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Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model.
S R Dtchetgnia Djeundam1, R Yamapi, T C Kofane
1Laboratory of Mechanics and Materials, Department of Physics, Faculty of Science, University of Yaoundé I, Box 812, Yaoundé, Cameroon.
Chaos (Woodbury, N.Y.)
|October 5, 2013
Summary
This study explores bifurcations in the Hindmarsh-Rose neuronal model, revealing chaotic transitions between bursting and spiking. Adding random signals introduces stochastic bifurcations, impacting neuronal dynamics even with noise.
Area of Science:
- Computational Neuroscience
- Nonlinear Dynamics
- Mathematical Biology
Background:
- The Hindmarsh-Rose model is a key mathematical model for neuronal excitability.
- Understanding bifurcations is crucial for characterizing complex neuronal dynamics.
- Neuronal models often exhibit transitions between different firing patterns like bursting and spiking.
Purpose of the Study:
- To analyze deterministic and stochastic bifurcations in the 3D Hindmarsh-Rose neuronal model.
- To investigate the impact of random Gaussian noise on neuronal firing patterns.
- To identify and classify different types of stochastic bifurcations.
Main Methods:
- Analysis of equilibrium solutions and their stability.
- Investigation of deterministic bifurcations.
- Introduction of random Gaussian signals to study stochastic bifurcations.
- Application of the asymptotical method for analyzing phenomenological bifurcations.
Main Results:
- Observed various bifurcations leading to chaotic transitions in neuronal activity.
- Identified chaotic transitions between periodic bursting and spiking solutions.
- Characterized two types of stochastic bifurcations: phenomenological (P-bifurcations) and dynamical (D-bifurcations).
- Found that spiking and bursting chaos persist for finite noise intensities.
Conclusions:
- The Hindmarsh-Rose model exhibits complex dynamics including chaotic transitions driven by bifurcations.
- Stochastic bifurcations significantly alter neuronal behavior when noise is present.
- Neuronal activity remains robust to noise up to certain intensity levels, maintaining spiking and bursting patterns.
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