Collective phase response curves for heterogeneous coupled oscillators

Phase response curves (PRCs) have become an indispensable tool in understanding the entrainment and synchronization of biological oscillators. However, biological oscillators are often found in large coupled heterogeneous systems and the variable of physiological importance is the collective rhythm resulting from an aggregation of the individual oscillations. To study this phenomena we consider phase resetting of the collective rhythm for large ensembles of globally coupled Sakaguchi-Kuramoto oscillators. Making use of Ott-Antonsen theory we derive an asymptotically valid analytic formula for the collective PRC. A result of this analysis is a characteristic scaling for the change in the amplitude and entrainment points for the collective PRC compared to the individual oscillator PRC. We support the analytical findings with numerical evidence and demonstrate the applicability of the theory to large ensembles of coupled neuronal oscillators.

Figure 1

The order parameter just before the perturbation is at $Z_0$. Just after the perturbation it is shifted to $\overline{Z}$. ${\mathrm{\Delta }}_0$ tracks the shift in the mean phase that occurs in the movement from $Z_0$ to $\overline{Z}$ and ${\mathrm{\Delta }}_R$ gives the relaxation phase shift of the collective oscillator. The isochrons here show the case where $\beta =-\frac{1}{2}$.

Figure 2

A representative plot of the prompt phase response curve ${\mathrm{\Delta }}_0$ for various values of R and $Q\left(\psi \right)=\sin \left(\psi \right)+\sin \left(4\psi \right)$ with $\epsilon =0.1$. The first harmonic is amplified and higher harmonics are dissipated in the collective PRC.

Figure 3

A representative plot of the amplitude response curve $\mathrm{\Lambda }$ for various values of the phase coherence with $Q\left(\psi \right)=\frac{1}{2}\sin \left(\psi \right)-\cos \left(\psi \right)$ and $\epsilon =0.1$. Perturbations around stable fixed points of $Q\left(\psi \right)$ give transient increases in the phase coherence and perturbations about unstable fixed points of $Q\left(\psi \right)$ give decreases in the phase coherence.

Figure 4

Change in the amplitude and entrainment points for a sinusoidal microscopic PRC. Here we set $Q\left(\psi \right)=\sin \left(\psi \right),\epsilon =0.1$, and $\beta =0.5$. The coupling strength $K_0$ was varied to produce phase distributions with differing phase coherence $\left(R\right)$ values in the synchronized state. Blue stars in (a) and (b) indicate the values of $R$ which are plotted in (c) and (d). (a) The amplitude of the collective phase response curve scales like $R+\frac{1}{R}$ with the phase coherence. (b) The shift in the zero at $\psi =\pi$ scales like $tan\left(\beta \right)\left(\frac{2}{R^2+1}-1\right)$. (c) Microscopic, predicted collective PRC and numerical collective PRC for $R=0.7$. (d) Microscopic, predicted collective PRC and numerical collective PRC when $R=0.5$.

Figure 5

Comparing theoretical predictions against numerical results for the collective PRC for various microscopic PRC with $\epsilon =0.1$ and $\beta =0.5$. The coupling strength $K_0$ was varied to produce phase distributions with differing phase coherence $\left(R\right)$ values in the synchronized state. Microscopic PRC (solid black), ${\mathrm{\Delta }}_{\infty }$ (dashed green), numerical simulation (red “+”). Let $H\left(\psi \right)$ be the heaviside step function. (a) $Q\left(\psi \right)=\sin \left(\psi \right)+\frac{1}{4}\sin \left(5\psi \right)$; (b) $Q\left(\varphi \right)=\sin \left(\psi \right)+\sin \left(4\psi \right)$; (c) $Q\left(\psi \right)=H(\psi -\pi )[-\sin (2\psi )-\sin (2\psi \left)\cos \right(2\psi \left)\right];$ (d) $Q\left(\psi \right)=H(-\psi -\pi )$.

Figure 6

Comparing theoretical predictions against numerical results for the collective PRC of Morris-Lecar neurons. Inset plots show individual neurons action potentials (mV) for 400 ms in the synchronized state for the two parameter regimes. Microscopic PRC (solid black), ${\mathrm{\Delta }}_{\infty }$ (dashed green), numerical simulation (red “+”). (a) Collective PRC for Type I Morris-Lecar neurons with $R=0.67$ and mean applied current of $50.0\frac{\mu \mathrm{A}}{{\mathrm{cm}}^2}$. (b) Type II Morris-Lecar system with $R=0.70$ and mean applied current of $95\frac{\mu \mathrm{A}}{{\mathrm{cm}}^2}$.