WIAS Preprint No. 969, (2004)

Analysis of nonlocal neural fields for both general and gamma-distributed connectivities


  • Hutt, Axel
  • Atay, Fatihcan M.

2010 Mathematics Subject Classification

  • 45J05 92C2

2008 Physics and Astronomy Classification Scheme

  • 02.50.Sk 05.45.Xt 05.10.-a


  • Nonlocal neural activity, space-dependent delay, stability analysis


This work studies the stability of spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation considers 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. In 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.

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