability, where N is the number of vesicles available for release (Dobrunz and Stevens, 1997, eq. 1.A):

(1)

Therefore, the univesicular release constraint implies a nonlinear dependence of the release probability on the number of available vesicles. For a pool of synapses with a similar value of _V, each with a different number of releasable vesicles N, the initial release probability p_r would increase with N according to equation (1) (see Fig. 5 of Dobrunz and Stevens, 1997).

View larger version (28K): [in this window] [in a new window]   Figure 5. Response of the model synapse to a spike train recorded from a neuron in the monkey prefrontal cortex during the delay period in an oculomotor delayed-response memory task (spike train provided by Matt Chafee and Patricia Goldman-Rakic). (A) Trial responses to a segment of the spike train, in the facilitation and depression regimes. Synaptic parameters same as in Figure 4A. Average stimulation rate is 36 Hz. Average release probabilities are pr = 0.182 in the facilitation and pr = 0.186 in the depression regimes. (B) Interspike interval histogram and (C) autocorrelation function for the experimental spike train. (D) The ratio of release probabilities for a spike in a burst (pB) and an isolated spike (pS), as a function of short-term synaptic plasticity. Conventions and parameters same as in Figure 4D. (E) Dependence of the pB/pS ratio, in the regime of strong facilitation, on the fast facilitation time constant F1, with other parameters kept fixed. This ratio shows a peak at F1 average burst duration (BD = 14.7 ms). The average ISI within a burst ISIB = 5.1 ms, and the average interval between single spikes and between isolated spikes and bursts ISIS = 50.7 ms.

 
We assume that the release site quickly recovers from the putative inhibition mechanism that prevents multivesicular release. As proposed by Dobrunz et al. (Dobrunz et al., 1997), such a ‘lateral inhibition’ mechanism may be the basis for the observed brief refractory period following a postsynaptic response, during which the probability for another response to happen is small (Stevens and Wang, 1995; Hjelmstad et al., 1997). Experimentally, one can distinguish between the relative and the absolute refractory times; for hippocampal synapses in culture, both values are close to 5 ms at room temperature (Stevens and Wang, 1995; Hjelmstad et al., 1997), and decrease to ~3–4 ms at 31°C (Stevens and Wang, 1995). For our simulations, we set both time constants to 3 ms. After a spike the vesicle fusion rate _V is set to zero for the duration of the absolute refractory time, after which _V recovers exponentially with a time constant equal to the relative refractory time. Since these time constants are very short, at physiological firing rates the refractory period should affect only the synaptic response to fast bursts of action potentials.

Short-term Facilitation

We introduce facilitation into our model by allowing the vesicle fusion rate to increase with stimulation: _V(t) = _V,0F(t), where F(t) is the facilitation factor which is incremented with each incoming action potential according to a deterministic rule; this implies that we neglect the stochasticity resulting from probabilistic opening of presynaptic calcium channels (Bertram et al., 1996; Bennett et al., 1997). We assume that facilitation arises as a result of stimulation-induced increase in the probability of release, due to a calcium-binding mechanism proposed by Bertram and colleagues (Bertram et al., 1996). According to this model, each release site is controlled by four independent calcium gates, consistent with the fourth-order cooperativity between presynaptic Ca²⁺ concentration and synaptic response (Dodge and Rahamimoff, 1967). In order for exocytosis to take place, each of the gates has to open by binding a Ca²⁺ ion. All gates are assumed to have different kinetics, which is suggested by evidence of stepwise increase in facilitation with increasing stimulus frequency at the squid giant synapse, accompanied by a decrease in the Ca²⁺ co-operativity of release (Stanley, 1986). Multiple facilitation time scales have also been observed at cortical synapses (Dobrunz et al., 1997; Thomson, 1997). One of the gates is assumed to have unbinding kinetics in the sub-millisecond range (Bertram et al., 1996), so it should not contribute to facilitation (at physiological firing rates). Facilitation thus involves only three gates. The probability of a gate of type j remaining open then evolves according to a simple equation

(2)

where Ca²⁺ influx is assumed to be brief, [Ca²⁺] = A_{Ca i} (t – t_i), with t_i the arrival time of the ith stimulus. The parameters k_j⁺ and k_j^– = 1/_Fj are respectively binding and unbinding kinetic coefficients for gate j. Time constants _Fj specify the decay times of the corresponding facilitation components. For simplicity, we assume that the vesicle release probability for a given action potential is determined by the states of release gates at the end of the spike. Let us denote by O_j(t_n⁺) the jth gating variable at the end of the nth spike, then the facilitation factor is F(t_n) = F₁(t_n)F₂(t_n)F₃(t_n), where F_j(t_n) = O_j(t_n⁺)/O_j(t₁⁺), j = 1,2,3. The vesicle fusion rate _V(t_n) = _V,0F(t_n), where _{V,0 V}(t₁) is the initial vesicle fusion rate.

