A varying-radius cable equation for the modelling of impulse propagation in excitable fibres

In this study, we present a varying-radius cable equation for nerve fibres taking into account the varying diameter along the neuronal segments. Finite element neuronal models utilising the classical (fixed-radius) and varying-radius cable formulations were compared using simple and realistic morphologies under intra- and extracellular electrical stimulation protocols. We found that the use of the classical cable equation to model intracellular neural electrical stimulation exhibited an error of 17% in a passive resistive cable model with abrupt change in radius from 1 to 2 μm, when compared to the known analytical solution and varying-radius cable formulation. This error was observed to increase substantially using more realistic neuron morphologies and branching structures. In the case of extracellular stimulation however, the difference between the classical and varying-radius formulations was less pronounced, but we expect this difference will increase under more complex stimulation paradigms such as high-frequency stimulation. We conclude that for computational neuroscience applications, it is essential to use the varying-radius cable equation for accurate prediction of neuronal responses under electrical stimulation.