Coherence resonance

The term coherence resonance (CR) refers to a phenomenon whereby addition of certain amount of noise in excitable system makes its oscillatory responses most coherent. Thus a coherence measure of stochastic oscillations attains an extremum at optimal noise intensity, hence the word “resonance”.

Introduction

Optimization of noise-induced oscillations by tuning noise was observed in a wide range of nonlinear systems and is due to various mechanisms. In particular, (Gang et al., 1993) considered a two-dimensional model close to saddle-node bifurcation perturbed by external noise. They found that noise induced a peak in the power spectrum. The height of noise-induced peak showed a maximum being plotted against noise intensity. Also, the position of noise-induced peak showed strong dependence on noise intensity. Subthreshold oscillations may result in noise-induced spikes and bursts (Braun et al., 1994; Longtin and Hinzer, 1996) those coherence can be optimized by tuning noise intensity (Longtin, 1997). Finally, noise-optimized oscillations can be found near period doubling and Andronov-Hopf bifurcations (Neiman et al., 1997; Ushakov et al., 2005).

In excitable systems noise-optimized coherence has a distinct mechanism which was first described and analyzed by Pikovsky and Kurths and termed coherence resonance (Pikovsky and Kurths, 1997). Coherence resonance was demonstrated in a number of experimental studies including electronic circuits (Postnov et al., 1999), optical laser systems (Giacomelli et al., 2000), Belousov – Zabotinsky reaction systems (Miyakawa and Isikawa, 2002; Beato et al., 2007), a neural pacemaker (Gu et al., 2002).

Coherence resonance in FitzHugh – Nagumo model

Coherence resonance was introduced with an example of FitzHugh-Nagumo model perturbed by noise as \[\tag{1} \varepsilon \dot{x}=x-\frac{x^3}{3}-y, \quad \dot{y}=x+a+D\,\xi(t), \]

Figure 1: Deterministic limit cycle in FitzHugh – Nagumo system for \(a=0.99\) and \(\varepsilon=10^{-3}\ .\) Red line shows nullcline, \(y=x-x^3/3\ .\)

Figure 2: Time traces for noise-perturbed FitzHugh – Nagumo model with \(a=1.05,\quad \varepsilon=10^{-3}\) and indicated values of noise amplitude.

where \(\varepsilon \ll 1\) is a small parameter separating time scales of fast activator variable \(x\) and slow inhibitor variable \( y\ ;\) \(\xi(t)\) is zero-mean white Gaussian noise with autocorrelation function \( \langle \xi(t) \xi(t+\tau) \rangle =\delta(\tau)\ ;\) parameter \(D\) sets noise amplitude.

In deterministic case (\(D=0\)) the system is excitable for \(a>1\) with a single stable equilibrium at \(x_0=-a,\quad y_0=a^3/3-a\ .\) This equilibrium undergoes Andronov-Hopf bifurcation at \(a=1\) and for \(a<1\) the system possesses a stable limit cycle. Due to separation of time scales controlled by \(\varepsilon \ll 1\) the motion along the limit cycle is highly non-uniform. That is, the limit cycle contains two parts of slow motion along stable branches of nullcline linked by fast jumps, as Fig. 1 indicates.

Coherence resonance occurs in excitable regime, \(a>1\ ,\) when noise is added. Random perturbation excites the system resulting in large excursions or spikes. Fig. 2 shows plots of \(x(t)\) for three different noise amplitudes. Small noise (upper row) results in rare spikes with variable interspike intervals, while large noise (bottom row) excites frequent spikes also occurring irregularly. The middle row in Fig.2 shows the case when noise-induced spiking is almost periodic. Thus, there is an optimal or resonant noise amplitude, \(D_{res}\ ,\) at which periodicity is maximal. In the phase space noise-induced limit cycle corresponding to optimal noise intensity lies in the vicinity of deterministic limit cycle shown in Fig. 1. Thus, similar to noisy precursors of bifurcation (Wiesenfeld 1985) noise induces a stochastic limit cycle which would appear in deterministic system with the control parameter \(a\) slightly beyond Andronov-Hopf bifurcation (DeVille et al., 2005).

Measures of coherence

Correlation time

In time domain coherence of stochastic oscillations can be assessed in terms of autocorrelation function (ACF), \(C(\tau)=\langle \tilde{x}(t)\tilde{x}(t+\tau) \rangle ,\quad \tilde{x}=x- \langle x \rangle \ ,\) where angular brackets denote ensemble average over realizations of noise or time average over long single realization of system variable \(x(t)\ .\) Periodic signal possesses non-vanishing ACF, while the opposite limit, e.g. white noise, is characterized by Dirak’s delta ACF. The decay rate of ACF is characterized by the correlation time (Stratonovich, 1963), defined as \[ \tau_{cor}=\frac{1}{\mathrm{var}(x)}\,\int_0^\infty C^2(\tau) d\tau, \] where \(\mathrm{var}(x)\) is the variance of \(x\ .\) A coherent oscillatory stochastic process possesses a large correlation time. Correlation time was used as coherence measure in the original paper on coherence resonance (Pikovsky and Kurths, 1997), where it was shown that \(\tau_{cor}\) is maximized at non-zero noise intensity.

