{"id":"becb49fe-a705-4d26-a375-ba1d8295be70","arxiv_id":"2501.03557","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Asymmetric excitatory-inhibitory coupling induces a rich high-order p/q frequency-locking structure in identical LIF neurons, with analytically computed existence, stability, and internal bifurcation boundaries.","lead":"A pair of identical leaky integrate-and-fire neurons, one excitatory and one inhibitory, synchronize at rational spike-count ratios despite having the same intrinsic firing rate. This work introduces an analytical event-driven map method that predicts which spike patterns exist and are stable, and how they bifurcate.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's analytical method is carefully constructed: it derives closed-form iterates of the event-driven map, imposes periodicity, solves the resulting threshold equations, and then filters unphysical solutions via the first-crossing conditions. The stability calculation is a standard linearization of the composite spike-to-spike map. I found no internal inconsistency in the derivation, and the paper is appropriately careful to label its broader structural claims as conjectural. The reader's weakest-assumption concern about model-specific reset and alpha-pulse synapses is not a correctness objection to the stated claim, which is explicitly about this LIF model; it is a scope limitation. The more defensible concern is that the analytical boundaries in Sec. V are produced by numerical continuation without code or data, so independent verification is needed before the fine-grained bifurcation structure can be fully trusted. However, that concern does not invalidate the central argument; it supports the existing CONDITIONAL verdict rather than requiring a stronger one.","tokens_in":31774,"tokens_out":19992,"duration_ms":204242,"concrete_test":"Independently reimplement the Sec. IV analytical procedure for the 2/13 tongue, including the three spike sequences {1,2^4,1,2^9}, {1,2^5,1,2^8}, and {1,2^6,1,2^7}, and recompute the existence/stability boundaries in Fig. 13; then compare those predicted boundaries against direct event-driven simulations at representative points in each region.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is internally consistent: the event-driven map reduction and the first-crossing conditions (Conditions 1 and 2 in Sec. IV.A) are necessary and sufficient for the scheduled spike sequence, and the stability criterion via the product of Jacobians in Eq. (34) is a standard monodromy construction for this class of reset maps. The conjectural statements—such as the universality of np/nq island families and mutual overlaps in Secs. III and V—are explicitly labeled as conjectures, and the analytical boundary results agree with simulations wherever checked. The principal limitation is reproducibility: no code or data is provided, and the boundary computations rely on unlisted continuation implementations, so an independent reimplementation is needed for full confirmation. This is a verification gap, not a detected flaw in the argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a pair of identical leaky integrate-and-fire neurons coupled by asymmetric excitatory-inhibitory pulse coupling, and demonstrates that high-order p/q frequency locking arises even though the uncoupled neurons have identical frequencies. The authors develop an analytical event-driven map method that determines existence and stability of any periodic spike sequence of a p/q frequency-locked state, including conditions (Conditions 1 and 2 in Sec. IV.A) that filter unphysical solutions arising from multiple threshold crossings of the inhibitory neuron. They use this method to compute existence and bifurcation boundaries for representative cases (0/1, 1/2, 2/13, 2/14, and others), showing that the p/q regions have internal bifurcation structure with saddle-node and grazing bifurcations, and that multistability occurs both within a spike sequence and between different p/q states, as well as between reducible np/nq islands. The simulation results for the (g, α) plane show a rich tongue-and-island structure with a devil's staircase and a Farey arrangement of spike sequences at large α.","tokens_in":31837,"tokens_out":7415,"duration_ms":73917,"significance":"If the analytical method is sound, this is a valuable contribution to the theory of pulse-coupled neurons: it provides a parameter-free framework to determine existence and stability of arbitrary periodic spike sequences in a pair of LIF neurons, and it demonstrates that high-order synchronization does not require frequency mismatch, but can be induced by coupling asymmetry alone. The paper also reports a novel Farey arrangement of spike sequences and identifies two new types of grazing bifurcations (Type-2b and Type-3). The analytical method is validated against direct simulations in several representative cases, which gives confidence in the central claim. The discovery of intra-sequence and inter-sequence multistability inside a single p/q tongue is an interesting and relatively unexplored phenomenon.","major_comments":[{"comment":"The event-driven map contains terms with factors 1/(α−1) and 1/(α−1)^2, e.g., in Hi(n) in Eq. (10) and in κ1, κ2 in Eq. (B15). The stated parameter range in Sec. III is 0 < α ≤ 30, which includes α = 1. For α = 1 the alpha-pulse f(u) = u e^{-u} is perfectly well defined, so the map should have a well-defined limiting form, but as written it diverges. The paper does not discuss this case or provide the limiting expression. Since