Dynamics in Neural Systems

Abstract

This chapter provides a dynamical systems viewpoint for various dynamics in neuronal systems, based on the assumption that there exist dynamical systems underlying neural dynamic behaviors. We treat the topological dynamical systems, where qualitative analyses such as phase plane analysis and bifurcation analysis are explained. These analyses can “qualitatively” solve the dynamical equations expressed by differential equations, thereby determining the geometry of trajectories, even if the analytical methods to solve the dynamical equations exactly are unknown. We explain how it is possible to reconstruct the dynamical trajectories from the observed data, by assuming that dynamical systems underlie the observed data. We explain the methods relating to the measurement processes. We also treat various attractors in dynamical systems and their potential functions in neural information processing. Furthermore, we describe dynamical modeling of neural dynamics from a single-neuron level to large-scale levels. We also introduce a globally coupled map as a generalized model, which efficiently describes the complex dynamics caused by heterogeneous interactions. Finally, we briefly review evolutionary dynamical models for nonstationary neural dynamics, which are typically observed in the brain’s developmental process.