Nonsmooth Multipattern Burster: a New Approach to Solvable Piecewise-Smooth Modeling of Neuron Activity

In this paper, we constructed a new model of the neuron membrane potential in the form of a piecewise-smooth (PWS) slow-fast system. Its fast part is a two-dimensional PWS subsystem specially “sewn” from the normal forms of a stable focus and a saddle-node bifurcation, and the slow one-dimensional equation serves to periodically force the fast one. Our approach to constructing PWS systems in this way allows them to qualitatively reproduce the phase space structure of their smooth counterpart and also provide explicit solutions. By obtaining an exact Poincaré map as a composition of solutions and examining it, we proved the complete bifurcation structure of the fast subsystem. By selecting three bifurcation routes and applying a time-periodic change of the subsystem parameter along them using the third slow equation, we obtained three different types of bursting patterns. Finally, we showed that, when coupled into a network, our PWS models form a synthetic central pattern generator that manifests a traveling bursting wave.