Abstract
Golomb, David. Models of neuronal transient synchrony during propagation of activity through neorcortical circuitry. J. Neurophysiol. 79: 1–12, 1998. Stereotypic paroxysmal discharges that propagate in neocortical tissues after electrical stimulations are used as a probe for studying cortical circuitry. I use modeling to investigate the effects of sparse connectivity, heterogeneity of intrinsic neuronal properties, and synaptic noise on synchronization of evoked propagating neuronal discharges in a network of excitatory, regular spiking neurons with spatially decaying connectivity. The global coherence of the traveling discharge is characterized by the correlation function between spike trains of neurons, averaged over all the pairs of neurons in the system at the same distance. Local coherence of two neurons is characterized by their correlation function averaged over many trials or, for persistent activity, over a long time interval. Spike synchronization between neurons emerges as a result of the transient activity; if activity is persistent, there is no synchrony, and crosscorrelation functions are flat. During discharge propagation, systemaverage crosscorrelation between neurons does not depend on their mutual distance except for a time shift. Spike synchronization occurs only when the average number of synapses M a cell receives is large enough. As M increases, there is a crossover from an asynchronized to a synchronized discharge. Synaptic depression appears to help synchrony; it reduces the M value at the crossover. The strengths of αamino3hydroxy5methyl4isoxazolepropionic acid (AMPA) and Nmethyldaspartate (NMDA) conductances affect synchrony only weakly. Spike synchronization is robust even with large levels of heterogeneity. Synaptic noise reduces synchrony, but strong synchrony is observed at a noise level that cannot evoke spontaneous discharges. Systemaverage spike synchronization is determined by the levels of sparseness, heterogeneity, and noise, whereas trialaverage spike synchronization is determined only by the noise level. Therefore, I predict that experiments will reveal local, twocell spike synchrony, but not global synchrony.
INTRODUCTION
Dynamics of cortical circuitry are determined, in a complicated way, by the intrinsic neuronal properties and the network architecture. Although experimental data about cortical dynamics have been collected recently using multielectrode recording (e.g., Gray 1994; Nicolelis et al. 1995, 1996; Singer 1993; Vaadia et al. 1995) and imaging (e.g., Grinvald et al. 1988; Prechtl et al. 1997) techniques, inferring the nature of cortical functional connectivity based on these experiments remains difficult. One possible approach to address this problem is to investigate a restricted region of cortex in various pharmacologically induced states. In cortical tissues, stereotypic propagating paroxysmal discharges are evoked by electrical stimulations when γaminobutyric acidA (GABA_{A}) inhibition is suppressed pharmacologically (Chervin et al. 1988; Connors 1984; Gutnick et al. 1982; Wadman and Gutnick 1993). These paroxysmal discharges have been studied experimentally in vitro, where they constitute models for interictal discharges in epilepsy. Investigating the discharge propagation range has led to conclusions regarding the range of connectivity in cortex (ChagnacAmitai and Connors 1989a,b; Chervin et al. 1988; Wadman and Gutnick 1993; I. A. Fleidervish, A. M. Binshtok, and M. J. Gutnick, unpublished data). Other properties of the circuitry, such as the sparseness of synaptic connectivity, heterogeneity of intrinsic neuronal properties and synaptic noise, mainly affect the neuronal spike synchronization during the discharge. Here, modeling is used to study these effects.
Whereas recording techniques usually have revealed information about the temporal dynamics of one or a few neurons, modern recording and imaging techniques enable researchers to study the spatiotemporal dynamics of neuronal ensembles. Using such techniques, cortical propagating neuronal activity in behaving animals was discovered (Nicolelis et al. 1995, 1996; Prechtl et al. 1997). The modeling approach developed here also should be applicable for studying the relationship between the anatomic connectivity patterns and the patterns of synchrony in such in vivo systems.
Heretofore, theoretical investigations have addressed the mechanism of largescale population synchrony in systems with longrange coupling^{1} (e.g., Hansel and Sompolinsky 1996; Hansel et al. 1993, 1995; Jefferys et al. 1996; Traub et al. 1996a; Wang et al. 1996; Whittington et al. 1995) and shortrange coupling (Traub et al. 1996b) during persistent activity, i.e., activity that does not stop for a long time. In the present study, synchrony is investigated in a transient state of a propagating discharge, which is a pulse of depolarization upon which rides a train of action potentials. In addition to the time scale of the whole event, there is a second, faster, time scale of the interval between spikes. In this respect, the discharge differs from other, wellstudied, propagating pulses in neurobiological systems, such as propagating action potentials in axons (Hodgkin and Huxley 1952), or in physical and chemical systems (Cross and Hohenberg 1993; Mikhailov 1990), that exhibit only one time scale. The typically large field potential recorded during paroxysmal discharges shows that the spikes of adjacent neurons are synchronized at the discharge time scale (Gutnick et al. 1982). However, on the basis of the existing experimental data, it is not clear whether the neuronal firing pattern is synchronized at the spike time scale.
