The boundary between multiple and distributed delays and noisy brain rhythms

This talk concerns three aspects of brain rhythms that are analyzed from a nonlinear stochastic dynamics perspective. The first discusses an interesting transition that occurs in a system with a large number of delays. We show that there is a non-monotonic behaviour of the complexity of the solutions as the number of delays increases. This complexity is characterized through a combination of lyapunov exponents and autocorrelation functions. It implies that more delayed loops, as seen in the brain, does not necessarily lead to more complexity, and in fact will eventually lead to simple (such as limit cycle) dynamics. We also consider the behaviour of the envelope of brain rhythms in the gamma and beta range. We apply a stochastic averaging method to a Wilson-Cowan model in a finite size system to produce a system of coupled envelope-amplitude (Z-phi)  equations, equivalent to that of a two-dimensional Ornstein-Uhlenbeck process. The analysis of this model and of the resulting Z-phi equations is used to characterize when the envelopes are bursty, as they should be in healthy conditions, and how to modify system parameters when bursting is out of the normal range. This approach can then be applied to multiple coupled quasi-cycles in the brain to study inter-areal synchronization. Finally we briefly consider, both numerically and analytically, reverberatory effects that appear when delayed systems receive transient harmonic input.