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Lyapunov exponents computation for hybrid neurons
Federico Bizzarri1, Angelo Brambilla, Giancarlo Storti Gajani
1Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, p.za. Leonardo da Vinci, n. 32, 20133, Milano, Italy, federico.bizzarri@polimi.it.
Journal of Computational Neuroscience
|March 7, 2013
Summary
This study introduces a variational model for hybrid neurons, enabling Lyapunov exponent calculation for non-smooth dynamical systems. This method accurately computes Lyapunov spectra for hybrid neurons and networks, even during synchronous firing.
Area of Science:
- Computational Neuroscience
- Dynamical Systems Theory
- Non-smooth Systems Analysis
Background:
- Lyapunov exponents are crucial for analyzing dynamical system behavior.
- Standard methods for computing Lyapunov exponents apply to smooth systems.
- Hybrid neurons, using integrate-and-fire mechanisms, present non-smooth dynamics, challenging traditional analysis.
Purpose of the Study:
- To extend the applicability of variational models to non-smooth dynamical systems, specifically hybrid neurons.
- To enable the computation of Lyapunov exponent spectra for hybrid neurons and their networks.
- To provide a numerical approach for analyzing the long-term behavior of these complex neural systems.
Main Methods:
- Development of a variational model for hybrid neurons incorporating saltation matrices.
- Application of the extended variational model to compute Lyapunov exponent spectra.
- Utilizing standard numerical approaches for analysis, even with synchronous neuron firing.
Main Results:
- A novel variational model is successfully defined for hybrid neurons.
- The Lyapunov exponent spectrum can now be computed for hybrid neurons and networks.
- The method remains effective even when neurons exhibit synchronous firing patterns.
Conclusions:
- The saltation matrix approach effectively extends variational models to non-smooth hybrid neuron dynamics.
- This work provides a robust method for characterizing the long-term behavior of neural networks composed of hybrid neurons.
- The developed technique facilitates deeper insights into the stability and dynamics of complex neural systems.
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