A mathematical model of psychotherapy: An investigation using dynamic non-linear equations to model the therapeutic relationship

Abstract

Mathematical models, such as the one developed by Gottman et al. (1998, 2000, 2002) to understand the interaction between husbands and wives, can provide novel insights into the dynamics of the therapeutic relationship. A set of nonlinear equations were used to model the changing emotional state of a therapist and client. The results suggest: (1) The person that is most responsive to the other achieves the most positive state, (2) the emotional state of the client oscillates before reaching its final state, (3) therapy is least successful when the therapist starts from a negative state, and (4) there is an inverse relationship between models that change only the influence parameter and models that change only the inertia parameter, creating a series of four basic models to work with. These theoretical models require further, empirical investigation to test the derived parameters. If validated, or revised based on observations of therapist-client relationships in development, they could provide specific direction in creating successful therapeutic relationships for training clinicians and those already in practice.

Abstract

Abstract

Os modelos matemáticos, tais como os desenvolvidos por Gottman et al (1998, 2000, 2002) para compreender a interacção entre maridos e mulheres, podem fornecer insight sobre as dinâmicas da relação terapêutica. Foi utilizado um conjunto de equações não lineares para modelar a mudança no estado emocional de cliente e terapeuta. Os resultado sugerem que: (1) a pessoa que é mais responsiva aos outros alcança estados mais positivos, (2) O estado emocional do cliente oscila antes de chegar ao estado final, (3) A terapia é pior sucedida quando o terapeuta começa num estado negativo e (4) existe uma relação inversa entre modelos que mudam apenas a o parâmetro de influência e modelos que mudam apenas o parâmetro de inércia, criando uma série de 4 modelos básicos de trabalho. Estes modelos teóricos requerem mais investigação empírica para testar os parâmetros derivados. Se validados ou revistos, baseados nas observações das relações terapeuta cliente em desenvolvimento, os modelos podem fornecer uma direção específica em criar relações terapêuticas de sucesso para formar clínicos e para aqueles que já se encontram na prática clinica.

Abstract

Les modèles mathématiques, tels ceux développés par Gottman et col. (1998, 2000, 2002) pour comprendre l'interaction entre maris et femmes, peut fournir de nouvelles perspectives sur la dynamique de la relation thérapeutique. Un ensemble d’équations non-linéaires ont été utilisées pour modéliser le changement de l’état émotionnel du thérapeute et du client. Les résultats montrent : (1) la personne qui est la plus responsive à l'autres atteint l’état le plus positif, (2) l’état émotionnel du client oscille avant d'atteindre son état final, (3) la thérapie est moins fructueuse lorsque le thérapeute commence par un état négatif, et (4) il y a une relation inverse entre les modèles qui changent uniquement le paramètre d'influence et les modèles qui changent uniquement le paramètre d'inertie, créant une série de quatre modèles de base avec lesquels travailler. Ces modèles théoriques requièrent de nouvelles investigations empiriques afin de tester les paramètres dérivés. S'ils sont validés, ou révisés en fonction d'observations sur la relation thérapeute-client en développement, ils peuvent fournir une direction spécifique dans la création de relations thérapeutiques fructueuses pour la formation des futurs cliniciens et ceux qui pratiquent déjà.

Abstract

I modelli matematici come quello sviluppato da Gottman et al. (1998, 2000, 2002) per capire le modalità di interazione tra mogli e mariti può fornire nuovi insight nelle dinamiche delle terapie relazionali. Un set di equazioni non lineari è stato utilizzato per rappresentare i cambiamenti degli stati emotivi del terapeuta e del suo cliente. I risultati suggeriscono: (1) la persona che è più responsiva verso l'altra conquista lo stato più positivo, (2) lo stato emotivo del cliente oscilla prima di raggiungere il suo stato finale, (3) la terapia è meno efficace quando il terapeuta parte con uno stato negativo e (4) c’è una relazione inversa tra i modelli che cambiano solo il parametro di influenza e i modelli che cambiano solo il parametro di inerzia creando una serie di quattro modelli di base con cui lavorare. Questi modelli teoretici richiedono inoltre analisi empiriche per testare i parametri derivati. Se validati o rivisti sulla base delle osservazioni della relazione paziente-terapeuta in evoluzione, possono provvedere specifiche direzioni nel creare relazioni terapeutiche efficaci per formare clinici in training o coloro che già esercitano.

