A Nonlinear Dynamic Systems Model of Psychotherapy: First Steps Toward Validation and the Role of External Input

Mathematical modeling and computer simulations are important means to understand the mechanisms of psychotherapy. The challenge is to design models which not only predict outcome, but simulate the nonlinear trajectories of change. Another challenge is to validate them with empirical data. We proposed a model on change dynamics which integrates five variables (order parameters) (therapeutic progress or success, motivation for change, problem severity, emotions, and insight) and four control parameters (capacity to enter a trustful cooperation and working alliance, cognitive competencies and mindfulness, hopefulness, behavioral resources). The control parameters modulate the nonlinear functions interrelating the variables. The evolution dynamics of the system is determined by a set of nine nonlinear difference equations, one for each variable and parameter. Here we outline how the model can be tested and validated by empirical time series data of the variables, by time series of the therapeutic alliance, and by assessing the input onto the system as it is perceived by the client. The parameters are measured by questionnaires at the beginning and at the end of the treatment. A key element of the validation algorithm is the adjustment of the parameter values as assessed by the questionnaires to model-specific parameter values by which the dynamics can be reproduced (calibration). The validation steps are illustrated by the data of a client who used an internet-based tool for high-frequency therapy monitoring (daily self-ratings). Especially after applying the input vector (interventions as experienced by the client) the similarity between the empirical and the model dynamics becomes evident. The averaged correlation between the empirical and the simulated dynamics across all variables is .41, after applying a short averaging mean window and eliminating an initial transient period, it is .62, varying between .47 and .81, depending on the variable. The discussion opens perspectives on the combination of mathematical modeling with real-time monitoring in order to realize data-driven simulations for short-term predictions and to estimate the effects of interventions before real interventions are applied.