{"id":"b3cf05b6-9a58-407f-9431-61278d4489e5","arxiv_id":"2411.15534","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stochastic phase resetting is added to the Kuramoto model with indirect coupling, yielding a four-dimensional Ott-Antonsen system with subsystem resetting that can induce noise-free transitions and slow/fast synchronization.","lead":"This paper adds random phase resets to a Kuramoto model where oscillators couple through a shared external medium, reducing the infinite system to four equations in which only the oscillator variables reset. It shows that this subsystem resetting can switch a bistable population between synchronized states and produces slow/fast behavior at low cell density.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"OA reduction under resetting is asserted, not proven: Eq. (4.7) omits the mode index n, and with the correct factor the resetting term does not close; the reduced system is only justified for Poisson-kernel reset densities, unverified by full-N simulations.","rationale":"I agree with the reader that the validity of the OA ansatz under resetting is the weakest link. The paper's derivation of the reduced equations is not just incomplete—it is incorrect as written: Eq. (4.7) is missing the factor n on the left, which follows from the standard Fourier expansion of the continuity equation. With the correct factor, the resetting source term h(t)(η0^n−η^n) cannot be reconciled with a single Riccati equation for η for all n; the reset must be treated as a jump, and the OA manifold is preserved only if the reset density itself is on the manifold. The paper explicitly restricts to ρ0 with a single real coefficient r0 independent of ω (the Poisson kernel), which is a very special class of initial phase distributions. The abstract and introduction, however, advertise the result as applying to 'stochastic phase resetting' in general. If a non-OA reset distribution is used, the four-dimensional system (4.15) is not a valid reduction, and the claimed noise-induced transitions and slow/fast dynamics may be artifacts of the ansatz. This is load-bearing because every subsequent result—subsystem-resetting-induced switching, slow/fast analysis, high-density recovery—is derived on this manifold. A direct N-particle simulation for the same parameters would either validate the reduction for the Poisson-kernel case and bound its failure for non-OA cases, settling the concern. Since the reader already flagged this assumption and assigned CONDITIONAL, my assessment leaves the verdict unchanged.","tokens_in":101,"tokens_out":29480,"duration_ms":436999,"concrete_test":"Run a direct N-particle simulation of Eqs. (2.3) with global resetting (θj→θ0,j at Poisson times) for the parameters of Fig. 5 (ω0−Ω=10, Δ=0.1, K=20, λ=0.5, σ0=0.05) and N=10^4, with initial phases drawn from the Poisson kernel p0(θ)=(1−r0²)/(2π(1−2r0 cosθ+r0²)) with r0=0.8, and Z(0)=0.8. Compare the empirical order parameter r(t),R(t) averaged over many realizations to the stochastic OA system (4.15) driven by the same reset times. Repeat with a non-OA initial/reset distribution (e.g., wrapped Gaussian with the same mean and variance as p0); if the reduced system fails to track the full model, the OA restriction is load-bearing and the central claim is limited to the Poisson-kernel reset protocol.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the reduction of the infinite-dimensional density to the four-dimensional OA system (4.15) under global resetting. The paper asserts this reduction holds 'provided that the set of initial phases also lies on the OA manifold' (Sect. 4), but it does not actually demonstrate the OA closure with resetting. The substitution step is wrong as written: Eq. (4.7) omits the mode index n on the advective term; the correct Fourier equation is n η^(n-1)[∂η/∂t + iωη + K/2(Zη²−Z*)] = h(t)(η0^n − η^n). With this factor, the resetting source cannot be absorbed into a single closed equation for η; the reset must be treated as an instantaneous jump ρ(θ,ω,T+)=ρ0(θ,ω). The paper does treat it as a jump in (4.9), but the stated derivation is therefore misleading. More importantly, the jump preserves the OA manifold only if ρ0 is exactly of OA form, i.e., the initial phase distribution is the Poisson kernel (4.6) with a single real coefficient r0 independent of ω. The physically motivated protocol 'reset to initial values' does not generally produce such a distribution. If the initial/reset phase distribution is not of this form, the 4D reduced system is not justified, and the noise-induced transitions and slow/fast dynamics of Figs. 5–8 could be artifacts of the OA ansatz. No full-N simulations are provided to validate the reduction in the presence of resetting, even for the OA-consistent initial conditions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper extends the Kuramoto model with indirect coupling through an external medium to include global stochastic phase resetting, where