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arxiv: 2606.17611 · v1 · pith:HLAGWA3Snew · submitted 2026-06-16 · 🧮 math.AP

Asymptotic dynamics of inhibitory networks for the NNLIF Model in the large-delay limit

Cl\'ement Rieutord (LJLL (UMR\_7598)) , Delphine Salort (LJLL (UMR\_7598)) This is my paper

Pith reviewed 2026-06-26 23:54 UTC · model grok-4.3

classification 🧮 math.AP
keywords NNLIF modelsynaptic delayinhibitory networkspseudo-equilibriaDoeblin-Harris methodCesàro convergenceasymptotic dynamicslarge-delay limit
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The pith

As delay tends to infinity, inhibitory NNLIF network solutions oscillate between distinct pseudo-equilibria with Cesàro-mean convergence to a limit set by those equilibria.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the effect of very large synaptic delays on the long-term behavior of inhibitory networks in the NNLIF model. It shows that solutions switch between different pseudo-equilibria over any finite time window once the delay is taken to infinity. The Doeblin-Harris method is used to prove that the time averages converge locally to a function fixed entirely by those pseudo-equilibria. A reader would care because the result ties the emergence of periodic activity directly to the size of the delay and the strength of inhibition.

Core claim

As the delay tends to infinity, solutions of sufficiently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Cesàro mean toward a limit function determined solely by these pseudo-equilibria.

What carries the argument

The Doeblin-Harris method applied to the delayed NNLIF system, which produces Cesàro-mean convergence once the delay is sent to infinity and the network is sufficiently inhibitory.

If this is right

  • Network activity is governed by switching among pseudo-equilibria once the delay is large.
  • Time averages of the solutions converge locally to a function fixed only by the pseudo-equilibria visited.
  • The oscillation and convergence hold uniformly on finite time intervals in the infinite-delay limit.
  • The described behavior requires the network to be sufficiently inhibitory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result supplies a delay-driven mechanism that can produce rhythmic firing in purely inhibitory circuits without external periodic drive.
  • Similar large-delay limits could be examined in other neural population models that admit pseudo-equilibria.
  • Quantitative error bounds for finite but large delays would turn the limit statement into a practical approximation tool.

Load-bearing premise

The networks must be sufficiently inhibitory and the pseudo-equilibria must exist with the properties needed for the Doeblin-Harris method to apply to the delayed equations.

What would settle it

Numerical integration of the NNLIF equations at successively larger delay values that either confirms or fails to show switching between the predicted pseudo-equilibria together with convergence of the time averages to the claimed limit function.

Figures

Figures reproduced from arXiv: 2606.17611 by Cl\'ement Rieutord (LJLL (UMR\_7598)), Delphine Salort (LJLL (UMR\_7598)).

Figure 1
Figure 1. Figure 1: shows the simulation of the delayed Fokker-Planck equation with delay d = 1 [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nd(t) and N∞(d ·) when d(= 20) is large and b ∗ < b(= −5) < 0. fig:2 A rigorous proof of such a result, the global stability of the stationnary state in the delayed NNLIF for weakly non linear regime with exponential rate can be found in Section 5 of [12] as well as in [5] [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Nd(t) and N∞(d ·) when d(= 20) is large and b(= −15) < b∗ < 0. fig:3 Finally, [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
read the original abstract

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Ces{\`a}ro mean toward a limit function determined solely by these pseudo-equilibria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that, as synaptic delay τ tends to infinity, solutions of sufficiently inhibitory NNLIF networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method on the delayed system, it establishes local convergence in the Cesàro mean to a limit function determined solely by these pseudo-equilibria, building on the notion introduced in prior references.

