Mutual Entrainment in Coupled Systems

• Mutual entrainment is the bidirectional adjustment among interacting oscillators that synchronizes phases and regulates amplitudes without relying on an external clock.
• It spans diverse fields such as musical rhythm, spoken dialog, neural coordination, quantum synchronization, and superfluid dynamics, highlighting broad applications.
• Researchers employ phase models, network analyses, and specialized measurement techniques to quantify entrainment and distinguish it from mere static synchrony.

Mutual entrainment denotes bidirectional adjustment among interacting systems. In dynamical-systems terms, two or more oscillators are “mutually entrained” when they exchange information and adjust their phases, and sometimes frequencies, so as to reduce their phase differences over time; in dialog research, “mutual entrainment refers to bi-directional adaptation, where each speaker both influences and is influenced by the other, leading to a stable, shared prosodic style” (Alderisio et al., 2016, Reichel et al., 2018). The term is used across markedly different domains: rhythmic synchronization in musical performance, prosodic and code-switching adaptation in spoken interaction, visually coupled motor coordination in groups, symbolic dynamics of joint brain states during dyadic coordination, cooperative and competitive quantum synchronization, and the Andreev–Bashkin effect in neutron–proton superfluids (Hennig, 2013, Bhattacharya et al., 2023, Pinto et al., 2024, Murtadho et al., 2023, Chamel et al., 2021).

1. Core dynamical formulations

Mutual entrainment is typically contrasted with external entrainment. External entrainment, or forced synchronization, occurs when a single limit-cycle oscillator locks its phase and frequency to an imposed periodic drive; mutual entrainment arises when two or more limit-cycle oscillators adjust their phases through bidirectional coupling, without any dominant external clock (Murtadho et al., 2023). In the classical phase-reduced setting, a generic network form is

$$\dot x_i = f_i(x_i)+\sum_j K_{ij}\,H(x_j,x_i),\qquad i=1,\dots,N,$$

and, in the hand-movement ensemble study, the phase-only Kuramoto-type model was written as

$$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$

with $\omega_k$ the intrinsic angular frequency, $a_{kh}$ the topology of visual coupling, and $c>0$ a global coupling strength (Alderisio et al., 2016).

For paired oscillators, the phase-locking condition appears in especially transparent form. In the circadian-clock analysis with diffusive coupling,

$$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$

so a locked solution requires

$|\Delta\omega|\le 2K_\text{diff}.$

That framework also distinguished diffusive coupling from mean-field coupling and showed that mean-field coupling leads to amplitude expansion of weak oscillators and, as a result, reduces the entrainment range, while coupling also determines the rigidity of the synchronized network through relaxation rates upon perturbation (Bordyugov et al., 2011).

A recurring implication is that mutual entrainment is not exhausted by phase coincidence. Depending on the model, it involves amplitude regulation, stability properties, and network topology in addition to phase pulling. This suggests that the phenomenon is better treated as a property of coupled dynamical organization than as a single scalar synchrony index.

2. Rhythmic interaction and spoken communication

In musical interaction, rhythmic synchronization is treated as the foundation of musical interaction. In the study of two humans playing rhythms, the interbeat intervals of both laypeople and professional musicians exhibit scale-free (power law) cross-correlations, and the next beat to be played by one person is dependent on the entire history of the other persons IBIs on time scales up to several minutes (Hennig, 2013). The proposed stochastic model for Mutually Interacting Complex Systems (MICS) suggests a physiologically motivated explanation for the occurrence of scale-free cross-correlations and was presented as potentially applicable to econophysics, physiological time series, and collective behavior of animal flocks (Hennig, 2013).

In spoken dialog, prosodic entrainment was operationalized through both convergence and synchrony. Convergence was defined as decreasing absolute feature-value differences, whereas synchrony was defined as co-variation relative to each speaker’s own long-term mean. The dialog-act study measured entrainment separately for several dialog acts in cooperative and competitive games, using intonation features derived from a superpositional intonation stylization as well as rhythm features. For each numeric feature $F$, the convergence distance was

$|F_A-F_B|,$

and the synchrony distance was

$|(F_A-\mu_A)-(F_B-\mu_B)|,$

with entrainment inferred when mean within-dialog distance was much smaller than mean across-dialog distance; two-sided $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$0-tests and linear mixed-effects models were then applied (Reichel et al., 2018).

The empirical pattern was strongly dialog-act dependent. In cooperative games, dialog acts with a high authority given by knowledge and with a high frequency showed the most entrainment. “Explain” and “Clarify” entrained strongly in cooperation but disentrained in competition, whereas “Instruct” disentrained in cooperation but entrained in competition. A linear mixed-effects model found a highly significant Dialog-Condition × Authority × Support interaction, $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$1, $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$2, in the cooperative subset (Reichel et al., 2018). The interpretation given in the source was that the selective entrainment or disentrainment by dialog-act function argues against purely automatic perception-production “mirroring” and supports hybrid models in which entrainment is partly automatic but also under conscious or goal-oriented control (Reichel et al., 2018).