the oscillators' phases reset simultaneously to their initial values at Poisson-distributed times while the external medium state Z(t) does not reset. The authors derive a stochastic continuity equation for the population density (Section 3), show that naive averaging fails to close, and then apply an Ott-Antonsen (OA) ansatz to reduce the infinite-dimensional description to a four-dimensional piecewise-deterministic system with subsystem resetting (Section 4). They use this reduced system to study the role of the cell density parameter σ0: in the high-density limit they recover the known OA dynamics of the classical Kuramoto model with global resetting, while in the low-density limit they report noise-induced transitions between coexisting states (Figures 5 and 6) and develop a slow/fast analysis with a nonequilibrium stationary state for the fast order parameter and a self-consistency condition for the slow amplitude (Section 5, Figure 8). The central claim is that subsystem resetting alone provides a mechanism for switching between basins of attraction without an external noise source.","tokens_in":16657,"tokens_out":11073,"duration_ms":90551,"significance":"If the OA reduction is valid under resetting, the paper offers a tractable low-dimensional description of a biologically motivated oscillator model with a non-trivial interplay between stochastic resetting and indirect coupling. The slow/fast analysis in Section 5 is carefully constructed, and the self-consistency check in Figure 8 provides nontrivial support for the averaging approximation. The paper explicitly recovers prior known results in the high-density limit, which serves as a useful consistency check. The main limitation is that the entire phenomenological picture — including the reported noise-induced transitions and slow/fast dynamics — rests on the OA ansatz being preserved by the resetting protocol, and this is asserted rather than demonstrated. No full-N simulations are reported to validate the reduction against the original finite-oscillator model, which is a significant gap given that the reset distribution in the original protocol is not generally of OA form.","major_comments":[{"comment":"The Fourier projection of the continuity equation is not correct as stated. The advective term in the Fourier-transformed equation carries a factor n, so the correct equation is n η^(n-1)(∂η/∂t + iωη + K/2(Zη²−Z*)) = h(t)(η0^n − η^n). This factor is essential: when it is included, the resetting source cannot be collapsed into a single closed ODE for η; instead, the reset must be described as an instantaneous jump of the density to ρ0. The paper does use the jump condition in (4.9), but the derivation as presented via (4.7) is therefore misleading and should be rewritten to state explicitly that between resets η satisfies (4.8) and at a reset η^n is set to η0^n so that η(T)=r0.","section":"Section 4, Eq. (4.7)"},{"comment":"The reduction to the four-dimensional system (4.15) is valid only when the reset density ρ0 is exactly on the OA manifold, i.e., ρ0(θ,ω)=g(ω)p0(θ) with p0 the Poisson kernel (4.6) and r0 real and independent of ω. The reset protocol defined in Eq. (2.8a), resetting each phase to its own initial value, does not guarantee such a distribution; in the continuum limit ρ0 is the joint empirical distribution of initial phases and frequencies, which in general is not of OA form. Since the central results — the noise-induced transitions in Figures 5 and 6 and the slow/fast behavior in Figures 7 and 8 — are obtained exclusively from the reduced system, the paper needs to either restrict its claims to the OA-form reset distribution and state this limitation prominently, or provide full-N simulations of the original model to confirm that the reduced dynamics faithfully reproduce the full system. No such simulations are reported.","section":"Section 4, Eqs. (4.4)–(4.9)"},{"comment":"The slow/fast reduction replaces Eq. (4.17b) by the averaged equation (5.20) and determines the asymptotic amplitude R* via the self-consistency condition (5.22). The numerical validation in Figure 8 covers only two values of λ for a single parameter set. Given that the slow/fast analysis is one of the paper's main contributions, the paper should provide a more systematic validation, for example by varying σ0, λ, r0, and the initial value of R, and should discuss the conditions under which the averaging approximation is expected to hold. As written, the claim rests on a small number of simulation runs, even though the qualitative findings appear plausible.","section":"Section 5.1, Eqs. (5.20)–(5.22)"}],"minor_comments":[{"comment":"There is a typo: the right-hand side reads \"− +i[ω0 − Ω]Z(t)\"; the plus sign should be removed.","section":"Eq. (2.3b)"},{"comment":"The phrase \"our analysis ts to project\" should read \"our analysis is to project\".","section":"Section 4, opening sentence"},{"comment":"The word \"coexsist\" should be \"coexist\".","section":"Section 4, after Eq. (4.17)"},{"comment":"The caption contains \"Eqs. Eqs. (4.14) and (4.15a-d)\"; the duplicated \"Eqs.