The facilitation factors are updated as follows, for an arbitrary input train: (i) at the time of spike arrival, facilitation factors are incremented according to F_j 1 + C_j F_j, where C_j = exp(–A_Cak_j⁺) (0 C_j 1); (ii) between spikes each F_j recovers to 1 with time constant _Fj (j = 1,2,3). This update rule is based on the analytic solution of equation (2), linking values of gating variables for two successive spikes, t_n and t_n+1 (Bertram et al., 1996):

(3)

where O_j(t₁⁺) = 1 – C_j is the value after the first spike. Dividing by O_j(t₁⁺), we obtain the update rule for the facilitation factors:

(4)

Note that the parameters C_j (j = 1,2,3) determine the facilitation strengths. From equation (4) follows that the paired-pulse facilitation (PPF) for very short interpulse intervals is given by (1 + C₁)(1 + C₂)(1 + C₃); thus, the maximal paired-pulse facilitation that can be achieved within this model is PPF_max = 2^p, where p = 3 is the number of facilitation gates.

From equation (4) one finds that with constant-frequency stimulation of rate r, facilitation exponentially approaches a stationary level equal to

(5)

The associated steady-state vesicle fusion rate is _V,ss = _V,0F_ss.

Parameters

An important parameter of the synapse model is the number of vesicles in the release-ready pool, N₀. The size of the release-ready pool varies across different types of central synapses (Zucker, 1996; Neher, 1998); we chose a range of values corresponding to hippocampal excitatory synapses, where recordings from individual boutons have been achieved (Bekkers and Stevens, 1990; Liu and Tsien, 1995; Forti et al., 1997). For the rat hippocampal synapses in slice and culture, Stevens and collaborators assessed the size of the releasable pool by measuring the number of postsynaptic responses elicited by a short, high-frequency electric stimulation (Stevens and Tsujimoto, 1995; Dobrunz and Stevens, 1997), or by a brief application of a hypertonic solution (Rosenmund and Stevens, 1996), as well as by optical monitoring of the amount of fluorescent dye uptaken and released during stimulation (Murthy et al., 1997; Murthy and Stevens, 1998) [see also (Ryan et al., 1997)]. The available pool size estimated in individual experiments varied between 2 and 25. Ultrastructural analysis of hippocampal synapses suggests that these numbers are consistent with the number of vesicles docked at single synaptic active zones (Forti et al., 1997; Schikorski and Stevens, 1997). In our simulations we choose N₀ = 3–10. For the vesicle refill time constant we choose a value of _D = 1–2 s, which agrees with the experimentally determined time of recovery of synaptic response from depression (Markram and Tsodyks, 1996; Dobrunz and Stevens, 1997; Varela et al., 1997). Decay time constants for the three facilitation components are _F1 = 35 ms, _F2 = 190 ms and _F3 = 2 s. Quantitatively, the values of _F1 and _F2 were deduced from the interpulse-interval dependence of facilitation measured at hippocampal synapses by Dobrunz and co-workers (Dobrunz et al., 1997) (see their Fig. 1). The value of the longer facilitation time constant _F3 agrees with the facilitation recovery time at cortical pyramid–interneuron connections studied by Thomson (Thomson, 1997).

In this form, the model is specified by nine parameters: the maximal size of the vesicle pool N₀, the depression recovery time constant _D, the initial fusion rate _V,0 [or, equivalently, the initial release probability p₀ = 1 – exp(–_V,0N₀)], and the facilitation parameters C_j and _Fj, j = 1,2,3.

The magnitude of the initial release probability has been shown to determine the tendency of a given synapse to exhibit facilitation or depression of response (Debanne et al., 1996; Dobrunz and Stevens, 1997; Tsodyks and Markram, 1997) [reviewed elsewhere (Korn and Faber, 1987; Zucker, 1989)]. Thus, we vary the values of p₀ (_V), N₀ and C_j to achieve regimes of strong facilitation (low p₀, high C_j) and strong depression (high p₀, low N₀). In the regime of strong depression, facilitation cannot play a significant role since vesicle fusion rate _V is already high; in this case we set C_j = 0 for the sake of simplicity.

