Radiophysics and Quantum Electronics, Vol. 44, Nos. 5–6, 2001 SOME PROBLEMS OF INFORMATION NEURODYNAMICS M. I. Rabinovich, 1,2 R. D. Pinto, 1 and R. Huerta 1,3 UDC 523.03 Will it ever happen that mathematicians will know enough about the physiology of the brain, and neurophysiologists enough of mathematical discovery, for efficient cooperation to be possible? Jacques Hadamard The goal of neural science is to understand the brain, how we perceive, move, think, and remember. All of these things are dynamical processes which are taking place in a complex, non-stationary and noisy environment. This means that these dynamical processes at all levels from small neural networks to behavior should be stable against perturbations but flexible and adaptive. The goal of neurodynamics is to formulate the main dynamical principles which can be a basis of such behavior and to predict the possible activities of neurons and neural ensembles using the tools of nonlinear dynamics. In this paper we discuss our last results related to the mostly challenging part of neurodynamics: information processing by dynamical neural ensembles. 1. INTRODUCTION 1.1. Neuroscience and nonlinear dynamics The last decades have seen increasing efforts of neurophysiologists and neuroscientists to use dynam- ical models as a key element to understand how different neural systems function [1–6]. It is clear now that detailed physiological data alone are not sufficient to understand how even simple neural systems work. Ex- perimentalists need a qualitative approach to the dynamical theory. Dynamical modeling can be important for prediction of the fundamental consequences of neural behavior. In fact, a new branch of science called neurodynamics was introduced in the last few years. Nowadays, after 100 years of intensive studies of the nervous system, both experimentalists and theoretician agree that nerve cells and synapses are functional elements that process information about the environment and control the behavior of the animals [7]. There is a lot of experimental evidence that neurons and neural ensembles behave as dynamical systems [8–11]. The major challenge of the dynamical approach arises from the diversity of neurons, synapses, network topologies, and functions. A single neuron stands in the midst of a controversy among modelers deciding the level of detail to address in their specific problem. Many scientists think that the details of a single neuron are superfluous when neurons operate in a large network. So, very simple input-output functions can be used to represent the neuron. At the other extreme, there are researchers who model very carefully the morphology and electrochemical details of neurons to reveal how the various kind of dendrites, axons, membrane channels, and synapses fit together to create a 1 Institute for Nonlinear Science University of California, San Diego, La Jolla, California, USA; 2 Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia; 3 E.T.S. Inform´ atica, Universidad Aut´ onoma de Madrid, Madrid, Spain. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 44, Nos. 5–6, pp. 439–463, May–June, 2001. Original article submitted February 26, 2001. 0033-8443/01/445-6-0403$25.00 c  2001 Plenum Publishing Corporation 403