CVPS: Complex-Valued Phase Synchrony

• CVPS is a framework that represents oscillatory states as complex numbers, encoding amplitude and phase as distinct yet interrelated features.
• It underpins diverse methodologies—from Kuramoto dynamics and neural binding to statistical phase estimation—ensuring accurate synchrony assessment.
• Applications span neural networks, object-centric models, and biological systems, highlighting both theoretical insights and practical control strategies in synchrony.

Searching arXiv for recent and foundational papers on complex-valued phase synchrony, Kuramoto embeddings, and synchrony-based representations. Complex-Valued Phase Synchrony (CVPS) can be understood as a family of formalisms in which oscillatory state, pairwise phase relation, or neural activation is represented in the complex domain so that phase is encoded by an argument and synchrony is assessed through phase alignment, complex interference, or complex pairwise structure. The label itself is not standardized: several closely related works do not use the acronym, whereas a later fMRI framework introduces “complex-valued phase synchrony” explicitly as a method that preserves both cosine and sine components of relative phase rather than collapsing synchrony to a scalar (Wiafe et al., 16 Sep 2025). Across oscillator theory, neural computation, and signal analysis, the recurring construction is to write a state as $z = r e^{i\phi}$ or to lift real phase variables into a complex state, with magnitude and phase then assigned distinct dynamical or representational roles [(Budzinski et al., 2021); (Reichert et al., 2013); (Stanić et al., 2023)].

1. Representational core and scope

A common representational template in CVPS-adjacent work is the complex state $z_i = r_i e^{i\phi_i}$, where magnitude and phase encode different kinds of information. In complex-valued deep networks inspired by neuronal synchrony, the magnitude $r_i = |z_i|$ is interpreted as firing-rate-like amplitude and the phase $\phi_i$ as spike timing or timing of maximal activity; synchrony is then expressed by similarity among phases rather than by a separate state variable (Reichert et al., 2013). In synchrony-based object-centric models, the same decomposition is repurposed: magnitude carries feature strength or content, while phase carries binding, grouping, or object identity (Stanić et al., 2023). A related division of labor appears in object-classification models with explicit Kuramoto dynamics, where activation amplitude indicates the presence of a feature and phase tags the group or object to which that feature belongs (Muzellec et al., 28 Feb 2025), and in recurrent attention models for tracking, where magnitude captures appearance and phase captures location (Muzellec et al., 2024).

This shared structure does not imply a single agreed metric. Several papers are explicit that they do not define a formal quantity called CVPS. Instead, synchrony is operationalized through phase alignment, constructive and destructive interference, or pairwise complex phase relations. In the neuronal synchrony formulation, presynaptic inputs are summed as phasors, $\zeta = \mathbf{w}\cdot \mathbf{z}$, so similar phases add constructively and phase disagreement reduces $|\zeta|$ (Reichert et al., 2013). In the object-binding literature, same-object features are encouraged to take similar phase values, while different objects are separated in phase space; grouping is later recovered by clustering decoder phases (Stanić et al., 2023). A plausible implication is that CVPS is less a single algorithm than a design pattern: retain phase in the complex domain long enough for synchrony, desynchrony, and grouping to remain geometrically explicit.

An important conceptual restriction recurs across this literature. Complex-valued representation does not by itself guarantee a new synchrony theory. Some works use complex states but still read out synchrony with classical quantities. In the complex reformulation of Kuramoto dynamics, for example, the explicit synchrony measure remains the standard order parameter

$$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$

with $\theta_j(t)$ obtained from the arguments of the complex states rather than from a new complex synchrony index (Budzinski et al., 2021).

