Delay asymmetry induces a functional separation between sensitivity and coordination

Delay asymmetry induces a functional separation between sensitivity and coordination | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Delay asymmetry induces a functional separation between sensitivity and coordination Nobuchika Yamaki, Tenna Churiki This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8896814/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 10 You are reading this latest preprint version Abstract Delayed interactions are ubiquitous in physical, biological, and engineered systems, yet delay is typically treated as a single parameter governing stability or synchronization. Here we demonstrate that delayed nonlinear systems generically exhibit a functional separation between sensitivity to delay asymmetry and coordination between interacting units. Using a minimal stochastic delayed system, we systematically varied delay asymmetry and quantified sensitivity via Fisher information and coordination using an information–phase composite index. We find that sensitivity to asymmetry emerges immediately upon symmetry breaking and peaks at very small normalized delay differences (Δτ/T₀ ≈ 0.07–0.08). In contrast, coordination develops more slowly and reaches its maximum at substantially larger asymmetries (Δτ/T₀ ≈ 0.30). No single delay configuration simultaneously optimizes both objectives. Mode-wise analysis reveals that this separation originates from a redistribution of information from collective to differential modes as asymmetry increases. Crucially, this phenomenon is absent in phase-only delayed Kuramoto models, indicating that delay geometry alone is insufficient. Instead, amplitude dynamics and amplitude–phase coupling are essential. Validation using delayed Stuart–Landau oscillators confirms that shear (non-isochronicity) amplifies early sensitivity by converting noise-driven amplitude fluctuations into phase information. Together, these results establish delay asymmetry as a functional resource, revealing a fundamental dynamical principle by which systems balance responsiveness and coordinated structure without fine-tuning to criticality. Biological sciences/Biophysics Physical sciences/Physics Delay asymmetry Sensitivity–coordination trade-off Nonlinear delayed systems Fisher information Amplitude–phase coupling Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction Delayed interactions are an unavoidable feature of many physical, biological, and engineered systems. Finite signal transmission times, processing delays, and distributed feedback loops arise naturally in neural circuits, sensorimotor control, genetic regulation, and networked control architectures (Izhikevich, 2007 ; Franklin et al., 2015 ; Deco et al., 2011 ). Even when the underlying dynamics are simple, the presence of delay can qualitatively alter system behavior, giving rise to oscillations, instabilities, and complex collective dynamics (Mackey & Glass, 1977 ; Farmer, 1982 ). Most theoretical studies of delayed systems have focused on stability, synchronization, and bifurcation structure (Hale & Verduyn Lunel, 1993; Erneux, 2009 ). Within this framework, delay is typically treated either as a destabilizing factor or as a tunable control parameter whose optimal value maximizes coherence or performance. Such approaches implicitly assume that the effects of delay can be summarized by a single notion of optimality. However, adaptive systems rarely optimize a single function. Instead, they must balance competing demands, such as maintaining coordinated collective behavior while remaining sensitive to small asymmetries or environmental changes (Ashby, 1956; Kelso, 1995 ). Sensitivity and coordination are conceptually distinct properties. Sensitivity concerns how strongly a system responds to small perturbations or parameter changes, whereas coordination reflects the degree to which components act coherently as a collective. In delay-coupled systems, there is no a priori reason for these two properties to peak at the same delay scale. Small asymmetries in delay may dramatically affect internal dynamics without immediately disrupting coordination, while larger asymmetries may stabilize collective structure at the expense of responsiveness. Despite its importance, this potential separation has rarely been quantified explicitly in delayed nonlinear systems. A further limitation of much existing work lies in the widespread use of phase-reduced models. Phase-only descriptions, such as the Kuramoto model, have provided powerful insights into synchronization phenomena (Kuramoto, 1984 ; Acebrón et al., 2005 ), but they neglect amplitude dynamics and therefore discard an important channel through which information can be encoded. In systems with delays and noise, amplitude fluctuations may carry critical information about asymmetry and perturbations (Pikovsky et al., 2001 ). Whether such information can be converted into phase-level responses—and how this conversion shapes sensitivity and coordination—cannot be addressed within phase-only frameworks. In this study, we investigate how delay asymmetry structures sensitivity and coordination in nonlinear delayed systems with amplitude dynamics. Using a minimal two-unit delayed nonlinear model, we systematically vary delay asymmetry and quantify both Fisher information, as a measure of sensitivity to asymmetry (Cover & Thomas, 2006 ), and coordination metrics derived from mutual information and phase dispersion. This approach allows us to ask whether sensitivity and coordination are intrinsically linked or whether they emerge at distinct delay scales. To assess the generality of the observed behavior, we perform two complementary validation analyses. First, we compare the results with a delayed Kuramoto baseline to isolate the role of amplitude dynamics. Second, we extend the analysis to delayed Stuart–Landau oscillators, which provide a canonical normal form for nonlinear oscillations (Stuart, 1960; Landau & Lifshitz, 1987 ) and allow controlled manipulation of amplitude–phase coupling through shear (Nakao, 2016 ). This extension enables us to test whether the separation between sensitivity and coordination persists in a broader class of oscillatory systems and to clarify the underlying physical mechanism. Our results demonstrate that sensitivity and coordination are not governed by a single optimal delay. Sensitivity rises sharply as soon as perfect delay symmetry is broken, reaching its maximum at very small asymmetries, whereas coordination peaks at substantially larger delays. This separation is robust across nonlinearity strengths and noise levels and cannot be reproduced by phase-only models. Instead, it relies on amplitude dynamics and amplitude–phase coupling, which act to transform asymmetry-induced amplitude fluctuations into informationally relevant phase responses. These findings reveal a previously unrecognized functional role of delay asymmetry. Rather than serving merely as a source of instability or degradation, delay asymmetry creates a structured landscape in which different functional objectives—sensitivity and coordination—are optimized at different scales. This separation provides a principled explanation for why adaptive systems may operate away from both perfect symmetry and maximal synchronization, and it suggests a general mechanism by which delayed nonlinear systems balance responsiveness and collective stability. 2. Methods 2.1 Model definition We investigated a minimal nonlinear delayed dynamical system consisting of two mutually coupled units. The state variables x1(t) and x2(t) represent generic nonlinear oscillatory amplitudes, without assuming a specific physical substrate such as membrane potentials or firing rates. The dynamics were defined as \(\:x1\_dot\left(t\right)\:=\:-x1\left(t\right)\:+\:tanh(kappa\:·\:x2(t\:-\:tau2\left)\right)\:+\:eta1\left(t\right)\) \(\:x2\_dot\left(t\right)\:=\:-x2\left(t\right)\:+\:tanh(kappa\:·\:x1(t\:-\:tau1\left)\right)\:+\:eta2\left(t\right)\) where kappa controls the strength of the nonlinear coupling, and tau1, tau2 ≥ 0 denote asymmetric transmission delays. The hyperbolic tangent introduces saturating nonlinearity while preserving odd symmetry. This formulation defines a generic delayed nonlinear oscillator and allows direct comparison with canonical amplitude–phase models, including the Stuart–Landau oscillator. 2.2 Numerical integration The system was integrated using the Euler–Maruyama scheme with a fixed time step dt = 0.001 s. All simulations included an initial burn-in period of 5 s to eliminate transient dynamics. Delayed terms were implemented using explicit history buffers. Sufficient pre-history was allocated in all simulations to guarantee that delayed indices were valid at every time step, and no interpolation was used. 2.3 Noise model The stochastic forcing terms eta1(t) and eta2(t) were modeled as independent Ornstein–Uhlenbeck processes with finite correlation time. The discrete-time update was given by $$\:eta(t\:+\:dt)\:=\:a\:·\:eta\left(t\right)\:+\:sigma\:·\:sqrt(1\:-\:a^2)\:·\:xi\left(t\right)$$ where a = exp(− dt / tau_noise), tau_noise = 0.05 s, and xi(t) is standard Gaussian noise. This choice yields temporally correlated perturbations and avoids unphysical white-noise forcing. 2.4 Delay asymmetry protocol The primary control parameter was the absolute delay asymmetry $$\:Delta\_tau\:=\:|tau1\:-\:tau2|.