The Journal of Neurophysiology Vol. 79 No. 1 January 1998, pp. 1-12
Copyright ©1998 by the American Physiological Society

Models of Neuronal Transient Synchrony During Propagation of Activity Through Neocortical Circuitry

David Golomb

Zlotowski Center for Neuroscience and Department of Physiology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

	ABSTRACT

Abstract Introduction Methods Results Discussion References

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 cross-correlation functions are flat. During discharge propagation, system-average cross-correlation 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 cross-over from an asynchronized to a synchronized discharge. Synaptic depression appears to help synchrony; it reduces the M value at the cross-over. The strengths of -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (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. System-average spike synchronization is determined by the levels of sparseness, heterogeneity, and noise, whereas trial-average spike synchronization is determined only by the noise level. Therefore, I predict that experiments will reveal local, two-cell spike synchrony, but not global synchrony.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

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 -aminobutyric acid-A (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 (Chagnac-Amitai 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 large-scale population synchrony in systems with long-range coupling¹ (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 short-range 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, well-studied, 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 one-dimensional 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 x/_D, where x is the distance between the neurons and _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, system-average correlation, which is affected by sparseness, heterogeneity, and synaptic noise, and local, trial-average 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 cross-correlation 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

Abstract Introduction Methods Results Discussion References

Model

Biophysical, conductance-based models of single regular spiking (RS) neurons and of -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (NMDA) synapses are used, as reported elsewhere (Golomb and Amitai 1997). In brief, the single cell dynamics are described by a single-compartment Hodgkin-Huxley type model, using a set of coupled differential equations

(1)

where C is the membrane capacitance, V_i(t) is the membrane potential of the ith neuron, and i = 1, ..., N is the neuron's index. Each RS cell possess a sodium current I_Na, a persistent sodium current I_NaP, a delayed-rectifier potassium current I_Kdr, an inactivating potassium current (A current) I_KA, a leak current I_L, a synaptic AMPA current I_AMPA, and a synaptic NMDA current I_NMDA. The conductances of these currents are denoted by g with the appropriate subscript. The slow potassium current I_K-slow(g_K-slow = 1 mS/cm²) represents voltage- and calcium-dependent potassium currents, with kinetics slower than the action-potential time scale, that are responsible for adaptation. When heterogeneity in the strengths of intrinsic conductances is introduced, it is modeled by choosing the conductances g_L, g_KA, and g_K-slow at random for each cell from a square distribution with a standard deviation that is multiplied by the mean, where 0 <  < 1/ because conductances must be positive. The mean values are: g_K-slow = 1 mS/cm², g_KA = 1.4 mS/cm², g_L = 0.02 mS/cm².

A cell receives fast, AMPA-mediated excitatory postsynaptic potentials (EPSPs) and slow, NMDA-mediated 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

(2)

where V_pre is the presynaptic potential, T_Glu is the normalized number of presynaptic glutamate vesicles, s(V) = {1 + exp[(V  _s)/_s]}¹, _s = 20 mV, _s = 2 mV, k_f = 1 ms¹, and k_r = 0.2 ms¹ for AMPA and 0.0067 ms¹ for NMDA. The variable T_Glu evolves according to

(3)

where k = 0.001 ms¹. The network is analyzed for two values of the parameter k_t that determines the strength of synaptic depression. At the reference value k_t = 1 s¹, synapses exhibit depression in response to a rapid train of action potentials. Some simulations were carried out without synaptic depression (k_t = 0).

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 , called the footprint length. The model is studied in the regime

(4)

There are many neurons within a footprint length

(5)

The maximal AMPA conductance (with all channels open) between neurons is taken to be g_AMPA w_ij, where g_AMPA is the total AMPA conductance a postsynaptic cell receives. Two architectures are examined. With a uniform architecture, the coefficients w_ij are

(6)

They are normalized such that _jw_ij = 1. With sparse architecture, it is assumed that the probability of cell i to get input from cell j is

(7)

where M is the average number of synaptic inputs a neuron receives.² The strength of each AMPA synapse is therefore g_AMPA/M. The uniform model is obtained from the sparse model for very large N and M.