2. Oscillator dynamics, complex lifting, and geometric interpretation

A major strand of CVPS concerns nonlinear oscillator networks, especially the Kuramoto model. One approach begins with the identical-frequency Kuramoto model with phase lag,

$$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$

moves to a rotating frame with $\omega=0$, and introduces a modified complex system

$z_i = r_i e^{i\phi_i}$0

With the complex state $z_i = r_i e^{i\phi_i}$1, this yields the linear system

$z_i = r_i e^{i\phi_i}$2

and hence

$z_i = r_i e^{i\phi_i}$3

The phase readout is $z_i = r_i e^{i\phi_i}$4, while the amplitude $z_i = r_i e^{i\phi_i}$5 introduces dynamics absent from the original phase-only Kuramoto model (Budzinski et al., 2021). The paper is explicit that this is not an exact conjugacy: the real-part phase equation acquires effective coupling weights $z_i = r_i e^{i\phi_i}$6, so the complex system is not identical to the original Kuramoto model. The proposed remedy is a short-window propagation scheme: initialize with unit modulus, propagate analytically with $z_i = r_i e^{i\phi_i}$7, extract arguments, reset to unit modulus, and repeat. Under that procedure, the resulting argument trajectories were reported to match the original Kuramoto model across synchrony, chimera, and traveling-wave regimes, including random initial conditions (Budzinski et al., 2021).

The same work gives a geometric, mode-based interpretation of synchrony. Writing

$z_i = r_i e^{i\phi_i}$8

with modal coefficients $z_i = r_i e^{i\phi_i}$9, phase synchrony corresponds to dominance of the first eigenmode, chimera states to balanced interplay between the synchronizing mode and higher wave-like modes, and traveling waves to localization on a single higher eigenmode (Budzinski et al., 2021). In circulant graphs these modes become Fourier modes, so CVPS is not merely “complex phases” but an eigenspace geometry in $r_i = |z_i|$0.

A second oscillator-theoretic route is analytic continuation of the Kuramoto model itself. In the complexified model,

$r_i = |z_i|$1

both the state variables and the coupling $r_i = |z_i|$2 may be complex (Lee et al., 2024). The real model persists as the invariant manifold $r_i = |z_i|$3, but the complexified system admits fixed points off that manifold. These are termed complex locked states and constitute the clearest mathematical candidate for intrinsic complex-domain synchrony. The paper shows that purely imaginary coupling produces conservative dynamics with asynchronous rotations or librations and no attractive locking even as $r_i = |z_i|$4, whereas generic complex coupling yields stable complex locked states and finite but arbitrarily large rotation numbers near the imaginary axis (Lee et al., 2024). This distinguishes existence of complex fixed points from asymptotic attraction to them.

A third development turns complex lifting into a control problem. In the control-theoretic framework,

$r_i = |z_i|$5

with $r_i = |z_i|$6, the argument dynamics reduce to the classical Kuramoto phase law exactly when all moduli are equal, because the ratio $r_i = |z_i|$7 in the phase equation then becomes one (Giordano et al., 8 Apr 2026). On this basis, regulating $r_i = |z_i|$8 to a common value becomes the central CVPS objective. The paper identifies earlier continuous and reset-based constructions as modulus-control strategies and proposes two switched controllers. A switched feedforward law preserves $r_i = |z_i|$9 for all time and gives exact phase correspondence under unit-modulus initialization. A feedforward plus sliding-mode law drives arbitrary initial magnitudes to one in finite time, with bound

$\phi_i$0

and after that time the argument dynamics coincide exactly with Kuramoto (Giordano et al., 8 Apr 2026). A separate non-autonomous MIMO sliding-mode controller enforces full phase locking at a prescribed frequency in finite time, even in heterogeneous regimes where the uncontrolled real-valued Kuramoto model does not synchronize (Giordano et al., 8 Apr 2026). This shifts CVPS from descriptive geometry to constructive control.

3. Phase synchronization as statistical estimation

A more classical mathematical usage of complex-valued phase synchrony concerns recovery of unit-modulus phases from noisy pairwise measurements. In the dense canonical model,

$\phi_i$1

the unknown state is $\phi_i$2 with $\phi_i$3, and the noiseless pairwise relation $\phi_i$4 encodes phase difference directly in the complex plane (Zhong et al., 2017). The maximum-likelihood estimator solves

$\phi_i$5

which is nonconvex but structurally natural: synchrony means recovering a globally consistent unit-modulus phase vector up to global phase (Zhong et al., 2017). The paper proves that an SDP relaxation is tight and that the generalized power method $\phi_i$6 converges linearly to the global optimum when

$\phi_i$7

with high-probability $\phi_i$8 and $\phi_i$9 error bounds for both the optimizer and the spectral initializer (Zhong et al., 2017). In this tradition, CVPS is a unit-circle estimation problem rather than a state-space dynamical embedding.

