Fractional calculus in neuronal electromechanics

Traumatic brain injuries (TBI) are among the leading causes of death and permanent disability worldwide. Recent experimental observations suggest that damage in brain tissue involves complex local as well as nonlocal chemomechanical interactions that happen on multiple spatiotemporal scales. Biomechanical models of TBI existing in the literature do not incorporate either electrochemical or multiscaling features. Given that neurons are the brain cells responsible for electrochemical signaling on multiplexed temporal scales we propose a novel mathematical model of neuronal electromechanics that uses a constrained Lagrangian formulation and Hamilton's principle to couple Newton's law of motion for a linear viscoelastic Kelvin-Voigt solid-state neuron and the classic Hodgkin-Huxley equations of the electronic neuron. We will use fractional order derivatives of variable order to model multiple temporal scales. Numerical simulations of possible damage dynamics in neurons due to mechanical trauma will be presented and discussed.