Coherence Resonance: Noise-Induced Order

• Coherence resonance is the noise-induced emergence of regular, quasi-periodic oscillations in nonlinear systems at an optimal noise level without any external periodic forcing.
• Key observables include interspike-interval variability, correlation time, and spectral sharpness that quantify how noise modulates system dynamics.
• Applications span excitable neurons, semiconductor devices, plasma systems, and quantum regimes, highlighting diverse deterministic structures and bifurcation mechanisms.

Searching arXiv for recent and foundational papers on coherence resonance, including provided ids. arxiv_search(query="coherence resonance arXiv coherence resonance chimera time delay nonlocal coupling quantum coherence resonance", max_results=10, sort_by="submittedDate") arxiv_search(query="(Rajhans et al., 2013) OR (Semenova et al., 2015) OR (Masoliver et al., 2017) OR (Korneev et al., 2024) OR (Kato et al., 2020) OR (Ryabov et al., 25 Mar 2026) coherence resonance", max_results=10, sort_by="relevance") Coherence resonance is the noise-induced emergence of maximally regular, quasi-periodic oscillations in an autonomous nonlinear system at an intermediate, nonzero noise intensity, in the absence of any periodic forcing. In the standard excitable picture, weak noise triggers rare and irregular excursions, strong noise destroys temporal order, and an optimal noise level produces the most coherent spike train. The phenomenon was first identified in simple bistable or excitable models and has since been studied in Hodgkin–Huxley and FitzHugh–Nagumo neurons, lasers, semiconductor devices, plasma systems, coupled oscillator networks, and quantum dissipative systems (Uchida, 2023, Ryabov et al., 25 Mar 2026, Kato et al., 2020). It is distinct from stochastic resonance because no external periodic signal is required; the resonance concerns regularization of the system’s own noise-induced dynamics (Tönjes et al., 2021).

1. Definition, observables, and diagnostic criteria

The standard operational definition of coherence resonance is nonmonotonic regularity as a function of noise intensity. Across the surveyed literature, this is quantified by three families of observables: interspike-interval variability, correlation measures, and spectral sharpness (Masoliver et al., 2017, Korneev et al., 2024, Shaw et al., 2014).

A widely used indicator is the coefficient of variation, or normalized variance, of interspike intervals:

$$R=\frac{\sigma_T}{\langle T\rangle} =\frac{\sqrt{\langle T^2\rangle-\langle T\rangle^2}}{\langle T\rangle}.$$

A pronounced minimum of $R$ as noise intensity increases signals coherence resonance. Some studies use the reciprocal convention, for example

$R=\frac{\langle T\rangle}{\sigma_T},$

so that coherence resonance appears as a maximum rather than a minimum; this is the convention adopted in the uni-junction transistor study (Rajhans et al., 2013).

A second class of indicators is based on temporal correlations. The correlation time is typically defined by

$\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$

or, in normalized form for a process $x_i(t)$ with autocorrelation $\Psi_i(s)$,

$$\tau_{c,i}=\frac{1}{\Psi_i(0)}\int_0^\infty |\Psi_i(s)|\,ds.$$

Coherence resonance corresponds to a maximum of $\tau_c$ at intermediate noise (Masoliver et al., 2017, Ryabov et al., 25 Mar 2026).

A third diagnostic is spectral concentration. The principal power-spectrum peak sharpens and narrows at coherence resonance; equivalently, its width at half-height is minimized, or its height or signal-to-noise ratio is maximized (Masoliver et al., 2017, Shaw et al., 2014, Mompo et al., 2020). In experimental plasma data, the spectrum at the resonance point sharpens around the dominant spike-repetition frequency, while in coupled non-excitable oscillators increasing the coupling radius narrows the main spectral peak without introducing new dominant frequencies (Shaw et al., 2014, Ryabov et al., 25 Mar 2026).

