Introduction

Rhythmic behaviors rely on properly functioning biological oscillators, and their dysregulation can lead to severe disorders, some of which are life-threatening1,2,3,4,5. Many of these oscillators operate through mutual inhibition, forming competitive oscillators that ensure stable rhythmic activity6,7,8,9,10. For example, tightly regulated gene expressions, which are critical for organismal development and cellular homeostasis, often involve transcriptional oscillators that are modulated by mutually inhibited gene expressions. Disruptions in these competitive oscillators are implicated in diseases such as cancer11,12. Similarly, neurons exhibit intrinsic oscillations, and neural circuits, which are governed by mutual inhibition, regulate vital behaviors like sleep and locomotion. Dysfunction in these competitive oscillators can result in sleep disorders, seizures, or motor deficits13,14. Thus, competitive oscillators play fundamental roles in maintaining proper physiological function.

Biological oscillators are often regulated by feedback loops, which play essential roles in their function. Positive feedback reinforces signals15, enhances synchronization16, whereas negative feedback promotes oscillation generation17,18, ensures robustness19 in most oscillatory biological behaviors. For example, in the mammalian circadian clock, RORa and RORb act as positive regulators, driving Bmal1 expression in the suprachiasmatic nuclei15. When biological oscillators are coupled with each other, synchronization is facilitated through positive feedback interactions16. In contrast, time-delayed negative feedback in protein synthesis can induce periodic oscillations18. Synthetic circuits also demonstrate that negative feedback enhances oscillator robustness19. The integration of both feedback types further fine-tunes dynamical control in biological oscillators20,21,22. For instance, combined positive and negative feedback not only improves noise resistance but also increases the tunability of key oscillation parameters22.

Competitive biological oscillators governed by mutual inhibition are dynamically modulated by both positive and negative feedback, giving rise to diverse rhythmic behaviors. Three primary circuit architectures have been demonstrated to modulate competitive oscillators: (i) purely positive feedback23,24,25, (ii) purely negative feedback26,27, and (iii) hybrid(positive-plus-negative) feedback9,10,28,29,30,31,32. For instance, in cell cycle regulation, the activation and inactivation of Cdk1 during mitotic entry and exit involve a positive feedback loop with Cdc25 and Wee1, forming a bistable trigger essential for cell cycle progression23. Conversely, in protein stress responses, AMPK, ULK1, and MTORC1 interact through negative feedback, generating high-amplitude oscillations in AMBRA1 phosphorylation under mild stress26. Hybrid feedback, however, is particularly widespread, appearing in key biological processes such as the cell cycle32, neuronal networks29, circadian rhythms30, gene transcriptional regulation28, predator-prey systems31, brain activity10, and locomotor control9. In the mammalian cell cycle, hybrid feedback precisely orchestrates the antagonistic relationship between CycB and Cdh1 to drive phase transitions32. Neuronal networks leverage hybrid feedback to create stable toggle switches, where mutual inhibition coupled with regulatory loops enables bifurcation-controlled state transitions29. Circadian clocks integrate multiple feedback loops, with Per2, Cry1, and Rev-erb-α forming a mutually repressive core that sustains endogenous oscillations30. Gene regulatory networks exploit interlinked feedback motifs to generate tuneable dynamics, functioning as bistable switches, oscillators, or excitable systems28. Ecological models reveal that hybrid feedback in predator-prey systems, combining parasitism and mutualism, shapes ecosystem stability and diversification31. In cortical microcircuits, excitatory-inhibitory interactions implement rate- and oscillation-based computations, producing synchronized state transitions across cortical layers10. Even locomotor control in Drosophila larvae arises from mechanical feedback loops, where interacting spring-like forces generate coordinated waves, oscillations, and chaotic deformations9. The widespread occurrence of hybrid feedback across biological scales suggests an evolutionary optimization for tuneable, robust oscillations. Yet, despite its prevalence, the precise mechanisms by which hybrid feedback shapes rhythmic patterns in competitive oscillators remain incompletely resolved, highlighting a critical gap in our understanding of biological dynamics.