Bursty Spike Train

To study the impact of short-term plasticity on synaptic response to bursts of spikes versus single spikes, we stimulate the model synapse with a spike train of high burst content. We generate such bursty spike train numerically, using a two-state pseudo-Markov process described by Ekholm and Hyvärinen (Ekholm and Hyvärinen, 1970) (Fig. 6A). In this process, firing alternates between two distinct modes or states: one of the states corresponds to a burst of spikes (high-frequency firing state), and the other corresponds to more sparsely spaced spikes between bursts (low-frequency firing state). This method produces spike sequences that are compatible with firing patterns observed in rabbit diencephalon and cat superior colliculus cells in vivo (Ekholm and Hyvärinen, 1970; Mandl 1993).

View larger version (31K): [in this window] [in a new window]   Figure 6. Response of the model synapse to a burst-rich spike train generated according to a two-state stochastic process (Ekholm and Hyvärinen, 1970). (A) Trial responses to a 2.9 s segment of the spike train. The average number of spikes in a burst is nburst = 6; average number of single spikes between bursts is nsingle = 5.7; average firing rate is 16 Hz. Conventions and parameters same as in Figure 4A. (B) Burst-discriminating ability as a function of synaptic parameters. Conventions same as in Figure 4D. Parameters are varied linearly left to right from strong facilitation values (N0 = 12, p0 = 0.07, C1 = 0.9, C2 = 0.95) to strong depression values (N0 = 3, p0 = 0.92, C1 = C2 = 0). Time constants are not varied (D = 2 s, F1 = 35 ms, F2 = 190 ms). (C) Burst-discriminating ability of a strongly facilitating synapse as a function of the facilitation time constant F1. Conventions same as in Figure 4E. Vertical lines mark the duration of the ISI within a burst (ISIB = 4.5 ms) and the average burst duration (BD = 22.5 ms). The average interval between stand-alone spikes and between a single spike and a burst is ISIS = 105 ms. (D) Burst-discriminating ability of a strongly facilitating (squares) and a strongly depressing (triangles) synapse as a function of the number of spikes per burst. Average firing rate varies from 10.9 to 19.5 Hz as nB is increased from 2 to 9. Open circles correspond to nB = 6 used in (A)–(C),(E). (E) Burst-discriminating ability of a strongly facilitating (squares) and a strongly depressing (triangles) synapse as a function of the average interval between isolated spikes, ISIS. Average firing rate varies from 53.5 to 5.5 Hz over the range of ISIS covered in this plot (ISIS = 30–315 ms). Open circles mark ISIS = 105 ms used in (A)–(D).

 
As in Ekholm and Hyvärinen (1970), the duration of a burst is determined by the number of interspike intervals (ISIs) within a burst, which obeys a binomial distribution P_B(n). The interburst interval is determined by the number of ISIs between two consecutive bursts, which is drawn from a geometric distribution P_S(n). The binomial distribution is given by

n = 0, . . ., m, with parameter values p_B = 0.5 and m = 8. The geometric distribution is defined by P_S(n + 1) = (1 – p_S)p_Sⁿ (n = 0, . . ., ) with p_S = 0.85. Both distributions are shifted by one so that there is at least one ISI separating two bursts, and at least one ISI within a burst (i.e. at least two spikes per burst).

Interspike intervals within a burst (ISI_B) and between isolated spikes (ISI_S) are drawn from gamma probability densities of index 2 with different time constants:

(6)

where _B = 1.2 ms corresponds to ISIs within bursts, and _S = 35 ms corresponds to ISIs between bursts (average value of ISI_S,B is equal to _S,B multiplied by a factor of 3). This choice of probability densities leads to a bimodal ISI distribution similar to one seen in in vivo spike trains (Fig. 4B). We impose a lower bound on the minimal ISI by adding a dead time of 1 ms to all intervals. The average spike rate for these parameter choices is equal to 16.25 Hz.

View larger version (29K): [in this window] [in a new window]   Figure 4. Response of the model synapse to a spike train recorded from the visual cortex of the awake monkey, in response to a grating stimulus (spike train provided by Charles Gray and Rony Azouz). (A) Trial responses to a segment of the spike train, in the facilitation and depression regimes. Thin lines: input spikes and synaptic release events; gray bars: release probability at the time of spike arrival. Parameters for the depression regime: N0 = 7, p0 = 0.95, D = 1 s; for the facilitation regime: N0 = 12, p0 = 0.11, D = 1 s, C1,2 = 0.9, 0.95; F1,F2 = 15, 190 ms. Average stimulation rate is 38 Hz. Average release probabilities are pr = 0.172 in the facilitation and pr = 0.170 in the depression regimes. (B) Interspike interval histogram (ISIH) for the experimental spike train exhibits bimodal character. (C) Autocorrelation function reveals oscillations around 32 Hz. (D) The ratio of release probabilities for a spike in a burst (pB) and an isolated spike (pS), as a function of short-term synaptic plasticity. Parameters are varied linearly from strong facilitation (N0 = 12, p0 = 0.05, C1 = 0.9, C2 = 0.7) to strong depression regime (N0 = 3, p0 = 0.95, C1 = C2 = 0). Time constants are not varied (D = 1 s, F1 = 15 ms, F2 = 190 ms). (E) Dependence of the pB/pS ratio, in the regime of strong facilitation, on the fast facilitation time constant F1, with other parameters kept fixed. This ratio shows a peak at F1 average burst duration (BD = 15.4 ms). The average ISI within a burst ISIB = 5.5 ms, and the average interval between single spikes and between isolated spikes and bursts ISIS = 42.2 ms.

