Dissertation/Thesis Abstract

Mathematical model of the dynamics of psychotherapy
by Norman, Michael Douglas, Ph.D., Florida Atlantic University, 2012, 176; 3536489
Abstract (Summary)

This is a novel attempt to produce a rigorous mathematical model of a complex system. The complex system under study is the relationship between therapists and their clients.

The success of psychotherapy depends on the nature of the relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model.

Previously Gottman et al. [26] used a very similar approach (physical-sciences paradigm) for modeling and making predictions about husband-wife relationships. They modeled interactions using difference equations and then compared the behavior of those equations to the experimental affect data coded from video of married couples in a 15-minute discussion. The parameters they determined in this way had high predictive value of whether the marriages were stable and also gave new insights into the dynamics of how couples interact. Since that novel approach shed light on the dyadic interaction between couples we thought that it also had the possibility to give us new insights into the relationship between therapist and client.

We describe the emotional state of both therapist and client with coupled, first order, nonlinear ordinary differential equations (ODE's). The rate of change of the emotional state of the therapist and client is proportional to their previous state, their uninfluenced state when alone, and an influence function which depends on the state of the other person. We formulated influence functions based on the research literature on psychotherapy and the therapeutic alliance. We then determined the critical points from the intersection of the nullclines and used a numerical ODE solver (Matlab ODE113) to compute the trajectories from different initial conditions.

To empirically validate this approach, 73 unique therapy sessions were video-recorded. Four of these interactions (chosen by our psychotherapy expert) were selected to be modeled and were coded using Gottman's Specific Affect Coding System. The results validate this prototypical approach to psychotherapy; we have shown that human interaction (in the context of psychotherapy) can be quantified and modeled using differential equations.

Indexing (document details)
Advisor: Blanks, Janet C.
Commitee: Dodel, Silke, Jirsa, Viktor K., Lieboritch, Larry S., Peluso, Paul R.
School: Florida Atlantic University
Department: Complex Systems and Brain Sciences
School Location: United States -- Florida
Source: DAI-B 74/06(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Applied Mathematics, Quantitative psychology, Systems science
Keywords: Complex systems, Dynamical systems theory, Ordinary differential equations, Psychotherapy
Publication Number: 3536489
ISBN: 978-1-267-94811-3
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