Here, we investigate how hybrid feedback modulates rhythmic dynamics in competitive oscillators across the above-mentioned seven classes of biological processes or systems. Analyzing models ranging from mammalian cell cycles to six additional biological oscillators, we discover two distinct modulation modes: higher-amplitude, lower-frequency oscillations or higher-frequency, lower-amplitude oscillations, depending on hybrid feedback strengths. Furthermore, we demonstrate that oscillation tunability scales with the asymmetry between positive and negative feedback loops, highlighting a key design principle for biological rhythm regulation. These findings provide new insights into oscillation regulation mechanisms and could guide therapeutic strategies for diseases related to rhythm disorders.

Results

Hybrid feedback regulation in mammalian cell cycle competitive oscillators

To investigate how competitive interactions and feedback loops regulate oscillatory dynamics, we developed a computational model based on ordinary differential equations representing two core mammalian cell cycle oscillators, as outlined in the prior influence diagram32. CycB and Cdh1, which are essential for the mammalian cell cycle, exhibit mutually inhibitory behavior: in the G1 steady state, the level of CycB is low while Cdh1 is active, whereas in the M steady state, CycB is active while Cdh1 is inactive. CycB and Cdh1 are further modulated by additional regulatory compounds. To recapture these dynamics, our model incorporates: (i) two primary negative feedback loops, [CycB → APC/Cdc20 ⊣ CycB] and [Cdh1 ⊣ Plk1 ⊣ Emi1 ⊣ Cdh1], as core oscillators; (ii) two positive feedback loops, [CycB → Cdc25 → CycB] and [Cdh1 → Cdc14 → Cdh1], for additional regulation; (iii) two secondary negative feedback loops, [CycB ⊣ E2F → CycB] and [Cdh1 → CycE ⊣ Cdh1], for modulation (Fig. 1a). The model effectively recaptures the antagonistic relationship between CycB and Cdh1, which are further fine-tuned by two positive-feedback and two negative-feedback loops, respectively.

Fig. 1: Competitive dynamics between CycB and Cdh1 in the mammalian cell cycle.

a Schematic representation of key regulatory interactions in the mammalian cell cycle. b Correlation heatmap showing phase relationship between CycB and Cdh1 across different mutual inhibition strengths (k1 and k2). c Time-course oscillations of CycB and Cdh1 protein levels under strong mutual inhibition (k1 = k2 = 1). Quantitative analysis of oscillation properties: d CycB and e Cdh1 amplitude and frequency as functions of competitive strength k, demonstrating an inverse relationship between amplitude and frequency scaling.

We quantified the mutual inhibition between CycB and Cdh1, assigning strength parameters k1 (Cdh1 inhibits CycB) and k2 (CycB inhibits Cdh1), while additional positive and negative feedback effects were represented by pos and neg, respectively. Initial simulations focused exclusively on competitive interactions by setting neg = 0 and pos = 0, with other parameters maintained at values known to produce stable oscillations from previous studies6,22,32. Systematically increasing k1 and k2 initially failed to produce correlated dynamics, with robust antiphase oscillations emerging only at higher inhibition strengths (Fig. 1b, c). To quantitatively assess how competition strength affects oscillator properties, we set k1 = k2 = k (where k ∈ [0,1]) and measured the resulting amplitude and frequency of these antiphase oscillations. Our analysis revealed that competition strength inversely modulates oscillatory properties: stronger competition (higher k values) led to increased oscillation amplitudes but decreased frequencies in both CycB and Cdh1 (Fig. 1d, e). These findings highlight how mutual inhibition tunes the trade-off between amplitude and frequency in coupled oscillators.

