Stochastic resonance

Post-publication activity

Curator: Gregoire Nicolis

Figure 1: Evolution of the global ice volume on earth during the last million years as inferred from oxygen isotope data.

Broadly speaking, stochastic resonance is a mechanism by which a system embedded in a noisy environment acquires an enhanced sensitivity towards small external time-dependent forcings, when the noise intensity reaches some finite level. As such it highlights the possibility that noise, a universal phenomenon and yet one considered traditionally to constitute a nuisance, may actually play a constructive role in large classes of both natural and artificially designed systems.

History

The concept of stochastic resonance was invented in 1981-82 in the rather exotic context of the evolution of the earth's climate. It has long been known that the climatic system possesses a very pronounced internal variability. A striking illustration is provided by the last glaciation which reached its peak some 18,000 years ago, leading to mean global temperatures of some degrees lower than the present ones and a total ice volume more than twice its present value. Going further back in the past it is realized that glaciations have covered, in an intermittent fashion, much of the Quaternary era. Statistical data analysis shows that the glacial-interglacial transitions that have marked the last $$10^6$$ years display an average periodicity of $$10^5$$ years, to which is superimposed a considerable, random looking variability (see Figure 1). This is intriguing, since the only known time scale in this range is that of the changes in time of the eccentricity of the earth's orbit around the sun, as a result of the perturbing action of the other bodies of the solar system. This perturbation modifies the total amount of solar energy received by the earth but the magnitude of this astronomical effect is exceedingly small, about $$0.1\%\ .$$ The question therefore arises, whether one can identify in the earth-atmosphere-cryosphere system mechanisms capable of enhancing its sensitivity to such small external time-dependent forcings. The search of a response to this question led to the concept of stochastic resonance. Specifically, glaciation cycles are viewed as transitions between glacial and interglacial states that are somehow managing to capture the periodicity of the astronomical signal, even though they are actually made possible by the environmental noise rather than by the signal itself. Starting in the late 1980's the ideas underlying stochastic resonance were taken up, elaborated and applied in a wide range of problems in physical and life sciences.

Classical setting

In its classical setting stochastic resonance deals with bistable systems, a class of nonlinear dynamical systems that are encountered in a wide range of phenomena across different scientific fields. More specifically, one considers one-variable bistable dynamical systems subjected simultaneously to noise and to a weak periodic forcing: $\tag{1} \frac{dx}{dt}=-\frac{\partial U}{\partial x}+F(t)+\epsilon h(x)\cos (\omega_0t+\phi)$

Here $$x$$ is the state variable (e.g., the global temperature or the global ice volume in the context of the Quaternary glaciations); $$U$$ is the "potential" driving the internal dynamics, taken to possess two minima $$x_+$$ and $$x_-$$ associated to the two stable states, separated by a maximum corresponding to an intermediate unstable state $$x_0\ ;$$ $$F(t)$$ is a "random force" accounting for internal variability or environmental noise and modeled classically as a Gaussian white noise of zero mean and strength equal to $$q^2\ ;$$ and $$\epsilon\ ,$$ $$\omega_0$$ and $$\phi$$ are, respectively, the amplitude, frequency and phase of the periodic forcing. Actually, the forcing contribution can be cast in a form similar to the first term in the right hand side of (1) by introducing a generalized time-dependent potential $\tag{2} W(x,t)=U(x)-\epsilon g(x) \cos (\omega_0 t+\phi)$

with $$dg(x)/dx=h(x)\ .$$

According to the theory of stochastic processes the stochastic differential equation for the random process $$x(t)\ ,$$ eq. (1), is equivalent to a Fokker-Planck equation for the probability distribution function $$P(x,t)$$ of values of $$x\ .$$ In the absence of periodic forcing this latter equation defines a particular type of Markov process known as diffusion process: The variable $$x$$ realizes, for most of the time, small scale excursions around $$x_+$$ or $$x_-\ ,$$ which are interrupted every now and then by noise-driven abrupt transitions from $$x_+$$ to $$x_-$$ or vice versa across the unstable state $$x_0\ ,$$ which constitutes a barrier of some sort. The kinetics of these transitions are determined by two quantities: The noise strength $$q^2$$ and the potential barrier $$\Delta U_{\pm}\ ,$$ defined by $\tag{3} \Delta U_{\pm}=U(x_0)-U(x_{\pm})$