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 s¹ 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 3-wk 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 _noise = 50 Hz. Each presynaptic spike at time t_o evokes an EPSC with an amplitude g_noisef (t  t_o) described by an extended  function (Destexhe et al. 1994; Rall 1967)

(8)

where the rise time is ₁ = 1 ms and the decay time is ₂ = 5 ms, corresponding to the AMPA kinetics.

Numerical integration was carried out with the fourth-order Runge-Kutta method with a time step of 0.03 ms and with  = 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/cm² for 18 ms into cells in an interval of length  on the left of the slice model. The synaptic noise variables were simulated 50 ms before the beginning of the simulation to reach equilibrium.

View larger version (11K): [in this window] [in a new window]   FIG. 6. Synchrony measure  as a function of the number of cells within a footprint length N/L. Reference value N = 4,096, N/L = 128 is denoted (). Simulations were carried out with the reference parameter set (kt = 1 s1); M = 128.

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

(9)

where t₁, t₂, ..., t_k are the times of the action potentials the ith neuron fires during a time interval T, and  is the Dirac function (Fig. 1). The cross-correlation c_i,j () between two spike trains recorded during one trial is defined as

(10)

When i = j, c is the autocorrelation function. The trial-average cross-correlation _i,j() is

(11)

where ... means average over trials. The system-average cross-correlation C_k() is defined as the average cross-correlation over all pairs at the same distance

(12)

When persistent activity is concerned, the cross-correlations are normalized by the time interval

(13)

In simulations, spikes of neurons close to the edges (at a distance <3) are not included in this average. Cross-correlations are calculated by binning (Abeles 1991), with a bin width of 0.1 ms. Global synchrony measures (see further) were calculated by averaging over 10 realizations for Fig. 6 and 3 realizations otherwise.

View larger version (9K): [in this window] [in a new window]   FIG. 1. Firing function of a neuron, (t) = k=1 (t  t) (Eq. 9). Dirac  function, (t  t), can be represented by a high, narrow rectangle centered around t (Tuckwell 1988), with a width t and a height 1/t, where t  0.

	RESULTS

Abstract Introduction Methods Results Discussion References

Sparse connectivity

I first study the effects of sparse connectivity only on the network dynamics (without heterogeneity or noise). The population-average maximal AMPA conductance a cell receives is g, and the standard deviation is g/. Therefore, I expect the firing pattern to be more ordered as the average number of synapses M increases. Rastergrams are shown in Fig. 2A. Stimulation of the left edge evokes a propagating discharge, during which each cell fires several action potentials. The discharge propagation velocity _D increases slightly with decreasing M, because with small M there are cells that receive relatively high synaptic input, fire faster, and transferthe information about the discharge more rapidly. With M =128 (Fig. 2A, left), the firing pattern looks more orderedthan with M = 16 (Fig. 2A, right), but there is no full coherence as occurs without sparseness (Golomb and Amitai 1997). The reduction of order with increasing sparseness is demonstrated by the system-average correlation function C_k(t) (Fig. 2B). Correlation functions are non-zero only for twice the discharge time, here ~30 ms. The autocorrelation function C_o is more oscillatory at larger M, implying larger resemblance between firing patterns of neurons. At M = 16, the nearest-neighbors cross-correlation C₁ has a peak near  = 0, and it decays monotonically as || increases. Therefore, neurons are synchronized at the discharge time scale, but there is no global spike-to-spike synchrony. At M = 128, C₁ has wiggles with a time period of a few milliseconds, reflecting some level of spike synchronization. The cross-correlation C₁₀₂₄ between neurons 8 apart is similar to C₁ for both M = 16 and M = 128, but with a time shift. Generally, the time shift of C_k is x/_D, where x = kL/N.

View larger version (26K): [in this window] [in a new window]   FIG. 2. Discharge propagation in sparse networks with average number of synapses M = 128 (left) or M = 16 (right). A, top: rastergrams. Spike times from every 32nd cell are shown. Bottom: voltage time trace of one neuron located at x = 0.5. B: system-average autocorrelations C0 (top), nearest-neighbors cross-correlations C1 (middle), and average cross-correlation between neurons 8 apart C1024 (bottom).

