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 |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
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 |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
![]() | (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 pr would increase with N according to equation (1)
(see Fig. 5
of Dobrunz and Stevens, 1997).
|
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+] = ACa i
(t ti), with ti the arrival time of the ith stimulus. The parameters kj+ and kj = 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 Oj(tn+) the jth gating variable at the end of the nth spike, then the facilitation factor is F(tn) = F1(tn)F2(tn)F3(tn), where Fj(tn) = Oj(tn+)/Oj(t1+), j = 1,2,3. The vesicle fusion rate
V(tn) =
V,0F(tn), 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 Fj 1 + Cj Fj, where Cj = exp(ACakj+) (0
Cj
1); (ii) between spikes each Fj 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, tn and tn+1 (Bertram et al., 1996
):
![]() | (3) |
where Oj(t1+) = 1 Cj is the value after the first spike. Dividing by Oj(t1+), we obtain the update rule for the facilitation factors:
![]() | (4) |
Note that the parameters Cj (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 PPFmax = 2p, 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,0Fss.
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 = 310. For the vesicle refill time constant we choose a value of
D = 12 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 pyramidinterneuron 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 Cj 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 Cj to achieve regimes of strong facilitation (low p0, high Cj) 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 Cj = 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
).
|
![]() |
n = 0, . . ., m, with parameter values pB = 0.5 and m = 8. The geometric distribution is defined by PS(n + 1) = (1 pS)pSn (n = 0, . . ., ) with pS = 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 (ISIB) and between isolated spikes (ISIS) 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 ISIS,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.
|
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 {ti} 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 Ki 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 TA and TB. Cut-offs TA and TB ensure that the spike rate r(t) remains finite for any value of ß.
We have chosen the following parameter values: ß = 0.9, TA = 2 ms, TB = 100 s, r0 = 0.2 Hz. Filter amplitude Ki is taken to be uniformly distributed between KA = 6 and KB = 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.
|
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)
= µT, 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 |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
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
pr
, 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
)].
|
In Figure 3A, the steady-state release probability
pr(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
pr(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
).
|
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 3035 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
pr
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 (pS) (i.e. fraction of single spikes that lead to a vesicle release) and that for a spike within a burst (pB). The ratio between these two values, pB/pS, 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 1015 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 pB/pS 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 pB/pS 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 pB/pS 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 pB/pS 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 23 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 pB/pS ratio of at most 30%, with significant increase taking place only under conditions of strong facilitation and large nB (simulation results not shown). This is because in the absence of facilitation an increase in
D causes comparable decrease in both pB (response to a burst spike) and pS (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.
|
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 loglog 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.
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
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 (Fig. 4
) and prefrontal cortices (Fig. 5
) of awake and behaving monkeys, and with artificially generated spike trains (Fig. 6
). We quantified the burst discrimination capability of the synapse by the ratio between the release probability for a spike within a burst (pB) and that for an isolated spike (pS). It was found that pB/pS can be as much as five times higher for a highly facilitating synapse than for a strongly depressing synapse (Fig. 6C
). We identified two quantitative conditions for optimal burst discrimination by a plastic synapse. First, the effect is maximized when the facilitation time constant matches the average burst duration (Figs 4E
, 5E, 6C). Second, burst discrimination can only be achieved for spike trains with a high number of spikes per burst (so that there is significant facilitation during a burst; see Fig. 6D
), and long time intervals between two consecutive bursts or between a burst and a single spike (so that facilitation decays away between two spikes not belonging to the same burst; see Fig. 6E
). The existence of several facilitation components with disparate decay times points to the possibility that a synapse may be tuned to detect temporal clustering of spikes in the presynaptic stimulation train at several distinct time-scales. How an optimal match between the facilitation kinetics (a synaptic property) and the characteristics of the bursty spike train (a neuronal property) could be achieved in a neural system remains an open question.
