Sinusoidal-signal detection by active, noisy oscillators on the brink of self-oscillation

The performance of a sinusoidal-signal detecter is enhanced if it operates quiescently in the proximity of a supercritical Hopf bifurcation [1, 2, 3]. The response to sinusoidal forcing of a quiescent deterministic system operating near a supercritical Hopf bifurcation is frequency-selective and nonlinear, with the response magnitude and the sharpness of frequency tuning increasing as the bifurcation is approached [1]. The range over which the input is nonlinearly compressed also rises as the system nears the bifurcation [2, 3]. To determine whether operation near a Hopf bifurcation is useful for signal detection, however, we must take into consideration the effects of noise, operation on the oscillatory side of the bifurcation, and subcritical as well as supercritical Hopf bifurcations.

Bifurcations in noisy systems often do not coincide with those in the corresponding deterministic systems [4]. Indeed, noise can create new bifurcations and destroy or change deterministic ones. For concreteness, we can study systems possessing bifurcations in the deterministic sense, to which noise has been added. We consider the normal forms of supercritical and subcritical Hopf oscillators in the presence of noise, which define noisy Hopf oscillators.

The effects of sinusoidal forcing on noisy oscillators have been described in many contexts. Stochastic oscillations, stochastic bifurcations, stochastic resonance, stochastic synchronization, and many other phenomena have been revealed through the study of specific systems such as van der Pol oscillators, integrate-and-fire models, and various phase oscillators [5, 6]. Often the response to sinusoidal forcing of a particular type of oscillator operating near a Hopf bifurcation has been investigated. For example, the FitzHugh-Nagumo model has been studied near a supercritical Hopf bifurcation whereas the behavior of the Noyes-Field-Thomson model has been discussed near a subcritical Hopf bifurcation [7, 8, 9].

Several authors have studied the behavior of noisy Hopf oscillators in the absence of deterministic forcing. The stationary Fokker-Planck equation corresponding to the normal form can be solved exactly for complex, additive, Gaussian white noise with uncorrelated components [10, 11] or with weakly correlated components in the supercritical case [12], or for particular types of multiplicative noise [13, 14, 15]. Indeed, the power spectrum for an isochronous oscillator can be calculated from the Fokker-Planck equation and is approximately Lorentzian when the oscillator is quiescent, in which case there is a closed-form approximation for the spectral width, and when the system spontaneously oscillates far from the bifurcation [10, 11]. The power spectrum of a non-isochronous, supercritical Hopf oscillator can be strongly non-Lorentzian, however, but can be expressed as an infinite sum of Lorentzian’s [16]. In general, noise in the normal form of a random dynamical system is multiplicative, colored, and cross-correlated [17, 4].

To understand the basic behavior of a sinusoidally forced noisy Hopf oscillator, one can consider the simplest type of noise. Complex Gaussian white noise, which possesses uncorrelated real and imaginary components, can be added to the sinusoidally-forced Hopf normal form [18]. If the amplitude of oscillation is approximately constant, the phase dynamics is equivalent to that of a Brownian particle in an inclined potential and the degree of entrainment can be found analytically [19]. More generally, the response function of isochronous, self-oscillating Hopf oscillators to sinusoidal forcing at or near the natural frequency is constant for weak forcing and declines compressively for stimuli of increasing amplitude, with an exponent for frequency-tuned forcing and weak noise that can be calculated analytically [20]. A self-oscillatory system exhibits strong compression far from a supercritical Hopf bifurcation or at any distance from a subcritical bifurcation. Moreover, for the supercritical Hopf normal form with a stimulus frequency close to the natural frequency and this kind of noise, the linear response amplitude possesses the same analytical form as the deterministic, quiescent case with renormalized parameters [21]. We lack a more complete understanding, however, of a noisy Hopf oscillator’s response as a function of the control parameter.