the analytical method and the simulations rely on these equations across the full (g, α) range, this is a load-bearing gap that must be fixed by either excluding α = 1 explicitly or deriving the appropriate α = 1 form.","section":"Sec. II, Eq. (10) and Appendix B"},{"comment":"The analytical boundaries and bifurcation diagrams are obtained by solving the system of equations (23) and by 'continuation algorithms', but the paper gives no details of these numerical methods and provides no code or data. Consequently, an independent reader cannot reproduce the reported boundaries, the eigenvalue computations, or the bifurcation classifications. This is a significant reproducibility gap for a paper whose central claim is the validity of the analytical method, and it weakens the verification of the results beyond the few cases that are explicitly compared with simulations.","section":"Secs. IV and V; Figs. 12–18"},{"comment":"The definitions of the new grazing bifurcations (Type-2b and Type-3) are given only through schematic figures and verbal conditions, such as ∂x_max/∂c_k = 0 for Type-2b. The paper does not state how these conditions are evaluated in terms of the state variables and parameters, nor how they are coded in the continuation procedure. Without explicit equations defining the tangency condition and the bifurcation manifold, the claimed classification of boundaries in Figs. 14–16 is not fully verifiable. This is especially important because the paper uses these types to explain the disappearance of solutions and the structure of the tongues.","section":"Sec. IV.C, Type-2b and Type-3"}],"minor_comments":[{"comment":"There is a typo: the second term should be 2α dE_i/dt, not 2α dE_i/dt^2.","section":"Sec. II, Eq. (3)"},{"comment":"The discussion of reducible np/nq islands and their 'mutual overlaps' is introduced as a conjecture; this is acceptable, but the text should more clearly separate the numerically observed overlaps from analytically established ones, since the analytical evidence is given only for specific families (e.g., n/6n and n/7n).","section":"Sec. III, Fig. 4 and related text"},{"comment":"The notation τ*_{r(m+1)}(m) is defined, but the text does not explicitly state that τ*_{r(m+1)}(m) is the first positive root of Eq. (11) for the neuron that does not fire; adding a sentence clarifying this would help readability.","section":"Sec. IV.A, Step 4"},{"comment":"The caption says 'yellow color indicates the numerically obtained 1/2 frequency-locked region', but the figure also shows a region 'A' and 'B'; the text explains these, but the caption could more clearly identify the comparison between analytical boundaries and the numerical region.","section":"Sec. V.B, Fig. 12"},{"comment":"The paper would benefit from a brief statement about the treatment of the α = 1 case in the simulations, even if it is only to say that this value was avoided or handled by a limiting procedure.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"This is a solid computational study with a well-conceived analytical framework, and I see no evidence of circularity or fitting. The main issues are technical: the singularity at α = 1 in the event-driven map, and the lack of code/data and numerical details that would allow independent verification. Both are fixable. If the authors can provide the α = 1 limiting form and make the numerical procedures reproducible (e.g., supplementary code or a detailed description of the continuation equations), the paper would be suitable for publication. The claimed novelty of the Farey arrangement of spike sequences and the new grazing bifurcations is plausible, but the verification gap currently tempers the strength of these claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a solid, specialist paper that does something genuinely new. It shows that in an identical E-I pair of LIF neurons, asymmetric coupling alone generates a full p/q frequency-locking structure—not just the one-off 1/2 state found earlier for I-I pairs. The analytical method (event-driven maps with first-crossing conditions) is the real contribution: it corrects the earlier firing-time approach that produced unphysical solutions for inhibitory coupling, and it gives a clean way to compute existence and stability of any scheduled spike sequence without weak-coupling constraints. The analytical boundaries match simulations in the cases they show (0/1, 1/2, 1/7, 2/13, 2/14), and there is no curve fitting or circular inference. That is genuine.\n\nThe soft spots are mostly about reproducibility and generality. No code or data is provided, and the bifurcation curves come from continuation algorithms that are described only sketchily, so an independent reimplementation is needed to confirm the intricacy of Figs. 14–16. The \"island families for all p/q\" claim is explicitly conjectural, and the mutual-overlap claim is based on two families plus analogy. That is honestly labeled, but it means the paper's most general statements are on weaker footing than the core tongue structure. Also, the results are tied to the specific LIF-alpha-pulse model with instantaneous reset; the quantitative boundaries won't transfer to more biological models, though the qualitative phenomenon probably does. That is a scope limitation, not a defect.