{"id":"baecd527-06d4-4fdc-a3e8-70abdfa06d5a","arxiv_id":"1909.00727","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For a two-dimensional bounded domain, the stochastic Hindmarsh-Rose equations with additive noise are claimed to possess a unique random pullback attractor in L2 space.","lead":"This mathematics paper claims to prove that the stochastic Hindmarsh-Rose neuron model, with additive random noise on a two-dimensional domain, always settles into a unique attracting state in the long run. The proof uses standard pullback attractor theory, but it contains a key erroneous inequality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The inequality-direction error in (2.13) is load-bearing: it supplies the quartic dissipation needed for the absorbing estimate, so Theorem 3.2 is not proved as written.","rationale":"I read the paper as aiming to prove existence of a unique random attractor in L^2(Omega,R^3) for the stochastic Hindmarsh-Rose system with additive noise, using the standard two-step route: pullback absorbing set plus pullback asymptotic compactness. For that route to work, the uniform energy estimates must provide genuine dissipation. The reader identified exactly the step where the proof attempts to convert the quartic expression of the shifted variable u = U + Gamma into quartic dissipation in U. I checked the algebra: the inequality used at (2.13) has the wrong direction, since (U+Gamma)^4 <= 8(U^4+Gamma^4), not >=. This is not a cosmetic typo; without it the coefficient -3/8 on u^4 cannot be turned into the large negative coefficient on U^4 that later produces the -2 integral U^4 term in (2.15). The subsequent Gronwall bounds, the absorbing radius R0(omega), the pullback absorbing set, and the uniform Gronwall step in Theorem 3.1 all depend on that estimate. Thus the submitted proof does not rigorously establish the central theorem, even though the overall strategy is standard and the result may well be repairable with a different estimate. I do not see a separate concern more load-bearing than this one; the other mathematical slips I noticed are secondary and would not by themselves overturn the proof. I am not treating absence of machine-checked verification as a flaw, and I am not attributing intent to the authors. My recommendation is to keep the reader's REJECT verdict; the concern is internal to the proof and requires revision of the estimate, not merely a change of interpretation.","tokens_in":21737,"tokens_out":5163,"duration_ms":53244,"concrete_test":"Check the asserted inequality in (2.13) with the pointwise test U=1, Gamma_1=0 on a domain of positive measure: (U+Gamma_1)^4=1 and 8(U^4+Gamma_1^4)=8, so -3/8 integral (U+Gamma_1)^4 dx = -3/8 |Omega| while -3 integral (U^4+Gamma_1^4) dx = -3 |Omega|. Since -3/8 > -3, the claimed bound -3/8 integral (U+Gamma_1)^4 <= -3 integral (U^4+Gamma_1^4) is false. Then attempt to re-derive (2.15) using only the valid bound -3/8 integral (U+Gamma_1)^4 >= -3 integral U^4 - 3 integral Gamma_1^4; verify whether the RHS still contains the -2 integral U^4 dissipation term that the Gronwall argument in (2.16)-(2.19) requires. If the dissipation term cannot be recovered, the pullback absorbing estimate and hence the main theorem lack proof as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3.2, and its proof rests on the pullback absorbing estimate of Lemma 2.1. The decisive step is the chain from (2.12) to (2.13). There the proof replaces -3/8 integral (U+Gamma_1)^4 dx by -3 integral (U^4 + Gamma_1^4) dx. This replacement requires the inequality (U+Gamma_1)^4 >= 8(U^4 + Gamma_1^4). The true convexity inequality has the opposite direction: (U+Gamma_1)^4 <= 8(U^4 + Gamma_1^4), with strict failure of the claimed direction e.g. at U=1, Gamma_1=0. Consequently the displayed estimate (2.13) is not valid, and the quartic dissipation -2 integral U^4 used in (2.15) is unsupported. Because (2.15) and its Gronwall consequence (2.19) feed directly into Lemma 2.2's absorbing radius R0(omega), Theorem 2.5's absorbing set K(omega), and Step 4 of Theorem 3.1's uniform Gronwall estimate, the existence of the pullback absorbing set and the asymptotic compactness argument are not established. The flaw is an inequality-direction error internal to the proof, not a matter of disagreement with an outside consensus. Secondary issues such as the suspicious integral limit in (2.21) and the informal pathwise OU representation do not change this assessment.