To determine the behaviors of noisy sinusoidal-signal detectors that are inherent to Hopf bifurcations, we study the response to sinusoidal forcing of supercritical and subcritical Hopf oscillators as a function of the control parameter. We discuss specific cyclostationary solutions to the Fokker-Planck equation describing a noisy Hopf oscillator. These solutions vary in time at the frequency of driving and describe the system’s behavior after transients have vanished. We find the response of the Hopf oscillators for both low and high noise levels and forcing amplitudes, and limit the forcing amplitude only to avoid the possibility of chaotic dynamics at very high levels of forcing [22, 23].

The correspondence between a system operating near a Hopf bifurcation and a Hopf oscillator may break down for sufficiently great forcing and noise levels. Nonetheless, the response of Hopf oscillators to strong forcing or their response in the presence of a substantial amount of noise serves to illustrate basic characteristics that we expect to be common for a large class of oscillators. To determine if a noisy Hopf oscillator’s ability to detect a sinusoidal signal is augmented when the system operates near the bifurcation, we focus on the phase-locked amplitude, the response function, and the vector strength. The phase-locked amplitude and response function quantify the oscillator’s entrained response to the stimulus, whereas the vector strength quantifies the degree of entrainment.

A Hopf bifurcation occurs when a fixed point changes stability at a critical value of a control parameter [24, 25, 26]. The dynamics of any system that possesses a Hopf bifurcation is two-dimensional in the vicinity of the critical point and can be described by the complex variable $z\equiv z_{\mathrm{R}}+iz_{\mathrm{I}}$. To linear order in $z$,

in which $\mathbf{x}$ is the state vector of the original system, $\mathbf{x^{*}}$ is its value at the fixed point corresponding to the bifurcation, and $\mathbf{c_{\mathrm{R}}}$ and $\mathbf{c_{\mathrm{I}}}$ are constant real vectors of the same dimension as $\mathbf{x}$ that depend on the system’s particular structure and parameters [27]. The angle $\phi$ satisfies $\tan\phi=z_{\mathrm{I}}/z_{\mathrm{R}}$. The dynamics of any component of the original system $x_{\mathrm{i}}$ is qualitatively similar to the dynamics of $z_{\mathrm{R}}$, $z_{\mathrm{I}}$, or $|z|$, which are described by the truncated normal form for a Hopf bifurcation.

We study the Hopf normal form in the presence of sinusoidal forcing with additive white noise possessing uncorrelated real and imaginary components:

in which $\mu$ is the control parameter, $\omega_{0}$ is the natural frequency, and $b$, $b^{\prime}$, $c$, and $c^{\prime}$ are coefficients defining the system’s nonlinearity. The complex forcing term $F(t)=fe^{i(\omega t+\theta)}$ has amplitude $f$, frequency $\omega$, and phase $\theta$. The noise $\eta(t)$ is complex and white and satisfies $\langle\eta(t)\rangle=0$, $\langle\eta(t)\eta(t^{\prime})\rangle=0$, and $\langle\eta(t)\eta^{\mathrm{\dagger}}(t^{\prime})\rangle=4d\delta(t-t^{\prime})$, in which $\langle\rangle$ represents the ensemble average, $\eta^{\mathrm{\dagger}}(t)$ is the complex conjugate of $\eta(t)$, and $d$ is the strength of the noise [18, 16, 21].

The correspondence between the Hopf oscillator defined by Eq. 3 and a system operating near a Hopf bifurcation generally relies on the forcing being weak and the noise being small, additive, and white [21, 28]. We can, however, also view Eq. 3 as defining the dynamics of a specific system possessing a Hopf bifurcation, namely, a Hopf oscillator.

In the absence of forcing and noise, a Hopf bifurcation occurs when $\mu=0$; the system oscillates when $\mu>0$ and possesses a stable fixed point for $\mu<0$. When $b<0$ and $c=0$ the amplitude of oscillation grows continuously from zero at the bifurcation, corresponding to a supercritical Hopf bifurcation. Oscillations of nonzero amplitude exist at a subcritical Hopf bifurcation that occurs when $b>0$ and $c<0$. The oscillations are generated by a saddle-node of limit cycles bifurcation at $\mu=b^{2}/4c<0$. For $b^{2}/4c<\mu<0$ a stable fixed point is encircled by an unstable limit cycle, which is in turn surrounded by a stable limit cycle. We refer to the situations with $\mu>0$ as the oscillatory sides of the bifurcations, the supercritical case of $\mu<0$ and subcritical case of $\mu<b^{2}/4c$ as the quiescent sides of the bifurcations, and the subcritical situation in which $b^{2}/4c<\mu<0$ as the coexistence region.

