Using dynamical systems mathematical modeling to examine the impact emotional expression on the therapeutic relationship: A demonstration across three psychotherapeutic theoretical approaches.

The purpose of this paper is to describe an approach to dynamical systems (DS) using a set of differential equations, and how an application of these equations can be used to address a critical element of the therapeutic relationship. Using APA's and videos, DS models were created for each of the six sessions with expert clinicians (Judith Beck, Leslie Greenberg, and Nancy McWilliams) from the three theoretical approaches. A second-by-second observational coding system of the emotional exchanges of the therapists and clients was used as the data for the equations. DS modeling allowed for a side-by-side comparison between the three approaches as well as between the two clients. Examining the graphs created by plotting the results of the DS equations (in particular, phase-space portraits) revealed that there were similarities among the three theoretical approaches, and there were notable differences between the two clients. DS modelling can provide researchers and clinicians with a powerful tool to investigate the complex phenomenon that is psychotherapy.