Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural “signal coupling,” ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

Figure 1

Response of two Hopf systems in Eq. (2) with characteristic frequencies ${\omega }_{1,2}/2\pi =180$ and 225 Hz (solid and dashed curves, respectively) to a periodic test signal with variable frequency $f=\omega /2\pi$ at three amplitudes $-20$, $-40$, and $-60\mathrm{dB}$. Uncoupled systems ($g=0$): (a) and (b). Coupled systems: (c) $g=0.12$ and (d) $g=0.24$.

Figure 2

Critical value $g_c$ for symmetric coupling $g=(g_{12}=g_{21})$, as a function of ${\omega }_2/{\omega }_1$, where ${\omega }_1/2\pi =180\mathrm{Hz}$, for four values of $\mu$. The lines single out the regions above which the coupled system starts to oscillate, based on elements that are individually below the bifurcation threshold. Red circle: Location of $g_c$ for ${\omega }_1/2\pi =180$ and ${\omega }_2/2\pi =225\mathrm{Hz}$.

Figure 3

Critical coupling $g_c$ (red line) and frequency of oscillation (contours) above the bifurcation for asymmetrical coupling (${\omega }_{1,2}/2\pi =180$ and 225 Hz, ${\mu }_{1,2}=-0.1$). Gray-shaded area: Small-signal amplification regime. Black dots: Location of the systems of Figs. 1, 1, and 1.

Figure 4

Response of $N=10$ systems, characteristic frequencies distributed around 200 Hz, to a test signal of amplitude $-60\mathrm{dB}$. (a) Uncoupled, $\mu =-0.2$, (b) signal-coupled (dashed curve, $\mu =-0.3$; solid curve, $\mu =-0.2$), exhibiting a coherent and sharply tuned response around $f_c\approx 200\mathrm{Hz}$. (c) $g_c+\mu$ as a function of $N$, for ${\omega }_{\text{ch},i}={\omega }_{\text{ch}}$ and ${\mu }_i=\mu =-0.1$. The same behavior (with a line shift) is obtained for reasonable variations of ${\omega }_{\text{ch},i}$ and ${\mu }_i$. The value of $\mu$ is reflected in the first data point obtained for $N=2$, which implies $g_c=-2\mu$ [cf. Eq. (5) and Fig. 2].

Figure 5

(a) $f_c$ as a function of the coupling $g$ beyond $g_c$ ($\mu =-0.1$ for all systems). (I) Two systems (180/225 Hz); (II) three systems (180, 200, and 300 Hz); (III) five systems (120, 160, 200, 240, and 300 Hz). (b) $f_c$ for $\tau$-delayed coupling (two systems at 180 and 225 Hz, ${\mu }_{1,2}=-0.1$).