$$ The mean delay (tau1 + tau2) / 2 was fixed at 0.20 s throughout all analyses. Delta_tau was varied directly in physical time units, without assuming the existence of a well-defined intrinsic oscillation period. This protocol avoids bias introduced by external normalization and allows delay-induced effects to be examined independently of imposed temporal scales. 2.5 Mode decomposition To separate collective and differential dynamics, the following mode decomposition was applied: \(\:s\left(t\right)\:=\:\left(x1\right(t)\:+\:x2(t\left)\right)\:/\:2\) \(\:d\left(t\right)\:=\:\left(x1\right(t)\:-\:x2(t\left)\right)\:/\:2\) The common mode s(t) captures synchronized activity, whereas the differential mode d(t) quantifies symmetry breaking between the two units. 2.6 Phase extraction and phase difference Instantaneous phases were extracted from the analytic signals of the demeaned state variables using the Hilbert transform. For each unit i in {1,2}, the instantaneous phase phi_i(t) was defined as the argument of the analytic signal of x_i(t) − mean(x_i). The phase difference was defined as the circular difference $$\:phi\left(t\right)\:=\:wrap(phi\_1(t)\:-\:phi\_2(t\left)\right)$$ where wrap(·) maps angles to the interval [− pi, pi]. 2.7 Coordination and information measures Statistical dependence between x1(t) and x2(t) was quantified using normalized mutual information, estimated via a histogram-based estimator and normalized by the joint entropy. Phase dispersion was quantified using circular variance of the phase difference phi(t). A coordination index was defined as $$\:C\:=\:I\_INT\:·\:Q$$ where I_INT denotes normalized mutual information and Q denotes circular variance. 2.8 Estimation of characteristic timescale A characteristic timescale T0 was estimated from the power spectral density of the common mode s(t) using Welch’s method. Spectral peaks were searched within a restricted frequency band (0.8–15 Hz) to exclude low-frequency drift. If no dominant peak was detected, T0 was treated as undefined for that condition. Normalized delay ratios Delta_tau / T0 were computed only after T0 had been empirically estimated for each condition. 2.9 Fisher information analysis Sensitivity to delay asymmetry was quantified using Fisher information with respect to Delta_tau. A Gaussian approximation of the joint state distribution was adopted. This approximation was chosen for numerical stability and to enable a clear separation between contributions arising from changes in the mean and those arising from changes in the covariance structure, which is essential for the mode-specific analysis presented in Section 3 . Under this approximation, Fisher information was computed from the mean vector and covariance matrix. Both full multivariate Fisher information and mode-specific Fisher information (common and differential modes) were evaluated. Derivatives with respect to Delta_tau were approximated using finite differences across adjacent delay conditions. 2.10 Parameter robustness analysis To assess robustness, the full delay-asymmetry sweep was repeated for multiple values of the nonlinearity parameter kappa (1.0, 2.0, and 4.0). For each kappa, the locations of the sensitivity peak (differential-mode Fisher information) and the coordination peak were identified and compared. 2.11 Baseline comparison: delayed phase model As a phase-only reference, a delayed Kuramoto model with imposed frequencies was analyzed using the same delay-asymmetry protocol. Phase sensitivity and dispersion were computed analogously to the main model to isolate the role of amplitude dynamics. 2.12 Stuart–Landau validation To test generality beyond the specific nonlinear system, delayed Stuart–Landau oscillators were analyzed. The Stuart–Landau model was implemented in its standard normal form, $$\:z\_dot\:=\:(lambda\:+\:i·omega)\:z\:-\:(1\:+\:i·c)\:\left|z\right|^2\:z\:+\:coupling\:+\:noise$$ where lambda controls limit-cycle stability, omega sets the intrinsic frequency, and c determines shear (non-isochronicity). Coupling between oscillators was chosen to be linear and symmetric, ensuring correspondence with the coupling structure of the x1–x2 model. Parameter values were selected such that the effective nonlinearity and coupling strength were comparable to those in the main delayed system. The same delay-asymmetry protocol and analysis pipeline were applied to ensure consistency across models. 3. Results 3.1 Main dynamics under symmetric delay We first analyzed the baseline dynamics of the two-unit nonlinear delayed system under symmetric coupling (τ₁ = τ₂ = 0.20 s). Both units exhibited sustained stochastic oscillations driven by delayed nonlinear feedback and colored noise (Fig. 1 a). The two state variables largely overlapped but showed small, noise-driven deviations. Decomposition into collective and differential coordinates revealed that the common mode s(t) = (x₁ + x₂)/2 dominated the dynamics, while the differential mode d(t) = (x₁ − x₂)/2 remained small and fluctuated around zero (Fig. 1 b). Phase differences fluctuated around zero with finite variance, indicating a near-synchronous but noise-perturbed state. Spectral analysis of the common mode showed a broad low-frequency structure rather than a sharply defined peak (Fig. 1 c). This confirms that, under noise-driven conditions, the system does not possess a strictly defined intrinsic period. Accordingly, we use delay asymmetry Δτ as the primary control parameter in subsequent analyses, rather than relying on an externally imposed reference frequency. 3.2 Sensitivity to delay asymmetry: emergence of a high-sensitivity regime Introducing a small delay asymmetry Δτ = |τ₁ − τ₂| immediately altered the system’s information-processing properties. At Δτ = 0, the differential mode carried negligible information, and the Fisher information associated with the antisymmetric mode F_d was minimal. However, even a very small asymmetry produced a rapid increase in F_d (Fig. 2 a). Across simulations, the Fisher information of the differential mode peaked sharply at a normalized delay ratio Δτ/T₀ ≈ 0.07–0.08. This peak occurred well before any substantial loss of synchrony or coordination was observed. Importantly, the increase in sensitivity was not gradual but emerged abruptly at the onset of symmetry breaking. Thus, the system is maximally sensitive not at large delays, but precisely at the transition from perfect symmetry to weak asymmetry. 3.3 Separation between sensitivity and coordination In contrast to the early sensitivity peak, the coordination index C = I_INT × Q exhibited a qualitatively different dependence on delay asymmetry. As shown in Fig. 2 a, C increased more slowly with Δτ and reached its maximum at a substantially larger delay ratio, around Δτ/T₀ ≈ 0.30. At this point, mutual information between units remained high while phase dispersion increased, indicating a balance between integration and differentiation. Notably, this coordination peak occurred well after the sensitivity peak identified in Section 3.2 . These results demonstrate a clear functional separation: the delay regime that maximizes sensitivity to perturbations is distinct from the regime that maximizes coordinated structure. No single value of Δτ simultaneously optimized both objectives. 3.4 Mode-wise redistribution of information To clarify the origin of this separation, we decomposed the Fisher information into common-mode (F_s) and differential-mode (F_d) contributions (Fig. 2 b). At very small Δτ, information was dominated by the common mode F_s, reflecting encoding through collective fluctuations. As delay asymmetry increased, F_s decreased monotonically, while F_d increased sharply and peaked near Δτ/T₀ ≈ 0.075. Beyond this point, F_d also declined, whereas coordination continued to improve due to increasing phase dispersion and sustained coupling. This demonstrates a transfer of informational dominance from collective to differential modes as symmetry is broken. Crucially, no special feature was observed near Δτ/T₀ ≈ 0.25; instead, the dominant transition occurred near the onset of symmetry breaking. 3.5 Comparison with delayed Kuramoto phase oscillators To assess whether the observed effects arise from generic phase geometry, we compared our results with a delayed Kuramoto model with imposed intrinsic frequencies (4 Hz and 8 Hz). In the Kuramoto system, phase sensitivity varied smoothly with Δτ/T₀ and followed a gradual, nearly sinusoidal dependence (Fig. 3 a). No sharp sensitivity peak at small Δτ was observed, and the separation between sensitivity and coordination regimes was absent. This comparison demonstrates that the early sensitivity peak observed in the nonlinear delayed system cannot be explained by phase delay alone and requires amplitude–phase coupling. 3.6 Robustness across nonlinearity strength (κ) We next examined whether the separation between sensitivity and coordination depends on the strength of nonlinearity κ. For κ = 1.0, 2.0, and 4.0, the qualitative ordering of peaks remained unchanged (Fig. 3 b). In all cases, the Fisher information associated with the differential mode peaked at smaller Δτ/T₀ than the coordination index C. While the exact peak locations shifted moderately with κ, the sequence “sensitivity first, coordination later” was preserved. This confirms that the separation is not a fine-tuned artifact of a specific parameter choice, but a robust feature of the system. 