Aims and Results - Summary

    Networks of coupled dynamical systems have been widely used to model excitable media, neural networks, genetic networks, electrical networks, computer networks and many other systems. In many biological, technological and physical networks, communication between elements occurs at discrete events. In this project we focus on  impulsive networks like spiking neural networks. One can expect that the techniques developed for impulsive neural networks could be extended to other systems.

    The numerical simulation of biological networks was severely limited because realistic models presented combinatorial and other problems beyond the capabilities of most computers. Specific numerical schemes have to be developed to simulate accurately and efficiently large neural networks. In the first part of this project, we have extended event-driven techniques to realistic integrate-and-fire networks (Tonnelier et al., Neural Computation, 2007). For the first time, an event-driven simulation of a network of nonlinear spiking neurons was done. However this approach is not generic and can not be applied to any models. Therefore, we have developed a novel numerical integration scheme that retains the advantage of event-driven simulations while allowing generic simulations (see Gang et al.). The resulting scheme is an implicit and adaptive time-stepping method where only active neurons are updated. This work opens promising perspectives on the numerical simulation of impulsive networks. Presently, works are in progress in the bipop project-team and a postdoctoral position will probably become available near mid-march 2008 on fast algorithms for impulsive networks simulation using the complementarity systems framework  (see proposal).