In the limit where $$q^2$$ is much smaller than $$\Delta U_{\pm}$$ the mean value of the transition time is given by the celebrated Kramers formula $\tag{4} \tau_{\pm}^{-1}=r_{\pm}=\frac{1}{2\pi}{(-U''(x_0)U''(x_{\pm}))}^{1/2} \ \exp(- \frac{\Delta U_{\pm}}{q^2/2})$

where the double prime designates the second derivative. The transitions themselves occur in an incoherent fashion, as their dispersion around the above mean value is comparable to the mean itself.

Figure 2: Amplitude of the periodic component of the response of a bistable system described by a symmetric quartic potential subjected simultaneously to noise and to a weak periodic forcing, against the variance $$q^2$$ of the noise strength. Parameter values are $$\lambda=1\ ,$$ $$\omega_0=2 \pi/10^5$$ and $$\epsilon=0.001\ .$$ The existence of a sharp maximum is one of the principal signatures of stochastic resonance.

When the periodic forcing is switched on $$U$$ is replaced by the generalized potential $$W$$ (eq. (2)). The corresponding barrier $$\Delta W_{\pm}$$ is now modulated in time leading periodically to situations where states $$x_{\pm}$$ are found at the bottom of wells that are, successively, less shallow and more shallow than those in the forcing-free system. One is thus led to expect that the transitions will be facilitated during a part of this cycle, provided the periodicity of the forcing matches somehow the Kramers time in eq. (4). As it turns out this intuitive idea is fully justified in the asymptotic limit of small $$q^2$$ in which the Fokker-Planck equation can be reduced, using an adiabatic approximation, to a closed equation for the probability $$p_{\pm}$$ to be in the attraction basin of state $$x_+$$ or $$x_-\ :$$ $\tag{5} \frac{dp_+(t)}{dt}=r_-(t)\ p_-(t)-r_+(t)\ p_+(t)$

with $$p_++p_-=1$$ and $$r_{\pm}$$ given by an expression similar to (4) in which $$U$$ is replaced by the generalized potential $$W\ .$$ This equation can be solved straightforwardly. In most of the quantitative studies of stochastic resonance the result is further expanded to the first non-trivial order in the forcing amplitude $$\epsilon\ .$$ A popular minimal model capturing the essence of the results is to set $$h(x)=1$$ (and hence $$g(x)=x$$) and to consider a symmetric quartic potential $$U(x)=-\lambda x^2/2+x^4/4$$ ($$\lambda >0$$), corresponding to $$x_{\pm}=\pm \lambda ^{1/2}$$ and $$x_0=0\ .$$ This leads to the following expression for the periodic component $$\delta p(t)$$ of the response, $\tag{6} \delta p(t)=A \cos(\omega_0 t+\phi+\psi)$

Here the amplitude $$A$$ and phase shift $$\psi$$ are given by $\tag{7} A=\epsilon\frac{\lambda}{q^2}\ \frac{r(q^2)}{{(r^2(q^2)+\omega_0^2/4)}^{1/2}} \ \ \ \ \psi=-\arctan (\frac{\omega_0}{2r})$

where $$r(q^2)=r_+=r_-=(\sqrt{2} \pi)^{-1}\lambda\exp{(-\lambda^2/(2q^2))}$$ for the symmetric potential model. The essential point is now that

• (a) the transitions across the barrier have been synchronized to follow, in the mean, the periodicity of the external forcing;
• (b), the response is negligible unless the period of the forcing comes close to the (noise intensity-dependent!) Kramers time; and
• (c), for given $$\omega_0$$ and $$\epsilon\ ,$$ $$A$$ goes through a sharp maximum for an intermediate (finite) value of $$q^2$$ (see Figure 2), thereby enhancing considerably the response to the (weak) periodic signal. This latter property is the principal signature of stochastic resonance and should be clearly differentiated from the mechanisms underlying classical resonance. More refined studies based on Floquet theory or on a spectral decomposition of the full Fokker-Planck equation confirm fully the validity of these conclusions.

