Transmembrane potential induced in a spherical cell model under low-frequency magnetic stimulation

Time-varying magnetic fields can induce electric fields in the neuronal tissue, a phenomenon that has been recently explored in clinical applications such as peripheral nerve stimulation and transcranial magnetic stimulation. Although the transmembrane potential induced during direct electric stimulation has already been the subject of a number of theoretical studies, an analytical solution for the magnetically induced transmembrane potential change is still unavailable. In addition, although several studies have analyzed the impact of stimulation parameters, including stimulation intensity and frequency, as well as coil design and position, on the amount of tissue polarization, the effects of tissue non-homogeneity on cell polarization have not been fully elucidated. In this study, we have derived an analytical expression for the transmembrane potential induced by a low-frequency magnetic field in a spherical neuronal structure. This model is representative of a spherical cell body or any neuronal structure of a similar shape. The model cell is located in an extracellular medium and possesses a low-conductive membrane and an internal cytoplasm. These three regions represent the basic tissue non-homogeneity of a neuron at a microscopic level. The sensitivity of the induced transmembrane potential to the coil position and to the geometrical and electrical parameters of the model structure was studied in a broad physiologically relevant range. Our results demonstrate that the structure is regionally polarized, with the pattern of polarization depending on the relative positioning between the model cell and the stimulation coil. In addition, both the geometrical and electrical parameters of the structure affect the amount of polarization. These results may be generalized to other neuronal tissues that possess similar non-homogenous properties, but different shapes, such as an axon. Our results support the idea that aside from coil design and position, tissue non-homogeneity could play an important role in determining the effects of magnetic stimulation.