chaos in models of neuronal excitabillity

Michael Zeller zeller at ks.uiuc.edu
Thu Sep 29 13:48:44 EST 1994

Chaotic dynamics in neural networks has been found both on the macroscopic and
on the microscopic level. On the singel cell level, experiments with squid giant axons as well as simulations with the HH equations or the Fitzhugh-Nagumo model (also called Bonhoeffer-van der Pol model) have revealed the complex behaviour of the membrane potential. 
Regarding the Bonhoeffer-van der Pol model, you only have to add an external sinusodial stimulation to find frequency lockings, followed by period doubling bifurcations to chaos. The observed dynamics shows similar properties as one-dimensional mappings and you can reduce the membrane potential dynamics of a single BvP neuron to a one-dimensional circle map.

In my opinion it is important to consider the complex temporal behaviour of single cells for future work on neural networks, e.g. the synchonisation of neural assemblies as found in the visual cortex of cats.

I would like to know, if anybody out there has some interesting results concerning 'real-world' applications of chaotic neural networks. 

Best regards,


Michael Zeller                              |  Theoretical Biophysics
============================================|  Beckman Institute
email: zeller at ks.uiuc.edu                   |  University of Illinois
phone: 217-244 1613                         |  405 N. Mathews Avenue
fax:   217-244 6078                         |  Urbana, IL 61801

Some interesting articles related to HH and BvP:

K. Aihara, G. Matsumoto. Chaotic oscillations and bifurcations in squid giant axons. In: Arun V. Holden (ed.). Chaos. Manchester University Press 1986.

K. Aihara, G. Matsumoto. Forced oscillations and routes to chaos in the Hodgkin-Huxley axons and squid gigant axons. In: H. Degn, A. V. Holden, L. F. Olsen (editors). Chaos in biological systems. NATO ASI Series A: Life Science Vol. 138, 1987.

P. Arrigo, L. Marconi, G. Morgavi, S. Ridella, C. Rolando, F. Scalia.
High sensitivity chaotic behaviour in sinusodially driven Hodgkin-Huxley
equations. In: H. Degn, A. V. Holden, L. F. Olsen (editors). Chaos in
biological systems. NATO ASI Series A: Life Science Vol. 138, 1987.

W. Wang. Bifurcations and chaos of the Bonhoeffer-van der Pol model. J.
Phys. A 22, L627-632 (1989).

S. Rajasekar, M. Lakshmanan. Period-doubling bifurcations, chaos, phase-locking and devil's staircase in a Bonhoeffer-van der Poloscillator. Physica D 32, 146-152 (1988).

M. Zeller, M. Bauer, W. Martienssen. Neural dynamics modeled by one-dimensional circle maps. Preprint (1994).

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