Further indicators of stochastic resonance

In addition to the periodic response (eq. (6)), stochastic resonance can also be characterized by a number of useful indicators related in one way or the other to quantities easily amenable to experimental visualizations.

Signal-to-noise ratio

Despite the periodicity of the response at the probability level (eq. (6)), the process itself (eq. (1)) contains a marked random component. The signal-to-noise ratio (SNR) provides a measure of the relative importance of the noisy and the systematic parts of the response. The basic quantity involved in SNR is the power spectrum $$G_{xx}(\omega)$$ of the variable $$x\ ,$$ essentially the Fourier transform of its time autocorrelation function. In the presence of a weak periodic input $$G_{xx}(\omega)$$ can be decomposed into a sum of contributions due to the noisy background, $$G_{xx}^{(0)}(\omega)$$ and to the signal itself, the latter being proportional to a sum of delta peaks centered at frequencies $$\pm \omega_0\ .$$ The SNR is then defined as the ratio of the total power $$G_{xx}(\omega)$$ on a narrow band $$\Delta \omega$$ surrounding $$\omega_0$$ and $$G_{xx}^{(0)}(\omega)$$ evaluated at the input frequency $$\omega_0\ ,$$ in the limit of $$\Delta \omega$$ tending to zero (practically of $$\Delta \omega$$ limited to the frequency bins around $$\omega_0$$). Much like the behavior in Figure 2, the SNR goes through a sharp maximum at some finite value of $$q^2\ .$$

Residence time distribution

Within the framework of the adiabatic approximation (eq. (5)) and in the absence of the periodic signal, the time $$\theta\ ,$$ spent by the system in the attraction basins of $$x_{\pm}$$ is a random quantity whose probability distribution decays exponentially with $$\theta\ .$$ The presence of the periodic signal induces a fine structure in the form of a sequence of Gaussian-like peaks, while the envelope of the overall distribution still decreases exponentially. Stochastic resonance is here manifested by the enhancement of the first peak (at half of the signal period), reflecting the phase synchronization of the switchings between $$x_+$$ and $$x_-$$ with the periodic signal.

Information theoretic measures

Inasmuch as stochastic resonance has to do with the quality of signal processing and, in particular, with the enhancement of information transmission through a system, it may be expected that information theoretic ideas should provide yet another natural characterization. A first quantity of this kind is mutual information, defined as the difference between the information entropies of an output time series per se and an output time series constrained by a given input time series. Of interest is also the Fisher information, defined as the amount of information carried by an observable (e.g., the output of a device) about a parameter (e.g., the strength of an input signal) and related closely to statistical estimation theory. Both quantities have been analyzed in systems in which noise allows an otherwise subthreshold (and thus undetectable) signal to overcome a prescribed threshold, and shown to exhibit (much like the SNR) an extremum at a well-defined finite value of the noise strength.

Stochastic resonance beyond the classical setting

Systems with coexisting attractors other than fixed points

The concept of stochastic resonance can be carried through to systems subjected to both noise and an external periodic forcing, and possessing coexisting stable states in the form of periodic or chaotic attractors. Theoretical analyses of these more involved situations draw on the existence, for such systems, of generalized potentials, not necessarily analytic in the state variables, possessing local minima on the corresponding attractors.

Stochastic resonance in deterministic chaotic systems

Dynamical systems in the regime of deterministic chaos evolve under certain conditions through a sequence of intermittent jumps between two preferred regions of phase space, without the intervention of a background noise. Such systems, which give rise to multimodal probability distributions, display an enhanced sensitivity to external periodic forcings through a stochastic resonance-like mechanism.