In previous computational work (Golomb and Amitai 1997), models of homogeneous onedimensional networks with spatially decaying synaptic strength were analyzed. It was shown that perfect spike synchrony was obtained between neurons along the slice model, except for a temporal phase shift Δx/ν_{D}, where Δx is the distance between the neurons and ν_{D} is the discharge velocity. Here I study the effects of sparseness, heterogeneity, and noise on spike synchrony by computing correlation functions. I discriminate between global, systemaverage correlation, which is affected by sparseness, heterogeneity, and synaptic noise, and local, trialaverage correlation, which is affected only by synaptic noise. Global synchronization, which can be achieved at low levels of sparseness and heterogeneity, emerges as a result of the transient, propagating characteristics of the discharge. The global synchrony between neurons does not depend on the distance between them, except for a phase shift. Synaptic depression helps synchrony. These results help to interpret crosscorrelation measurements during propagation and lead to predictions regarding the level of local and global synchrony expected in future experiments. Combining these modeling result with future experimental results will help to determine the level of sparseness, heterogeneity, and synaptic noise in cortex.
METHODS
Model
Biophysical, conductancebased models of single regular spiking (RS) neurons and of αamino3hydroxy5methyl4isoxazolepropionic acid (AMPA) and Nmethyldaspartate (NMDA) synapses are used, as reported elsewhere (Golomb and Amitai 1997). In brief, the single cell dynamics are described by a singlecompartment HodgkinHuxley type model, using a set of coupled differential equations
A cell receives fast, AMPAmediated excitatory postsynaptic potentials (EPSPs) and slow, NMDAmediated EPSPs from other cells. Both types of synapses exhibit synaptic depression. A gating variable s for an AMPA or an NMDA receptor, representing the fraction of open channels, is modeled according to
The N neurons are distributed along an interval of length L, representing the slice length. The position of the ith neuron is x_{i}
= iL/N. Synaptic coupling decays exponentially with the distance between neurons with a characteristic length λ, called the footprint length. The model is studied in the regime
Synapses show spontaneous release (Fleidervish et al. 1996; Salin and Prince 1996). Typically, such release occurs randomly in terminals innervating cortical pyramidal cells with a rate of ∼20–50 s^{−1} per neuron. (I. A. Fleidervish and M. J. Gutnick, unpublished data) (recording of spontaneous events from rat slices at 33°C, where the rats were ≥3wk old). Synaptic noise [spontaneous excitatory postsynaptic conductances (EPSCs)] is modeled by a linear synapse on each neuron (without saturation or depression) receiving a Poisson spike train (Golomb et al. 1994a) at a rate ν_{noise} = 50 Hz. Each presynaptic spike at time t
_{o} evokes an EPSC with an amplitude g
_{noise}
f (t − t
_{o}) described by an extended α function (Destexhe et al. 1994; Rall 1967)
Numerical integration was carried out with the fourthorder RungeKutta method with a time step of 0.03 ms and with λ = L/32. Except for Fig. 6, N = 4096. At t = 0, all the cells are at their resting state. Evoked discharges were initialized by injecting a current pulse of 6 μA/cm^{2} for 18 ms into cells in an interval of length λ on the left of the slice model. The synaptic noise variables were simulated 50 ms before the beginning of the simulation to reach equilibrium.
Correlation functions
Activity of each neuron is often described by the time series of the occurrence of action potentials a neuron fires. The spike train of the ith neuron is represented by the spiking function defined as
RESULTS
Sparse connectivity
I first study the effects of sparse connectivity only on the network dynamics (without heterogeneity or noise). The populationaverage maximal AMPA conductance a cell receives is g, and the standard deviation is g/
To demonstrate the collective nature of the discharges, the crosscorrelations between neighboring neurons averaged over four groups of pairs separately is calculated (Fig. 3). In the first group, neurons are not coupled monosynaptically; in the second group, the left neuron projects to the right neuron; in the third group, the right neuron projects to the left one; and in the forth group, the two neurons projects to each other. No significant difference was found between the average crosscorrelations, indicating that the coherence between neurons is not a result of whether there is a monosynaptic coupling between them. This is expected from a system in a synchronized state (Ginzburg and Sompolinsky 1994).
