Traveling Waves and Synchrony in an Excitable Large-Scale Neuronal Network with Asymmetric Connections

Abstract. We study (i) traveling wave solutions, (ii) the formation and spatial spread of synchronous oscillations, and (iii) the effects of variations of threshold in 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. In the integral component of the equation for u the firing rate function is the heaviside function, and the coupling function w is positive. Thus, there is no inhibition in the system. There is a critical value of the parameter β (β∗ > 0) that appears in the equation for v, at which the eigenvalues μ of the linearization of the system around the rest state (u, v) = (0, 0) change from real to complex. We focus on the range β > β∗ where μ are complex, and analyze properties of wave fronts, 1-pulse and 2-pulse waves when the connection function w is asymmetric. For wave fronts we demonstarte how an initial stimulus evolves into two solutions which propagate in opposite directions with different speeds and shapes. For 1-pulse waves our main theoretical result (Theorem 4.2) shows that there is a range of β > β∗ where two families of waves exist, each consisting of infinitely many solutions. The waves in these two families also propagate in opposite directions with different speeds and shapes. There is a critical value θ > 0 such that if θ > θ then 1-pulse waves can only propagate in one direction. In addition, there is a second critical β value, β > β∗, where bulk oscillations come into existence and the system becomes bistable. When β ≥ β∗ we show how an initial stimulus evolves into a solution with large amplitude oscillations that spread out uniformly from the point of stimulus. The asymmetry in w causes the rate of spread of the ‘region of synchrony’ to be more rapid to the right of the point of stimulus than to the left. When θ > θ we construct a ’uni-directional’ circuit where synchronization in one region can trigger synchronization in a distant, second region. However, when synchronization is initially triggered in the second region, it cannot spread to the first region.