The transition between square-wave and pseudo-plateau bursting

Abstract

Electrically excitable cells such as neurons may exhibit different types of bursting activities due to a difference in timescales between changes in electrical potential across the cell membrane and changes in concentration of ions inside the cells. Some of these bursting activities can be observed in mathematical models with multiple timescales. We focus on two types of bursting patterns that appear to be very similar in the observed experimental data but are different in terms of the mathematical mechanism by which they are generated; these are so-called squarewave and pseudo-plateau bursting. In this thesis, we study the mathematical structure of square-wave and pseudo-plateau bursting patterns in a polynomial system of ordinary differential equations with one slow and two fast variables. We use the technique of freezing the single slow variable and relating the resulting bifurcation diagram of the fast subsystem to the dynamics of the full system. This polynomial system can exhibit both types of bursting, depending on the choice of parameters. We are interested in a continuous deformation from square-wave to pseudo-plateau bursting, which we study by numerical continuation. We find that a direct transition between the two bursting patterns through a single connected family can be obtained via a variation of a single bifurcation parameter. As the pseudo-plateau bursting pattern transforms into the square-wave bursting pattern, a transitional bursting pattern arises. This transitional bursting pattern can be viewed as a mix between square-wave and pseudo-plateau bursting. Based on our understanding gained from studying the polynomial model, we propose precise definitions for square-wave, pseudo-plateau and transitional bursting patterns in generic systems with one slow and two fast variables. Our classifications of the different bursting patterns depend mainly on the underlying bifurcation diagram of the fast subsystem, the location of the nullcline of the slow variable and the timescale ratio between the variables.