Generation of periodic and chaotic bursting in an excitable cell model

There are interesting oscillatory phenomena associated with excitable cells that require theoretical insight. Some of these phenomena are: the threshold low amplitude oscillations before bursting in neuronal cells, the damped burst observed in muscle cells, the period-adding bifurcations without chaos in pancreatic β-cells, chaotic bursting and beating in neurons, and inverse period-doubling bifurcation in heart cells. The three variable model formulated by Chay provides a mathematical description of how excitable cells generate bursting action potentials. This model contains a slow dynamic variable which forms a basis for the underlying wave, a fast dynamic variable which causes spiking, and the membrane potential which is a dependent variable. In this paper, we use the Chay model to explain these oscillatory phenomena. The Poincaré return map approach is used to construct bifurcation diagrams with the ‘slow’ conductance (i.e., gK,C) as the bifurcation parameter. These diagrams show that the system makes a transition from repetitive spiking to chaotic bursting as parameter gK,C is varied. Depending on the time kinetic constant of the fast variable (λn), however, the transition between burstings via period-adding bifurcation can occur even without chaos. Damped bursting is present in the Chay model over a certain range of gK,C and λn. In addition, a threshold sinusoidal oscillation was observed at certain values of gK,C before triggering action potentials. Probably this explains why the neuronal cells exhibit low-amplitude oscillations before bursting.