Deterministic Coherence Resonance

• Deterministic Coherence Resonance is defined by the emergence of regular oscillations through deterministic control parameters rather than noise.
• In delay-coupled FitzHugh–Nagumo networks, synchronized periodic spiking arises via a saddle-node bifurcation of limit cycles, independent of system size and topology.
• In coupled chaotic Lorenz oscillators, tuning the coupling strength reorganizes chaotic intermittency, producing both coherence and anti-coherence resonance in different observables.

Searching arXiv for papers on deterministic coherence resonance and closely related coherence-resonance formulations. I’ll look up relevant arXiv records to ground the article in the cited literature. Deterministic coherence resonance denotes the emergence or optimization of temporally regular oscillatory behavior by varying deterministic parameters rather than by tuning noise intensity. In the literature, the term is not used uniformly. In a delay-coupled excitable FitzHugh–Nagumo network, the noise-free system can generate regular, synchronized, self-sustained oscillations solely because time-delayed coupling creates periodic dynamics (Masoliver et al., 2017). In two coupled identical chaotic Lorenz oscillators, a nonmonotonic dependence of coherence on coupling strength appears with no external noise and no parameter mismatch (Komkov et al., 14 Sep 2025). Both usages differ from the usual sense of coherence resonance, in which an excitable system shows the most regular spike trains at an intermediate noise level (Masoliver et al., 2017).

1. Definition and terminological scope

The strict stochastic meaning of coherence resonance is noise-induced regularization: noise kicks a system out of a stable rest state, and an optimal noise intensity minimizes timing irregularity. Deterministic coherence resonance departs from that construction by replacing the noise-control parameter with a deterministic control parameter, or by considering coherent oscillations that are generated in a noise-free setting by deterministic coupling or delay (Masoliver et al., 2017).

Two formulations are especially prominent. The first is a delay-driven formulation in excitable media: coherent periodic spiking is produced without noise because delayed diffusive coupling creates a stable periodic attractor (Masoliver et al., 2017). The second is a coupling-driven formulation in chaotic systems: intrinsic chaotic irregularity is reorganized by changing a deterministic coupling parameter, yielding a nonmonotonic coherence curve in the absence of externally applied noise (Komkov et al., 14 Sep 2025).

A recurring source of confusion is that several nearby literatures study deterministic structure without studying deterministic coherence resonance in the strict sense. Time-delayed feedback in a subcritical Hopf normal form, state-dependent excitable potential wells, intrinsic-noise plasma spikes, nonlocal control of noisy FitzHugh–Nagumo or generalized Van der Pol ensembles, and pathwise coherence measures in stochastic Hodgkin–Huxley dynamics all retain noise as an essential ingredient of the resonance itself (Geffert et al., 2014, Bogatenko et al., 2018, Shaw et al., 2014, Ryabov et al., 28 Jun 2025, Ryabov et al., 25 Mar 2026, Uchida, 2023).

2. Delay-induced coherent oscillations in excitable FitzHugh–Nagumo networks

A canonical deterministic setting is the ring network of $N$ FitzHugh–Nagumo elements with $P$ nearest neighbors on each side and delayed diffusive coupling,

$$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$

$$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$

Here $u_i$ is the activator, $v_i$ the inhibitor, $\epsilon=0.01$, $a$ is the excitability parameter, $\sigma$ the coupling strength, $\tau$ the delay, and $P$0 the noise intensity. In the deterministic regime one sets $P$1; the isolated FitzHugh–Nagumo system has a Hopf bifurcation at $P$2, and $P$3 corresponds to the excitable regime with a stable fixed point (Masoliver et al., 2017).

In that noise-free regime, the network can undergo delay-induced oscillations even though each isolated unit is excitable and quiescent. For certain $P$4, the system develops synchronized periodic spiking. The oscillations arise through a saddle-node bifurcation of limit cycles, and the oscillation period is approximately

$P$5

The resulting dynamics are highly regular and synchronized across the ring (Masoliver et al., 2017).

The principal deterministic result is that the delay-induced oscillation regime is independent of the number of nearest neighbors $P$6 and the system size $P$7. In the synchronized state,

$P$8

so the network reduces to a single delay equation that contains no $P$9 or $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$0. This reduction explains why the deterministic delay-induced oscillation region does not depend on topology or size (Masoliver et al., 2017).

The location of that oscillatory region depends strongly on the excitability parameter $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$1. For $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$2, delay-induced oscillations appear for a smaller range of delay and coupling values. For $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$3, one needs larger $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$4 and $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$5 to induce oscillations. Moving farther from the Hopf bifurcation therefore makes delay-induced coherent oscillations harder to generate (Masoliver et al., 2017).

