Novel cable equation model for myelinated nerve fiber

The cable equation is capable of handling analytically linear and non-linear non-myelinated axon models. Unfortunately, it hasn't been extended yet analytically for myelinated axon model, which is crucially important for applications involving vertebrates. Herein, the well known non-myelinated axon model is extended analytically by incorporating periodic membrane conductivity. The classical cable equation is thereby modified into a linear second order ordinary differential with periodic coefficient, known as Hill's equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent non-myelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Furthermore, the conductivity of the equivalent axon is the precise average of the periodic myelinated axon conductivity. Hill's formulation is further reduced into Mathieu's equation for the specific choice of sinusoidal conductivity, thereby resulting in explicit closed form expression for the transmembrane potential.