Amphibian sacculus and the forced Kuramoto model with intrinsic noise and frequency dispersion

The amphibian sacculus (AS) is an end organ that specializes in the detection of low-frequency auditory and vestibular signals. In this paper, we propose a model for the AS in the form of an array of phase oscillators with long-range coupling, subject to a steady load that suppresses spontaneous oscillations. The array is exposed to significant levels of frequency dispersion and intrinsic noise. We show that such an array can be a sensitive and robust subthreshold detector of low-frequency stimuli, though without significant frequency selectivity. The effects of intrinsic noise and frequency dispersion are contrasted. Intermediate levels of intrinsic noise greatly enhance the sensitivity through stochastic resonance. Frequency dispersion, on the other hand, only degrades detection sensitivity. However, frequency dispersion can play a useful role in terms of the suppression of spontaneous activity. As a model for the AS, the array parameters are such that the system is poised near a saddle-node bifurcation on an invariant circle. However, by a change of array parameters, the same system also can be poised near an emergent Andronov-Hopf bifurcation and thereby function as a frequency-selective detector.

Figure 1

Solutions of Eq. (1), with ${\omega }_0=5$ and with a load $f_0\left(\tau \right)=0.044\tau$ that is slowly ramped in time, obtained without noise (a, $D_{\theta }=0$) and with noise (b, $D_{\theta }=1$). The red arrow indicates the saddle-node bifurcation threshold $f_0={\omega }_0=5$.

Figure 2

Superposition of 400 subthreshold, uncoupled hair bundles that obey Eq. (1) with $D_{\theta }=0.05,{\omega }_0=1,f_0=1.15,\mathrm{\Omega }=0.25$, and $f_{\mathrm{\Omega }}=0.1$. The bottom panels show the number of oscillators $N$ that undergo a phase slip event in a fixed time window. Blue line (Black Line): stimulus. (a) No frequency dispersion is introduced ($\mathrm{\Delta }=0$). (b) The oscillators exhibit frequency dispersion ($\mathrm{\Delta }=0.5$).

Figure 3

Superposition of 400 coupled hair bundles that obey Eq. (3) with $\mathrm{\Delta }=0.5,f_0=1.15,\mathrm{\Omega }=0.25,f_{\mathrm{\Omega }}=0.1$, and $K=15$. The noise intensity increases: (a) $D_{\theta }=0.5$, (b) $D_{\theta }=2.5$, and (c) $D_{\theta }=8$. Light blue (thick light gray) lines show the phase of the order parameter. Blue (black) lines indicate the sinusoidal stimulus. The time-averaged order parameter amplitude $r_{\mathrm{ave}}$ is shown separately in (d). The solid line indicates a numerical fit of the data.

Figure 4

Solutions of the DMFT equations in the absence of frequency dispersion by the spectral method (top row), as compared with numerical solution of the equations of motion for 400 coupled noisy Adler phase oscillators (bottom row). System parameters: $K=4,D_{\theta }=1$, and $f_{\mathrm{\Omega }}=0$. The load $f_0\left(\tau \right)$ slowly increases in time ($d{f}_0/d\tau =1/100$). The natural frequency is (a) ${\omega }_0=1$, (b) ${\omega }_0=1.8$, and (c) ${\omega }_0=3.5$. Depending on the value of the parameter, the system undergoes different types of bifurcation: (a) and (d) a SNIC bifurcation, (c) and (f) an Andronov-Hopf bifurcation, and (b) and (e) a Bogdanov-Takens bifurcation. The three bifurcation points described here are also indicated in Fig. 6.

