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Neural field models with transmission delays and diffusion.

Len Spek1, Yuri A Kuznetsov2,3, Stephan A van Gils2,3

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We introduce a diffusion term into neural field models to study neuron electrical connections. Diffusion suppresses non-synchronized states and promotes synchronized oscillations in neural activity.

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Area of Science:

  • Computational neuroscience
  • Mathematical biology
  • Dynamical systems theory

Background:

  • Neural field models describe large-scale neuronal activity.
  • Previous models lacked direct electrical connection modeling.
  • Incorporating diffusion represents direct neuronal connections.

Purpose of the Study:

  • Extend neural field models to include diffusion.
  • Analyze spectral properties and bifurcations in these extended models.
  • Investigate the impact of diffusion on neuronal synchrony.

Main Methods:

  • Developed novel sun-star calculus for delay differential equations with diffusion.
  • Characterized the essential spectrum of the extended neural field model.
  • Computed spectral properties and the first Lyapunov coefficient for Hopf bifurcations.
  • Utilized numerical simulations to validate theoretical findings.

Main Results:

  • The addition of diffusion to neural field models was mathematically tractable.
  • Diffusion was shown to suppress non-synchronized steady-states.
  • Synchronized oscillatory modes were favored by the inclusion of diffusion.
  • Spectral properties and bifurcation analysis provided insights into system dynamics.

Conclusions:

  • Diffusion terms significantly alter neural field dynamics.
  • The model extension provides a richer understanding of neuronal network behavior.
  • Findings suggest diffusion plays a key role in promoting synchronized neuronal activity.