\" should be removed.","section":"Figure 6 caption"},{"comment":"The term \"noise-induced transition\" is used for transitions caused by the Poisson resetting process. Since resetting is a jump process rather than a continuous noise source, the nomenclature should be clarified in the text.","section":"General terminology"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cond-mat.stat-mech and addresses a topic of current interest. The main risk is the correctness and scope of the OA reduction under resetting; the missing factor in Eq. (4.7) needs correction, and the absence of any full-N simulation leaves the central claims unvalidated. These issues are fixable within the scope of a revision. I do not see problems with citation or novelty beyond what is discussed in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on Bressloff's paper. It's a competent extension of the stochastic-resetting Kuramoto program to the indirectly coupled case. The genuinely new pieces are the four-dimensional OA reduction with subsystem resetting and the slow/fast NESS analysis in the low-density limit. The self-consistency check in Fig. 8 supports the averaging, and the high-density limit recovers the known direct-coupling results, which is a good sign. The proposed mechanism—resetting only the oscillator variables can switch a deterministic multistable system between basins—is plausible and is the kind of thing people will want to explore.\n\nThe soft spots are real but not fatal. First, Eq. (4.7) is not correct as written: the left side is missing the factor n in the Fourier expansion, and the resetting term cannot be absorbed into the single eta equation. The paper treats the reset as an instantaneous jump, which is the right way to go, but the stated derivation is misleading and should be fixed. Second, the OA invariance under resetting is assumed, not proven. The physical protocol 'reset to initial values' produces an empirical distribution that is generally not a Poisson kernel; the reduction only holds if the reset distribution lies on the OA manifold. This is stated explicitly, but it's a strong restriction, and there are no full-N simulations to check whether the reduced system faithfully represents the oscillator population. The noise-induced transitions in Figs. 5 and 6 are shown as single trajectories, so we have no statistics to back the claim of irreversibility or transition rates.\n\nNone of this torpedoes the paper. The mathematics between resets is standard, the slow/fast analysis is internally consistent, and the author is transparent about the OA assumption. But the central claim—that subsystem resetting induces transitions in the full model—rests on that assumption, so the paper is conditional until verified.\n\nWho is this for? Researchers working on stochastic resetting in oscillator networks, and to a lesser extent on quorum-sensing models. It deserves a serious referee. My recommendation: send it out, but ask for a corrected derivation (or an explicit jump-based statement), full-N simulations for at least the OA-consistent reset distribution, and some ensemble statistics for the transitions. With those, it would be a solid contribution.","headline":"A solid OA-based extension of resetting to indirectly coupled Kuramoto oscillators, but the OA invariance under resetting is assumed rather than tested, and Eq. (4.7) needs fixing.","tokens_in":17196,"tokens_out":4620,"would_cite":true,"duration_ms":40945,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","60G55","92B25"],"pacs":["05.45.Xt","05.40.-a"],"model":"deepseek-v4-flash","headline":"Global phase resetting in an indirectly coupled Kuramoto model reduces to a four-dimensional system in which only the oscillator variables reset, and this subsystem resetting can flip a bistable population between synchronized states even…","keywords":["Kuramoto model","stochastic resetting","external medium","quorum sensing","Ott-Antonsen ansatz","subsystem resetting","slow-fast dynamics","noise-induced transition"],"falsifier":"Simulate the full phase-oscillator model with many oscillators and global phase resetting, then check whether the second and higher circular moments of the phase distribution stay locked to the powers of a single complex number predicted by the reduced system; if they drift away after resets, the four-dimensional equations do not represent the population and the predicted transitions are artifacts.","tokens_in":16127,"feed_emoji":"🔄","tokens_out":8900,"duration_ms":69960,"temperature":0.7,"pith_summary":"The paper asks what happens when a population of phase oscillators that interact indirectly through a common external medium is subjected to global stochastic phase resetting, with the medium itself left untouched. It derives the continuity equation for the population density under such resetting and shows that the