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Abstract
Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization 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, the BT bifurcation exerts its immediate influence 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 an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.
Highlights
Different states of macroscopic network dynamics are a hallmark of complex systems such as the brain
Starting at a saddle node on an invariant cycle (SNIC) bifurcation in planar general neuron models, we demonstrate that a variation in the separation of time scales provokes a generic sequence of firing onset bifurcations
At a Bogdanov-Takens bifurcation, which is classically regarded as the transition point between type I and type II excitability, stable limit cycles are not directly involved
Summary
Different states of macroscopic network dynamics are a hallmark of complex systems such as the brain. We present a general case where a specific variation in single-neuron properties can drastically switch network synchronization properties, provided the cells’ parameters are close to a critical transition point: the saddle-node-loop (SNL) bifurcation. While this bifurcation is not unknown [3], our results demonstrate that its substantial, qualitative consequences for neural dynamics have so far not been sufficiently acknowledged. We investigate an alternative transition, which switches the spike onset from a saddle node on an invariant cycle (SNIC) bifurcation to a saddle homoclinic orbit (HOM) bifurcation [Fig. 1(b)] This transition is organized by a codimension-two bifurcation: the SNL bifurcation [21,22]. V, where we prove that SNL bifurcations generically occur in planar neuron models
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