Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations

Abstract

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

Internal inconsistencies in models of electrical stimulation in neural tissue

“…Mathematically, consistency in the calculated membrane potential between current density and voltage boundary conditions is obtained if and only if the extracellular voltage is related to the extracellular current density by the transimpedance of the neurite plus its thin extracellular space [9]. From the transimpedance expression, it can be seen that the conductivity of the required volume conductor is not only anisotropic but also has non-trivial spatial and temporal properties.…”

“…A consistent membrane potential results if and only if the extracellular voltage is related to the extracellular current density (normal to the membrane surface) by the transimpedance of the neurite (see Eqs. (33) and (34) in [9]). These relations do not hold for general volume conductors in the absence of embedded neurites.…”

“…The equations describing the subthreshold membrane potential under voltage and current density boundary conditions are given below and were derived in [9], [10]. They involve two modes of stimulation: a longitudinal mode, which is the conventional mode described by a classical cable equation, and a transverse mode, which is often neglected and is described by an ordinary differential equation in time.…”

“…They involve two modes of stimulation: a longitudinal mode, which is the conventional mode described by a classical cable equation, and a transverse mode, which is often neglected and is described by an ordinary differential equation in time. These modes of stimulation are illustrated schematically in Figure 2 (see [9] for a full discussion). It should be emphasised that our basic conclusions are unaltered if we consider only the conventional longitudinal mode.…”

“…We do this by considering the simple example of a cylindrical neurite in a homogeneous volume conductor with stimulation by a point source electrode. The two stage model described above is followed: Stage 1 uses the standard expression for the electrical potential in a homogeneous volume conductor; Stage 2 uses the equations derived in [9], [10] for subthreshold membrane potential under the two types of boundary conditions.…”