---
title: Anti-Coherence Resonance
url: https://www.emergentmind.com/topics/anti-coherence-resonance
type: topic
---

# Anti-Coherence Resonance

Searching arXiv for recent and foundational papers on anti-coherence resonance and closely related suppression of coherence resonance.
Anti-coherence resonance denotes a nonmonotonic loss of temporal regularity at an intermediate value of a control parameter, typically noise intensity or coupling strength, in systems that otherwise display coherence resonance or related noise-induced ordering. Across the recent literature, the term is used operationally rather than through a single universal definition: in stochastic excitable networks it is identified by a maximum of an interspike-interval variability measure, whereas in deterministic coupled chaotic oscillators it is identified by a minimum of correlation time. Closely related work also studies suppression of coherence resonance without always naming a distinct anti-coherence-resonance regime, which makes the topic partly terminological and partly metric-dependent [2009.09830, 2009.08884, 2509.11189, 2506.22909].

## 1. Conceptual definition

In the stochastic FitzHugh–Nagumo literature, coherence resonance is the appearance of maximally regular noise-induced spiking at an intermediate noise level, and anti-coherence resonance is its opposite in the sense that spiking becomes maximally irregular at an intermediate control parameter. In the multiplex neural-network formulation, the distinction is stated explicitly: coherence resonance corresponds to a pronounced minimum of the coefficient of variation of the interspike intervals, while anti-coherence resonance corresponds to a maximum of the same quantity as the noise intensity in another layer is varied [2009.09830].

In the deterministic Lorenz literature, the same conceptual opposition is retained but the observable changes. There, deterministic coherence resonance is a local maximum of correlation time, whereas deterministic anti-coherence resonance is a local minimum of correlation time as the coupling strength is varied. The phenomenon is therefore defined through coherence loss, not through the presence of external noise [2509.11189].

This usage should be distinguished from two neighboring notions. First, “suppression of coherence resonance” may refer to a flattening or weakening of the coherence-resonance curves without the formal introduction of a separate anti-coherence-resonance branch. Second, “anti-resonance” in a parametrically excited Van der Pol oscillator refers to a deterministic minimum of oscillation amplitude at a special modulation frequency, not to a minimum of temporal coherence or a maximum of interspike irregularity [2506.22909, 1202.6592].

## 2. Diagnostic measures and operational criteria

The literature uses a small set of regularity measures. In stochastic excitable systems, the central observable is the coefficient of variation of interspike intervals,
\[
R_k = \frac{\sigma^{ISI}_k}{\langle ISI \rangle_k},
\]
or related population-averaged variants. In deterministic chaotic systems, the principal measure is the correlation time,
\[
t_{\text{cor}}=\frac{1}{\Psi(0)}\int_0^\infty \left| \Psi(s) \right|ds,
\]
sometimes normalized as
\[
\tilde{t}_{\text{cor}}(K)=\frac{t_{\text{cor}}(K)}{t_{\text{cor}}(K=0)}.
\]
A third class of work quantifies suppression of coherence resonance through a decrease of peak correlation time, an increase of the minimum normalized interspike-interval deviation, or a less pronounced spectral peak, but does not necessarily label the effect anti-coherence resonance [2009.09830, 2009.08884, 2509.11189, 2506.22909].

| System class | Regularity measure | Anti-coherence signature |
|---|---|---|
| Multiplex FHN networks | \(R_k=\sigma^{ISI}_k/\langle ISI\rangle_k\) | Maximum of \(R_k\) |
| Global FHN mean-field/network model | \(R(D)\) from ISI variability | Maximum of \(R(D)\) |
| Coupled Lorenz oscillators | \(t_{\text{cor}}\), \(\tilde t_{\text{cor}}\) | Minimum of correlation time |
| Nonlocal-coupled FHN control | \(\overline{t}_{\mathrm{cor}}\), \(\overline{R}_{ISI}\), \(\overline{S}(\omega)\) | Suppression of CR, not a formal ACR definition |

A direct implication is that anti-coherence resonance is not tied to a single formula. It is tied to a local extremum of a system-specific regularity functional, with the extremum having the sign opposite to the one used for coherence resonance.

