Ordinarily, functional complexity in neural networks is held as stemming from the interaction of large numbers of functionally simple neuron-like processing elements. This paper focuses on complexity on the single neuron level as elucidate by a nonlinear dynamical systems approach to the analysis of the integrate-and-fire model neuron. The resulting dynamics, described by an iterated phase-transition map (PTM), suggest that a wide-range of complex firing modalities can be produced by a dendritic neuron when its dendrites are subjected to correlated arriving spike trains that give rise to periodic activation potential of its excitable membrane. The dynamical approach leads to the bifurcating neuron concept and model which combines functional complexity, in its spiking behavior, approaching that of the biological neuron with structural simplicity and power efficiency. The bifurcating model neuron is well suited for the modeling, simulation, and construction of a new generation of artificial neural networks in which synchronicity, bifurcation, and chaos can play a role in realizing higher-level functions. The theory and characterization of a photonic embodiment of the bifurcating neuron are discussed and it is proposed that bifurcating neuron dynamics offer a plausible basis for the mechanism subserving transient correlations in local field potentials observed at different widely separated cortical areas of Cat and Monkey.