Circular Cumulants in Phase Oscillators

• Circular cumulants are defined as normalized cumulants of e^(iϕ), providing a concise description of phase oscillator distributions and their deviation from the Ott–Antonsen (OA) manifold.
• They enable perturbative expansions in regimes with weak intrinsic noise or near-OA states, linking theoretical models with experimental observations in neural and oscillator systems.
• Unlike ordinary cumulants, finite truncation beyond the OA case is unphysical, necessitating a hierarchical infinite series approach to accurately capture synchronization dynamics.

Circular cumulants are cumulant coordinates for probability measures on the circle, defined from the Fourier or Kuramoto–Daido order parameters $Z_m=\langle e^{im\varphi}\rangle$ or $a_j=\langle e^{ij\varphi}\rangle$, rather than from moments of a real-valued random variable. In the phase-oscillator literature, they provide a compact description of departures from the Ott–Antonsen (OA) manifold and a perturbative language for partially synchronized ensembles, especially when direct Fourier-mode descriptions become inconvenient in strongly synchronized regimes (Goldobin, 2018, Goldobin et al., 2019). A central point is that these are cumulants of the complex quantity $e^{i\varphi}$, not cumulants of the phase $\varphi$ itself (Goldobin et al., 2019).

1. Definition and normalization conventions

For an ensemble of phases $\varphi_k$, the basic Fourier moments are

$a_j=\langle e^{ij\varphi}\rangle .$

One formulation introduces the generating functions

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$

and then defines the rescaled cumulants

$\varkappa_j \equiv \frac{K_j}{(j-1)!}.$

These $\varkappa_j$ are called the circular cumulants in that normalization (Goldobin, 2018).

A second formulation starts from the Kuramoto–Daido order parameters

$a_j=\langle e^{ij\varphi}\rangle$0

and defines

$a_j=\langle e^{ij\varphi}\rangle$1

together with

$a_j=\langle e^{ij\varphi}\rangle$2

In this notation, the first cumulants are

$a_j=\langle e^{ij\varphi}\rangle$3

The literature summarized here therefore uses two closely related normalizations, one based on $a_j=\langle e^{ij\varphi}\rangle$4 and one based on $a_j=\langle e^{ij\varphi}\rangle$5 (Goldobin et al., 2019).

The formal analogy with ordinary cumulants is explicit, but the object is different. Circular cumulants are not cumulants of the phase $a_j=\langle e^{ij\varphi}\rangle$6 itself; they are cumulants of $a_j=\langle e^{ij\varphi}\rangle$7 and therefore inherit the boundedness constraints characteristic of circular variables (Goldobin et al., 2019).

2. Relation to Ott–Antonsen dynamics

For globally sinusoidally coupled phase oscillators,

$a_j=\langle e^{ij\varphi}\rangle$8

the phase density obeys

$a_j=\langle e^{ij\varphi}\rangle$9

and in Fourier space this yields

$e^{i\varphi}$0

In the thermodynamic-limit notation of the Watanabe–Strogatz/Ott–Antonsen analysis, the same hierarchy is written as

$e^{i\varphi}$1

Ott and Antonsen found the special solution

$e^{i\varphi}$2

equivalently

$e^{i\varphi}$3

which defines the OA manifold (Goldobin, 2018, Goldobin et al., 2019).

In circular-cumulant language, the OA manifold is characterized by the vanishing of all higher cumulants: $e^{i\varphi}$4 or, in the $e^{i\varphi}$5-notation,

$e^{i\varphi}$6

Thus higher circular cumulants quantify the “non-OA-ness” of the distribution (Goldobin, 2018).

The corresponding OA phase distribution is the wrapped Cauchy distribution,

$e^{i\varphi}$7

This makes the OA manifold the unique one-cumulant closure in the circular setting (Goldobin et al., 2019).

A common misconception is that the Fourier hierarchy itself is the most convenient perturbative description near OA. The circular-cumulant approach was introduced precisely because, for near-OA states, higher cumulants are small and often form a hierarchy, whereas the direct series $e^{i\varphi}$8 can converge poorly as $e^{i\varphi}$9 (Goldobin, 2018).

