PHASE TRANSITION FOR THE MCKEAN-VLASOV EQUATION OF WEAKLY COUPLED HODGKIN-HUXLEY OSCILLATORS

A phase-reduced model for weakly coupled Hodgkin-Huxley neurons, in particular its thermodynamic limit - the McKean-Vlasov equation - is considered. Synaptic interactions require that the phase interaction function consists of at least two competing Fourier modes representing an attractive and a repulsive contribution. The stationary equation permits a finite-dimensional reduction in terms of generalized modified Bessel functions. For the case of two competing contributions that are in phase, it is shown that the system undergoes a single continuous phase transition as in the case of one-mode interaction; by contrast, two attractive contributions permit multiple continuous and discontinuous phase transitions. When the two contributions are out of phase, the picture that emerges stands in sharp contrast to the discrete model for a large number of globally coupled neurons, for which the existing numerical results show a complex dynamical landscape featuring a variety of cluster states, as well as periodic phase motion. For the thermodynamic limit with an asymmetric two-mode interaction, it will be shown that no coherent steady states are possible for any interaction strength implying a transition from the incoherent state to irregular or chaotic phase motion as the former loses stability.