3.7 Stuart–Landau validation: role of amplitude dynamics and shear To test generality beyond the specific nonlinear model, we analyzed a delayed Stuart–Landau oscillator pair. When shear (non-isochronicity) was absent, sensitivity and coordination showed weaker separation (Fig. 4 a). Introducing shear substantially enhanced the early sensitivity peak while leaving the coordination peak at larger Δτ. Under noise, amplitude fluctuations became strongly correlated with increases in phase sensitivity near the symmetry-breaking regime. Analysis of amplitude variance and the covariance contribution to Fisher information revealed that, although amplitude variance remained small, the covariance term dominated the information content (Fig. 4 b). This indicates that shear enables amplitude variability to act as an information carrier, effectively converting noise-induced amplitude fluctuations into phase sensitivity. The same qualitative separation between sensitivity and coordination was observed across shear signs and noise levels, confirming that the phenomenon is not model-specific. 3.8 Summary of empirical findings Across all analyses, three empirical facts consistently emerged: Sensitivity to delay asymmetry peaks at very small Δτ, immediately after perfect symmetry is broken (Δτ/T₀ ≈ 0.07–0.08). Coordination is maximized at substantially larger delay asymmetry (Δτ/T₀ ≈ 0.30). This separation persists across nonlinearity strength, noise, and model class, but disappears in phase-only systems. Together, these results establish that delayed nonlinear systems do not possess a single optimal delay. Instead, sensitivity and coordination are maximized in distinct delay regimes, reflecting fundamentally different functional roles of delay asymmetry. 4. Discussion 4.1 Summary of main findings This study demonstrates that delayed nonlinear systems exhibit a robust and nontrivial separation between sensitivity to delay asymmetry and coordination between units. Across a broad range of delay differences, Fisher information associated with the differential mode peaked at substantially smaller delay asymmetries than the coordination index, indicating that the system becomes maximally sensitive to asymmetry before strong coordination is established. This separation was consistently observed in the main nonlinear delayed model, persisted across variations in the nonlinearity parameter, and was absent in a phase-only delayed Kuramoto baseline (Kuramoto, 1984 ; Acebrón et al., 2005 ). Crucially, the same qualitative separation emerged in delayed Stuart–Landau oscillators, confirming that the phenomenon is not model-specific but reflects a broader class of nonlinear oscillatory dynamics (Stuart, 1960; Nakao, 2016 ). 4.2 Why sensitivity and coordination must separate The observed separation between sensitivity and coordination is not an accidental numerical outcome but a structural consequence of delayed nonlinear coupling. At exact symmetry (Δτ = 0), the system possesses an exchange symmetry between units, suppressing differential responses. However, introducing even an infinitesimal delay asymmetry breaks this symmetry and immediately activates the differential mode, a general consequence of symmetry breaking in coupled dynamical systems (Golubitsky & Stewart, 2002 ). As a result, sensitivity—as quantified by Fisher information—rises sharply as soon as symmetry is perturbed (Cover & Thomas, 2006 ). In contrast, coordination requires the buildup of statistical dependence and phase structure across time. This process depends on sustained nonlinear interaction and therefore develops over larger delay asymmetries. Thus, sensitivity and coordination are governed by distinct dynamical mechanisms and cannot peak simultaneously. From an adaptive or engineering perspective, this separation enables systems to detect small asymmetries without sacrificing global coherence. Excessive sensitivity would destabilize collective behavior, whereas excessive coordination would suppress responsiveness to environmental changes, echoing classical trade-offs in adaptive control and biological regulation (Ashby, 1956; Kelso, 1995 ). 4.3 Relation to edge-of-chaos arguments The results resonate with, but are not reducible to, classical edge-of-chaos arguments (Langton, 1990; Bertschinger & Natschläger, 2004 ). Rather than tuning a single parameter to maximize computational capacity, the system exhibits a geometric separation in delay space: small asymmetries maximize information about change, while larger asymmetries stabilize coordinated dynamics. Importantly, this separation does not rely on proximity to a bifurcation or critical point. It arises from the interplay between delay-induced phase geometry and nonlinear amplitude dynamics, even in regimes where the system remains globally stable (Erneux, 2009 ). 4.4 Absence of a fixed intrinsic period A key feature of the system is the absence of a sharply defined intrinsic oscillation period across all conditions. While an emergent timescale could be estimated from spectral peaks in some regimes, this timescale was neither universal nor stable under parameter variation, a common property of noisy nonlinear systems with delay (Pikovsky et al., 2001 ). Rather than treating this as a limitation, we emphasize that delay asymmetry was controlled in absolute time units. Normalized ratios were computed only after empirical estimation of characteristic timescales, ensuring that conclusions do not hinge on the assumption of a well-defined period. This distinguishes the present results from analyses that rely on externally imposed frequencies or strictly periodic dynamics. 4.5 Comparison with phase-only delayed models The delayed Kuramoto baseline exhibited smooth, monotonic changes in phase dispersion and phase sensitivity, but no sharp separation between sensitivity and coordination. This contrast highlights the critical role of amplitude dynamics. Phase-only models lack the degrees of freedom necessary to encode asymmetry-induced information in differential modes (Kuramoto, 1984 ; Acebrón et al., 2005 ). As a result, sensitivity and coordination collapse onto a single timescale. The failure of the Kuramoto model to reproduce the observed separation demonstrates that the phenomenon cannot be attributed to delay alone, but requires nonlinear amplitude–phase interactions. 4.6 Role of amplitude dynamics and shear The Stuart–Landau validation clarifies the mechanistic origin of the separation. In these oscillators, the shear parameter couples amplitude fluctuations to phase dynamics, a well-established mechanism in nonlinear oscillator theory (Nakao, 2016 ). When shear was absent, sensitivity peaks were weak or absent. Introducing shear amplified the separation by enabling amplitude fluctuations—enhanced by delay asymmetry and noise—to be converted into phase variability. In this sense, shear acts as a catalyst, transforming amplitude information into phase information without being consumed. This mechanism explains why sensitivity peaks occur at small asymmetries: even minimal delay differences generate amplitude fluctuations that shear immediately converts into phase-sensitive signals. 4.7 Noise-induced amplification of sensitivity Noise played a constructive role in the observed dynamics. Near the sensitivity peak, amplitude fluctuations carried significant information about delay asymmetry, which was then transferred to phase dynamics through nonlinear coupling, consistent with noise-enhanced information processing in nonlinear systems (Gammaitoni et al., 1998 ). This noise-induced amplification did not destabilize coordination, because the coordination peak remained separated at larger asymmetries. Thus, noise enhances detectability without destroying collective structure. 4.8 Robustness across nonlinearity strength Varying the nonlinearity parameter κ shifted the absolute locations of sensitivity and coordination peaks but preserved their ordering. Sensitivity consistently peaked at smaller normalized delay asymmetries than coordination. This robustness indicates that the separation is not fine-tuned to a specific parameter regime, but reflects a general property of delayed nonlinear systems. 4.9 Implications The results suggest a general design principle for adaptive systems: delay-induced symmetry breaking enables early sensitivity, while nonlinear interaction consolidates coordination at larger scales. This principle applies across biological, physical, and engineered systems in which delays, noise, and nonlinear coupling coexist (Ashby, 1956; Franklin et al., 2015 ). Rather than being a nuisance, delay asymmetry emerges as a functional resource for information processing. 4.10 Limitations and future directions The present analysis focused on two-unit systems to isolate fundamental mechanisms. Extensions to larger networks may reveal how local sensitivity–coordination separation scales with system size and topology (Strogatz, 2001). Further exploration of parameter regimes near bifurcation boundaries may clarify how the separation interacts with criticality, but the phenomenon itself does not depend on critical tuning. 