Separating Long-Term Equilibrium Adaptation from Short-Term Self-Regulation Dynamics Using Latent Differential Equations

Self-regulating systems change along different timescales. Within a given week, a depressed person's affect might oscillate around a low equilibrium point. However, when the timeframe is expanded to capture the year during which they onboarded antidepressant medication, their equilibrium and oscillatory patterns might reorganize around a higher affective point. To simultaneously account for the meaningful change processes that happen at different time scales in complex self-regulatory systems, we propose a single model that combines a second-order linear differential equation for short timescale regulation and a first-order linear differential equation for long timescale adaptation of equilibrium. This model allows for individual-level moderation of short-timescale model parameters. The model is tested in a simulation study which shows that, surprisingly, the short and long timescales can fully overlap and the model still converges to the reasonable estimates. Finally, an application of this model to self-regulation of emotional well-being in recent widows is presented and discussed.