To investigate how supplementary feedback loops modulate CycB and Cdh1 dynamics, we systematically varied positive (pos) and negative (neg) feedback strengths across the range [0,1], while maintaining constant competition strength (k1 = k2 = 1) to preserve antiphase oscillations. Our analysis revealed two distinct regulatory patterns: (i) in single-feedback versions, negative feedback alone (pos = 0) generated higher-frequency oscillations compared to positive feedback alone (neg = 0), whereas positive feedback alone yielded larger amplitudes than negative feedback alone; (ii) in the hybrid-feedback version, combining positive and negative feedback recapitulated the full range of frequencies and amplitudes as seen in single-feedback systems (Fig. 2a, b), yet also exhibited additional properties. Scatter plot analysis revealed superior tunability beyond a simple additive effect of individual feedback loops (Fig. 2c), with CycB oscillation displaying particularly enhanced control over both frequency and amplitude (Fig. 2d). Cdh1 oscillations exhibited analogous modulation patterns with CycB (Supplementary Fig. 1), confirming consistent regulatory effects across competitive oscillators. Thus, hybrid feedback enables dual modulation, permitting higher-amplitude, lower-frequency oscillations and higher-frequency, lower-amplitude oscillations, while significantly expanding the tuneable parameter space in both dimensions.

Fig. 2: Modulation of CycB oscillation properties by feedback loops.

a Frequency distributions for three feedback configurations: positive-only (neg = 0), negative-only (pos = 0), and hybrid (positive-plus-negative) feedback. b Corresponding amplitude distributions, with violin plot width representing density. c The amplitude/frequency scatters across all feedback conditions. d Normalized tunability ranges for CycB frequency (left) and amplitude (right) under each feedback configuration, calculated by (maximum - minimum)/reference value, where the reference value is the baseline (pos = 0, neg = 0) of frequency/amplitude.

Hybrid feedback regulation in six additional competitive oscillator models

To evaluate the generalizability of enhanced tunability through hybrid feedback in competitive oscillators, we analyzed six additional biological oscillator models: (i) Fitzhugh-Nagumo, a well-studied model in neuronal networks29,33,34, (ii) Goodwin, an oscillator relevant to circadian rhythms35,36,37,38,39, (iii) Repressilator, a tripe-negative feedback loop in transcriptional regulation40,41, (iv) Predator-prey, a model employed in ecological systems42,43,44, (v) Van der Pol, a model could describe brain activity10,45,46, (vi) Neuromechanical oscillator, a model proposed for motor control47,48. These models represent diverse biological systems where competitive oscillators naturally arise, often incorporating additional feedback loops to fine-tune their dynamics. In the neuronal networks, for example, neurons frequently couple through mutual repression and are embedded in motifs containing both positive and negative feedback loops29. Similarly, in the plant circadian clock, the mutually inhibited genes LHY and TOC1, which are modulated by additional feedback loops, regulate the rhythm at the morning phase and evening phase, respectively39. Gene regulatory networks also commonly feature mutually inhibited transcription factors under the influence of supplementary feedback loops41. Ecological systems exhibit analogous dynamics, where predator populations may compete through mutual inhibition while also engaging in mutualism and parasitism44. In brain regions, PV1 and PV2 exhibit mutual inhibition with additional feedback modulation10. In motor control, muscles often show mutual inhibition because of structural constraints, further regulated by neuron-muscle or muscle-muscle feedback47.

To systematically compare these systems, we incorporated competitive interactions (mutual inhibition) and additional positive and negative feedback loops (Supplementary Fig. 2). To establish a consistent baseline, we initialized each system with high mutual inhibition strengths (k1, k2) to guarantee antiphase oscillations. Specific strengths for the positive (pos) and negative (neg) feedback loops were then applied. The transient and equilibrium dynamics, visualized through phase portraits and time-series plots (Supplementary Fig. 3), reveal the core behavior of each system. Furthermore, we assessed the robustness of these dynamics by conducting parameter sweeps of the pos and neg values (Supplementary Fig. 4). This analysis consistently showed that higher positive feedback strength increases amplitude while decreasing frequency, whereas negative feedback strength produces the inverse effect.

