We study synchronization in large populations of N identical neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses M a cell receives is larger than a critical value Mc. Below Mc   , the system is in an asynchronized state.
In the limit of weak coupling, we use the averaging method to reduce the network model into a model of phases which are coupled via a function of their phase differences. Using mean field theory, we show that the stability of the asynchronized state can be determined exactly in the asymptotic limit 1<< Mc << N. In particular we establish that in this limit Mc/ N = O(1/N). When the condition Mc >> 1 is not satisfied, our theory is approximate but provides us with an estimate for Mc as a function of the single neuron and synaptic properties.
We apply our analytical theory to study integrate-and-fire neurons with inhibitory coupling. We find that Mc varies non-monotonously with the level of the external input on the neuron, that it is a strongly decreasing function of the the synaptic rise time and of the duration of the refractory period of the neuron. For typical inhibitory synapses and refractory period of 2-5 msec we find that Mc is of the order of 100 synaptic connections per neuron. Numerical simulations are performed to show that our theory provides very good results also for finite Mc (Mc of the order of few tens) and mildly strong coupling. Finally we study numerically the strong coupling regime: in particular our simulations indicate that Mc is an increasing function of the coupling strength.
We conclude by discussing the relevance of our theory to understand the mechanisms of synchrony in neocortex and hippocampus and by proposing experiments to test it.
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