 
To summarize, the spike train is generated by repeating the following sequence of steps: (i) the number of ISIs in a burst, m_B, is chosen according to probability distribution P_B; (ii) burst is formed by generating m_B ISIs according to probability density _B; (iii) the number of long ISIs corresponding to single spikes between bursts, m_S, is drawn from probability distribution P_S; (iv) m_S ISIs are chosen based on distribution _S.

Fractal Spike Trains

To study the response of the synapse model to inputs with long-term temporal autocorrelations, we stimulate the model with numerically generated fractal spike trains. We generate such spike trains using the fractal shot-noise driven doubly stochastic Poisson process described by Lowen and Teich (Lowen and Teich, 1991). According to this process, probability of a spike occurring at time t is determined by a stochastically varying firing rate r(t); namely, the probability of a spike occurrence within time interval [t, t + t] is equal to r(t)t. The rate function r(t) is constructed using another (primary) Poisson process of some constant rate r₀. The event times {t_i} of the primary Poisson process are passed through a linear filter h(t), yielding rate function r(t) of the fractal process:

(7)

where amplitudes K_i are in general stochastic quantities. It is the power-law form of the filter function h(t) that leads to long-term temporal correlations and the fractal nature of the process. Statistical quantities such as the autocorrelation function and the Fano factor (see definitions below) exhibit power-law temporal behavior for time-scales between T_A and T_B. Cut-offs T_A and T_B ensure that the spike rate r(t) remains finite for any value of ß.

We have chosen the following parameter values: ß = 0.9, T_A = 2 ms, T_B = 100 s, r₀ = 0.2 Hz. Filter amplitude K_i is taken to be uniformly distributed between K_A = 6 and K_B = 8. To prevent events from occurring too close to each other, an absolute refractory time of 1.5 ms and a relative refractory time of 2 ms are imposed. Average event rate for these parameter choices is 14.7 Hz. The statistical properties of the resulting fractal spike train are shown in Figure 8.

View larger version (35K): [in this window] [in a new window]   Figure 8. Interval distribution and variability of a fractal signal before and after transmission through a model synapse. Both linear and log–log plots are given for each curve. Thick solid lines: presynaptic spike train; thin solid and dashed lines: output release event sequences for facilitation and depression regimes respectively. (A) Comparison of ISI distribution with the interrelease event interval distributions. (B) Fano factors for the fractal input and the synaptic outputs. For the fractal input, away from the upper and lower cut-offs the Fano factor behaves as F(T) ~ 1 + cT2(1 – ß), where ß = 0.9, and c is a constant. In the depression regime the Fano factor of the output release train is close to 1, as expected for a totally uncorrelated Poisson train. (Synaptic parameters are N0 = 5, D = 2 s, p0 = 0.9 for the depression regime, and N0 = 8, D = 2 s, p0 = 0.02; F1,2,3 = 35 ms, 190 ms, 2 s; C1,2,3 = 0.9, 0.95, 0.8 for the facilitation regime.)

 
Statistical Analysis: Temporal Autocorrelation

For a discrete (point) process such as a spike train, or a train of release events, autocorrelation function G() characterizes the likelihood of observing two events separated by a time interval equal to . It is defined by

(8)

where µ is the average event rate. In this normalization the autocorrelation function is therefore equal to the difference between the conditional probability rate of observing an event at (or close to) time t + , given an event at (or close to) time t, and the average (unconditional) event rate µ. Here we assume that the process is stationary, so neither G() nor µ depend on t. Autocorrelation function approaches zero as ® ¥, since the correlation between the occurrences of two events should decrease as the time between the events grows.

Sometimes it may be convenient to normalize the autocorrelation by the average event rate; the resultant quantity is referred to as the coincidence rate: g() = G()/µ + 1. The advantage of such a correlation measure is that it does not depended on the overall level of activity, i.e. it will not change if the event rate is modified by a constant factor.