To quantify the global level of spike synchrony, the synchrony measure χ_{k} is defined as follows (see Fig. 4
A). The values of the crosscorrelation C
_{k}(τ) at the central three peaks are denoted, from left to right, by P
_{−1}, P
_{0}, and P
_{1}. The values of C
_{k}(τ) at the central two troughs are denoted by R
_{−1} and R
_{1}. The synchrony measure χ_{k} is defined as
Persistent activity
The emergence of global synchrony, and, in particular, that global synchrony between neurons does not depend on distance, is related to the propagating and transient characteristics of the discharge and does not hold during persistent activity. To demonstrate this, a model with no synaptic depression (k_{t} = 0) and reduced slow potassium conductance (g _{Kslow} = 0.4 mS/cm^{2}) is simulated. The system parameters are tuned such that the discharge is not terminated, and instead continues indefinitely. A rastergram of neurons in a network reaching a persistent activity state is shown in Fig. 5 A, together with the voltage time trace of one neuron. The spiking patterns of the first few spikes of neurons are similar to those of neurons in Fig. 2 A (M = 128). The slow potassium current I _{Kslow} builds up within ∼10 spikes. It is not strong enough to terminate the activity, and instead it reduces the firing rate to a steady state level. The firing pattern of each neuron is close to periodic, as indicated by the rastergram, the voltage trace, the damped oscillation seen in the systemaverage autocorrelation C _{0} (Fig. 5 B, top) and the periodic nature of autocorrelation of a specific neuron calculated during a long time interval (29.7 s) (Fig. 5 C, top). The firing times of neurons do not show any special order. The nearestneighbor systemaverage crosscorrelation C _{1} (Fig. 5 B, bottom), or any other crosscorrelation, is flat. Similarly, the crosscorrelation between two specific neurons (Fig. 5 C, bottom) is close to being flat, with fluctuations that decay with the length of the time interval over which correlation is calculated, indicating an asynchronous state. The patterns and microdomains that are seen in the rastergram (Fig. 5 A) mean only that there is locking in a brief time period, as expected in an asynchronized state when neurons have similar time periods (compare with Fig. 6 in Golomb et al. 1994b).
The appearance of an asynchronized state in a onedimensional neuronal network during persistent activity is expected based on the following argument. In many systems that exhibit persistent firing with alltoall coupling, excitation leads to an asynchronized state (Hansel et al. 1995). In other systems with longrange coupling, even if a homogeneous system exhibits a certain level of synchrony, small levels of sparseness, heterogeneity or noise shift the system into an asynchronous state^{3} (D. Hansel, personal communication). An alltoall network with the parameter set of Fig. 5 will go to an almostasynchronized state, with the population voltage (defined as in Golomb et al. 1994b) exhibiting lowamplitude oscillations (data not shown). Even a low sparseness level (e.g., N = 4096, M = 512) shifts the system into an asynchronized state, in which the population voltage is almost constant in time. The global synchrony during persistent activity in a onedimensional system is expected to be not larger than the synchrony in a system in which the coupling does not decay with distance.
If the initial conditions of the neurons in a network with spatially independent coupling are close to a synchronized state, dephasing can occur within a few cycles. A similar effect is seen here in networks with spatially decaying, onedimensional coupling, in which an asynchronized state is reached after a transient of ∼10 cycles.
Dependence of synchrony on N and M
Most simulations are carried out with N = 4096, and the number of cells within a footprint is Nλ/L = 128. In an actual slice, the number of cells is much larger. Therefore I investigate the dependence of the synchrony measure χ for propagating discharges as N varies between 2^{11} = 2,098 and 2^{17} = 13,1072 (Fig. 6). The reference parameter set (k _{t} = 1 s^{−1}) is used; M = 128. As N increases, χ decreases; the decrease is weaker as log (N) increases. These simulations indicate that χ approaches an asymptotic value at large N. In the example, χ at the reference value (4,096) is ∼30% higher than its value at N = 13,1072. Hence, I expect the numerical calculations of χ to be qualitatively correct for N in the order of the number of cells in the slice^{4} (of order 10^{6}), but quantitatively somewhat lower. The trialtotrial variability of systemaverage magnitudes such as χ is very small at large N. As a result, only one large realization is necessary for calculating trialaverage of such magnitudes.
