{"id":"83cd48fc-94de-43a1-afd6-23e0a565bf2d","arxiv_id":"2505.05972","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Noise allows the Hindmarsh-Rose neuron model to switch between two bursting patterns even outside its deterministic bistability region, via saddle-node ghost states.","lead":"This paper shows that a standard mathematical model of neuron bursting, the Hindmarsh-Rose model, can randomly switch between two different burst patterns even outside the parameter range where both patterns are normally stable. The authors explain this with a simple toy model and derive exact formulas for where the switching happens.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"HR stochastic-birhythmicity boundaries rest on a 2% cutoff over finite windows; near a saddle-node ghost, finite simulations can mimic stationary coexistence, so the reported boundaries may be artifacts.","rationale":"The reader's weakest assumption is essentially the same concern I identify: the HR stochastic-birhythmicity region is defined by a 2% cutoff over finite simulations, with no demonstration that the observed two-state behavior reflects the stationary distribution rather than finite-time ghost transients. The paper does offer genuine independent support for the qualitative picture: the analytic toy model has an exact stationary probability distribution with two local maxima outside the deterministic saddle-node, and Fig. 4 shows deterministic ghost transients that make the finite-time issue concrete. However, the toy model cannot by itself validate the HR phase diagram because HR has no analytic stationary distribution and the numerical classification is operational. A careful convergence and threshold-sensitivity check would settle whether the reported boundaries are physical or artifacts. Since the reader already conditioned the verdict on exactly this issue, my stress-test does not change the verdict: it remains CONDITIONAL. I therefore recommend UNCHANGED rather than a new verdict, and I agree with the reader's identification of the weakest assumption.","tokens_in":12151,"tokens_out":3843,"duration_ms":43825,"concrete_test":"Recompute the Fig. 6 boundaries at representative parameter points (b = 2.906 with epsilon = 4e-3, b = 2.924 with epsilon = 8e-3, and the bottleneck region near b = 2.916 with epsilon = 1e-3) using 10x longer simulations and classification thresholds 0.8, 0.9, and 1.0 for the z-amplitude. Plot the fraction of 2-cycle oscillations versus observation time; if the fraction converges to a time-independent value that is largely insensitive to the threshold, the boundary is stationary, whereas monotonic decay or strong threshold dependence would indicate a finite-time ghost artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that noise extends birhythmicity in the Hindmarsh-Rose model is plausible and supported by the analytic toy model, but in HR itself it rests almost entirely on Fig. 6, where a parameter point is counted as birhythmic if each z-amplitude class exceeds 2% of roughly 20,000 observed oscillations, with bursts classified by a fixed z-amplitude threshold of 0.9. The paper itself admits that the 2% proportion is 'a rather arbitrary cutoff' (Sec. II) and notes for the toy model that 'longer simulations would allow for the less likely stable states to be eventually observed' (Sec. III). This is the load-bearing weakness: near a saddle-node bifurcation, deterministic ghosts have long, parameter-dependent dwell times (Fig. 4), so a finite noisy simulation can exhibit both amplitude classes even if the stationary distribution is unimodal. The observed 2% boundary will then shift with integration time and cutoff choice, and the reported narrowing-then-widening behavior, as well as the specific noise intensities at which the boundaries cross the deterministic bifurcations, are not established as stationary properties. This does not refute the qualitative phenomenon, but it leaves the quantitative HR phase diagram unsubstantiated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Hindmarsh-Rose (HR) neuron model with additive white noise on the slow variable z. In the deterministic case, the model has two coexisting bursting orbits (2-cycle and 3-cycle) in a narrow interval of the control parameter b between two saddle-node bifurcations of limit cycles (about b=2.9082 and 2.9231). The authors report that for b outside this interval but close to it, noise induces random switching between the unique stable bursting attractor and the ghost of the other attractor, so that the effective birhythmic region is extended by noise. They map this stochastic birhythmicity region in the (b,ε) plane using numerical simulations and a 2% occurrence cutoff. To put this on an analytical footing, they introduce a simple axially symmetric two-variable model with a periodic-solution saddle-node bifurcation, solve the stationary Fokker-Planck equation, derive