4.11 Conclusion Delayed nonlinear systems generically separate sensitivity to asymmetry from coordination. This separation is structurally enforced by symmetry breaking, amplified by amplitude–phase coupling, and robust to noise and parameter variation. The convergence of results across distinct models establishes this separation as a fundamental dynamical property rather than a model-specific artifact. Declarations Conflict of Interest The author declares no conflicts of interest related to this work. Ethics Statement Not applicable .This study involved no human or animal subjects. Funding This study received no external funding. Author Contribution The author designed the study, performed all analyses, drafted the manuscript, and approved the final version. A co-author developed the simulation code and contributed to the implementation of the computational framework. Data Availability All simulation code and analysis scripts supporting the findings of this study are publicly available on Zenodo: https://doi.org/10.5281/zenodo.18501631 References Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77 (1), 137–185. https://doi.org/10.1103/RevModPhys.77.137 (2005). Ashby, W. R. An Introduction to Cybernetics . Chapman & Hall. (1956). https://doi.org/10.5962/bhl.title.5851 Bertschinger, N. & Natschläger, T. Real-time computation at the edge of chaos in recurrent neural networks. Neural Comput. 16 (7), 1413–1436. https://doi.org/10.1162/089976604323057443 (2004). Cover, T. M. & Thomas, J. A. Elements of Information Theory (2nd ed.). Wiley. (2006). https://doi.org/10.1002/047174882X Deco, G., Jirsa, V. K. & McIntosh, A. R. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8896814","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":599246213,"identity":"d1f7bc2b-5d69-4819-9f5d-34616d5e832e","order_by":0,"name":"Nobuchika Yamaki","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYFACHgYGxgYJAygvgYcfxiJei2QbkDpAWAsDXAuDwTECWszZe49J/NxhYczfwGP46UZNmozx/d6Dtz9U2OUxsDcffIBFi2XPuTTJ3jMSZhIHeIylc47l8Jgd40u2OHAmuZiB51iyARYtBjdyzKQZ2yRsGA7wGEjnsFUAtfCYSRxsO5DYIJFjJoFPizzQlt85/yp4jNtAWv4R1mJmcIDHTDq3LYfHgA2kpQGPljPnki172ySMDQ+zlVnn9qXxSBzLMbY4cyw5sQ2XX473Hrzxs63OcN7x5s23c74l2/M3nzG8UVFjl9iPI8QQgJkDYSTYPWx4lYMBO8JIbF4YBaNgFIyCkQsAmIthmvIjDLMAAAAASUVORK5CYII=","orcid":"","institution":"TNQ Tech","correspondingAuthor":true,"prefix":"","firstName":"Nobuchika","middleName":"","lastName":"Yamaki","suffix":""},{"id":599246217,"identity":"3fe35fbf-7989-4248-81cb-381dd32b89b1","order_by":1,"name":"Tenna Churiki","email":"","orcid":"","institution":"TNQ Tech","correspondingAuthor":false,"prefix":"","firstName":"Tenna","middleName":"","lastName":"Churiki","suffix":""}],"badges":[],"createdAt":"2026-02-17 00:53:09","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8896814/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8896814/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103939383,"identity":"f5b00ccd-9605-48a8-985a-0b5af29a8e57","added_by":"auto","created_at":"2026-03-04 18:48:50","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":286010,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eBasic dynamics and mode structure under symmetric delay.\u003c/strong\u003e\u003cbr\u003e\n(a) Time series of the two state variables x1(t) and x2(t) under symmetric delayed coupling. The trajectories largely overlap but exhibit small, noise-driven deviations.\u003cbr\u003e\n(b) Decomposition into the common mode s(t) = (x1 + x2)/2 and the differential mode d(t) = (x1 − x2)/2. The dynamics are dominated by the common mode, while the differential mode fluctuates around zero.\u003cbr\u003e\n(c) Power spectrum of s(t), showing a broad low-frequency structure rather than a sharp spectral peak. This indicates a noise-driven, weakly synchronized state without a well-defined intrinsic period T0.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8896814/v1/33386a02a241d2c35eebdeb5.png"},{"id":104779275,"identity":"7654fb37-4470-4933-8abf-812501d7c499","added_by":"auto","created_at":"2026-03-17 07:38:02","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":265790,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSeparation of sensitivity and coordination across delay scales.\u003c/strong\u003e\u003cbr\u003e\n(a) Differential-mode Fisher information Fd (left axis) and coordination index C (right axis) as functions of normalized delay Δτ/T0. Sensitivity exhibits a sharp peak at small delays, whereas coordination increases more gradually and peaks at larger delays. Error bars represent standard deviations across realizations.\u003cbr\u003e\n(b) Redistribution of Fisher information across modes. At small delays, information is predominantly carried by the common mode Fs, while increasing delay shifts information toward the differential mode Fd. Together, these results demonstrate that sensitivity and coordination are optimized at distinct delay scales.\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-8896814/v1/f96dab0304294c5093a5b8ad.png"},{"id":103939386,"identity":"7eb05fe1-d4da-4df4-b2a2-200fd0e4237a","added_by":"auto","created_at":"2026-03-04 18:48:50","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":209596,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eComparison with phase-only models and robustness to coupling strength.\u003c/strong\u003e\u003cbr\u003e\n(a) Phase sensitivity d⟨Δφ⟩/d(Δτ/T0) in the Kuramoto model (lines) compared with the main amplitude–phase model (symbols). Phase-only dynamics yield smooth and weak sensitivity, whereas the proposed model exhibits a localized and sharp response.\u003cbr\u003e\n(b) Peak locations of sensitivity (Fd) and coordination (C) as functions of coupling strength κ. The separation between the two peaks persists across κ, demonstrating robustness and indicating that the observed effect cannot be explained by phase geometry alone.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-8896814/v1/98d13af122e1c227ee1ae024.png"},{"id":103939385,"identity":"cbefe5d6-90ad-4627-94ec-85d15bcad08c","added_by":"auto","created_at":"2026-03-04 18:48:50","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":297908,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eShear-induced amplitude–phase coupling as the mechanism of information transfer.\u003c/strong\u003e\u003cbr\u003e\n(a) Total Fisher information Ftotal as a function of normalized delay Δτ/T0 for different shear values in the Stuart–Landau model. Increasing shear amplifies sensitivity while preserving the qualitative dependence on delay.\u003cbr\u003e\n(b) Noise-driven amplitude variance and the covariance contribution ratio Fcov/Ftotal (shear = 1). Although amplitude variance remains small, the covariance term dominates the Fisher information, indicating that noise-induced amplitude fluctuations are converted into phase sensitivity through shear.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-8896814/v1/3f1efe937cda16e964d9ec17.png"},{"id":104783671,"identity":"a7859341-462c-45ac-9121-a6188ce7b337","added_by":"auto","created_at":"2026-03-17 08:03:14","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1906399,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8896814/v1/f9bf47fe-ec93-4a55-b6ec-d619b1f2fe8b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Delay asymmetry induces a functional separation between sensitivity and coordination","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eDelayed interactions are an unavoidable feature of many physical, biological, and engineered systems.\u003c/p\u003e \u003cp\u003eFinite signal transmission times, processing delays, and distributed feedback loops arise naturally in neural circuits, sensorimotor control, genetic regulation, and networked control architectures (Izhikevich, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Franklin et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Deco et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Even when the underlying dynamics are simple, the presence of delay can qualitatively alter system behavior, giving rise to oscillations, instabilities, and complex collective dynamics (Mackey \u0026amp; Glass, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Farmer, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1982\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eMost theoretical studies of delayed systems have focused on stability, synchronization, and bifurcation structure (Hale \u0026amp; Verduyn Lunel, 1993; Erneux, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Within this framework, delay is typically treated either as a destabilizing factor or as a tunable control parameter whose optimal value maximizes coherence or performance. Such approaches implicitly assume that the effects of delay can be summarized by a single notion of optimality. However, adaptive systems rarely optimize a single function. Instead, they must balance competing demands, such as maintaining coordinated collective behavior while remaining sensitive to small asymmetries or environmental changes (Ashby, 1956; Kelso, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1995\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eSensitivity and coordination are conceptually distinct properties. Sensitivity concerns how strongly a system responds to small perturbations or parameter changes, whereas coordination reflects the degree to which components act coherently as a collective. In delay-coupled systems, there is no a priori reason for these two properties to peak at the same delay scale. Small asymmetries in delay may dramatically affect internal dynamics without immediately disrupting coordination, while larger asymmetries may stabilize collective structure at the expense of responsiveness. Despite its importance, this potential separation has rarely been quantified explicitly in delayed nonlinear systems.