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the longtime pullback dynamics of the stochastic Hindmarsh-Rose equations with additive noise on a bounded domain of dimension at most two. The authors convert the stochastic PDE into a random PDE via an Ornstein-Uhlenbeck transformation, prove the existence of a pullback absorbing set and pullback asymptotic compactness in L², and conclude via a standard existence theorem that the associated random dynamical system has a unique random attractor. The main result is Theorem 3.2, which asserts such an attractor for arbitrary positive parameters and any c in R.","tokens_in":22058,"tokens_out":8815,"duration_ms":71857,"significance":"If the proof were valid, the result would be a meaningful extension of the authors' earlier multiplicative-noise result to additive noise for a widely used neurodynamics model, and it would provide a clean example of the additive-transformation plus uniform Gronwall approach. The paper is clearly structured, defines all random dynamical system concepts, and uses a standard criterion (Theorem 1.8) for attractor existence. However, the proof as written contains several load-bearing mathematical errors, so the central claim is not established.","major_comments":[{"comment":"The step from `-3/8 ∫ (U+Γ_1^h)^4 dx` to `-3 ∫ (U^4 + (Γ_1^h)^4) dx` is invalid. It requires `(U+Γ_1^h)^4 ≥ 8(U^4 + (Γ_1^h)^4)`, whereas the true convexity inequality is the opposite, `(U+Γ_1^h)^4 ≤ 8(U^4 + (Γ_1^h)^4)`. The claimed direction fails, for example, at U = 1, Γ_1^h = 0. Consequently the quartic dissipation `-2∫ U^4` in (2.15) is unsupported, and the Gronwall estimate (2.19), the absorbing radius in (2.27), the absorbing set in Theorem 2.5, and Step 4 of Theorem 3.1 all rely on this estimate. The existence of the pullback absorbing set is therefore not proved.","section":"§2.1, Eq. (2.13)"},{"comment":"The limits of the integral after setting t = -1 are wrong. From (2.19) with t = -1, the convergent integral should be `∫_{-∞}^{-1} e^{σ(s+1)}(...) ds`, but (2.21) displays `∫_{-1}^{∞} e^{σ(1+s)}(...) ds`, which diverges because the exponential has positive exponent. This makes the bound on `‖G(-1,θ_τ ω; τ, g0)‖²` and the definition of r0(ω) in (2.23) invalid; the same expression is reused in the proof of Theorem 2.5.","section":"§2.2, Eq. (2.21)"},{"comment":"The proof invokes the Sobolev embedding H¹(Ω) ֒→ L^∞(Ω) for dim(Ω) ≤ 2. This embedding is false in dimension two; H¹ of a bounded domain in R² embeds into L^p for every finite p, but not into L^∞. Since this embedding is used to control `−2β∫ u∇u·∇v dx` and the cubic terms via `‖u‖_{L^∞}`, the estimate (3.4) and hence the asymptotic compactness argument of Theorem 3.1 are not justified as written.","section":"§3.1, Eq. (3.4)"},{"comment":"The Ornstein-Uhlenbeck process is written as `−κ∫_{-∞}^{0} e^{κs}(θ_t ω_i)(s) ds`, treating a Brownian path as if it were a density against Lebesgue measure. Since Brownian paths are almost surely not of bounded variation, this pathwise Lebesgue-integral representation is not well-posed. The definition can be made rigorous only through the stochastic integral or through a suitable integration-by-parts formulation. The informality should be repaired, although it is not the main obstruction to the proof.","section":"§1.2, Eq. (1.15)"}],"minor_comments":[{"comment":"The displayed integral in (2.20) is written as `∫_0^{-1}` but the intended interval is [-1,0]; it should be `∫_{-1}^{0}`.","section":"§2.1, Lemma 2.2"},{"comment":"There are numerous typos and typesetting artifacts, for example 'compldeted' in Lemma 2.2, 'weal solution' in Section 2, '/interleave/' artifacts, and 'spacial' in Theorem 3.2; these should be corrected.","section":"Throughout"},{"comment":"The constants `(C2+32η)` in (3.21) and (3.25) should be `(C1+32η)` to match the definitions in (3.14)-(3.15).","section":"§3.1, Step 4"},{"comment":"In (3.24), the argument of the supremum is written inconsistently as `D(θ_tω)` and `D(θ_{-t}ω)`; it should be `D(θ_{-t}ω)` throughout.","section":"Theorem 3.1 proof"}],"recommendation":"reject","confidential_remarks":"The paper shares a framework with the authors' earlier works [23,24] and the overall