{"id":"baf65690-f109-4f38-9714-6cfa16ec62bd","arxiv_id":"2505.09199","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A nonlinear predictive coding network supports bistable traveling waves whose speed sign determines upward or downward propagation, with numerically found thresholds for stimulus strength and duration.","lead":"Neural networks with predictive coding feedback are modelled as nonlinear wave systems; the paper shows when an input spreads upward through layers, downward, or fails to spread at all. The results connect mathematical wave speed signs to theories of perception and psychiatric conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The semi-infinite propagation thresholds s0* and tau* are numerical observations, not established results; the paper provides no proof of the sharp stagnation/propagation dichotomy or of convergence to the stationary solutions it names.","rationale":"The reader's weakest assumption is precisely the unproved numerical threshold, and my read confirms that this is the main load-bearing gap. The bi-infinite traveling-wave construction is on much firmer ground: the comparison principle is proved in the appendix and the front existence is tied to a known theorem, so I do not see a correctness issue there. The semi-infinite sections, however, define propagation via convergence to stationary solutions whose existence and uniqueness are not proved, and the claimed sharp thresholds rest entirely on unspecified numerical simulation. This is a genuine gap between the strength of the claim (an exact dichotomy with precise thresholds) and the evidence presented (plots plus a sentence stating that numerical investigations show a threshold). I therefore agree with CONDITIONAL: the paper should be published only if the threshold results are either proved or explicitly downgraded to numerical observations with error analysis, and the abstract's 'precisely determine' should be qualified accordingly. My critique does not add a new concern beyond the reader's, so the verdict is unchanged.","tokens_in":25814,"tokens_out":9749,"duration_ms":105017,"concrete_test":"Reproduce the constant-input threshold for one representative case, e.g. (theta,mu,p)=(0.5,16,0.1), q=0.35 in (3.10)-(3.11). Use a high-accuracy integrator (rtol=atol=10^-12) on domains J=200,400,800 and final times T=10^3,10^4,10^5. For each s0 near the reported s0*, track a level set of the interface and measure its velocity in the last half of the simulation. If the bisection estimate s0*(T,J) changes systematically with T or J, or if s0 just below the reported threshold shows a positive interface velocity that decays only algebraically in T, then the sharp threshold is an artifact of finite-horizon numerics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rigorous core of the paper is the bi-infinite traveling-wave analysis: the comparison principles in Lemmas A.1 and A.2 are proved, and the front existence follows from Mallet-Paret's theorem. The load-bearing gap is in Sections 3.2 and 3.3. The abstract promises precise conditions for propagation versus failure in the semi-infinite network, but the sharp threshold s0* is introduced with the statement 'Our numerical investigations ... show the existence of a sharp threshold' (Section 3.2.1), while the flashed-input threshold tau* is only 'numerically computed' (Sections 3.2.2 and 3.3.2). For the claimed dichotomy to be true one needs: (i) existence and uniqueness of the stationary solutions xd(s0) and xu(s0) defined by (3.12)-(3.13); (ii) actual convergence of the trajectory to one of them, not merely comparison bounds; and (iii) a guarantee that the numerically observed boundary between the two basins is not a finite-time or finite-domain artifact. Near the threshold the relevant front speed tends to zero, so any finite-time simulation will misclassify a very slow front as stagnation unless the time horizon is controlled with care. No numerical scheme, tolerances, domain size, or convergence checks are reported, so the claimed sharpness of s0* and tau* is not supported by the evidence presented. This matters because the abstract's 'precisely determine' and the biological conclusions in Sections 3.4 and 4 rely on the exact threshold dichotomy.