\u003c/p\u003e \u003cp\u003eA further limitation of much existing work lies in the widespread use of phase-reduced models. Phase-only descriptions, such as the Kuramoto model, have provided powerful insights into synchronization phenomena (Kuramoto, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1984\u003c/span\u003e; Acebr\u0026oacute;n et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2005\u003c/span\u003e), but they neglect amplitude dynamics and therefore discard an important channel through which information can be encoded. In systems with delays and noise, amplitude fluctuations may carry critical information about asymmetry and perturbations (Pikovsky et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Whether such information can be converted into phase-level responses\u0026mdash;and how this conversion shapes sensitivity and coordination\u0026mdash;cannot be addressed within phase-only frameworks.\u003c/p\u003e \u003cp\u003eIn this study, we investigate how delay asymmetry structures sensitivity and coordination in nonlinear delayed systems with amplitude dynamics. Using a minimal two-unit delayed nonlinear model, we systematically vary delay asymmetry and quantify both Fisher information, as a measure of sensitivity to asymmetry (Cover \u0026amp; Thomas, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2006\u003c/span\u003e), and coordination metrics derived from mutual information and phase dispersion. This approach allows us to ask whether sensitivity and coordination are intrinsically linked or whether they emerge at distinct delay scales.\u003c/p\u003e \u003cp\u003eTo assess the generality of the observed behavior, we perform two complementary validation analyses. First, we compare the results with a delayed Kuramoto baseline to isolate the role of amplitude dynamics. Second, we extend the analysis to delayed Stuart\u0026ndash;Landau oscillators, which provide a canonical normal form for nonlinear oscillations (Stuart, 1960; Landau \u0026amp; Lifshitz, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1987\u003c/span\u003e) and allow controlled manipulation of amplitude\u0026ndash;phase coupling through shear (Nakao, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). This extension enables us to test whether the separation between sensitivity and coordination persists in a broader class of oscillatory systems and to clarify the underlying physical mechanism.\u003c/p\u003e \u003cp\u003eOur results demonstrate that sensitivity and coordination are not governed by a single optimal delay. Sensitivity rises sharply as soon as perfect delay symmetry is broken, reaching its maximum at very small asymmetries, whereas coordination peaks at substantially larger delays. This separation is robust across nonlinearity strengths and noise levels and cannot be reproduced by phase-only models. Instead, it relies on amplitude dynamics and amplitude\u0026ndash;phase coupling, which act to transform asymmetry-induced amplitude fluctuations into informationally relevant phase responses.\u003c/p\u003e \u003cp\u003eThese findings reveal a previously unrecognized functional role of delay asymmetry. Rather than serving merely as a source of instability or degradation, delay asymmetry creates a structured landscape in which different functional objectives\u0026mdash;sensitivity and coordination\u0026mdash;are optimized at different scales. This separation provides a principled explanation for why adaptive systems may operate away from both perfect symmetry and maximal synchronization, and it suggests a general mechanism by which delayed nonlinear systems balance responsiveness and collective stability.\u003c/p\u003e"},{"header":"2. Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Model definition\u003c/h2\u003e \u003cp\u003eWe investigated a minimal nonlinear delayed dynamical system consisting of two mutually coupled units.\u003c/p\u003e \u003cp\u003eThe state variables x1(t) and x2(t) represent generic nonlinear oscillatory amplitudes, without assuming a specific physical substrate such as membrane potentials or firing rates.\u003c/p\u003e \u003cp\u003eThe dynamics were defined as\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:x1\\_dot\\left(t\\right)\\:=\\:-x1\\left(t\\right)\\:+\\:tanh(kappa\\:\u0026middot;\\:x2(t\\:-\\:tau2\\left)\\right)\\:+\\:eta1\\left(t\\right)\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:x2\\_dot\\left(t\\right)\\:=\\:-x2\\left(t\\right)\\:+\\:tanh(kappa\\:\u0026middot;\\:x1(t\\:-\\:tau1\\left)\\right)\\:+\\:eta2\\left(t\\right)\\)\u003c/span\u003e \u003c/span\u003e \u003c/p\u003e \u003cp\u003ewhere kappa controls the strength of the nonlinear coupling, and tau1, tau2\u0026thinsp;\u0026ge;\u0026thinsp;0 denote asymmetric transmission delays.\u003c/p\u003e \u003cp\u003eThe hyperbolic tangent introduces saturating nonlinearity while preserving odd symmetry.\u003c/p\u003e \u003cp\u003eThis formulation defines a generic delayed nonlinear oscillator and allows direct comparison with canonical amplitude\u0026ndash;phase models, including the Stuart\u0026ndash;Landau oscillator.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Numerical integration\u003c/h2\u003e \u003cp\u003eThe system was integrated using the Euler\u0026ndash;Maruyama scheme with a fixed time step dt\u0026thinsp;=\u0026thinsp;0.001 s.\u003c/p\u003e \u003cp\u003eAll simulations included an initial burn-in period of 5 s to eliminate transient dynamics.\u003c/p\u003e \u003cp\u003eDelayed terms were implemented using explicit history buffers.\u003c/p\u003e \u003cp\u003eSufficient pre-history was allocated in all simulations to guarantee that delayed indices were valid at every time step, and no interpolation was used.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Noise model\u003c/h2\u003e \u003cp\u003eThe stochastic forcing terms eta1(t) and eta2(t) were modeled as independent Ornstein\u0026ndash;Uhlenbeck processes with finite correlation time.\u003c/p\u003e \u003cp\u003eThe discrete-time update was given by\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:eta(t\\:+\\:dt)\\:=\\:a\\:\u0026middot;\\:eta\\left(t\\right)\\:+\\:sigma\\:\u0026middot;\\:sqrt(1\\:-\\:a^2)\\:\u0026middot;\\:xi\\left(t\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere a\u0026thinsp;=\u0026thinsp;exp(\u0026minus;\u0026thinsp;dt / tau_noise), tau_noise\u0026thinsp;=\u0026thinsp;0.05 s, and xi(t) is standard Gaussian noise.\u003c/p\u003e \u003cp\u003eThis choice yields temporally correlated perturbations and avoids unphysical white-noise forcing.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Delay asymmetry protocol\u003c/h2\u003e \u003cp\u003eThe primary control parameter was the absolute delay asymmetry\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:Delta\\_tau\\:=\\:|tau1\\:-\\:tau2|.$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe mean delay (tau1\u0026thinsp;+\u0026thinsp;tau2) / 2 was fixed at 0.20 s throughout all analyses.\u003c/p\u003e \u003cp\u003eDelta_tau was varied directly in physical time units, without assuming the existence of a well-defined intrinsic oscillation period.\u003c/p\u003e \u003cp\u003eThis protocol avoids bias introduced by external normalization and allows delay-induced effects to be examined independently of imposed temporal scales.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Mode decomposition\u003c/h2\u003e \u003cp\u003eTo separate collective and differential dynamics, the following mode decomposition was applied:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:s\\left(t\\right)\\:=\\:\\left(x1\\right(t)\\:+\\:x2(t\\left)\\right)\\:/\\:2\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:d\\left(t\\right)\\:=\\:\\left(x1\\right(t)\\:-\\:x2(t\\left)\\right)\\:/\\:2\\)\u003c/span\u003e \u003c/span\u003e \u003c/p\u003e \u003cp\u003eThe common mode s(t) captures synchronized activity, whereas the differential mode d(t) quantifies symmetry breaking between the two units.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.6 Phase extraction and phase difference\u003c/h2\u003e \u003cp\u003eInstantaneous phases were extracted from the analytic signals of the demeaned state variables using the Hilbert transform.\u003c/p\u003e \u003cp\u003eFor each unit i in {1,2}, the instantaneous phase phi_i(t) was defined as the argument of the analytic signal of x_i(t) \u0026minus; mean(x_i).\u003c/p\u003e \u003cp\u003eThe phase difference was defined as the circular difference\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:phi\\left(t\\right)\\:=\\:wrap(phi\\_1(t)\\:-\\:phi\\_2(t\\left)\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere wrap(\u0026middot;) maps angles to the interval [\u0026minus;\u0026thinsp;pi, pi].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e2.7 Coordination and information measures\u003c/h2\u003e \u003cp\u003eStatistical dependence between x1(t) and x2(t) was quantified using normalized mutual information, estimated via a histogram-based estimator and normalized by the joint entropy.\u003c/p\u003e \u003cp\u003ePhase dispersion was quantified using circular variance of the phase difference phi(t).\u003c/p\u003e \u003cp\u003eA coordination index was defined as\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$\\:C\\:=\\:I\\_INT\\:\u0026middot;\\:Q$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere I_INT denotes normalized mutual information and Q denotes circular variance.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e2.8 Estimation of characteristic timescale\u003c/h2\u003e \u003cp\u003eA characteristic timescale T0 was estimated from the power spectral density of the common mode s(t) using Welch\u0026rsquo;s method.