strategy is standard. However, the central estimate in (2.13) is flawed in a direction that invalidates the absorption argument, the integral in (2.21) has divergent limits, and the proof of Theorem 3.1 relies on a false H¹↪L∞ embedding in dimension two. These are not merely typographical; they require new mathematical arguments. A substantial revision might succeed, but the current manuscript does not establish its main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper does something real: it extends the authors' prior multiplicative-noise random-attractor theorem for the Hindmarsh-Rose SPDE to additive noise, using the standard Ornstein-Uhlenbeck transformation, and the main theorem is clearly stated: for n <= 2 and all listed parameters positive, the random dynamical system has a unique random attractor in L^2. That result is not in the literature. Second, the proof as written fails at a load-bearing inequality. In the chain (2.12) to (2.13) they replace -3/8 integral (U+Gamma_1)^4 dx with -3 integral (U^4+Gamma_1^4) dx. The required inequality (U+Gamma_1)^4 >= 8(U^4+Gamma_1^4) has the wrong direction; convexity gives <=, and U=1, Gamma_1=0 kills it. The -3 integral U^4 term is exactly the quartic dissipation that closes (2.15), (2.19), Lemma 2.2, Theorem 2.5, and Step 4 of Theorem 3.1. Without a corrected estimate there is no pullback absorbing set and no asymptotic compactness proof. This is not a matter of interpretation; the central argument is unsupported as written. Secondary issues are real but smaller: (2.21) writes integral from -1 to infinity where the convergent estimate needs integral from -infinity to -1; and (1.15)'s pathwise representation of the OU process mixes Ito integrals with pathwise integrals in a way that is at best informal. I would call those cosmetic relative to (2.13). What the paper does well: the introduction and setup are honest, the additive-noise extension is genuinely new, the self-citations to [23,24] are legitimate companion work and not citation padding, and the authors openly flag n=3 as open. The overall architecture is conventional and likely repairable, so I do not think the theorem is false. But the submitted proof is not rigorous enough to check in. Who it is for: people working on random attractors for stochastic neural-field or reaction-diffusion systems will find a useful template and a plausible theorem, but they should not cite the current version as a proof. My recommendation: the paper deserves a serious referee in the sense that the topic and claimed result are worth one. I would send it out, but with the expectation of a reject-or-revise: the authors need to repair (2.13) and clean up the integral-limit and OU-representation issues. If the inequality can be fixed with a small change, the paper is acceptable; if not, the main claim needs a different argument.","headline":"A legitimate additive-noise extension of the authors' earlier random-attractor result, but the proof has a load-bearing inequality with the wrong direction at (2.13), so the central theorem is not established as written.","tokens_in":820,"tokens_out":850,"would_cite":false,"duration_ms":61550,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","35Q80","37L30","37L55","37N25","35B40","60H15","92B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes a unique random attractor for the stochastic Hindmarsh–Rose equations with additive noise in $L^2$ on bounded $n\\le 2$ domains.","keywords":["stochastic Hindmarsh-Rose equations","random attractor","additive noise","pullback absorbing set","pullback asymptotic compactness","Ornstein-Uhlenbeck process","neuronal bursting","random dynamical system"],"falsifier":"Evaluate the displayed inequality at a single point: with $U=1$ and $\\Gamma_1^h=0$, the claimed bound $(U+\\Gamma_1^h)^4\\ge 8(U^4+(\\Gamma_1^h)^4)$ reads $1\\ge 8$, which is false. A direct check of whether the subsequent chain from (2.12) to (2.15) can be repaired with a different constant, or with a term like $-\\frac38\\int_\\Omega U^4 dx$ kept instead of the sum, would settle whether the absorbing-ball conclusion survives.","tokens_in":21521,"feed_emoji":"🧠","tokens_out":7849,"duration_ms":70362,"temperature":0.7,"pith_summary":"The paper seeks to show that the stochastic Hindmarsh–Rose equations—three coupled reaction–diffusion equations