Statistical Analysis: the Fano Factor

The Fano factor characterizes the fluctuations of a point process, and is defined by the ratio of the variance and the mean of the number of events in a given time duration T (Fano, 1947):

(9)

The Fano factor is equal to 1 for a Poisson process for any time interval T: in this case var[n(T)] = n(T) = µ_T, where µ is the average event rate. F(T) is <1 for a process more regular than Poisson, and is >1 for a process with fluctuations larger than those in a Poisson process.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Notes References

 
Synaptic Response to Constant-frequency Stimulation

In response to a stimulus train, the model synapse may display either facilitation or depression of response, depending on values of model parameters. We choose two sets of parameter values, corresponding to regimes of strong facilitation and strong depression (Fig. 2). In each of these two regimes, the release probability and release event sequence are shown in Figure 2A for two sample trials of constant-frequency stimulation. Since only one release is allowed per action potential, synaptic output is a binary event sequence (release/failure). Toward the end of the traces there are periods of zero release probability, which are the times where the vesicle pool is completely depleted. The trial-averaged release probability p_r, which represents the average synaptic response per stimulus, is shown in Figure 2B. These time courses of short-term plasticity are similar to those observed experimentally in cortical synapses [cf. Fig. 2 in (Dobrunz and Stevens, 1997)].

View larger version (28K): [in this window] [in a new window]   Figure 2. Response of the model to constant-frequency stimulation. (A) An example with a stimulus rate of 20 Hz; (a) stimulus train, (b) response in the facilitation regime, (c) response in the depression regime. In (b) and (c), two sample trials are shown. Black vertical bars represent release events; the height of the thick gray bars denotes the release probability at the time of arrival of a spike. Initial release probabilities are p0 = 0.1 and p0 = 0.9 for the facilitation and depression regimes respectively. For both cases, N0 = 8 and D = 2 s. Facilitation parameters in (b) are F1,2,3 = 35 ms, 190 ms, 2 sec; C1,2,3 = 0.9, 0.95, 0.8. (B) Trial-averaged release probability as a function of time in the facilitation (full circles) and depression (open circles) regimes. Same parameter values as in (A).

 
In contrast to the depression regime where p_r decreases monotonically, in the case of facilitation p_r initially grows, until vesicle depletion takes over; for both regimes, release probability eventually approaches a stationary state, p_rss. The characteristic time of response decay in the depression regime depends both on the recovery time constant _D and the rate of stimulation r, and is typically much shorter than _D. It is smaller with larger _V or higher stimulation rate r (Wang, 1999; Matveev and Wang, 2000).

In Figure 3A, the steady-state release probability p_r(r)_ss is plotted as a function of the stimulation rate. In the facilitation regime, this dependence is non-monotonic, displaying a maximum near 6 Hz (Markram et al. 1998). However, in both cases the synaptic response rate, given by the product of p_r(r)_ss and the stimulation rate r, increases monotonically and approaches a plateau at high stimulation frequencies (Fig. 3B). The saturation of the response rate implies that the steady-state release probability decays as 1/r at high rates, due to vesicle depletion (Liley and North, 1952). Therefore, the response rate becomes insensitive to the frequency of sustained presynaptic stimulation at high input rates (Abbott et al., 1997; Tsodyks and Markram, 1997).

View larger version (22K): [in this window] [in a new window]   Figure 3. Dependence of steady-state synaptic response on the stimulation frequency. Curves with filled and open circles correspond to facilitation and depression regimes respectively. (A) Steady-state release probability. In the facilitation regime, release probability is non-monotonic, reaching a maximum at ~6 Hz. Parameter values are the same as in Figure 2. (B) Steady-state synaptic response rate, given by the product of the average release probability and stimulation rate. In both regimes, response amplitude reaches a plateau at high firing rates. The value of the plateau is insensitive to initial release probability and facilitation, but depends on N0 and D: prr N0/D.

 
Response to Bursty Spike Trains

Our synapse model can display various degrees of short-term facilitation and depression, depending on the choice of parameters. As we have seen above, in response to a constantfrequency input train, the behavior of the synapse in regimes of strong facilitation and depression differs dramatically only during the initial few stimuli, but not in the steady state (Figs 2B, 3B). The situation, however, is different for more complex input patterns. When the stimulation train possesses a rich temporal structure, the input rate is constantly changing in time, which unceasingly modifies the internal state of the synapse due to activity-dependent, short-term plasticity, and the output is expected to be different at a strongly facilitating synapse compared to a strongly depressing synapse. Here we test this idea using real spike trains recorded in vivo from cortical cells, as well as spike sequences generated numerically.