\u003c/p\u003e \u003cp\u003eSpectral peaks were searched within a restricted frequency band (0.8\u0026ndash;15 Hz) to exclude low-frequency drift.\u003c/p\u003e \u003cp\u003eIf no dominant peak was detected, T0 was treated as undefined for that condition.\u003c/p\u003e \u003cp\u003eNormalized delay ratios Delta_tau / T0 were computed only after T0 had been empirically estimated for each condition.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e2.9 Fisher information analysis\u003c/h2\u003e \u003cp\u003eSensitivity to delay asymmetry was quantified using Fisher information with respect to Delta_tau.\u003c/p\u003e \u003cp\u003eA \u003cb\u003eGaussian approximation\u003c/b\u003e of the joint state distribution was adopted.\u003c/p\u003e \u003cp\u003eThis approximation was chosen for numerical stability and to enable a clear separation between contributions arising from changes in the mean and those arising from changes in the covariance structure, which is essential for the mode-specific analysis presented in Section \u003cspan refid=\"Sec15\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eUnder this approximation, Fisher information was computed from the mean vector and covariance matrix.\u003c/p\u003e \u003cp\u003eBoth full multivariate Fisher information and mode-specific Fisher information (common and differential modes) were evaluated.\u003c/p\u003e \u003cp\u003eDerivatives with respect to Delta_tau were approximated using finite differences across adjacent delay conditions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e2.10 Parameter robustness analysis\u003c/h2\u003e \u003cp\u003eTo assess robustness, the full delay-asymmetry sweep was repeated for multiple values of the nonlinearity parameter kappa (1.0, 2.0, and 4.0).\u003c/p\u003e \u003cp\u003eFor each kappa, the locations of the sensitivity peak (differential-mode Fisher information) and the coordination peak were identified and compared.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e2.11 Baseline comparison: delayed phase model\u003c/h2\u003e \u003cp\u003eAs a phase-only reference, a delayed Kuramoto model with imposed frequencies was analyzed using the same delay-asymmetry protocol.\u003c/p\u003e \u003cp\u003ePhase sensitivity and dispersion were computed analogously to the main model to isolate the role of amplitude dynamics.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e2.12 Stuart\u0026ndash;Landau validation\u003c/h2\u003e \u003cp\u003eTo test generality beyond the specific nonlinear system, delayed Stuart\u0026ndash;Landau oscillators were analyzed.\u003c/p\u003e \u003cp\u003eThe Stuart\u0026ndash;Landau model was implemented in its standard normal form,\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\:z\\_dot\\:=\\:(lambda\\:+\\:i\u0026middot;omega)\\:z\\:-\\:(1\\:+\\:i\u0026middot;c)\\:\\left|z\\right|^2\\:z\\:+\\:coupling\\:+\\:noise$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere lambda controls limit-cycle stability, omega sets the intrinsic frequency, and c determines shear (non-isochronicity).\u003c/p\u003e \u003cp\u003eCoupling between oscillators was chosen to be linear and symmetric, ensuring correspondence with the coupling structure of the x1\u0026ndash;x2 model.\u003c/p\u003e \u003cp\u003eParameter values were selected such that the effective nonlinearity and coupling strength were comparable to those in the main delayed system.\u003c/p\u003e \u003cp\u003eThe same delay-asymmetry protocol and analysis pipeline were applied to ensure consistency across models.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Main dynamics under symmetric delay\u003c/h2\u003e \u003cp\u003eWe first analyzed the baseline dynamics of the two-unit nonlinear delayed system under symmetric coupling (τ₁ = τ₂ = 0.20 s).\u003c/p\u003e \u003cp\u003eBoth units exhibited sustained stochastic oscillations driven by delayed nonlinear feedback and colored noise (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe two state variables largely overlapped but showed small, noise-driven deviations.\u003c/p\u003e \u003cp\u003eDecomposition into collective and differential coordinates revealed that the common mode\u003c/p\u003e \u003cp\u003es(t) = (x₁ + x₂)/2 dominated the dynamics, while the differential mode\u003c/p\u003e \u003cp\u003ed(t) = (x₁ \u0026minus; x₂)/2 remained small and fluctuated around zero (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003ePhase differences fluctuated around zero with finite variance, indicating a near-synchronous but noise-perturbed state.\u003c/p\u003e \u003cp\u003eSpectral analysis of the common mode showed a broad low-frequency structure rather than a sharply defined peak (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ec).\u003c/p\u003e \u003cp\u003eThis confirms that, under noise-driven conditions, the system does not possess a strictly defined intrinsic period.\u003c/p\u003e \u003cp\u003eAccordingly, we use delay asymmetry Δτ as the primary control parameter in subsequent analyses, rather than relying on an externally imposed reference frequency.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Sensitivity to delay asymmetry: emergence of a high-sensitivity regime\u003c/h2\u003e \u003cp\u003eIntroducing a small delay asymmetry Δτ = |τ₁ \u0026minus; τ₂| immediately altered the system\u0026rsquo;s information-processing properties.\u003c/p\u003e \u003cp\u003eAt Δτ\u0026thinsp;=\u0026thinsp;0, the differential mode carried negligible information, and the Fisher information associated with the antisymmetric mode F_d was minimal.\u003c/p\u003e \u003cp\u003eHowever, even a very small asymmetry produced a rapid increase in F_d (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAcross simulations, the Fisher information of the differential mode peaked sharply at a normalized delay ratio Δτ/T₀ \u0026asymp; 0.07\u0026ndash;0.08.\u003c/p\u003e \u003cp\u003eThis peak occurred well before any substantial loss of synchrony or coordination was observed.\u003c/p\u003e \u003cp\u003eImportantly, the increase in sensitivity was not gradual but emerged abruptly at the onset of symmetry breaking.\u003c/p\u003e \u003cp\u003eThus, the system is maximally sensitive not at large delays, but precisely at the transition from perfect symmetry to weak asymmetry.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Separation between sensitivity and coordination\u003c/h2\u003e \u003cp\u003eIn contrast to the early sensitivity peak, the coordination index\u003c/p\u003e \u003cp\u003eC\u0026thinsp;=\u0026thinsp;I_INT \u0026times; Q exhibited a qualitatively different dependence on delay asymmetry.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea, C increased more slowly with Δτ and reached its maximum at a substantially larger delay ratio, around Δτ/T₀ \u0026asymp; 0.30.\u003c/p\u003e \u003cp\u003eAt this point, mutual information between units remained high while phase dispersion increased, indicating a balance between integration and differentiation.\u003c/p\u003e \u003cp\u003eNotably, this coordination peak occurred well after the sensitivity peak identified in Section \u003cspan refid=\"Sec17\" class=\"InternalRef\"\u003e3.2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThese results demonstrate a clear functional separation:\u003c/p\u003e \u003cp\u003ethe delay regime that maximizes sensitivity to perturbations is distinct from the regime that maximizes coordinated structure.\u003c/p\u003e \u003cp\u003eNo single value of Δτ simultaneously optimized both objectives.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Mode-wise redistribution of information\u003c/h2\u003e \u003cp\u003eTo clarify the origin of this separation, we decomposed the Fisher information into common-mode (F_s) and differential-mode (F_d) contributions (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eAt very small Δτ, information was dominated by the common mode F_s, reflecting encoding through collective fluctuations.\u003c/p\u003e \u003cp\u003eAs delay asymmetry increased, F_s decreased monotonically, while F_d increased sharply and peaked near Δτ/T₀ \u0026asymp; 0.075.\u003c/p\u003e \u003cp\u003eBeyond this point, F_d also declined, whereas coordination continued to improve due to increasing phase dispersion and sustained coupling.\u003c/p\u003e \u003cp\u003eThis demonstrates a transfer of informational dominance from collective to differential modes as symmetry is broken.\u003c/p\u003e \u003cp\u003eCrucially, no special feature was observed near Δτ/T₀ \u0026asymp; 0.25; instead, the dominant transition occurred near the onset of symmetry breaking.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Comparison with delayed Kuramoto phase oscillators\u003c/h2\u003e \u003cp\u003eTo assess whether the observed effects arise from generic phase geometry, we compared our results with a delayed Kuramoto model with imposed intrinsic frequencies (4 Hz and 8 Hz).\u003c/p\u003e \u003cp\u003eIn the Kuramoto system, phase sensitivity varied smoothly with Δτ/T₀ and followed a gradual, nearly sinusoidal dependence (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eNo sharp sensitivity peak at small Δτ was observed, and the separation between sensitivity and coordination regimes was absent.\u003c/p\u003e \u003cp\u003eThis comparison demonstrates that the early sensitivity peak observed in the nonlinear delayed system cannot be explained by phase delay alone and requires amplitude\u0026ndash;phase coupling.