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

Several health and disease brain processes rely on neuronal synchronization. The functions and synchronization of neurons, mechanical behavior of neurons, and electromechanical interactions of neurons with their shared microenvironment are intricately connected. Notably, neuronal functions are very sensitive to neuronal volumetric variations. Interactions of neurons with the extracellular fluid govern the Regulatory Volume Decrease or Increase (RVD/I) mechanisms preserving neuronal volumes. Published mathematical models focus either on function–volume dynamics of one neuron or on functional synchronization of neurons. These models ignore neuronal electromechanics. This paper presents a novel, proof-of-concept, mathematical model linking two functionally synchronized and volumetrically coupled mathematical neurons belonging to nearby synaptic paths. The electromechanics of one mathematical neuron is described by a generalized Hodgkin–Huxley model in which ion channels behave mechanically as fractional Maxwell linear viscoelastic materials. The mathematical neurons share the same microenvironment, electrical properties, initial conditions and shapes, and external current. Their ion channels, however, have distinct viscoelastic memories. This is a one-way coupled model since electric activities are assumed to be independent of volume changes. An imposed master–slave synchronization is proposed. The coupled volumetric dynamics is described by modified Lotka–Volterra equations with voltage-dependent growth rates and Michaelis–Menten-like terms modeling active RVD/I mechanisms. Computer simulations show volumetric variations resembling the patterns of associated action potentials and a slightly delayed volumetric synchronization triggered by the functional synchronization. The findings suggest that, depending on changes in resource availability and functional–structural integrity, these mathematical neurons may experience uncoupled, mutualism, competition, and predator–prey regimes.