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Fractional-order neural field models reveal how memory properties influence brain waves preceding seizure termination. The order of the fractional derivative (α) impacts wave speed and width, suggesting neuronal memory shapes cortical activity.

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Epilepsy Research

Background:

  • Cortical waves precede seizure termination, but their dynamics are not fully understood.
  • Fractional-order derivatives incorporate memory and hereditary properties, relevant to biological systems.
  • Previous models often lack the capacity to capture complex temporal dynamics observed in neural activity.

Purpose of the Study:

  • To develop and analyze a fractional-order neural field model to understand the role of fractional calculus in cortical wave dynamics.
  • To investigate how different fractional orders (0 < α < 2) affect key wave features like speed and width.
  • To model *in vivo* cortical wave characteristics observed before seizure termination.

Main Methods:

  • Developed a fractional-order neural field model using Caputo's fractional derivative.
  • Employed Mittag-Leffler functions for initial wave dynamics analysis.
  • Utilized the Adomian decomposition method for approximating pulse solutions over longer durations and a wider range of fractional orders.

Main Results:

  • Fractional order significantly influences initial wave speed and width, with distinct effects for α < 1 versus α > 1.
  • The model successfully replicates *in vivo* cortical wave propagation shapes by adjusting wave speed.
  • Wave width modulation by fractional order is dependent on synaptic threshold and connectivity.

Conclusions:

  • Fractional-order derivatives, representing system memory, are crucial for accurately modeling cortical wave features.
  • Neuronal collective memory, as modeled by fractional calculus, actively modifies cortical wave propagation dynamics.
  • The findings provide a novel mathematical framework for understanding seizure termination mechanisms.