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Ill-Posed Point Neuron Models.

Bjørn Fredrik Nielsen1, John Wyller2

  • 1Department of Mathematical Sciences 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.

Point-neuron models using Heaviside firing rates can be ill-posed, causing numerical inaccuracies. Smooth firing rates are well-posed, but steepness can amplify errors, requiring careful error control in simulations.

Keywords:
Ill posedNumerical solutionPoint-neuron models

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Area of Science:

  • Computational neuroscience
  • Numerical analysis

Background:

  • Point-neuron models are fundamental in computational neuroscience.
  • The choice of firing rate function impacts model well-posedness and numerical stability.

Purpose of the Study:

  • To investigate the well-posedness of point-neuron models with different firing rate functions.
  • To analyze the impact of firing rate steepness on numerical solution accuracy.

Main Methods:

  • Theoretical analysis of initial-condition-to-solution maps.
  • Numerical simulations using finite precision arithmetic.
  • Examination of error amplification in steep firing rate regimes.

Main Results:

  • Point-neuron models with Heaviside firing rate functions can become ill-posed due to discontinuous maps.
  • Smooth firing rate functions ensure well-posedness according to ODE theory.
  • Steep firing rate functions, while theoretically well-posed, can exhibit significant error amplification.

Conclusions:

  • Heaviside firing rate models pose challenges for accurate numerical solutions.
  • Steep firing rate functions necessitate robust error control mechanisms to mitigate simulation errors.
  • Careful selection of firing rate functions and numerical methods is crucial for reliable point-neuron simulations.