Courbage M.

Spiking-bursting activity is a common feature of the temporal organization in many neural firing patterns. Spiking-bursting dynamics can be regular or chaotic depending on the concentration of neuromodulators, currents and other control parameters. In order to understand the emergence of chaotic oscillations in a neurones network, we use a two variables FitzHugh-Nagumo model of the membrane potential of an isolated neural cell. For appropriate values of the parameters, the system possesses a stable fixed point, a focus, surrounded by a separatrix such that for large enough perturbations, the system responds by an excitation spike which decays to the stable focus. Hence, introducing a Poincaré section by a half-line we study the time series corresponding to occurrence of cluster of spikes (bursts). Thus, we obtain a one-dimensional piece-wise smooth map with one discontinuity point representing the excitation threshold. However, bursting activity appears only in coupled network. Then, we consider a coupling of electrical type, and we develop a detailed study only between two identical neurones, for simplicity. It is found that for suitable values of the parameters, this two-dimensional system develop sulf-sustained oscillations, locally hyperbolic having discontinuity treshold- lines and a strange attractor. The interesting fact is that these lines define regions in the phase space corresponding to the thresholds of excitability of only one of these neurones, or none of them. A suitable symbolic dynamics represents then the bursting activity of the neurones. It is then possible to develop a statistical analysis of time series generated by the system, namely through Perron-Frobenius Operator. This allows to characterize some important features of neural networks, like anti-synchronization and memory. M. COURBAGE, V.B. KAZANTSEV, V.I. NEKORKIN, M. SENNERET Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons, CHAOS, 14, 1148-1156, (2004)