density equation is itself stochastically reset, so the mean-field average does not close. Projecting onto the Ott-Antonsen manifold yields a four-dimensional piecewise-deterministic system in which only the oscillator order parameter $r$ and average phase $\\phi$ reset, while the medium amplitude $R$ and phase $\\Phi$ continue smoothly. The paper then uses this reduced system to show that at high cell densities one recovers the classical Kuramoto model with global resetting, while at low densities subsystem resetting strongly affects synchronization, enabling noise-induced transitions in bistable regimes and slow/fast dynamics. A sympathetic reader would care because the model is aimed at bacterial quorum-sensing networks and other indirectly coupled systems, where resetting only the cells' phases could act as a control that switches collective behavior.","feed_headline":"Resetting only oscillator phases can flip multistable populations","feed_subtitle":"After each reset the system starts from a fresh basin of attraction, so switching needs no extra noise.","key_machinery":"The load-bearing object is the Ott-Antonsen ansatz for the oscillator phase density, which assumes all Fourier coefficients in the angle expansion are powers $\\eta^n$ of a single complex function $\\eta(\\omega, t)$; combined with a Lorentzian natural-frequency distribution, the contour integral collapses the infinite hierarchy to closed equations for $z(t)$ and $Z(t)$, and then to the four real variables $r, R, \\phi, \\Phi$. The resetting enters as a jump condition on $\\eta$ (equivalently on $z = r e^{i\\phi}$) at Poisson times, which is why only half the coordinates reset. For the frequency-locked case $\\omega_0 = \\Omega$ the equations reduce further to the planar system (4.17), whose piecewise-deterministic stationary density $q^*(r; R_0)$ is obtained explicitly and then used in a slow/fast averaging argument to determine the slow medium amplitude $R(t)$.","core_discovery":"On the paper's own terms, the central discovery is that global phase resetting with a non-resetting external medium does not destroy the low-dimensional Ott-Antonsen description; it simply turns it into a four-dimensional piecewise-deterministic system $(r, R, \\phi, \\Phi)$ with subsystem resetting $r \\to r_0$, $\\phi \\to 0$ at Poisson times. Because the medium is not reset, the post-reset state is a new initial condition lying in a potentially different basin of attraction, so the stochastic resetting acts as a source of noise even though no thermal or extrinsic noise is present. In the low-density regime this manifests as noise-induced transitions between coexisting coherent states or from incoherence to coherence, and in the frequency-locked case $\\omega_0 = \\Omega$ the reduced planar system exhibits slow/fast dynamics whose stationary distribution for the fast order parameter can be computed exactly and matched to the slow evolution of the medium amplitude. The paper presents numerical simulations of the reduced OA system supporting these transitions, plus a slow/fast averaging calculation whose self-consistency condition $R^* = E[r|R^*]$ agrees with the simulated asymptotes.","pith_inferences":["The mechanism suggests a practical control protocol for quorum-sensing or coupled-laser systems: deliberately resetting the oscillator phases at a tunable rate could move the population between a low-activity and a high-activity synchronized state without changing coupling or adding noise, if the OA reduction survives finite-size effects.","One could test the transition-rate statistics against the reduced system: the mean time to switch should scale with the resetting rate and with the distance of the reset point from the separatrix; such a scaling law is not derived in the paper.","The same slow/fast averaging strategy should apply to other mean-field oscillator models with resetting in only a subset of variables, provided the fast subsystem has a unique ergodic nonequilibrium stationary state; this is an unproven extension.","A quantitative falsifier of the OA-invariance assumption would be to measure the second and third circular moments of the finite-$N$ density immediately after resets: if they deviate from $\\eta^n$ scaling, the four-dimensional system overstates the switching capability."],"forward_implications":["At high cell densities ($\\sigma_0 \\to \\infty$), the medium amplitude locks to the order parameter, $R(t) \\to r(t)$, and the dynamics reduces to the one-dimensional classical Kuramoto model with global resetting, reproducing the known nonequilibrium stationary state.","At low densities, the external amplitude becomes a slow variable; averaging the fast resetting order parameter against its nonequilibrium stationary state yields a self-consistency equation for the asymptotic medium amplitude, and the paper's numerics confirm this approximation.","Subsystem resetting can induce transitions between coexisting states in a bistable deterministic