The dependence of χ on M is accessed quantitatively in Fig. 7 A for networks with strong synaptic depression (k_{t} = 1 s^{−1}), including one with the reference parameter set, and in Fig. 7 B for networks without synaptic depression (k _{t} = 0). In both cases, two values of g _{AMPA} are examined. One g _{AMPA} value is near the value for g _{AMPA,c}, which is the threshold value for propagation with no sparseness [i.e., 0.28 mS/cm^{2} for the parameter set without synaptic depression and 0.54 mS/cm^{2} for the parameter set with synaptic depression (Golomb and Amitai 1997)]. The second g _{AMPA} value is larger. For each g _{AMPA} value, simulations were carried out with and without NMDA conductance. In all cases, two behavioral regimes are observed. At small M, χ is low, implying no global spike synchronization. At large M, χ increases with M, and the dependence of χ on M is close to linear. There is a smooth crossover between the regimes. The number of inputs, M _{co}, at which the crossover between the regimes occurs, depends only weakly on the values of g _{AMPA} and g _{NMDA}, despite the fact that the number of spikes and the duration of the discharge both depend strongly on these parameters. Synaptic depression reduces M _{co} and therefore increases synchrony: M _{co} ≈ 35 for k _{t} = 1 s^{−1} (Fig. 4 A), whereas M _{co} ≈ 65 for k _{t} = 0 (Fig. 4 B). A possible explanation for this result is that in most cases, prolonged excitation eventually precludes synchrony (Ermentrout 1996; Hansel et al. 1993, 1995; van Vreeswijk et al. 1994; see previous text). Because synaptic depression reduces the strength of consecutive EPSCs, the initial spikes evoke strong EPSCs and help to recruit more cells into the discharge, whereas subsequent spikes evoke weaker EPSCs, such that the excitatory synaptic input that leads to desynchronization diminishes.
Heterogeneity and synaptic noise
The neuronal population is heterogeneous with respect to the intrinsic properties of individual neurons. In particular, the strengths of various ionic conductances vary from cell to cell. I model heterogeneity by assuming that each of the three ionic conductances—g
_{L}, g
_{KA}, and g
_{Kslow}—is chosen separately, at random, from a square distribution with a ratio α between the standard deviation and the mean. The A current is deinactivated when the neuron is hyperpolarized and thus is effective at the beginning of the discharge. Heterogeneity in g
_{KA} affects the timing of the first spikes. In contrast, I
_{Kslow} becomes effective after the neuron has fired several spikes; heterogeneity in g
_{Kslow} affects the duration of the firing period and the timing of the later spikes. Heterogeneity in g
_{L} affects the timing of all the spikes. A rastergram for α = 0.3 is shown in Fig. 8
A. Despite the relatively high heterogeneity, the first four spikes of each neuron are well synchronized with the first four spikes of any other neuron, with a time shift that depends on the distance between them. Subsequent spikes (after the first four spikes) are much less synchronized. The dependence of χ on α is shown in Fig. 8
B. As expected, χ decreases with α. However, even at a maximal value of α (1/
Synaptic noise has two effects. When it is weak, it reduces synchrony, and if it is strong enough, it also induces spontaneous discharges. I simulated the system with two values of ν, 20 and 50 s^{−1} (Fig. 8, C and D), corresponding to the borders of the experimental range for spontaneous synaptic input. A rastergram for g _{noise} = 0.1 mS/cm^{2} and ν = 50 s^{−1} is shown in Fig. 8 C. All the spikes are very well synchronized (with a time shift). The dependence of χ on g _{noise} is shown in Fig. 8 D. The arrows denote the g _{noise} levels above which spontaneous discharges were generated; the dependencies of χ on the noise level below these values are shown. Increasing ν or g _{noise} reduces χ. However, in the regime where no spontaneous discharges are observed, χ > 14 ms^{−1}, i.e., it is much higher than the χ values obtained with sparseness for M < 200 and is higher than the χ values with large heterogeneity level α. Hence, synaptic noise contributes to synchrony reduction much less than sparseness and heterogeneity.
As with sparseness, the function C_{k} of a system with noise or heterogeneity depends on k only via a phase shift, while the value of w_{k} remains unchanged.
