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

Abstract

Objective: 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 Three Approaches to Psychotherapy with a Female Client: The Next Generation and Three Approaches to Psychotherapy with a Male Client: The Next Generation 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. Method: A second-by-second observational coding system of the emotional exchanges of the therapists and clients was used as the data for the equations. Results: 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. Conclusions: DS modelling can provide researchers and clinicians with a powerful tool to investigate the complex phenomenon that is psychotherapy.

Keywords:

• : alliance
• cognitive behavior therapy
• emotion in therapy
• process research
• psychoanalytic/psychodynamic therapy
• experiential/existential/humanistic psychotherapy
• statistical methodology
• technology in psychotherapy research & training
• philosophical/theoretical issues in therapy research

Supplemental data

Supplemental data for this article can be accessed https://doi.org/10.1080/10503307.2021.1921303

Notes

1 These two parameters are combined to create uninfluenced steady state combining the inertia parameter (resistance to change), and the initial state (the individual’s dispositional characteristics) using the formula: b₁/m.

2 For specific information on the derivation of parameters, please see Gottman et al. (2002)

3 In contrast, for example, with Li and Kivlighan (2020) and Li et al. (2020), who appeared to use constant values in the description of their influence functions that seemed to mimic Gottman et al.’s ojive influence functions.

4 For a graphic demonstration of the timing of the repair switch (early vs. late), and the strength of the repair on influence functions (strong vs. weak) and phase-space portraits, please see Strawinska-Zanko et al. (2018).

5 Note, that there is also the same formula for a repair function for client, R_C(T), which will not be detailed as it would be repetitive

6 However, it is noteworthy that unlike Safran and others’ work, this term does not necessarily identify specific incidents of therapeutic ruptures in the relationship at this time, but rather what would likely happen in the event of a rupture, where in the negative affect the repair would happen, and how strong the repair would be for both the therapist and the client. Using this parameter to study actual ruptures and repairs in session it is an area for future exploration

7 This model is a deterministic model (J. Gottman, Personal Communication, January 3, 2021), meaning that given the inputs all elements of the system can be defined, including its final state (attractors). As a result it does not incorporate stochastic elements (that can fluctuate randomly, often modeled as “noise” or an error term, see Tschacher and Haken (2020) for a more thorough description of this concept). A critique of this approach is that the model created is not “real world” since there are no error terms to correct for variance not accounted for in the model. However, this is a mathematical model, not a statistical one (L. Liebovitch, Personal Communication, January 4, 2021), and the variables in the model are accounted for by it. Its validity and applicability to “real world” scenarios are determined when the model is compared to actual data (see below).

8 We use MATLAB, though any ODE solver in R, Python or other software would get similar results.

9 More detailed information on the coding procedure, including definitions of the SPAFF codes, are included in the supplemental materials.

10 We used the original model with repair and damping and the combination model to derive the necessary parameters.

11 An analysis of the influence functions for all six sessions is included in the supplemental material.

12 The phase-space portraits can also be rendered in a three-dimensional phase-space portraits of the dynamic nonlinear models using time steps as the z-axis. These are available upon request.

13 While there is a weak attractor in the positive-positive quadrant for Judith Beck’s session with Kevin, the attractor in the therapist positive/client negative quadrant is more pronounced and likely the better outcome for the session.