text:
\section{Introduction\label{intro}}
The performance of a sinusoidal-signal detecter is enhanced if it operates quiescently in the proximity of a supercritical Hopf bifurcation \cite{choe98,eguiluz00,camalet00}. 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 \cite{choe98}. The range over which the input is nonlinearly compressed also rises as the system nears the bifurcation \cite{eguiluz00, camalet00}. 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 \cite{arnold03}. 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 \cite{lindner04,pikovsky03}. 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 \cite{longtin93,pei95,hohmann96}.

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 \cite{hempstead67,ushakov05} or with weakly correlated components in the supercritical case \cite{baras82}, or for particular types of multiplicative noise \cite{graham82a,zheng85,triana08}. 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 \cite{hempstead67,ushakov05}. 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 \cite{gleeson06}. In general, noise in the normal form of a random dynamical system is multiplicative, colored, and cross-correlated \cite{coullet85,arnold03}.

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 \cite{stratonovich67b}. 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 \cite{haken67}. 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 \cite{lindner09}. 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 \cite{julicher09}. 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 \cite{perez82,rajasekar88}.

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.

\section{Noisy Hopf Oscillators\label{Hopf}}
A Hopf bifurcation occurs when a fixed point changes stability at a critical value of a control parameter \cite{kuznetsov,guckenheimer83,strogatz}. 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 $z \equiv z_\mathrm{R} + i z_\mathrm{I}$. To linear order in $z$,
\begin{eqnarray}
\mathbf{x} &=& \mathbf{x^*} + \mathbf{c_\mathrm{R}}z_\mathrm{R} + \mathbf{c_\mathrm{I}}z_\mathrm{I}\\
&=& \mathbf{x^*} + \left[\mathbf{c_\mathrm{R}}\cos(\phi) + \mathbf{c_\mathrm{I}}\sin(\phi)\right]|z| \mathrm{,}
\end{eqnarray}
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 \cite{ipsen98}. 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:
\begin{equation}
\label{HopfNF}
\dot{z} = (\mu+i\omega_0)z+(b+ib')|z|^2 z+(c+ic')|z|^4 z+fe^{i(\omega t+\theta)}+\eta \mathrm{,}
\end{equation}
in which $\mu$ is the control parameter, $\omega_0$ is the natural frequency, and $b$, $b'$, $c$, and $c'$ 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 $\langle \eta(t) \rangle=0$, $\langle\eta(t)\eta(t')\rangle = 0$, and $\langle\eta(t)\eta^\mathrm{\dagger}(t')\rangle = 4d\delta(t-t')$, 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 \cite{stratonovich67b,gleeson06,julicher09}.

The correspondence between the Hopf oscillator defined by Eq.~\ref{HopfNF} and a system operating near a Hopf bifurcation generally relies on the forcing being weak and the noise being small, additive, and white \cite{julicher09,zhang11}. We can, however, also view Eq.~\ref{HopfNF} 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 \ref{HopfNF} can be rewritten in the rotating frame of the forcing as
\begin{equation}
\label{HopfNFrot}
\dot{y} = (\mu-i\delta\omega)y+(b+ib')|y|^2 y+(c+ic')|y|^4 y+f+\eta e^{-i(\omega t+\theta)} \mathrm{,}
\end{equation}
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:
\begin{eqnarray}
\label{FProt}
\partial_\mathrm{t} P
&=&-\partial_\mathrm{R}\left[\left(\mu y_\mathrm{R}+\delta\omega y_\mathrm{I}+(by_\mathrm{R}-b'y_\mathrm{I})\rho^2+(cy_\mathrm{R}-c'y_\mathrm{I})\rho^4+f\right)P-d\partial_\mathrm{R}P\right]\nonumber\\
&&-\partial_\mathrm{I}\left[\left(\mu y_\mathrm{I}-\delta\omega y_\mathrm{R}+(by_\mathrm{I}+b'y_\mathrm{R})\rho^2+(cy_\mathrm{I}+c'y_\mathrm{R})\rho^4\right)P-d\partial_\mathrm{I}P\right]\mathrm{,}
\end{eqnarray}
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 $\rho(y_\mathrm{R},y_\mathrm{I}) \equiv \sqrt{y_\mathrm{R}^2+y_\mathrm{I}^2}$ \cite{stratonovich67b,gardiner}. Owing to the additive nature of the noise, the Stratonovich and It\^o forms of the Fokker-Plank equation are identical. A special case of Eq.~\ref{FProt} corresponding to the unforced, isochronous, supercritical Hopf oscillator ($f = \delta \omega = b' = c = c' = 0$) has been discussed previously \cite{hempstead67}.

The parameters $b'$ and $c'$ define a nonlinear relationship between the frequency of spontaneous oscillations and the control parameter. In the supercritical case the response to sinusoidal forcing when $b'$ is nonzero can be multivalued and the forcing generates a large number of bifurcations \cite{zhang11,wiser15}. To capture the most basic effects of noise on a forced Hopf oscillator, we restrict further analysis to the case of $b'=0$ and $c'=0$, which defines an isochronous Hopf oscillator \cite{pikovsky03}.

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
\begin{equation}
\label{sdist}
P_\mathrm{s} = N\mathrm{exp}\left(\frac{\mu\rho^2}{2d}+\frac{b\rho^4}{4d}+\frac{c\rho^6}{6d}+\frac{fy_\mathrm{R}}{d}\right)\mathrm{,}
\end{equation}
in which $N$ is a normalization constant, as reported previously for the supercritical case \cite{stratonovich67b}. Although no closed-form solution for Eq.~\ref{FProt} 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.~\ref{sdist}, \ref{numericalFPApp}).

To find the average response, we consider the Fourier amplitude of $z(t)$ at the frequency $\omega_\mathrm{n} = 2\pi n/T$
\begin{equation}
\label{FT}
\tilde{z}(\omega_\mathrm{n})\equiv\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} e^{-i\omega_\mathrm{n}t}z(t)\,dt\mathrm{,}
\end{equation}
in which $T$ is the observation time for $z(t)$ and $n$ is an integer.
As $T \to \infty$, the average Fourier amplitude $\langle\tilde{z}(\omega')\rangle_\mathrm{zs}$ is nonzero only if $\omega'=\omega$, in which $\langle \rangle_\mathrm{zs}$ denotes the long-time ensemble average (\ref{chimagApp}).
The magnitude of the average response amplitude at the frequency of driving, which we call the phase-locked amplitude, is then defined as
\begin{equation}
\label{PLres}
R \equiv |\langle \tilde{z}(\omega) \rangle_\mathrm{zs}| = |\langle y\rangle_\mathrm{s}|\mathrm{,}
\end{equation}
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 (\ref{chimagApp}). 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
\begin{equation}
\label{chimag}
|\tilde{\chi}(\omega)| \equiv \frac{|\langle \tilde{z}(\omega) \rangle_\mathrm{zs}|}{f}=\frac{|\langle y\rangle_\mathrm{s}|}{f}\mathrm{.}
\end{equation}
The degree of entrainment is quantified by the vector strength
\begin{equation}
\label{VS}
V \equiv \left|\lim_{T\to\infty} \frac{1}{T}\int_0^\mathrm{T} e^{i\psi(t)}\,dt\right| = \left|\langle e^{i\psi}\rangle_\mathrm{s}\right|\mathrm{,}
\end{equation}
in which $\psi \equiv \phi - \omega t - \theta$ is the phase difference between the response and driving.
The right-hand side of Eq.~\ref{VS} 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 $f\neq0$.

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.~\ref{sdist}) when the forcing is tuned, or from the numerical solution to Eq.~\ref{FProt} 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 (\ref{amplmomApp}). 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.
\begin{figure}[t]
\centering
\includegraphics{ChiandVS_vsmu_tuned}
\caption{The phase-locked amplitude and vector strength are shown as functions of the control parameter $\mu$ at the natural frequency near supercritical (a, b) and subcritical (c, d) Hopf bifurcations. The noise level is $d=0$ (black, dashed), $d=0.01$ (red), $d=0.1$ (green), or $d=1$ (blue). In the deterministic subcritical case in panel (c), two stable responses lines exist for a limited range of control-parameter values. The Hopf bifurcations occur at $\mu=0$ (gray, dashed) and a saddle-node of limit cycles bifurcation occurs at $\mu =  b^2/4c = -0.25$ (magenta, dashed) in the subcritical case. The forcing amplitude is $f=0.01$. Forcing has little effect on the phase-locked amplitude for large control parameter values, for which the amplitude increases as $\mu^{1/2}$ in the supercritical case and as $\mu^{1/4}$ for a subcritical oscillator. For all figures in this manuscript the natural frequency $\omega_0 = 1$, $b = -1$ and $c = 0$ in the supercritical case, or $b = 1$ and $c = -1$ in the subcritical case. \label{chiVSvsmu_fig}}
\end{figure}
\subsection{Sinusoidal-signal detection as a function of the control parameter}
\subsubsection{For tuned forcing, the phase-locked amplitude and vector strength grow monotonically as the control parameter increases.}
In the presence or absence of noise, the phase-locked amplitude of a resonantly forced Hopf oscillator grows as the control parameter increases (Fig.~\ref{chiVSvsmu_fig}). The closer the system operates to the Hopf bifurcation, however, the greater is the influence of noise. Owing to the rise in the amplitude of spontaneous oscillations as the control parameter grows, the effects of noise diminish. Noise likewise does not affect the phase-locked amplitude significantly in the limit of very negative values of the control parameter, for which the response to both deterministic and stochastic input is small.