3. Coupling-induced deterministic coherence resonance in coupled Lorenz oscillators

A stricter realization of deterministic coherence resonance is provided by two identical chaotic Lorenz oscillators coupled diffusively and symmetrically through the $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$6-variables,

$$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$7

with $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$8, $$\epsilon \dot{u}_i = u_i-\frac{u_i^3}{3}-v_i + \frac{\sigma}{2P} \sum_{j=i-P}^{i+P} \big[u_j(t-\tau)-u_i(t)\big],$$9, and $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$0. No external noise is added, and there is no parameter mismatch (Komkov et al., 14 Sep 2025).

As the coupling strength $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$1 increases, the system passes through weakly coupled asynchronous chaos, an on-off intermittency regime, and eventually complete synchronization. In the numerical model the onset of on-off intermittency occurs at $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$2, while complete synchronization occurs at $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$3. Before the synchronization threshold, the coupled system exhibits hyperchaotic dynamics associated with on-off intermittency (Komkov et al., 14 Sep 2025).

Within this intermittency window, chaotic dynamics acts as an internal source of irregularity. Tuning $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$4 reorganizes that irregularity. For the observables $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$5 and $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$6, the normalized correlation time first increases, reaches a maximum, and then decreases. This nonmonotonic regularization is the signature of deterministic coherence resonance. The optimal coupling is around $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$7 in the numerical model and $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$8 in experiment (Komkov et al., 14 Sep 2025).

The same system simultaneously displays deterministic anti-coherence resonance in the $$\dot{v}_i = u_i + a + \sqrt{2D}\,\xi_i(t), \qquad i=1,\dots,N.$$9 variables. There the normalized correlation time shows a minimum at intermediate coupling, around $u_i$0 numerically and $u_i$1 experimentally. This observable dependence is central: the same deterministic parameter can make some variables more regular while making another less regular (Komkov et al., 14 Sep 2025).

The on-off intermittency interpretation is supported by standard scaling laws. At fixed $u_i$2, the laminar phase length distribution obeys

$u_i$3

and the mean laminar phase length satisfies

$u_i$4

These are characteristic signatures of on-off intermittency and tie deterministic coherence resonance to intermittency-mediated reorganization of chaotic motion (Komkov et al., 14 Sep 2025).

4. Quantification and observable dependence

Deterministic coherence resonance is identified through regularity measures rather than through a universal bifurcation criterion. In the delay-coupled FitzHugh–Nagumo network, coherence is quantified by the normalized standard deviation of the interspike interval generalized to the network,

$u_i$5

Smaller $u_i$6 means more regular, coherent oscillations, and the mean oscillation period is

$u_i$7

Although this metric is mainly used later for noisy simulations, in the deterministic setting the appearance of periodic synchronized oscillations corresponds to highly coherent dynamics (Masoliver et al., 2017).

In the Lorenz system the principal measure is the correlation time,

$u_i$8

where $u_i$9 is the autocorrelation function and $v_i$0 its variance. The paper normalizes it as

$v_i$1

Larger $v_i$2 means more regular dynamics, and a nonmonotonic dependence on $v_i$3 reveals deterministic coherence resonance or deterministic anti-coherence resonance depending on the variable considered (Komkov et al., 14 Sep 2025).

Observable dependence is not a secondary detail. In the Lorenz pair, $v_i$4 and $v_i$5 display deterministic coherence resonance while $v_i$6 displays deterministic anti-coherence resonance at the same parameter values. The phenomenon is therefore not captured by a single scalar notion of “system coherence”; it is variable-specific and tied to how each observable responds to intermittency, synchronization episodes, and the geometry of the attractor (Komkov et al., 14 Sep 2025).

A broader methodological context is provided by stochastic coherence-resonance studies that compare time averages with ensemble averages. In a stochastic Hodgkin–Huxley neuron, numerical evidence suggests that in the stationary regime a given noise sample path uniquely determines the dynamics, and time-averaged coherence measures become sample-path independent and equal to ensemble-averaged measures (Uchida, 2023). This does not establish deterministic coherence resonance in the strict sense, but it clarifies why pathwise regularity measures remain meaningful in nearby literatures.

5. Relation to stochastic coherence resonance and adjacent mechanisms

Several mechanisms are closely related to deterministic coherence resonance but should not be conflated with it. In a subcritical Hopf normal form with noise and time-delayed feedback, coherence resonance is associated with the ghost of a saddle-node bifurcation of periodic orbits. Deterministic bifurcation structure organizes where resonance can occur, but noise is necessary to activate excursions into the ghost region (Geffert et al., 2014). In a state-dependent excitable potential well with positive dissipation, the deterministic system supports type-II excitability and self-oscillation, yet coherence resonance is still produced when noise triggers excursions from a stable equilibrium (Bogatenko et al., 2018).