Equation 3 can be rewritten in the rotating frame of the forcing as

in which $y\equiv ze^{-i(\omega t+\theta)}$ and $\delta\omega\equiv\omega-\omega_{0}$ is the frequency detuning. Because we focus on the response of the oscillator relative to the forcing, the factor $e^{-i(\omega t+\theta)}$ has no effect on the results. In the case of complex Gaussian noise, for which the real and imaginary components are Gaussian distributed with variance $2d$, the probability density $P(y,t)$ for $y$ satisfies the corresponding Fokker-Planck equation:

in which $y\equiv y_{\mathrm{R}}+iy_{\mathrm{I}}$, $\partial_{\mathrm{R}}\equiv\partial/\partial y_{\mathrm{R}}$, $\partial_{\mathrm{I}}\equiv\partial/\partial y_{\mathrm{I}}$, and the radial coordinate $\rho(y_{\mathrm{R}},y_{\mathrm{I}})\equiv\sqrt{y_{\mathrm{R}}^{2}+y_{\mathrm{I}}^{2}}$ [18, 29]. Owing to the additive nature of the noise, the Stratonovich and Itô forms of the Fokker-Plank equation are identical. A special case of Eq. 5 corresponding to the unforced, isochronous, supercritical Hopf oscillator ($f=\delta\omega=b^{\prime}=c=c^{\prime}=0$) has been discussed previously [10].

The parameters $b^{\prime}$ and $c^{\prime}$ define a nonlinear relationship between the frequency of spontaneous oscillations and the control parameter. In the supercritical case the response to sinusoidal forcing when $b^{\prime}$ is nonzero can be multivalued and the forcing generates a large number of bifurcations [28, 30]. To capture the most basic effects of noise on a forced Hopf oscillator, we restrict further analysis to the case of $b^{\prime}=0$ and $c^{\prime}=0$, which defines an isochronous Hopf oscillator [6].

The solution for the steady-state distribution $P_{\mathrm{s}}(y)$ of $y$ in the case of zero detuning, $\delta\omega=0$, is given by

in which $N$ is a normalization constant, as reported previously for the supercritical case [18]. Although no closed-form solution for Eq. 5 is apparent when the detuning $\delta\omega$ is nonzero, we can find the steady-state distribution numerically using the exact solution for zero detuning to set the initial conditions and to approximate the boundary conditions (Eq. 6, A).

To find the average response, we consider the Fourier amplitude of $z(t)$ at the frequency $\omega_{\mathrm{n}}=2\pi n/T$

in which $T$ is the observation time for $z(t)$ and $n$ is an integer. As $T\to\infty$, the average Fourier amplitude $\langle\tilde{z}(\omega^{\prime})\rangle_{\mathrm{zs}}$ is nonzero only if $\omega^{\prime}=\omega$, in which $\langle\rangle_{\mathrm{zs}}$ denotes the long-time ensemble average (B). The magnitude of the average response amplitude at the frequency of driving, which we call the phase-locked amplitude, is then defined as

in which $|\langle\tilde{z}(\omega)\rangle_{\mathrm{zs}}|$ is finite for all values of $T$ including $T\to\infty$ and the right-hand side is a consequence of $P_{\mathrm{s}}$ being stationary (B). The symbol $\langle\rangle_{\mathrm{s}}$ denotes the stationary ensemble average in the rotating frame. The magnitude of the corresponding response function, also known as the sensitivity, is

The degree of entrainment is quantified by the vector strength

in which $\psi\equiv\phi-\omega t-\theta$ is the phase difference between the response and driving. The right-hand side of Eq. 10 is based on the assumption that the system is ergodic in the frame that rotates at the frequency of driving. In the presence of noise, the phase-locked amplitude and the vector strength are nonzero only if the forcing amplitude $f\neq 0$.