In the subcritical case, a single averaged response replaces the two stable responses seen in the deterministic limit when $9 b^2/20 c < \mu < 0$ and $f_\mathrm{1}(\mu,b,c) < f < f_\mathrm{2}(\mu,b,c)$ (\ref{perfectlockApp}). For some noise levels, the phase-locked amplitude is greater than the deterministic response, but this enhancement depends on the control parameter's value.

In the absence of noise, the system is perfectly entrained by the stimulus for all values of the control parameter. In contrast to the deterministic limit, the vector strength of a noisy oscillator rises as the control parameter is increased. Noise thus introduces a qualitative change in the response of the system to sinusoidal stimuli.

For tuned forcing, $\langle y_\mathrm{R}\rangle_\mathrm{s} > 0$ and $\langle y_\mathrm{I}\rangle_\mathrm{s} = 0$ such that $|\langle y \rangle_\mathrm{s}| = \langle y_\mathrm{R} \rangle_\mathrm{s}$. Thus we find
\begin{eqnarray}
\frac{\partial |\langle y \rangle_\mathrm{s}|}{\partial \mu} &=& \frac{\partial \langle y_\mathrm{R}\rangle_\mathrm{s}}{\partial \mu}\nonumber\\
&=& \int^\infty_{-\infty} \int^\infty_{-\infty} \frac{\partial P_\mathrm{s}}{\partial \mu} y_\mathrm{R} dy_\mathrm{R} dy_\mathrm{I}\nonumber\\
&=& \frac{\langle \rho^2 y_\mathrm{R} \rangle_\mathrm{s} - \langle \rho^2 \rangle_\mathrm{s}\langle y_\mathrm{R} \rangle_\mathrm{s}}{2d}\nonumber\\
&=& \frac{\mathrm{cov}(y_\mathrm{R}, \rho^2)}{2 d}\mathrm{,}
\end{eqnarray}
in which $\mathrm{cov}()$ is the covariance under the distribution $P_\mathrm{s}$. It can be shown similarly that
\begin{eqnarray}
\frac{\partial |\langle e^{i\psi} \rangle_\mathrm{s}|}{\partial \mu} &=& \frac{\mathrm{cov}(y_\mathrm{R}/\rho, \rho^2)}{2 d}\mathrm{.}
\end{eqnarray}
The rise in the responsiveness measures as a function of $\mu$ is a consequence of the specific form of the distribution $P_\mathrm{s}$, for which $\mathrm{cov}(y_\mathrm{R}, \rho^2) > 0$ and $\mathrm{cov}(y_\mathrm{R}/\rho, \rho^2) > 0$.

Because the phase-locked amplitude and the vector strength grow as the control parameter is increased, a resonantly forced, noisy, self-oscillating Hopf oscillator performs better as a detector of sinusoidal forces the \emph{further} it operates from a Hopf bifurcation. The oscillator may be required to detect a range of stimulus frequencies, however, leading us to examine its response to detuned forcing.

\subsubsection{For detuned forcing, the phase-locked amplitude and the stochastic system's vector strength peak near a Hopf bifurcation.}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{ChiandVS_vsmu_detuned}
\caption{The phase-locked amplitude and vector strength are shown as functions of the control parameter for detuned forcing at $f=0.01$. (a, b, e, f) The response for forcing at a frequency below the natural frequency $\omega=0.9\,\omega_0$ is show for a noise level of $d = 0$ (dashed black), $d = 0.01$ (red), $d=0.1$ (green), or $d= 1$ (blue). (c, d, g, h) The response in the presence of noise ($d=0.01$) is shown for different amounts of detuning $|\delta\omega| = 0$ (black), $|\delta\omega| = 0.05\,\omega_0$ (red), or $|\delta\omega| = 0.1\,\omega_0$ (green). (c, d) The supercritical response in the absence of noise is shown for the detuning $|\delta\omega| = 0$ (black, dashed), $|\delta\omega| = 0.05\,\omega_0$ (red, dashed), or $|\delta\omega| = 0.1\,\omega_0$ (green, dashed). In the subcritical case, Hopf bifurcations occur at $\mu=0$ (gray, dashed) and a saddle-node of limit cycles bifurcation occurs at $\mu =  b^2/4c$ (magenta, dashed).
\label{chiVSvsmu_detuned_fig}}
\end{figure*}

A Hopf oscillator responds to detuned forcing in a qualitatively different manner than it does to tuned forcing. In the presence of noise, the phase-locked amplitude and the vector strength peak as functions of the control parameter for all non-zero levels of noise and detuning (Fig.~\ref{chiVSvsmu_detuned_fig}). For both types of bifurcation, the peaks move to larger values of the control parameter $\mu$ and their magnitudes decrease as the noise level increases. Owing to the limited range of $\mu$ shown, it may appear as if the phase-locked amplitude and vector strength plateau for large values of the control parameter and that they do not peak at finite value of $\mu$. We argue, however, that $R \to 0$ and $V \to 0$ as $\mu \to \pm \infty$ (\ref{CPlimApp}). Consequently, the responsiveness measures for detuned forcing must peak at finite values of the control parameter.

The mechanism that creates the peak in the phase-locked amplitude is clearest for a supercritical Hopf bifurcation without noise. Starting on the quiescent side of the bifurcation, the system is perfectly entrained by the stimulus and, owing to increasing proximity to the bifurcation, the phase-locked amplitude grows as the control parameter is raised. The response continues to rise with the control parameter after the system passes through the Hopf bifurcation until entrainment declines owing to a Hopf or saddle-node bifurcation of the forced system (\ref{perfectlockApp}). The phase-locked amplitude thereafter decreases as detuned forcing fails to entrain the oscillator. It is increasingly difficult to entrain an oscillator to a detuned stimulus as the amplitude of spontaneous motion rises. As for the tuned case, the phase-locked amplitude is affected by noise only when the system is close to a supercritical Hopf bifurcation.

For the deterministic case, the vector strength drops monotonically once the control parameter exceeds a critical value. In contrast, the vector strength peaks in the presence of noise owing to its decline for increasingly negative values of the control parameter. The oscillator's response to sinusoidal forcing decreases and is consequently more corrupted by noise. We also note that the maximum in the vector strength occurs at a value of the control parameter different from that at which the phase-locked amplitude peaks.

In the subcritical case, the stochastic phase-locked amplitude and vector strength peak near the saddle node of limit cycles bifurcation that is associated with the subcritical Hopf bifurcation.

Finally, the positions and sizes of the peaks for the phase-locked amplitude and vector strength depend only on the magnitude of the detuning and not on its sign (Fig.~\ref{chiVSvsmu_detuned_fig}). The phase-locked amplitude and vector strength for detuned forcing asymptotically approach the tuned limit, however, as the control parameter declines.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{ChiandVS_vsw}
\caption{(a-f) The phase-locked amplitude and vector strength are shown as functions of the driving frequency for a force of $f=0.01$ and values of the control parameter $\mu = 1$ (red), $\mu = 0$ (green), or $\mu = -1$ (blue). (a, b) The deterministic supercritical case. (c, d) The stochastic supercritical case. (e, f) The stochastic subcritical case. (g) The quality factor is shown as a function of control parameter for the supercritical case with (solid line) and without (dashed line) noise. The Hopf bifurcation occurs at $\mu=0$ (gray, dashed). (h) The quality factor is shown as a function of the control parameter for the subcritical bifurcation in the presence of noise. A Hopf bifurcation occurs at $\mu=0$ (gray, dashed) and a saddle-node of limit cycles bifurcation occurs at $\mu =  b^2/4c$ (magenta, dashed). In all panels, the noise level for the stochastic cases is $d=0.01$.
\label{chiVSvsw_fig}}
\end{figure*}
\subsection{Phase-locked amplitude and entrainment as functions of the stimulus frequency}
The phase-locked amplitude and vector strength depend on the driving frequency for supercritical and subcritical Hopf oscillators (Fig.~\ref{chiVSvsw_fig}). In the absence of noise, the phase-locked amplitude of a supercritical system displays a maximum as a function of frequency for all values of the control parameter. In contrast, the vector strength is unity for all frequencies on the quiescent side of the bifurcation, but is frequency-tuned on the oscillatory side. In the presence of noise, however, both quantities peak at the natural frequency for all values of $\mu$.

There is a qualitative difference between frequency tuning on the oscillatory and quiescent sides of a Hopf bifurcation. For a quiescent oscillator, the phase-locked amplitude increases at all stimulus frequencies the closer the oscillator is to the bifurcation. In contrast, the phase-locked amplitude of a self-oscillating system at stimulus frequencies far from the resonant frequency \emph{decreases} as the control parameter increases. This difference is preserved when noise is taken into account.