The same criteria can be formulated through time averages rather than ensemble averages. For the noisy Hodgkin–Huxley model, numerical evidence shows that in the stationary state a given noise sample path uniquely determines the dynamics, and ergodicity implies that time-averaged irregularity and time-averaged correlation functions become independent of the particular noise sample and coincide with ensemble averages (Uchida, 2023). This establishes coherence resonance as a pathwise as well as ensemble-level phenomenon.

2. Deterministic structures and noise-induced excursion mechanisms

The canonical mechanism of coherence resonance is threshold crossing in an excitable or near-bifurcation system. In deterministic conditions, the system rests at a stable node or focus; noise intermittently pushes trajectories across an excitability threshold; the ensuing excursion is largely governed by the deterministic return dynamics; and the interplay between activation time and refractory time yields maximal regularity at intermediate noise (Shaw et al., 2014, Bogatenko et al., 2018).

This mechanism appears in several deterministic settings. In FitzHugh–Nagumo-type descriptions, the control parameter places the system just beyond threshold, so that without noise it remains at rest, whereas additive noise produces rare spikes for small noise, quasi-periodic spikes for intermediate noise, and overdriven irregular behavior for large noise (Shaw et al., 2014, Korneev et al., 2024). In the “excitable potential well” model, the system is interpreted as motion in a state-dependent potential

$$U(x,y)=\frac{1}{2\varepsilon}x^2+\frac{a}{\varepsilon}x-\frac{g}{\varepsilon}\arctan(x-y),$$

with strictly positive dissipation. There, type-II excitability arises from the geometry of the potential well rather than from negative dissipation, and coherence resonance is marked by an optimal noise intensity $D^*\approx 2.5\times 10^{-5}$ at which the correlation time is maximal and the interspike-interval variability minimal (Bogatenko et al., 2018).

Coherence resonance also occurs in non-excitable systems near global bifurcations. In the subcritical Hopf normal form with additive white noise and delayed feedback,

$R$0

noise excites the ghost of the saddle-node bifurcation of periodic orbits. The resulting resonance can be tracked not only by correlation time and spectral linewidth, but also by the “ghost weight”

$R$1

whose slope peaks near the same noise intensity as the correlation time (Geffert et al., 2014).

In semiconductor superlattices, the deterministic structure is high dimensional: the excitable state is a stable stationary transport state, and noise-induced current self-oscillations arise from repeated nucleation and propagation of charge-dipole waves. The critical event is that the total current falls below a device-dependent critical current determined by the intersection of the static sequential tunneling curve and the contact load line (Mompo et al., 2020). In glow-discharge plasma, an anharmonic oscillator for ion-acoustic oscillations can be reduced to a FitzHugh–Nagumo-like system in which the discharge voltage controls both excitability and intrinsic noise amplitude, producing the same rare-spike $R$2 regular-spike $R$3 irregular-spike sequence (Shaw et al., 2014).

3. Information-theoretic and statistical formulations

One explicit information-theoretic interpretation treats coherence resonance as simultaneous optimization of differential entropy and mutual information in a noise-driven oscillator (Rajhans et al., 2013). For a continuous variable $R$4 with density $R$5,

$R$6

and for continuous variables $R$7 with joint density $R$8,

$R$9

The same quantity can be written as $R=\frac{\langle T\rangle}{\sigma_T},$0 (Rajhans et al., 2013).

The central variational argument is that, among all distributions with fixed mean and variance, the Gaussian maximizes differential entropy, and for fixed second-order moments mutual information is maximized for jointly Gaussian variables. For a linear channel $R=\frac{\langle T\rangle}{\sigma_T},$1 with noise covariance $R=\frac{\langle T\rangle}{\sigma_T},$2,

$R=\frac{\langle T\rangle}{\sigma_T},$3

In the uni-junction transistor relaxation oscillator, external additive white Gaussian noise is then interpreted as shaping the output statistics so that differential entropy of the output voltage and mutual information between injected noise and output voltage both rise from low-noise values, peak sharply at an optimal noise intensity, and decline thereafter (Rajhans et al., 2013).