To demonstrate the collective nature of the discharges, the cross-correlations 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 cross-correlations, 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).

View larger version (24K): [in this window] [in a new window]   FIG. 3. System-average nearest-neighbors cross-correlations of a sparse network. Averaging is done over pairs of neurons from the following 4 groups: without monosynaptic coupling (top); with left-to-right projection (top middle); with right-to-left projection (bottom middle); and with projection in both direction (bottom). M = 128 = N/L, and therefore the probability of a pair to be in each one of the groups is ~0.25 (Eq. 7).

To quantify the global level of spike synchrony, the synchrony measure _k is defined as follows (see Fig. 4A). The values of the cross-correlation C_k() at the central three peaks are denoted, from left to right, by P₁, P₀, and P₁. The values of C_k() at the central two troughs are denoted by R₁ and R₁. The synchrony measure _k is defined as

(14)

The measure _k is larger when there is more global spike synchronization. With a uniform architecture (no sparseness), it is infinity, because the C_k is a sum of  peaks. With low M (high sparseness), it is very small and is governed by the numerical noise. The dependence of _k on the distance between the neurons in a pair is shown in Fig. 4B. Except for numerical fluctuations,  remains constant with the distance, i.e., the global correlation between neurons does not depend on distance, except for a phase shift.

View larger version (16K): [in this window] [in a new window]   FIG. 4. A: scheme of a typical cross-correlation. Synchrony measure is defined as:  = P1 + 2P0 + P1  2 (R1 + R1) (Eq. 14), where the Ps and the Rs are the cross-correlation values at the 3 central peaks and the 2 central troughs, respectively. B: synchrony measure  as a function of the distance between neurons in the pairs x for M = 128 () and M = 16 (···). Synchrony measure remains constant with the distance, except for numerical fluctuations.

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_K-slow = 0.4 mS/cm²) 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. 5A, 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. 2A (M = 128). The slow potassium current I_K-slow 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 system-average autocorrelation C₀ (Fig. 5B, top) and the periodic nature of autocorrelation of a specific neuron calculated during a long time interval (29.7 s) (Fig. 5C, top). The firing times of neurons do not show any special order. The nearest-neighbor system-average cross-correlation C₁ (Fig. 5B, bottom), or any other cross-correlation, is flat. Similarly, the cross-correlation between two specific neurons (Fig. 5C, 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. 5A) 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).

View larger version (62K): [in this window] [in a new window]   FIG. 5. Persistent activity is achieved with no synaptic depression (kt = 0) and with reduced IK-slow conductance (gK-slow = 0.4 mS/cm2); M = 128. A: after the front has passed, the neurons fire tonically as seen in the rastergram of spike times from every 32nd cell (top) and the voltage time course of one neuron at x = 0.5 (i = 2,048, bottom). B: system-average autocorrelation Cper0 (top) and nearest-neighbor cross-correlation Cper1 (bottom). Correlations are calculated during a period of 300 ms starting at t = 300 ms, i.e., after the decaying of transient effects. Central peak of the autocorrelation is clipped. Flat cross-correlation shows that there is no global synchrony in the system. C: autocorrelation of one neuron's spike train cperi,i (top) and the cross-correlation between spike trains of two neurons cperi,i+1 (bottom) calculated during the time interval 0.3 s  t  30 s; i = 2,048. Central peaks of the autocorrelation are clipped. Cross-correlation is close to being flat, with fluctuations that decay as the square root of the time interval over which correlations are calculated, indicating that, despite the nearly periodic nature of the firing pattern, there is no synchrony, even between two specific, adjacent neurons.