For all models, we quantified oscillation properties (amplitude and frequency) under three feedback configurations: positive feedback only, negative feedback only, and hybrid feedback. This systematic approach tests whether hybrid feedback universally enhances tunability across competitive biological oscillators. The amplitude-frequency scatter plots revealed consistent patterns across models: positive feedback alone generated low-frequency, high-amplitude oscillations, whereas negative feedback alone produced high-frequency, low-amplitude oscillations, with hybrid feedback spanning the full range between these extremes (Fig. 3). Given that biological oscillators operate across periods ranging from seconds to days, resulting in substantial frequency variation, while amplitudes remained relatively stable, except in the Repressilator, where amplitude varied nearly eight folds (Fig. 3c). Further analysis of three feedback configurations demonstrated distinct tuning capabilities across six oscillator models. Although positive and negative feedback differentially modulated the frequency and amplitude, hybrid feedback consistently exhibited broader normalized tuning ranges for both frequency and amplitude compared to single-feedback systems (Fig. 4). Thus, hybrid feedback offered two key advantages: (i) expanded coverage of amplitude-frequency combinations beyond the simple additive effects of single-feedback modes, and (ii) significantly enhanced tunability in both frequency and amplitude compared to single-feedback configurations.

Fig. 3: Amplitude-frequency relationships of oscillation properties across six competitive biological models.

a Fitzhugh-Nagumo model (neuronal activity). b Goodwin oscillator (circadian rhythm). c Repressilator (genetic transcriptional network). d Predator-prey system (ecological dynamics). e Van der Pol oscillator (brain region activity). f Neuromechanical oscillator (motor control).

Fig. 4: Comparative analysis of oscillation tunability ranges.

Normalized tunability ranges for frequency (left) and amplitude (right) across six competitive biological models: a Fitzhugh-Nagumo. b Goodwin. c Repressilator. d Predator-prey. e Van der Pol. f Neuromechanical oscillator. Tunability is calculated as (maximum - minimum)/reference value, where reference represents the no-feedback condition (pos = 0, neg = 0).

Enhanced tunability through asymmetric feedback loops

To investigate how feedback loop asymmetry affects oscillation tunability, we systematically varied the relative strengths of positive and negative feedback while maintaining antiphase oscillations in competitive oscillators. Our analysis revealed that hybrid feedback modulates oscillations through mechanisms distinct from the simple superposition of individual feedback effects. We further examined whether the relative strength of positive-to-negative or negative-to-positive feedback influences these additional tuned domains. By varying the feedback ratios across seven biological oscillator models, we calculated the frequency and amplitude for each condition and quantified the proportion of extra-tuned domains relative to all achievable amplitude-frequency ranges under hybrid feedback.

Our initial experiments, conducted using the Fitzhugh-Nagumo and Van der Pol models for their symmetrical feedback structures, revealed that imbalanced feedback strengths, whether positive- or negative-dominant, significantly expanded the accessible tuning domains (Fig. 5). In the Fitzhugh-Nagumo model, the proportion of extra tuned domains surged from near zero to almost 100% under positive-dominant conditions and to near 60% under negative-dominant conditions as the relative strength of the dominant feedback increased, eventually plateauing in both regimes. The Van der Pol model exhibited a sharp initial increase from zero to 60%, followed by slower growth. Extending this analysis to five additional models with distinct feedback structures consistently demonstrated enhanced tunability under asymmetric feedback conditions (Fig. 5). Specifically, increasing the relative strength of positive-to-negative feedback boosted tunability in the cell cycle oscillator (from ~85.5% to ~91%), the Repressilator (from ~94.8% to ~97%), the Neuromechanical oscillator (from ~98.55% to ~98.7%), and the Goodwin model (from ~95% to ~96.5%). Conversely, increasing negative-to-positive feedback initially reduced tunability in the cell cycle oscillator (to ~85%) before a rise to 89%, and produced declines in the Neuromechanical oscillator (to ~98%), the Goodwin model (to ~92.5%), and the Repressilator (to ~94.5%), followed by a slight recovery in the latter. In contrast, the Predator-prey model was dominated by negative feedback, with tunability increasing alongside negative-to-positive feedback strength (from ~96.9% to ~97.1%) and decreasing with enhanced positive feedback (to ~96.1%). Together, these findings demonstrate that while a balance exists between the two feedback types, moving beyond this equilibrium toward either form of asymmetry generally enhances tunability across a wide range of competitive oscillatory models.

