Catastrophe Theory Enables Moves to be Detected Towards and Away from Self-Organization: The Example of Epileptic Seizure Onset

Abstract

Macroscopic systems with many interacting subunits, when driven far from equilibrium, exhibit self-organization, for example when a pathological rhythm appears suddenly in an epileptic patient. Sudden changes occurring while conditions vary smoothly have, in cases of interest, underlying mathematics that are the subject of Thom’s catastrophe theory. The assumption made herein that the system’s state variables, akin to order parameters, reduce in practice to only one single real variable, ensures that the system derives from a potential function, and warrants recourse to the catastrophe theory. The order parameter is, furthermore, interpreted as a measure of the electropathophysiological activity in the brain, increasing monotonously with the degree of neuronal synchronism. With two neuronal influences, excitatory and inhibitory, as control parameters, the catastrophe is the archetypal cusp. Implementation of catastrophe theory leads to equations showing that fluctuations in a system’s dynamics may be utilised for signalling steps precursory to oncoming catastrophes. Pre-seizure dynamics in epileptic patients exhibit steps towards and away from catastrophe; the steps away are interpreted in terms of homeostatic feedback, consequent on changing patterns of neuronal activity. A number of characteristics of epileptic seizures of differing types merely follow from the geometry of the cusp equilibrium surface. In particular, types of seizures are distinguished by their angle of final approach to onset in parameter space. The measurable parameters by which approach to catastrophe is characterized, may be of use in investigations of the organism’s plasticity in epileptic patients, and in tests of therapeutic means for preventing seizures. There is no need to resort to a model, in the usual sense of the word, and therefore no differential equation needs to be set up.