{"id":"f72a2956-dd83-4167-9c00-88cc4b1d9436","arxiv_id":"2608.03454","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In a bistable FitzHugh-Nagumo neuron, genuine inverse stochastic resonance occurs when the quasi-potential barrier for leaving the spiking state exceeds that for leaving the resting state, provided the fitted noise exponent is negative.","lead":"This paper studies a brain-cell model where an intermediate amount of noise silences spiking. It separates real noise-induced silencing from artifacts of short simulations and predicts which parameters produce genuine suppression.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Monte Carlo MFET estimates in the fitting interval appear censored: for M1, ΔS/σ^2≈8.5e4 at σ=1e-5 makes E[τ_LC] unobservable within T=2.5e5, so fitted β and σ* are not validated.","rationale":"The reader's weakest assumption is the unproved metastable two-state Markov reduction. My concern is related but distinct: even if the reduction holds, the numerical test that claims to validate it cannot be performed by direct Monte Carlo at the stated noise levels because one of the escape processes is exponentially rare. This is not a dispute about the existence of an invariant measure (Theorem 5.1 is rigorous and well supported) nor about the qualitative mechanism. The load-bearing defect is that the fitted exponent β and the reported agreement in Tables 1 and 2 rest on mean-exit-time estimates that are censored at the simulation horizon. The paper itself acknowledges censoring for log σ < −6, but the same reasoning—using the paper's own ΔS values—shows censorship extends through most of the fitting interval. The concrete test (rare-event simulation) would resolve whether the fits are artifacts. Therefore the appropriate verdict remains CONDITIONAL: the qualitative escape-balance mechanism is plausible, but the quantitative predictions require rare-event simulations or other unbiased estimators before they can be accepted.","tokens_in":24093,"tokens_out":17609,"duration_ms":157773,"concrete_test":"Reproduce the Monte Carlo setup for parameter point M1 (a = −0.055, ε = 0.02751, b = 1.0, c = 2.0) with T = 2.5×10⁵ and 300 realizations at σ = 10⁻⁵, 10⁻⁴·⁵, and 10⁻⁴. Starting on the limit cycle Γ_LC, record the first exit time from B_LC; starting from the fixed point, record the first exit from B_FP. If the mean/median exit time from B_LC equals T (no exits in any realization), the data point is censored. Compare the measured ratio E[τ_FP]/E[τ_LC] with the prediction of Eq. (47) using the fitted parameters in Table 1. If they differ by orders of magnitude, the fit is invalid. Independently estimate E[τ_LC] at these noise levels using a rare-event method (e.g., AMS or importance sampling) and check whether the Arrhenius scaling with the gMAM barrier ΔS holds. This settles whether the reported β and σ* reflect the true escape balance or censoring artifacts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative validation in Section 7 depends on direct Monte Carlo estimates of the mean first-exit times E[τ_FP] and E[τ_LC] over the fitting interval log σ ∈ [−6.0,−3.3]. For the reported barrier differences (ΔS ≈ 5–8.5×10⁻⁶), the weak-noise ansatz itself implies astronomically rare escapes at the lower end of this interval. For M1 at σ = 10⁻⁵, ΔS/σ² ≈ 8.5×10⁴, so E[τ_LC]/E[τ_FP] ~ exp(8.5×10⁴); even with E[τ_FP] of order unity, E[τ_LC] vastly exceeds the simulation horizon T = 2.5×10⁵. No escape from the limit-cycle basin can be observed. The paper excludes only log σ < −6 as 'too rare', but the same censoring applies throughout most of the fitting interval; the fitted β is therefore extracted from a mixture of censored values (E[τ] → T) and possibly a few resolved points. The claimed collapse of occupation curves from the two initial basins over log σ ∈ [−6,−3.3] is also suspect: if no transitions occur, the finite-time occupation probabilities remain at 1 and 0, respectively, rather than collapsing. Consequently, the sign criterion and the predicted σ* in Eq. (46) are not empirically established by the reported Monte Carlo data.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies inverse stochastic resonance (ISR) in a bistable FitzHugh–Nagumo neuron driven by additive voltage noise. After identifying a bistable wedge in the (a,ε) parameter plane via deterministic bifurcation analysis, it shows that finite-time Monte Carlo occupation curves of the limit-cycle basin can depend on the initial basin, and proves that this dependence disappears asymptotically: the SDE admits a unique invariant probability measure (Theorem 5.1, Appendix). The paper then computes quasi-potential barriers S_FP and S_LC with a degenerate-noise geometric minimum action method, and uses their difference ΔS = S_LC − S_FP to partition the wedge into two escape-dominated regimes. A