Join virtual roomNeurons and other cells use electrical signals for intra and intercellular communication. Accurate mathematical models have been developed to describe the spatial-temporal dynamics of their voltage in response to input current. Best known is the cable equation, which describes voltage propagation along the length of a passive cable in response to current injection [1]. However, this model is not appropriate for cells with geometries other than the cylinder. Here we study the flow of electrical currents in cells with a spherical geometry in which only a thin shell close to the surface conducts. Such geometries arise for instance in white adipocytes (fat cells), which are cells consisting of an insulating lipid droplet core surrounded by a thin conductive cytoplasm shell, and might also be relevant for some types of spherically shaped peripheral neurons.
First, we construct a circuit model based on the nature of the passive membrane, and derive the equivalent of the cable equation for spherical geometries. We derive the steady-state solution analytically and show that the shape of the voltage profile depends on a single parameter that describes its electrotonic compactness. Furthermore, we show that, in contrast to the cable equation, the voltage profile across the cell is sensitive to the electrode geometry.
Next, we numerically explore the time-dependent solution to step input currents. In particular, we find that the charging and discharging are much faster than one would expect from the membrane time constant, which is important when one aims to extract fundamental membrane properties from experimental recordings.
Finally, we consider voltage-clamping experiments, often used to measure input current of cells and examine the distortions arising from imperfect space- clamp.
In conclusion, our study yields an equivalent of the cable equation for spherical geometries, which can facilitate further investigations of the electrical signals on cells with spherical structures.
[1] Koch, Christof. Linear Cable Theory. In _Biophysics of Computation: Information Processing in Single Neurons_. Oxford UP. Pages 25–47. 2004.