\n\nWho this is for: people working on pulse-coupled integrate-and-fire networks, synchronization in neuronal pairs, or non-smooth bifurcations in threshold systems. It is not a broad-audience paper. But it deserves a serious referee. I would ask for code/data and a more detailed account of the continuation procedure, and for the conjectures to be clearly separated (they mostly already are). I'd be comfortable with conditional acceptance after those additions.","headline":"Solid analytical and numerical study showing asymmetric E-I coupling alone produces rich p/q locking in identical LIF neurons; main gaps are reproducibility and the conjectural status of the island-family claims.","tokens_in":32385,"tokens_out":1760,"would_cite":true,"duration_ms":17088,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","37G15","92C20","37N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that two identical leaky integrate-and-fire neurons, when coupled asymmetrically as an excitatory-inhibitory pair, can synchronize in high-order p/q frequency-locked states, and it provides an event-driven map method that…","keywords":["high-order synchronization","p/q frequency locking","leaky integrate-and-fire neurons","excitatory-inhibitory pair","event-driven maps","grazing bifurcations","multistability","Farey tree"],"falsifier":"Simulate the full two-neuron ODE system with the same alpha-pulse synapses and first-crossing reset at a parameter point inside the analytically predicted 1/2 tongue while scanning many initial conditions; if no trajectory settles into the {1,2,2} spike sequence with stable linearized map, the central existence claim fails. Equivalently, hardware or biological E-I pairs that show no predicted p/q locking at the corresponding parameters would refute the model's quantitative relevance.","tokens_in":31532,"feed_emoji":"🧠","tokens_out":4711,"duration_ms":44603,"temperature":0.7,"pith_summary":"This paper establishes that two leaky integrate-and-fire neurons with identical intrinsic frequencies can lock into high-order p/q frequency ratios, where p spikes of one neuron accompany q spikes of the other, provided the coupling is asymmetric. The asymmetry is the natural one in an excitatory-inhibitory pair: both neurons send equal-strength pulses, but one pulse excites and the other inhibits. The authors show these p/q states organize into Farey-arranged tongues in the (g, alpha) parameter plane, carry their own internal bifurcation structure, and overlap to produce multistability. They introduce an event-driven map method that determines existence and linear stability of any periodic spike sequence and explicitly handles non-smooth grazing bifurcations, without restricting coupling strength or requiring frequency mismatch.","feed_headline":"Equal-frequency neuron pairs lock into high-order p/q rhythms","feed_subtitle":"An event-driven map predicts every spike sequence and its stability, exposing multistability inside each locking tongue.","key_machinery":"The central object is the event-driven map of the two-neuron network, which integrates the voltage x_i and the synaptic variables E_i, Q_i between consecutive network spikes, replacing the infinite sum over past alpha pulses with a differential equation for the synaptic current. A periodic spike sequence is found by expressing the state after p+q spikes in terms of p+q unknown interspike intervals, imposing threshold-crossing equations, and then filtering solutions with Condition 1, that reset happens at the first threshold crossing of the firing neuron, and Condition 2, that the non-firing neuron does not reach threshold earlier. Stability is computed from the product of Jacobian matrices over one period, and grazing bifurcations are identified by tangency of the excitatory neuron's extended voltage trajectory to the threshold.","core_discovery":"A pair of coupled LIF neurons with identical frequencies and asymmetric pulse coupling can exhibit high-order p/q frequency-locked states. In an E-I pair, equal reciprocal synaptic strengths produce asymmetry naturally: the neuron receiving inhibitory current has a non-monotonic voltage trajectory that can cross threshold multiple times, and the physical reset occurs at the first crossing. Using event-driven maps, the authors derive self-consistent equations for the interspike intervals of any scheduled periodic spike sequence, impose first-crossing and ordering conditions to reject unphysical roots, and construct a product Jacobian for linear stability. This yields the existence and stability of irreducible p/q tongues and reducible np/nq islands, including their internal bifurcations, with boundaries set by saddle-node bifurcations and three types of grazing bifurcations, two of which are new to this study.","pith_inferences":["If the mechanism is generic, other pairs of identical threshold oscillators with sign-asymmetric or directional coupling, not only chemical E-I neurons, should show similar p/q locking; this could be tested in electronic or optomechanical oscillator pairs.","The appearance of reducible np/nq regions only as small-alpha islands suggests a period-inflation cascade in parameter space, and mapping the full n-dependence might reveal scaling near the firing-death boundary. This is an editorial inference, not a claim of the paper.","Because intra-sequence and inter-sequence multistability involve different spike patterns with the same firing ratio, neural codes could in principle carry information in which p/q spike sequence is selected, not only in the firing rate. This extension is mine, not the paper's."],"forward_implications":["High-order p/q synchronization does not require two different intrinsic frequencies; coupling asymmetry alone can produce it in identical E-I neuron pairs.","The analytical method gives existence and stability of any p/q