reduced two-state Markov approximation leads to the occupation formula μσ(B_LC) = [1 + P σ^β exp(−ΔS/σ²)]^{-1}, the sign criterion ΔSβ < 0 for genuine ISR, and the predicted ISR-minimizing noise amplitude σ*² = −2ΔS/β. These predictions are compared with Monte Carlo simulations at three parameter points with ΔS > 0 and three with ΔS < 0.","tokens_in":24429,"tokens_out":9120,"duration_ms":87874,"significance":"The invariant-measure theorem is a solid and useful contribution: it cleanly separates finite-time initial-condition effects from asymptotic behavior and is proved with standard tools (Lyapunov function, Hörmander bracket condition, strong Feller property, irreducibility). The gMAM quasi-potential computation is methodologically appropriate and is logically independent of the later fitting step. The reduced escape-balance formula is simple, falsifiable, and leads to an interpretable partition of parameter space. If the quantitative validation were reliable, the paper would provide a valuable mechanistic account of ISR in bistable excitable systems. However, as detailed below, the fitting-based validation of the sign criterion and of σ* is currently not solid enough to support the paper's central quantitative claims.","major_comments":[{"comment":"The reported fits are not supported by the stated Monte Carlo data over the full fitting interval. The paper excludes log σ < −6 because mean exit times saturate at T = 2.5×10^5, but for M1, ΔS = 8.547×10^-6, so at log σ = −5 (σ = 10^-5) one has ΔS/σ² ≈ 8.5×10^4; the ratio E[τ_LC]/E[τ_FP] is then of order exp(8.5×10^4) up to polynomial prefactors, so the limit-cycle exit is completely unresolved at that noise level. The same censoring that motivates the cutoff at −6 therefore extends into the interior of the stated fitting interval [−6.0, −3.3]. A least-squares fit of Eq. (48) over the full stated interval would be dominated by censored points with log(E[τ_FP]/E[τ_LC]) = 0 and a rapidly growing correction ΔS/σ², which is inconsistent with the clean linear fits displayed. In addition, the predicted minima σ* in Table 1 (≈ 0.0057, 0.0047, 0.0037) and the observed minima σ_min (≈ 0.0067, 0.0055, 0.0040) all lie above the upper end of the fitting interval (σ = 10^-3.3 ≈ 5×10^-4), so the claimed agreement requires extrapolating the fitted power law outside the range where Eq. (42) is asserted to hold. The fitted β, and hence the sign criterion and Eq. (46), are therefore not empirically established by the present data.","section":"Section 7, Tables 1–2 and Figs. 7–8"},{"comment":"The two-state reduction used for Eq. (42) is an assumption rather than a consequence of the cited exponential exit law. Ref. [54] (Eq. (35) in the manuscript) gives marginal exponentiality of each exit time, but the independence of successive waiting times, the exponentiality conditional on the previous state, and especially the asserted uniform distribution of the post-exit state Y_k on {FP, LC} are additional structural assumptions. The uniform post-exit claim is delicate for an asymmetric pair of basins separated by an unstable limit cycle, and no numerical diagnostic (e.g., empirical distribution of return states or autocorrelation of the sequence Y_k) is provided. Since Eq. (42), the sign criterion ΔSβ < 0, and the formula for σ* all rest on this reduction, the central quantitative prediction is conditional on an unverified assumption.","section":"Section 7, after Eq. (40)"},{"comment":"The validation protocol is partially circular. The parameters β and log P are obtained by least-squares fitting Eq. (47) to Monte Carlo estimates of E[τ_FP] and E[τ_LC] at the same parameter points whose occupation curves are then used to read off σ_min; the predicted σ* is thus not an independent, out-of-sample prediction. Furthermore, no sensitivity analysis with respect to the arbitrarily chosen fitting interval endpoints (−6.0 and −3.3) is reported; since β is an effective exponent, different endpoints could change its sign or magnitude and hence the sign criterion. An independent validation, such as estimating β from resolved events only or predicting an unseen parameter point, is needed before the semiquantitative agreement in Tables 1–2 can be taken as evidence for the mechanism.","section":"Section 7, Eq. (48)"}],"minor_comments":[{"comment":"The notation σβFP, σβLC, and Pσβ should be typeset as σ^{β_FP}, σ^{β_LC}, and P σ^β; as written the expressions can be misread as products with β.","section":"Eqs. (34) and (42)"},{"comment":"The inequality '0 = V1 < V−' is inconsistent with the immediately following sentence and with the condition V− < V1 = 0 used in Proposition 3.6; this appears to be a typographical reversal.","section":"Section 3, paragraph after Eq. (16)"},{"comment":"The symbol T_LC is used before it is defined; it should be defined at first use in Eq. (21) or immediately