The phase-locked amplitude and vector strength signify two different aspects of a system’s response to sinusoidal driving. The phase-locked amplitude increases as entrainment to the stimulus grows, but can rise without limit after perfect entrainment has been achieved. Perfect entrainment is equivalent to a constant phase difference between the stimulus and the response. The vector strength is a bounded measure of entrainment, in which a value of unity indicates perfect entrainment and a value of zero implies none. The phase-locked amplitude and vector strength can be calculated numerically using the closed-form expression for the steady-state probability density (Eq. 6) when the forcing is tuned, or from the numerical solution to Eq. 5 when the detuning is nonzero.

We note that we can also calculate the average of the amplitude’s magnitude at the frequency of driving $\langle|y|\rangle_{\mathrm{s}}=\langle\rho\rangle_{\mathrm{s}}\geq|\langle y\rangle_{\mathrm{s}}|$ from the probability density. More generally, higher-order moments of the amplitude’s magnitude can be derived systematically from only three functions (C). For an unforced supercritical Hopf oscillator, analytical expressions for the moments can be derived iteratively. Because all moments of the amplitude’s magnitude can be nonzero in the absence of forcing, we focus in the remainder of this manuscript on the phase-locked amplitude and the vector strength.

Many studies of oscillators employ power spectra, residence-time distributions, and information metrics to demonstrate a system’s response to sinusoidal forcing [32, 5, 6]. Here we study two comparatively neglected measures, the phase-locked amplitude and the vector strength, that can be calculated directly from the probability density.

Because the phase-locked amplitude and the vector strength are zero in the absence of deterministic forcing, they allow a system to clearly distinguish sinusoidal forcing from fluctuations owing to noise. In contrast, the average of the amplitude’s magnitude $\langle\rho\rangle_{\mathrm{s}}$ is non-zero in the absence of input. If the goal of a detector is to determine whether a sinusoidal signal is present, the phase-locked amplitude and the vector strength suffice, whereas the average of the amplitude’s magnitude does not differentiate between the signal and the noise.

In the deterministic limit, the response of a supercritical Hopf oscillator is qualitatively different from that of a subcritical Hopf oscillator. A supercritical Hopf oscillator exhibits a single stable response to sinusoidal forcing. In contrast, a subcritical Hopf oscillator possesses two stable responses for a range of negative control-parameter values, and the response amplitude changes discontinuously at each end of this range. Bivalued or parametrically discontinuous responses are not desirable properties for a signal detector. In the presence of noise, the subcritical Hopf oscillator responds in a manner qualitatively similar to the supercritical Hopf oscillator: there is a single, averaged response at all values of the control parameter. The response within the deterministically-bivalued region is temporally irregular, however, because noise induces transitions between the stable loci. For this reason, a noisy oscillator may perform better as a detector of sinusoidal signals and be more robust to perturbations in parameter values when it operates near a supercritical Hopf rather than a subcritical Hopf bifurcation.

For simplicity, we specialize to the case in which the resonant frequency is independent of the forcing amplitude ($b^{\prime}=0$ and $c^{\prime}=0$). When $b^{\prime}\neq 0$ in the supercritical case and in the absence of forcing, an analytical solution to the Fokker-Planck equation exists, which can be used to demonstrate the non-Lorentzian form of the power spectrum [16]. For weak forcing and small detuning, an analytical solution to the Fokker-Planck equation permits the calculation of the linear response of a supercritical oscillator [21]. More generally, the Fokker-Planck equation (Eq. 5) in the instance of nonzero $b^{\prime}$ and $c^{\prime}$ can be solved numerically through the strategy that we employ here to solve the equation for detuned forcing. The exact solution that we report (Eq. 6) can be used to set the initial and boundary conditions. Although we postpone a description of this more general solution to a subsequent publication, we offer some comments here. In the deterministic limit for a self-oscillating supercritical Hopf oscillator, there can exist many stable responses at a single driving frequency when $b^{\prime}$ is sufficiently large [28, 30]. The response is undoubtedly even more complicated in the case of a subcritical Hopf oscillator when $b^{\prime}$ and $c^{\prime}$ are nonzero. By definition, however, the averaged response in the presence of noise is single-valued. Nonetheless, the stochastic dynamics of a Hopf oscillator in the most general case likely involves transitions between all stable responses.