spike sequence at arbitrary coupling strength, removing unphysical firing-time roots that earlier approaches could produce for inhibitory coupling.","p/q frequency-locked regions in this system are not structure-less Arnold tongues; they contain internal bifurcations, spike-exchange boundaries, and multiple coexisting solutions.","The apparent abrupt endings of tongues and reducible np/nq islands in simulations are explained by multistability: the analytically computed regions continue smoothly, but other attractors win for the chosen initial condition.","At large alpha values, spike sequences obey a Farey arrangement: the sequence for a ratio P/Q is a concatenation of the sequences of its Farey parents."],"supporting_citations":[{"why":"Supplies the prior analytical formulation for high-order frequency locking in coupled integrate-and-fire neurons, which this paper extends by adding conditions for non-smooth bifurcations.","marker":"[47]"},{"why":"Reports the earlier single 1/2 frequency-locking observation in identical inhibitory neurons, the phenomenon this paper generalizes to a rich structure of p/q states.","marker":"[48]"},{"why":"Provides the firing-time map approach for pulse-coupled integrate-and-fire neurons, including the stability analysis that the new event-driven method builds on and improves.","marker":"[59]"},{"why":"Documents earlier examples of high-order p/q frequency locking in coupled LIF neurons under chemical pulse coupling, establishing the baseline this study expands.","marker":"[66]"},{"why":"Supplies the classification of Type-1 and Type-2 grazing bifurcations in forced integrate-and-fire neurons, which the paper adapts and extends with Type-2b and Type-3.","marker":"[86]"},{"why":"Shows how to convert the infinite sum of alpha-pulse synaptic currents into a differential equation, enabling the event-driven map formulation used throughout the paper.","marker":"[94]"},{"why":"Demonstrates linear stability analysis from event-driven maps for splay states, the technique the paper adapts to p/q frequency locking where spike order is not preserved.","marker":"[92]"}],"fun_headline_variants":["Asymmetric coupling creates p/q locks in identical neurons","E-I neuron pair: no frequency mismatch needed for p/q sync","Event-driven maps decode stable spike sequences in neurons","Multistability and Farey order in coupled neuron pairs","Identical-frequency neurons lock via asymmetric synapses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All quantitative boundaries assume the idealized leaky integrate-and-fire dynamics with instantaneous reset at the first threshold crossing and alpha-pulse synaptic currents; if real synapses or reset rules deviate from these, the exact p/q boundaries would shift.","fun_headline_variants_meta":{"raw":{"variants":["Asymmetric coupling creates p/q locks in identical neurons","E-I neuron pair: no frequency mismatch needed for p/q sync","Event-driven maps decode stable spike sequences in neurons","Multistability and Farey order in coupled neuron pairs","Identical-frequency neurons lock via asymmetric synapses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3498,"prompt_tokens":1058,"completion_tokens":2440,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":2363}},"tokens_in":674,"tokens_out":2440,"duration_ms":18313,"temperature":1.0,"reasoning_tokens":2363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:53:09.469121+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full two-neuron ODE system with the same alpha-pulse synapses and first-crossing reset at a parameter point inside the analytically predicted 1/2 tongue while scanning many initial conditions; if no trajectory settles into the {1,2,2} spike sequence with stable linearized map, the central existence claim fails. Equivalently, hardware or biological E-I pairs that show no predicted p/q locking at the corresponding parameters would refute the model's quantitative relevance.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prior analytical formulation for high-order frequency locking in coupled integrate-and-fire neurons, which this paper extends by adding conditions for non-smooth bifurcations."},{"cited_title":"Montoya, M","cited_arxiv_id":null,"evidence_quote":"Reports the earlier single 1/2 frequency-locking observation in identical inhibitory neurons, the phenomenon this paper generalizes to a rich structure of p/q states."},{"cited_title":"Ermentrout and D","cited_arxiv_id":null,"evidence_quote":"Provides the firing-time map approach for pulse-coupled integrate-and-fire neurons, including the stability analysis that the new event-driven method builds on and improves."},{"cited_title":"Chartrand, M","cited_arxiv_id":null,"evidence_quote":"Documents earlier examples of high-order p/q frequency locking in coupled LIF neurons under chemical pulse coupling, establishing the baseline this study expands."},{"cited_title":"Jain and S","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of Type-1 and Type-2 grazing bifurcations in forced integrate-and-fire neurons, which the paper adapts and extends with Type-2b and Type-3."},{"cited_title":"Tattini, S","cited_arxiv_id":null,"evidence_quote":"Shows how to convert the infinite sum of alpha-pulse synaptic currents into a differential equation, enabling the event-driven map formulation used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates linear stability analysis from event-driven maps for splay states, the technique the paper adapts to p/q frequency locking where spike order is not preserved."}],"review_version":1}