the stochastic equilibrium branches and the stochastic saddle-node boundaries, and locate the cusp where the boundaries meet at (b,ε)=(-1/3,4/(3√3)). They show that in the toy model the stochastic birhythmicity region extends beyond the deterministic saddle-node, but not beyond the Hopf bifurcation, analogous to the HR result.","tokens_in":12428,"tokens_out":9337,"duration_ms":82261,"significance":"The analytical part is a genuine strength: the stationary Fokker-Planck solution, the branch equations (Eqs. (11)-(14)), and the cusp calculation are derived without fitting parameters, giving a clean demonstration that an additive-noise-driven system can have two stable stochastic equilibrium states beyond a saddle-node bifurcation, with the boundaries computable in closed form. The HR observation is qualitatively plausible and well illustrated by the time series and histograms (Fig. 3) and by the deterministic ghost simulations (Fig. 4). If the quantitative phase diagram were reliable, the paper would provide a useful map of noise-induced birhythmicity in a widely used neuronal model. However, the HR phase diagram in Fig. 6 rests on an arbitrary 2% cutoff and finite-time simulations, which undermines the precision of the reported boundaries, though not the qualitative phenomenon. The toy model cannot be used to calibrate the HR cutoff, so the HR claims need additional support to be fully convincing.","major_comments":[{"comment":"The stochastic birhythmicity region in the HR model is defined by a 2% occurrence cutoff over about 20,000 oscillations per parameter point, a threshold that the authors themselves call 'rather arbitrary.' The deterministic ghost has a long, parameter-dependent lifetime (Fig. 4), so finite-time simulations can show both z-amplitude classes even if the stationary distribution is unimodal. The paper's own discussion in Sec. III ('Longer simulations would allow for the less likely stable states to be eventually observed') shows that finite-time statistics underrepresent rare states. Consequently, the reported boundaries—including the narrowing before widening and the crossing points at approximately ε=2×10^-3 and 3×10^-3—are not established as stationary properties and may shift with the cutoff, the simulation length, and the initial-condition protocol. The authors should either base the boundaries on an estimated stationary distribution (with convergence checks) or show a sensitivity analysis of Fig. 6 with respect to these choices, and temper the claim that the noise 'effectively extends the birhythmicity region' to reflect the operational nature of the definition.","section":"Section II, Fig. 6 and following text"},{"comment":"The classification of bursts into 2-cycle and 3-cycle uses a fixed z-amplitude threshold of 0.9. The z-amplitude distributions broaden with increasing noise (Fig. 5), and for large ε the amplitude ranges of the two classes appear to overlap, as the authors note that amplitudes 'fluctuate greatly from their average values.' This raises the possibility that the 2% proportions in Fig. 6 are sensitive to the chosen threshold. The authors should either count spikes directly to classify bursts or demonstrate that the phase diagram in Fig. 6 is robust to reasonable variation of the threshold.","section":"Section II, caption of Fig. 3 and text following Fig. 5"},{"comment":"The sentence 'In all cases, the system exhibits stochastic birhythmicity, without distinguishing between actual attractors and their ghosts' blends two distinct notions. In the toy model, stochastic birhythmicity is rigorously defined through the bimodality of the stationary Fokker-Planck distribution. In the HR model, the evidence outside the coexistence region consists of finite-time observations of two amplitude classes, which can also arise from long transients associated with a ghost in a system with a single attractor. The paper should clarify whether stochastic birhythmicity in the HR model is intended as a stationary, bimodal property (and if so, provide evidence from long-time stationary histograms) or as a finite-time switching phenomenon (and if so, avoid the implication of two coexisting stable states).","section":"Section II, paragraph after Fig. 4"}],"minor_comments":[{"comment":"The symbol r denotes both the two-dimensional vector and its Euclidean norm; please use a distinct symbol for the radial coordinate (e.g., ρ) to avoid ambiguity.","section":"Sec. III, Eq. (3)"},{"comment":"The phrase 'replaced by an new notion' should be 'replaced by a new notion.'","section":"Sec. III, text after Eq. (7)"},{"comment":"The term 'sadle-node' is a typo for 'saddle-node.'","section":"Sec. III, below Eq. (14)"},{"comment":"The caption states that the saddle-node branches emerge from the deterministic saddle-node and Hopf bifurcations at (b,ε)=(0,0) and (1,0); please label which branch corresponds to each deterministic bifurcation