Global and local synchrony
Sparseness and heterogeneity affect the global (systemaverage) synchrony of the traveling discharge. They do not, however, affect the local (trialaverage) synchrony between two neurons, because they are not stochastic in time. If the network model is stimulated again and again exactly in the same manner, the two neuron will respond exactly the same, and the trialaverage crosscorrelation will be a sum of δ functions. Synaptic noise, which is stochastic, affects both global and local synchrony. Moreover, for a large noisy network with no heterogeneity and uniform architecture (no sparseness, Eq. 6 ), the trialaverage crosscorrelation between two neurons at a distance k neurons apart is equal to the systemaverage crosscorrelation C_{k} (Fig. 9). In a cortical slice, there is sparseness, heterogeneity, and noise. Here the trialaverage and systemaverage correlations are calculated for a system that has both sparseness and synaptic noise.
Examples of trialaverage correlation functions between two adjacent neurons for a noisy and sparse network(g _{noise} = 0.1 mS/cm^{2}, v = 50 s^{−1}, M = 48) are shown in Fig. 10 A, and the corresponding systemaverage correlations are shown in Fig. 10 B. Synaptic noise reduces synchrony only moderately, and hence trialaverage autocorrelations show very sharp peaks, and crosscorrelations reveal a decaying oscillatory pattern. The relative phase of firing varies from one pair of neurons to another. For example, the crosscorrelation presented in Fig. 10 A, middle, indicates that those particular two cells fire almost in antisynchrony. The central peak of the crosscorrelation presented in Fig. 10 A, bottom, is broader and the phase relationship between those two cell is less well defined. In general, even if the phase relationship between two neurons in a pair is well defined, it varies considerably from pair to pair. With a relatively low value of M (high sparseness), it is distributed almost uniformly. As a result, the systemaverage crosscorrelation exhibits almost no wiggles in the time scale of the interspike interval, indicating that there is only a very small global (systemaverage) spike synchronization in the network.
DISCUSSION
The main results of this work are as follows. 1) Global spike synchronization can emerge during propagating discharges because of their propagating and transient nature. 2) This synchronization is a collective effect; it does not depend on monosynaptic connections between neurons. 3) The global correlation between neurons does not depend on the distance between them, except for a phase shift. 4) Global synchrony is obtained if the number of inputs a cell receives on average is above a certain crossover value M _{co}, of order 30–70. 5) Synaptic depression appears to help synchrony. The crossover value M _{co} is reduced by synaptic depression, but is only weakly affected by the strength of AMPA and NMDA conductances, g _{AMPA} and g _{NMDA}. 6) Global synchrony can exist even in very heterogeneous neuronal populations. 7) and, synaptic noise reduces synchrony only mildly.
Transient and persistent synchrony
As long as discharges propagate, they are always synchronous at the discharge time scale, and this synchrony does not decay with distance between neurons for any level of sparseness, heterogeneity and noise.^{5} An unexpected result of this work is that, in addition to discharge synchronization, neurons also can exhibit spike synchronization that does not decay with distance. Intuitively, this can be viewed as if the firing pattern of each neuron is locked to the pulse through the envelope of the discharge, defined by averaging the voltage of many neurons at the same location, which propagates at a constant velocity and maintains its shape. In two limiting conditions, this result could be predicted. In the homogeneous case (Golomb and Amitai 1997), there is perfect synchrony (with a time shift Δx/ν_{D}). With a small average number of synapses on a cell M, there is no spike synchronization. This modeling work shows that a certain (but not full) level of spike synchronization independent of distance exists in an intermediate regime, where M is above M _{co} but does not need to be extremely high.
As M increases there is a crossover between an asynchronized region and a region of spike synchronization. Another unexpected result of this work is that the crossover value between asynchrony and synchrony, M _{co}, depends only weakly on the strength of the synaptic coupling, but its reduction by synaptic depression is apparent. AMPA conductance strength, which strongly affects the discharge velocity (Golomb and Amitai 1997), has only a weak effect on M _{co}. NMDA conductance strength has a negligible effect. Therefore, synchrony is much more affected by the kinetics of the synapses than by their strength. Synaptic depression decreases M _{co} because it reduces the excitatory strength at long times and thereby removes desynchronizing forces. In the synchronized regime, the spike synchrony χ increases linearly with M for M > M _{co}. Variability in the synaptic strength of existing synapses (not considered in the model) and heterogeneity in intrinsic neuronal properties are expected to reduce synchrony experimentally.
