Strength–Duration Relationship for Extracellular Neural Stimulation: Numerical and Analytical Models

Abstract

The strength–duration relationship for extracellular stimulation is often assumed to be similar to the classical intracellular stimulation model, with a slope asymptotically approaching 1/τ at pulse durations shorter than chronaxy. We modeled extracellular neural stimulation numerically and analytically for several cell shapes and types of active membrane properties. The strength–duration relationship was found to differ significantly from classical intracellular models. At pulse durations between 4 μs and 5 ms stimulation is dominated by sodium channels, with a slope of −0.72 in log-log coordinates for the Hodgkin–Huxley ion channel model. At shorter durations potassium channels dominate and slope decreases to −0.13. Therefore the charge per phase is decreasing with decreasing stimulus duration. With pulses shorter than cell polarization time (∼0.1–1 μs), stimulation is dominated by polarization dynamics with a classical −1 slope and the charge per phase becomes constant. It is demonstrated that extracellular stimulation can have not only lower but also upper thresholds and may be impossible below certain pulse durations. In some regimes the extracellular current can hyperpolarize cells, suppressing rather than stimulating spiking behavior. Thresholds for burst stimuli can be either higher or lower than that of a single pulse, depending on pulse duration. The modeled thresholds were found to be comparable to published experimental data. Electroporation thresholds, which limit the range of safe stimulation, were found to exceed stimulation thresholds by about two orders of magnitude. These results provide a biophysical basis for understanding stimulation dynamics and guidance for optimizing the neural stimulation efficacy and safety.