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Inflection, canards and excitability threshold in neuronal models.

M Desroches1, M Krupa, S Rodrigues

  • 1INRIA Paris-Rocquencourt Research Centre, Domaine de Voluceau, Rocquencourt BP 105, Le Chesnay Cedex,  78153, France, mathieu.desroches@inria.fr.

Journal of Mathematical Biology
|September 5, 2012
PubMed
Summary
This summary is machine-generated.

Researchers developed a novel technique using differential geometry to find neuronal excitability thresholds. This method identifies inflection sets, accurately approximating firing thresholds in both resonator and integrator neuron models.

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Neuronal excitability is fundamental to neural function, but precisely defining the threshold for electrical activity in models remains challenging.
  • Existing models, like FitzHugh-Nagumo and Hodgkin-Huxley, exhibit complex dynamics near the excitability threshold, including Hopf bifurcations and canard explosions.
  • Distinguishing between sub-threshold and spiking activity is crucial for understanding neuronal behavior and network dynamics.

Purpose of the Study:

  • To introduce a novel technique based on differential geometry to determine the excitability threshold of planar neuronal models.
  • To define and compute the zero-curvature (inflection) set as a discriminator between sub-threshold and spiking neuronal activity.
  • To assess the accuracy of the inflection set approximation for the excitability threshold across different neuronal model types (Type I and Type II).

Main Methods:

  • Applied differential geometry of planar curves to identify regions of zero local curvature in neuronal model phase planes.
  • Utilized invariant manifold theory and singularity theory to compute inflection sets and analyze their topological changes with parameter variation.
  • Compared the computed inflection sets against the known excitability thresholds for both canard and non-canard regimes in resonator and integrator models.

Main Results:

  • Successfully computed inflection sets for various planar neuronal models, including FitzHugh-Nagumo and reduced Hodgkin-Huxley.
  • Demonstrated that the inflection set accurately approximates the excitability threshold in both Type I (integrator) and Type II (resonator) neurons.
  • Showcased the robustness of the inflection set approximation across different dynamical regimes, including those involving canard explosions.

Conclusions:

  • The zero-curvature (inflection) set provides a robust and geometrically intuitive method for defining the excitability threshold in planar neuronal models.
  • This technique offers a unified approach to understanding threshold phenomena in diverse neuronal types, simplifying analysis of complex dynamics.
  • The findings have implications for computational neuroscience, enabling more accurate simulations and theoretical analyses of neural excitability.