Fig. 5: Enhanced tunability through asymmetrical feedback loops in competitive biological oscillators.

Percentage of additional accessible states (vertical axis) as a function of relative strength (horizontal axis) of feedback loops for seven competitive oscillators. The percentage of additional states was calculated as (N**h – N**s)/N**h × 100%, where N**h and N**s represent the number of accessible states under hybrid feedback and single feedback, respectively.

Discussion

Our study elucidates a fundamental design principle governing biological rhythmicity: hybrid feedback, when coupled with competitive mutual inhibition, dramatically expands the tunability of oscillatory dynamics. We demonstrated this across seven distinct biological oscillator models, revealing that the relative asymmetry between feedback strengths is a key determinant of the accessible amplitude-frequency space. This enhanced tunability provides a versatile mechanism for biological systems to generate a wide repertoire of rhythms from a core competitive circuit, a capability crucial for adapting to internal and external demands.

A foundational observation from our analysis of the mammalian cell cycle model was that increasing the strength of mutual inhibition (parameter k) led to an increase in oscillation amplitude but a concurrent decrease in frequency. This inverse relationship can be understood through the lens of nonlinear dynamics. Stronger mutual inhibition deepens the potential wells defining the system’s stable states (e.g., G1 and M phases), increasing the energy barrier between them. Transitions between these states, driven by the slower, integrating action of the supplementary feedback loops, thus become less frequent (lower frequency) but more decisive, resulting in sharper, more pronounced switches (higher amplitude). This trade-off establishes a baseline dynamic range upon which additional feedback acts.

The core of our findings lies in the superior performance of hybrid feedback over purely positive or negative configurations. The mathematical principle behind this is the introduction of multi-timescale dynamics and nonlinear interactions. Positive feedback typically acts as an accelerant, promoting rapid, all-or-nothing transitions that favor high-amplitude outputs but can lead to bistability rather than oscillation if unchecked22. Negative feedback provides the necessary restorative force to create a stable limit cycle, often leading to faster, smaller-amplitude oscillations28. When combined, they do not merely act additively. Their interaction can create complex nullclines and bifurcation structures. This complexity manifests as a broader range of stable oscillatory regimes, allowing the system to access amplitude-frequency combinations that are unreachable by either feedback type alone. This explains the “extra tuned domains” we observed in the scatter plots, which represent novel dynamic states emergent from the nonlinear coupling of the two feedback types.

A particularly significant finding is that this enhanced tunability is maximized under conditions of asymmetric feedback strength. This phenomenon, prominently observed in the Fitzhugh-Nagumo and Van der Pol models, can be explained by breaking the symmetry of the system’s phase space. In symmetric or balanced feedback conditions, the system’s dynamics are often constrained to a narrower corridor of possible states. Introducing asymmetry, whether positive- or negative-dominant, effectively warps the phase space, creating new basins of attraction and altering the shape and location of the limit cycle. This expands the envelope of achievable amplitudes and frequencies. Biologically, this suggests that evolvable oscillatory circuits are not necessarily optimized for balanced feedback but rather for a tuneable imbalance, allowing one type of feedback to be preferentially recruited to shift rhythms between a high-amplitude/low-frequency state and a low-amplitude/high-frequency state.

Our comparative analysis across diverse models, Fitzhugh-Nagumo, Van der Pol, Repressilator, Predator-Prey, Goodwin, and the Neuromechanical oscillator, confirms that this is a general principle, though its specific manifestation depends on the intrinsic dynamics of each system. For instance, the Repressilator’s core triple-negative-feedback and Goodwin’s negative-feedback designs are inherently oscillatory but often produce low-amplitude oscillations35,36,40,41. Our results show that incorporating asymmetric hybrid feedback can increase its amplitude range, making it a far more robust and tuneable genetic clock. Conversely, the Predator-Prey model, which is naturally dominated by negative feedback, namely consumption and mortality42,43,44, showed its greatest tunability when this negative dominance was reinforced. This indicates that the universal applicability of asymmetric feedback lies not in a single recipe that always favors positive feedback, but in the general strategy of breaking symmetry to access a wider parameter space. The specific optimal asymmetry is a property of the underlying oscillator’s structure.

