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Non-Markovian approach to globally coupled excitable systems
T Prager1, M Falcke, L Schimansky-Geier
1Institute of Physics, Humboldt-University of Berlin, Newtonstrasse15, 12489 Berlin, Germany.
Summary
This study models stochastic excitable units using non-Markovian dynamics. Collective oscillations emerge in coupled ensembles, persisting beyond Hopf bifurcations, and are validated against simulations.
Area of Science:
- Computational neuroscience
- Nonlinear dynamics
- Statistical physics
Background:
- Stochastic excitable units are fundamental in modeling complex systems.
- Existing models often rely on Markovian assumptions, limiting their applicability.
- A non-Markovian approach is needed for a more comprehensive description of excitable dynamics.
Purpose of the Study:
- To develop a non-Markovian framework for stochastic excitable units.
- To investigate the emergence and stability of collective oscillations in globally coupled ensembles.
- To compare theoretical predictions with numerical simulations.
Main Methods:
- Modeling excitable units with three discrete states and state-dependent waiting time density functions.
- Deriving non-Markovian mean-field equations for large ensembles.
- Analyzing the stability of steady states using bifurcation theory.
- Comparing results with simulations of discrete units and coupled FitzHugh-Nagumo systems.
Main Results:
- Collective oscillations emerge in globally coupled ensembles with excitatory coupling.
- Non-Markovian mean-field equations describe ensemble dynamics.
- Oscillations persist over a wide parameter range, including beyond supercritical and subcritical Hopf bifurcations.
- Theoretical results are consistent with numerical simulations.
Conclusions:
- The non-Markovian approach provides a robust framework for studying excitable systems.
- Collective oscillations are a robust phenomenon in these systems, even with non-Markovian dynamics.
- The derived mean-field equations offer valuable insights into the behavior of large ensembles.
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