Physica D 203 (2005) 30–54 Analysis of nonlocal neural fields for both general and gamma-distributed connectivities Axel Hutt a, ∗ , Fatihcan M. Atay b a Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany b Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany Received 28 May 2004; received in revised form 22 February 2005; accepted 7 March 2005 Communicated by C.K.R.T. Jones Abstract This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation includes synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation, namely the general reaction–diffusion equation and the Swift–Hohenberg equation. Hence, the discussed integro- differential equation generalizes these PDEs. In the case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition–lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results. © 2005 Published by Elsevier B.V. PACS: 02.30.Rz; 87.18.Hf Keywords: Neuronal populations; Synaptic connectivity; Bifurcation analysis 1. Introduction Understanding the basic mechanisms of neural activity is supposed to yield insights to major brain functions such as cognitive processes [1], motor coordination [2], or perception [3]. In addition, the understanding of pathological ∗ Corresponding author: Institute of Physics, Humboldt University of Berlin, Newtonstr. 15, 12489 Berlin, Germany. E-mail addresses: Axel.Hutt@physik.hu-berlin.de (A. Hutt); atay@member.ams.org (F.M. Atay). 0167-2789/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.physd.2005.03.002