Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: (i) Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. (ii) In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. (iii) Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. (iv) The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. (v) Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience?

The idea that cognitive activity can be understood using nonlinear dynamics has been intensively discussed at length for the last 15 years. One of the popular points of view is that metastable states play a key role in the execution of cognitive functions. Experimental and modeling studies suggest that most of these functions are the result of transient activity of large-scale brain networks in the presence of noise. Such transients may consist of a sequential switching between different metastable cognitive states. The main problem faced when using dynamical theory to describe transient cognitive processes is the fundamental contradiction between reproducibility and flexibility of transient behavior. In this paper, we propose a theoretical description of transient cognitive dynamics based on the interaction of functionally dependent metastable cognitive states. The mathematical image of such transient activity is a stable heteroclinic channel, i.e., a set of trajectories in the vicinity of a heteroclinic skeleton that consists of saddles and unstable separatrices that connect their surroundings. We suggest a basic mathematical model, a strongly dissipative dynamical system, and formulate the conditions for the robustness and reproducibility of cognitive transients that satisfy the competing requirements for stability and flexibility. Based on this approach, we describe here an effective solution for the problem of sequential decision making, represented as a fixed time game: a player takes sequential actions in a changing noisy environment so as to maximize a cumulative reward. As we predict and verify in computer simulations, noise plays an important role in optimizing the gain. The modeling of the temporal structure of cognitive processes is a key step for understanding cognition. Cognitive functions such as sequential learning, short-term memory, and decision making in a changing environment cannot be understood using only the traditional view based on classical concepts of nonlinear dynamics, which describe static or rhythmic brain activity. The execution of many cognitive functions is a transient dynamical process. Any dynamical mechanism underlying cognitive processes has to be reproducible from experiment to experiment in similar environmental conditions and, at the same time, it has to be sensitive to changing internal and external information. We propose here a new dynamical object that can represent robust and reproducible transient brain dynamics. We also propose a new class of models for the analysis of transient dynamics that can be applied for sequential decision making.

A1 Functional advantages of cell-type heterogeneity in neural circuits

We report on experimental studies of synchronization phenomena in a pair of analog electronic neurons (ENs). The ENs were designed to reproduce the observed membrane voltage oscillations of isolated biological neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus. The ENs are simple analog circuits which integrate four-dimensional differential equations representing fast and slow subcellular mechanisms that produce the characteristic regular/chaotic spiking–bursting behavior of these cells. In this paper we study their dynamical behavior as we couple them in the same configurations as we have done for their counterpart biological neurons. The interconnections we use for these neural oscillators are both direct electrical connections and excitatory and inhibitory chemical connections: each realized by analog circuitry and suggested by biological examples. We provide here quantitative evidence that the ENs and the biological neurons behave similarly when coupled in the same manner. They each display well defined bifurcations in their mutual synchronization and regularization. We report briefly on an experiment on coupled biological neurons and four-dimensional ENs, which provides further ground for testing the validity of our numerical and electronic models of individual neural behavior. Our experiments as a whole present interesting new examples of regularization and synchronization in coupled nonlinear oscillators.