for neuronal membrane potential, spiking, and bursting, driven by additive white noise—have a unique random attractor in the space $H=L^2(\\Omega,\\mathbb{R}^3)$ on bounded domains of dimension $n\\le 2$. If the proof is correct, the noisy system is dissipative in the pullback sense even though the cubic term $u^3$ injects energy: every bounded tempered set of initial data is eventually drawn into a bounded absorbing ball, and the dynamics compacts to an invariant random set that attracts all such data. The argument works for all positive parameter values and any real reference potential $c$, so the conclusion is not confined to a special parameter regime. The main theorem is stated as Theorem 3.2 and rests on a two-step strategy: subtract the Ornstein–Uhlenbeck process to convert the SPDE into a random PDE, then prove pullback absorbing and pullback asymptotic compactness via weighted energy estimates.","feed_headline":"Random attractor exists for noisy neuron equations","feed_subtitle":"Even with white noise, the neuron equations settle onto one compact invariant set over long times.","key_machinery":"The load-bearing object is the abstract Ornstein–Uhlenbeck process $\\Gamma^h(\\theta_t\\omega)=(h_1\\Gamma_1,h_2\\Gamma_2,h_3\\Gamma_3)$ defined by $d\\Gamma_i=-\\kappa\\Gamma_i dt+dW_i$. Subtracting it from the solution $(u,v,z)$ converts the stochastic PDE into the random PDE (2.2)–(2.4), which is then studied pathwise. The estimates are organized around a weighted $L^2$-energy with weight $c_1=\\frac1b(2\\beta^2+\\frac{11}{8})$ on the $U$-component; the cubic term $au^2-bu^3$ is turned into quartic dissipation, which produces the pullback absorbing ball. Uniform Gronwall estimates on $\\|\\nabla G\\|^2$ then give a pullback bound in $E=H^1$, and the compact embedding $E\\hookrightarrow H$ supplies the pullback asymptotic compactness needed for the attractor criterion (Theorem 1.8). The reduction to $n\\le 2$ enters through the Sobolev embedding $H^1(\\Omega)\\hookrightarrow L^\\infty(\\Omega)$, used to control the term $-2\\beta\\int_\\Omega u\\nabla u\\cdot\\nabla v\\,dx$.","core_discovery":"The paper's central claim is Theorem 3.2: for a bounded domain $\\Omega\\subset\\mathbb{R}^n$ with $n=\\dim\\Omega\\le 2$, with $h_i\\in W^{2,4}(\\Omega)$ noise coefficients and $W_i$ independent two-sided Wiener processes, and for arbitrary positive $d_1,d_2,d_3,a,b,\\alpha,\\beta,q,r,J$ and arbitrary $c\\in\\mathbb{R}$, the random dynamical system $\\Phi$ generated by the additive-noise Hindmarsh–Rose equations has a unique random attractor $A(\\omega)$ in $H=L^2(\\Omega,\\mathbb{R}^3)$ with respect to the universe of tempered random sets. The attractor is invariant, compact, and pullback-attracts every bounded tempered set. The authors' proof obtains it by verifying the two hypotheses of the standard random-attractor criterion: a closed pullback absorbing ball in $H$, and pullback asymptotic compactness coming from a uniform $H^1$ estimate and the compact embedding $H^1\\hookrightarrow L^2$.","pith_inferences":["The paper does not ask whether the random attractor supports a unique invariant measure whose statistics match the bursting and spiking patterns observed in neuron models; if it does, the attractor would be the natural object for studying stochastic bifurcations.","The same OU-subtraction and weighted-energy scheme could be tested on other three-component excitable systems with cubic recovery nonlinearities, where a similar quartic-dissipation argument may produce absorbing balls.","The strategy is likely adaptable to multiplicative noise, where the same weighted energy and quartic dissipation would have to absorb a random coefficient rather than a random shift."],"forward_implications":["If Theorem 3.2 is correct, the additive-noise Hindmarsh–Rose SPDE is globally dissipative: every bounded tempered set of initial data is pulled into a fixed random ball in finite time.","The random attractor $A(\\omega)$ is invariant under the cocycle and compact in $L^2$, so long-time paths remain confined to a bounded random set despite the continual injection of noise.","The existence holds for all positive values of the model parameters and any real $c$, so no parameter tuning is required for the attractor to exist.","The paper's estimates for pullback absorption are claimed to