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Regularization of Ill-Posed Point Neuron Models.

Bjørn Fredrik Nielsen1

  • 1Faculty of Science and Technology, Norwegian University of Life Sciences, P.O. Box 5003, Ås, 1432, Norway. bjorn.f.nielsen@nmbu.no.

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Summary
This summary is machine-generated.

Neural point neuron models using steep firing rate functions are well-posed. However, their solutions may not always converge to the ill-posed Heaviside limit, especially if the solution reaches the firing threshold multiple times.

Keywords:
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Area of Science:

  • Computational neuroscience
  • Mathematical modeling of neural systems

Background:

  • Point neuron models are fundamental in computational neuroscience.
  • Heaviside firing rate functions can lead to ill-posed models, causing mathematical instability.
  • Well-posed approximations are needed for reliable numerical simulations.

Purpose of the Study:

  • To investigate the well-posedness of point neuron models with steep, Lipschitz continuous firing rate functions.
  • To determine if solutions of these well-posed models converge to solutions of ill-posed Heaviside models.
  • To analyze the conditions under which convergence occurs.

Main Methods:

  • Employing Lipschitz continuous, steep firing rate functions as approximations.
  • Utilizing the Arzelà-Ascoli theorem to analyze solution convergence.
  • Investigating the properties of the limit function, specifically its threshold simplicity.

Main Results:

  • Models with steep firing rate functions are well-posed, allowing approximate solutions with finite precision arithmetic.
  • Convergence of regularized solutions to the ill-posed limit is not guaranteed for all cases.
  • Convergence depends on the "threshold simplicity" of the limit function, meaning the set of times at which components hit the firing threshold must have zero Lebesgue measure.

Conclusions:

  • While steep firing rate functions create well-posed models, their solutions may not fully capture the behavior of ill-posed Heaviside models.
  • The existence of multiple solutions for Heaviside models complicates convergence analysis.
  • The "threshold simplicity" assumption is crucial for ensuring convergence of regularized solutions to the Heaviside limit.