The One- and Two-Dimensional Fokker-Planck Equation

Abstract

The Fokker-Planck model of psychotherapy can be developed in one or two dimensions. We put forward minimal models for both cases. In the one-dimensional minimal model, we formalize how the therapist can influence client states. We use the slaving principle of synergetics, which claims that slow variables entrain (i.e., “enslave”) the behavior of the quick variables during pattern formation. Applied to therapist and client, this means that the therapist should be “slow,” i.e., have long relaxation times. This is consistent with findings of psychotherapy research suggesting that resilient, stable, and nonjudgmental therapists are actually more effective. In our model, therapist personality has an essential impact of client states. Preferably, the therapist should not be overly influenced by random events, and the coupling of the client to the therapist should be higher than vice versa. The two-dimensional minimal model allows modeling the therapist-client relationship by a limit cycle, i.e., a periodic attractor. This model dispenses with the slaving principle and allows for both oscillations and synchronized phase-shifted coupling of therapist and client behavior, thus linking the theoretical model with empirical findings on interactional synchrony. Synchrony research often applies cross-correlational methods, and we give a short summary of this research. The phase shift of the minimal model can be related to the empirically found cross-correlations. We conclude the structural-mathematical modeling of the process of psychotherapy by a two-dimensional Fokker-Planck equation.