Multi-frequency phase synchronization extends this estimator by using higher harmonics $\zeta = \mathbf{w}\cdot \mathbf{z}$0, $\zeta = \mathbf{w}\cdot \mathbf{z}$1, instead of only first-order phase differences (Gao et al., 2019). The proposed nonconvex objective,

$\zeta = \mathbf{w}\cdot \mathbf{z}$2

enforces consistency across multiple frequency channels generated by the same underlying phases (Gao et al., 2019). The paper develops a two-stage procedure—Periodogram Peak Extraction with Spectral Methods and the Multi-Frequency Generalized Power Method—that first estimates each harmonic spectrally and then fuses them through harmonic retrieval. Under the stated sub-Gaussian model, pairwise phase error scales as $\zeta = \mathbf{w}\cdot \mathbf{z}$3 and final correlation loss as $\zeta = \mathbf{w}\cdot \mathbf{z}$4 (Gao et al., 2019). A plausible implication is that, in estimation settings, CVPS can benefit from treating phase relations as a hierarchy of consistent complex moments rather than as a single-frequency object.

These statistical formulations clarify a central distinction. In dynamical CVPS, complex states often carry both amplitude and phase, and synchrony emerges through evolution. In estimation-theoretic phase synchronization, amplitudes are fixed to one and the task is to infer phases from a noisy complex relation matrix. The two lines are mathematically adjacent but operationally different.

4. Neural synchrony, binding, and object-centric representation

In neural-network research, complex-valued phase synchrony has been developed primarily as a mechanism for binding and selective communication. An early formulation replaces a real-valued neuron state by a complex state $\zeta = \mathbf{w}\cdot \mathbf{z}$5, computes complex input $\zeta = \mathbf{w}\cdot \mathbf{z}$6, sets output phase to $\zeta = \mathbf{w}\cdot \mathbf{z}$7, and uses a mixed magnitude rule

$\zeta = \mathbf{w}\cdot \mathbf{z}$8

The synchrony term $\zeta = \mathbf{w}\cdot \mathbf{z}$9 is phase-sensitive, the classic term $|\zeta|$0 is phase-blind, and the combined rule is meant to avoid pathological cancellation and instability from negative weights (Reichert et al., 2013). Within this framework, synchrony serves two functions: binding distributed features into soft assemblies and dynamically gating communication between phase-matched and phase-mismatched groups. The paper’s evidence is qualitative rather than benchmark-oriented, but bars, corners, shapes, and MNIST-like mixtures show phase clusters corresponding to objects or object parts (Reichert et al., 2013).

Object-centric autoencoders later made phase-coded binding more explicit. In CAE-family models, complex activations “store and process information about object instances in the phases of complex activations,” while constructive and destructive interference “pressurizes the network to use similar phase values for all patches belonging to the same object while separating those associated with different objects” (Stanić et al., 2023). Operationally, a CAE layer applies the same real-valued map to real and imaginary parts, decomposes the result into magnitude and phase, adds separate biases, preserves phase, and applies nonlinear processing only to magnitude so that “phase flips are prevented” (Stanić et al., 2023). Grouping is obtained by K-means clustering on decoder phases, after masking low-magnitude pixels and mapping phases to the unit circle. The contrastive variant CtCAE uses magnitudes as “addresses” and phases as “features,” with an InfoNCE-like loss shaping phase geometry so that visually similar regions become phase-similar and dissimilar regions become phase-separated (Stanić et al., 2023). The paper reports that this was the first synchrony-based object-discovery model to work on multi-object color datasets and to represent more than three objects simultaneously, although it also notes that synchrony-based models remain below Slot Attention on CLEVR and still rely on K-means with the ground-truth number of clusters (Stanić et al., 2023).