The appearance of an asynchronized state in a one-dimensional neuronal network during persistent activity is expected based on the following argument. In many systems that exhibit persistent firing with all-to-all coupling, excitation leads to an asynchronized state (Hansel et al. 1995). In other systems with long-range 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 state³ (D. Hansel, personal communication). An all-to-all network with the parameter set of Fig. 5 will go to an almost-asynchronized state, with the population voltage (defined as in Golomb et al. 1994b) exhibiting low-amplitude 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 one-dimensional 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, one-dimensional 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 N/L = 128. In an actual slice, the number of cells is much larger. Therefore I investigate the dependence of the synchrony measure  for propagating discharges as N varies between 2¹¹ = 2,098 and 2¹⁷ = 13,1072 (Fig. 6). The reference parameter set (k_t = 1 s¹) is used; M = 128. As N increases,  decreases; the decrease is weaker as log (N) increases. These simulations indicate that  approaches an asymptotic value at large N. In the example, at the reference value (4,096) is ~30% higher than its value at N = 13,1072. Hence, I expect the numerical calculations of  to be qualitatively correct for N in the order of the number of cells in the slice⁴ (of order 10⁶), but quantitatively somewhat lower. The trial-to-trial variability of system-average magnitudes such as  is very small at large N. As a result, only one large realization is necessary for calculating trial-average of such magnitudes.

The dependence of  on M is accessed quantitatively in Fig. 7A for networks with strong synaptic depression (k_t = 1 s¹), including one with the reference parameter set, and in Fig. 7B 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/cm² for the parameter set without synaptic depression and 0.54 mS/cm² 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,  is low, implying no global spike synchronization. At large M,  increases with M, and the dependence of  on M is close to linear. There is a smooth cross-over between the regimes. The number of inputs, M_co, at which the cross-over 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 s¹ (Fig. 4A), whereas M_co  65 for k_t = 0 (Fig. 4B). 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.

View larger version (15K): [in this window] [in a new window]   FIG. 7. A: synchrony measure  as a function of the average number of synapses on a pyramidal cell M for strong synaptic depression (kt = 1 s1), 2 values of gAMPA (0.6 and 0.9 mS/cm2) and 2 values of gNMDA (0.0 and 0.9 mS/cm2). Graph for the reference parameter set is denoted  Synchrony level is at a low level at small M, reflecting no global spike synchrony. It increases linearly with M at large M. B: synchrony measure  as a function of M with no synaptic depression (kt = 0), 2 values of gAMPA (0.31 and 0.4 mS/cm2) and 2 values of gNMDA (0.0 and 0.25 mS/cm2). Note that at the higher -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid level, N-methyl-D-aspartate conductance has almost no effect on .

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 conductancesg_L, g_KA, and g_K-slowis chosen separately, at random, from a square distribution with a ratio  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_K-slow becomes effective after the neuron has fired several spikes; heterogeneity in g_K-slow 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  = 0.3 is shown in Fig. 8A. 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  on  is shown in Fig. 8B. As expected,  decreases with . However, even at a maximal value of  (1/) the value of  is ~4 ms¹. This value is well above the value of  at the cross-over to spike synchrony M_co with the same parameter set ( < 1 ms¹); to reach it in the sparse coupling case, M shouldbe >120. Therefore, heterogeneity in the cell intrinsic properties is a less severe cause of synchrony reduction than sparseness.

View larger version (29K): [in this window] [in a new window]   FIG. 8. Discharge propagation in networks with heterogeneity (A and B) or synaptic noise (C and D). Architecture is uniform (no sparseness, Eq. 6). Heterogeneity is introduced in gL, gKA, and gK-slow (see METHODS). A and C: rastergrams. Spike times from every 32nd cell are shown for heterogeneity level  = 0.3 (A) or for synaptic noise level gnoise = 1.0 mS/cm2 and  = 50 s1 (C). B: synchrony measure  as a function of the heterogeneity parameter . *, heterogeneity level in A. D: synchrony measure  as a function of the synaptic noise level gnoise. Synaptic noise was simulated with rate  = 50 s1() and  = 20 s1 (- - -). , gnoise levels above which spontaneous discharges are created for  = 50 s1 (left) and  = 20 s1 (right). *, represents the synaptic noise level in C.