Abstract

Los modelos matemáticos, como el que desarrollaron Gottman y cols (1998, 2000, 2002) para estudiar las interacciones entre esposos pueden proveer una manera novedosa de comprender la relación terapéutica. Un conjunto de ecuaciones no lineales fueron utilizadas para modelizar los cambiantes estados de animo que se daban en cliente y terapeuta al o largo de la relación. Los resultados sugieren que : 1) aquél de los dos que es más reactivo en la relación es que logra el estado mas positivo; 2) el estado emocional del cliente oscila antes de llegar a su estado final; 3) la terapia es menos exitosa cuando el terapeuta comienza el tratamiento partiendo de un estado negativo; y 4) hay una relación inversa entre los modelos que cambian solo el parámetro de influencia y los modelos que cambian sólo el parámetro de inercia, creando una serie de cuatro modelos con los que se puede trabajar. Estos modelos de investigación requieren mayores desarrollos, especialmente en el terreno de la investigación empírica para evaluar los parámetros derivados. Si son validados o revisados en base a observaciones de la relación terapeuta-cliente en movimiento, podrían proveer recomendaciones útiles en relación a mejorías en la relación terapéutica para ser utilizados tanto en la formación de terapeutas como por aquéllos que ya ejercen la función terapéutica.

Keywords:

• dynamical systems
• alliance
• emotion in therapy
• process research
• psychotherapist training/supervision/development
• statistical methodology
• technology in psychotherapy research and training

Notes

1. Once the equations were completed, Gottman and his associates sought to test their models on couples. Couples were videotaped, and the quality of their interaction was coded using the Specific Affect Coding System (SPAFF; Gottman et al, 1996). The SPAFF codes specific emotional behavior of a husband and wife in real time, and can be used in any conversation. According to Gottman et al., the SPAFF focuses solely on the affects expressed by the participants, drawing on facial expressions, vocal tones, and speech content to characterize the emotions that are displayed. SPAFF coders “categorized the affects displayed using five positive codes (interest validation, affection, humor and joy), 10 negative affect codes (disgust, contempt, belligerence, domineering, anger, fear/tension, defensiveness, whining, sadness, stonewalling), and a neutral affect code” (2002, p. 179). The final weighted scale ranged from 24 to +24, giving equal weighting to the five positive codes and the 10 negative codes. Couples were asked to have a 15-minute discussion about an area of ongoing conflict, which was videotaped for coding using the SPAFF. Each videotape was coded in its entirety by two independent observers, and then used to determine the parameters of the previously formulated nonlinear equations of the mathematical model. The results of their analyses yielded support for the predictive ability of the parameters of the model to discriminate between three separate criterion groups (happy stable couples, unhappy stable couples, and divorced couples), as well as discover the importance of each of the four parameters (husband and wife's influence and uninfluenced steady states) and the ability to intervene therapeutically (for a full discussion of these results, please see Gottman et al., 2002).

2. Critical points can be “stable” or “unstable.” If small changes in T and C bring both the therapist and client back to their values at the critical point, then that critical point is a stable steady state. This is a point of equilibrium that the relationship would gravitate towards (or be attracted to). It is therefore called an “attractor” of the relationship. However, if small changes in T and C always push the therapist and client further away from the critical point, then it is unstable and does not represent a final steady state. We will see in the computer simulations below that the critical point at the intersection of the influence functions in the upper right of Figure 1(c) is a stable steady state, while the critical point at the intersection of the influence functions in the lower left of Figure 1(c) is unstable. This has important (and potentially useful) implications for the therapeutic relationship.

3. We used computer software (MATLAB ODE113) to calculate how the values of T and C, for the behavior state of the therapist and client respectively, in equations (1a) and (1b), change in time when they start from many different initial conditions. We then plot these trajectories on the phase-space. These simulations create a picture of how the therapeutic relationship might change given different initial conditions of the client and the therapist. These plots allow for a visual representation of the dynamics within the system, and form the basis for predictions within the model.

4. In fact, if you overlay both Figure 1(c) and Figure 2, the intersection of the client and therapist influence function exactly correlates with the steady states in the phase space of Figure 2.