Patterns and Features of Families of Traveling Waves in Large-Scale Neuronal Networks

We study traveling wave solutions of a system of integro–differential equations which describe the activity of large scale networks of excitatory neurons on spatially extended domains. The independent variables are the activity level u of a population of excitatory neurons which have long range connections, and a recovery variable v. We have found a critical value of the parameter β (β∗ > 0) that appears in the equation for v, at which the eigenvalues of the linearization of the system around the rest state (u, v) = (0, 0) change from real to complex. In contrast to previous studies which analyzed properties of traveling waves when the eigenvalues are real, we examine the range β > β∗ where the eigenvalues are complex. In this case we show that there is a range of parameters over which families of wave fronts, 1-pulse and more general N-pulse waves can coexist as stable solutions. In two space dimensions our numerical experiments show how single-ring and multi-ring waves form in response to a gaussian shaped stimulus. With a spatially invariant coupling function outwardly propagating waves can be periodically produced when a ring-shaped wave receives an appropriately timed perturbation. When the coupling is inhomogeneous the periodic production of waves eventually breaks down and a stable, one-armed rotating spiral wave emerges and fills the entire domain. It is noteworthy that all of these phenomena can be initiated at any point in the medium, that they are not driven by an underlying time dependent periodic pacemaker, and they do not depend on the persistent or periodic presence of an external input.