# Talk:Chaos in neurons

## Contents |

## Editor-in-Chief

This article was renamed from "Chaos in Nerve Membranes" to "Chaos in Neurons".

## Reviewer A

Historically important contribution to bridge between neuro science and dynamical systems theory !

## Reviewer B

This article does not take into account the enormous experimental litterature on chaotic dynamics (see Korn and Faure for a review). Moreover, some mention of the difficulties in distinguishing between stochastic and chaotic dynamics in experimental data should be included. Finally, the following paragraph should be added

Chaotic dynamics have also been observed in bursting systems (Korn and Faure, 2003). In particular, chaotic burst firing was observed in compartmental models of electrosensory pyramidal neurons (Doiron et al. 2001, 2002). Although experimental data suggested that chaotic dynamics may be observed in *in vitro* recordings from these neurons (Doiron et al. 2003a) through scaling of the interspike intervals, actually determining whether variability in experimental series results from a deterministic process or a stochastic one is complicated at best (Eckmann and Ruelle, 1992; Theiler et al. 1992).

references:

Korn H, Faure P. Is there chaos in the brain? II. Experimental evidence and related models. C R Biol. 2003 Sep;326(9):787-840.

Eckmann JP, and Ruelle D. Fundamental limitations for estimating dimensions and lyaponov exponents in dynamic systems. Phys D 56: 185-187, 1992

Doiron B, Longtin A, Turner RW, Maler L (2001) Model of gamma frequency burst discharge generated by conditional backpropagation. J Neurophysiol 86:1523-1545.

Doiron B., Laing C., Longtin A., and Maler L. (2002) Ghostbursting: a novel neuronal burst mechanism, J. Comput. Neurosci. 12:5-25. neurons

Doiron B, Noonan L, Lemon N, Turner RW (2003a) Persistent Na+ current modifies burst discharge by regulating conditional backpropagation of dendritic spikes. J Neurophysiol 89:324-337.

Doiron B, Chacron MJ, Maler L, Longtin A, Bastian J (2003b) Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli. Nature 421:539-543.

Izhikevich E.M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, Cambridge, MA.

Izhikevich E.M. (2000) Neural Excitability, Spiking, and Bursting. International Journal of Bifurcation and Chaos, 10:1171-1266.

Izhikevich E.M., Desai N.S., Walcott E.C., Hoppensteadt F.C. (2003) Bursts as a unit of neural information: selective communication via resonance. Trends in Neuroscience, 26:161-167.

Theiler J, Eubank S, Longtin A, Galdrikian B, and Farmer JD. Testing for nonlinearity in time seriesthe method of surrogate data. Phys D 58: 77-94, 1992

## Reviewer C

Professor Aihara's article provides a concise introduction to the phenomenon of chaos in nerve membranes. I have a few general suggestions which I hope will help make the article more accessible and useful for a general audience, particularly for beginning students.

1. I think some mention of often-used theoretical tools for analyzing mathematical models and empirical data would be quite useful (I understand that within the context of an encyclopedia article, it is neither possible nor even desirable to discuss these things at length). Some examples which come to mind include Lyapunov exponents, attractor reconstruction, and phase response curve.

2. In the section "Chaos and Bifurctions," the typical routes to chaos deserve references -- either to appropriate Scholarpedia articles or to the literature at large, so that interested readers can find out more about these scenarios.

3. It is not entirely clear how the "chaotic neuron model" fits within the scope of the article -- can it be derived in a systematic way from a biophysical model, e.g. Hodgkin-Huxley? What general insights do we gain from such a map, either biologically or mathematically? Perhaps the relevance of the map could be made more clear by providing a few words on its interpretation and relation to biophysical models.

## Reviewer D

The article give a good introduction to chaos in nerve membrane. I have however two comments

1. The short section about quasiperiodicity is not clear. The ratios of the period will be irrational and the motion quasiperiodic, but it is still "periodic" in a sense that small changes in the initial condition are not amplified during the evolution of the system. It is surprising that sensibility to initial conditions( perhaps the simplest properties of chaotic system) is not discuss in the article.

2. I would be also useful to give references to review papers or books that adress the problem of identification of chaos in biological data: that is the reconstruction of attractor, estimation of various parameters and surrogate technics.