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

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.

Figure 1

The transition from rest to spiking in response to an increase in input current $I_{\text{DC}}$ requires (a) that the resting state loses stability (illustrated are fold and subcritical Hopf bifurcations) and (b) the creation of a limit cycle [illustrated are saddle homoclinic orbit (HOM) and SNIC bifurcations]. The membrane capacitance $C_{\text{m}}$ makes it possible to switch between these bifurcations. The separation function, $\text{sep}$, marked in red, measures the distance between the stable and unstable manifold of the saddle. The overlap of both, i.e., $\text{sep}=0$, results in a homoclinic orbit.

Figure 2

(a) Spike raster plot of two small globally coupled networks of five Wang-Buzsaki models (see Appendix pp1), one close to the SNL bifurcation with $C_{\mathrm{m}}=1.47\mu \mathrm{F}/{\mathrm{cm}}^2$, the other at a SNIC bifurcation with $C_{\mathrm{m}}=1 \mu \mathrm{F}/{\mathrm{cm}}^2$. Synaptic connections are modeled as voltage perturbations of $\varepsilon =0.39$ mV. The frequency detuning of the neurons is approximately equally spaced between 5 and $11\mathrm{Hz}$. (b) Phase-locking index, calculated between pairwise neurons $i$ and $j$ with phase ${\varphi }_{i,j}$ as $\langle e^{i2\pi ({\varphi }_i-{\varphi }_j)}\rangle$, where brackets denote temporal averaging. Error bars denote standard deviations. (Top) Locking index for the network when the capacitance is changed [as in panel (a)]. (Bottom) Locking index for the network when the leak conductance $g_{\text{L}}$ is changed (instead of capacitance). $\varepsilon =0.08$ mV, $C_{\mathrm{m}}=1 \mu \mathrm{F}/{\mathrm{cm}}^2$, and SNIC at $g_{\text{L}}=0.1$ mS/${\mathrm{cm}}^2$ and SNL at $g_{\text{L}}\approx 0.57$ mS/${\mathrm{cm}}^2$; all other parameters are identical to those in the top panel. Locking at both SNL bifurcations exceeds locking at the corresponding SNIC bifurcation.

Figure 3

(Top to bottom) (a) Schematic illustration of the orbits at small SNL bifurcation, nondegenerated SNIC bifurcation, and big SNL bifurcation, with semistable (small single arrow) and strongly stable manifold (double arrows). These bifurcations occur in the Wang-Buzsaki model for $I_{\text{DC}}\approx 0.16 \mu \mathrm{A}/{\mathrm{cm}}^2, C_{\text{m}}\approx [1.47,1,0.09]$ $\mu \mathrm{F}/{\mathrm{cm}}^2$. (b) The associated phase-response curves measured for $I_{\text{DC}}$ 2% above the fold bifurcation.

Figure 4

Phase-response curve (left) and phase plot around the saddle node (right) for (a) a nondegenerated SNIC bifurcation with $C_{\text{m}}=1 \mu \mathrm{F}/{\mathrm{cm}}^2$ and (b) a small SNL bifurcation with $C_{\text{m}}\approx 1.47$ $\mu \mathrm{F}/{\mathrm{cm}}^2$ in the Wang-Buzsaki model with the limit-cycle period fixed in both cases to $2\mathrm{Hz}$.

Figure 5

Variation of the membrane capacitance $C_{\text{m}}$ for the Wang-Buzsaki model with input fixed at 2% above limit-cycle onset at $I_{\text{DC}}\approx 0.16 \mu \mathrm{A}/{\mathrm{cm}}^2$. (a) Limit-cycle period (black) and relative limit-cycle (LC) stability (gray) given by the ratio of the limit-cycle attraction time (inverse of the Floquet exponent) and the period. A small LC stability supports the validity of the phase description [36]. (b) Maximal amplitude of the odd part of the PRC, corresponding to the entrainment range when normalized by the coupling strength assuming $\delta$ coupling (abbreviated sync).

Figure 6

(a) Bifurcation diagram of the Wang-Buzsaki model under variation of membrane capacitance $C_{\text{m}}$ and input current $I_{\text{DC}}$. With $C_{\text{m}}=1 \mu \mathrm{F}/{\mathrm{cm}}^2$, the limit cycle arises from a SNIC bifurcation. Increasing $C_{\text{m}}$ leads to the small SNL at $C_{\text{m}}\approx 1.47 \mu \mathrm{F}/{\mathrm{cm}}^2$. Hatched areas mark bistability. (b) Decreasing $C_{\text{m}}$ leads to the big SNL and then to a Bogdanov-Takens (BT) bifurcation. Note that a change of stability in the big HOM branch, not shown here, follows from Ref. [42]. (c) Schematic illustration for the limit $C_{\text{m}}\to 0$, in which the system corresponds to a relaxation oscillator. Drawn in the state space of gating variable $n$ versus voltage $v$, the solid line with an inverted N shape represents the voltage nullcline, and the dashed line represents the gating nullcline. At some $I_{\text{DC}}$, the resting state loses stability and a big HOM orbit around all fixed points (green) is created.