Berlin 2005 – scientific programme

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SYBN: Biological and Social Networks

SYBN 3: Biologische und Soziale Netzwerke, Postersitzung

SYBN 3.29: Poster

Monday, March 7, 2005, 14:00–15:30, Poster TU E

Pattern Formation in the Visual Cortex: A Numerical Study Based on Newton-Krylov Methods — •D. Heide¹, M. Kaschube¹, S. Löwel², and F. Wolf¹ — ¹MPI für Strömungsforschung, Bunsenstrasse 10, 37073 Göttingen — ²Leibniz Institute for Neurobiology, Brenneckestr. 6, 39118 Magdeburg

In the visual cortex, neurons respond strongest to a visual stimulus of their preferred orientation. Parallel to the cortical surface, these preferred orientations form a complex pattern, called the orientation map. Orientation maps presumably emerge by activity-dependent self-organisation of the cortical network during development [1]. This process of dynamic pattern formation can be described by a phenomenological model based on biological plausible symmetry assumptions [2]. The dynamics of the orientation map is given by a Swift-Hohenberg type nonlinear integro-differential equation. Solving the model poses a substantial numerical challenge, because the equations of motion are stiff and one is interested in the asymptotic behaviour of large systems.

Here, we present a fully implicit parallel numerical integrator for this model. Discretisation in time is done using a Crank-Nicolson scheme. Spatially, the equation is discretised in real space, using finite difference stencils for the Swift-Hohenberg operator. The incomplete Newton-Krylov method is applied to solve the resulting nonlinear equation for every time step. We find that this fully implicit scheme is numerically much more stable than previous semi-implicit integrators.

[1] F. Wolf & T. Geisel. Nature, 395:73 (1998). [2] F. Wolf. In D. Hansel et al., Methods and Models in Neurophysics. Elsevier (in press).