Analyzing multidimensional formal dynamic models in psychology: A tutorial using graphical tools.

Formal theories translate verbal theories into a mathematical representation, such as a coupled differential equation or other dynamical systems, intending to strengthen the deductive power of (clinical) theories and to formulate testable and novel hypotheses. Work in clinical formal theories mainly relies on simulations, which is an intuitive method for evaluating overall model performance, but may fall short of establishing a precise link between the mathematical properties of the model and the dynamic properties of its outcome. Moreover, when the model's outcome contradicts clinical observations, it is unclear where the discrepancy lies and how to improve the model. In this article, we introduce formal mathematical techniques for graphical model analysis, including phase plane analysis, which allows identifying a system's stable and unstable equilibria, and bifurcation analysis, a framework to delineate parameter regimes corresponding to qualitatively different dynamical outcomes for a model. Using two formal dynamic models in psychology (one for panic disorder and one for suicidal ideation), we illustrate those methods through an easy-to-use R package, deBif, with a graphical user interface. These examples demonstrate the importance of using graphical tools to investigate the hypothesized mechanisms of psychological systems.