Within that framework, coherence resonance is not merely regular timing; it is reduction of uncertainty in the output once the noise is known. The proposed physical interpretation is that at optimal noise the oscillator is repeatedly kicked onto essentially the same limit-cycle trajectory, so that the marginal output distribution approaches a maximum-entropy form compatible with its second moments while the conditional uncertainty in the output given the noise is minimized (Rajhans et al., 2013).

The same paper also states clear limitations. The argument assumes that first and second moments capture the relevant statistics, that the optimal output distribution is Gaussian, and that piecewise-linear ordinary differential equations or one-dimensional maps adequately describe the device. Real waveforms are only approximately Gaussian in the noise-driven regime, and higher-dimensional device physics and parasitic capacitances are neglected (Rajhans et al., 2013). These caveats are representative of a broader issue in coherence-resonance theory: many analytical descriptions depend on near-bifurcation reductions, Gaussian closures, or spectral linearizations that are accurate only in restricted regimes.

4. Spatially extended systems and coherence-resonance chimeras

In networks, coherence resonance can acquire a simultaneously temporal and spatial character. A chimera state is a regime in which identical units with symmetric coupling split into coherent and incoherent spatial domains. A coherence-resonance chimera is defined by the simultaneous occurrence of temporally regular noise-induced spiking and spatial coexistence of coherent and incoherent regions (Semenova et al., 2015, Khatun et al., 2022).

The first such states were reported in nonlocally coupled excitable FitzHugh–Nagumo rings with additive Gaussian white noise. For very low noise the network remains in the homogeneous rest state; for an intermediate noise window it develops coherence-resonance chimeras; at slightly larger noise it becomes spatially incoherent but temporally nearly periodic; and at very large noise it is incoherent in both space and time (Semenova et al., 2015). The local order parameter

$R=\frac{\langle T\rangle}{\sigma_T},$4

distinguishes coherent regions, where $R=\frac{\langle T\rangle}{\sigma_T},$5, from incoherent ones, where $R=\frac{\langle T\rangle}{\sigma_T},$6 (Semenova et al., 2015).

A distinctive property of these states is alternating switching of the coherent and incoherent domains. In the FitzHugh–Nagumo ring, the incoherent domain acts as the seed of noise-induced spiking; two pulses propagate coherently in opposite directions, annihilate on the far side of the ring, and a new incoherent patch forms at the antipode. The resulting alternation has been noted as potentially relevant to unihemispheric sleep (Semenova et al., 2015).

The same phenomenon has now been extended to type-I excitability. In nonlocally coupled SNIPER units, coherence-resonance chimeras occur over an optimum range of noise intensity, with coherent and incoherent domains periodically swapping position along the ring. In addition, a mixed or hybrid coherence-resonance chimera appears in which three spatial blocks coexist: a spatially coherent region, a fully spatiotemporally incoherent region, and a spatially incoherent but temporally regular region (Khatun et al., 2022). The intermediate block exhibits an arc-shaped mean-phase-velocity profile,

$R=\frac{\langle T\rangle}{\sigma_T},$7

which is a standard signature of a classical chimera embedded into a coherence-resonance background (Khatun et al., 2022).

These results establish that coherence resonance is not restricted to uniform temporal order. In spatially extended excitable media, the optimal noise level can regularize spike timing while simultaneously inducing symmetry breaking and structured incoherence.

5. Control by topology, delay, multiplexing, and heterogeneous architecture

A major development in the subject is the shift from description to control. In time-delayed FitzHugh–Nagumo rings, coherence resonance depends jointly on noise intensity, coupling delay, and coupling range. Without delay, increasing the number of nearest neighbors lowers the optimal noise level and improves coherence. With delay, the effect depends on topology: for a local ring, representative delays such as $R=\frac{\langle T\rangle}{\sigma_T},$8 or $R=\frac{\langle T\rangle}{\sigma_T},$9 slightly weaken coherence resonance, whereas for nonlocal or global coupling, appropriate delays can either enhance or suppress it (Masoliver et al., 2017). The mechanism is that indirect multi-hop pathways create effective delays of $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$0, so enhancement occurs when the total effective delay is commensurate with the intrinsic period and suppression occurs when it is not (Masoliver et al., 2017).

