Deterministic chaos in mathematical model of pacemaker activity in bursting neurons of snail, Helix pomatia

Chaotic regimes in a mathematical model of pacemaker activity in the bursting neurons of a snail, Helix pomatia, have been investigated. The model includes a slow-wave generating mechanism, a spike-generating mechanism, an inward Ca current, intracellular Ca ions, [Ca2+](in), their fast buffering and uptake by intracellular Ca stores, and a [Ca2+](in)-inhibited Ca current. Chemosensitive voltage-activated conductance, g(B)*, responsible for termination of the spike burst, and chemosensitive sodium conductance, g(Na)*, responsible for the depolarizing phase of the slow-wave, were used as control parameters. These conductances in intact snail bursting neuron are regulated by neuropeptides. Time courses of the membrane potential and [Ca2+](in) were employed to analyse different regimes in the model. Histograms of interspike intervals, autocorrelograms, spectral characteristics, one-dimensional return maps, phase plane trajectories, positive Lyapunov exponent and especially cascades of period-doubling bifurcations demonstrate that approaches to chaos were generated. The bifurcation diagram as a function of g(B)* and the ([Ca2+](in)-V) phase diagram of initial conditions reveal fractal features. It has been observed that a short-lasting depolarizing current or elevation of [Ca2+](in) may evoke transformation of chaotic activity into a regular bursting one. These kinds of transitions do not require any changes in the parameters of the model. The results demonstrate that chaotic regimes of neuronal activity modulated by neuropeptides may play a relevant role in information processing and storage at the level of a single neuron. (C) 1996 Academic Press Limited