Local Exponents of Nonlinear Compression in Periodically Driven Noisy Oscillators

Nonlinear compression of periodic signals is a key feature of the active amplifier in inner ear organs of all vertebrates. Different exponents ${\alpha }_0\in [-0.88,-0.5]$ of the sensitivity vs forcing amplitude $|\chi |\sim f^{{\alpha }_0}$ have been observed. Here we calculate analytically the local exponent for a generic oscillator, the normal form of a Hopf bifurcation driven by noise and a periodic signal. For weak noise and sufficient distance from the bifurcation on the unstable side, the exponent may be close to $-1$ for moderate forcing amplitudes beyond linear response. Such strong compression is also found in a model of hair bundle motility.

Figure 1

Sensitivity (top) and local exponents of compression (bottom) vs input amplitude. Shown: in (a) data from [3] for the basilar membrane vibration amplitude with respect to sound pressure vs normalized sound pressure ($p_0=20\mu \mathrm{Pa}$); in (b) for the Hopf oscillator ($r=-1$, $\omega ={\omega }_0=1$) for the supercritical [in (b) with $B=1$, $C=0$] and subcritical case [in (b) with $B=-1$, $C=1$]; simulations (symbols) compared to theory [Eq. (4, 5), lines]. Selected exponents (0, $-2/3$, $-4/5$, and $-1$) are indicated; enclosed straight lines in (a) are power laws and exponents with $\alpha =-0.75$.

Figure 2

Sensitivity (a) and local exponent (c) vs forcing amplitude for a subcritical Hopf oscillator ($B=-1$, $C=1$) for relative detuning as indicated; sensitivity vs relative detuning for different values of the force at noise intensity $\epsilon =0.002$ (b) and for different $\epsilon$ for forces in the linear response regime (d). Stochastic simulations (symbols) compared to theory [Eq. (4), lines]. Local exponents are determined by numerical differentiation using Eq. (4).

Figure 3

Sensitivity (top) and local exponent of compression (bottom) of the hair bundle model vs forcing amplitude at operation points 1 (a) and 2 (b) for various $\epsilon$ as indicated ($\epsilon =1$ corresponds to parameters of a single uncoupled hair bundle [8]). Sensitivities and exponent were determined numerically from the response to a driving at the characteristic frequency (peak frequency of the spontaneous power spectrum).

Figure 4

Simulation results for the hair bundle model at OP2. Sensitivity (a) and local exponent (c) vs forcing amplitude for different values of the relative detuning as indicated; sensitivity vs relative detuning for different values of the force (b) and in the linear response regime for different $\epsilon$ (d). In (a)–(c), noise-reduction factor $\epsilon =0.01$.