origin, since both branches meet at the cusp.","section":"Figure 10 caption"},{"comment":"The PACS numbers listed (02.02.30.Oz, 02.02.50.Ey, 05.05.10.Gg) appear malformed; they should be checked against the standard PACS scheme.","section":"Title page, PACS numbers"}],"recommendation":"major_revision","confidential_remarks":"The analytical toy model is a solid, self-contained contribution and should be publishable once the HR numerical claims are strengthened. The paper is within the scope of nonlinear dynamics journals. I see no citation or novelty concerns; the prior work by Slepukhina et al. is properly cited and clearly extended. The main risk is overclaiming for the HR model based on an operational cutoff rather than stationary properties; that needs to be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result—that additive noise lets the Hindmarsh-Rose model switch between 2-spike and 3-spike bursting outside the deterministic bistable strip (b < 2.9082 or b > 2.9231)—is credible. The direct evidence in Fig. 3, bimodal z-amplitude histograms for b outside the coexistence region, is hard to explain away. The deterministic ghost transients in Fig. 4 make the mechanism clear. And the toy model in Sec. III gives a genuinely clean analytical foundation: exact stationary Fokker-Planck solution, stochastic saddle-node branches in Eq. (14), and a correctly derived cusp at (-1/3, 4/(3*sqrt(3))). I checked the math; it holds together.\n\nThe paper is also honest. It explicitly calls the 2% cutoff rather arbitrary, and it acknowledges that in the toy model longer simulations would reveal the less likely stable state. That is better than many papers in this area.\n\nThe soft spot is real but localized. The quantitative stochastic-birhythmicity boundaries in Fig. 6 rest on a 2% occurrence threshold over about 20,000 oscillations, a fixed z-amplitude threshold of 0.9, and no error bars. Near a saddle-node ghost, deterministic dwell times are long and parameter-dependent, so finite simulation windows can underrepresent rare states. The boundaries shown, including the reported narrowing-then-widening, should not be treated as stationary properties without longer runs or convergence tests. The authors own finite-time remark about the toy model applies to the HR simulations too. No code or data is provided, which is a real omission for a numerically heavy figure.\n\nNone of this undermines the qualitative claim. The bimodal histograms outside the deterministic region are direct evidence, and the toy model shows analytically that stochastic branches cross the deterministic saddle-node. The gap is between the solid qualitative result and the specific quantitative phase diagram, which is a de facto observation-window-dependent map rather than an asymptotic phase boundary.\n\nWho gets value from this paper: anyone working on noise-induced dynamics in neuron models, stochastic bifurcations, or ghost effects. It deserves a serious referee. I would send it to review and, if I were refereeing, ask for error estimates or longer simulations on Fig. 6, a statement about observation-window dependence, and ideally code release. With moderate revision it is a solid addition to the literature.","headline":"Noise-induced birhythmicity beyond deterministic coexistence in the Hindmarsh-Rose model is credible and the analytic toy model is clean; the quantitative HR phase diagram is cutoff-dependent, but that soft spot does not sink the main claim.","tokens_in":691,"tokens_out":1639,"would_cite":true,"duration_ms":59150,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34F05","37G15","60H10","92C20"],"pacs":["02.30.Oz","02.50.Ey","05.10.Gg"],"model":"deepseek-v4-flash","headline":"Noise extends neuron birhythmicity: the Hindmarsh-Rose model switches between two bursting rhythms even outside the deterministic coexistence interval.","keywords":["Hindmarsh-Rose model","stochastic birhythmicity","additive noise","ghost attractor","saddle-node bifurcation","bursting oscillations","Fokker-Planck equation","bistability extension"],"falsifier":"Run the Hindmarsh-Rose model at $b=2.906$ with $\\varepsilon$ in the range $1\\times 10^{-3}$ to $6\\times 10^{-3}$ for at least $10^6$ bursting oscillations and count how often a 2-cycle (with $z$-amplitude below 0.9) occurs: if the proportion stays below the 2% cutoff for all $\\varepsilon$ below the claimed crossing, the extension of the birhythmicity region is a finite-time artifact. For the simple model, measure the radial distribution at $\\varepsilon=0.9$: since 0.9 exceeds the cusp value $4/(3\\sqrt{3})\\approx 0.77$, theory predicts a single maximum everywhere, so observing two well-separated peaks for any $b$ would refute the analytic saddle-node curves.","tokens_in":11974,"feed_emoji":"🧠","tokens_out":8379,"duration_ms":77667,"temperature":0.7,"pith_summary":"The paper argues that adding white noise to the Hindmarsh-Rose neuron model