Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation

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.

Phase‐response curves of ion channel gating kinetics

“…One such bifurcation, the codimension‐1 Saddle node on invariant cycle (SNIC) bifurcation, ranks prominently as the onset of spiking in many neuronal models . Introducing a second parameter that controls the time‐scale separation between voltage and kinetic equations in Equation , it was conjectured (and shown for planar systems) that this SNIC bifurcation is always enclosed by two codimension‐2 SNL bifurcations . They result in a small‐ and big‐homoclinic loop.…”

“…General properties of the big‐SNL and small‐SNL flip bifurcations have been discussed . For neuron models, it was shown previously that increasing the relative time scale between the current‐balance equation and the gating kinetics, the following sequence of bifurcations is traversed: big‐SNL → SNIC → small‐SNL . For example, increasing the membrane capacitance, C m , can induce this sequence, such that the interval of capacitance values that give rise to a SNIC bifurcation is encapsulated by the two SNL points, .…”

“…The flips switch the approach to the SN from the semistable manifold to the strongly stable manifold, leaving the departure unaltered . This breaks the symmetry of the dynamics in the center manifold as well as the PRC and has stark consequences for neuronal dynamics and coding …”

“…One may confirm the symmetric PRC at the SNIC onset bifurcation (Figure A) and the asymmetric PRC at the small‐SNL point (Figure B). Some implications of this symmetry breaking have been discussed previously . With the general shape and properties of the PRCs in the center manifold of the SNIC and SNL bifurcations established by Observation , we now turn to the case where perturbations are given along the different gating dimensions.…”

Neuronal firing and DNA dynamics in a neural network

“…The main dynamical behavior of a neural network that has received much attention from neuroscientists is the neuronal synchronization phenomenon [11,15,16,19,[21][22][23][24][25]. In fact, also intrinsic properties of a neuron (its ion channel composition, morphology, capacitance and temperature) change its susceptibility to synchronization in a network [49]. As a threshold phenomenon, neuronal synchronization is known to be very sensitive to the fluctuations of some parameters such as: coupling strength [11,16,17], long-range parameter [11], time delay [15], spatial in homogeneity stimulus [7] and noise [8,17].…”