INSTITUTE OF PHYSICS PUBLISHING NETWORK: COMPUTATION IN NEURAL SYSTEMS Network: Comput. Neural Syst. 14 (2003) 351–368 PII: S0954-898X(03)55351-9 Pattern formation in intracortical neuronal fields Axel Hutt 1,3 , Michael Bestehorn 2 and Thomas Wennekers 1 1 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, D-04103 Leipzig, Germany 2 Brandenburgische Technische Universit¨ at Cottbus, Lehrstuhl f¨ ur Theoretische Physik II, Erich-Weinert-Strasse 1, D-03044 Cottbus, Germany Received 25 October 2002, in final form 13 March 2003 Published 17 April 2003 Online at stacks.iop.org/Network/14/351 Abstract This paper introduces a neuronal field model for both excitatory and inhibitory connections. A single integro-differential equation with delay is derived and studied at a critical point by stability analysis, which yields conditions for static periodic patterns and wave instabilities. It turns out that waves only occur below a certain threshold of the activity propagation velocity. An additional brief study exhibits increasing phase velocities of waves with decreasing slope subject to increasing activity propagation velocities, which are in accordance with experimental results. Numerical studies near and far from instability onset supplement the work. (Some figures in this article are in colour only in the electronic version) 1. Introduction Measured brain activity by means of encephalography is supposed to originate from collective neuronal behaviour [1]. Current source density studies indicate that, in some cortical layers, neurons are oriented in a special way to generate a large net activity measured as encephalograms (see [2] and references therein). Suitable basic models for a coherent activity are networks of single coupled neurons on the one side and continuous neuronal fields in space on the other. Both approaches have been studied in numerous works (see, e.g., [3–6] and references therein) with different interests. A major aspect of modelling neuronal activity is the interaction range examined of neural excitation and inhibition. It turns out that delay effects occur easily in local interaction models, in contrast to long-range interactions. In recent years, many works have examined delay effects in integrate-and-fire networks [7–10], spiking neuron networks [11] and pulse coupled networks [12, 13]. However, the study of pattern formation being subject to delay in short-range continuous fields is rare [14]. The present contribution is based on an existing neuronal field model for long-range connections [15–18]. However, in order to formulate a short-range, i.e. intracortical, field 3 Present address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany. 0954-898X/03/020351+18$30.00 © 2003 IOP Publishing Ltd Printed in the UK 351