Modelling extracellular electrical stimulation: III. Derivation and interpretation of neural tissue equations

OBJECTIVE A common approach in modelling extracellular electrical stimulation is to represent neural tissue by a volume conductor when calculating the activating function as the driving term in a cable equation for the membrane potential. This approach ignores the cellular composition of tissue, including the neurites and their combined effect on the extracellular potential. This has a number of undesirable consequences. First, the two natural and equally valid choices of boundary conditions for the cable equation (i.e. using either voltage or current) lead to two mutually inconsistent predictions of the membrane potential. Second, the spatio-temporal distribution of the extracellular potential can be strongly affected by the combined cellular composition of the tissue. In this paper, we develop a mean field volume conductor theory to overcome these shortcomings of available models. APPROACH This method connects the microscopic properties of the constituent fibres to the macroscopic electrical properties of the tissue by introducing an admittivity kernel for the neural tissue that is non-local, non-instantaneous and anisotropic. This generalizes the usual tissue conductivity. A class of bidomain models that is mathematically equivalent to this class of self-consistent volume conductor models is also presented. The bidomain models are computationally convenient for simulating the activation map of neural tissue using numerical methods such as finite element analysis. MAIN RESULTS The theory is first developed for tissue composed of identical, parallel fibres and then extended to general neural tissues composed of mixtures of neurites with different and arbitrary orientations, arrangements and properties. Equations describing the extracellular and membrane potential for the longitudinal and transverse modes of stimulation are derived. SIGNIFICANCE The theory complements our earlier work, which developed extensions to cable theory for the micro-scale equations of neural stimulation that apply to individual fibres. The modelling framework provides a number of advantages over other approaches currently adopted in the literature and, therefore, can be used to accurately estimate the membrane potential generated by extracellular electrical stimulation.