Toward a dynamical systems account of behavior networks: Integrating matching, disequilibrium theory, momentum, and Brunt-Väisälä frequency.

Behavior science has produced robust quantitative accounts of individual processes such as matching, disequilibrium, and behavioral momentum. However, research on these behavioral processes has largely developed independently. This paper attempts a step toward a unifying, dynamical systems framework that integrates these three traditions and extends them with an analogy drawn from fluid dynamics (the Brunt-Väisälä frequency, a measure of oscillatory return rate in stratified media). In the proposed framework, the matching law defines equilibrium geometry as the baseline allocation surface toward which behavior tends; disequilibrium theory provides the restoring vector field that arises when behavior is displaced from that equilibrium; behavioral momentum supplies the inertial term governing each responses' resistance to change; and the Brunt-Väisälä analogy characterizes the rate and stability of return to equilibrium relative to reinforcer density gradients. Because contingencies operating on one response can propagate through an organism's repertoire, the framework further incorporates network coupling among stimuli, responses, and relations among stimuli-responses. All components are combined into a single second-order differential equation governing behavior allocation over time. An implementation strategy based on supra-adjacency tensors and temporal-difference learning is described, and numerical simulations of an ABABABAB reversal design illustrate how the framework captures transitions between reinforcement conditions, the persistence conferred by reinforcement history, and the effects of inter-response-stimulus coupling.