The sharpness of frequency tuning can be quantified by calculating the quality factor $Q \equiv \omega_\mathrm{max}/\Delta\omega$, in which $\omega_\mathrm{max}$ is the frequency at which the phase-locked amplitude peaks at $R_\mathrm{max}$ and $\Delta\omega$ is the width of the frequency range over which $R>R_\mathrm{max}/\sqrt{2}$. The larger the quality factor, the greater the frequency selectivity. With or without noise, the sharpness of frequency tuning grows as the control parameter $\mu$ increases. In the supercritical case, noise decreases the frequency selectivity at all control-parameter values. The effects of noise are greatest, however, near the Hopf bifurcation. Frequency tuning approaches the deterministic limit for very positive or negative values of the control parameter.

\begin{figure}[t]
\centering
\includegraphics{TH_BW}
\caption{The threshold bandwidth peaks as a function of the control parameter. (a) The phase-locked amplitude for a deterministic supercritical Hopf bifurcation depends on the stimulus frequency for a control-parameter value of $\mu = 1$ (orange), $\mu = 0$ (cyan), or $\mu = -1$ (purple). The threshold bandwidth $B$ is the range of frequencies for which the phase-locked amplitude exceeds a threshold $R_\mathrm{T}$. The amplitude does not exceed the threshold of 0.1 for $\mu = -1$ (purple), but the threshold bandwidth is smaller when the control parameter is 1 (orange) than when it is 0 (cyan). (b) The threshold bandwidth in the absence of noise peaks near a supercritical Hopf bifurcation (gray, dashed) for a threshold of 0.1 (red), 0.2 (green), or 0.4 (blue). (c, d) The threshold bandwidth peaks near a Hopf bifurcation (gray, dashed) when the noise level $d=0.01$ for a threshold of 0.025 (red), 0.05 (green), or 0.1 (blue). (d) A saddle node of limit cycles bifurcation occurs at $\mu =  b^2/4c$ (magenta, dashed). The stimulus force $f=0.01$ for all panels.
\label{BWmu_fig}}
\end{figure}

\subsection{Maximal threshold bandwidth near a Hopf bifurcation.}
As we have seen, a Hopf oscillator responds best to sinusoidal stimuli with frequencies near its resonant frequency. The range of stimulus frequencies for which the stimulus is detected depends on the control parameter's value. We suppose that an oscillator detects a stimulus if the magnitude of its phase-locked amplitude equals or exceeds a threshold $R \geq R_\mathrm{T}$. We define the \textit{threshold bandwidth} $B$ to be the size of the frequency range for which detection occurs, that is, $B \equiv |\{\omega: R \geq R_\mathrm{T}\}|$ (Fig.~\ref{BWmu_fig}a).

In the presence or absence of noise, the threshold bandwidth is greatest when a supercritical Hopf system operates close to and on the oscillatory side of the bifurcation (Fig.~\ref{BWmu_fig}). As the threshold increases, the bandwidth declines and the bandwidth's peak moves to larger values of the control parameter. When we compare the bandwidth for a specific threshold in the absence of noise to that in the presence of noise, we see that adding noise decreases the size of the peak and shifts it to greater values of the control parameter. The threshold bandwidth eventually falls to zero for sufficiently positive or negative control parameters.

The threshold bandwidth of a stochastic subcritical Hopf system depends on the control parameter in a manner similar to that of a system possessing a supercritical Hopf bifurcation. The main difference is that the bandwidth peak occurs near the saddle node of limit cycles bifurcation, that is, at the boundary of the region possessing spontaneous oscillations. On the side of the peak closer to the oscillatory region, the threshold bandwidth initially rises slowly as the control parameter increases further. The bandwidth finally falls to zero, however, as the system moves far from the bifurcation in either the quiescent or the oscillatory direction.
\begin{figure}[H]
\centering
\includegraphics{ChiandVS_vsf_tuned}
\caption{The response function, vector strength, phase-locked amplitude, and an approximation's relative error for forcing at the natural frequency near supercritical (a--d) and subcritical (e--h) bifurcations are shown as functions of the forcing amplitude. Stable responses are depicted in the absence of noise (dashed lines) and in the presence of noise ($d = 0.01$, solid lines).  (a, b, c, d) The supercritical control parameter is $\mu=1$ (red), $\mu=0$ (green), or $\mu=-1$ (blue). (e, f, g, h) The subcritical control parameter is $\mu=1$ (red), $\mu=-0.2$ (green), or $\mu=-1$ (blue). (a, e) The black dotted lines labeled by the number $\alpha$ indicate the range of forces corresponding to the power law $|\tilde{\chi}|\sim f^\alpha$. (e) In the absence of noise, two stable responses exist for low forcing amplitudes (green dashed lines). (b, f) Because the vector strength is unity for all forcing amplitudes and control-parameter values in the deterministic cases, it is shown as a black dashed line. (c, g) The phase-locked amplitude (solid lines) and an approximation based on the vector-strength (Eqs.~\ref{chilinandVSlin} and \ref{chiandVSinfty}, solid points) are shown. (d, h) The error of the approximation to the phase-locked amplitude relative to the exact calculation is illustrated (Eq.~\ref{yfromVSerror}).
\label{chiVSvsf_fig}}
\end{figure}
In all cases, the threshold bandwidth is very sensitive to changes in the control parameter on the side of the peak closer to the quiescent region. This sensitivity arises from the failure of the system to achieve threshold for any stimulus frequency when it operates on the quiescent side of and far from a Hopf bifurcation. The gradual drop on the oscillatory side of the peak reflects the loss of entrainment that occurs as the system's spontaneous oscillations grow in amplitude. To ensure a large threshold bandwidth while maintaining robustness of the bandwidth to parameter changes, it is consequently best to position the operating point of a Hopf system near to but on the larger-$\mu$ side of the bandwidth's peak. In summation, the range of frequencies that an oscillator can detect is maximized when it spontaneously oscillates near a Hopf bifurcation.

\subsection{Responsiveness as a function of the forcing amplitude}
\subsubsection{Noise allows an oscillator to detect small tuned forces.}
Many of the phenomena we have discussed thus far rely on the forcing amplitude being small. We now examine how sinusoidal-signal detection depends on the signal's amplitude. The response function captures the sensitivity of an oscillator to a change in the forcing amplitude (Eq.~\ref{chimag}). For tuned sinusoidal forcing with $f \gg \sqrt{2d}$, the system's responsiveness depends little on the control parameter and is similar to the deterministic limit (Fig.~\ref{chiVSvsf_fig}). The vector strength is close to unity and the response function decreases with a power law characteristic of the bifurcation type, $|\tilde{\chi}|\sim f^{-2/3}$ in the supercritical case and $|\tilde{\chi}|\sim f^{-4/5}$ in the subcritical case \cite{eguiluz00,camalet00,lindner09}. Far from the bifurcation, noise has little effect on the response function of a quiescent oscillator, but the vector strength grows to unity as the forcing rises.

On the oscillatory sides of the bifurcations, noise reduces the response function and the vector strength and changes the power law for weak forces from $|\tilde{\chi}|\sim f^{-1}$ to $|\tilde{\chi}|\sim f^{0}$. This linear-response regime is associated with a decline in entrainment owing to noise and is coincident with a range of forces for which the vector strength increases linearly. In the subcritical case, the two deterministic responses observed for $9 b^2/20 c < \mu < 0$ and $f_\mathrm{1}(\mu,b,c) < f < f_\mathrm{2}(\mu,b,c)$ are replaced by a single averaged response (\ref{perfectlockApp}). Our calculations agree with analytical results for autonomously oscillating systems based on small fluctuations in $\rho$ owing to noise, in which case the compressive exponents can also be calculated analytically \cite{lindner09}. For weakly forced quiescent oscillators, however, noisy fluctuations in $\rho$ cannot be neglected, but we find the sensitivity's dependence on the forcing amplitude to be similar to that on the oscillatory sides of the bifurcations.

The decrease in the phase-locked amplitude with greater forcing allows the oscillator to compress many orders of magnitude in input into a significantly smaller output range. This compressive region extends to smaller forces as the control parameter increases. In the deterministic case, compression following the power law $|\tilde{\chi}|\sim f^{-1}$ is not useful for signal detection, for the oscillator does not respond to the stimulus. In the stochastic instance, however, weak forces can be discriminated, because the vector strength and the phase-locked amplitude increase linearly with the forcing amplitude.

Analytical expressions for the response function and the vector strength on either side of the bifurcation can be found for weak forcing by expanding the exact solution for the steady-state distribution (Eq.~\ref{sdist}) to linear order in $f$, yielding
\begin{equation}
\label{chilin}
|\tilde{\chi}_\mathrm{l}| \equiv \frac{\left|\langle y_\mathrm{R}y\rangle_\mathrm{s0}\right|}{d}
\end{equation}
and
\begin{equation}
\label{VSlin}
V_\mathrm{l} \equiv \frac{\left|\langle y_\mathrm{R}e^{i\psi}\rangle_\mathrm{s0}\right|f}{d}\mathrm{,}
\end{equation}
in which $\langle\rangle_\mathrm{s0}$ represents averaging over the steady-state ensemble in the absence of forcing (\ref{chimagApp}). Equation \ref{chilin} relates the linear response $\tilde{\chi}_\mathrm{l}$ to the covariance of the unstimulated system $\langle y_\mathrm{R}y\rangle_\mathrm{s0}$ and is essentially a fluctuation-dissipation relation for Hopf oscillators \cite{landau_lifshitzStatPhys}. This expression accords with previous estimates that either assume that the noise has little effect on $\rho$ or specialize to the supercritical bifurcation \cite{lindner09,julicher09}.