In general, the effects of noise are small when $f\gg\sqrt{2d}$ and are great when $f\ll\sqrt{2d}$. The presence of any level of noise, however, changes the behavior of a Hopf oscillator qualitatively. For example, noise creates a peak in the vector strength as a function of the control parameter. To illustrate the effects of noise clearly in many figures, we choose $f\ll\sqrt{2d}$. This choice demonstrates that an oscillator can detect sinusoidal input in the presence of large amounts of noise. Such a regime may correspond to the threshold of hearing, for the sensory detectors of the auditory system apparently function as noisy Hopf oscillators driven by weak periodic forces owing to quiet sounds [33, 34].

A deterministic oscillator responds to sinusoidal forcing in a qualitatively different manner when self-oscillating than when quiescent. A quiescent oscillator is entrained by forcing of any amplitude and frequency, whereas a self-oscillating system is entrained only when the forcing is large and the detuning is small. Any level of noise eliminates this difference. The state diagrams of a forced Hopf oscillator operating on the quiescent side of the bifurcation are qualitatively similar to the diagrams corresponding to the oscillatory side.

Because the phase-locked amplitude at resonance and the sharpness of tuning of a quiescent supercritical Hopf oscillator increase as the system approaches the bifurcation, it has been suggested that sinusoidal-signal detectors, such as those in the auditory system, should operate near and on the quiescent side of a supercritical Hopf bifurcation [1, 2, 3]. We find, however, that the phase-locked amplitude at resonance and tuning sharpness continue to grow as the control parameter is raised even after the system crosses such a bifurcation. If the phase-locked amplitude and frequency selectivity of a sinusoidal-signal detector are required to be as great as possible, then an oscillator should be set to spontaneously oscillate far from a Hopf bifurcation. Moreover, noise has the largest effect on the phase-locked amplitude when the system operates near a Hopf bifurcation. These requirements and the effect of noise seem to imply that a detector of sinusoidal signals does not perform best when it operates near a Hopf bifurcation.

Detectors of sinusoidal signals, nevertheless, gain other advantages by operating close to a Hopf bifurcation. In the presence of noise, the phase-locked amplitude and the vector strength corresponding to detuned forcing peak when the system is close to the bifurcation. In addition, the range of stimulus frequencies that can be detected is maximized near the Hopf bifurcation. This frequency bandwidth is most robust to changes in the value of the control parameter on the larger-$\mu$ side of the bandwidth’s peak. For sufficiently large noise levels, an oscillator can detect detuned sinusoidal stimuli best when it spontaneously oscillates near a Hopf bifurcation.

In the presence of noise and detuning, the phase-locked amplitude, range of compression, degree of entrainment, and threshold bandwidth peak at different values of the control parameter near the critical point. A compromise must consequently be made depending on which features of the signal detector are essential for its function. For example, if very strong compression in a supercritical Hopf oscillator is desired then a spontaneously oscillating system that is operating far from the bifurcation should be chosen [20]. The oscillator’s detection bandwidth would be small, however, limiting its response to detuned stimuli. In contrast, if an oscillator is required to detect sinusoidal signals within a specific frequency bandwidth and to be robust to changes in parameter values, then the detector should be set to spontaneously oscillate near a Hopf bifurcation. Under these conditions, the oscillator is sharply tuned—but not too sharply tuned—and displays a wide dynamic range owing to compression of the input amplitude. For stimulus frequencies within the detection bandwidth, the detector’s phase-locked amplitude and degree of entrainment are large but are not maximized as functions of the control parameter.

In conclusion, we show that an active, noisy oscillator can detect sinusoidal stimuli best when it spontaneously oscillates near a Hopf bifurcation. To detect sinusoidal signals in a noisy environment, operation near criticality has apparently evolved in biological systems [35], might be enforced by homeostatic mechanisms [36], and can be employed in the construction of artificial devices [37].