Formal Derivation of Lotka–Volterra–Haken Amplitude Equations of Task-Related Brain Activity in Multiple, Consecutively Performed Tasks

Abstract

The Lotka–Volterra–Haken equations have been frequently used in ecology and pattern formation. Recently, the equations have been proposed by several research groups as amplitude equations for task-related patterns of brain activity. In this theoretical study, the focus is on the circular causality aspect of pattern formation systems as formulated within the framework of synergetics. Accordingly, the stable modes of a pattern formation system inhibit the unstable modes, whereas the unstable modes excite the stable modes. Using this circular causality principle it is shown that under certain conditions the Lotka–Volterra–Haken amplitude equations can be derived from a general model of brain activity akin to the Wilson–Cowan model. The model captures the amplitude dynamics for brain activity patterns in experiments involving several consecutively performed multiple-choice tasks. This is explicitly demonstrated for two-choice tasks involving grasping and walking. A comment on the relevance of the theoretical framework for clinical psychology and schizophrenia is given as well.