  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.