The expressions for the linear response (Eqs.~\ref{chilin} and \ref{VSlin}) can be used to relate the magnitude of the phase-locked amplitude for weak forcing $|\langle y\rangle_\mathrm{sl}|$ to $V_\mathrm{l}$ as
\begin{equation}
\label{chilinandVSlin}
|\langle y\rangle_\mathrm{sl}| = \frac{\left|\langle y_\mathrm{R}y\rangle_\mathrm{s0}\right|V_\mathrm{l}}{\left|\langle y_\mathrm{R}e^{i\psi}\rangle_\mathrm{s0}\right|}\mathrm{.}
\end{equation}
For large forcing a similar relationship holds between the phase-locked amplitude and the vector strength because the system's response becomes essentially deterministic. From Eq.~\ref{VS},
\begin{eqnarray}
\label{chiandVSinfty}
\left|\langle y\rangle_\mathrm{s\infty}\right|
&=& \frac{\left|\langle y\rangle_\mathrm{s\infty}\right|\left |\langle y_\mathrm{R}\rangle_\mathrm{s\infty}\right|V_\mathrm{\infty}}{\left|\langle y_\mathrm{R}\rangle_\mathrm{s\infty}\right|\left|\langle e^{i\psi}\rangle_\mathrm{s\infty}\right|}\nonumber\\
&=& \frac{\left|\langle y_\mathrm{R} y \rangle_\mathrm{s\infty}\right|V_\mathrm{\infty}}{\left|\langle y_\mathrm{R} e^{i\psi}\rangle_\mathrm{s\infty}\right|} \mathrm{,}
\end{eqnarray}
in which the subscript $\infty$ denotes averaging over the steady-state ensemble in the limit of arbitrarily large forcing. An approximation for $R$ is thus
\begin{equation}
\label{RfromV}
R_\mathrm{V} = \frac{\left|\langle y_\mathrm{R}y\rangle_\mathrm{s}\right|V}{\left|\langle y_\mathrm{R}e^{i\psi}\rangle_\mathrm{s}\right|}\mathrm{.}
\end{equation}
The magnitude of the phase-locked amplitude agrees with $R_\mathrm{V}$ independent of the value of the control parameter for small (Eq.~\ref{chilinandVSlin}) and large (Eq.~\ref{chiandVSinfty}) forcing, but the agreement is worse for intermediate forces at which the approximation based on $V$ systematically exceeds the true phase-locked amplitude (Fig.~\ref{chiVSvsf_fig}).
The relative error is given by
\begin{eqnarray}
\label{yfromVSerror}
\frac{R_\mathrm{V}-R}{R}\mathrm{,}
\end{eqnarray}
whose peak rises as the control parameter declines. We thus encounter an unusual situation in which an approximation is best in the limits of small and large inputs, but is least accurate for intermediate values. The relation between $R$ and $V$ can be expressed formally, however, as
\begin{eqnarray}
R &=&\frac{\left\langle \rho I_\mathrm{1} \left(\frac{f\rho}{d}\right) \right\rangle_\mathrm{s0} V}{\left\langle I_\mathrm{1} \left(\frac{f\rho}{d}\right) \right\rangle_\mathrm{s0}}\mathrm{,}
\label{PLvVSBessel}
\end{eqnarray}
in which $I_\mathrm{n}(x)$ is the modified Bessel function of the first kind of integer order $n$ (\ref{chimagApp}).

\begin{figure}[t]
\centering
\includegraphics{ChiandVS_vsf_detuned}
\caption{The response function and vector strength are shown as functions of the forcing amplitude at a frequency below the natural frequency, $\omega=0.9\,\omega_0$, near supercritical (a, b) and subcritical (c, d) bifurcations. The stochastic curves ($d = 0.01$, solid lines) are plotted only for the forcing levels $f\leq1$ because they are indistinguishable from the deterministic curves (dashed lines) for greater forcing. (a, b) The supercritical control parameter is $\mu=1$ (red), $\mu=0$ (green), or $\mu=-1$ (blue). The deterministic vector strength is shown using a black dashed line when $\mu=0$ and $\mu=-1$ because it is unity for all forces. (c, d) The subcritical control parameter is $\mu=1$ (red), $\mu=-0.2$ (green), or $\mu=-1$ (blue). The deterministic lines are shown only for $f\geq1$; two stable responses exist for lower levels of forcing. (d) For $f\geq1$, the vector strength is essentially unity (black dashed). (a, c) The black dotted lines labeled by the number $\alpha$ indicate the range of forces corresponding to the power law $|\tilde{\chi}|\sim f^\alpha$.
\label{chiVSvsf_detuned_fig}}
\end{figure}

\subsubsection{Small detuned forces are best detected near a Hopf bifurcation.}
When the forcing level is large, the responses to frequency-detuned forcing are similar to those to tuned forcing (Fig.~\ref{chiVSvsf_detuned_fig}). There are, however, several qualitative differences for weak forcing. On the oscillatory side of a deterministic supercritical Hopf bifurcation, the phase-locked amplitude can possess a peak as a function of forcing level owing to a decrease in entrainment at a critical forcing amplitude (\ref{perfectlockApp}). In addition, the system possesses a linear response on the oscillatory side of the bifurcations in both the deterministic and stochastic cases. With or without noise, the vector strength increases linearly with the forcing amplitude on the oscillatory side of the bifurcation and is larger when the system operates near a Hopf bifurcation than when it operates further away. The deterministic and stochastic systems are consequently most sensitive to small forces when they operate near a Hopf bifurcation.

\begin{figure}[t]
\centering
\includegraphics{ArnoldTongueDet}
\caption{State diagrams of the driven deterministic system as a function of the forcing amplitude and frequency. (a, c) Wedge-shaped regions of perfect entrainment are bounded below by bifurcation lines of the forced system for different control-parameter values. Thick lines denote Hopf bifurcations and thin lines correspond to saddle-node bifurcations. The  full set of bifurcation curves is not shown \cite{pikovsky03}. (a) The value of the control parameter is $\mu = 1$ (red), $\mu = 0.5$ (green), or $\mu = 0.1$ (blue). Thick lines represent supercritical Hopf bifurcations. (c) The control-parameter value is $\mu = 1$ (red), $\mu = 0.1$ (green), or $\mu = -0.2$ (blue, cyan). When $\mu = 1$ or $\mu = 0.1$, thick lines demarcate supercritical Hopf  bifurcations. When $\mu = -0.2$, the system is imperfectly locked in the belts delimited by contours of supercritical Hopf (blue thick), subcritical Hopf (cyan thick), and saddle-node bifurcations (blue thin). At this value of the control parameter, a perfectly phase-locked amplitude coexists with an imperfectly locked response for the regions demarcated above by saddle-node bifurcation lines and lines of subcritical Hopf bifurcations. (b, d) The vector strength is shown as a function of the stimulus amplitude and frequency in the case of $\mu = 1$. The analytical boundaries for perfect phase locking (red) bound the area for which $V = 1$.\label{ArnoldTongueDet_fig}}
\end{figure}
\subsection{Entrainment as a function of the stimulus amplitude and frequency\label{tongue_sect}}
Quiescent deterministic systems can be perfectly entrained by sinusoidal forcing of any amplitude and frequency. In contrast, self-oscillatory systems exhibit perfect entrainment for a limited range of stimulus amplitudes and frequencies illustrated by state diagrams of the forced system (Fig.~\ref{ArnoldTongueDet_fig} and \ref{perfectlockApp}). The region of perfect entrainment can be calculated analytically and is demarcated by lines corresponding to bifurcations of the forced system. For large forces, the width of this wedge-shaped region is less than that of the one-to-one Arnold tongue over which the phase difference $\psi$ between the system's response and stimulus is bounded \cite{pikovsky03}. On the oscillatory sides of the Hopf bifurcations, the wedge is delimited by lines of saddle-node bifurcations for small forcing amplitudes and detuning, and by lines of supercritical Hopf bifurcations for large forcing and detuning.

For a set of control-parameter values near a subcritical Hopf bifurcation, there can be two stable responses for low forcing amplitudes and sufficiently small detuning. In the case of large detuning, a stable response can coexist with a stable limit cycle, a situation that corresponds to less-than-perfect entrainment. Depending on the initial conditions, $\psi$ is either constant, corresponding to $V=1$, or time-dependent, in which case $V<1$. For both types of Hopf oscillator, however, the width of the perfect-entrainment sector decreases as the control parameter grows, yielding sharper frequency selectivity at all forcing amplitudes.