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e3.6 Robustness across nonlinearity strength (κ)\u003c/h2\u003e \u003cp\u003eWe next examined whether the separation between sensitivity and coordination depends on the strength of nonlinearity κ.\u003c/p\u003e \u003cp\u003eFor κ\u0026thinsp;=\u0026thinsp;1.0, 2.0, and 4.0, the qualitative ordering of peaks remained unchanged (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eIn all cases, the Fisher information associated with the differential mode peaked at smaller Δτ/T₀ than the coordination index C.\u003c/p\u003e \u003cp\u003eWhile the exact peak locations shifted moderately with κ, the sequence \u0026ldquo;sensitivity first, coordination later\u0026rdquo; was preserved.\u003c/p\u003e \u003cp\u003eThis confirms that the separation is not a fine-tuned artifact of a specific parameter choice, but a robust feature of the system.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e3.7 Stuart\u0026ndash;Landau validation: role of amplitude dynamics and shear\u003c/h2\u003e \u003cp\u003eTo test generality beyond the specific nonlinear model, we analyzed a delayed Stuart\u0026ndash;Landau oscillator pair.\u003c/p\u003e \u003cp\u003eWhen shear (non-isochronicity) was absent, sensitivity and coordination showed weaker separation (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIntroducing shear substantially enhanced the early sensitivity peak while leaving the coordination peak at larger Δτ.\u003c/p\u003e \u003cp\u003eUnder noise, amplitude fluctuations became strongly correlated with increases in phase sensitivity near the symmetry-breaking regime.\u003c/p\u003e \u003cp\u003eAnalysis of amplitude variance and the covariance contribution to Fisher information revealed that, although amplitude variance remained small, the covariance term dominated the information content (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eThis indicates that shear enables amplitude variability to act as an information carrier, effectively converting noise-induced amplitude fluctuations into phase sensitivity.\u003c/p\u003e \u003cp\u003eThe same qualitative separation between sensitivity and coordination was observed across shear signs and noise levels, confirming that the phenomenon is not model-specific.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec23\" class=\"Section2\"\u003e \u003ch2\u003e3.8 Summary of empirical findings\u003c/h2\u003e \u003cp\u003eAcross all analyses, three empirical facts consistently emerged:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eSensitivity to delay asymmetry peaks at very small Δτ, immediately after perfect symmetry is broken (Δτ/T₀ \u0026asymp; 0.07\u0026ndash;0.08).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCoordination is maximized at substantially larger delay asymmetry (Δτ/T₀ \u0026asymp; 0.30).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThis separation persists across nonlinearity strength, noise, and model class, but disappears in phase-only systems.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eTogether, these results establish that delayed nonlinear systems do not possess a single optimal delay.\u003c/p\u003e \u003cp\u003eInstead, sensitivity and coordination are maximized in distinct delay regimes, reflecting fundamentally different functional roles of delay asymmetry.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cdiv id=\"Sec25\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Summary of main findings\u003c/h2\u003e \u003cp\u003eThis study demonstrates that delayed nonlinear systems exhibit a robust and nontrivial separation between sensitivity to delay asymmetry and coordination between units.\u003c/p\u003e \u003cp\u003eAcross a broad range of delay differences, Fisher information associated with the differential mode peaked at substantially smaller delay asymmetries than the coordination index, indicating that the system becomes maximally sensitive to asymmetry before strong coordination is established.\u003c/p\u003e \u003cp\u003eThis separation was consistently observed in the main nonlinear delayed model, persisted across variations in the nonlinearity parameter, and was absent in a phase-only delayed Kuramoto baseline (Kuramoto, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1984\u003c/span\u003e; Acebr\u0026oacute;n et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2005\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eCrucially, the same qualitative separation emerged in delayed Stuart\u0026ndash;Landau oscillators, confirming that the phenomenon is not model-specific but reflects a broader class of nonlinear oscillatory dynamics (Stuart, 1960; Nakao, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Why sensitivity and coordination must separate\u003c/h2\u003e \u003cp\u003eThe observed separation between sensitivity and coordination is not an accidental numerical outcome but a structural consequence of delayed nonlinear coupling.\u003c/p\u003e \u003cp\u003eAt exact symmetry (Δτ\u0026thinsp;=\u0026thinsp;0), the system possesses an exchange symmetry between units, suppressing differential responses.\u003c/p\u003e \u003cp\u003eHowever, introducing even an infinitesimal delay asymmetry breaks this symmetry and immediately activates the differential mode, a general consequence of symmetry breaking in coupled dynamical systems (Golubitsky \u0026amp; Stewart, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2002\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAs a result, sensitivity\u0026mdash;as quantified by Fisher information\u0026mdash;rises sharply as soon as symmetry is perturbed (Cover \u0026amp; Thomas, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2006\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eIn contrast, coordination requires the buildup of statistical dependence and phase structure across time.\u003c/p\u003e \u003cp\u003eThis process depends on sustained nonlinear interaction and therefore develops over larger delay asymmetries.\u003c/p\u003e \u003cp\u003eThus, sensitivity and coordination are governed by distinct dynamical mechanisms and cannot peak simultaneously.\u003c/p\u003e \u003cp\u003eFrom an adaptive or engineering perspective, this separation enables systems to detect small asymmetries without sacrificing global coherence.\u003c/p\u003e \u003cp\u003eExcessive sensitivity would destabilize collective behavior, whereas excessive coordination would suppress responsiveness to environmental changes, echoing classical trade-offs in adaptive control and biological regulation (Ashby, 1956; Kelso, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1995\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec27\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Relation to edge-of-chaos arguments\u003c/h2\u003e \u003cp\u003eThe results resonate with, but are not reducible to, classical edge-of-chaos arguments (Langton, 1990; Bertschinger \u0026amp; Natschl\u0026auml;ger, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2004\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eRather than tuning a single parameter to maximize computational capacity, the system exhibits a geometric separation in delay space: small asymmetries maximize information about change, while larger asymmetries stabilize coordinated dynamics.\u003c/p\u003e \u003cp\u003eImportantly, this separation does not rely on proximity to a bifurcation or critical point.\u003c/p\u003e \u003cp\u003eIt arises from the interplay between delay-induced phase geometry and nonlinear amplitude dynamics, even in regimes where the system remains globally stable (Erneux, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2009\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec28\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Absence of a fixed intrinsic period\u003c/h2\u003e \u003cp\u003eA key feature of the system is the absence of a sharply defined intrinsic oscillation period across all conditions.\u003c/p\u003e \u003cp\u003eWhile an emergent timescale could be estimated from spectral peaks in some regimes, this timescale was neither universal nor stable under parameter variation, a common property of noisy nonlinear systems with delay (Pikovsky et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2001\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eRather than treating this as a limitation, we emphasize that delay asymmetry was controlled in absolute time units.\u003c/p\u003e \u003cp\u003eNormalized ratios were computed only after empirical estimation of characteristic timescales, ensuring that conclusions do not hinge on the assumption of a well-defined period.\u003c/p\u003e \u003cp\u003eThis distinguishes the present results from analyses that rely on externally imposed frequencies or strictly periodic dynamics.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec29\" class=\"Section2\"\u003e \u003ch2\u003e4.5 Comparison with phase-only delayed models\u003c/h2\u003e \u003cp\u003eThe delayed Kuramoto baseline exhibited smooth, monotonic changes in phase dispersion and phase sensitivity, but no sharp separation between sensitivity and coordination.\u003c/p\u003e \u003cp\u003eThis contrast highlights the critical role of amplitude dynamics.