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives a continuous-time lattice neural-network model motivated by predictive coding, reduces it to the scalar one-population equation (3.1), and analyzes propagation in three settings: bi-infinite networks, semi-infinite networks with constant boundary input, and semi-infinite networks with flashed boundary input. For the bi-infinite bistable regime it claims existence, uniqueness, and monotonicity of traveling fronts by invoking Mallet-Paret's theorem, with wave speeds whose signs are computed numerically over the parameter set P. For the semi-infinite settings it claims that a sharp input-amplitude threshold s0* separates stagnation from upward or downward propagation and that a flashed input has a sharp presentation-time threshold tau*, with the threshold values determined numerically. The paper closes with biological interpretations relating the parameter regions to predictive-coding phenomena and to computational psychiatry.","tokens_in":26088,"tokens_out":7053,"duration_ms":72227,"significance":"If the semi-infinite threshold claims were established, the paper would provide a useful nonlinear extension of the authors' earlier linear analysis [21], with a clean parameter-space classification of bottom-up and top-down propagation and falsifiable predictions about propagation failure. The rigorous parts are genuine strengths: the comparison principles in Lemmas A.1 and A.2 are proved, the parameter regime P is explicit, and the bistable traveling-wave existence is delegated to a verifiable external theorem. However, the central semi-infinite claims--the sharp dichotomy at s0* and tau*--are numerical observations rather than theorems, and the manuscript contains no numerical-convergence evidence supporting their sharpness. The contribution is therefore conditionally significant: the rigorous bi-infinite core is solid, while the semi-infinite propagation results currently outrun their proof.","major_comments":[{"comment":"The sharp threshold dichotomy for s0* is not proved. The text states: \"Our numerical investigations ... show the existence of a sharp threshold\" and then asserts that for all s0 in [xd(θ,µ),s0*) there is stagnation and for all s0>s0* there is propagation. These two statements require, respectively, existence and uniqueness of the stationary solutions xd(s0) and xu(s0) in (3.12)-(3.13), and actual convergence of the trajectory to one of them. The comparison principle in Lemma A.2 only yields the a priori interval bound vj(t) in [xd,xm] for s0 in [xd,xm]; it does not establish convergence to xd(s0), nor does it rule out alternative asymptotic states. Near the threshold the associated bi-infinite wave speed tends to zero, so any finite-time simulation can misclassify a very slow front as stagnation; the manuscript reports no time horizon, domain truncation, or convergence criterion. Since the abstract promises to \"precisely determine\" the propagation conditions, this gap is load-bearing.","section":"Section 3.2.1 (Eqs. (3.12)-(3.13))"},{"comment":"The flashed-input threshold tau* is only \"numerically computed\"; there is no theorem asserting that the boundary between propagation failure and propagation is sharp. The classification into propagation failure, stacked interface, and front propagation is made from the sign ordering of cu→d and cd→u, but no proof is given that these sign conditions are sufficient for the stated convergence statements on the semi-infinite domain. The critical equal-speed case 0s0*, implies convergence of the semi-infinite trajectory to the stationary solution yu(s0) in (3.17). The definitions of stagnation and propagation in this subsection also assume without proof the existence and uniqueness of the stationary solutions yd(s0) and yu(s0). Since the top-down results are presented as a complement to the bottom-up analysis and feed into the biological interpretations in Section 3.4, the missing convergence proof is not merely cosmetic.","section":"Section 3.3.1"},{"comment":"The numerical evidence for the thresholds is not verifiable as reported. The manuscript specifies parameter values (µ=16, p=0.1 and selected q,θ) but gives no discretization of (3.10), no time-stepping scheme, no domain size, no tolerances, and no criterion for deciding that a solution has converged to a stationary state versus propagating with a very small speed. Because the sharpness of s0* and tau* is the central novel claim for the semi-infinite model, the authors should either provide a reproducible numerical protocol with convergence checks or reformulate the statements as numerical