Dynamic Homeostasis in Relaxation and Bursting Oscillations

Abstract. Homeostasis traditionally describes the regulation of a stable internal state despite external perturbations, but many biological systems are intrinsically dynamic, exhibiting oscillations across multiple timescales. In this work, we introduce a theoretical framework for dynamic homeostasis (or homeodynamics) in systems with fast-slow dynamics, showing that the temporal average of a slow variable can remain robust under parameter changes, even when fast variables oscillate wildly. We first apply this to the FitzHugh–Nagumo model to illustrate relaxation oscillations, and then extend to models of electrical bursting and calcium oscillations in pancreatic [Formula: see text]-cells. In bursting models with multiple slow components, only the variable driving the current bursting regime demonstrates homeostatic invariance. Our results highlight that homeostasis can manifest in the averaged behavior of slow variables, rather than in fast oscillatory dynamics, suggesting a new way to understand regulatory stability in excitable biological systems. Relevance to Life Sciences. Homeostasis, broadly speaking, refers to the maintenance of a stable internal state when faced with external or internal stimuli. Failure to manage these regulatory processes can lead to different diseases or death. Most physiologists and cell biologists around the world agree that homeostasis is a fundamental tenet of their disciplines. Nevertheless, a precise definition of homeostasis is hard to come by. Often times, homeostasis is simply defined as “you know it when you see it.” Mathematical Content. Mathematical treatments of homeostasis involve studying equilibria of dynamical systems that are relatively invariant with respect to parameters. However, physiological processes are rarely static and often involve dynamic processes such as oscillations. In such dynamic environments, quantities such as average values may be relatively invariant with respect to parameters. This has been referred to as “homeodynamics.” We present a general framework for homeodynamics involving systems with two or more timescales that elicits homeostasis in the temporal average of a species. The key point is that homeostasis manifests when measuring the slow variable responsible for driving oscillations and is not apparent in the fast variables. We demonstrate this in the FitzHugh–Nagumo model for relaxation oscillations and then in two models for electrical bursting activity and calcium oscillations in pancreatic [Formula: see text]-cells. One of these models has multiple slow variables, each driving the bursting oscillations in different parameter regimes, but homeodynamics is only present in the variable currently engaged in this role.