Figure 5

Arnold tongue: Order parameter obtained by the spectral method for $D_{\theta }=1$ in the absence of frequency dispersion. (a) $K=1.8$ and (b) $K=4$. The horizontal axis is the natural frequency ${\omega }_0$, and the vertical axis is the steady load $f_0$. The color scale represents the amplitude of the static order parameter. Red (bright): high amplitude; blue (dark): low amplitude. Inside the yellow-red (gray) wedge, the static order parameter is stable. The black line in the (a) represents the bifurcation line of a single Adler equation. For $K<K_c$, the order parameter undergoes an Andronov-Hopf bifurcation. The lines in (b) represent the different bifurcations obtained from the linear response theory with the stationary solution of the mean-field Fokker-Planck equation. H indicates an Andronov-Hopf bifurcation [red (dotted line)], SN a SNIC bifurcation (black), and BT a Bogdanov-Takens bifurcation. The three arrows correspond to the three time traces in Fig. 4.

Figure 6

Mode-locking diagram for $K=4$ and $D_{\theta }=1$ in the absence of frequency dispersion, as obtained by the spectral method. The horizontal axis is the mean natural frequency, and the vertical axis is the steady load. The color map represents time average of the time derivative ($\langle dr/d\tau \rangle$) of the order parameter amplitude $r$, starting from $r=1$. Dark blue (dark gray) indicates that the order parameter does not decay with time, and lighter shades indicate increasing rates of decay of the order parameter. The lines indicate the bifurcations obtained by the linear-response theory, while the three arrows correspond to the three time traces of Fig. 4.

Figure 7

Kuramoto synchronization in the presence of noise (a) and frequency dispersion (b). The color map represents the time derivative of the order parameter amplitude. Dark blue (dark gray) indicates that the order parameter is time independent, and lighter shades indicate that the order parameter is increasingly time dependent. Red (gray curves between two regions) lines indicate instability onset regions where $0<\langle dr/d\tau \rangle <0.08$. $K=4$ in both cases. (a) No frequency dispersion with ${\omega }_0=1$. Horizontal axis: noise level $D_{\theta }$. Vertical axis: steady load $f_0$. (b) No noise. Horizontal axis: frequency dispersion $\mathrm{\Delta }$. Vertical axis: steady load $f_0$.

Figure 8

Stochastic resonance near a SNIC bifurcation point in the absence of frequency dispersion. The response of a coupled DNKA array, poised near a saddle-node bifurcation, and subject to a sinusoidal stimulus. The system parameters are $K=15,{\omega }_0=1.0$, and $f_0=1.15$. The stimulus frequency is $\mathrm{\Omega }=0.25$, and the stimulus amplitude is $f_{\mathrm{\Omega }}=0.1$. The top panels [(a–c)] represent the time-dependent real part of the order parameter, computed at three different values of the phase diffusion coefficient $D_{\theta }$: (a) $D_{\theta }=0.5$, (b) $D_{\theta }=3$, and (c) $D_{\theta }=8$. The bottom panels [(d–f)] show the corresponding power spectra. (g) Power output $S(\omega =\mathrm{\Omega })$ as a function of the noise level $D_{\theta }$ and coupling constant $K$. Red (gray) dots in this panel represent $K$ and $D_{\theta }$ values of plots shown in (a)–(f), as indicated by the arrows. The black line in (g) represents the Kuramoto transition line. The system exhibits clear stochastic resonance.

Figure 9

Frequency selectivity near the SNIC bifurcation. (a) Power output $S(\omega =\mathrm{\Omega })$ as a function of the stimulus frequency $\mathrm{\Omega }$ and stimulus amplitude $f_{\mathrm{\Omega }}$, for $D_{\theta }=1,\mathrm{\Delta }=0,K=15,f_0=1.15$. (b) Power output $S(\omega =\mathrm{\Omega })$ as a function of the stimulus frequency $\mathrm{\Omega }$, for various values of $f_{\mathrm{\Omega }}$ (0.001,0.01,0.1) and with $K=15,D_{\theta }=2.5$, and $f_0=1.15$. For $\mathrm{\Delta }=0$, the results are shown in bright green, bright magenta, and bright blue [light shades of gray, curves with slightly higher $S\left(\mathrm{\Omega }\right)$ values for a given stimulus amplitude]. For $\mathrm{\Delta }=1.5$, the curves are shown in dark green, dark magenta, and dark blue [dark shades of gray, curves with slightly lower $S\left(\mathrm{\Omega }\right)$ values for a given stimulus amplitude]. The coupled DNKA array clearly shows high SNR at low-frequency range. The frequency dispersion degrades the responsiveness, but the stochastic resonance is still present at low frequency.

