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David.Hansel{at}biomedicale.univ-paris5.fr, Laboratoire de Neurophysique et de Physiologie du Système Moteur, Université René Descartes, 75270 Paris Cedex 06, France, and Interdisciplinary Center for Neural Computation, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
matog{at}cab.cnea.gov.ar, Comisión Nacional de Energía Atómica and CONICET, Centro Atómico Bariloche and Instituto Balseiro (CNEA and UNC), 8400 San Carlos de Bariloche, R.N., Argentina
We investigate theoretically the conditions for the emergence of synchronous activity in large networks, consisting of two populations of extensively connected neurons, one excitatory and one inhibitory. The neurons are modeled with quadratic integrate-and-fire dynamics, which provide a very good approximation for the subthreshold behavior of a large class of neurons. In addition to their synaptic recurrent inputs, the neurons receive a tonic external input that varies from neuron to neuron. Because of its relative simplicity, this model can be studied analytically. We investigate the stability of the asynchronous state (AS) of the network with given average firing rates of the two populations. First, we show that the AS can remain stable even if the synaptic couplings are strong. Then we investigate the conditions under which this state can be destabilized. We show that this can happen in four generic ways. The first is a saddle-node bifurcation, which leads to another state with different average firing rates. This bifurcation, which occurs for strong enough recurrent excitation, does not correspond to the emergence of synchrony. In contrast, in the three other instability mechanisms, Hopf bifurcations, which correspond to the emergence of oscillatory synchronous activity, occur. We show that these mechanisms can be differentiated by the firing patterns they generate and their dependence on the mutual interactions of the inhibitory neurons and cross talk between the two populations. We also show that besides these codimension 1 bifurcations, the system can display several codimension 2 bifurcations: Takens-Bogdanov, Gavrielov-Guckenheimer, and double Hopf bifurcations.
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