observations rather than exact thresholds.","section":"Figures 7, 8, 11, 13, 16"}],"minor_comments":[{"comment":"The invocation of Mallet-Paret's theorem in [28] should be made checkable by stating the precise theorem and explicitly verifying its hypotheses. The manuscript checks ∂uN>0 and ∂wN>0 and the bistable structure of Fp, but the reader is left to infer that these are the exact conditions required by the cited result.","section":"Section 3.1.3"},{"comment":"In the top-down stationary equations the limit condition \"yj → j→+∞ xd(θ,µ)\" should read j→−∞; the sequence is indexed over j≤−1, so the far-field limit is at −∞. This appears twice and is a mathematical typo that should be corrected.","section":"Eqs. (3.16)-(3.17)"},{"comment":"The initial conditions in (3.11) and (3.15) write vj(0)=ud(θ,µ), but the stable stationary states constructed in Section 3.1.1 are denoted xd(θ,µ) and xu(θ,µ). The notation should be unified.","section":"Section 3.2.1 and 3.3.1"},{"comment":"Several lines in the proof of Lemma A.1 contain malformed expressions, e.g. \"p(S′(vℓ(t))−S′(wℓ(Tϵ))wℓ−1(Tϵ)\" has mismatched parentheses and a stray t, and a similar expression appears later. These should be rewritten as p(S′(vℓ(Tϵ))−S′(wℓ(Tϵ)))wℓ−1(Tϵ) or the intended equivalent. As written, this part of the proof cannot be parsed unambiguously.","section":"Appendix A, Lemma A.1 proof"},{"comment":"The symbol q0 is used with different meanings in the bottom-up case (smallest q with cu→d=0) and the top-down case (largest q with cd→u=0). This overloaded notation is confusing when Figures 7 and 13 are compared; distinct symbols would improve clarity.","section":"Sections 3.2.1 and 3.3.1"}],"recommendation":"major_revision","confidential_remarks":"The core issue is that the abstract promises precise determination of propagation thresholds while the semi-infinite sharp-threshold results are numerical observations with no proof and no reported convergence checks. This is a mismatch between claims and evidence in a mathematics journal. The bi-infinite traveling-wave core appears sound, so the paper could become acceptable if the authors either prove the threshold dichotomy in a tractable regime or carefully restrict all claims to numerically observed behavior with full numerical protocols."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does something real. It takes the linear predictive-coding hierarchy model from the authors' earlier paper, adds a sigmoidal nonlinearity, and treats the resulting lattice equation as a bistable discrete reaction-diffusion system. The rigorous core is the application of Mallet-Paret's theorem to get existence and uniqueness of monotone traveling fronts between the down and up states on the bi-infinite lattice, plus a genuine comparison principle proved in Appendix A. That part is solid. The wave speed sign is then computed numerically as a function of parameters, showing propagation failure (pinning) regions, which is the expected behavior for bistable lattice equations. The numerical sign diagrams are plausible, and the symmetry cu->d(theta) = cd->u(1-theta) is a nice observation.\n\nThe genuinely new contribution is the nonlinear version, and the biological interpretation in terms of bottom-up versus top-down propagation and the schizophrenia-autism continuum is speculative but clearly hedged. I don't see a circularity problem: the new results are not re-derivations of the linear model; they extend it.\n\nWhere the paper is soft is exactly where the stress-test lands. The semi-infinite threshold results (s0*, tau*) are numerical observations, not theorems. The abstract says “precisely determine” for both bi-infinite and semi-infinite, but the semi-infinite part is explicitly only “numerically demonstrated” later. The paper states “Our numerical investigations ... show the existence of a sharp threshold” — that's honest, but it is not proof. Near the threshold, the front speed approaches zero, so any finite-time simulation will misclassify a very slow front as stagnation unless the time horizon is carefully controlled. No numerical scheme, tolerances, domain size, or convergence checks are reported. That matters because the biological conclusions rely on the existence of a sharp boundary.