A network of excitatory cells will generally not exhibit spike synchrony during persistent firing even with spatially independent synaptic strength, in particular in the presence of sparseness, heterogeneity, and noise (Ermentrout 1996; Hansel et al. 1993, 1995; van Vreeswijk et al. 1994). Hence, it is not surprising to find that the global crosscorrelations are flat during persistent activity even between neighboring pairs. If neurons are active for a long time [e.g., with strong NMDA conductance (Golomb and Amitai 1997)], and the sparseness level is low, the first spikes of each neuron are synchronized, but the level of synchrony gradually decreases until, within a few cycles, the system becomes asynchronous.
When neuronal firing is terminated by I _{Kslow} or by synaptic depression, the propagating discharge exhibits global transient spike synchronization for a large enough M and not too strong heterogeneity. This synchronization emerges both from the propagating and the transient characteristics of the discharge. The synchrony of the first few spikes neurons fire results from the recruitment process. Resting neurons in front of the discharge are affected by the excitatory synaptic field moving toward them. Adjacent neurons will tend to rise together from rest and fire at least the first few spikes in a synchronous manner, unless M is too small (and in that case the two neurons may be affected by too different synaptic fields) or the heterogeneity level is very high (and then each neuron may respond differently). Distant neurons will be affected by the same discharge but at a time difference that is equal to the distance between them divided by the discharge velocity. Therefore, they will fire their first spikes synchronously with the corresponding time shift. Subsequent spikes tend to be less synchronized because of the excitation effect, but the discharge is terminated before the neurons have enough time to fire in the desynchronized state.
Axonal velocity is estimated to be ∼1 m/s in somatosensory cortex (Gil and Amitai 1996) and olfactory cortex (Haberly 1990). This value is large in comparison to the typical discharge propagation velocity, about 0.16 m/s (Golomb and Amitai 1997), and therefore axonal velocity is neglected in this work. Variability in axonal delays may be another desynchronizing factor. More experimental and computational work is needed for estimating this effect quantitatively.
It was estimated (Golomb and Amitai 1997) that the total AMPA synaptic strength g_{AMPA} is ∼5 times stronger than the conductance needed to evoke a spike in a synchronized manner. A spike reaching a presynaptic terminal of a synapse with that conductance evokes an EPSP of ∼20 mV, which is the voltage difference between the resting potential and the threshold. Assuming a typical EPSP of 1–2 mV (Thomson et al. 1993) ∼10–20 presynaptic cells need to fire spikes together to evoke a postsynaptic spike. Therefore, the number of presynaptic cells that send input to a postsynaptic cell is ∼50–100. This number is much smaller than the number of synapses on a pyramidal cell of a rat, ∼10^{4} (Abeles 1991; Colonnier 1981; Schuz and Palm 1989), because a cell may form more than one synaptic contact on another cell (Markram et al. 1997; Peters and Proskauer 1980; Thomson et al. 1996) and because many connections are cut during the slicing procedure. The number of synaptic inputs estimated here is close to the crossover values obtained by the model. Hence, it is expected that in real rat cortical slices, only a low amount of global synchrony will be recorded, if any.
Comparison with other models
The synchrony measure χ used here (Fig. 4
A) is derived from the global correlation function, which is the sum of correlations between the firing patterns of neuronal pairs at the same distance (Eq. 12
). In general, the correlation can be measured between other functions of the neuronal activity, denoted here by f_{i}, such as the membrane potential and the synaptic variables. When the coupling does not depend on space, it is natural to sum over all the pairs. In this case, The global correlation will be
The synchrony measure χ in my model increases continuously with M, and there is a smooth crossover between the asynchronized and synchronized regions. In models with spatially independent architecture, the transition between the two regimes is sharp (Barkai et al. 1990; Wang and Buzsáki 1996; Wang et al. 1995). In thalamic networks, composed of excitatory and inhibitory populations (Wang et al. 1995), synchrony is achieved if M is >2–5, whereas in hippocampal networks of inhibitory neurons, the transition occurs at M of order 100 (Wang and Buzsáki 1996), a number that is similar to what is obtained here. This may indicate that synchronous activity is more robust in a network that includes coupled excitatory and inhibitory populations (D. Hansel, unpublished data). Synchrony between neurons in inhibitory networks does not depend on whether there is a monosynaptic coupling between them (Wang and Buzsáki 1996), as obtained here for global synchrony during transient discharges.