creates birhythmicity — random switching between a two-spike and a three-spike bursting rhythm — even for parameter values where the deterministic model has only one stable bursting rhythm. The mechanism is the ghost of the lost attractor near a saddle-node bifurcation: noise keeps kicking the trajectory into a state that would be stable in a nearby parameter regime, where it lingers before escaping. The authors map this noise-induced birhythmic region in the (noise intensity, control parameter) plane of the Hindmarsh-Rose model, and they reproduce and explain the effect in a simpler axially symmetric model whose stationary Fokker-Planck distribution can be solved analytically. The upshot is that noise effectively widens the birhythmic region beyond the deterministic coexistence interval.","feed_headline":"Noise extends neuron birhythmicity past the noiseless limit","feed_subtitle":"White noise keeps neurons switching between 2- and 3-spike bursts even where only one rhythm exists.","key_machinery":"The load-bearing object is the ghost of a limit cycle near a saddle-node bifurcation: a transiently attracting, long-lived orbit that exists on the side of the bifurcation where the periodic solution has disappeared. In the Hindmarsh-Rose model the ghost is triggered by additive noise on the slow variable $z$, and the paper detects it through the $z$-amplitude, whose value separates two-spike bursts (below 0.9) from three-spike bursts (above 0.9). The analytical backbone is a simple axially symmetric model with radial dynamics $f(r) = -r\\left((r^2-1)^2-b\\right)$ and a constant rotation term; for this model the stationary Fokker-Planck equation has the exact solution $\\rho_s(r) = C \\exp(-2u(r)/\\varepsilon^2)$ with $u(r) = \\tfrac12\\left[\\tfrac13(r^2-1)^3 - b r^2\\right]$, so stochastic equilibrium states are the extrema of the radial distribution, given by $b(r) = -\\varepsilon^2/(2r^2)+(r^2-1)^2$. The folds of this branch are stochastic saddle-node bifurcations, parameterized by $b=(1-r^2)(1-3r^2)$ and $\\varepsilon = 2r^2\\sqrt{1-r^2}$, and they merge in a cusp at $(b,\\varepsilon)=(-1/3,\\,4/(3\\sqrt{3}))$, the analogue of a stochastic pitchfork.","core_discovery":"On the paper's own terms, the central discovery is that stochastic birhythmicity is not confined to the deterministic bistable interval $2.9082 \\le b \\le 2.9231$ of the Hindmarsh-Rose model. For $b$ just outside this interval, where only the 2-cycle or only the 3-cycle is a stable bursting attractor, sufficiently strong additive noise induces recurrent transitions between that attractor and the ghost of the other bursting cycle, so the system spends time in both rhythms and the histogram of $z$-amplitudes shows two distinct peaks. The paper therefore treats the noise as extending the birhythmicity region, with the left boundary crossing the deterministic saddle-node near $\\varepsilon \\approx 2\\times 10^{-3}$ and the right boundary near $\\varepsilon \\approx 3\\times 10^{-3}$. The same qualitative behavior is established in a simple two-variable model with a radial potential and a single saddle-node bifurcation; there, stationary probability maxima are computed from the exact Fokker-Planck solution and the boundaries of stochastic birhythmicity are given analytically as a parametric curve ending in a cusp.","pith_inferences":["Beyond what the paper states, the same ghost-based mechanism should operate near every saddle-node fold in the bursting branch, not just the 2-cycle/3-cycle fold studied here; other spike-count transitions in the Hindmarsh-Rose parameter space should exhibit similar noise-driven switching.","The simple model's exact stationary distribution suggests a quantitative scaling: the width of the stochastic birhythmic region in the control parameter should grow roughly like a power of $\\varepsilon$ near each deterministic fold, so measuring that width in the Hindmarsh-Rose model could test whether a universal shape law holds.","Biologically, this implies that channel-noise variability could let a neuron express a rhythm not encoded in its deterministic parameters, effectively adding an extra functional mode that could be either computational flexibility or a source of variability, depending on the context."],"forward_implications":["In the Hindmarsh-Rose model there is a de facto stochastic birhythmicity region in the $(b,\\varepsilon)$ plane that is wider than the deterministic one once $\\varepsilon$ exceeds roughly $2\\times 10^{-3}$ on the left and $3\\times 10^{-3}$ on the right; for large noise, both bursting rhythms occur with comparable probability.","The stochastic birhythmicity region narrows before it widens as noise increases: near $\\varepsilon \\approx 10^{-3}$ it shrinks to about a third of the deterministic interval because transitions preferentially fall into the lower-energy 2-cycle state.","In the simple model, noise creates a large-orbit stochastic