Slowly varying parameters

In many systems, the dynamics in the absence of both noise and forcing is controlled by a number of parameters $$\lambda$$ describing the constraints acting from the external world. Ordinarily these parameters are assumed to remain constant, but there are situations in which this constitutes an oversimplification (gradual switching on of a device, man-biosphere-climate interactions, etc.). In the absence of external periodic forcing the simultaneous action of noise and of a slow variation of $$\lambda$$ in the form of a ramp, $$\lambda=\lambda_0+\mu t$$ $$(0<\mu <<1)\ ,$$ may lead to the freezing of the system on a preferred state by practically quenching the transitions across the barrier. The interaction between stochastic resonance and the action of the ramp provides a new method for the control of the transition rates by allowing the system to perform, transiently, a certain number of transitions (depending on the forcing frequency and the noise strength) prior to quenching.

Complex signals, aperiodic stochastic resonance

Stochastic resonance-like enhancements of the response of a noisy system have also been established when the signal possesses a complex spectrum as is the case in many real situations (multiperiodic signals, aperiodic signals with a finite bandwidth around a preferred frequency). In each case, the crux is the existence of an optimum of a suitably defined measure of the response, attained for some intermediate (finite) value of the noise strength.

Non-dynamical stochastic resonance

The concept of stochastic resonance can also be extended to situations where the system itself does not derive from a deterministic dynamics. For instance, the sole feature to be retained can be that a threshold (or a series of thresholds) is (are) somehow generated by a mechanism that need not be specified in its details. The result is then, again, that in the presence of noise of moderate strength the detection of a weak signal can be optimized.

Spatial couplings

Spatially extended systems of coupled bistable elements are widely spread in nature and technology, from neurophysiology to computer science. Under the presence of noise and an external periodic forcing the stochastic resonance that would be observed in the limit of totally independent elements is further enhanced by the spatial coupling, the enhancement being at a maximum for a certain well-defined finite coupling strength. Analytic studies show that under the assumption of isotropy and translational invariance the response of each individual unit is in the form of eq. (7) where the Kramers rate is now scaled by a factor related to the coupling strength.

Quantum stochastic resonance

Quantum mechanics allows a system to tunnel through a barrier separating two states without going over it. Such contributions enhance the classical stochastic resonance. They eventually dominate the thermally activated transitions below some crossover temperature, which in certain systems can be as large as $$1000^oK\ .$$ Typically, the Quantum Stochastic Resonance phenomenon requires some amount of asymmetry. For symmetric systems it can occur nevertheless but then requires sufficiently strong quantum friction.

Experimental aspects, simulations and applications

Stochastic resonance has been observed in a wide variety of experiments involving electronic circuits, chemical reactions, semiconductor devices, nonlinear optical systems, magnetic systems and superconducting quantum interference devices (SQUID). Of special interest are the neurophysiological experiments on stochastic resonance, three popular examples of which are the mechanoreceptor cells of crayfish, the sensory hair cells of cricket and human visual perception.

Numerical solutions are an important tool in studies of stochastic resonance, since even in the classical setting of eq. (1) and the associated Fokker-Planck equation there is no full analytic solution available. Direct simulation is another useful tool. Of special interest are analog simulations based on electronic circuits modeling the nonlinearities involved in bistable systems, which lie in fact at the frontier between simulation and experiment. Their advantage is that the parameters can be easily tuned over a wide range of values and the response can be followed straightforwardly.

Stochastic resonance is a generic phenomenon. It has to do with the fact that adding noise to certain types of nonlinear systems possessing several simultaneously stable states may improve their ability to process information. As such, it is at the origin of intense interdisciplinary research at the crossroads of nonlinear dynamics, statistical physics, information and communication theories, data analysis, life and medical sciences. It opens tantalizing perspectives, from the development of new families of detectors to brain research. From the fundamental point of view it is still a largely open field of research. Its microscopic foundations have been hardly addressed, its quantum counterpart needs to be further elucidated, and its relevance in global bifurcations and in complex transition phenomena in spatially extended systems remains to be explored.