before it.","section":"Section 4, Eq. (21)"},{"comment":"No numerical details of the gMAM implementation (discretization, number of nodes, stopping criteria, verification against known test cases) are provided, so the quasi-potential values in Tables 1–2 are not reproducible from the text.","section":"Section 6"},{"comment":"The quantity plotted on the vertical axis in panels (a2)–(c2) is not fully specified in the captions; the text should state explicitly that the plotted quantity is y + ΔS/σ² versus log σ.","section":"Figures 7 and 8"},{"comment":"The name 'Strook's theorem' should be 'Stroock's theorem'.","section":"Appendix 8.6"},{"comment":"References [57] and [60] are the same article by Hörmander and should be merged or cross-referenced.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The invariant-measure theorem and the conceptual framework are solid, and the paper should not be rejected solely on the basis of the quantitative validation. However, Section 7 needs substantial reworking: the authors should either restrict the fits to demonstrably resolved rare-event data, use a censored-data or rare-event simulation method, or clearly present the results as an extrapolation with a sensitivity analysis. Without this, the central claim of a validated prediction for σ* is not substantiated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"My take on 2608.03454: the conceptual contribution is real, but the quantitative validation is not load-bearing as presented. I agree with the conditional verdict.\n\nWhat is actually new: the two-parameter picture of the FHN bistable wedge bounded by subcritical Hopf and SNLC bifurcations, the explicit proof of a unique invariant measure for the degenerate-noise FHN process, and the gMAM computation of quasi-potential barriers S_FP(a,ε) and S_LC(a,ε). The partition of the wedge by ΔS = S_LC − S_FP into two escape-dominated regimes is a genuinely useful way to think about where ISR should be asymptotic and where finite-time dips are just rare-event artefacts. The appendix proof is a correct application of standard hypoellipticity plus Lyapunov and irreducibility arguments; no new technique, but it is solid and worth having on record.\n\nThe escape-balance reduction leading to Eq. (42) is a natural ansatz, and the sign criterion ΔSβ<0 is elementary calculus once you grant it. The soft spot is the validation. The two-state Markov reduction is assumed, not proved, and at the noise levels actually used (log σ ∈ [−6,−3.3]) the exponential exit law is a hope, not a theorem. More concretely, the stress-test note about censoring lands. At M1, ΔS/σ² is about 8.5×10⁴ at σ=10⁻⁵ and still about 34 at the upper end of the fitting interval, so E[τ_LC] is many orders of magnitude beyond the simulation horizon T=2.5×10⁵. The paper excludes log σ<−6 for exactly this reason, but the same censoring covers most of [−6,−3.3]. If no transitions are observed, the \"collapse\" of occupation curves from the two initial basins cannot be evidence for convergence to the invariant measure, and the fitted β and log P are not empirically established. No error bars appear in the Monte Carlo estimates, and no code or data are shipped, so the numbers are not independently checkable.\n\nNone of this kills the central mechanism as a hypothesis. The qualitative direction—genuine dips only on the ΔS>0 side when β<0—might survive a correct validation. But the quantitative σ* formula and Tables 1–2 should be treated as tentative until the rare-event problem is addressed, for example with a dedicated rare-event estimator or a narrower, genuinely resolved noise window.\n\nRecommendation: send it to peer review, with a request for major revision focused on Section 7's validation. I would bring it to a reading group with that caveat.","headline":"A useful quasi-potential partition and a correct invariant-measure proof, but the headline ISR-minimum prediction rests on a validation fit that is largely censored and needs major revision before the quantitative claims stand.","tokens_in":24947,"tokens_out":4769,"would_cite":false,"duration_ms":43875,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34F05","37N25","60H10","92C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A sign criterion decides when noise truly silences a bistable neuron's spiking.","keywords":["inverse stochastic resonance","FitzHugh-Nagumo neuron","bistability","quasi-potential barrier","geometric minimum action method","timescale separation","invariant probability measure","escape-balance mechanism"],"falsifier":"Run the same Monte Carlo protocol for a green test point such as $G_1$, where $\\Delta S < 0$ and the fitted $\\beta < 0$, with a substantially longer horizon than $T = 2.5\\times10^5$; if an interior minimum in $\\mu_\\sigma(B_{\\mathrm{LC}})$ persists and deepens as $T$ grows, the claim that