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Neural Field Models with Threshold Noise.

Rüdiger Thul1, Stephen Coombes2, Carlo R Laing3

  • 1Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. ruediger.thul@nottingham.ac.uk.

Journal of Mathematical Neuroscience
|March 4, 2016
PubMed
Summary
This summary is machine-generated.

Adding noise to neural field models reveals how spatial correlations impact wave patterns. The average speed of traveling fronts depends on spatial covariance, not distribution shape, under weak noise conditions.

Keywords:
BumpsFrontsInterface dynamicsNon-Gaussian quenched disorderStochastic neural field

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Complex Systems

Background:

  • The Wilson-Cowan neural field model traditionally uses a sigmoidal firing-rate function, representing averaged behavior of neural elements.
  • This sigmoidal function arises from assuming a distribution of switching thresholds in neural populations.
  • Previous models often relied on averaging, potentially obscuring the effects of individual neural element properties.

Purpose of the Study:

  • To investigate the impact of threshold noise on neural field models without relying on averaging.
  • To analyze how spatial correlations and non-Gaussian distributions affect wave and pattern dynamics.
  • To analytically determine wave speeds and stability of solutions in the presence of noise.

Main Methods:

  • Exploration of threshold noise effects in continuum models, focusing on spatial correlations.
  • Analysis of Gaussian versus non-Gaussian stationary distributions for a given spatial covariance function.
  • Application of an interface approach to analytically calculate wave speeds for traveling fronts with exponentially decaying interactions.
  • Utilizing interface stability arguments to identify stable solution branches for localized bump solutions.

Main Results:

  • Spatial correlations significantly influence the behavior of waves and patterns in continuum neural models.
  • For traveling fronts with weak noise, the spatially averaged speed is determined by the covariance function, independent of the stationary distribution's shape.
  • In systems with Mexican-hat connectivity, noise can generate localized bump solutions, with potential for multiple stable states.

Conclusions:

  • Threshold noise, when considered without averaging, introduces complex dynamics and spatial correlations are crucial.
  • The analytical findings for traveling fronts highlight the robustness of speed dependence on covariance under weak noise.
  • The emergence of multiple stable bump solutions demonstrates noise-induced pattern formation and stability phenomena in neural fields.