remain valid in dimension $n=3$, but the compactness step uses $H^1\\hookrightarrow L^\\infty$, which holds only for $n\\le 2$; the authors leave $n=3$ as a conjecture.","This extends the deterministic global-attractor result for diffusive Hindmarsh–Rose equations to the additive-white-noise setting."],"supporting_citations":[{"why":"Establishes the companion multiplicative-noise random-attractor result that this paper extends to additive noise.","marker":"[23]"},{"why":"Gives the deterministic global attractor whose stochastic analogue is the target here.","marker":"[24]"},{"why":"Provides the omega-limit-set measurability and random-attractor existence criterion invoked as Theorem 1.8.","marker":"[40]"},{"why":"Supplies the uniform Gronwall lemma and parabolic regularity used to get the $H^1$ absorbing estimate.","marker":"[29]"},{"why":"Supplies the Wiener process growth and Ornstein–Uhlenbeck process construction used in the additive transformation.","marker":"[22]"},{"why":"Sets up the universe/tempered-random-set framework and pullback absorbing definitions used throughout.","marker":"[8]"}],"fun_headline_variants":["Noisy neurons settle on random attractor","Additive noise still yields random attractor","Stochastic neuron model has compact attractor","Random attractor proven for Hindmarsh-Rose","Neuron noise tamed: random attractor exists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the quartic estimate in step (2.13): the proof replaces $-\\frac38\\int_\\Omega (U+\\Gamma_1^h)^4 dx$ with $-3\\int_\\Omega (U^4+(\\Gamma_1^h)^4) dx$ by asserting $(U+\\Gamma_1^h)^4\\ge 8(U^4+(\\Gamma_1^h)^4)$, which is false already at $U=1,\\Gamma_1^h=0$. Without a corrected inequality that still yields the quartic dissipation, the pullback absorbing ball and hence the attractor are not established.","fun_headline_variants_meta":{"raw":{"variants":["Noisy neurons settle on random attractor","Additive noise still yields random attractor","Stochastic neuron model has compact attractor","Random attractor proven for Hindmarsh-Rose","Neuron noise tamed: random attractor exists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1211,"prompt_tokens":826,"completion_tokens":385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":315}},"tokens_in":442,"tokens_out":385,"duration_ms":3963,"temperature":1.0,"reasoning_tokens":315,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:38:55.069875+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the displayed inequality at a single point: with $U=1$ and $\\Gamma_1^h=0$, the claimed bound $(U+\\Gamma_1^h)^4\\ge 8(U^4+(\\Gamma_1^h)^4)$ reads $1\\ge 8$, which is false. A direct check of whether the subsequent chain from (2.12) to (2.15) can be repaired with a different constant, or with a term like $-\\frac38\\int_\\Omega U^4 dx$ kept instead of the sum, would settle whether the absorbing-ball conclusion survives.","supporting_citations":[{"cited_title":"Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise","cited_arxiv_id":"1908.01220","evidence_quote":"Establishes the companion multiplicative-noise random-attractor result that this paper extends to additive noise."},{"cited_title":"Global Attractors for Hindmarsh-Rose Equations in Neurodynamics","cited_arxiv_id":"1907.13225","evidence_quote":"Gives the deterministic global attractor whose stochastic analogue is the target here."},{"cited_title":"You, Random dynamics of stochastic reaction-diffusion systems w ith additive noise , Journal of Dynamics and Differential Equations, 29 (2017), 83-112","cited_arxiv_id":null,"evidence_quote":"Provides the omega-limit-set measurability and random-attractor existence criterion invoked as Theorem 1.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the uniform Gronwall lemma and parabolic regularity used to get the $H^1$ absorbing estimate."},{"cited_title":"Øksendal, Stochastic Differential Equations , 6th edition, Springer-Verlag, Berlin, 2003","cited_arxiv_id":null,"evidence_quote":"Supplies the Wiener process growth and Ornstein–Uhlenbeck process construction used in the additive transformation."},{"cited_title":"Chueshov, Monotone Random Systems Theory and Applications , Lect","cited_arxiv_id":null,"evidence_quote":"Sets up the universe/tempered-random-set framework and pullback absorbing definitions used throughout."}],"review_version":1}