In spontaneous code-switched speech, mutual entrainment was defined directionally: if both $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$3 and $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$4 were statistically significant, the pair was judged to exhibit mutual entrainment on that feature. The corpus consisted of 39 dyadic code-switched conversations selected from Bangor Miami, and the analysis covered lexical features, acoustic-prosodic features, and code-switching features such as presence of CSW, amount of CSW, and CSW strategy (Bhattacharya et al., 2023). Lexical entrainment was measured by

$$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$5

Among the reported findings, Top-100 words and affirmative cues showed significant entrainment in all 39 conversations, 29 of 39 conversations showed bidirectional entrainment on overall language when excluding OOVs, and code-switching presence showed turn-level proximity in 29 of 39 conversations with $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$6 (Bhattacharya et al., 2023).

3. Motor coordination, joint brain states, and neuro-motor coupling

In human ensembles, mutual entrainment has been formalized as synchronization in networks of coupled heterogeneous oscillators. In the study of visually coordinated oscillatory hand motion, two groups of seven participants were tested under four imposed coupling topologies: Complete graph, Ring graph, Path graph, and Star graph. Group homogeneity was measured by the coefficient of variation of the natural frequencies, with $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$7 for Group 1 and $$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$8 for Group 2 (Alderisio et al., 2016). Synchronization was quantified through the Kuramoto order parameter

$$\dot\theta_k=\omega_k+\frac{c}{N}\sum_{h=1}^N a_{kh}\,\sin(\theta_h-\theta_k),$$9

the individual synchronization index $\omega_k$0, and the group synchronization index $\omega_k$1. Complete and Star graphs produced the highest $\omega_k$2 and $\omega_k$3 across both groups, whereas Ring and Path yielded lower entrainment, especially for the more heterogeneous group; in the Path graph, the less homogeneous group synchronized poorly, with $\omega_k$4 versus $\omega_k$5 in the more homogeneous group (Alderisio et al., 2016).

The joint-brain-state analysis extended mutual entrainment to simultaneous EEG. Individual brain states were defined from 448-ROI source-space time series, converted into unsigned correlation matrices over 2 s windows with 50% overlap, embedded by MDS in 3D chosen by BIC, and clustered with a Gaussian Mixture Model. Joint symbols were then defined as ordered pairs from the two subjects’ symbol streams, and the resulting sequence was analyzed through recurrence methods and transition networks (Pinto et al., 2024). The dissimilarity between correlation matrices $\omega_k$6 and $\omega_k$7 was

$\omega_k$8

The recurrence analysis focused on dwell time and motif length, while the transition-network analysis used edge probabilities $\omega_k$9, edge costs $a_{kh}$0, average shortest path length $a_{kh}$1, and average betweenness centrality $a_{kh}$2 (Pinto et al., 2024).

The effect of mutual feedback depended sharply on task. In synchronization, increasing feedback promoted stability with longer dwell times and motif length, and stronger coupling stabilized a few states restricting the pattern of flow between states while preserving a core-periphery structure of the joint brain states. In syncopation, Leader-Follower interactions enhanced stability, but Mutual feedback dramatically reduced stability; the network analysis then showed a more distributed flow amongst a larger set of joint brain states and reduced dominance of core joint brain states (Pinto et al., 2024). This supports the more general point that mutual entrainment can reorganize a metastable landscape rather than simply increase a generic synchrony measure.

A related result appears in collaborative dance improvisation. Using 3D motion capture and hyperscanning EEG, synchronization events were quantified by Motif Synchronization for biomechanical data and multilayer Time-Varying Graphs for neural data. Training produced an opposite trend: inter-brain synchronization increased, particularly within the frontal lobe, while interpersonal motor synchrony decreased (Ramos et al., 7 Jan 2026). The source interprets this as a “coupling-decoupling paradox,” in which togetherness emerges not from identical motor outputs but from shared neural intentionality distributed across multilayer interaction networks (Ramos et al., 7 Jan 2026).

4. Quantum synchronization, cooperation, and competition

Quantum studies use the same distinction between forced entrainment and mutual synchronization, but the mechanisms are expressed through coherences, Liouville spectra, and degeneracies. In collectively driven degenerate quantum thermal machines, the interplay between external entrainment and mutual synchronization was shown to exhibit competition and cooperation due to phase pulling and phase repulsion; the balance depends on whether the machine operates as a refrigerator or an engine (Murtadho et al., 2023). The semiclassical phase equations presented in the source have the form

$a_{kh}$3

$a_{kh}$4

so that, for the phase difference $a_{kh}$5,

$a_{kh}$6

When the two mechanisms pull the same way they cooperate; when they pull in opposite directions they compete and may enforce anti-phase locking (Murtadho et al., 2023). In the thermodynamic limit of degeneracy, mutual synchronization dominates (Murtadho et al., 2023).