Experimental intrinsic-noise-induced coherence resonance in a glow discharge plasma is explicitly interpreted as noise-driven rather than deterministic; the regularity of spikes is maximal at an intermediate discharge voltage, and a noisy FitzHugh–Nagumo-like reduction reproduces the effect (Shaw et al., 2014). Likewise, an alternative excitable-system model in which noise perturbs only the control parameter explains coherence resonance, constant coherence resonance, and stochastic resonance, but the resonance remains noise-induced even though the post-trigger excursion is deterministic (Nurujjaman, 2009).

Nonlocal coupling can control the strength of noise-driven coherence resonance without creating deterministic coherence resonance. In noisy FitzHugh–Nagumo ensembles, increasing the coupling radius can enhance or suppress coherence resonance depending on coupling strength, and the effect saturates as the system approaches the global-coupling limit (Ryabov et al., 28 Jun 2025). In non-excitable generalized Van der Pol ensembles near a saddle-node bifurcation of limit cycles, increasing the coupling radius weakens the suppression caused by local coupling and enhances coherence resonance, but the mechanism remains stochastic because the oscillators are explicitly driven by Gaussian white noise (Ryabov et al., 25 Mar 2026).

The same caution applies to more interpretive frameworks. An information-theoretic account based on differential entropy and mutual information proposes a noise-assisted explanation of coherence resonance in chaotic relaxation oscillators, not a purely deterministic mechanism (Rajhans et al., 2013). A useful summary is therefore the following:

Formulation	Deterministic substrate	Role of noise
Delay-induced coherent oscillations in excitable networks	Delay and coupling create periodic dynamics	Absent in the deterministic regime
Coupling-induced coherence in chaotic Lorenz oscillators	Coupling reorganizes chaotic intermittency	No external noise
Classical coherence resonance and its controlled variants	Excitable or bifurcation structure organizes response	Essential for resonance

6. Conceptual status, misconceptions, and significance

A common misconception is that any coherent oscillation generated without periodic forcing should be called coherence resonance. The literature does not support such a broad equivalence. In the FitzHugh–Nagumo ring, the authors explicitly use the phrase in a broader, delay-driven sense, while also noting that in the deterministic setting the phenomenon is not “resonance” in the strict stochastic sense; rather, delay creates a self-sustained periodic attractor in an excitable network (Masoliver et al., 2017). In the Lorenz pair, by contrast, the designation is stricter: regularity is optimized by a deterministic coupling parameter in a system with no external noise (Komkov et al., 14 Sep 2025).

A second misconception is that deterministic architecture controlling a stochastic resonance is itself deterministic coherence resonance. The nonlocal-coupling studies on FitzHugh–Nagumo and generalized Van der Pol ensembles explicitly do not claim deterministic coherence resonance without noise; they show that deterministic topology can enhance or suppress a resonance that remains noise-driven (Ryabov et al., 28 Jun 2025, Ryabov et al., 25 Mar 2026).

A third misconception concerns “pathwise determinism.” In stationary stochastic Hodgkin–Huxley dynamics, a fixed noise sample path can uniquely determine the long-time trajectory, giving a pathwise interpretation of coherence measures. This remains a stochastic phenomenon because the resonance depends on noise amplitude and on the particular noise realization used to generate the stationary trajectory (Uchida, 2023).

The significance of deterministic coherence resonance lies in the fact that temporal regularity need not be organized exclusively by externally imposed stochastic forcing. Delay-coupled excitable networks show that coherent periodic spiking can be created by delayed coupling alone, with an oscillatory region independent of topology and system size in the synchronized regime (Masoliver et al., 2017). Coupled chaotic oscillators show that intrinsic chaotic irregularity can itself play the role of an effective irregular drive, and that varying coupling strength can produce coherence resonance and anti-coherence resonance simultaneously in different observables before complete synchronization is reached (Komkov et al., 14 Sep 2025).

This suggests that deterministic coherence resonance is best understood as a family of parameter-induced regularization phenomena rather than as a single canonical mechanism. In one case the organizing structure is a delay-generated periodic attractor in an excitable medium; in another it is the coupling-controlled intermittency of identical chaotic oscillators. What these cases share is a nontrivial increase in temporal order produced by deterministic control, but the underlying dynamical mechanisms remain system-specific.