Characterization of Dynamical Stability of Neuronal Networks

The classical theory and algorithm about the largest Lyapunov exponent (LE) cannot address the issue of long-time stability of non-smooth degenerate dynamical systems, e.g., integrate-and-fire (I&F) type neuronal networks. Through correct renormalization of the voltage variable, we extend the classical LE theory to the I&F like network dynamics and demonstrate that the I&F network dynamics could be chaotic due to interactions among neurons.

We further present a new theoretical framework and stable numerical method for the accurate evaluation of the whole spectrum of Lyapunov exponents for integrate-and-fire (I&F) like neuronal systems, e.g., linear I&F networks, exponential I&F networks, and I&F networks with adaptive threshold.

We propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation of Hodgkin-Huxley neuronal networks. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network.

It is hypothesized that chaotic dynamics in the balanced state may be the underlying mechanism for the irregularity of neural activity. We investigate networks of current-based integrate-and-fire neurons with delta-pulse coupling and show that the balanced state robustly persists in this system within a broad range of parameters. We mathematically prove that the largest Lyapunov exponent of this type of neuronal networks is negative. Therefore, the irregularity of balanced neuronal networks need not arise from chaos.