Turing instability and pattern formation in a two-population neuronal network model

A two-population firing-rate model describing the dynamics of excitatory and inhibitory neural activity in one spatial dimension is investigated with respect to formation of patterns, in particular stationary periodic patterns and spatiotemporal oscillations. Conditions for existence of spatially homogeneous equilibrium states are first determined, and the stability properties of these equilibria are investigated. It is shown that the nonlocal synaptic interactions may promote a finite bandwidth instability in a way analogous to diffusion effects in the classical Turing instability for reaction–diffusion equations and modulational instability in the theory of nonlinear waves in nonlocal defocusing Kerr media. Our analysis relies on the wave-number dependent invariants of the 2×2-matrix representing the spatially Fourier-transformed linearized evolution equations. The generic picture which emerges is an instability consisting of a finite set of well-separated unstable bands in wave-number space (gain bands). The case with symmetrical, exponentially decaying connectivity functions is investigated in detail, allowing for a more comprehensive analysis of the gain-band structure, and, in particular, conditions for the excitation of a single gain band through a Turing–Hopf bifurcation with the relative inhibition time constant as control parameter. Two typical situations emerge depending on the thresholds and inclinations of the sigmoidal firing-rate functions: (i) A single gain-band is excited through a Turing–Hopf bifurcation, and the resulting state is a spatiotemporally oscillating pattern, or (ii) the instability develops into a stationary periodic pattern, i.e. a set of equidistant bumps. The dependence of instability-type on the inclinations of the firing-rate function and the time constant are comprehensively investigated, demonstrating, for example, that only stationary patterns can be generated for sufficiently small inhibitory time constants. The nonlinear development of the gain-band instabilities is further elucidated by direct numerical simulations.