Outside the area of perfect entrainment for a self-oscillating system, the vector strength decreases as the forcing falls or the detuning rises. Wedge-shaped regions in which entrainment exceeds a threshold can still be defined, however, by contours of constant vector strength (Fig.~\ref{ArnoldTongueDet_fig}). These wedges asymptotically collapse onto the region of perfect entrainment as the forcing amplitude declines to zero. In the coexistence region near a subcritical Hopf oscillator, the contours of constant vector strength depend on the choice of initial conditions.
\begin{figure}
\centering
\includegraphics[height=6in]{ArnoldTongueStoch}
\caption{State diagrams of the driven stochastic system ($d = 0.01$) as a function of the forcing amplitude and frequency. (a, b) Contours of constant vector strength are shown for $\mu = 1$. The boundaries for perfect phase locking in the deterministic limit are shown (red). (c, d) Contours for which the phase-locked amplitude is constant are shown for $\mu = 1$. (e, f) Contours defined by a constant value for the response function are shown for $\mu = 1$. The $|\tilde{\chi}| = 2$ contours are approximated by lines of constant detuning from $f=0.1$ to $f=0.001$ (black dashed). (g) Contours of constant vector strength are shown for $\mu = -0.5$. (h) Contours of constant vector strength are shown for $\mu = -0.2$, at which there is coexistence between a stable fixed point and a stable limit cycle. The intensity scale for the vector strength above panels a and b also applies to panels g and h. Small irregularities are evident for some contours owing to numerical error.\label{ArnoldTongueStoch_fig}
}
\end{figure}
For the stochastic cases, larger forcing amplitudes and less detuning are required to achieve a threshold level of entrainment as the noise increases (Fig.~\ref{ArnoldTongueStoch_fig}). Unlike the situation for the deterministic limit, entrainment is not perfect on the quiescent sides of the bifurcations. Noise simplifies the state diagrams corresponding to the coexistence region near a subcritical Hopf bifurcation: there is a single entrainment wedge for each value of the stochastic system's vector strength. As for the cases without noise, the entrainment wedges broaden and rise to larger forcing amplitudes as the control parameter decreases.

The state diagrams for the phase-locked amplitude have a structure similar to those corresponding to the vector strength. The phase-locked amplitude increases if either the forcing grows or the detuning falls. In contrast, response-function state diagrams possess contours that change shape as the value of $|\tilde{\chi}|$ declines. For small detuning, raising the forcing amplitude decreases the response function. Sufficiently large detuning, however, creates a peak in the response function at a specific forcing amplitude (Fig.~\ref{chiVSvsf_detuned_fig}), corresponding to an increase in the width of the region enclosed by a contour as the forcing rises.

\section{Discussion\label{discussion}}
Many studies of oscillators employ power spectra, residence-time distributions, and information metrics to demonstrate a system's response to sinusoidal forcing \cite{gammaitoni98,lindner04,pikovsky03}. 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 ($b' = 0$ and $c' = 0$). When $b' \neq 0$ 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 \cite{gleeson06}. 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 \cite{julicher09}. More generally, the Fokker-Planck equation (Eq.~\ref{FProt}) in the instance of nonzero $b'$ and $c'$ 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.~\ref{sdist}) 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 $b'$ is sufficiently large \cite{zhang11,wiser15}. The response is undoubtedly even more complicated in the case of a subcritical Hopf oscillator when $b'$ and $c'$ 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 \cite{salvi15,salvi16}.

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 \cite{choe98,eguiluz00,camalet00}. 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 \emph{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 \cite{lindner09}. 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 \emph{spontaneously oscillates} near a Hopf bifurcation. To detect sinusoidal signals in a noisy environment, operation near criticality has apparently evolved in biological systems \cite{omaoileidigh12}, might be enforced by homeostatic mechanisms \cite{milewski17}, and can be employed in the construction of artificial devices \cite{martignoli07}.

redpajama_set_name: RedPajamaArXiv


text:
OPEL reports rise in quoting activity for industrial-scale CPV systems
Despite an uneven economic recovery and the lingering effects of the global recession on virtually every industry worldwide, OPEL Solar Inc of Shelton, CT, USA (a subsidiary of OPEL International Inc of Toronto, Canada) is reporting a sharp increase in requests for quotes on industrial-scale solar power systems in both North America and abroad, according to CEO Leon (Lee) M. Pierhal.
"With the worst hopefully behind us, there is no question that both governments and investors alike are reaffirming their interest and commitment to solar power in a big way," says Pierhal. "Since late 2008 when the credit markets first crashed, we have seen a significant increase in requests for quotes on both our HCPV [high-concentration photovoltaic] solar panels and our ground-based and rooftop tracker systems," he adds.
OPEL Solar claims that its gallium arsenide-based HCPV panels can generate up to 40% more kilowatt-hours than conventional flat-plate silicon solar panels. On 28 April, the firm also announced plans for US-based manufacturing of its utility-scale TF-800, a ground-mounted single-axis tracker (developed with FEiNA) that allows ease of installation and avoids shadowing from adjacent trackers (due to its reverse tracking ability).
Pierhal attributes the increase in quote activity to OPEL Solar's completion (with Spanish partner BETASOL) of a 330kW HCPV utility-grade power plant in Spain last October, as well as the firm's agreement in April to participate in a 1MW Portuguese HCPV installation with Tecneira S.A of Porto Salvo, Portugal, the renewable energy firm of power plant developer ProCME Group (and one of the largest renewable energy construction firms in Portugal). The latter is Portugal's first use of CPV technology, and will also use a dual-axis tracking system. Both installations, which are eligible for feed-in-tariffs (FIT), will provide investors with a guaranteed rate of return, OPEL assures.
Pierhal also notes an especially sharp and recent focus in the marketplace on solar power installations of at least 1MW in size. Also in first-quarter 2010, OPEL Solar was selected (via a partnership with French tracker firm Exosun) for an initial 1MW HCPV power plant in the South of France, with installation starting mid-year. Again, the plant will be grid connected, selling electricity to the country's utility. Deliveries on both the Portuguese and French 1MW projects will begin in second-half 2010.
"We are also partnering with one of the largest engineering and construction companies in the world to build utility-scale solar power farms in North America," Pierhal says.
"Neither the private sector nor forward-looking countries want to mortgage their future or straitjacket their ability to compete globally by relying exclusively on more traditional fossil fuel sources."
On 20 May, OPEL International Inc reported revenue for first-quarter 2010 up on both last quarter and a year ago. In addition, the firm says that the increased quoting activity on large-scale, utility-grade projects during first-quarter 2010 is due to its continued focus on forging relationships with large engineering, procurement and construction (EPC) companies and power producers. "With our new relationships and increased backlog, we are poised to significantly increase our revenue and shareholder value in 2010," asserts chief financial officer Michael McCoy.
*On 25 May, OPEL International said that it had mailed proxy materials to shareholders prior to their Annual Meeting in Toronto on 18 June at which they will be asked to consider a special resolution authorizing (subject to regulatory approval) a change of the firm's name to OPEL Solar International Inc. The firm says that, as it gains market share in solar energy, the name change better reflects its primary business and heightens awareness of OPEL's purpose.
See related item:
OPEL's revenues shrink 60% in Q4/2009 while focusing on demonstration projects
Search: OPEL Solar CPV
Visit: www.opelinc.com

redpajama_set_name: RedPajamaCommonCrawl


Keep us healthy as many are working in places that are unhygienic, slums as well as working with many sick people.
God given wisdom as we expand and go into new areas, direction on which way to go.
The grace to be full of compassion, love, kindness and patience as we work together.


text:
Palestyna pod panowaniem Imperium Osmańskiego znajdowała się w latach 1517-1917.

Historia

Rys ogólny
Imperium Osmańskie powstało na przełomie XIII i XIV wieku. Zostało założone przez tureckiego wodza Osmana I, który początkowo był wasalem sułtanatu Seldżuków, a w 1299 roku przyjął tytuł sułtana.

W następnych latach Imperium Osmańskie rosło w siłę, podbijając kolejne terytoria. W okresie od XIV do XX wieku obejmowało terytorium części południowo-zachodniej Azji, północną Afrykę i południowo-wschodnią Europę.

Było to jedno z najpotężniejszych politycznie i militarnie państw na świecie.

Palestyna
W 1517 roku sułtan Selim I Groźny pokonał wojska egipskich mameluków w bitwach pod Marj Dabiq i al-Raydaniyya, a następnie zajął Syrię, Palestynę oraz Egipt. W ten sposób Palestyna znalazła się w Imperium Osmańskim.

Sułtan Selim I prowadził politykę przychylną Żydom i zezwalał na żydowskie osadnictwo w Palestynie. Jednak sama Palestyna została zdegradowana do roli majątku dzierżawionego powoływanym na okres roku poborcom podatkowym. Czerpano dochody łupiąc chrześcijańskich pielgrzymów, przy czym nic nie robiono dla promowania rozwoju gospodarczego.

W 1538 roku sułtan Sulejman Wspaniały wybudował mury obronne Jerozolimy, które do dzisiaj otaczają Stare Miasto. Sułtan nadał Żydowi, Józefowi Nassi, ziemię nad Jeziorem Tyberiadzkim, z przeznaczeniem odbudowania starego żydowskiego miasta Tyberiady. Józef Nassi otrzymał wyraźne zezwolenie, że w odbudowanym mieście będą mogli mieszkać tylko Żydzi.

