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 and 1-pulse and 2-pulse waves when the connection function w is asymmetric. For wave fronts we demonstrate 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 propagate only 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 "unidirectional" 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.

Spatially structured oscillations in a two-dimensional excitatory neuronal network with synaptic depression

“…Note that taking the firing rate to be a linear function close to threshold is consistent with the observation that spike frequency adaptation tends to linearize the firing frequency-input current curve [12,58]. In the limit that σ → ∞, we recover the Heaviside step function used in Amari's original work on scalar networks [2] and most analytical studies of the Pinto-Ermentrout model [42,43,50,14,16,15,54,55]:…”

“…As opposed to the usual Pinto-Ermentrout formulation of negative feedback [42,14,54,55] in spatially extended neural fields, here we take negative feedback to depend on output firing rate ∂u(r, t) ∂t = −u(r, t) + w * (qf (u)) (r, t) (2.1a)…”

Response of traveling waves to transient inputs in neural fields

“…Asymmetric coupling has been studied before as a model of direction selectivity [40] and a unidirectional circuit for the spread of synchrony [52]. Studying (1) in the absence of inputs (I(x, t) = 0), we now consider traveling pulse solutions.…”

“…However, there has yet to be a substantial study of transient inputs on their behavior. Recently, the effect of spatial inhomogeneities in parameters was considered as a model of epileptic tissue [52]. Conceivably, the traveling pulses generated by regions of cortex prone to seizures could be controlled by some transient input to reduce pathological effects of such rogue activity [53].…”

“…We assume here that the amplitude of the stimulus satisifies the inequality (52). Weight amplitude A = 0.5.…”