Power spectrum measures

Noise-induced oscillations are characterized by a peak at a frequency \(\omega_p\) in the power spectral density (PSD). This noise-induced peak can be characterized by its width measured usually at half maximal power, \(\Delta \omega\ .\) Narrower peak refers to more coherent oscillations and \(\Delta \omega \propto 1/\tau_{cor}\)(Stratonovich, 1963). A widely used measure of the sharpness of a spectral peak is the quality factor defined as \[Q=\frac{\omega_p}{\Delta \omega}\ .\] Large quality factor of a peak in PSD refers to coherent oscillatory process. Thus, coherence resonance corresponds to a maximum of \(Q\)versus noise intensity.

Many studies of coherence resonance used another spectral measure, introduced first in (Gang et al., 1993) as the “signal-to-noise” ratio of the spectral peak height, \(h\ ,\) and its relative width: \[\beta=\frac{h}{\Delta \omega /\omega_p}=hQ\ .\] A maximum of the coherence measure \(\beta\) with respect to noise intensity was considered as a signature of coherence resonance. However, this measure must be used with caution, because a maximum of \(\beta\) does not necessary implies maximization of coherence of noise-induced oscillations. For example, spectral peaks of noisy precursors of period-doubling or Andronov-Hopf bifurcations demonstrate bell-curve shape of \(\beta\) versus noise (Neiman et al., 1997). However, this effect is due to competition of noise dependences of peak height (increases with noise) and the quality factor (decreases with noise). Thus, coherence of the processes (assessed with the quality factor or correlation time) does not improve. Rather, \(\beta\) characterizes how well a spectral peak is expressed with respect to noise background.

Figure 3: Probability density of interspike intervals (A) and power spectrum (B) of noise-perturbed FitzHugh – Nagumo model with \(a=1.05,\quad \varepsilon=10^{-3}\) and indicated values of noise amplitude.

Figure 4: Coefficient of variation (A) and effective diffusion constant (B) versus noise amplitude for FitzHugh – Nagumo model with \(a=1.05,\quad \varepsilon=10^{-3}\ .\)

Interspike interval histogram, coefficient of variation and effective diffusion coefficient

Often only timing of spikes is of interest, so that continuous dynamics of the excitable system is mapped onto stochastic point process of sequences of spike times, \(\{t_i\}\ ,\) that is, moments of time at which spikes occur. Interspike intervals (ISI), \(T_i=t_i-t_{i-1}\ ,\) represent instantaneous period and their distribution known as interspike interval histogram (ISIH) used widely in neuroscience as characteristic of spiking process. Fig. 3A shows probability densities of ISIs of spike trains generated by stochastic FitzHugh-Nagumo model (1) for the same three values of noise amplitude as in Fig. 2. Both weak and strong noises result in wide distributions. Intermediate noise leads to a narrow distribution, indicating high degree of coherence of noise-induced spiking. A dimensionless measure of spike train variability is the coefficient of variation (CV) defined as the ratio of ISIs standard deviation to the mean ISI, \[CV=\frac{\sqrt{ \langle T^2 \rangle -\langle T \rangle ^2}}{\langle T \rangle}\ .\] \(CV=1\) for Poisson processes. Coherence resonance is manifested as a pronounced minimum in the dependence of CV versus noise strength.

A spike train can be represented as a sum of delta functions centered at spike times\[s(t)=\sum_i \delta(t-t_i)\ .\] ACF and power spectrum of the spike train can then be calculated. For renewal stochastic point processes, such as generated by stochastic FitzHugh-Nagumo and leaky integrate and fire models with white noise, PSD is related directly to the probability density of ISIs (Stratonovich, 1963). In Fig. 3B power spectrum shows sharpest peak for intermediate noise intensity (green line), consistent with the behavior of ISIs distribution shown in Fig. 3A.

By integrating spike train function \(s(t)\ ,\) from 0 to t we obtain the number of spikes in a window (0, t) which is called the spike count, \(n(t)=\int_0^t s(t') dt'\ .\) The spike count is a stochastic variable that undergoes diffusive spreading: its variance grows linearly in time. The slower is this growth the more coherent is corresponding spike train, as for pure periodic spike sequence the variance of \(n(t)\) does not grow at all. The rate of the diffusive spreading is given by the effective diffusion constant, \(D_{\mathrm{eff}}=\lim_{t\to\infty}\frac{\langle [n(t)-\langle n(t) \rangle ]^2 \rangle }{2t}\ ,\) which serves as another measure of coherence. Calculation of the effective diffusion constant is significantly simplified for renewal processes, providing the following relation (Lindner and Schimansky-Geier, 1999), \[D_{\mathrm{eff}}=\frac{CV^2}{2\langle T \rangle}\ ,\] and \(D_{\mathrm{eff}}\propto 1/\tau_{cor}\ .\) Coherence resonance is characterized by a minimum of \(D_{\mathrm{eff}}\) plotted versus noise strength.

