A Fractional Hodgkin-Huxley-Lotka-Volterra Model of Neuronal Synchronized Electromechanics

Introduction. Neuronal synchronization is critical to various brain processes, such as memory formation and motor control, and to brain disorders such as epilepsy. Neuronal functions and synchronization are achieved through the dynamics of action potentials and chemical neurotransmitters, which are intricately coupled with the mechanical responses of neurons and depend on neuronal structure and interactions with their shared environment. Mathematically, the propagation of action potentials can be described by the Hodgkin-Huxley model, and neuronal synchronization may be modelled using Hodgkin-Huxley models coupled by prescribed controller-based synchronizers. These mathematical models depict the transport of specific ions across neuronal membranes, but ignore neuronal mechanics. In particular, various studies have shown that neuronal functions are highly sensitive to changes in neuronal volume. The homeostatic mechanisms that conserve neuronal volume are facilitated by the mechanical interactions of neurons with the cerebrospinal fluid surrounding them. This work aims to develop a mathematical model that links the functions and volumes of synchronized neurons.

Methods. A mathematical model of two adjacent neurons, functionally synchronized and volumetrically coupled, is proposed. Neuronal functions are described by a fractional Hodgkin-Huxley model where the ion channels are assumed to behave mechanically as variable-order fractional Maxwell linear viscoelastic materials. Neuronal synchronization is characterized by a synchronization time, and the coupled volumetric dynamics are modeled by a modified coupled Lotka-Volterra model in which the rates depend on action potentials, and regulatory volumetric processes are represented as Michaelis–Menten-like terms.

Results. Computer simulations in MATLAB show oscillatory volumetric dynamics similar to those of action potentials, and the return of volumes to their initial values when regulatory mechanisms are present.

Conclusion. The proposed model allows the study of neuronal function-volume dynamics of brain health and disease that may help with the design of better therapies for various disorders.