Neuronal spike patterns recorded in a variety of neural systems were shown to possess self-similar temporal structure, characterized by long-lasting correlations (Teich, 1992; Lowen et al., 1997
). We analyzed the effects of short-term plasticity on transmission of such fractal inputs using numerically generated spike trains. We found that facilitation enhances correlations present in the presynaptic stimulation pattern, at short time-scales. On the other hand, depression drastically reduces correlations in the release sequence at all time-scales and destroys the power-law scaling of the output autocorrelation (Fig. 8
). This result agrees with and extends the conclusions of the previous work (Goldman et al., 1999
), which reported decorrelation by a depressing synapse model of an input train with a characteristic correlation time constant of a few hundreds of milliseconds. Statistical analysis revealed that sensory inputs from the external world display correlations at all scales according to fractal-like scaling laws (Ruderman, 1994
; van Hateren, 1997
). It has been suggested that neural coding efficiency of sensory inputs could be enhanced by a reduction in input redundancy (i.e. strong correlations) (Barlow, 1961
; Atick, 1992
; Goldman et al., 1999
). The present work demonstrates that short-term synaptic depression is able to remove temporal correlations at all scales and whiten fractal-like inputs. Correlations could be preserved or even enhanced at short time-scales if a synapse also displays activity-dependent facilitation. Therefore, decorrelation and redundancy reduction may not necessarily exclude the presence of correlations at shorter time scales (from a few to a few hundreds of milliseconds) which is often seen in cortical neurons (Abeles et al., 1994
; Gray, 1999
; Singer, 1999
).
Our theoretical predictions could be tested by using fractal-like stimulation train in studies of synaptic transmission in cortex. A more indirect approach would be to compare temporal autocorrelations of two monosynaptically connected neurons along a sensory pathway. Such a comparison was done for the cat retinal ganglion cells and neurons in the lateral geniculate nucleus, during spontaneous discharges (Teich et al., 1997). It was found that fractal-like temporal statistics are similar in both cell populations, suggesting minimal decorrelation effect at the retino-geniculate synapses. On the other hand, the study by Teich and colleagues also indicates that fractal-like long-term correlations could be generated intrinsically in the visual system, since the activities recorded were spontaneous in the absence of visual stimuli (Teich et al., 1997
). Long-term correlations could be introduced by internal cellular mechanisms acting either at the synaptic level or at the level of spike generation (Teich, 1992
). Indeed, analysis of exocytic events at neuromuscular junctions and at the rat hippocampal synapses in culture provided evidence for fractal-like scaling in the rate of spontaneous release events (Lowen et al., 1997
). If the self-similar behavior is caused by intracellular mechanisms acting predominantly at the synaptic level, this could indicate that short-term plasticity itself displays fractal properties, and the decay of some facilitation and depression components could be power-law rather than exponential in time.
There is evidence that depression mechanisms beyond vesicle depletion contribute to the short-term depression observed at central synapses (Bellingham and Walmsley, 1999). For instance, depression may result from calcium-dependent inactivation of exocytosis machinery (Hsu et al., 1996
; Matveev and Wang, 2000
). Such an effect would further decrease the pB/pS ratio in the response of the depressing synapse to bursty spike trains, and would strengthen the decorrelation effect of short-term depression described here. We have chosen not to incorporate this inactivation mechanism into our model since it would not affect the main conclusions of this work, and would require including parameters with values that are currently not constrained by experimental data.
![]() |
Notes |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Address correspondence to Xiao-Jing Wang, Volen Center for Complex Systems, Brandeis University, Waltham, MA 02454, USA. Email: xjwang{at}volen.brandeis.edu.
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Abeles M, Prut Y, Bergman H and Vaadia E (1994) Synchronization in neuronal transmission and its importance for information processing. Progr Brain Res 102:395404.[ISI][Medline]
Ali AB, Thomson AM (1998) Facilitating pyramid to horizontal oriens-alveus interneurone inputs: dual intracellular recordings in slices of rat hippocampus. J Physiol 507:185199.
Ali AB, Deuchars J, Pawelzik H, Thomson AM (1998) CA1 pyramidal to basket and bistratified cell EPSPs: dual intracellular recordings in rat hippocampal slices. J Physiol 507:201217.