Discharge propagation in CA3 hippocampal slices with sparse and spatially decaying architecture was studied by Traub and colleagues (Miles et al. 1988; Traub et al. 1993). In these studies, all the cells received exactly the same number of inputs, and therefore their networks are expected to be much more synchronous than what is presented here. The intrinsic bursting properties of CA3 neurons introduce three time scales of synchrony (entire discharges, bursts, and spikes). Synchrony in the discharge time scale was observed in their model, but synchrony in the burst and spike time scale was not analyzed.
Experimental verifications of the model
The sparseness level, expected to be found in the neocortical slice, reduces synchrony to a small level or abolishes it completely. Heterogeneity further degrades synchrony. Both factors affect the global, systemaverage synchrony. Stochastic noise affects synchrony only mildly; this is the only factor that affects both the systemaverage and trialaverage crosscorrelation. As a consequence of these results I predict that in cortical coronal or sagittal slices, strong local, trialaverage spike synchronization will be measured but that global, systemaverage spike synchrony will be low or absent. Local synchrony between two distinct neurons can be measured easily experimentally in neocortical slices by repetitively eliciting events in the slice and measuring the trialaverage correlation between the spike patterns. To measure global synchrony, many pairs with a constant distance between the neurons in each pair should be recorded with a multielectrode array. The model presented here applies mainly to regimes for which the connectivity patterns and the intrinsic neuronal properties do not change with space, for example, areas of order 0.2–1 mm in coronal slices (Chervin et al. 1988; Wadman and Gutnick 1993) or larger areas in sagittal slices (M. J. Gutnick, unpublished data).
If strong systemaverage spike synchrony is found experimentally, it probably will mean that the connectivity pattern is less sparse than what is assumed in the model. Even if M is small, an ordered connectivity pattern (for example, if each cell is coupled to exactly M/2 cells on its right and M/2 cells on its left) will lead to high synchrony. If only weak trialaverage spike synchrony is found experimentally, it will mean that the level of stochastic noise is higher than what is expected from the model. For example, low probability of release can be another source of stochastic noise.
Another prediction is that the systemaverage crosscorrelation of neurons does not depend on the distance between them. This prediction is expected to be valid in each cortical region separately, but systematic changes in the intrinsic and synaptic conductances of neurons, as may be found between adjacent cortical areas in a single slice (Chervin et al. 1988; Wadman and Gutnick 1993), may modify it.
Prechtl et al. (1997) have discovered plane waves, spirallike waves and waves with more complex patterns in the turtle cortex. A model similar to the one reported here, but with a twodimensional architecture, can be used for studying these waves.
Acknowledgments
I am grateful to M. J. Gutnick and I. A. Fleidervish for introducing me to the subject of cortical discharges and for a careful reading of this manuscript. I thank E. Barkai, B. W. Connors, G. B. Ermentrout, Z. Gil, D. Hansel, N. Kopell, M.A.L. Nicolelis, J. Rinzel, and E. L. White for helpful discussions and to D. Hansel also for a careful reading of this manuscript.
This research was supported by a grant from the Israel Science Foundation.
Footnotes

1 In models with longrange coupling, the coupling between neurons does not decay, or decays only weakly, with the distance between them. In models with shortrange coupling, the coupling between neurons decays with the distance between them with a characteristical decay length much smaller than the system length (see Eq. 4 ).

2 Because the model neuron has only one compartment, it is assumed that one cell can send output to a second cell only through 0 or 1 synapses. If there are more synapses between two cells, they can be modeled as one stronger synapse.

3 An asynchronized state is obtained, in general, in networks with spatially independent excitatory coupling with even small levels of disorder (sparseness, heterogeneity, or noise), when only one time scale is involved (here the time scale of the interspike interval). If two time scales are involved and the neurons burst, either because of the neuronal intrinsic properties (Pinsky and Rinzel 1994), or because of population effects (Goldberg et al. 1996), then the bursts are often synchronized (although generally not fully synchronized), but the spikes within bursts are not.

4 The slice dimensions are ∼15 × 2 × 0.4 mm (Golomb and Amitai 1997), and the density of cells in rat neocortex is ∼10^{5} mm^{−3} (Abeles 1991).

5 We study systems with M ≥ 8. If M is very small (≳2), propagation may stop because of lack of connectivity. Such extreme sparseness is not expected to occur in neocortex.
 Copyright © 1998 the American Physiological Society