Section: New Results

Propagation in space-discrete excitable systems

Participant : Arnaud Tonnelier.

We introduce a simplified model of excitable media where the response of an isolated element to an incoming signal is given by a fixed pulse-shape function. When the total activity of one element reaches a given threshold, a signal is sent to its $N$ nearest neighbors. We show that an excitable chain supports the propagation of a set of simple traveling waves where the interval between the emitting time of two successive elements remains constant. We propose a classification of travelling waves that depends on the number of signals that are received by an element. Results on stability of travelling signals are derived. We also discussed the global shape of the speed curve (velocity of the wave with respect to the global coupling strength). In particular, we show that for a given network connectivity, different wave velocities can be obtained, i.e., depending on initial conditions, the network may propagate different signals. A comprehensive study is done for a transmission line with $N=2$ and $N=3$. Some necessary conditions for multistationarity are derived for an arbitrary $N$ and for different network connectivities.