Arancio O, Korn H, Gulyas A, Freund T, Miles R (1994) Excitatory synaptic connections onto rat hippocampal inhibitory cells may involve a single transmitter release site. J Physiol 481:395405.[Abstract]
Atick JJ (1992) Could information theory provide an ecological theory of sens ory processing? Network 3:213251.[ISI]
Bair W, Koch C, Newsome W, Britten K (1994) Power spectrum analysis of bursting cells in area MT in the bahaving monkey. J Neurosci 14:28702892.[Abstract]
Barlow HB (1961) Possible principles underlying the transformations of sensory messages. In: Sensory communication (Rosenblith WA, ed.), pp. 217234. Cambridge, MA: MIT Press.
Bekkers JM, Stevens CF (1990) Presynaptic mechanism for long-term potentiation in the hippocampus. Nature 346:724729.[ISI][Medline]
Bellingham MC, Walmsley B (1999) A novel presynaptic inhibitory mechanism underlies paired pulse depression at a fast central synapse. Neuron 23:159170.[ISI][Medline]
Bennett MR, Gibson WG, Robinson J (1997) Probabilistic secretion of quanta and the synaptosecretosome hypothesis: evoked release at active zones of varicosities, boutons, and endplates. Biophys J 73:18151829.[Abstract]
Bertram R, Sherman A, Stanley EF (1996) Single-domain/bound calcium hypothesis of transmitter release and facilitation. J Neurophysiol 75:19191931.
Bishop PO, Levick WR, Williams WO (1964) Statistical analysis of the dark discharges of lateral geniculate neurones. J Physiol 170:598612.[ISI]
Buhl EH, Tamas G, Szilagyi T, Stricker C, Paulsen O, Somogyi P (1997) Effect, number and location of synapses made by single pyramidal cells onto aspiny interneurones of cat visual cortex. J Physiol 500:689713.[Abstract]
Buonomano DV (2000) Decoding temporal information: a model based on short-term synaptic plasticity. J Neurosci 20:11291141.
Buonomano DV, Merzenich MM (1995) Temporal information transformed into a spatial code by a neural network with realistic properties. Science 267:10281030.[ISI][Medline]
Canepari M, Cherubini E (1998) Dynamics of excitatory transmitter release: analysis of synaptic responses in CA3 hippocampal neurons after repetitive stimulation of afferent fibers. J Neurophysiol 79:19771988.
Castro-Alamancos MA, Connors BW (1997) Distinct forms of short-term plasticity at excitatory synapses of hippocampus and neocortex. Proc Natl Acad Sci USA 94:41614166.
Cattaneo A, Maffei L, Morrone C (1981) Two firing patterns in the discharge of complex cells encoding different attributes of the visual stimulus. Exp Brain Res 43:115118.[ISI][Medline]
Chafee MV and Goldman-Rakic PS (1998) Neuronal activity in macaque prefrontal area 8a and posterior parietal area 7ip related to memory guided saccades. J Neurophysiol 79:29192940.
Debanne D, Guérineau NC, Gähwiler BH, Thompson SM (1996) Paired-pulse facilitation and depression at unitary synapses in rat hippocampus: quantal fluctuation affects subsequent release. J Physiol 491(part 1):163176.[Abstract]
DeBusk BC, DeBruyn EJ, Snider RK, Kabara JF, Bonds AB (1997) Stimulus-dependent modulation of spike burst length in cat striate cortical cells. J Neurophysiol 78:199213.
Dittman JS, Regehr WG (1998) Calcium dependence and recovery kinetics of presynaptic depression at the climbing fiber to Purkinje cell synapse. J Neurosci 18:61476162.
Dittman JS, Kreitzer AC, Regehr WG (2000) Interplay between facilitation, depression, and residual calcium at three presynaptic terminals. J Neurosci 20:13741385.
Dobrunz LE, Stevens CF (1997) Heterogeneity of release probability, facilitation, and depletion at central synapses. Neuron 18:9951008.[ISI][Medline]
Dobrunz LE, Stevens CF (1999) Response of hippocampal synapses to natural stimulation patterns. Neuron 22:157166.[ISI][Medline]
Dobrunz LE, Huang EP, Stevens CF (1997) Very short-term plasticity in hippocampal synapses. Proc Natl Acad Sci USA 94:1484314847.