References

Original papers

R. Benzi, A. Sutera and A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A14, L453-L457 (1981).

R. Benzi, G. Parisi, A. Sutera and A. Vulpiani, Stochastic resonance in climate change, Tellus, 34, 10-16 (1982).

C. Nicolis, Solar variability and stochastic effects on climate, Sol. Phys. 74, 473-478 (1981).

C. Nicolis, Stochastic aspects of climatic transitions-response to a periodic forcing, Tellus 34, 1-9 (1982).

Early applications in physics and biology

S. Fauve and F. Heslot, Stochastic resonance in a bistable system, Phys. Lett. 97A, 5-7 (1983).

P. Jung and P. Hanggi, Amplification of small signals via Stochastic Resonance, Phys. Rev. A 44, 8032-8042 (1991).

B. McNamara and K. Wiesenfeld, Theory of Stochastic Resonance, Phys. Rev. A 39, 4854-4869 (1989).

Indicators of stochastic resonance

Signal-to-noise ratio

B. McNamara, K. Wiesenfeld and R. Roy, Observation of Stochastic Resonance in a Ring Laser, Phys. Rev. Lett. 60, 2626-2629 (1988).

Residence time distribution

L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta and S. Santucci, Stochastic Resonance in Bistable Systems, Phys. Rev. Lett. 62, 349-352 (1989).

Information theoretic measures

P. Greenwood, L. Ward and W. Wefelmeyer, Statistical analysis of stochastic resonance in a simple setting, Phys. Rev. E60, 4687-4695 (1999).

T. Munakata, A. Sato and T. Hada, Stochastic Resonance in a Simple Threshold System from a Static Mutual Information Point of View, J. Phys. Soc. Japan 74, 2094-2098 (2005).

Reviews

A. Anischenko, V. Astakhov, A. Neiman, T. Vadivasova and L. Schimansky-Geier, Nonlinear dynamics of chaotic and stochastic systems, Springer, Berlin (2002).

A. Bulsara, P. Hänggi, F. Marchesoni, F. Moss and M. Shlesinger, eds, Stochastic Resonance in Physics and Biology, J. Stat. Phys. 70, 1-512 (1993).

M. Dykman and P. Mc Clintock, Stochastic Resonance, Science Progress 82, 113-134 (1999).

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic Resonance, Rev. Mod. Phys. 70, 223-287 (1998).

P. Hänggi, Stochastic resonance in biology - How noise can enhance detection of weak signals and help improve biological information processing, ChemPhysChem 3, 285-290 (2002).

M. D. McDonnell and D. Abbott, What Is Stochastic Resonance? Definitions, Misconceptions, Debates, and Its Relevance to Biology, PLoS Computational Biology 5:e1000348 (2009).

M. D. McDonnell and L. M. Ward, The benefits of noise in neural systems: bridging theory and experiment, Nature Reviews Neuroscience 12, 415-426 (2011).

F. Moss, L. Ward and W. Sannita, Stochastic resonance and sensory information processing: a tutorial and review of application, Clinical Neurophysiology 115, 267-281 (2004).

C. Nicolis, Long-Term Climatic Transitions and Stochastic Resonance, J. Stat. Phys. 70, 3-14 (1993).

Th. Wellens, Y. Shatokhin and A. Buchleitner, Stochastic Resonance, Rep. Progr. Phys. 67, 45-105 (2004).

Recent developments

D. Alcor, V. Croquette, L. Jullien and A. Lemarchand, Molecular sorting by stochastic resonance, Proc. Nat. Acad. Sci. U.S.A. 101, 8276-8280 (2004).

R. Alley et al, Abrupt climatic change, Science 299, 2005-2010 (2003).

A. Ganopolski and S. Rahmtorf, Abrupt glacial climatic changes due to stochastic resonance, Phys. Rev. Letters 88, 038501 (2002).

J. Harry, J. Niemi, A. Priplata and J. Collins, Balancing Act, IEEE Spectrum 42, 36-41 (2005).

T. Mori and S. Kai, Noise-induced entrainment and stochastic resonance in human brain waves, Phy. Rev. Letters 88, 218101 (2002).

C. Rao, D. Wolf and A. Arkin, Control, exploitation and tolerance of intracellular noise, Nature 420, 231-237 (2002).

Internal references