The dynamics of quantifiable homeostasis. II. Characterization of linear processes

This paper deals with the representation of physiological homeostatic processes by mathematical models that lead to lag-differential equations. It deals exclusively with feedback processes that are linear. It is surmised that in the response to standard perturbations from equilibrium, in many systems the strength at which the restorative force works (denoted by the constant b) is a genetic characteristic, while the average or homing value may or may not be. The class of equations with linear feedback is solved. The solution provides predictions against which experimental data may be tested. Three types of displacement from the homing value are recognized: drifts, catastrophes, and saccades. Several methods of evaluating the mathematical functions are described. They are forward recursion, backward recursion, numerical integration, steady-state solutions, and inversion of the Heaviside expansion of the Laplace transform. Conditions for homeostatic processes that result in damped nonoscillating responses, damped oscillating responses, stable oscillations, and uncontrolled oscillations are derived. The concept of an associated penalty that depends on the degree of displacement from the homing value is discussed. Penalties are evaluated by numerical integration for linear, quadratic, and cubic costs. Costs for polynomial functions may be found by linear weighting. The optimum strength of response (ie, the one that produces the minimum cost) is calculated for each of these cost functions.