A complementary formulation was given through Liouville-space perturbation theory. Starting from

$a_{kh}$7

the dynamics are written as $a_{kh}$8, with right and left eigenvectors of the non-Hermitian Liouville superoperator characterizing steady states and oscillating coherences. The first-order correction to the steady-state eigenvector was written as

$a_{kh}$9

The central condition derived in the source is that if $c>0$0 is non-degenerate and $c>0$1, then $c>0$2 is diagonal in the $c>0$3 eigenbasis and cannot generate the off-diagonal steady-state coherences required for mutual phase synchronisation; conversely, degeneracies permit an energy-conserving interaction to act off-diagonally within the degenerate block and thereby generate the coherences associated with synchronization (Solanki et al., 2021).

This theoretical line places mutual entrainment in quantum systems within a broader algebra of conservation laws and symmetry constraints. A plausible implication is that, in quantum settings, “mutuality” is often best diagnosed through the structure of coherences and spectrum rather than through a direct analogue of classical phase locking alone.

5. Neutron–proton superfluids and the Andreev–Bashkin effect

In nuclear many-body theory, mutual entrainment is the nondissipative coupling by which the mass current of one species depends on the superflow of the other. In a mixture of neutron and proton superfluids, the local or spatially averaged mass currents can be written as

$c>0$4

so that a pure neutronic superflow drags along a fraction of the protons, and vice versa (Chamel et al., 2021). This is the Andreev–Bashkin effect. The cold-matter treatment derived exact expressions for the mass currents directly from the time-dependent Hartree–Fock equations with no further approximation and showed the equivalence with the Fermi-liquid expression (Chamel et al., 2021). The finite-temperature extension, based on the time-dependent Hartree–Fock–Bogoliubov equations, showed that the local mass currents in homogeneous or inhomogeneous nuclear systems have the same formal expression as the ones found earlier in the absence of pairing at zero temperature, and provided analytical expressions for the entrainment matrix valid for arbitrary temperatures and currents (Allard et al., 2020).

The source gives an explicit parametrization in terms of the entrainment matrix and the isovector effective mass. Because of Galilean invariance,

$c>0$5

and the dimensionless determinant

$c>0$6

satisfies

$c>0$7

An equivalent parametrization is

$c>0$8

$c>0$9

$$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$0

with $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$1 (Chamel et al., 2021).

The neutron-star application is explicit. The formulas were presented for superfluid neutron-star cores and for Brussels–Montreal functionals. At nuclear saturation density, $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$2, the isovector effective mass lies in the range $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$3, with BSk24 and BSk25 giving $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$4, and consequently $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$5 (Chamel et al., 2021). The off-diagonal coefficient $$\dot\theta=\Delta\omega-2K_\text{diff}\sin\theta,\qquad \Delta\omega=\omega_1-\omega_2,$$6 determines how strongly neutron vortices carry proton mass and hence the magnitude of the magnetic field trapped in a vortex core and the electron-vortex drag; entrainment also modifies the two-fluid sound speeds and the spectrum of global oscillation modes (Chamel et al., 2021).

6. Measurement strategies, interpretation, and recurrent misconceptions

The measurement of mutual entrainment depends strongly on domain. In dialog studies, it is common to distinguish convergence from synchrony, and to compare within-interaction distances against across-interaction controls (Reichel et al., 2018). In code-switched speech, mutuality is explicitly directional and requires significance in both directions (Bhattacharya et al., 2023). In neural deconfounding work, the central methodological issue is separating entrainment from consistency, defined as the tendency of a speaker to adhere to their own baseline style. Two neural measures were proposed for that purpose: Deep Residualization (DR) and Adversarial deconfounding (A). On Fisher English test data, fake-session discrimination accuracy was reported as 95.1% (0.7) for DR and 94.4% (0.6) for A; on the Games corpus, 85.8% (12.8) for DR and 80.3% (12.6) for A (Weise et al., 2020). The study further reported that the direction and existence of correlations with social variables can flip once consistency is removed (Weise et al., 2020).

One persistent misconception is that mutual entrainment can be inferred from any apparent recovery of synchronization at a distance. The study of “apparent remote synchronization of amplitudes” showed that the observed dip and recovery were caused by a demodulation and interference effect: local destructive interference occurred when lower-sideband and demodulated baseband components became nearly opposite in phase, while transfer entropy and the auxiliary system approach showed that synchronization and causality actually decrease with distance monotonically (Minati et al., 2018). The conclusion drawn there was that remoteness is arguably only apparent (Minati et al., 2018).

Another misconception is that togetherness requires identical outputs or zero-lag locking. The dance-improvisation study states explicitly that entrainment is neither a static zero-lag locking of identical signals nor a mere by-product of shared stimuli, and its principal result was the dissociation between increased inter-brain synchronization and decreased interpersonal motor synchrony after training (Ramos et al., 7 Jan 2026). In the same vein, the dialog-act study argues against purely automatic “mirroring” and favors hybrid accounts in which entrainment is partly automatic but also under conscious or goal-oriented control (Reichel et al., 2018).

Taken together, these results indicate that mutual entrainment is not a unitary observable but a family of bidirectional coupling phenomena whose empirical signature depends on timescale, observable, topology, task demands, and the distinction between true adaptation and static similarity. This suggests that cross-domain comparison is most rigorous when the analysis preserves directionality, controls for confounds such as consistency, and distinguishes local synchrony measures from the underlying dynamical mechanism.