Neuronal firing patterns recorded from different cortical areas of both anesthetized and behaving animals reveal a rich temporal structure: periods of rapid firing alternate with periods of relative inactivity, and bursts of closely spaced spikes are often observed along with spikes separated by longer time intervals (Bair et al. 1994; Gray and McCormick, 1996). It is conceivable that short-term plasticity could allow the synapse to select specific temporal features from the input spike train for transmission to the postsynaptic neuron. For instance, Lisman suggested that facilitation enables synapses to respond reliably to bursts of spikes, which might contain most of the information carried by the spike train, while filtering out stand-alone tonic spikes that could represent unwanted noise (Lisman, 1997). An alternative possibility is that bursts and single spikes could code for different features of the same stimulus (Cattaneo et al., 1981; DeBusk et al., 1997).

Here we study quantitatively the ability of the model synapse to detect bursts by analyzing its response to burst-rich stimulus trains. As a specific example, we drive the model with a spike train recorded in the visual cortex of the awake monkey, in response to a grating visual stimulus (Fig. 4A). This cell displays ‘chattering’ behavior (Gray and McCormick, 1996; Wang, 1999), firing bursts and single spikes rhythmically; the ISI histogram is bimodal (Fig. 4B) and the autocorrelation function for the given cell shows a pronounced oscillatory component in the 30–35 Hz frequency range (Fig. 4C). As seen in Figure 4A, in the facilitation regime the release probability is substantially enhanced within a burst of spikes. By contrast, in the depression regime the release probability is typically reduced within a burst due to vesicle depletion. In Figure 4A the time-averaged release probability p_r is about the same in the depressing and facilitating cases.

To characterize the ability of the synapse to detect bursts, we calculate separately the release probability for a single spike (p_S) (i.e. fraction of single spikes that lead to a vesicle release) and that for a spike within a burst (p_B). The ratio between these two values, p_B/p_S, is calculated for different model parameters covering a continuous range from the strong facilitation regime to the strong depression regime. A burst spike is defined as a spike that is preceded or followed by another spike within a short time interval of 10–15 ms; this interval corresponds to the trough in the bimodal ISI distribution (such as one seen in Fig. 4B). In this definition burst spikes correspond to the short ISI mode in the ISI distribution.

Since we are mostly interested in spike trains with relatively high spike rates (>10 Hz), we reduce the number of facilitation processes to two, assuming that the slowest facilitation component is close to saturation at high firing frequencies, and does not significantly affect the character of synaptic response. As can be seen in Figure 4, we find that in the case of strong facilitation, the synapse can be 50% more likely to respond to an incoming spike if it belongs to a burst. Conversely, in the strong depression case, synapse is almost twice as likely to respond to a single spike than a spike within a burst, since vesicle depletion makes multiple release events during a single burst less probable. For a certain intermediate plasticity regime, facilitation balances depression, and the release probability is the same for any spike.

Naturally, this ability of the synapse to discriminate the bursts depends crucially on the facilitation time constants, especially the shortest one _F1. The effect is expected to be optimal if _F1 is shorter than the average interval between single spikes and between a single spike and a burst (so that facilitation decays away between single spikes), but significantly longer than the ISIs within a burst (so that facilitation accumulates during a burst). This is demonstrated in Figure 4E, where the behavior of the release probability ratio p_B/p_S is shown as a function of _F1. One can see that the maximal burst discrimination is achieved when the facilitation decay time matches the average burst duration. For the same reason, the p_B/p_S ratio will be greater if the second facilitation time constant, _F2, is smaller and closer to the average burst duration.

A second example is a bursty spike train recorded from the monkey prefrontal cortex during the delayed period of an oculomotor delayed response task (Fig. 5A); it thus represents mnemonic neuronal activity correlated with working memory (Chafee and Goldman-Rakic, 1998). This cell shows a strong propensity to fire brief bursts of spikes, as evidenced by visual inspection of the spike train shown in Figure 5A and by the peak in the ISIH at very short intervals (Fig. 5B). This cell displays a strong positive autocorrelation at short temporal scale (Liu et al., 1998), as demonstrated by the large peak in the autocorrelogram (Fig. 5C), but does not exhibit oscillatory behavior. Similarly to the case of the chattering cell from the visual cortex, we found that for a facilitating synapse the release probability is significantly higher for a spike belonging to a burst than for an isolated spike; the opposite is true for a depressing synapse (Fig. 5A). The p_B/p_S ratio is 1.5 in the strongly facilitating regime, and 0.55 in the strongly depressing regime (Fig. 5D). Again, the burst detectability is optimal if there is a match between the time constant of short-term facilitation and the mean burst duration (Fig. 5E).