KomplexNet makes the synchrony mechanism explicit rather than emergent. Each activation is represented as $|\zeta|$1, a Kuramoto-style phase dynamics is applied to first-layer phases,

$|\zeta|$2

and a cluster synchrony loss regularizes intra-cluster synchrony and inter-cluster desynchrony (Muzellec et al., 28 Feb 2025). Same-object units are encouraged to align in phase, different objects to occupy distinct phase clusters, and the synchronized phase structure is then propagated through complex-valued layers. Feedforward and feedback versions outperformed real-valued baselines and complex-valued models with random phases on overlapping digits, noise, and out-of-distribution object counts (Muzellec et al., 28 Feb 2025). Unlike the CAE line, this framework uses explicit Kuramoto recurrence and synchrony supervision.

A recurrent attention variant, the CV-RNN, uses synchrony to preserve location under appearance change. Its central claim is that “the magnitude of neurons captures object appearances, and the phase captures object locations,” while the recurrent state learns to tag the target with a phase value distinct from distractors (Muzellec et al., 2024). The complex attention module evolves a recurrent hidden state $|\zeta|$3, extracts a phase map $|\zeta|$4, and trains with a synchrony loss defined over three phase groups—target, distractors, and background—using within-group circular variance and an inter-group spreading term (Muzellec et al., 2024). The model behaved similarly to humans on the FeatureTracker benchmark, and ablations showed that removing the synchrony loss or randomizing phases across time destroyed reliable tracking (Muzellec et al., 2024). The paper is explicit, however, that this remains a computational proof-of-concept rather than a claim that synchrony is the only explanation of human behavior.

A neighboring but not fully complex-valued development is phase synchrony component self-organization for EEG. This framework learns data-dependent spatial filters whose paired narrowband outputs exhibit discriminative synchrony patterns, but it avoids direct complex phase computation inside the network by introducing a real-valued phase-to-amplitude transcoder (Niu et al., 2023). Its strongest CVPS-relevant lesson is architectural rather than representational: learned source-like components can expose far stronger synchrony than manual channel selection. The paper reports an average PLV of $|\zeta|$5 for a learned tongue-motor-imagery component pair in the $|\zeta|$6–$|\zeta|$7 Hz band, with lower quartile $|\zeta|$8, median $|\zeta|$9, and upper quartile $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$0 (Niu et al., 2023). Yet the same paper is explicit that it is not a genuinely complex-valued synchrony model.

5. Signal-derived synchrony measures, warping, and directional coupling

Another branch of CVPS begins not from a dynamical model or neural architecture but from complex analytic signals derived from measured time series. For a real signal $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$1, the analytic signal

$$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$2

yields instantaneous amplitude $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$3 and phase $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$4, and standard phase locking value is

$$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$5

Warped phase coherence modifies this by translating the analytic signal in the complex plane before taking the angle,

$$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$6

and then defining

$$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$7

This construction makes the resulting synchrony statistic sensitive to amplitude fluctuations and average phase offset, because the warped angle depends on both $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$8 and $$R(t)=\frac{1}{N}\left|\sum_{j=1}^{N}e^{i\theta_j(t)}\right|,$$9 (Minati et al., 2019). The paper is explicit that the measure is no longer “pure phase only,” and equally explicit about its limits: when $\theta_j(t)$0, the relation between $\theta_j(t)$1 and $\theta_j(t)$2 may be non-monotonic, so the warped angle can no longer be considered a proper phase; the measure then has empirical rather than strict phase-synchrony status (Minati et al., 2019). Within those limits, the method improved inference of structural couplings in oscillator data and improved EEG motor-imagery classification.

An explicit CVPS framework for fMRI takes the preservation of complex phase information one step further. Phases are estimated with a complex Gabor wavelet,

$\theta_j(t)$3

using $\theta_j(t)$4 Hz and $\theta_j(t)$5 Hz in the reported experiments (Wiafe et al., 16 Sep 2025). Pairwise complex-valued phase synchrony is then defined as