## 3. Noise-induced anti-coherence resonance in excitable networks

A direct stochastic realization appears in the two-layer multiplex neural-network model of excitable FitzHugh–Nagumo neurons. Each layer is a nearest-neighbor ring, all neurons are in the excitable regime with \(a=1.05\) and \(\epsilon=0.01\), and the two layers are weakly multiplexed with \(\sigma_{12}\ll \sigma\). Layer 1 is driven by supra-threshold noise \(D_1\), whereas layer 2 receives sub-threshold noise \(D_2=2.5\cdot 10^{-6}\) and is silent when \(\sigma_{12}=0\). In this setting, varying only \(D_1\) can induce in layer 2 a sequence of coherence resonance, anti-coherence resonance, and inverse stochastic resonance. The anti-coherence branch is identified by a maximum of
\[
R_2=\frac{\sigma^{ISI}_2}{\langle ISI\rangle_2},
\]
at a higher \(D_1\) than the coherence-resonance minimum [2009.09830].

The same paper attributes this behavior to a competition among three ingredients: noise-induced excitation in layer 1, weak signal transfer across layers, and the intrinsic excitability of the otherwise silent layer 2. At intermediate transferred drive, the perturbations arriving from layer 1 can induce comparatively regular spikes in layer 2; at other values of \(D_1\), especially higher ones, the transferred fluctuations are strong enough to induce firing but too uneven to produce temporally regular responses. Anti-coherence resonance therefore denotes maximally irregular induced spiking, not simply weak firing.

The phenomenon is reported from the smallest multiplex pair up to large rings. Demonstrated network sizes include \(N=1,3,50,100,500\). It is also robust to sparse inter-layer connectivity: for \(N=500\), coherence resonance and anti-coherence resonance in layer 2 persist even when up to \(80\%\) of inter-layer links are removed at random, and the extended material reports that for \(\sigma=0.1\) and \(\sigma_{12}=0.01\), coherence resonance and anti-coherence resonance can appear even with just one inter-layer link [2009.09830].

A second stochastic setting is the globally coupled FitzHugh–Nagumo population studied through both a network model and a mean-field limit. In the extended globally coupled formulation with noise in both equations, the regularity measure \(R(D)\) shows an ordinary coherence-resonance minimum at low or intermediate noise and an anti-coherence-resonance maximum at larger noise. For \(a=1.05\), the minimum is reported near \(D=0.001\) and the maximum near \(D=1.58489\); for \(a=1.3\), the coherence-resonance minimum shifts to \(D=0.01259\), while the anti-coherence maximum again occurs near \(D=1.58489\). The proposed mechanism is that strong noise destroys the refractory-time structure of excitable firing and produces very small interspike intervals, so that firing becomes fluctuation-driven rather than governed by the intrinsic excursion-and-recovery cycle [2009.08884].

That result is not universal across all FHN reductions. The same paper states that the anti-coherence effect is robust in the extended globally coupled model, is not established for the locally coupled reduction, and is not a genuine network effect in the original globally coupled one-noise formulation, where a mean-field maximum is judged artificial [2009.08884].

## 4. Deterministic anti-coherence resonance

A qualitatively different realization appears in two bidirectionally coupled identical Lorenz oscillators,
\[
\frac{dx_{1,2}}{dt} = \sigma(y_{1,2}-x_{1,2})+K(x_{2,1}-x_{1,2}),
\]
\[
\frac{dy_{1,2}}{dt} = x_{1,2}(\rho-z_{1,2})-y_{1,2},
\]
\[
\frac{dz_{1,2}}{dt} = -\beta z_{1,2}+x_{1,2}y_{1,2},
\]
with \(\sigma=10\), \(\rho=28\), and \(\beta=8/3\). Here there is no external noise. The control parameter is the coupling strength \(K\), and anti-coherence resonance is identified through a minimum of the correlation time of the \(z\)-oscillations, while \(x(t)\) and \(y(t)\) simultaneously show deterministic coherence resonance through a maximum of correlation time [2509.11189].

The effect occurs in the interval bounded by the onset of intermittent temporary synchronization at \(K_1^{\text{crit}}\approx 0.5\) and the onset of complete synchronization at \(K_2^{\text{crit}}\approx 3.92\). This is the on-off intermittency window, and the abstract characterizes it as hyperchaotic dynamics associated with the intermittency. Within that interval, the strongest deterministic coherence resonance in \(x\) and \(y\) occurs near \(K_1^{\text{peak}}\approx 1.8\), whereas the strongest deterministic anti-coherence resonance in \(z\) occurs near \(K_2^{\text{peak}}\approx 2.3\). Experiments on an analog electronic model confirm the same qualitative structure, with the extrema shifted to \(K_1^{\text{peak}}\approx 2.8\) and \(K_2^{\text{peak}}\approx 3.8\) [2509.11189].

This variable dependence is a distinctive feature. The same coupled chaotic attractor is reported as becoming more coherent in the \(x\)- and \(y\)-projections while becoming less coherent in the \(z\)-projection. The paper is explicit, however, that the theoretical reasons for the occurrence of deterministic coherence resonance and deterministic anti-coherence resonance are not clear and remain for further study [2509.11189].

