Coquaternion Transformations in Nonequilibrium Dynamics of Biologic Systems

Imaginary coquaternions cℍ can be represented by matrices of negative feedback N−, positive feedback P+, and reciprocal links R±. An added environmental element E± endows biologic systems with the structure of cℍ module. Although cℍ representation links base patterns with the geometric structure of the pseudo-Euclidean R24 space, unknown physiologic aspects of relationships between base elements may add new functional features to the structure of a functional module. Another question is whether achieving and remaining in the equilibrium state provides stability for a biologic system. Considering the property of a biologic system to return deviated conditions to the equilibrium, the system of ordinary differential equations describing the behavior of a mechanical pendulum was modified and used as a basic tool to find the answers. The results obtained show that in evolving systems, the regulatory patterns are organized in a sequence NPRN of base elements, allowing the system to perform a high amount of energy-consuming functions. In order to keep dissipating energy at the same level, the system bifurcates and finalizes its regulatory cycle in R± by splitting P+ after which the next cycle may begin. Obtained flows are continuous pathways that do not interfere with equilibrium states, thus providing a homeostasis mechanism with nonequilibrium dynamics. Functional transformations reflect changes in the geometry and metric index of the coquaternion. Related coquaternion dynamics show the transformation of a hyperbolic hyperboloid into the closed surface, which is the fusion of the portions of the hyperbolic hyperboloid and two spheres.