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 , 20 and 50 s¹ (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/cm² and  = 50 s¹ is shown in Fig. 8C. All the spikes are very well synchronized (with a time shift). The dependence of  on g_noise is shown in Fig. 8D. The arrows denote the g_noise levels above which spontaneous discharges were generated; the dependencies of  on the noise level below these values are shown. Increasing  or g_noise reduces . However, in the regime where no spontaneous discharges are observed,  > 14 ms¹, i.e., it is much higher than the  values obtained with sparseness for M < 200 and is higher than the  values with large heterogeneity level . 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 (system-average) synchrony of the traveling discharge. They do not, however, affect the local (trial-average) 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 trial-average cross-correlation will be a sum of  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 trial-average cross-correlation between two neurons at a distance k neurons apart is equal to the system-average cross-correlation C_k (Fig. 9). In a cortical slice, there is sparseness, heterogeneity, and noise. Here the trial-average and system-average correlations are calculated for a system that has both sparseness and synaptic noise.

View larger version (25K): [in this window] [in a new window]   FIG. 9. Correlations in a homogeneous system with synaptic noise and uniform architecture (no sparseness, Eq. 6). A: trial-average cross-correlation i,i+1 between one neuron at x = 0.5 (i = 2,048) and its neighbor. Averaging is done over 200 trials. B: system-average nearest-neighbor cross-correlation C1. There is no significant difference between the two cross-correlations.

Examples of trial-average correlation functions between two adjacent neurons for a noisy and sparse network(g_noise = 0.1 mS/cm², v = 50 s¹, M = 48) are shown in Fig. 10A, and the corresponding system-average correlations are shown in Fig. 10B. Synaptic noise reduces synchrony only moderately, and hence trial-average autocorrelations show very sharp peaks, and cross-correlations reveal a decaying oscillatory pattern. The relative phase of firing varies from one pair of neurons to another. For example, the cross-correlation presented in Fig. 10A, middle, indicates that those particular two cells fire almost in antisynchrony. The central peak of the cross-correlation presented in Fig. 10A, 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 system-average cross-correlation exhibits almost no wiggles in the time scale of the interspike interval, indicating that there is only a very small global (system-average) spike synchronization in the network.

View larger version (23K): [in this window] [in a new window]   FIG. 10. Correlations in a system with both sparseness (M = 48) and synaptic noise ( = 50 s1, gnoise = 0.1 mS/cm2). A: trial-average correlations: autocorrelation i,i of one neuron at x = 0.5 (i = 2,048; top), cross-correlation i,i+1 between that neuron and its neighbor (middle), and cross-correlation i+1,i+2 (bottom). Averaging is done over 200 trials. B: system-average correlation: autocorrelation C0 (top) and nearest-neighbor cross-correlation C1 (bottom). Although the trial-average cross-correlation shows strong spike synchrony, the system-average spike synchrony is very small (note the small wiggles around  = 0).

	DISCUSSION

Abstract Introduction Methods Results Discussion References

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 cross-over value M_co, of order 30-70. 5) Synaptic depression appears to help synchrony. The cross-over 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 x/_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 cross-over between an asynchronized region and a region of spike synchronization. Another unexpected result of this work is that the cross-over 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  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 cross-correlations 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_K-slow 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, ~10⁴ (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 cross-over 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  used here (Fig. 4A) 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

(15)

For  = 0, C(0) =  dt [_if²_i(t)]². From this value, a synchrony measure is constructed by subtracting the square of the average firing rate and then normalizing it (Golomb and Rinzel 1994). Therefore, the main difference between the synchrony measures used for a one-dimensional system and a spatially independent system is the selection of the neuronal pairs that are included in the summation.

The synchrony measure  in my model increases continuously with M, and there is a smooth cross-over 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, system-average synchrony. Stochastic noise affects synchrony only mildly; this is the only factor that affects both the system-average and trial-average cross-correlation. As a consequence of these results I predict that in cortical coronal or sagittal slices, strong local, trial-average spike synchronization will be measured but that global, system-average 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 trial-average 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 multi-electrode 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 system-average 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 trial-average 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 system-average cross-correlation 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, spiral-like waves and waves with more complex patterns in the turtle cortex. A model similar to the one reported here, but with a two-dimensional architecture, can be used for studying these waves.

	ACKNOWLEDGEMENTS

  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 long-range coupling, the coupling between neurons does not decay, or decays only weakly, with the distance between them. In models with short-range 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 ~10⁵ mm³ (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.

  Received 16 May 1997; accepted in final form 4 September 1997.

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