Recurrence-based analysis and controlling switching between synchronous silence and bursting states of coupled generalized FitzHugh-Nagumo models driven by an external sinusoidal current

We investigate the response characteristics of a generalized FitzHugh-Nagumo model under an external sinusoidal current and the synchronization of two neurons coupled with a gap junction. In the autonomous case, we find analytically by the Lindsted's method that the system can admit tristable activities in silence, subthreshold, and nerve pulse; depending on the conductance parameters and the state of ionic conductance. In the presence of an external sinusoidal current, we find by numerical simulations that neurons can exhibit a coexistence between different spiking patterns and periodic waves, which are well observed in the structure of the recurrence plot. We further study the synchronization between coupled neurons each admitting bistable activities, such as a coexistence between chaotic (active) and silence (inactive) regimes. We apply recurrence analysis tool to reveal the range of the coupling parameter where synchronization occurs, as well as the dynamical transitions between the synchronous coexisting states (hysteresis phenomenon). The coupling strength is an indicator of the phenomenon of synchronization that can also bring the system to any of the desired synchronous attractors. These phenomena of synchronization and the control between synchronous states can be improved by the presence of an external electrical field. The switching of the coupled neurons to bursting patterns or to periodic waves explains the well-known properties of excitatory (switching on) or inhibitory (switching off) synaptic coupling, respectively; while the unstable signal separating the two stable synchronous signals can be taken as the synaptic threshold. Rather, this study adds to our theoretical understanding of the topic and poses new challenges for investigation. Experimental investigations are required to validate these conclusions in real-world settings, and biological implications must be evaluated within the particular framework of the modeling that was done.