Dodge FA Jr, Rahamimoff R (1967) Co-operative action of calcium ions in transmitter release at the neuromuscular junction. J Physiol 193:419432.[ISI][Medline]
Ekholm A, Hyvärinen J (1970) A pseudo-Markov model for series of neural spike events. Biophys J 10:773796.[ISI][Medline]
Fano U (1947) Ionization yield of radiations. II. The fluctuations of the number of ions. Phys Rev 72:2629.[ISI]
Fisher SA, Fischer TM, Carew TJ (1997) Multiple overlapping processes underlying short-term synaptic enhancement. Trends Neurosci 20:170177.[ISI][Medline]
Forti L, Bossi M, Bergamaschi A, Villa A, Malgaroli A (1997) Loose-patch recordings of single quanta at individual hippocampal synapses. Nature 388:874878.[ISI][Medline]
Galarreta M, Hestrin S (1998) Frequency-dependent synaptic depression and the balance of excitation and inhibition in the neocortex. Nature Neurosci 1:587594.[ISI][Medline]
Goldman MS, Nelson SB, Abbott LF (1999) Decorrelation of spike trains by synaptic depression. Neurocomputing 26/27:147153.
Gray CM (1999) The temporal correlation hypothesis of visual feature integration: still alive and well. Neuron 24:3147.[ISI][Medline]
Gray CM, McCormick DA (1996) Chattering cells: superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. Science 274:109113.
Grüneis F, Nakao M, Mizutani Y, Yamamoto M, Meesmann M, Musha T (1993) Further study on 1/f fluctuations observed in central single neurons during REM sleep. Biol Cybern 68:193198.[ISI][Medline]
Hjelmstad GO, Nicoll RA, Malenka RC (1997) Synaptic refractory period provides a measure of probability release in the hippocampus. Neuron 19:13091318.[ISI][Medline]
Hsu S-F, Augustine GJ and Jackson MB (1996) Adaptation of Ca2+-triggered exocytosis in presynaptic terminals. Neuron 17:501512.[ISI][Medline]
Korn H, Faber DS (1987) Regulation and significance of probabilistic release mechanisms at central synapses. In: Synaptic function (Edelman GM, Gall WE, Cowan WM, eds), pp. 57108. New York: John Wiley.
Korn H, Faber DS (1991) Quantal analysis and synaptic efficacy in the CNS. Trends Neurosci 14:439445.[ISI][Medline]
Korn H, Sur C, Charpier S, Legendre P, Faber DS (1994) The one-vesicle hypothesis and multivesicular release. In: Molecular and cellular mechanisms of neurotransmitter release (Stjärne L, Greengard P, Grillner S, Hökfelt T, Ottoson D, eds), pp. 301322. New York: Raven Press.
Kreitzer AC, Regehr WG (2000) Modulation of transmission during trains at a cerebellar synapse. J Neurosci 20:13481357.
Kuffler SW, FitzHugh R, Barlow HB (1957) Maintained activity in the cat's retina in light and darkness. J Gen Physiol 40:683702.
Liley AW, North KAK (1952) An electrical investigation of effects of repetitive stimulation of mammalian neuromuscular junction. J Neurophysiol 16:509527.[ISI]
Lisman JE (1997) Bursts as a unit of neural information: making unreliable synapses reliable. Trends Neurosci 20:3843.[ISI][Medline]
Liu G, Tsien RW (1995) Properties of synaptic transmission at single hippocampal synaptic boutons. Nature 375:404408.[ISI][Medline]
Liu YH, Wang X-J, Williams GV, Rao S and Goldman-Rakic PS (1998) Analysis of temporal structure of spike trains recorded from monkey prefrontal cortex during working memory tasks. Soc Neurosci Abstr 24:1426.
Livingstone MS, Freeman DC, Hubel DH (1996) Visual responses in V1 of freely viewing monkeys. Cold Spring Harb Symp Quant Biol 61:2737.[ISI][Medline]
Lowen SB and Teich MC (1991) Doubly stochastic Poisson point process driven by fractal shot noise. Phys Rev A 43:41924215.[ISI][Medline]
Lowen SB, Teich MC (1996) The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains. J Acoust Soc Am 99:35853591.[ISI][Medline]
Lowen SB, Cash SS, Poo M-M, Teich MC (1997) Quantal neurotransmitter secretion rate exhibits fractal behavior. J Neurosci 17:56665677.
