Unstable Periodic Orbits at Tonic-to-Bursting Bifurcations in Neuronal Discharges.

Transitions from neuronal single-spike discharges to impulse groups, i.e. tonic-to-bursting bifurcations, are of particular physiological and pathophysiological relevance for major autonomous and cognitive functions. Experimental recordings have shown that such transitions are often associated with distinct irregularities of the firing patterns as we have seen, for example, in peripheral cold receptors and hypothalamic neurons (Braun et al. Pflügers Arch 386:1-9, 1980; Dewald et al. J Thermal Biol 24: 339-345, 1999). Recently developed methods for the detection of unstable periodic orbits (UPOs) indicate that the irregular firing is not only caused by stochastic components ("noise") but partly results from "deterministic chaos" (e.g. Braun et al. J Comp Neurosci 7: 17-32, 1999).

For a better understanding of these neuronal pattern generators we use a Hodgkin-Huxley type computer model which consists of a minimal set of ionic conductances for spike-generation and subthreshold oscillations (Braun et al., Int J Bifurc & Chaos 8: 881-889, 1998). With addition of noise it successfully mimicks the experimentally observed impulse patterns and their stimulus dependent modifications including the occurrence of UPOs. Moreover, simulation runs without noise clearly exhibited a broad range of deterministic chaos at tonic-to-bursting bifurcations (W Braun et al., Phys Rev E 62: 6352-6360, 2000; Feudel et al., Chaos 10: 231-239). These chaotic dynamics can be seen as the result of "critical" interactions between the two oscillatory subsystems, i.e. the subthreshold oscillations and the spike-generating processes (Braun et al. Neurocomputing 32: 51-59, 2000). However, in "noisy" simulations the unstable regime seems to be considerably extended towards the deterministically regular spiking range (Braun et al. Biosystems 62: 99, 112, 2001) which eventually allows the anticipation of tonic-to-bursting bifurcations and their control.