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 explains 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 the membrane interface of the axonal cellular system can be mathematically described by a nonlinear system with several small parameters. This allows for the rigorous derivation of an accurate yet simpler nonlinear model following the formal small parameter expansion. The resulting action potential model exhibits a smooth, continuous transition from the linear wave oscillatory regime to the nonlinear 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 overcomes the limitations of the Hodgkin-Huxley model, such as the inability to explain extracellular spiking, efficient brain synchronization, saltatory conduction along myelinated axons, and a variety of other observed coherent macroscopic brain electrical phenomena. We also show that the standard cable axon theory can be recovered by our approach, using the very crude 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.