Oscillatory Dynamics in Mathematical Models of Neural Networks

Abstract: Oscillations are ubiquitous in the brain and occur at different scales. I will introduce some neuronal models and I will show how tools from dynamical systems theory, such as the parameterization method for invariant manifolds can be used to provide a thorough analysis of the oscillatory dynamics. In particular, I will present a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle) using the concept of isochrons. Using the isochronous sections of the oscillator, we can control changes in phase and amplitude variables. I will show some examples of the computations we have carried out for some well-known biological models and its possible implications for neural communication.