no genuine asymptotic ISR occurs on the $\\Delta S < 0$ side fails. Independently, at a $\\Delta S > 0$ point, an extended fit of $\\beta$ over a wider weak-noise window that yields a positive slope would leave Eq. (46) with no positive solution and contradict the predicted minimum.","tokens_in":23903,"feed_emoji":"⚡","tokens_out":11213,"duration_ms":92296,"temperature":0.7,"pith_summary":"This paper tries to establish when inverse stochastic resonance—the effect where an intermediate noise amplitude suppresses neuronal spiking more than weak or strong noise—is a genuine long-time phenomenon in a bistable FitzHugh-Nagumo neuron, and when it is only a finite-time artifact of too-short simulations. The central claim is that the long-time occupation of the spiking basin is controlled by the difference of two quasi-potential escape barriers, and that a genuine ISR minimum occurs exactly when this difference and the effective noise exponent have opposite signs. The paper proves that the stochastic system has a unique invariant probability measure, so any apparent dependence of firing statistics on the initial basin disappears asymptotically. A closed-form weak-noise formula for the basin occupation then predicts the noise amplitude at which the ISR dip occurs, and Monte Carlo simulations support that prediction semiquantitatively.","feed_headline":"A sign rule tells when noise truly silences a neuron","feed_subtitle":"Barrier imbalance plus the noise exponent fix where inverse stochastic resonance appears.","key_machinery":"The mechanism that carries the argument is the reduced metastable two-state description of the bistable neuron: the trajectory spends long random times near the fixed point or near the limit cycle and occasionally crosses the separating unstable cycle, after which it forgets its history. Two computed ingredients enter. The first is the quasi-potential barrier difference $\\Delta S(a,\\varepsilon) = S_{\\mathrm{LC}} - S_{\\mathrm{FP}}$, obtained from a degenerate-noise geometric minimum action method that finds the least noise-action path from each attractor to the separatrix. The second is the weak-noise form of the mean exit times, $E[\\tau] \\sim P\\,\\sigma^\\beta \\exp(S/\\sigma^2)$, whose ratio produces the occupation formula, the sign criterion $\\Delta S\\,\\beta < 0$, and the explicit minimizer $\\sigma_*^2 = -2\\Delta S/\\beta$.","core_discovery":"The central claim, stated on the paper's own terms, is this. In the bistable wedge of the $(a,\\varepsilon)$ parameter plane where a stable quiescent fixed point and a stable spiking limit cycle coexist, the weak-noise occupation probability of the limit-cycle basin has the closed form\n\n$$\\mu_\\$\\sigma$(B_\\mathrm{LC}^\\delta) = \\left[1 + P\\,\\$\\sigma$^\\$\\beta$ \\exp\\left(-\\frac{\\$\\Delta$ S}{\\$sigma^{2}$}\\right)\\right]^{-1},$$\n\nwhere $\\Delta S = S_{\\mathrm{LC}} - S_{\\mathrm{FP}}$ is the difference between the quasi-potential escape barriers to the separating unstable cycle. From this expression the paper derives the sign criterion $\\Delta S\\,\\beta < 0$ for non-monotone dependence on $\\sigma$ and, in the fitted regime where $\\beta < 0$, predicts a genuine ISR minimum only on the $\\Delta S > 0$ side of the wedge, at $\\sigma_*^2 = -2\\Delta S/\\beta$. It also proves uniqueness of the invariant measure for the stochastic FitzHugh-Nagumo system, so the apparent dependence of finite-time ISR curves on the initial basin is not asymptotic. Tables 1 and 2 report semiquantitative agreement between the predicted minimizer and the Monte Carlo occupation minima at three $\\Delta S > 0$ points, and absence of an asymptotic dip at three $\\Delta S < 0$ points.","pith_inferences":["Extending the paper's logic, the sign criterion should be testable in other two-variable excitable neuron models: only the quasi-potential barrier difference and the fitted exponent are needed, since the formula does not use the specific polynomial drift of the FitzHugh-Nagumo system.","A natural reading the paper leaves implicit is that the ISR dip is a minimum of a ratio of two Arrhenius-type rates, where the power-law prefactor $\\sigma^\\beta$ tilts the exponential competition; this explains why the sign of $\\beta$ matters as much as the ordering of the barriers.","A practical consequence for network-scale simulations: because one escape direction can be exponentially rarer than the other, reported ISR dips in finite-horizon studies may disappear when the simulation time is increased, even in parameter regions where the two-state formula would predict a genuine minimum."],"forward_implications":["If the formula and sign criterion are correct, the existence and location of the ISR dip in a bistable neuron are fixed by two computable quantities: the quasi-potential barrier difference and the effective noise exponent of the escape-time