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Numerical continuation of travelling waves and pulses in neural fields

Authors
Journal
BMC Neuroscience
1471-2202
Publisher
Springer (Biomed Central Ltd.)
Publication Date
Identifiers
DOI: 10.1186/1471-2202-14-s1-p70
Keywords
  • Poster Presentation
Disciplines
  • Mathematics

Abstract

Numerical continuation of travelling waves and pulses in neural fields POSTER PRESENTATION Open Access Numerical continuation of travelling waves and pulses in neural fields Hil GE Meijer1*, Stephen Coombes2 From Twenty Second Annual Computational Neuroscience Meeting: CNS*2013 Paris, France. 13-18 July 2013 We study travelling waves and pulses in neural fields. Neural fields are a macroscopic description of the activity of brain tissue, which mathematically are formulated as integro-differential equations. While linear and weakly nonlinear analysis can describe instabilities and small amplitude patterns, numerical techniques are needed to study the nonlinear properties of travelling waves and pulses. Here we report on progress of such an approach using an excitatory neural field with linear adaptation. First, we find and analyse an anti-pulse, where the whole field is active except for a moving region with lowered activity. Such an anti-pulse may be relevant in modelling spreading depression. Second, we consider dynamics for relatively shallow, smooth activation functions where the neural field behaves as an excitable medium. We com- pute numerically the dispersion curves for travelling waves, i.e. the wavespeed as a function of the spatial per- iod. This allows a kinematic analysis, see [3], that may be useful for analysing spreading epileptiform activity. Third, we present a numerical continuation method for periodic orbits of integro-differential equations based on fast Fourier transforms. Hence, neural fields with biophy- sically realistic mechanisms may be analysed beyond linearisation. First we vary the strength of adaptation and observe that travelling fronts and pulses are organized by a heteroclinic cycle: a codimension 2 bifurcation. The unfolding uncovers a new type of solution that we call anti-pulse. This is a simi- lar solution as in [2] but in a much simpler model. Follow- ing the construction as in [1] for pulses in the case of a Heaviside firing rate, we obtai

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