Figure 10

Stochastic resonance in the numerical simulation of a 400-element coupled DNKA array and comparison to the mean-field DNKA array. Response of a coupled DNKA array to a sinusoidal stimulus of amplitude $f_{\mathrm{\Omega }}=0.1$ and frequency $\mathrm{\Omega }=0.25$. The parameters of the array are $D_{\theta }=1,\mathrm{\Delta }=0.5,K=15,f_0=1.15$. Left column, parts (a), (c), and (e): the time-dependent real part of the order parameter (vertical axis) vs $\tau /2\pi$. The noise levels are (a) $D_{\theta }=0.5$, (c) $D_{\theta }=3$, and (e) $D_{\theta }=8$. Red (thin light gray) lines show the DMFT result, and the blue (thick dark gray) lines show the numerically computed order parameter, for an array of 400 oscillators. Right column, parts (b), (d), and (f): corresponding power spectra of the order parameter.

Figure 11

Power output vs frequency dispersion, computed using the spectral method. Power at the drive frequency ($\mathrm{\Omega }=0.25$) is plotted as a function of the frequency dispersion $\mathrm{\Delta }$, for $f_{\mathrm{\Omega }}=0.1,D_{\theta }=0.5$, and $f_0=1.15$, for different values of the coupling constant $K$. The system does not exhibit stochastic resonance with varying frequency dispersion.

Figure 12

Response of the coupled DNKA model near a Andronov-Hopf bifurcation. An array of coupled oscillators, with large frequency dispersion, is subject to a stimulus of frequency $\mathrm{\Omega }=0.9$ and amplitude $f_{\mathrm{\Omega }}=0.02$. The system parameters are $f_0=0.5$ and $\mathrm{\Delta }=0.16$. For panels (a)–(i), the coupling constant is fixed at $K=0.8$; in panel (j), it is varied. First row: Time-dependent real part of the order parameter, without stimulus. Second row: Time-dependent real part of the order parameter, with stimulus. Third row: Corresponding power spectra, with stimulus. The columns correspond to different phase diffusion coefficients; (a), (d), and (g): $D_{\theta }=0.02$; (b), (e), and (h): $D_{\theta }=0.2$; (c), (f), and (i): $D_{\theta }=1$. Arrows in (g), (h), and (i) indicate the spontaneous frequency of the system. (j) Power output $S(\omega =\mathrm{\Omega })$ as a function of the noise level $D_{\theta }$ and coupling constant $K$. The black line denotes the Kuramoto-Hopf bifurcation line. The red (gray) dots show the $K$ values used in top panels, as indicated by the arrows.

Figure 13

Frequency selectivity near the Kuramoto-Hopf bifurcation. (a) Power output $S(\omega =\mathrm{\Omega })$ as a function of the stimulus frequency $\mathrm{\Omega }$ and stimulus amplitude $f_{\mathrm{\Omega }}$, for $D_{\theta }=0.2,\mathrm{\Delta }=0.16,K=0.8$, and $f_0=0.5$. (b) Power output $S(\omega =\mathrm{\Omega })$ as a function of the stimulus frequency $\mathrm{\Omega }$, for various values of $f_{\mathrm{\Omega }}$ ($0.0001,0.001,0.01$), computed with $K=0.8,D_{\theta }=0.2,f_0=0.5$, and $\mathrm{\Delta }=0.16$.

Figure 14

Schematic diagram of 2D membrane.