The wave nature of the action potential

An alternative to the standard Hodgkin-Huxley model for the action potential in axons is presented. It is based on our recently developed theory of electric field wave propagation in anisotropic and inhomogeneous brain tissues, which has been shown to explain a broad range of observed coherent synchronous brain electrical processes. We demonstrate that this theory also explains the spiking behavior of single neurons, thereby bridging the gap between the fundamental element of brain electrical activity-the neuron-and large-scale coherent synchronous electrical activity. We demonstrate that our recently developed theory of electric field wave propagation in anisotropic and inhomogeneous brain tissues, which has been shown to explain a broad range of observed coherent synchronous brain electrical processes, also applies to the spiking behavior of single neurons, thus bridging the gap between the fundamental element of brain electrical activity (the neuron) and large-scale coherent synchronous electrical activity. Our analysis indicates that a non-linear system with several small parameters can mathematically describe the membrane interface of the axonal cellular system. This enables the rigorous derivation of an accurate yet simpler non-linear model through the formal small-parameter expansion. The resulting action potential model exhibits a smooth, continuous transition from the linear wave oscillatory regime to the non-linear spiking regime, as well as a critical transition to a non-oscillatory regime. These transitions occur with changes in the criticality parameter and include several different bifurcation types, representative of the various experimentally detected neuron types. This new theory addresses the limitations of the Hodgkin-Huxley model, including its inability to explain extracellular spiking, efficient brain synchronization, saltatory conduction along myelinated axons, and various other observed coherent macroscopic brain electrical phenomena. We also demonstrate that our approach recovers the standard cable axon theory, utilizing the relatively simple assumptions of piece-wise homogeneity and isotropy. However, the diffusion process described by the cable equation is not capable of supporting action potential propagation across a wide range of experimentally reported axon parameters.