A General Framework for Closed Loop Negative Feedback Multivariable Physiological Control Systems: Existence, Uniqueness, and Stability of Homeostatic Equilibrium Points

The study of homeostatic equilibrium is a key concern in several fields, from physiology and biology to medicine and biomedical engineering. Control theory approaches can provide effective strategies to model physiological control systems, helping in understanding the dynamics of bio- and physio-logical regulation processes. However, the intrinsic complexity of living systems makes it difficult to identify unified biomodels that can represent a wide variety of physiological systems. In this context, the present work proposes a general framework to model the dynamics and describe the behavior of a wide class of multivariable physiological control systems, from the molecular to the whole-organ scale. The framework adopts a structure based on a closed-loop topology taking into account multiple inputs and outputs and with the negative feedback action intrinsically embedded within the model. The development of such a general model has at least three important repercussions: the first concerns the possibility of better understanding the basic mechanisms common to many physiological systems; the second is to develop a common theoretical framework to enable effective approaches to the analysis and design of synthetic biological control systems; finally, the investigation of the structural properties of the model in a general context, allows a guided and simplified application to specific cases. To this regard, in this paper, the existence, possible uniqueness and stability properties of the homeostatic equilibrium points of the general model are investigated; the theoretical framework is then illustrated through two real-world case-studies: (i) the PI3K/AKT/mTOR pathway nonlinear dynamics, a critical regulator of cellular growth, proliferation, and survival; (ii) the control mechanism of the neuromuscular stretch reflex, among the prime triggers implicated in postural control. Results proved the capability of the proposed framework to capture the intricate dynamics of multivariable physiological systems at different scales, highlighting the existence of asymptotically stable homeostatic equilibrium and allowing the study of the impact of transmission delays on the system’s stability. At the best of authors’ knowledge, following the paper Ponsiglione et al. (2023) where monovariable systems where dealt with, the proposed methodology is the first attempt to represent and investigate homeostasis from the molecular up to systemic level by exploiting a unified multivariable biomodeling architecture, which makes it a novel approach to understanding homeostatic control from a broader perspective.