Therefore, our conclusion about the optimal facilitation time constant for burst discrimination is rather general and is not limited to a particular type of burst-containing spike train. To further confirm this point, we also considered artificial random bursty spike trains generated numerically according to a pseudo-Markov stochastic process (see Materials and Methods, and Fig. 6A). In this case the burst-discriminating ability of the synapse in the facilitating regime is significantly higher (Fig. 6B,C), and the release probability for a spike within a burst is almost twice as high as that for an isolated spike. As in the case of spike trains recorded in vivo, the p_B/p_S ratio is maximized when the dominant facilitation time constant matches the average burst duration (Fig. 6C). The greater burst discrimination is realized because of the larger average number of spikes within a burst (6 compared to 2–3 for the chattering cell spike train), and longer average interval between bursts and standalone spikes. Thus, the average number of spikes in a burst and the stimulation ‘duty cycle’ are parameters that critically determine the ability of the synapse to detect bursts in the given stimulation pattern. This is demonstrated in Figure 6, where the ratio of release probabilities for a burst spike and a single spike is shown to increase monotonically as a function of the number of spikes per burst (Fig. 6D), and the length of the interval between isolated spikes (Fig. 6E).

Variation of the depression recovery time parameter has a much weaker effect on the burst discrimination ability of the model synapse. Changing _D from 1 to 4 s leads to an increase in the p_B/p_S ratio of at most 30%, with significant increase taking place only under conditions of strong facilitation and large n_B (simulation results not shown). This is because in the absence of facilitation an increase in _D causes comparable decrease in both p_B (response to a burst spike) and p_S (response to a single spike), but strong facilitation partially compensates for stronger depression during a burst.

Response to Fractal Spike Trains

It has been traditionally assumed that a sequence of action potentials produced by a firing neuron can be accurately represented by a ‘memoryless’ stochastic Poisson process, in which individual ISIs are statistically independent of each other (Mueller, 1954; Kuffler et al., 1957; Bishop et al., 1964; Smith and Smith, 1965). However, it has been established that long sequences of action potentials recorded in a variety of neural systems exhibit considerable long-term autocorrelations and reveal fractal (self-similar) temporal structure, characterized by the power-law scaling of autocorrelation with time and 1/f behavior of the power spectrum. This effect has been observed in visual and auditory systems of vertebrates and invertebrates (Teich, 1989, 1992; Turcott et al., 1995; Lowen and Teich, 1996; Teich et al., 1997), in somatosensory cortex (Wise, 1981), and reticular formation neurons (Grüneis et al., 1993). Thus, it appears that this property of neural firing is common and it is therefore of interest to study how the statistics of such self-similar signals are modified by short-term synaptic dynamics. For this purpose we have generated a fractal spike train according to the fractal shot-noise driven doubly stochastic Poisson model (see Materials and Methods), and used it as an input to the model synapse. As shown in Figure 7, in response to such a fractal spike train, the output of the synapse model is dramatically different in the facilitation and depression regimes. For a fair comparison, the overall average release probability is adjusted to be the same in these two cases, so that the distinct statistics of the output patterns must be accounted for by the difference in the synaptic temporal dynamics rather than in the average transmission efficiency. For a facilitating synapse, the release probability is very small for an isolated spike, but is greatly increased during a cluster of spikes, whereas for a depressing synapse the release probability is significant for an isolated spike, but usually decreases to zero during a cluster of spikes due to vesicle depletion. Therefore, facilitation is expected to enhance temporal autocorrelation of the release event sequence at relatively short term scales (e.g. within a cluster), whereas depression should reduce the autocorrelation.

View larger version (17K): [in this window] [in a new window]   Figure 7. A sample spike train generated according to the fractal shot-noise driven doubly stochastic Poisson process (Lowen and Teich, 1991) and synaptic responses for the strong facilitation and depression regimes. Average input spike rate is r = 14.7 Hz. Synaptic parameters are N0 = 7, D = 2 s, p0 = 0.95 for the depression regime, and N0 = 12, D = 2 s, p0 = 0.11; F1,2 = 35, 190 ms; C1,2 = 0.9, 0.95 for the facilitation regime. Average release probabilities are pr = 0.211 for the facilitation regime and pr = 0.230 for the depression regime. The inset shows a cluster in the input spike train with greater resolution. Note the dramatic increase of the release probability during a cluster in the facilitation regime.

