Synchronization of embedded FitzHugh-Nagumo motor-neurons through phase reduction theory

Synchronization is one of the main studied behaviors in networks of oscillators. Particular synchronization manifolds are desired when designing Central Pattern Generators as a locomotion control strategy where, depending on the desired gait to implement, particular phase shifts must be imposed among the oscillators. Phase reduction theory provides a simple yet effective strategy to reduce the complexity of the problem to its essential elements. Thanks to the approximations of the piecewise linear (PWL) nonlinearity and slow-fast behavior of the FitzHugh-Nagumo (FHN) neural oscillator, harnessing singular perturbation analysis is possible to derive a closed-form solution of both the asymptotic phase and phase sensitivity function of the system, which is rarely possible. Using this approach, it is possible to relate the complex network of FHN neurons to a simple Kuramoto model, where synchronization conditions are already established.