\u003c/p\u003e \u003cp\u003ePhase-only models lack the degrees of freedom necessary to encode asymmetry-induced information in differential modes (Kuramoto, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1984\u003c/span\u003e; Acebr\u0026oacute;n et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2005\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAs a result, sensitivity and coordination collapse onto a single timescale.\u003c/p\u003e \u003cp\u003eThe failure of the Kuramoto model to reproduce the observed separation demonstrates that the phenomenon cannot be attributed to delay alone, but requires nonlinear amplitude\u0026ndash;phase interactions.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec30\" class=\"Section2\"\u003e \u003ch2\u003e4.6 Role of amplitude dynamics and shear\u003c/h2\u003e \u003cp\u003eThe Stuart\u0026ndash;Landau validation clarifies the mechanistic origin of the separation.\u003c/p\u003e \u003cp\u003eIn these oscillators, the shear parameter couples amplitude fluctuations to phase dynamics, a well-established mechanism in nonlinear oscillator theory (Nakao, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eWhen shear was absent, sensitivity peaks were weak or absent.\u003c/p\u003e \u003cp\u003eIntroducing shear amplified the separation by enabling amplitude fluctuations\u0026mdash;enhanced by delay asymmetry and noise\u0026mdash;to be converted into phase variability.\u003c/p\u003e \u003cp\u003eIn this sense, shear acts as a catalyst, transforming amplitude information into phase information without being consumed.\u003c/p\u003e \u003cp\u003eThis mechanism explains why sensitivity peaks occur at small asymmetries: even minimal delay differences generate amplitude fluctuations that shear immediately converts into phase-sensitive signals.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec31\" class=\"Section2\"\u003e \u003ch2\u003e4.7 Noise-induced amplification of sensitivity\u003c/h2\u003e \u003cp\u003eNoise played a constructive role in the observed dynamics.\u003c/p\u003e \u003cp\u003eNear the sensitivity peak, amplitude fluctuations carried significant information about delay asymmetry, which was then transferred to phase dynamics through nonlinear coupling, consistent with noise-enhanced information processing in nonlinear systems (Gammaitoni et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1998\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThis noise-induced amplification did not destabilize coordination, because the coordination peak remained separated at larger asymmetries.\u003c/p\u003e \u003cp\u003eThus, noise enhances detectability without destroying collective structure.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec32\" class=\"Section2\"\u003e \u003ch2\u003e4.8 Robustness across nonlinearity strength\u003c/h2\u003e \u003cp\u003eVarying the nonlinearity parameter κ shifted the absolute locations of sensitivity and coordination peaks but preserved their ordering.\u003c/p\u003e \u003cp\u003eSensitivity consistently peaked at smaller normalized delay asymmetries than coordination.\u003c/p\u003e \u003cp\u003eThis robustness indicates that the separation is not fine-tuned to a specific parameter regime, but reflects a general property of delayed nonlinear systems.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec33\" class=\"Section2\"\u003e \u003ch2\u003e4.9 Implications\u003c/h2\u003e \u003cp\u003eThe results suggest a general design principle for adaptive systems:\u003c/p\u003e \u003cp\u003edelay-induced symmetry breaking enables early sensitivity, while nonlinear interaction consolidates coordination at larger scales.\u003c/p\u003e \u003cp\u003eThis principle applies across biological, physical, and engineered systems in which delays, noise, and nonlinear coupling coexist (Ashby, 1956; Franklin et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2015\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eRather than being a nuisance, delay asymmetry emerges as a functional resource for information processing.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec34\" class=\"Section2\"\u003e \u003ch2\u003e4.10 Limitations and future directions\u003c/h2\u003e \u003cp\u003eThe present analysis focused on two-unit systems to isolate fundamental mechanisms.\u003c/p\u003e \u003cp\u003eExtensions to larger networks may reveal how local sensitivity\u0026ndash;coordination separation scales with system size and topology (Strogatz, 2001).\u003c/p\u003e \u003cp\u003eFurther exploration of parameter regimes near bifurcation boundaries may clarify how the separation interacts with criticality, but the phenomenon itself does not depend on critical tuning.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec35\" class=\"Section2\"\u003e \u003ch2\u003e4.11 Conclusion\u003c/h2\u003e \u003cp\u003eDelayed nonlinear systems generically separate sensitivity to asymmetry from coordination.\u003c/p\u003e \u003cp\u003eThis separation is structurally enforced by symmetry breaking, amplified by amplitude\u0026ndash;phase coupling, and robust to noise and parameter variation.\u003c/p\u003e \u003cp\u003eThe convergence of results across distinct models establishes this separation as a fundamental dynamical property rather than a model-specific artifact.\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eConflict of Interest\u003c/h2\u003e \u003cp\u003eThe author declares no conflicts of interest related to this work.\u003c/p\u003e \u003c/p\u003e\u003cp\u003e \u003ch2\u003eEthics Statement\u003c/h2\u003e \u003cp\u003eNot applicable .This study involved no human or animal subjects.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThis study received no external funding.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThe author designed the study, performed all analyses, drafted the manuscript, and approved the final version. A co-author developed the simulation code and contributed to the implementation of the computational framework.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eAll simulation code and analysis scripts supporting the findings of this study are publicly available on Zenodo: https://doi.org/10.5281/zenodo.18501631\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAcebr\u0026oacute;n, J. A., Bonilla, L. L., P\u0026eacute;rez Vicente, C. J., Ritort, F. \u0026amp; Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. \u003cem\u003eRev. Mod. 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Here we demonstrate that delayed nonlinear systems generically exhibit a functional separation between sensitivity to delay asymmetry and coordination between interacting units. Using a minimal stochastic delayed system, we systematically varied delay asymmetry and quantified sensitivity via Fisher information and coordination using an information\u0026ndash;phase composite index.\u003c/p\u003e \u003cp\u003eWe find that sensitivity to asymmetry emerges immediately upon symmetry breaking and peaks at very small normalized delay differences (Δτ/T₀ \u0026asymp; 0.07\u0026ndash;0.08). In contrast, coordination develops more slowly and reaches its maximum at substantially larger asymmetries (Δτ/T₀ \u0026asymp; 0.30). No single delay configuration simultaneously optimizes both objectives. Mode-wise analysis reveals that this separation originates from a redistribution of information from collective to differential modes as asymmetry increases.\u003c/p\u003e \u003cp\u003eCrucially, this phenomenon is absent in phase-only delayed Kuramoto models, indicating that delay geometry alone is insufficient. Instead, amplitude dynamics and amplitude\u0026ndash;phase coupling are essential. Validation using delayed Stuart\u0026ndash;Landau oscillators confirms that shear (non-isochronicity) amplifies early sensitivity by converting noise-driven amplitude fluctuations into phase information. Together, these results establish delay asymmetry as a functional resource, revealing a fundamental dynamical principle by which systems balance responsiveness and coordinated structure without fine-tuning to criticality.\u003c/p\u003e","manuscriptTitle":"Delay asymmetry induces a functional separation between sensitivity and coordination","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-04 18:48:45","doi":"10.21203/rs.3.rs-8896814/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-03-27T15:06:33+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-25T23:41:29+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-08T08:03:54+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"84368161029571276246695396573718206173","date":"2026-03-01T03:57:25+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"217205533289312663599763637969864498246","date":"2026-02-28T07:27:16+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-02-27T13:14:05+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-02-26T10:35:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-17T06:37:37+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-17T06:32:38+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-02-17T00:36:59+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b2560d24-bea3-469b-b1dd-46a98185154d","owner":[],"postedDate":"March 4th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[{"id":63765490,"name":"Biological sciences/Biophysics"},{"id":63765491,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-03-27T15:10:32+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-04 18:48:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8896814","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8896814","identity":"rs-8896814","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}