Maass W, Zador AM (1999) Dynamic stochastic synapses as computational units. Neural Comput 11:903917.
Magleby KL (1987) Short-term changes in synaptic efficacy. In: Synaptic function (Edelman G, Gall W, Cowan W, eds), pp. 2156. New York: Wiley.
Mandl G (1993) Coding for stimulus velocity by temporal patterning of spike discharges in visual cells of cat superior colliculus. Vis Res 33:14511475.[ISI][Medline]
Markram H, Tsodyks M (1996) Redistribution of synaptic efficacy between neocortical pyramidal neurons (Letter). Nature 382: 807810.[ISI][Medline]
Markram H, Wang Y, Tsodyks M (1998) Differential signaling via the same axon of neocortical pyramidal neurons. Proc Natl Acad Sci USA 95:53235328.
Matveev V, Wang X-J (2000) Implications of all-or-none synaptic transmission and short-term depression beyond vesicle depletion: a computational study. J Neurosci 20:15751588.
Melkonian DS, Kostopoulos GK (1996) Stochastic particle formulation of the vesicle hypothesis. Relevance to short-term phenomena. NeuroReport 7:937942.[ISI][Medline]
Mueller CG (1954) A quantitative theory of visual excitation for the single photoreceptor. Proc Natl Acad Sci USA 40:853863.[ISI]
Murthy VN, Stevens CF (1998) Synaptic vesicles retain their identity through the endocytic cycle. Nature 392:497501.[ISI][Medline]
Murthy VN, Sejnowski TJ, Stevens CF (1997) Heterogeneous release properties of visualized individual hippocampal synapses. Neuron 18:599612.[ISI][Medline]
Neher E (1998) Vesicle pools and Ca2+ microdomains: new tools for understanding their roles in neurotransmitter release. Neuron 20:389399.[ISI][Medline]
Neher E, Zucker RS (1993) Multiple calcium-dependent processes related to secretion in bovine chromaffin cells. Neuron 10:2130.[ISI][Medline]
O'Donovan MJ, Chub N (1997), Population behavior and self-organization in the genesis of spontaneous rhythmic activity by developing spinal networks. Semin Cell Dev Biol 8:2128.[ISI]
O'Donovan MJ, Rinzel J (1997) Synaptic depression: a dynamic regulator of synaptic communication with varied functional roles. Trends Neurosci 20:431433.[ISI][Medline]
Quastel DMJ (1997) The binomial model in fluctuation analysis of quantal neurotransmitter release. Biophys Journal 72:728753.[Abstract]
Redman SJ (1990) Quantal analysis of synaptic potentials in Neurons of the central nervous system. Physiol Rev 70: 165198.
Reyes A, Lujan R, Rozov A, Burnashev N, Somogyi P, Sakmann B (1998) Target-cell-specific facilitation and depression in neocortical circuits. Nature Neurosci 1:279285.[ISI][Medline]
Rosenmund C, Stevens CF (1996) Definition of the readily releasable pool of vesicles at hippocampal synapses. Neuron 16:11971207.[ISI][Medline]
Ruderman DL (1994) The statistics of natural images. Network 5:517548.[ISI]
Ryan TA, Reuter H, Smith SJ (1997) Optical detection of a quantal presynaptic membrane turnover. Nature 388:478482.[ISI][Medline]
Schikorski T, Stevens CF (1997) Quantitative ultrastructural analysis of hippocampal excitatory synapses. J Neurosci 17:58585867.