$\theta_j(t)$6

Its real part recovers cosine synchrony, while its imaginary part preserves the sign of the lag and therefore directional lead-lag information (Wiafe et al., 16 Sep 2025). In the paper’s simulation, cosine-only synchrony could not distinguish $\theta_j(t)$7 from $\theta_j(t)$8 phase offsets ($\theta_j(t)$9), whereas the angle of the complex synchrony vector separated them with $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$0 (Wiafe et al., 16 Sep 2025). In a non-stationary phase-estimation simulation, the Gabor-wavelet phase estimate achieved RMSE $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$1 versus $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$2 for a Hilbert-based method (Wiafe et al., 16 Sep 2025). Applied to resting-state fMRI, the mean dwell time of one dynamic state was positively associated with chlorpromazine-equivalent dose, $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$3, $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$4, whereas cosine-only relative phase and sliding-window Pearson correlation baselines were not significant (Wiafe et al., 16 Sep 2025). The paper interprets the resulting directional structure as occipital-to-parietal and prefrontal-to-striatal or thalamic lead-lag flow, while also cautioning that fMRI phase offsets reflect integrated neuro-hemodynamic timing rather than pure neural conduction delays (Wiafe et al., 16 Sep 2025).

A recurring misconception is that a scalar synchrony magnitude is sufficient whenever phases are available. The directional fMRI formulation is an explicit rebuttal: reducing $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$5 to $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$6 or to a magnitude destroys the sign of the lag, because $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$7 (Wiafe et al., 16 Sep 2025). Conversely, the warped-coherence paper shows that keeping more complex-domain structure is not automatically desirable either, because strong warping can move the measure away from representing phase locking at all (Minati et al., 2019). CVPS is therefore not synonymous with “more complex features”; it is a question of which complex information is preserved and for what purpose.

6. Spatially extended biological systems and field-based synchrony

In spatially extended active matter, CVPS takes the form of a complex phase field rather than a finite-dimensional vector or pairwise relation matrix. In confluent pulsatile epithelia, the experimentally observed oscillatory quantity is the divergence of the velocity field, $$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$8, and the continuum theory introduces the explicit complex order parameter

$$\dot{\theta}_i=\omega_i+\epsilon\sum_{j=1}^{N}A_{ij}\sin(\theta_j-\theta_i-\phi),$$9

The reduced density-coupled dynamics are

$\omega=0$0

$\omega=0$1

$\omega=0$2

so local density adapts to the phase pattern and feeds back onto both phase and amplitude (Tang et al., 22 Jul 2025). Experimentally, local phase is extracted from the detrended divergence signal by a quarter-period delay embedding,

$\omega=0$3

and temporal synchrony is quantified through the autocorrelation of the phase field, with persistence time $\omega=0$4 defined by the first zero crossing (Tang et al., 22 Jul 2025). The paper finds that synchrony increases with cell density, peaks at intermediate density, and is lost at higher density; the same non-monotonic trend appears in spatial correlation length, while defect density varies oppositely (Tang et al., 22 Jul 2025). Extending the analysis to breast cancer cell lines, more malignant cells show longer phase persistence and fewer topological defects (Tang et al., 22 Jul 2025). In this setting, CVPS is not merely a pairwise synchrony calculation but a continuum complex field whose coherence is shaped by gradients, density coupling, and defect topology.

A biologically motivated but methodologically more conventional application is GnRH-induced neuronal phase synchrony. There the analytic signal

$\omega=0$5

is obtained from the Hilbert transform, and synchrony is assessed through constancy of the phase difference $\omega=0$6, time evolution of phase differences, and recurrence plots (Alam, 2016). The paper reports phase-transition-like behavior separating synchronized and desynchronized regimes in a mean-field coupled population of GnRH-secreting neurons (Alam, 2016). This is not an explicitly complex-valued synchrony formalism in the stronger sense of complex state-space dynamics or complex neural representation, but it illustrates the analytic-signal route by which many biological applications enter CVPS-like analysis.

Taken together, these biological examples show two distinct scales of complex-valued phase synchrony. At the smaller scale, CVPS can mean extracting phase from single time series and assessing pairwise or population locking. At the larger scale, it can mean treating the system itself as a complex oscillatory field and studying persistence, gradients, and defects as the geometry of synchrony. The latter view suggests that, in spatial systems, coherent phase organization may be more faithfully characterized by correlation length and defect statistics than by any single global synchrony scalar.