An important factor not explicitly explored in our current models but critical to biological realism is time delay. In many of the systems studied, from transcriptional feedback in the Repressilator and Goodwin models to Neuromechanical signaling, feedback loops are not instantaneous35,36,40,41. Time delays in negative feedback are a classic mechanism for generating oscillations, as they can cause an overshoot in the system’s response to a change. In a hybrid setup, delays can introduce crucial phase lags between the positive and negative signals. For example, in the Neuromechanical oscillator, a delay in the sensory feedback from a muscle’s stretch could ensure that excitatory (positive) feedback peaks just as the muscle is at its optimal length for contraction, while delayed inhibitory (negative) feedback arrives to terminate the contraction and initiate the antagonist’s movement47,48. This orchestration of phase via delay could be a key mechanism for precisely tuning both the frequency and waveform of the resulting rhythm, and its integration with hybrid feedback represents a vital avenue for future research.

While our study provides broad insights, several limitations must be acknowledged. First, our models are necessarily simplifications of a vastly more complex biological reality. We represent feedback with single strength parameters (pos, neg), whereas in vivo, these loops involve multi-step biochemical reactions with their own nonlinearities and delays. Second, we focused on deterministic models, whereas biological systems operate in a noisy environment. Another limitation is that our approach complements these summaries but relies solely on statistical metrics. Future work could investigate how stochastic fluctuations interact with hybrid feedback to affect the robustness and reliability of the tuned oscillations.

In conclusion, our work establishes asymmetric hybrid feedback as a powerful and general mechanism for enhancing the tunability of competitive biological oscillators. By providing a unified framework for understanding rhythm regulation across cellular, neural, genetic, and ecological domains, these findings offer a new perspective on the design principles of biological clocks. This knowledge could inform the development of therapeutic strategies for rhythm disorders by suggesting novel targets for manipulating feedback asymmetry to restore healthy dynamics, as well as guide the engineering of synthetic biological oscillators with precisely specified properties.

Methods

We numerically solved the ordinary differential equations (ODEs) for each oscillator model using Runge-Kutta integration in MATLAB. For each model, we initialized parameters from published values known to produce stable limit cycle oscillations, and we determined the (k1, k2) parameter space, maintaining antiphase oscillations between competitive oscillators. We systematically varied positive (pos) and negative (neg) feedback strengths while computing limit cycle solutions and recording amplitude and frequency characteristics. Finally, we modulated the relative feedback strength ratio (N/P and P/N) to assess its impact on oscillation properties. The complete set of ODE formulations and parameter values used for each oscillator model is provided below.

Cell cycle oscillator model

We constructed a mammalian cell cycle oscillator using a system of ten ODEs. As outlined in the prior influence diagram32, we abstract the core model includes two interlocked negative feedback loops that generate autonomous oscillations, mutual inhibition between the CycB and Cdh1 oscillators, and experimentally validated parameters that reproduce established oscillation patterns for CycB and Cdh1 activity. Parameter nomenclature follows established conventions. k**s is the synthesis rate. k**a is the activation rate. k**i is the inactivation rate. k**d is the degradation rate. The model also incorporates four additional regulatory elements that implement two positive-feedback loops and two negative-feedback loops (Fig. 1a), which interact to fine-tune the overall dynamics. Specifically, Cdc25 and E2F enhance CycB synthesis, while Cdc20 and Cdh1 promote degradation of CycB. Conversely, Cdc14 increases Cdh1 synthesis, while Emi1, CycE, and CycB accelerate the degradation of Cdh1. The model further depicts activation chains [CycB → APC → Cdc20] and inactivation chains [Cdh1 → Plk1 → Emi1]. Regulatory crosstalk is also captured; for instanc