Significance. If the Doeblin-Harris application is verified, the result would give a rigorous asymptotic description of periodic dynamics emerging from large delays in inhibitory networks, extending the pseudo-equilibria framework with a probabilistic convergence tool. The explicit use of Cesàro mean convergence provides a concrete, falsifiable prediction for the large-delay regime.

major comments (2)
  1. [Proof of the main theorem (Doeblin-Harris section)] The application of the Doeblin-Harris theorem requires verifying that the Markov semigroup on the augmented state space L¹(ℝ) × C([−τ,0]; L¹(ℝ)) satisfies uniform minorization and irreducibility conditions independent of τ. The abstract asserts this yields Cesàro convergence, but the infinite-dimensional delay component makes the minorization constant non-obvious; without an explicit check that the constant remains positive in the τ → ∞ limit, the central convergence claim does not follow.
  2. [Statement of assumptions and main result] The oscillation between pseudo-equilibria in the large-delay limit is asserted for 'sufficiently inhibitory' networks, but the precise quantitative threshold (in terms of the inhibition strength parameter) is not restated from the referenced works [11] and [14]; this condition is load-bearing for the oscillation statement and must be made self-contained.
minor comments (2)
  1. [Abstract] Abstract contains the typo 'sufficitently' (should be 'sufficiently').
  2. [Model formulation] The notation for the history segment and the precise function space for the delayed system should be introduced earlier, before the application of Doeblin-Harris, to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for providing constructive feedback. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Proof of the main theorem (Doeblin-Harris section)] The application of the Doeblin-Harris theorem requires verifying that the Markov semigroup on the augmented state space L¹(ℝ) × C([−τ,0]; L¹(ℝ)) satisfies uniform minorization and irreducibility conditions independent of τ. The abstract asserts this yields Cesàro convergence, but the infinite-dimensional delay component makes the minorization constant non-obvious; without an explicit check that the constant remains positive in the τ → ∞ limit, the central convergence claim does not follow.

    Authors: We thank the referee for highlighting this important point. The proof in Section 3 establishes the required minorization and irreducibility conditions for the Markov semigroup on the augmented state space, with the minorization constant chosen independently of τ by exploiting the uniform positive lower bound on transition probabilities induced by the inhibitory coupling and the firing rate function. This bound holds uniformly because the pseudo-equilibria are attractive in the large-delay regime independently of the precise delay value. To make the uniformity explicit and address the concern about the infinite-dimensional component, we will add a short lemma or remark in the Doeblin-Harris section verifying that the constant remains bounded away from zero as τ → ∞. revision: yes

  2. Referee: [Statement of assumptions and main result] The oscillation between pseudo-equilibria in the large-delay limit is asserted for 'sufficiently inhibitory' networks, but the precise quantitative threshold (in terms of the inhibition strength parameter) is not restated from the referenced works [11] and [14]; this condition is load-bearing for the oscillation statement and must be made self-contained.

    Authors: The referee is correct that the manuscript should be self-contained on this load-bearing assumption. We will revise the statement of the main result (Theorem 1.1) and the assumptions section to explicitly restate the quantitative threshold on the inhibition strength parameter, quoting the precise range from [11] and [14] for which the oscillation between distinct pseudo-equilibria holds. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for pseudo-equilibria; central large-delay result independent

specific steps
  1. self citation load bearing [Abstract]
    "Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficiently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval."

    The load-bearing objects (pseudo-equilibria) originate in a citation whose authors overlap with the present paper; however the overlap is limited to the definition step and does not propagate into a self-referential reduction of the new large-delay convergence statement.

full rationale

The derivation introduces the large-delay oscillation and Cesàro convergence via Doeblin-Harris on the delayed NNLIF system. Pseudo-equilibria are taken from prior references [11] and [14] (the latter by the same authors), but these serve only as external inputs whose existence and properties are assumed; the new asymptotic statements and convergence proof do not redefine or fit quantities in terms of themselves. No fitted-input-called-prediction, self-definitional, or ansatz-smuggling steps appear. The single self-citation is not load-bearing for the novel limit claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the NNLIF model equations, the prior definition and properties of pseudo-equilibria, and the applicability of the Doeblin-Harris theorem to the delayed system; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The NNLIF model equations and the notion of pseudo-equilibria hold as defined in the referenced works.
    The analysis is performed inside the NNLIF framework and explicitly builds on the pseudo-equilibria introduced in [11].
  • domain assumption The Doeblin-Harris method applies to the delayed NNLIF system under the stated inhibitory conditions.
    The proof technique is invoked without further justification in the abstract.

pith-pipeline@v0.9.1-grok · 5654 in / 1410 out tokens · 33764 ms · 2026-06-26T23:54:52.193323+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 1 linked inside Pith

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