Nonlocal coupling itself acts as a control parameter. In an ensemble of coupled FitzHugh–Nagumo oscillators, increasing the coupling radius can either enhance or suppress coherence resonance depending on the coupling strength: at weak coupling, the peak correlation time increases and the interspike-interval deviation decreases with radius, while at strong coupling the opposite trend occurs (Ryabov et al., 28 Jun 2025). In the generalized Van der Pol ensemble near the saddle-node bifurcation of limit cycles, increasing the nonlocal coupling radius produces a monotonic increase of the peak global correlation time and a sharpening of the main spectral peak, with the optimal noise intensity remaining approximately $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$1 (Ryabov et al., 25 Mar 2026).

Architecture can also replace homogeneity. In influencer networks of phase oscillators, only a small subset of highly connected hubs is subjected to tunable noise. The long-time follower synchrony, measured by the global order parameter magnitude $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$2, becomes maximal at an intermediate influencer noise intensity. The reduced description yields an effective diffusion transfer of the form

$\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$3

which is maximal at $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$4 and explains why both too little and too much influencer noise fail to synchronize the network (Tönjes et al., 2021).

Multiplexing creates another control channel. In two weakly coupled noisy FitzHugh–Nagumo rings, one layer can be kept at sub-threshold noise so that it would remain silent in isolation, while a second layer is driven by supra-threshold noise. Weak inter-layer coupling then induces coherence resonance, anti-coherence resonance, and inverse stochastic resonance in the otherwise silent layer, and only a small number of randomly distributed inter-layer links may suffice to produce these effects (Masoliver et al., 2020).

Mean-field analysis clarifies which aspects of such control are topology dependent. For globally coupled FitzHugh–Nagumo populations, the mean-field model reproduces the network’s coherence-resonance curves closely across noise and coupling ranges; for locally coupled populations, the low-noise regime is captured, but larger-noise dynamics with traveling spike waves breaks the Gaussian approximation underlying the local mean-field closure (Baspinar et al., 2020). This suggests that analytical tractability is strongest when coupling enforces strong self-averaging.

6. Experimental realizations and non-Gaussian or quantum extensions

The phenomenon is experimentally documented in several physical media. In glow-discharge plasma, the floating potential remains at a stable fixed point around a discharge voltage of approximately $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$5 V, develops irregular excitable spikes above threshold, reaches maximal regularity near $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$6 V, and becomes more irregular again beyond that point. The normalized variance has a clear minimum there, the Hurst exponent peaks near $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$7, and Wiener filtering indicates that the intrinsic noise amplitude increases monotonically with discharge voltage (Shaw et al., 2014).

In weakly coupled semiconductor superlattices, noise induces nearly periodic current self-oscillations even though the deterministic state is stationary. For a representative dc bias of $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$8 V, self-oscillations appear only when the external noise rms exceeds approximately $\tau_c=\int_0^\infty |C(\tau)|\,d\tau,$9 mV, and the normalized interval standard deviation reaches a pronounced minimum near $x_i(t)$0–$x_i(t)$1 mV (Mompo et al., 2020). In that same system, when a weak ac voltage is added in the coherence-resonance frequency band, phase locking occurs for intermediate noise amplitudes and the output signal-to-noise ratio exhibits a classical stochastic-resonance curve (Mompo et al., 2020).