Two measures of coherence for stochastic FitzHugh–Nagumo model (1) are summarized in Fig.4. For weak noise CV is close to 1 indicating that statistics of rarely occurring spikes is close to Poissonian. With the increase of noise CV decreases reaching minimum at and then increases again. Similarly, effective diffusion constant is minimal at the same noise strength, indicating that diffusive spreading of spike count is slowest for that noise.

It is a good practice to employ several measures of coherence, not just one, when studying noise-induced dynamics and its coherence. In particular, several studies have shown limitations of the coefficient of variation as a coherence measure for noise-induced dynamics (Shuai et al., 2002; Lindner et al., 2003).

Mechanism of coherence resonance

Although mechanisms of appearance of noise-induced limit cycle may be different as discussed in details in (DeVille et al., 2005), an extremum of a coherence measure can be explained qualitatively based on distinct behavior of characteristic timescales of excitable systems on noise strength (Pikovsky and Kurths, 1997). A period of noise-induced limit cycle can be decomposed onto two parts: activation time (\(T_a\)), needed for the phase trajectory to escape from the stable equilibrium to the excited state, and excursion time (\(T_e\)), needed to return to equilibrium. The activation time follows Arrhenius law as a function of noise intensity, \(<T_a>\propto \exp(\Delta/D^2)\ ,\) where \(\Delta\) is a threshold of excitation. On the contrary, the decay of unstable excited state given by the excursion time, shows weak dependence on noise. For weak noise, \(D^2\ll\Delta\ ,\) the period of noise-induced oscillations is dominated by activation time, and spiking follows Poissonian statistics. Thus, fluctuations of interspike intervals are large for weak noise and the coefficient of variation approaches 1. On the other hand, for large noise activation time becomes very short and the period is dominated by the excursion time. Although the mean excursion time depends weakly on noise intensity, its variance is proportional to \(D^2\ .\) Thus, for large noise fluctuations of period are mainly due to a jitter of excursion times and the coefficient of variation increases with noise as \(CV \propto D\ .\) Coherence resonance (a minimum of CV) occurs at intermediate noise intensity: noise is large enough so that activation time is short and the period of noise-induced oscillations is dominated by excursion time, but not large enough to result in large fluctuations of excursion time.

Analytical calculations of CV have been done in the original paper (Pikovsky and Kurths, 1997) for a simplified model using the mean first passage time approach. In the limit \(\varepsilon \to \infty\) analytical treatment is possible for the FitzHugh–Nagumo model using the Fokker-Planck equation and the mean first passage time formalisms (Lindner and Schimansky-Geier, 1999; Lindner and Schimansky-Geier, 2000) which allowed calculations of CV, effective diffusion constant and PSD. Several analytical studies showed the existence of coherence resonance in a leaky integrate and fire model (Pakdaman et al., 2001; Lindner et al., 2002).

Coherence resonance in biophysical neuron models due to internal noise

In contrast to formal models such as FitzHugh–Nagumo or integrate and fire where noise is just introduced to the right hand side of the equations, in biophysical conductance-based Hodgkin-Huxley type models noise is intrinsic and is due to internal thermal fluctuations of membrane conductance mediated by random opening and closing of ion channels. Intrinsic noise due to ion channel fluctuations can be modeled explicitly using various kinetic schemes (Clay and DeFelice, 1983; Chow and White, 1996) or implicitly using Langevin equations approach (Fox and Lu, 1994). The intensity of intrinsic fluctuations is inversely proportional to the number of ion channels in a membrane patch. Thus, varying the size of ion channels cluster one can study effect of intrinsic fluctuations on statistics of spontaneously generated action potentials. Deterministic dynamics corresponds to the limit of large number of channels.

Coherence resonance due to intrinsic ion channel fluctuations was studied for stochastic Hodgkin-Huxley model using kinetic approach in (Jung and Shuai, 2001; Zeng and Jung, 2004) and using Langevin approach in (Schmid et al., 2001; Schmid and Hanggi, 2007). Both methods showed the existence of an optimal size of ion channels cluster (~ 1\(\mu m^2\)) which maximized the coherence of spontaneous trains of action potential. Another important manifestation of coherence resonance was found in theoretical studies on intracellular calcium signaling, where optimal clustering of ion channels resulted in most coherent calcium oscillations (Shuai and Jung, 2002; Shuai and Jung, 2003).

References

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Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Frances K. Skinner (2006) Conductance-based models. Scholarpedia, 1(11):1408.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• Catherine Rouvas-Nicolis and Gregoire Nicolis (2007) Stochastic resonance. Scholarpedia, 2(11):1474.
• Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.

See also

Excitable System, Oscillator, Resonance, Stochastic Resonance, Synchronization