regime without any additional noise source; in the examples shown the transitions are irreversible.","The resetting density equation cannot be closed by a mean-field expectation, because the medium amplitude is a linear functional of the stochastic density; the stochastic resetting must be retained in the continuum limit.","The qualitative conclusions generalize to any piecewise-deterministic ODE with subsystem resetting: if only a subset of variables resets, the post-reset point can land in a different basin of attraction, enabling noise-induced transitions."],"supporting_citations":[{"why":"Provides the deterministic Ott-Antonsen reduction of the Kuramoto model coupled through an external medium and the bistability phase diagram that the resetting analysis extends.","marker":"[10]"},{"why":"Supplies the Ott-Antonsen ansatz that collapses the infinite Fourier hierarchy to low-dimensional equations.","marker":"[16]"},{"why":"Establishes the global density-equation approach with stochastic resetting for phase oscillator systems, used to derive Eq. (3.12).","marker":"[24]"},{"why":"Provides the classical Kuramoto global-resetting NESS that the high-density limit recovers.","marker":"[23]"},{"why":"Defines subsystem resetting in the Kuramoto model, the phenomenon this paper realizes through a non-resetting medium.","marker":"[26]"},{"why":"Underlies the low-density mapping to a Kuramoto model with bimodal frequency distribution, used to frame the low-density regime.","marker":"[17]"}],"fun_headline_variants":["Phase-only resets turn Kuramoto medium into noise source","Resetting phases, not medium, creates noise-like transitions","Subsystem resetting enables noise-free switching in Kuramoto","Global phase resets trigger transitions without external noise","Resetting phases only flips basins in Kuramoto medium"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole reduction rests on an unproven assumption that the special low-dimensional family of phase distributions used in the analysis stays closed under stochastic resetting whenever the reset phases also come from that family, so if repeated resets push the population outside this family the predicted transitions need not occur in the full system.","fun_headline_variants_meta":{"raw":{"variants":["Phase-only resets turn Kuramoto medium into noise source","Resetting phases, not medium, creates noise-like transitions","Subsystem resetting enables noise-free switching in Kuramoto","Global phase resets trigger transitions without external noise","Resetting phases only flips basins in Kuramoto medium"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1559,"prompt_tokens":1027,"completion_tokens":532,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":450}},"tokens_in":643,"tokens_out":532,"duration_ms":5329,"temperature":1.0,"reasoning_tokens":450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:11:19.853280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full phase-oscillator model with many oscillators and global phase resetting, then check whether the second and higher circular moments of the phase distribution stay locked to the powers of a single complex number predicted by the reduced system; if they drift away after resets, the four-dimensional equations do not represent the population and the predicted transitions are artifacts.","supporting_citations":[{"cited_title":"Kuramoto model with coupling through an external medium","cited_arxiv_id":null,"evidence_quote":"Provides the deterministic Ott-Antonsen reduction of the Kuramoto model coupled through an external medium and the bistability phase diagram that the resetting analysis extends."},{"cited_title":"Low dimensional behavior of large systems of globally coupled oscillators","cited_arxiv_id":null,"evidence_quote":"Supplies the Ott-Antonsen ansatz that collapses the infinite Fourier hierarchy to low-dimensional equations."},{"cited_title":"Global density equations for interacting particle systems with stochastic resetting: from overdamped Brownian motion to phase synchronization","cited_arxiv_id":null,"evidence_quote":"Establishes the global density-equation approach with stochastic resetting for phase oscillator systems, used to derive Eq. (3.12)."},{"cited_title":"Synchronization in the Kuramoto model in presence of stochas- tic resetting","cited_arxiv_id":null,"evidence_quote":"Provides the classical Kuramoto global-resetting NESS that the high-density limit recovers."},{"cited_title":"Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it","cited_arxiv_id":null,"evidence_quote":"Defines subsystem resetting in the Kuramoto model, the phenomenon this paper realizes through a non-resetting medium."},{"cited_title":"Exact results for the Kuramoto model with a bimodal frequency distribution","cited_arxiv_id":null,"evidence_quote":"Underlies the low-density mapping to a Kuramoto model with bimodal frequency distribution, used to frame the low-density regime."}],"review_version":1}