Dynamics of a structured neuron population

We study the dynamics of assemblies of interacting neurons. For large fully connected networks, the dynamics of the system can be described by a partial differential equation reminiscent of age-structure models used in mathematical ecology, where the 'age' of a neuron represents the time elapsed since its last discharge. The nonlinearity arises from the connectivity J of the network.

We prove some mathematical properties of the model that are directly related to qualitative properties. On the one hand, we prove that it is well-posed and that it admits stationary states which, depending upon the connectivity, can be unique or not. On the other hand, we study the long time behaviour of solutions; both for small and large J, we prove the relaxation to the steady state describing asynchronous firing of the neurons. In the middle range, numerical experiments show that periodic solutions appear expressing re-synchronization of the network and asynchronous firing.