Miasto Tyberiada zostało odbudowane. Zasadzono tam drzewa morwowe dla hodowli jedwabnika i założono przędzalnie jedwabiu. Planowano uczynić z Tyberiady miasto fabryczne, które mogłoby współzawodniczyć z Wenecją, jednakże plany te upadły.

W latach 1538–1541 w mieście Safed działał Sanhedryn, w którym zgromadziło się 25 rabinów pod przywództwem Rabina Jakuba Beraba. Miasto Safed stało się nowym centrum życia żydowskiego w Palestynie. Rabin Józef Karo otworzył tutaj jesziwę.

Około 1569 roku w Safedzie osiedlił się Rabin Izaak Luria. To za jego sprawą Safed stał się ważnym ośrodkiem kabalistycznym na świecie.

W 1759 roku wielkie trzęsienie ziemi zniszczyło Safed zabijając 2 tys. ludzi, w tym 190 Żydów.

Francuska kampania syryjska

Podczas wyprawy Napoleona do Egiptu w 1798 roku flota brytyjska zniszczyła francuską flotę w bitwie pod Abukirem. Turcy spodziewali się, że oznacza to koniec Napoleona Bonaparte i przygotowali ofensywę, którą miały przeprowadzić dwie armie uderzające na Egipt z kierunku Syrii i Palestyny.

Napoleon Bonaparte uprzedził działania Turków, wezwał Żydów z Afryki i Azji, by podjęli przy jego wsparciu dzieło odbudowy Jerozolimy i w styczniu 1799 roku przeprowadził wyprzedzające uderzenie na Syrię i Palestynę.

Francuzi zdobyli Gazę oraz Jafę, 15 marca 1799 miała miejsce bitwa pod Nablusem, 16 kwietnia 1799 bitwa pod Górą Tabor. Po niepowodzeniu przy oblężeniu Akka, Francuzi wycofali się w maju do Egiptu.

Początek osadnictwa żydowskiego

W 1841 roku Żydzi zakupili w Palestynie pierwszy gaj pomarańczowy. Uczyniono to za pieniądze brytyjskiego finansisty sir Mosesa Montefiore, a za namową Rabina Cwiego Hirsza Kaliszera.

W 1870 roku Żydzi z francuskiej organizacji Alliance Israélite Universelle wykupili w Palestynie mały obszar ziemi (nazwany Mikwat Israel), na którym założyli szkołę rolniczą. Do tego miejsca przybywali Żydzi z Europy Środkowej i Wschodniej, zwłaszcza z terenów Rosji.

W 1879 roku organizacja Alliance Israélite Universelle wykupiła mały obszar ziemi w pobliżu Jafy. Utworzono w tym miejscu kolonię rolniczą Petach Tikwa.

W latach 1881-1903 do Palestyny napłynęła pierwsza fala żydowskich emigrantów. Była to tzw. Pierwsza Alija. Szacuje się, że w tym okresie osiedliło się w Palestynie ok. 25 tys. Żydów.

W latach 1904-1913 podczas Drugiej Aliji do Palestyny przybyło ok. 40 tys. Żydów.

W 1914 roku w Palestynie istniały już 43 żydowskie kolonie rolnicze, z czego 14 było kibucami.

I wojna światowa

Gdy w Europie rozpoczęła się I wojna światowa (1914-1918) naczelny wódz wojsk tureckich w Syrii, Djemel Pasza, rozpoczął prześladowania Żydów w Palestynie. Między innymi nakazał ich ewakuację z Jafy i Tel Awiwu. Prześladowania, wysiedlenia, ewakuacje, nędza, głód i choroby wstrząsnęły podstawami żydowskiej kolonizacji w Palestynie. W takich okolicznościach żydowscy działacze Josef Trumpeldor i Władimir Żabotyński wystosowali rezolucję do władz brytyjskich wzywającą do stworzenia Legionu Żydowskiego w Egipcie, który znajdował się pod brytyjską władzą.

W 1915 w Palestynie powstała żydowska organizacja szpiegowska NIL (Netzah Israel Lo Yeshaker) działająca przeciwko Turcji na rzecz Wielkiej Brytanii. Założycielami byli Awszalom Feinberg i Aaron Aaronson. W 1917 roku Turcy częściowo rozbili organizację, aresztując i zabijając wielu jej członków.

W 1916 roku doszło do zawarcia tajnego układu francusko-brytyjskiego (M. Sykes-S. Picot) o podziale Bliskiego Wschodu na sfery wpływów. Porozumienie zakładało uznanie niezależnego państwa arabskiego lub konfederacji państw arabskich w Palestynie, ale z rządem międzynarodowym. Nie uwzględniono postulatów środowisk syjonistycznych, a jedynie memorandum komitetu żydowsko-angielskiego, który żądał równouprawnienia dla Żydów w Palestynie.

W październiku 1916 roku żydowscy syjoniści przedstawili rządowi brytyjskiemu oddzielne memorandum, w którym przedstawili własne stanowisko w sprawie Palestyny. Postulowali uznanie przez rząd Wielkiej Brytanii siedziby narodowego żydostwa w Palestynie. Realizacja siedziby miałaby nastąpić na drodze autonomii narodowej, poprzez wspieranie emigracji i kolonizacji w Palestynie. Szczegóły miałyby zostać sformułowane na podstawie oddzielnych porozumień między rządem brytyjskim i przedstawicielami Światowej Organizacji Syjonistycznej. W lutym 1917 roku nowy rząd brytyjski spotkał się z przedstawicielami Światowej Organizacji Syjonistycznej: Rothschildem, Weizmanem i Sokołowem. Postulowali oni, aby Wielka Brytania objęła protektorat nad Palestyną oraz udzieliła państwowego czarteru dla żydowskiej imigracji i kolonizacji. Chaim Weizman, przewodniczący Światowej Organizacji Syjonistycznej, uzyskał poparcie brytyjskie dla projektów żydowskich w postaci słynnej Deklaracji Balfoura (2 listopada 1917 roku).

W pierwszej połowie 1916 Brytyjczycy rozszerzyli swoje pozycje obronne na Synaj i odrzucili wszystkie ataki tureckie (główna bitwa pod Ruamani – 3 sierpnia 1916). Następnie armia brytyjska osiągnęła El Arish. 8-9 stycznia 1917 w wyniku bitwy pod Magruntein oddziały tureckie zostały całkowicie wyparte z Synaju. Tym sposobem Anglicy mogli teraz swobodnie wkroczyć do Palestyny. Po dwóch nieudanych atakach na Gazę (26 marca i 17-19 kwietnia 1917), nowy dowódca wojsk brytyjskich w Palestynie, gen. Allenby, otrzymał rozkaz zdobycia Jerozolimy.

Po reorganizacji struktury dowodzenia Allenby'emu udało się wreszcie zdobyć Gazę (31 października 1917) i zmusić Turków do odwrotu. Pomimo rozpaczliwej obrony tureckiej, którą tworzył niemiecki gen. von Falkenhayn, Anglicy zdobyli Jerozolimę 9 grudnia 1917 r. Dalsze postępy Brytyjczyków zostały zahamowane, bowiem z armii wydzielono duże siły wysłane na front zachodni. Ale we wrześniu 1918 Allenby przeprowadził kolejną ofensywę. Atakowi temu miała się przeciwstawić silna obrona utworzona na linii sięgającej od Jafy do rzeki Jordan. Anglicy zdecydowali się uderzyć od strony wybrzeża, natarcie powiodło się i przełamano pozycje tureckie, a przez wytworzoną w ten sposób lukę szybko wdarła się kawaleria, która parła w głąb terytorium przeciwnika. Piechota natomiast niszczyła ostatnie punkty oporu w bitwie pod Megiddo (19-21 września 1918). Pościg za uciekającymi oddziałami wroga zakończył się kapitulacją Turków 30 października 1918.