\n\nThis doesn't sink the paper. The bi-infinite analysis is a real contribution, and the threshold phenomena are plausible and well-illustrated. But the authors should either prove the threshold dichotomy in a tractable regime, state it explicitly as a conjecture supported by numerics, or at minimum report their numerical method and convergence checks. As written, the abstract oversells the semi-infinite results.\n\nI would send this to a serious referee. The math is nontrivial, the literature is engaged, and the comparison-principle proof is verifiable. With a revision that clearly separates theorem from numerical observation and strengthens the numerics reporting, this could be a good paper for a mathematical biology or applied analysis journal.","headline":"A genuinely useful nonlinear extension of predictive-coding lattice dynamics with a rigorous bi-infinite wave front analysis, but the semi-infinite sharp thresholds s0* and tau* are numerical observations that the abstract oversells.","tokens_in":26639,"tokens_out":1971,"would_cite":true,"duration_ms":20105,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35C07","37L60","92B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a nonlinear predictive-coding network, the direction of activity propagation is controlled by the signs of two unique traveling-wave speeds, and input thresholds separate propagation from stagnation.","keywords":["predictive coding","hierarchical neural networks","traveling waves","bistable dynamics","propagation failure","sigmoid nonlinearity","input threshold","lattice dynamical systems"],"falsifier":"Fix a parameter set in $\\mathcal{P}$ with $c_{u\\to d}>0$, run the semi-infinite system (3.10)-(3.11) with constant input for amplitudes $s_0$ in a fine grid straddling the reported $s_0^*$, and check the long-time limit; the sharp-threshold claim fails if any $s_0s_0^*$ yields uniform convergence to $x_d(s_0)$.","tokens_in":25635,"feed_emoji":"🧠","tokens_out":9095,"duration_ms":81879,"temperature":0.7,"pith_summary":"This paper analyzes a continuous-time, nonlinear version of a hierarchical predictive-coding network and asks when a sensory input at the bottom layer propagates upward, when top-down activity propagates downward, and when propagation fails. In the bistable regime, where each layer has a stable down state and a stable up state, the authors show that the outcome is governed by two unique traveling-wave speeds, $c_{u\\to d}$ and $c_{d\\to u}$, whose signs determine the direction of invasion; in open parameter regions both speeds vanish and no propagation occurs. For semi-infinite networks receiving constant input at the first layer, they numerically identify a sharp amplitude threshold $s_0^*$ above which the up state propagates and below which activity stays localized. For flashed inputs, a minimal presentation time $\\tau^*$ separates failure from propagation, with the up state traveling as a front, a stacked interface, or a pulse depending on the relative speeds. The results map onto predictive-coding theories and single out parameter combinations that could correspond to over-reliance on sensory or endogenous activity.","feed_headline":"Wave speeds decide whether predictive-coding signals travel up or down","feed_subtitle":"Unique front speeds and input thresholds predict when a stimulus propagates through the hierarchy or dies out.","key_machinery":"The central machinery is the scalar lattice equation (3.1) with the three-point nonlinearity $N(u,v,w)=(1-p-q)(S(u)-v)+pS'(v)(u-S(v))+q(S(w)-v)$ acting between neighboring layers, where $S(x)=1/(1+e^{-\\mu(x-\\theta)})$ is the sigmoid activation. On a bi-infinite lattice the equation is bistable in the parameter regime $\\mathcal{P}$: the homogeneous stationary states are exactly the down state $x_d$, the unstable middle state $x_m$, and the up state $x_u$, and the dynamics between them is carried by monotone traveling fronts $\\Phi_{u\\to d}$ and $\\Psi_{d\\to u}$. A theorem of Mallet-Paret applied to this discrete bistable system supplies existence, uniqueness, and continuous dependence of the front speeds $c_{u\\to d}$ and $c_{d\\to u}$; the sign of those speeds, rather than their magnitude, is the quantity that determines the long-time behavior. A comparison principle (Lemmas A.1 and A.2) then extends the wave picture to semi-infinite domains and input-driven initiation, where the stationary solutions $x_d(s_0)$ and $x_u(s_0)$ of the boundary-value problems (3.12) and (3.13) serve as the stagnation and propagation targets.","core_discovery":"The central claim is that, for the one-population model obtained by