 
Statistical properties of the input stimulus train and output release trains are compared in Figure 8. In Figure 8A the interstimulus interval histogram (ISIH) is superimposed with the output interrelease interval histogram (IRIH). Note the much larger probability of short IRIs in the facilitation regime than in the depression regime. Also, the probability is larger for IRIs than for ISIs at long time intervals, because synaptic transmission is not reliable (overall average release probability is 0.15 and 0.17 for the facilitating and depressing regimes respectively) so that the mean IRI is longer than the mean ISI. Figure 8B shows the Fano factor, which characterizes the temporal fluctuations of event counts at different time-scales. For the fractal input train, the Fano factor grows with time, which reflects the presence of spike-count variations at all time-scales. The Fano factor of the release event sequence remains close to unity at all time-scales in the depression regime, which is expected for a Poisson process with spike count variance equal to its mean. Even in the facilitation regime, the Fano factor is greatly reduced compared to its value for the input spike train, especially at time-scales of 1 s and beyond. Therefore, synaptic depression leads to the reduction of high variability present in the input. We note that these results also apply to the statistical properties of combined response of several synaptic connections, since release events at different synapses are statistically independent. For instance, the temporal autocorrelation of the postsynaptic response will be equal to the sum of response autocorrelations of individual synapses.

The decorrelation effect of short-term synaptic plasticity is quantified by comparing the temporal correlation and power spectrum of the synaptic output to those of the fractal input (Fig. 9). For the fractal input train, both temporal correlation and power spectrum display power laws in time (manifested by the linear regions in log–log plots, Fig. 9). As expected, short-term synaptic facilitation leads to an increase in autocorrelation magnitude at short time-scales, while depression dramatically reduces correlations (Fig. 9A). The dip in the millisecond time range results from refractoriness of vesicle release. Even in the facilitation regime, the long-term temporal correlations that are a hallmark of fractal signals are reduced at time-scales longer than several hundred milliseconds. The power spectrum of the output train is virtually flat for both facilitating and depressing synapses (Fig. 9B), in this sense we can say that short-term synaptic depression can effectively ‘whiten’ the input, and reduce strong redundancies present in the inputs in the form of temporal correlations. Goldman and colleagues have previously shown a decorrelation effect by synaptic depression in the case where the input train has a correlation time of a few hundreds of milliseconds (Goldman et al., 1999). Here, it is demonstrated that this synapse-specific mechanism can even decorrelate fractal-like inputs with correlations at all time-scales.

View larger version (33K): [in this window] [in a new window]   Figure 9. Temporal correlation and power spectrum of a fractal signal before and after transmission through a model synapse. Conventions and parameters same as in Figure 8. (A) Autocorrelation for the fractal signal and the synaptic output. Sharp decline at small time scales is due to the refractory time. (B) Power spectral density for the fractal input and the synaptic outputs. For the fractal input, away from the upper and lower cut-offs the autocorrelation is approximated by g() 1 + c1 – 2ß, and the power spectrum behaves as S(f) 1 + cf–2(1 – ß), where ß = 0.9, and c, c are constants. In the depression regime, the input signal is decorrelated by the synaptic dynamics, and the power spectrum of the output release train becomes flat (‘whitened’).

 

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Notes References

 
The present model study was partly motivated by the recent experimental finding that short-term synaptic plasticity is differentially expressed in cortex: some synapses show strong depression while others express pronounced facilitation (Thomson, 1997; Thomson and Deuchars, 1997; Markram et al., 1998; Reyes et al., 1998; Varela et al., 1999). To shed light on the computational implications of this differential short-term plasticity, we used a stochastic model of short-term synaptic dynamics and investigated how a facilitating or depressing synapse would respond to complex stimuli similar to those occurring in the intact brain. Our synapse model includes a vesicle turnover process and a facilitation mechanism, with the constraint that at most one vesicle can be released per stimulus (Triller and Korn, 1982; Redman, 1990; Korn and Faber, 1991; Korn et al., 1994; Stevens and Wang, 1995).

It has been proposed that bursts of spikes and isolated spikes in a neuronal spike train could differ in the extent and kind of information that they provide about the external stimulus. For example, it has been reported that orientation of a visual stimulus is encoded in the burst component of the firing discharges in visual cortical neurons, while the isolated spike component is correlated with the contrast of the stimulus (Cattaneo et al., 1981; Livingstone, 1996; DeBusk et al., 1997). For motion-sensitive visual cells of the cat superior colliculus, evidence suggests that stimulus velocity is encoded in relative durations of bursting versus ‘resting’ (low-frequency) episodes (Mandl, 1993). If bursts and isolated spikes encode different types of information, then it would be important for a synapse to be able to respond differently to isolated spikes and spikes within a burst, thereby selecting the type of information that is transmitted to the postsynaptic neuron. To study this possibility, we have analyzed the ability of a facilitating synapse to respond preferentially to bursts of action potentials (Lisman, 1996; Thomson, 1997; Wang, 1999). This was done by driving the synapse model with burst-rich spike trains recorded from visual