Coherence resonance is not confined to Gaussian forcing. In the FitzHugh–Nagumo system driven by Lévy noise, the correlation time and interspike-interval coefficient of variation depend not only on the noise intensity but also on the stability index $x_i(t)$2 and skewness $x_i(t)$3. As $x_i(t)$4 decreases from the Gaussian value $x_i(t)$5, heavy-tailed jumps progressively suppress coherence resonance; positive skewness enhances it, while negative skewness suppresses it (Korneev et al., 2024). An analog electronic FitzHugh–Nagumo circuit driven by real-time synthesized Lévy noise reproduces the same qualitative trends, with quantitative discrepancies attributed to amplitude clipping of large jumps, component tolerances, and finite sampling rate (Korneev et al., 2024).

A further extension is quantum coherence resonance. In a squeezed quantum van der Pol oscillator governed by a Lindblad master equation, the regularity of the oscillatory response is maximized at an optimal intensity of quantum fluctuations, measured by spectral sharpness or by the coefficient of variation of spike-like events in quantum trajectories (Kato et al., 2020). In the semiclassical regime this first resonance peak has a direct classical interpretation in terms of noise-excited excursions near a bistable excitable structure; in a stronger quantum regime, a second peak appears that the paper identifies as a strong quantum effect associated with dynamics involving only a few energy states (Kato et al., 2020).

7. Scope, limitations, and conceptual issues

Several recurrent misconceptions are explicitly contradicted by the literature. First, coherence resonance is not stochastic resonance: the former requires no periodic signal, whereas the latter concerns optimal response to a weak external periodic input (Tönjes et al., 2021, Nurujjaman, 2009). Second, coherence resonance is not restricted to excitable systems in the narrow sense; it also occurs in non-excitable oscillators near the saddle-node bifurcation of limit cycles, in subcritical Hopf normal forms, and in quantum dissipative systems (Ryabov et al., 25 Mar 2026, Geffert et al., 2014, Kato et al., 2020). Third, it is not exclusively an ensemble-averaged phenomenon; in the noisy Hodgkin–Huxley model it persists for time-averaged pathwise measures under ergodic stationary dynamics (Uchida, 2023).

The role of noise is constructive but not uniformly so. Gaussian white noise can regularize firing, but Lévy noise may either enhance or suppress coherence depending on tail index and skewness, and coupling topology may either increase or decrease coherence depending on delay, radius, or inter-layer architecture (Korneev et al., 2024, Masoliver et al., 2017, Ryabov et al., 28 Jun 2025). This suggests that “optimal noise” is not a scalar property of a system alone; it is conditional on dynamical class, coupling organization, and noise law.

Theoretical limitations are equally explicit. The information-theoretic explanation based on maximizing differential entropy and mutual information assumes that first and second moments are sufficient and that near-optimal waveforms are approximately Gaussian; the device-level description of the uni-junction transistor is reduced to crude piecewise equations or a one-dimensional map (Rajhans et al., 2013). In neuronal populations, local mean-field reductions fail once the state distribution becomes strongly non-Gaussian and spike waves propagate (Baspinar et al., 2020). In the quantum case, the semiclassical stochastic differential equation becomes invalid in the strong-quantum regime where third-order Wigner terms and few-level dynamics dominate (Kato et al., 2020).

An alternative modeling proposal sharpens the distinction between constructive and destructive noise action. In the “constant coherence resonance” framework, noise perturbs the control parameter only to trigger threshold crossing; once a spike begins, the noise is frozen and the system executes one deterministic limit cycle. This construction reproduces standard coherence resonance, explains constant coherence resonance as a plateau of low normalized variance at high noise, and also recovers stochastic resonance when a subthreshold periodic signal is added (Nurujjaman, 2009). A plausible implication is that part of the diversity of coherence-resonance phenomenology across models reflects different assumptions about whether noise acts continuously during the excursion or only at spike initiation.

Taken together, these results define coherence resonance not as a single mechanism but as a family of noise-induced regularization effects organized around a common empirical signature: an intermediate noise level maximizes temporal order. What varies from system to system is the underlying deterministic scaffold—excitable threshold, ghost of a periodic orbit, charge-domain nucleation, hub-mediated diffusion transfer, or quantum few-level dynamics—and the corresponding control parameters through which coherence can be enhanced, suppressed, or spatially structured.