Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos

Year: 2007

Authors: Innocenti G., Morelli A., Genesio R., Torcini A.

Autors Affiliation: Dipartimento di Sistemi ed Informatica, Universitá di Firenze, via S. Marta 3, 50139 Firenze, Italy;
Centro Interdipartimentale per lo Studio di Dinamiche Complesse, Universitá di Firenze, I-50019 Sesto Fiorentino, Italy;
CNR – Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, I-50125 Firenze, Italy;
CNR, Istituto dei Sistemi Complessi, via Madonna del Piano, 10, I-50019 Sesto Fiorentino, Italy; Istituto di Fisica Nucleare, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy

Abstract: The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts. (c) 2007 American Institute of Physics.

Journal/Review: CHAOS

Volume: 17 (4)      Pages from: 43128-1  to: 43128-11

More Information: We thank G. Giacomelli and L. Citi for their collaboration in the initial stages of this study. Useful discussions with J.-M. Ginoux, R. Lima, and M. Lefranc during the Journées de Dynamique Non Linéaire that took place in Marseille, France in November 2006, are warmly acknowledged, as well as clarifications and hints received by S. Yanchuk and S. Lepri. J. Rubin is acknowledged for his critical reading of the unpublished manuscript. This work has been partially supported by the Italian Ministry of University and Research (MIUR), under the Project PRIN 2005 No. 2005098133-003 “Nonlinear dynamic networks: techniques for robust analysis of deterministic and stochastic models.”
KeyWords: bifurcation; bioelectric phenomena; chaos; Lyapunov methods; neurophysiology
DOI: 10.1063/1.2818153

ImpactFactor: 2.188
Citations: 130
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