Yuu Miino, Tetsushi Ueta, Structurally unstable synchronization and border-collision bifurcations in the twocoupled Izhikevich neuron model, World Scientific, International Journal of Bifurcation and Chaos, Vol.(in press), pp.699-702, 2023.
Abstract: This study investigates a structurally unstable synchronization phenomenon observedin the two-coupled Izhikevich neuron model. As the result of varying the systemparameter in the region of parameter space close to where the unstablesynchronization is observed, we find significant changes in the stability of its periodicmotion. We derive a discrete-time dynamical system that is equivalent to the originalmodel and reveal that the unstable synchronization in the continuous-time dynamicalsystem is equivalent to border-collision bifurcations in the corresponding discrete-timesystem. Furthermore, we propose an objective function that can be used to obtain theparameter set at which the border-collision bifurcation occurs. The proposed objectivefunction is numerically differentiable and can be solved using Newton's method. Wenumerically generate a bifurcation diagram in the parameter plane, including theborder-collision bifurcation sets. In the diagram, the border-collision bifurcation setsshow a novel bifurcation structure that resembles the `strike-slip fault' observed ingeology. This structure implies that, before and after the border-collision bifurcationoccurs, the stability of the periodic point discontinuously changes in some cases butmaintains in other cases. In addition, we demonstrate that a border-collision bifurcationsets successively branch at distinct points. This behavior results in a tree-like structurebeing observed in the border-collision bifurcation diagram; we refer to this structure asa border-collision bifurcation tree. We observe that a periodic point disappears at theborder-collision bifurcation in the discrete-time dynamical system and is simultaneouslyreplaced by another periodic point; this phenomenon corresponds to a change in thefiring order in the continuous-time dynamical system.