state for $b<0$, beyond the deterministic saddle-node bifurcation, and the stochastic saddle-node branches delimit a pointy birhythmicity region with a cusp; above the cusp noise the two stochastic states merge into one.","Since the effect depends only on saddle-node ghosts, any neuron model whose bursting branch folds at a saddle-node bifurcation should show analogous noise-driven two-rhythm switching near the fold."],"supporting_citations":[{"why":"Defines the Hindmarsh-Rose model and its bursting behavior, providing the starting point of the study.","marker":"[3]"},{"why":"Supplies the fixed parameter values and the prior observation of noise-induced transitions within the deterministic birhythmic region.","marker":"[6]"},{"why":"Maps the homoclinic and saddle-node organization of the Hindmarsh-Rose bursting branches, grounding the claim that branches fold and add one spike per fold.","marker":"[7]"},{"why":"Establishes the ghost phenomenon near saddle-node bifurcations, the mechanism invoked for the extended birhythmicity region.","marker":"[1]"},{"why":"Provides the saddle-node normal form and ghost behavior, plus the energy-based reasoning used to explain transition asymmetries.","marker":"[8]"},{"why":"Supplies the Fokker-Planck stationary-solution framework used to derive the exact radial probability distribution in the simple model.","marker":"[10]"}],"fun_headline_variants":["Noise resurrects ghost rhythm outside deterministic bistable zone","White noise births two burst rhythms in neuron model","Stochastic birhythmicity appears beyond classical saddle-node limits","Noise-induced two-rhythm switching beyond bifurcation boundary","Ghost attractor revealed by noise in Hindmarsh-Rose"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that the numerically estimated boundaries of stochastic birhythmicity — based on a 2% occurrence cutoff over about 20,000 oscillations and a $z$-amplitude threshold of 0.9 — represent the asymptotic stationary behavior; the simulations in the simple model show that finite observation time can hide rare states, so the Hindmarsh-Rose boundaries could shift with longer runs or a different cutoff.","fun_headline_variants_meta":{"raw":{"variants":["Noise resurrects ghost rhythm outside deterministic bistable zone","White noise births two burst rhythms in neuron model","Stochastic birhythmicity appears beyond classical saddle-node limits","Noise-induced two-rhythm switching beyond bifurcation boundary","Ghost attractor revealed by noise in Hindmarsh-Rose"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001354,"raw_usage":{"total_tokens":5490,"prompt_tokens":930,"completion_tokens":4560,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":4491}},"tokens_in":546,"tokens_out":4560,"duration_ms":35425,"temperature":1.0,"reasoning_tokens":4491,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:51:42.207257+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Hindmarsh-Rose model at $b=2.906$ with $\\varepsilon$ in the range $1\\times 10^{-3}$ to $6\\times 10^{-3}$ for at least $10^6$ bursting oscillations and count how often a 2-cycle (with $z$-amplitude below 0.9) occurs: if the proportion stays below the 2% cutoff for all $\\varepsilon$ below the claimed crossing, the extension of the birhythmicity region is a finite-time artifact. For the simple model, measure the radial distribution at $\\varepsilon=0.9$: since 0.9 exceeds the cusp value $4/(3\\sqrt{3})\\approx 0.77$, theory predicts a single maximum everywhere, so observing two well-separated peaks for any $b$ would refute the analytic saddle-node curves.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Hindmarsh-Rose model and its bursting behavior, providing the starting point of the study."},{"cited_title":"Slepukhina, I","cited_arxiv_id":null,"evidence_quote":"Supplies the fixed parameter values and the prior observation of noise-induced transitions within the deterministic birhythmic region."},{"cited_title":"Barrio, S","cited_arxiv_id":null,"evidence_quote":"Maps the homoclinic and saddle-node organization of the Hindmarsh-Rose bursting branches, grounding the claim that branches fold and add one spike per fold."},{"cited_title":"Izhikevich, Dynamical systems in neuroscience: the geometry of excitability and bursting","cited_arxiv_id":null,"evidence_quote":"Establishes the ghost phenomenon near saddle-node bifurcations, the mechanism invoked for the extended birhythmicity region."},{"cited_title":"Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering","cited_arxiv_id":null,"evidence_quote":"Provides the saddle-node normal form and ghost behavior, plus the energy-based reasoning used to explain transition asymmetries."},{"cited_title":"Gardiner","cited_arxiv_id":null,"evidence_quote":"Supplies the Fokker-Planck stationary-solution framework used to derive the exact radial probability distribution in the simple model."}],"review_version":1}