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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Strongly nonlinear age-structured equation, time-elapsed model and large delays.

Benoît Perthame1, Clément Rieutord2, Delphine Salort3

  • 1Sorbonne Université, CNRS, Université de Paris Cité, Inria, Laboratoire Jacques-Louis Lions, LJLL, EPC MUSCLEES, F-75005, Paris, France. benoit.perthame@sorbonne-universite.fr.

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This study proves that inhibitory neural networks without delay stabilize activity. Including a delay, however, allows both inhibitory and excitatory networks to generate periodic solutions, advancing neural modeling.

Keywords:
Age-structured equationsNeural networksPopulation dynamics

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Neural assemblies exhibit complex dynamics influenced by network structure and delays.
  • Understanding the stability and activity patterns in neural networks is crucial for neuroscience.
  • Age-structured models provide a framework for analyzing neural population dynamics.

Purpose of the Study:

  • To rigorously prove the stabilization of inhibitory neural networks without delay.
  • To investigate the emergence of periodic solutions in neural networks with delays.
  • To introduce a new formalism for analyzing age-structured neural models.

Main Methods:

  • Analysis of a nonlinear age-structured equation for neural assemblies.
  • Application of a non-expansion property and a novel strict nonlinearity condition.
  • Development of a new formalism to rigorously establish periodic solutions for large delays.

Main Results:

  • Proved that inhibitory networks without delay converge to a unique steady state, ensuring stable network activity.
  • Demonstrated that including delays can lead to periodic solutions in both inhibitory and excitatory networks.
  • Established a fundamental contraction property applicable to other age-structured systems.

Conclusions:

  • Neural network models with age structure can exhibit diverse dynamic behaviors based on delay parameters.
  • The findings offer rigorous mathematical insights into neural synchrony and network stability.
  • The developed formalism provides a powerful tool for analyzing complex neural dynamics.