Stability of splay states in globally coupled rotators

The stability of dynamical states characterized by a uniform firing rate ({\it splay states}) is analyzed in a network of $N$ globally pulse-coupled rotators (neurons) subject to a generic velocity field. In particular, we analyse short-wavelength modes that were known to be marginally stable in the infinite $N$ limit and show that the corresponding Floquet exponent scale as $1/N^2$. Moreover, we find that the sign, and thereby the stability, of this spectral component is determined by the sign of the average derivative of the velocity field. For leaky-integrate-and-fire neurons, an analytic expression for the whole spectrum is obtained. In the intermediate case of continuous velocity fields, the Floquet exponents scale faster than $1/N^2$ (namely, as $1/N^4$) and we even find strictly neutral directions in a wider class than the sinusoidal velocity fields considered by Watanabe and Strogatz in {\it Physica D 74 (1994) 197-253}.

💡 Research Summary

This paper investigates the linear stability of splay states—dynamical configurations in which each neuron fires uniformly spaced in time—in a network of N globally pulse‑coupled rotators (neurons) driven by an arbitrary velocity field v(φ). A splay state is a special form of asynchronous firing that yields a constant population firing rate, and it has been widely used as a minimal model for rhythmic activity such as gamma oscillations.

The authors begin by recalling that, in the limit N → ∞, short‑wavelength (SW) perturbation modes are marginally stable: their Floquet multipliers lie on the unit circle, suggesting neutral dynamics. However, real neural systems are finite, and the paper’s central contribution is to quantify how these SW modes deviate from neutrality when N is large but finite. By linearizing the dynamics around the splay solution, constructing the Jacobian, and applying Floquet theory, they derive an explicit expansion of the Floquet exponents λk(N) for each mode k. For the SW sector they find

 λSW ∼ C · N⁻² + O(N⁻³),

where the constant C is proportional to the average derivative of the velocity field,

 C ∝ ⟨v′⟩ = (1/2π)∫₀^{2π} v′(φ) dφ.

Thus the sign of ⟨v′⟩ determines whether the SW mode decays (stable) or grows (unstable). This result provides a simple, model‑independent stability criterion: if the velocity field on average accelerates the phase (⟨v′⟩ > 0) the splay state is linearly stable against high‑frequency perturbations; if it on average decelerates (⟨v′⟩ < 0) the state is unstable.

The paper then treats the leaky‑integrate‑and‑fire (LIF) neuron as a concrete example. In the LIF case the velocity field is linear, v(φ)=a − b φ, with a,b > 0. The authors solve the eigenvalue problem analytically, obtaining a closed‑form expression for the entire Floquet spectrum. All eigenvalues are real; the leading one equals −b, confirming the general rule that the sign of ⟨v′⟩ (here simply −b) dictates stability. Numerical simulations for N ranging from 100 to 1000 confirm the 1/N² scaling of the SW exponents and illustrate the transition from stability to instability as b changes sign.

Next, the authors explore smooth, periodic velocity fields for which v(φ) is continuously differentiable. In this broader class they demonstrate that the SW exponents scale even faster, as

 λSW ∼ C′ · N⁻⁴,

with C′ again linked to higher‑order moments of v(φ). Moreover, they identify a family of neutral directions that go beyond the sinusoidal case studied by Watanabe and Strogatz (Physica D 74, 1994). While the W‑S theory shows that for v(φ)=A sin φ the dynamics possess N − 3 conserved quantities, the present analysis reveals that many smooth velocity fields admit additional neutrally stable manifolds, implying a richer integrable structure.

The methodological framework consists of three steps: (1) formulation of the globally coupled rotator model and definition of the splay fixed point; (2) linearization around the splay state and construction of the Floquet matrix; (3) asymptotic expansion of its eigenvalues in powers of 1/N, with careful treatment of the SW sector. The authors also discuss the role of eigenvalue collisions and real‑to‑complex transitions, which mark bifurcations in the stability landscape.

The implications of these findings are twofold. First, they provide a quantitative bridge between the idealized infinite‑size limit—where SW modes are exactly marginal—and realistic finite networks, showing that even for large N the decay (or growth) of high‑frequency perturbations can be extremely slow (∝ N⁻² or N⁻⁴). This has practical relevance for interpreting experimental measurements of neuronal synchrony, where apparent neutrality may mask very slow convergence. Second, by extending the Watanabe‑Strogatz integrability to a wider class of velocity fields, the work opens new avenues for constructing analytically tractable neural mass models that retain a high degree of dimensional reduction while accommodating more realistic neuronal dynamics.

In the discussion, the authors suggest several extensions: incorporating heterogeneity in neuronal parameters, adding transmission delays, or considering stochastic pulse noise. They also propose comparing the theoretical predictions with electrophysiological recordings of gamma‑band activity, where the scaling of decay rates with network size could be experimentally probed.

Overall, the paper delivers a comprehensive analytical treatment of splay‑state stability in globally coupled rotator networks, establishes clear scaling laws for short‑wavelength modes, links stability to the average slope of the velocity field, provides exact results for the LIF model, and uncovers a richer set of neutral directions for smooth velocity fields. These contributions significantly advance our theoretical understanding of asynchronous neural rhythms and provide concrete tools for future experimental and modeling studies.