Stability of Periodic Oscillations in a Network of Neurons with Time Delay

Oscillatory behavior is investigated in the case of a network of oscillatory pairs of neurons with time delayed connections. The uniform periodic solution is obtained from the numerical integration of a system reduced to a single pair of neurons. The stability of this periodic orbit in the network is estimated from a discretization of the time axis. It is shown that the periodic solution become unstable as the size of the system is increased. This loss of stability results in the apparition of spatiotemporally structured behavior, such as travelling waves. 2 Oscillatory activity is a common feature of natural 1,2] and artiicial 3] networks of neu-rons. Oscillations can arise as an intrinsic property of individual neurons 4], or alternatively, from the synaptic interactions between neurons that would not oscillate otherwise 2,5]. In this paper, we investigate oscillatory phenomena of the latter kind. The role of synchronized oscillations in the cerebral cortex was subject to much attention these last years (reviewed in ref. 6]). Models of neural networks with time-delayed interactions were proposed to account for this oscillatory activity 7{10] and typically displayed robust oscillations for some conditions of input. Solvable models without time-delay were also investigated in the same context 11{13]. Such models consider densely connected networks for which synchronized oscillations are usually seen. The present model addresses oscillatory behavior in the case of networks with local connectivity, which is closer to the connectivity of cerebral cortical neurons 14]. The type of model decribed here is derived from passive membrane equations with sigmoid-type of synaptic coupling. This is a simpliied model of synaptic interaction which relies on the hypothesis that the output activity of the synapse can be expressed as a sig-moid function of the presynaptic membrane potential, with a time delay characteristic of propagation and release of transmitter. Similar types of models have been introduced previously 15{17] and were shown to display oscillatory behavior. Limit cycle oscillations and bursting behavior were shown to happen in a single pair of neurons interconnected with excitatory and inhibitory synapses 17]. This system was described by a two-variable delay diierential equation (DDE) in which limit cycle oscillations arose through a Hopf bifurcation and turn into bursting oscillations via a homoclinic bifurcation 17]. We investigate here the oscillatory behavior of a locally-connected network constituted of pairs of neurons described by the same equations. We show that such a network displays spatially uniform oscillations which …