3. Watanabe–Strogatz variables and the circular-cumulant representation

The Watanabe–Strogatz (WS) theory introduces auxiliary phases $\varphi$0 and a complex parameter $\varphi$1, together with the constraint

$\varphi$2

If

$\varphi$3

denote the Fourier amplitudes of the WS distribution $\varphi$4, then the relation between WS variables and circular cumulants can be written as

$\varphi$5

where

$\varphi$6

The coefficients $\varphi$7 are polynomial combinations of the cumulants with the first cumulant removed: $\varphi$8 This gives an explicit map from the circular cumulants of the original phases to the Fourier content of the WS-variable distribution (Goldobin, 2018).

The WS Möbius parameter $\varphi$9 is fixed by the WS constraint $\varphi_k$0, yielding

$\varphi_k$1

An iterative scheme begins with

$\varphi_k$2

In general, $\varphi_k$3 is not the order parameter, and extracting $\varphi_k$4 from $\varphi_k$5 and $\varphi_k$6 is nontrivial; the OA manifold is the special WS case

$\varphi_k$7

This identifies OA as the uniform-WS distribution (Goldobin, 2018).

The hierarchy of circular cumulants has a direct WS interpretation. For

$\varphi_k$8

the WS Fourier modes satisfy

$\varphi_k$9

whereas

$a_j=\langle e^{ij\varphi}\rangle .$0

The paper also states that, when such a hierarchy exists,

$a_j=\langle e^{ij\varphi}\rangle .$1

This gives a precise meaning to the statement that near-OA states correspond to weakly nonuniform WS distributions (Goldobin, 2018).

4. Finite truncation and the uniqueness of the OA closure

A central structural theorem is that, unlike cumulants for real-valued variables, circular cumulants do not admit nontrivial exact finite truncations beyond the OA case. A truncation means assuming

$a_j=\langle e^{ij\varphi}\rangle .$2

and reconstructing the full sequence $a_j=\langle e^{ij\varphi}\rangle .$3 from the remaining finitely many cumulants. In that situation the generating function becomes

$a_j=\langle e^{ij\varphi}\rangle .$4

For two cumulants one obtains

$a_j=\langle e^{ij\varphi}\rangle .$5

The asymptotic analysis in the phase-oscillator setting shows that if $a_j=\langle e^{ij\varphi}\rangle .$6, then for sufficiently large $a_j=\langle e^{ij\varphi}\rangle .$7, $a_j=\langle e^{ij\varphi}\rangle .$8; more generally, if $a_j=\langle e^{ij\varphi}\rangle .$9 and higher cumulants vanish, then $$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$0 for large enough $$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$1 (Goldobin et al., 2019).

Since

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$2

such finite truncations are unphysical. Therefore the only admissible finite truncation is the single-cumulant case, namely OA. This is the exact circular analogue of the statement that the wrapped Cauchy/OA manifold is the only nontrivial exact finite closure (Goldobin et al., 2019).

The contrast with the linear case is decisive. For a real-valued variable $$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$3, one may have

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$4

which yields the Gaussian distribution. That closure is admissible because moments on the line need not remain bounded by $$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$5. For circular variables, the boundedness of the order parameters destroys all finite truncations beyond OA (Goldobin et al., 2019).

This resolves a frequent misunderstanding generated by the formal resemblance between ordinary and circular cumulants: the circular cumulant series may be asymptotically truncatable, but, except on the OA manifold, it is not exactly finitely truncatable (Goldobin et al., 2019).

5. Hierarchical regimes, asymptotic reductions, and applications

The physically relevant alternative to exact truncation is a hierarchical infinite cumulant series. The circular-cumulant literature repeatedly identifies the regime

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$6

or, in the alternative notation,

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$7

Such hierarchies arise in weak intrinsic noise, perturbed OA solutions, wrapped Gaussian phase distributions, and noisy Kuramoto ensembles (Goldobin, 2018, Goldobin et al., 2019).

