2001 Special issueDistributed synchrony in a cell assembly of spiking neurons
Introduction
Consider the process of formation of a Hebbian cell assembly. Conventional wisdom would proceed along the following line of reasoning: start out with a group of neurons that are interconnected, using both excitatory and inhibitory cells. Feed them with a common input that is strong enough to produce action potentials, and let the excitatory synapses grow until a consistent firing pattern can be maintained even if the input is turned off. Using theoretical models of neuronal and synaptic dynamics, we follow this procedure and study the resulting firing modes. Although the model equations may be an oversimplification of true biological dynamics, the emerging firing patterns are intriguing and may connect to existing experimental observations.
Recent studies of firing patterns by Brunel (1999) have shown in simulations, and in mean-field calculations, that large scale sparsely connected neuronal networks can fire in different modes. Whereas strong excitatory couplings lead to full synchrony, weaker couplings will usually lead to asynchronous firing of individual neurons that can exhibit either oscillatory or non-oscillatory collective behavior. For fully connected networks, there exists evidence of the possibility of cluster formations, where the different neurons within a cluster fire synchronously. This phenomenon was analyzed by Golomb, Hansel, Shraiman and Sompolinsky (1992) in a network of phase-coupled oscillators, and was studied in networks of pulse-coupled spiking neurons by van Vreeswijk (1996) and by Hansel, Mato and Meunier (1995).
In contrast to previous studies, the present investigation concentrates on the study of a network storing patterns via Hebbian synapses. We mainly concentrate on a single Hebbian cell-assembly, where full connectivity is assumed between all excitatory neurons. We employ synaptic dynamics that are based on the recent experimental observations of Markram et al., 1997, Zhang et al., 1998. They have shown that potentiation or depression of synapses connecting excitatory neurons occurs only if both pre- and post-synaptic neurons fire within a critical time window of approximately 20 ms. If the pre-synaptic neurons fires first, potentiation will take place. Depression is the rule for the reverse order. The regulatory effects of such a synaptic learning curve on the synapses of a single neuron that is subjected to external inputs were investigated by Song, Miller and Abbott (2000) and by Kempter, Gerstner and van Hemmen (1999). We investigate here the effect of such a rule within an assembly of neurons that are all excited by the same external input throughout a training period, and are allowed to influence one another through their resulting sustained activity. We find that this synaptic dynamics facilitates the formation of clusters of neurons, thus splitting the Hebbian cell-assembly into subassemblies and producing the firing pattern that we call distributed synchrony (DS).
In the next section we present the details of our model. It is based on excitatory and inhibitory spiking neurons. The synapses among excitatory neurons undergo learning dynamics that follow an asymmetric temporal rule of the kind observed by Markram et al., 1997, Zhang et al., 1998. We study the resulting firing patterns and synaptic weights in 3 Dynamical attractors in Hebbian assemblies, 4 Stability of a cycle. The phenomenon of distributed synchrony is displayed and discussed. To understand it better, we perform in 5 The two-neurons synaptic matrix, 6 Analysis of a cycle a theoretical analysis of the influence of an ordered firing pattern on the development of the synaptic couplings. This is derivable in a two-neuron model, and is compared with the results of simulations on a network of neurons. In Section 7 we proceed to demonstrate that similar types of dynamics may appear also in the presence of multiple memory states. A first version of our model was presented in Horn, Levy, Meilijson and Ruppin (2000).
Section snippets
The model
We study a network composed of NE excitatory and NI inhibitory integrate-and-fire neurons. Each neuron in the network is described by its subthreshold membrane potential Vi(t) obeyingwhere τn is the neuronal membrane decay time constant. A spike is generated when Vi(t) reaches the threshold θ, upon which a refractory period of τR sets in and the membrane potential is reset to Vreset where 0<Vreset<θ. For simplicity we set the level of the rest potential to 0. Ii (t) is
Dynamical attractors in Hebbian assemblies
We start by studying the behavior of the network described in the previous section using numerical simulations. We look at the types of dynamical attractors the excitatory network flows into, starting from random firing induced by stochastic inputs. We find that in addition to synchronous and asynchronous dynamical attractors, a mode of distributed synchrony (DS) emerges. In this state, the network breaks into n groups, or subassemblies, of neurons, each of which fires synchronously.
Fig. 2(a)
Stability of a cycle
A stable DS cycle can be simply understood when a single synaptic delay sets the basic step, or phase difference, of the cycle. When several delay parameters exist, a situation that probably more accurately represents the α-function character of synaptic transmission in cortical networks, distributed synchrony may still be obtained. In this case, however, the cycle may destabilize and regrouping may occur by itself as time goes on, because different synaptic connections that have different
The two-neurons synaptic matrix
The values of synaptic connections between excitatory neurons are governed by the kernel function K(tlj−tki) and by the temporal firing patterns of the two neurons. In this section, the synaptic matrix of a two-neuron system is analyzed in terms of these variables. We look at neurons i and j and at the synaptic connections wij and wji between them. The stationary joint density function f(wij,wji) of the two synaptic connections is calculated. This function is the probability of finding synaptic
Analysis of a cycle
As shown in the previous section, the phase shift between the firing times of two neurons characterizes their synaptic connections. These phase shifts are determined by the firing pattern of the network. By evaluating all of them, the synaptic distribution function for a network of NE neurons can be constructed. Assessing all the phase shifts for an arbitrary firing state may be difficult, but for the case of distributed synchrony, when these phase shifts take several distinct values, the
Overlapping cell assemblies
So far, we have followed the procedure, stated at the beginning of the Introduction, of formation of a Hebbian cell-assembly. We noted that it can break into several subassemblies forming a cycle of DS. If such a cell-assembly should represent some memory in an associative memory model, we have to consider the problem of encoding of multiple memories. As a first step toward answering this question, we will show in this section that overlapping DS synaptic matrices can be employed in a retrieval
Discussion
The asymmetric temporal nature of synaptic learning curves among excitatory neurons, as observed by Markram et al., 1997, Zhang et al., 1998, naturally leads to asymmetric and, to some extent antisymmetric, synaptic matrices. This is manifested in our various simulations, starting with Fig. 3, and in our analytic results. The main point that we make in this article is that this asymmetry helps to engrave and stabilize a cyclic firing pattern that we call distributed synchrony.
The system that we
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