Evidence for roles of the activating function in electric stimulation

Modeling extracellular electrical neural stimulation: from basic understanding to MEA-based applications

Extracellular electrical stimulation of neural networks has been widely used empirically for decades with individual electrodes. Since recently, microtechnology provides advanced systems with high-density microelectrode arrays (MEAs). Taking the most of these devices for fundamental goals or developing neural prosthesis requires a good knowledge of the mechanisms underlying electrical stimulation. Here, we review modeling approaches used to determine (1) the electric potential field created by a stimulation and (2) the response of an excitable cell to an applied field. Computation of the potential field requires solving the Poisson equation. While this can be performed analytically in simple electrode-neuron configurations, numerical models are required for realistic geometries. In these models, special care must be taken to model the potential drop at the electrode/tissue interface using appropriate boundary conditions. The neural response to the field can then be calculated using compartmentalized cell models, by solving a cable equation, the source term of which (called activating function) is proportional to the second derivative of the extracellular field along the neural arborization. Analytical and numerical solutions to this equation are first presented. Then, we discuss the use of approximated solutions to intuitively predict the neuronal response: Either the "activating function" or the "mirror estimate", depending on the pulse duration and the cell space constant. Finally, we address the design of optimal electrode configurations allowing the selective activation of neurons near each stimulation site. This can be achieved using either multipolar configurations, or the "ground surface" configuration, which can be easily integrated in high-density MEAs. Overall, models highlighting the mechanisms of electrical microstimulation and improving stimulating devices should help understanding the influence of extracellular fields on neural elements and developing optimized neural prostheses for rehabilitation.

Stimulatory current at the edge of an inactive conductor in an electric field: role of nonlinear interfacial current-voltage relationship

Cardiac electric field stimulation is critical for the mechanism of defibrillation. The presence of certain inactive epicardial conductors in the field during defibrillation can decrease the defibrillation threshold. We hypothesized this decrease is due to stimulatory effects of current across the interface between the inactive conductor and the heart during field stimulation. To examine this current and its possible stimulatory effects, we imaged transmittance of indium-tin-oxide (ITO) conductors, tested for indium with X-ray diffraction, created a computer model containing realistic ITO interfacial properties, and optically mapped excitation of rabbit heart during electric field stimulation in the presence of an ITO conductor. Reduction of indium decreased transmittance at the edge facing the anodal shock electrode when trans-interfacial voltage exceeded standard reduction potential. The interfacial current-voltage relationship was nonlinear, producing larger conductances at higher currents. This nonlinearity concentrated the interfacial current near edges in images and in a computer model. The edge current was stimulatory, producing early postshock excitation of rabbit ventricles. Thus, darkening of ITO indicates interfacial current by indium reduction. Interfacial nonlinearity concentrates current near the edge where it can excite the heart. Stimulatory current at edges may account for the reported decrease in defibrillation threshold by inactive conductors.

The "mirror" estimate: an intuitive predictor of membrane polarization during extracellular stimulation

Achieving controlled extracellular microstimulation of the central nervous system requires understanding the membrane response of a neuron to an applied electric field. The "activating function" has been proposed as an intuitive predictor of membrane polarization during stimulation, but subsequent literature raised several limitations of this estimate. In this study, we show that, depending on the space constant lambda, the steady-state solution to the passive cable equation is theoretically well approximated by either the activating function when lambda is small, or the "mirror" image of the extracellular potential when lambda is large. Using simulations, we then explore the respective domain of both estimates as a function of lambda, stimulus duration, fiber length, and electrode-fiber distance. For realistic lambda (>50-100 microm), the mirror estimate is the best predictor for either long electrode-fiber distances or short distances (<20-30 microm) when stimulus durations exceed a few tens of microseconds. For intermediate distances, the mirror estimate is all the more valid that the stimulus duration is long and the fiber is short. We also illustrate that this estimate correctly predicts the steady-state membrane polarization of complex central nervous system arborizations. In conclusion, the mirror estimate can often be preferred to the activating function to intuitively predict membrane polarization during extracellular stimulation.

Use of translucent indium tin oxide to measure stimulatory effects of a passive conductor during field stimulation of rabbit hearts

Biomathematical models and experiments have indicated that passive extracellular conductors influence field stimulation. Because metallic conductors prevent optical mapping under the conductor, we have evaluated a passive translucent indium tin oxide (ITO) thin-film conductor to allow mapping of transmembrane potential (V(m)) and stimulatory current under the conductor. A 1-cm ITO disk was patterned photolithographically and positioned between 0.3-cm(2) mesh shock electrodes on the ventricular epicardium of isolated perfused rabbit hearts stained with 4-{2-[6-(dibutylamino)-2-naphthylenal]ethenyl}-1-(3-sulfopropyl)-, hydroxide, inner salt (di-4-ANEPPS). For a 1-A, 10-ms shock during the action potential plateau, optical maps from fluorescence collected using emission ratiometry (excitation at 488 nm and emissions at 510-570 and >590 nm) indicated that the disk altered V(m) by as much as the height of an action potential. DeltaV(m) became more positive near the edge of the disk, where the ITO conductance gradient was parallel to applied current, and more negative near the opposite edge, where the gradient was not parallel to current. For diastolic shocks, the disk expedited membrane excitation at the sites of positive DeltaV(m) in the heart and in a cardiac model with realistic ITO disk surface and interfacial conductances. Optical maps of ITO transmittance and the model indicated that the disk introduced anodal and cathodal stimulatory current at opposite edges of the disk. Thus ITO allows study of the stimulatory effects of a passive conductor in an electric field.

Comments on "accuracy limitations of chronaxie values"

Some additions/corrections are offered to Geddes, 2004. Stimulation is initiated by the second spatial derivative of the voltage along the nerve (activating function) rather than current density. Chronaxie values change with distance from the electrode. Anodic stimulation can excite via anodic break excitation, or via virtual cathodes around the anode.

Field Stimulation of 2-D Sheets of Excitable Tissue

This paper develops equations for the transmembrane potentials (Vm) that occur in two-dimensional (2-D) sheets of tissue in response to field stimulation from an electrode near but not on the surface of the tissue. Comparison of results with those for one dimension shows that an additional term is present in the 2-D equations that influences the evolution of Vm in the interval between the end of the stimulus and the active propagation that may follow. The results provide an analytical framework for understanding Vm in response to field stimulation in two dimensions, both during the tissue's critical linear phase and thereafter.