For the standard phase dynamics,

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$8

the circular cumulants satisfy

$$F(\zeta)\equiv \left\langle \exp(\zeta e^{i\varphi_k})\right\rangle = \sum_{j=0}^\infty a_j \frac{\zeta^j}{j!},$$9

With intrinsic noise $$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$0, this becomes

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$1

A canonical example is intrinsic white noise, which yields

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$2

This makes higher cumulants small in a controlled way and supports perturbation theory around OA (Goldobin, 2018).

In this regime one uses asymptotic expansions rather than fictitious exact finite closures. For the macroscopic observable $$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$3, the first two asymptotic orders are

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$4

and

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$5

The correct procedure is therefore to truncate the asymptotic expansion at the desired order in $$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$6, not to truncate the exact cumulant series (Goldobin et al., 2019).

An important application is the quadratic integrate-and-fire (QIF) neuron population,

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$7

with the phase transform

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$8

Here

$$\ln F(\zeta)=\sum_{j=1}^\infty K_j\frac{\zeta^j}{j!},$$9

so that

$\varkappa_j \equiv \frac{K_j}{(j-1)!}.$0

On the OA manifold,

$\varkappa_j \equiv \frac{K_j}{(j-1)!}.$1

But for any finite truncation beyond OA, the exact series for $\varkappa_j \equiv \frac{K_j}{(j-1)!}.$2 diverges; in the two-cumulant truncation one finds

$\varkappa_j \equiv \frac{K_j}{(j-1)!}.$3

which diverges for any nonzero $\varkappa_j \equiv \frac{K_j}{(j-1)!}.$4 (Goldobin et al., 2019).

The hierarchical regime is also supported by model distributions and data. Wrapped Gaussian, von Mises, and a wrapped heavy-tailed non-Cauchy distribution all exhibit rapidly decaying, approximately geometric cumulant progressions. Experimental phase data from single-cell mammalian suprachiasmatic nucleus (SCN) oscillators and from coupled electrochemical oscillators show that the first 15 circular cumulants decay rapidly and approximately geometrically; during regime transitions, the progression multiplier changes and the disturbance propagates to higher cumulants (Goldobin et al., 2019).

6. Broader cumulant frameworks and terminological scope

Circular cumulants of phase oscillators sit within a broader landscape of cumulant calculi, but they are not the only object that has been described as “circular” or “cyclic.” In non-commutative probability, a general framework defines cumulants as convolution logarithms in graded Hopf algebras, with moment–cumulant relations encoded by polynomial generating maps. In that setting, symmetric independences yield additive cumulants, whereas non-symmetric independences produce Campbell–Baker–Hausdorff corrections (Manzel et al., 2016). The same source explicitly states that it does not discuss circular cumulants specifically, although its cumulant framework is broad enough to encompass additive cumulants and their Lie-algebraic deformation behavior (Manzel et al., 2016).

A separate usage appears in cyclic-conditional freeness. There, the “circular” or “cyclic” cumulants are the cumulants adapted to cyclic-conditional additive convolution. They are constructed by first building a cyclic companion functional via a multivariate inverse Markov–Krein transform, then defining cumulants for cyclic freeness, and finally pulling these back to the cyclic-conditional setting. The resulting cumulants are characterized by vanishing of mixed cumulants and are combinatorially controlled by noncrossing partitions, cyclic partitions, Kreweras complements, and type-$\varkappa_j \equiv \frac{K_j}{(j-1)!}.$5 structures (Arizmendi et al., 2023).

This suggests a terminological overlap rather than a single universal notion. In the phase-oscillator literature, circular cumulants are coordinates for distributions of $\varkappa_j \equiv \frac{K_j}{(j-1)!}.$6 and measure deviations from OA; in cyclic-conditional freeness, they are cumulants adapted to a non-commutative independence structure (Goldobin, 2018, Arizmendi et al., 2023). What is shared across these settings is the cumulant principle itself: moment data are reorganized into coordinates in which the relevant independence, reduction, or perturbative structure becomes sparse or additive.