Zobacz też
 historia Izraela
 starożytny Izrael

Historia Palestyny

redpajama_set_name: RedPajamaWikipedia


Closeup of Blueberry cannabis strain grown indoors under optimum conditions by expert growers in BC Canada.
Blueberry cannabis buds are some of the most delicious on earth! I love taking a big hoot of some blueberry.
Blueberry cannabis is amazing. It has won the Cannabis Cup as best strain.
Whenever I have the opportunity to smoke some blueberry I get big time excited! It tastes great and has a wonderful uplifting high.
If grown well and flushed of nutrients properly Blueberry cannabis is very pungent. The smell is remarkable.
I especially like this strain if grown outdoors. Cannabis grown indoors in a small controlled environment can be very well cared for and produce amazing results. However, I believe the moon and an organic outdoor environment provide the best tasting Blueberry cannabis buds.
Find this strain or 100 cross varieties at most collectives and dispensaries across the US and Canada. Blueberry has also become very popular in Europe.


text:
Medical Textbooks Overwhelmingly Use Pictures of Young White Men
"You shouldn't have a 20-year-old with coronary artery disease."
by Rose Eveleth
May 9 2019, 1:30pm
It's rare to see women and people of color in medical textbooks, researcher Rhiannon Parker says. Illustration: Xavier Lalanne-Tauzia
There are lots of conversations about the lack of diversity in science and tech these days. In response, people constantly ask, "So what? Why does it matter?" There are many ways to answer that question, but perhaps the easiest is this: because a homogenous team produces homogenous products for a very heterogeneous world.
This is Design Bias, a Motherboard column in which writer Rose Eveleth explores the products, research programs, and conclusions made not necessarily because any designer or scientist or engineer sets out to discriminate, but because to them the "normal" user always looks exactly the same. The result is a world that's biased by design. -the Editor
Before it was a television show, Grey's Anatomy was a textbook. Published in 1858, Gray's Anatomy (spelled with an "a") quickly became the gold standard in medical illustration, featuring detailed diagrams of everything from the tiny bones in the hand to the internal structure of the eye.
Gray's Anatomy is still in print today, in its 41st edition, but if you open the book up and flip through the pages you might notice something. Or really, the dearth of something: women. And it turns out it's not just Gray's Anatomy that has this problem—almost all medical textbooks are heavily biased towards depicting male bodies.
In 2014, Rhiannon Parker, a researcher at the University of Wollogong in Australia, set out to quantify just how bad that bias actually is. By analyzing more than 6,000 images from 17 anatomy textbooks published between 2008 and 2013, Parker and her colleagues found that only 36 percent of the anatomical images with an identifiable sex were female.
Even more discouraging is the results weren't all that different from a study done in 1994, in which 32 percent of images represented female bodies. "I expected there to be a much bigger improvement on representation," Parker told me.
Not all books were equally biased. General Anatomy, 2nd ed., had the highest proportion of male bodies at 5:1, while Human Anatomy and Physiology, 9th ed. was the only textbook to have the same proportion of male and female bodies. Gray's Anatomy for Students—a condensed version of Gray's Anatomy—has nearly three times more men as women.
"The textbook with the worst ratio was edited solely by men"
Interestingly, the editors of Human Anatomy, the textbook with parity, are Elaine N. Marieb and Katja Hoehn, both women, while the textbook with the worst ratio was edited solely by men.
Madelene Hyde, the vice president for content and education at Elsevier, which publishes several top anatomy books including Gray's Anatomy, said they are "trying more and more to be as balanced as possible with our textbook images."
"We take gender and racial diversity seriously and are working to include more diverse images in our textbooks," she told Motherboard in an email.
Read More: The Design Bias of Heart Failure
In some cases, showing a female body makes sense, if the content is specifically about female health. But Parker found that even in cases where there is no reason to show one sex over another, men are more likely to be depicted as the "normal body." This lines up with previous research from 1992 that found that even when it comes to medical imagery around reproduction, men outnumbered women in textbooks 2.5 to 1.
"For the anatomy titles that do not solely focus on surgical anatomy (interior rather than exterior), we do our best to provide images of diverse subjects," Hyde said in her email. "When possible, we try to replace older images of rare clinical conditions. As a provider of a significant volume of global health content, we try more and more to be as balanced as possible with our textbook images."
Parker also saw other trends in her dataset. Even though she was looking for images where the body being depicted might be identified by readers as male or female (whether by genitalia or by traditional gender markers), she also found that bodies were also overwhelmingly white, slim, and young. Of the images of women she did find, 86 percent of them were white (compared to 76 percent of identifiably male bodies).
Medical students who saw gender-biased images were more likely to show implicit gender bias. Illustration: Xavier Lalanne-Tauzia
Male bodies are almost always muscular, while female bodies are drawn thin. Only 2.7 percent of the images the team analyzed depicted visibly disabled bodies. And only 2.2 percent of the textbook images depicted elderly patients. "You don't have any elderly in these textbooks even though the elderly need more healthcare," Parker said. "I think that was quite surprising to me, that those types of representations weren't really in there."
Parker has also researched the impact the images might have on medical students. In a survey of 456 anatomy students studying at the School of Medicine at the University of Wollongong in 2018, she found that the disparity in bodies didn't affect their explicit bias, but it increased their scores on implicit bias tests that measure more subtle attitudes.
"People know what they should be saying, what they should be thinking" when it comes to gendered assumptions, Parker said. "But that doesn't stop the insidious impact of biased images. The way they normalize certain images and body types, you can't necessarily stop that from impacting your biases."
There is plenty of research to suggest that biases held by doctors have real, negative impacts on patients. "Coronary heart disease is seen as a male disease," Parker said. Another study found that many textbooks don't address the differences in the way coronary heart disease presents in women, opting to show men only, which could contribute to women being misdiagnosed at higher rates than men.
It's not just coronary heart disease, either. Parker notes that fatphobia among doctors leads to people not seeking medical attention when they need it, and that Black women in America are three times more likely to die in childbirth than white women, which many experts say is largely due to doctor bias against Black women.
Even beyond the bias it could engender, depicting the same body over and over again as white, male, and athletic isn't the best way to teach future doctors. "I think we all fall into [the idea] we want to draw pretty people; everybody is attracted to pretty people," said Jill Gregory, a medical illustrator and associate director of instructional technology at the Icahn School of Medicine at Mount Sinai. "But especially when you're illustrating a disease of older people, you shouldn't have a 20-year-old with coronary artery disease."
Anatomy images on stock images sites are also overwhelmingly male, white, slim, and young. Image: Shutterstock
Stock image sites and online medical resources fall into the same trap. Gregory was recently asked to do a series of medical illustrations for an anatomy class at Mount Sinai. To depict breast health, "the professor had gotten random images off of Google," she said, which mostly showed a very specific kind of breast—perky, white, mostly with implants. "They're like these little balls on your chest. Most people don't have breasts like that," Gregory said.
So Gregory created an illustration of an elderly breast instead, collapsed with the nipple facing down. She said that students in the course were much more likely to encounter breasts that looked like her illustrations than the ones commonly depicted in books or stock images.
"It takes time for people to say, 'I should use diverse skin tones, maybe someone in a wheelchair'"
And sometimes illustrators are explicitly told to not include certain kinds of bodies. Parker said that in her research, she heard stories from illustrators who reported being asked not to show female nipples unless the illustration had something to do with breast health, for example.
By using past books as examples, illustrators are also perpetuating the bias. "Frank Netter, the father of medical illustration—his work is 100 percent white people," said Gregory. "It's a matter of not being lazy thinkers and starting to be, like, 'I shouldn't just default to the same skin tone.'"
This is exacerbated by the fact that the majority of medical illustrators are white (85 percent of people who responded to an Association of Medical Illustrators survey in 2018 identified as such), a problem that the association started tackling head on a few years ago with a special diversity committee (which Gregory is a member of).
The solution lies in the hands of publishers and illustrators. Gregory says that it's crucial to explain to people working in the field why this bias is harmful. "It's really about awareness. It takes time for people to say, 'Oh right, I should use diverse skin tones, maybe someone in a wheelchair, different kinds of people.'"
But are things getting better? Parker isn't so optimistic. She points out that doctors have a really rigid idea of what a healthy body looks like, and that's a hard thing to change. "No, I don't think it's changing," she told me. "I do think that maybe the gender disparity is slowly changing. But the intersections with ethnicity and body type, I don't think they're being paid attention to, and are getting worse."
Gregory, on the other hand, sees progress, even if it's been slow. She says that even she has changed her methods around this issue over the years. "Ten or 15 years ago I was maybe throwing something in for the heck of it, but now every time I approach an illustration I think about how I can include diversity in my images." Between that and recruiting a more diverse set of illustrators in the field, Gregory has hopes that things will get better.
Correction: In the original version of the story, Frank Netter's name was spelled incorrectly.
Design Bias