setting the feedforward and feedback weight matrices to the identity, the propagation properties of the hierarchy are completely described by two monotone traveling fronts and their speeds. For every parameter set in the bistable regime $\\mathcal{P}$, there exist a unique decreasing front $\\Phi_{u\\to d}$ connecting the up state $x_u$ to the down state $x_d$ with speed $c_{u\\to d}$, and a unique increasing front $\\Psi_{d\\to u}$ connecting $x_d$ to $x_u$ with speed $c_{d\\to u}$. The sign of $c_{u\\to d}$ decides whether an up-state region invades a down-state background, and the sign of $c_{d\\to u}$ decides the reverse; when both speeds are zero the wave is pinned and the network cannot transmit activity. On semi-infinite networks, the same front structure produces a threshold: a constant input of amplitude $s_0$ below $s_0^*$ leaves the network in a stagnating state converging to $x_d(s_0)$, while above $s_0^*$ the up state propagates; a flashed input of duration $\\tau$ below $\\tau^*$ fades back to the down state, while longer flashes produce stacked-interface or front propagation. The speeds obey the symmetry $c_{u\\to d}(\\theta)=c_{d\\to u}(1-\\theta)$, so $\\theta=1/2$ is the balanced point where both directions behave identically.","pith_inferences":["Editorial inference: the traveling-pulse family at $\\theta=1/2$ with width controlled by $\\tau$ suggests a concrete coding scheme in which stimulus duration is represented by the spatial extent of a propagating pulse; a rigorous existence proof for this family is left open in the paper.","Editorial inference: the sharp-threshold dichotomy is a natural target for a theorem; comparison-principle monotonicity in $s_0$ might establish a unique critical amplitude $s_0^*$ at which the stable manifold of $x_u(s_0)$ appears, replacing the current numerical evidence.","Editorial inference: the symmetry $c_{u\\to d}(\\theta)=c_{d\\to u}(1-\\theta)$ hints at an up/down duality that could persist in two-population or asymmetric-weight extensions, where it could be tested by computing front speeds numerically.","Editorial inference: with the spike-frequency-adaptation term the authors propose, the permanent up state would become transient, and the bistable fronts studied here would be expected to turn into traveling pulses whose existence could be analyzed with the same profile equations augmented by a slow variable."],"forward_implications":["Upward propagation of a sensory input requires $c_{u\\to d}>0$: outside that parameter region the up state cannot invade the down state, no matter how strong the input.","In the pinning regions where $c_{u\\to d}=c_{d\\to u}=0$, the network exhibits intrinsic propagation failure, so information cannot travel from layer to layer.","For flashed inputs, the ordering of the two speeds selects the spatial pattern: a front when $c_{d\\to u}\\le 00$, run the semi-infinite system (3.10)-(3.11) with constant input for amplitudes $s_0$ in a fine grid straddling the reported $s_0^*$, and check the long-time limit; the sharp-threshold claim fails if any $s_0s_0^*$ yields uniform convergence to $x_d(s_0)$.","supporting_citations":[{"cited_title":"Mallet-Paret","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that guarantees existence, uniqueness, and continuous dependence of monotone traveling fronts and their speeds for the bistable lattice equation."},{"cited_title":"Chen, J-S Guo, and C-C Wu","cited_arxiv_id":null,"evidence_quote":"Proves exponential convergence of lattice solutions toward shifted traveling waves, linking the sign of the speed to long-time dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes stability of traveling wavefronts for the discrete Nagumo equation, underpinning the convergence result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the linear precursor of this model; the continuous-time nonlinear equations here are obtained by the same formal limit."},{"cited_title":"Erneux and G","cited_arxiv_id":null,"evidence_quote":"Frames the propagation-failure regions as the discrete-bistable pinning phenomenon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classical analysis of propagation failure in coupled discrete excitable systems, cited for the pinning intervals where wave speeds vanish."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Justifies the sigmoidal activation function as the canonical nonlinearity in neural-mass models."}],"review_version":1}