## 5. Suppression of coherence resonance and anti-coherence-like regimes

Several papers provide nearby phenomena that are directly relevant even when they avoid the term anti-coherence resonance. In a ring of \(N=100\) noisy excitable FitzHugh–Nagumo oscillators with nonlocal coupling,
\[
f_i=\frac{\sigma}{2R}\sum_{j=i-R}^{i+R}(x_j-x_i),
\]
increasing the coupling radius can either enhance or suppress coherence resonance. The crucial case is strong coupling, \(\sigma=2\), where growth of the coupling radius suppresses coherence resonance: the peak of the averaged correlation time decreases, the minimum of the averaged normalized interspike-interval deviation increases, and the spectral peak becomes less favorable. The paper presents this as topology-induced suppression of coherence resonance rather than as a formally distinct anti-coherence-resonance state [2506.22909].

A similar constructive-to-destructive crossover appears in the excitable semiconductor superlattice driven by global voltage noise. There coherence resonance is identified by a minimum of
\[
R_{T_a}=\frac{\sqrt{\langle T_a^2\rangle-\langle T_a\rangle^2}}{\langle T_a\rangle},
\]
as external noise induces regular current self-oscillations through repeated nucleation and propagation of charge dipole waves from the injector. The paper does not use the term anti-coherence resonance, but it shows that after the coherence-resonance optimum, \(R_{T_a}\) increases again, the spectra become less sharply organized, and in the ac-driven case the oscillations eventually lose frequency locking. These are anti-coherence-like deterioration branches beyond the optimum [2011.09327].

An information-theoretic treatment of coherence resonance in a uni-junction transistor relaxation oscillator is relevant mainly by contrast. It associates the resonance regime with maximized differential entropy,
\[
h(X)=-\int_{-\infty}^{+\infty} f_X(x)\log(f_X(x)),
\]
and maximized mutual information. That paper does not mention anti-coherence resonance explicitly. It only suggests, by implication, that departure from the resonance window would reduce periodicity, predictability, and effective information transfer [1312.0203].

## 6. Mechanisms, neighboring concepts, and unresolved issues

The mechanisms proposed for anti-coherence resonance differ by system class. In multiplex excitable networks, the phenomenon is linked to the mismatch between a noisy source layer and a silent target layer under weak multiplexing: transferred fluctuations can be regularizing at one noise level and maximally irregular at another. In globally coupled stochastic FHN populations, the anti-coherence branch is tied to strong-noise destruction of refractory-time organization and the appearance of very small interspike intervals. In coupled Lorenz oscillators, the relevant background is on-off intermittency and hyperchaotic dynamics below complete synchronization, but the paper explicitly leaves the detailed theory unresolved [2009.09830, 2009.08884, 2509.11189].

Several common misconceptions are corrected by these results. Anti-coherence resonance is not merely “large noise gives disorder,” because the defining feature is a nonmonotonic extremum of a regularity measure. It is not necessarily a noise phenomenon, because deterministic anti-coherence resonance has been reported in the Lorenz system. It is also not identical to inverse stochastic resonance: in the multiplex FHN model, anti-coherence resonance is the maximum of \(R_k\), whereas inverse stochastic resonance is a maximum of \(\langle ISI\rangle_k\), equivalently a minimum of firing rate [2009.09830].

It should also be separated from deterministic anti-resonance in parametrically excited nonlinear oscillators. In the Van der Pol system with parametrically excited nonlinearity,
\[
\ddot{x}+ [1+\gamma\cos(\Omega t)](x^2-1)\dot{x}+x=0,
\]
anti-resonance means amplitude suppression of the stable self-oscillation at \(\Omega=2\), with
\[
A=\sqrt{4-2|\gamma|}\quad (|\gamma|<1), \qquad A=\sqrt{2}\quad (|\gamma|\ge 1).
\]
That is an amplitude minimum of a deterministic limit cycle, not a minimum of temporal coherence or a maximum of interspike irregularity [1202.6592].

Taken together, the literature indicates that anti-coherence resonance is best understood as a family of coherence-loss phenomena defined operationally by the extremum of a regularity measure. The most mature direct realizations are the multiplex and globally coupled FitzHugh–Nagumo studies and the deterministic Lorenz study. The neighboring literature on suppression of coherence resonance, high-noise deterioration, and deterministic anti-resonance clarifies the boundaries of the concept, but also shows that its nomenclature remains system-dependent and still lacks a single canonical formulation [2506.22909, 2011.09327, 1202.6592].

Source: https://www.emergentmind.com/topics/anti-coherence-resonance