text:
MTG Arena
Signature Spellbook - Gideon
Mythic Championship Team Series
Pro Players Club
Planeswalker Points
Casual Events
Friday Night Magic
Draft Weekend
Sealed Deck
New to Magic
Card Database
Banned & Restricted List
NEWS ARCHIVES - ARTICLES Latest Developments
Creatures On Another Level
Posted in Latest Developments on April 2, 2010
By Tom LaPille
Bio Archive
Tom LaPille makes things. Some of the things he makes are card sets, like Dark Ascension and Born of the Gods. Sometimes he makes stories, too. Sometimes he makes unexpected things, like 16th-century Japanese clothing. He's probably making something right now.
Welcome to the end of the first week of Rise of the Eldrazi previews. I've got a preview card today, but before I get into Rise of the Eldrazi, I want to give a proper farewell to what came before it. So, let's talk about...
Zendikar Limited is the the fastest Limited environment I remember playing in years. That's not a surprise, but I'll let you in on a little secret: Zendikar Limited is faster than we thought it was.
This is not to say that we did not know that it was fast, because we did know that. Landfall and Allies are two of the set's defining mechanics in Limited. Under normal circumstances, landfall ensures that your creatures are stronger on your turn than on your opponent's turn. Allies get stronger when you play other Allies, which once again mostly happens on your turn. Because your creatures are better on your own turn, attacking is just going to be better than blocking, leading to an aggressive environment.
We understood that Zendikar was fast, so we built Worldwake to slow things down a little bit. Its landfall creatures don't get +2/+2, fewer of its Allies get +1/+1 counters, and more of its cards can actually block Steppe Lynxes and Plated Geopedes.
Our misperception, then, was one of degree. Strong players know that they need to curve out and attack early in the game in Zendikar Limited, so they value early plays, and especially two-mana creatures, very highly. Cards like Highland Berserker and Goblin Shortcutter routinely are taken above cards that are more powerful but more expensive. I'm pretty sure that very few of us third-picked Goblin Shortcutters before the set left the building, but the collective wisdom of thousands of Magic Online drafters and the best players on the Magic Pro Tour has shown that we were wrong.
I don't think this means we failed at all. There are tens of thousands of Magic players out there, and there are only twenty or so of us in Magic R&D. If we understand a format perfectly when we are done with development, it's too easy to figure out. We spend about as long working with each set as you do, but you all have the advantage of talking to each other while you try to figure out what to do with a set's cards. We do not have that luxury. Statistically speaking, we are some of the best Magic players in the world, but by necessity we work in a vacuum, so our understanding will only be so good. As long as our understanding was close to correct and the formats we create are fun to play, I think we're doing our jobs right.
This is one of the more remarkable aspects of working on Magic. We play with a set for many months before any of you get to see it, but that doesn't mean that we have nothing more to learn when the sets release. We like playing Limited with real cards, and it's not uncommon to see drafts take place outside of work hours because we're nowhere near done having fun with the sets when we have to stop working on them. As we do this, we keep learning. For example, I found a deck on my desk recently that I drafted soon after Worldwake's release, and it looks nothing like anything I drafted while Zendikar or Worldwake was in development. I can say this with confidence because it has nine two-drops in it.
The Akoum Battlesinger is a little questionable, but I had eight other Allies in the deck and was just slightly short on cards. I wasn't that sad about it, though. I knew that it would be pretty strange if my games lasted past turn six or seven, so the liability of a drawing a random 1/1 creature late in the game was pretty low compared to how likely the Battlesinger was to help an aggressive Ally draw outrace an opponent or punch through defenses.
But enough about Zendikar and Worldwake. It's time for...
Rise of the Eldrazi is a very different world from Zendikar. It's the same plane, but some truly enormous monsters are ravaging the world. Those monsters look something like this:
I'm sure you've thought about this already, but I will call your attention once more to the rarity symbols on these cards. Ulamog's Crusher is a common. It costs eight mana. It is an 8/8. And it is a common. Artisan of Kozilek is an uncommon. These are not the only common and uncommon colorless Eldrazi. You should expect to see these guys across the table when you go to the Rise of the Eldrazi Prerelease.
Eight-mana creatures would not be very strong in Zendikar Limited. Rise of the Eldrazi, however, is not Zendikar. We built Rise of the Eldrazi so that you would actually cast these guys in a typical Limited game.
Rise of the Eldrazi Limited games are luxurious affairs, more like three-hour seven-course meals than the vicious feeding frenzies that are Zendikar Limited games. You can expect them to take a while. You'll have plenty of lands for lots of turns in a row. You'll sit behind walls like Overgrown Battlement and Vent Sentinel while you build up all that mana. And then, hopefully, you'll start crashing through your opponent's defenses with some enormous colorless creatures. Don't be surprised if this takes ten turns, or even fifteen. It won't take long for the game to be over once the Eldrazi get going, but you'll have time to do plenty of things before that happens.
This has serious consequences for creatures that aren't enormous or colorless and don't have annihilator. Remember Highland Berserker from Zendikar? It was a great turn-two play in that format. If you didn't attack early and often, your opponent would and you would die. Even better for the Highland Berserkers of the world, games usually took around six to eight turns, so there wasn't a lot of opportunity for you to draw a random Highland Berserker late in the game when it wasn't relevant.
Rise of the Eldrazi Limited is a different world, and not a friendly one for Highland Berserker. Games don't last ten to fifteen turns by accident. They do that when there is a mess of creatures on the ground, creatures that make attacking with small creatures unprofitable or downright impossible. By turn eight or nine, the stars of the show are the big creatures. They might be naturally huge like the Eldrazi or have grown to huge sizes with totem armor Auras, but they will make a random 2/1 Highland Berserker with little hope of advancement close to irrelevant. The cheap creatures that stand up in this world are the ones that have important effects even late into the game. Trying to win the game by turn six is a noble goal, but it's unlikely to work in Rise of the Eldrazi Limited.
I just said something about creatures that matter late into the game. You might be wondering to yourself what such a creature might look like. Let me show you one that's already in the Visual Spoiler in the Rise of the Eldrazi product section.
In Rise of the Eldrazi, those creatures have level up. And, as you have seen in the Visual Spoiler, there are plenty of them. But let's investigate this card on its own for a moment.
On turn two, Knight of Cliffhaven is a 2/2. These are stats that you expect to see on the cheap beatdown creatures in any Limited format. If that's all it had to offer, it might not be such a strong card, as it would get stuck behind Overgrown Battlements. But that's not all we have here.
As soon as walls or bigger ground creatures make it impossible to attack with your Knight of Cliffhaven, you can take it to the air, where the defensive ground creatures have nothing better to do than watch as it flies over and hits their controller for two a turn. As the game goes on, though, you might run out of cards to play from your hand. When that happens, you can put some of that extra mana into the Knight, and you end up with ...
... a Serra Angel, which trumps most flying creatures. The key here is to not think of Knight of Cliffhaven as a fourteen-mana Serra Angel. Think of it as a creature that is exactly the creature you want exactly when you want it. It's a two drop on turn two, then it's a flying creature as soon as that isn't good enough, and then when the game stalls out and you don't have much else to do, it's a big flying creature.
I suppose I should show you a preview card to further illustrate this concept. Let's do that now.
Remember Figure of Destiny? That card was the inspiration for the level-up mechanic, but how could we print level-up without making a call-back to it? Student of Warfare is that card. We got to know it quite well during Future Future League playtesting, where it became good friends with both Elite Vanguard and Ranger of Eos. It's also a very strong Limited card, as anyone who opens it and plays white at the Prerelease will be able to attest.
Not coincidentally, levelers are what you want in Constructed for exactly the same reason you want them in Limited: they are exactly the card you need every time you draw them. Aggressive decks need cheap creatures to start the game off right with. Playing too many cheap creatures, though, hampers their ability to function in the late game, when one- or two-mana 2/2 creatures just can't compare to things like Baneslayer Angels and Broodmate Dragons. Student of Warfare and Kargan Dragonlord, however, do not have this problem. They are fine plays on turn one and two, and just as fine to play on turn six.
Although you've never met her before, Student of Warfare is a veteran of countless playtest matches. She's seen every kind of game of Magic, from slow Rise of the Eldrazi Sealed Deck games to blazingly fast Future Future League Constructed games. Whatever kind of game you happen to find yourself in, she'll be ready to help you win it.
I won't be writing an article for next week, but I'm leaving you in the capable hands of Pro Tour Hall of Fame member and Magic developer Mike Turian, who has his own spicy preview card. I'll see you again in two weeks, and keep enjoying the Rise of the Eldrazi previews while I'm gone!
Last Week's Poll
What do you think of Kozilek, Butcher of Truth?
I love it! 1852 30.5%
I like it. 1702 28.0%
I'm okay with it. 1528 25.1%
I don't like it. 632 10.4%
I hate it! 365 6.0%
Total 6079 100.0%
Awesome. The Rise of the Eldrazi Prerelease is only two weeks away, so you will get to meet him for yourself soon enough!
Which of the following is your favorite keyword / ability word?
Flash 911 11.9%
Other 754 9.9%
Ninjutsu 714 9.4%
Storm 650 8.5%
Flashback 627 8.2%
Cycling 599 7.8%
Buyback 511 6.7%
Morph 405 5.3%
Exalted 403 5.3%
Wither 365 4.8%
Persist 354 4.6%
Scry 306 4.0%
Epic 273 3.6%
Imprint 217 2.8%
Splice 202 2.6%
Devour 177 2.3%
Bloodthirst 116 1.5%
Radiance 50 0.7%
Magic at San Diego Comic-Con 2019! Wizards of the Coast
We're celebrating all things M20 and Chandra at SDCC (San Diego Chandra-Con?). Read all about what we're up to next weekend.
HOW TO PLAY LIMITED
Core Set 2020 Draft First Picks Luis Scott-Vargas
Hall of Fame pro Luis explains his tiers for the best first-pick cards in Core Set 2020.
Latest Latest Developments Articles
Changes by, Sam Stoddard
Hello and welcome to another edition of Latest Developments! Today I'm going to talk about several kinds of changes within R&D and how we deal with those. Card Changes From the day ...
Things I've Learned by, Sam Stoddard
Hello, and welcome to another edition of Latest Developments! This week is the five-year anniversary of me joining Wizards of the Coast as a contractor on the development team. My officia...
Latest Developments Archive
Consult the archives for more articles!
Convention Support
Wizards Play Network
esrb-world
Terms Code of Conduct Privacy Policy Customer Support Cookies Fan Content Policy
简体中文 繁體中文 Français Deutsch Italiano 日本語 한국어 Português (Brasil) Русский Español English
We use cookies on this site to personalize content and ads, provide social media features and analyze web traffic. By clicking YES, you are consenting for us to set cookies.	(Learn more about cookies)