Senn W, Wyler K, Streit J, Larkum J, Lüscher H-R, Mey H, Müller L, Steinhauser D, Vogt K, Wannier Th (1996) Dynamics of a random neural network with synaptic depression. Neural Networks 9:575588.[ISI]
Singer W (1999) Neuronal synchrony: a versatile code for the definition of relations? Neuron 24:4965.[ISI][Medline]
Smith DR, Smith GK (1965) A statistical analysis of the continual activity of single cortical neurones in the cat unanaesthetized isolated forebrain. Biophys J 5:4774.[ISI]
Somogyi P, Tamas G, Lujan R, Buhl EH (1998) Salient features of synaptic organisation in the cerebral cortex. Brain Res Rev 26:113135.[ISI][Medline]
Stanley EF (1986) Decline in calcium cooperativity as the basis of facilitation at the squid giant synapse. J Neurosci 6:782789.[Abstract]
Stevens CF, Tsujimoto T (1995) Estimates for the pool size of releasable quanta at a single central synapse and for the time required to refill the pool. Proc Natl Acad Sci USA 92:846859.[Abstract]
Stevens CF, Wang Y (1995) Facilitation and depression at single central synapses. Neuron 14:795802.[ISI][Medline]
Tank DW, Regehr WG, Delaney KR (1995) A quantitative analysis of presynaptic calcium dynamics that contribute to short-term enhancement. J Neurosci 15:79407952.[Abstract]
Teich MC (1989) Fractal character of the auditory neural spike train. IEEE Trans Biomed Engng 36:150160.[ISI][Medline]
Teich MC (1992) Fractal neuronal firing patterns. In: Single neuron computation (McKenna T, Davis J, Zornetzer S, eds), pp. 589625. Boston, MA: Academic Press.
Teich MC, Heneghan C, Lowen SB, Ozaki T, Kaplan E (1997) Fractal character of the neural spike train in the visual system of the cat. J Opt Soc Am A 14:529546.[ISI][Medline]
Thomson AM (1997) Activity-dependent properties of synaptic transmission at two classes of connections made by rat neocortical pyramidal axons in vitro. J Physiol 502:131147.[Abstract]
Thomson AM, Deuchars J (1997) Synaptic interactions in neocortical local circuits: dual intracellular recordings in vitro. Cereb Cortex 7:510522.[Abstract]
Thomson AM, West DC, Hahn J, Deuchars J (1996) Single axon IPSPs elicited in pyramidal cells by three classes of interneurones in slices of rat neocortex. J Physiol 496:81102.[Abstract]
Triller A, Korn H (1982) Transmission at a central inhibitory synapse. III. Ultrastructure of physiologically identified and stained terminals. J Neurophysiol 48:708736.
Tsodyks MV, Markram H (1997) The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability. Proc Natl Acad Sci USA, 94:719723.
Turcott RG, Barker PDR, Teich MC (1995) Long-duration correlation in the sequence of action potentials in an insect visual interneuron. J Statist Comput Simul 52:253271.[ISI]
Varela JA, Sen K, Gibson J, Fost J, Abbott LF, Nelson SB (1997) A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex. J Neurosci 17:79267940.
Varela JA, Song S, Turrigiano GG, Nelson SB (1999) Differential depression at excitatory and inhibitory synapses in visual cortex. J Neurosci, 19:42934304.
Vere-Jones D (1966) Simple stochastic models for the release of quanta of transmitter from a nerve terminal. Austr J Statist 8:5363.
van Hateren JH (1997) Processing of natural time series of intensities by the visual system of the blowfly. Vis Res 37:34073416.[ISI][Medline]
Walmsley B, Alvarez FJ, Fyffe RE (1998) Diversity of structure and function at mammalian central synapses. Trends Neurosci 21:8188.[ISI][Medline]
Wang X-J (1999) Fast burst firing and short-term synaptic plasticity: a model of neocortical chattering neurons. Neuroscience 89:347362.[ISI][Medline]
Wise ME (1981) Spike interval distributions for neurons and random walks with drift to a fluctuating threshold. Statistical distributions in scientific work, Vol. 6 (Taillie CEA, ed.), pp. 211231. Boston, MA: Reidel.
Wu LG, Betz WJ (1998) Kinetics of synaptic depression and vesicle recycling after tetanic stimulation of frog motor nerve terminals. Biophys J 74:30033009.
Zucker RS (1989) Short-term synaptic plasticity. Annu Rev Neurosci 12:1331.[ISI][Medline]
Zucker RS (1996) Exocytosis: a molecular and physiological perspective. Neuron 17:10491055.[ISI][Medline]