Qualitative changes in spike-based neural coding and synchronization at the saddle-node loop bifurcation

Abstract: Information processing in the brain crucially depends on encoding properties of single neurons, with particular relevance of the spike-generation mechanism. The latter hinges upon the bifurcation type at the transition point between resting state and limit cycle spiking. Prominent qualitative changes in encoding have previously been attributed to a specific switch of such a bifurcation at the Bogdanov-Takens (BT) point. This study unveils another, highly relevant and so far underestimated transition point: the saddle-node loop bifurcation. As we show, this bifurcation turns out to induce even more drastic changes in spike-based coding than the BT transition. This result arises from a direct effect of the saddle-node loop bifurcation on the limit cycle and hence spike dynamics, in contrast to the BT bifurcation, whose immediate influence is exerted upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle-node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails close to optimal coding and synchronization properties in event-based information processing units, such as neurons. The saddle-node loop bifurcation leads to a peak in synchronization range and provides an attractive mechanism for the so far unresolved facilitation of high frequencies in neuronal processing. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators. On the experimental side, our theory applies to optical stimulation of nerve cells, and reveals that these techniques could manipulate a variety of information processing characteristics in nerve cells beyond pure activation.