prefactors.","Near the curve $\\Delta S = 0$, the predicted minimizer $\\sigma_*$ approaches zero, so the model exhibits full inverse stochastic resonance: arbitrarily close to the balance curve, an arbitrarily weak noise can drive the spiking-basin occupation toward zero.","Because a unique invariant measure guarantees that initial-condition dependence vanishes asymptotically, finite-time Monte Carlo studies of ISR must sample both escape directions sufficiently before a noise-induced dip can be attributed to genuine ISR.","The semiquantitative agreement in Tables 1 and 2 suggests the same escape-balance formula can be used as a parameter-dependent diagnostic for ISR in slow-fast excitable models of the FitzHugh-Nagumo type."],"supporting_citations":[{"why":"Introduces inverse stochastic resonance in Hodgkin-Huxley neurons and the mean-first-exit-time asymmetry that the escape-balance formula builds on.","marker":"[1]"},{"why":"Provides the quasi-potential calculation and minimum action method for limit cycles used to obtain S_LC.","marker":"[47]"},{"why":"Supplies the Freidlin-Wentzell action functional and the quasi-potential framework used to define the escape barriers.","marker":"[48]"},{"why":"Supplies the geometric minimum action method for degenerate noise used to compute the barrier difference Delta S.","marker":"[49]"},{"why":"Gives the Arrhenius weak-noise form of the mean exit time from a stable limit cycle through an unstable periodic orbit, the basis for E[tau_LC].","marker":"[51]"},{"why":"Gives the corresponding escape behavior through an unstable limit cycle for a stable fixed point, the basis for E[tau_FP].","marker":"[53]"},{"why":"Provides the exponential exit law used to justify the asymptotically exponential waiting times of the two-state Markov reduction.","marker":"[54]"},{"why":"Supplies the hypoelliptic analysis of the FitzHugh-Nagumo SDE used in the proof of the unique invariant probability measure.","marker":"[56]"},{"why":"Supplies the exponential ergodicity criterion applied to prove uniqueness of the invariant measure and convergence to it.","marker":"[58]"}],"fun_headline_variants":["Noise silencing? It's all about barrier balance","A sign flips the switch on neuron silencing","Quasi-potentials decide when noise kills spikes","Barrier imbalance predicts noise-induced silence","Sign of barrier gap tells noise-silencing outcome"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, at the small noise levels used in the fits, the neuron's stays in the quiet and spiking states are independent random waits whose average sizes follow the stated exponential-and-power-law escape formula, so the two-state jump process faithfully represents the real dynamics.","fun_headline_variants_meta":{"raw":{"variants":["Noise silencing? It's all about barrier balance","A sign flips the switch on neuron silencing","Quasi-potentials decide when noise kills spikes","Barrier imbalance predicts noise-induced silence","Sign of barrier gap tells noise-silencing outcome"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1684,"prompt_tokens":1096,"completion_tokens":588,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":517}},"tokens_in":712,"tokens_out":588,"duration_ms":5422,"temperature":1.0,"reasoning_tokens":517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:49:33.476155+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same Monte Carlo protocol for a green test point such as $G_1$, where $\\Delta S < 0$ and the fitted $\\beta < 0$, with a substantially longer horizon than $T = 2.5\\times10^5$; if an interior minimum in $\\mu_\\sigma(B_{\\mathrm{LC}})$ persists and deepens as $T$ grows, the claim that no genuine asymptotic ISR occurs on the $\\Delta S < 0$ side fails. Independently, at a $\\Delta S > 0$ point, an extended fit of $\\beta$ over a wider weak-noise window that yields a positive slope would leave Eq. (46) with no positive solution and contradict the predicted minimum.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces inverse stochastic resonance in Hodgkin-Huxley neurons and the mean-first-exit-time asymmetry that the escape-balance formula builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quasi-potential calculation and minimum action method for limit cycles used to obtain S_LC."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Freidlin-Wentzell action functional and the quasi-potential framework used to define the escape barriers."},{"cited_title":"Berglund, B","cited_arxiv_id":null,"evidence_quote":"Gives the Arrhenius weak-noise form of the mean exit time from a stable limit cycle through an unstable periodic orbit, the basis for E[tau_LC]."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the hypoelliptic analysis of the FitzHugh-Nagumo SDE used in the proof of the unique invariant probability measure."}],"review_version":2}