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arxiv: 2606.21508 · v1 · pith:IZB2R4IKnew · submitted 2026-06-19 · 🧬 q-bio.NC

Adaptive conduction delays and phase locking in spiking Haken Lighthouse networks

Stephen Coombes , R\"udiger Thul , Stefan Ruschel , Rachel Nicks This is my paper

Pith reviewed 2026-06-26 12:38 UTC · model grok-4.3

classification 🧬 q-bio.NC
keywords phase lockingconduction delaysmyelination plasticityspiking networkssynchronyHaken Lighthouse modeladaptive delays
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The pith

Activity-dependent myelination selects commensurate delay-period relationships to organize phase locking in spiking networks.

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

This paper establishes a theory for phase-locked activity in delayed spiking networks using the analytically tractable Haken Lighthouse model. For fixed delays, it provides self-consistency conditions and stability analysis in terms of spike-time perturbations. Introducing an activity-dependent plasticity rule for myelination that changes conduction delays leads to a slow-fast system. In this system, the plasticity selects delay-period ratios that are commensurate, enabling the emergence of synchrony, frequency-locked states, and slow switching between patterns.

Core claim

The activity-dependent white matter plasticity rule, in which myelination modulates axonal conduction speed, leads to a slow-fast system with state-dependent delays. Frozen phase-locked branches of the fast subsystem organize the adaptive dynamics, and the rule selects commensurate delay-period relationships.

What carries the argument

The activity-dependent plasticity rule for axonal conduction delays that creates a slow-fast dynamical system organized by phase-locked states.

If this is right

  • The emergence of synchrony and other frequency-locked states.
  • Slow switching between competing phase-locked patterns.
  • The organisation of heterogeneous delays into discrete delay-period classes.
  • Reshaping of the attractor structure of delayed spiking networks.
  • Generation of long-timescale transitions in network activity.

Where Pith is reading between the lines

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

  • This mechanism could underlie how neural systems achieve temporal coordination through activity-dependent changes in white matter.
  • The framework may extend to explaining observed slow switching in brain dynamics.
  • It suggests that myelin plasticity acts as a regulator of communication through coherence.

Load-bearing premise

The plasticity rule produces a slow-fast system with state-dependent delays whose attractors are organized by the frozen phase-locked branches of the fast subsystem.

What would settle it

A simulation of the adaptive network that fails to show selection of commensurate delay-period relationships or the predicted phase-locked attractors would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.21508 by Rachel Nicks, R\"udiger Thul, Stefan Ruschel, Stephen Coombes.

Figure 1
Figure 1. Figure 1: Bifurcation structure of the Haken light [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bifurcation diagram for a reciprocal two [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Branch structure and quasi-static sweep simulation for an 11-node ring Haken network [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical simulation of the reciprocal two-node Haken model with white matter plastic￾ity on the symmetric subspace τ1 = τ = τ2, where τ1 = τ12 and τ2 = τ21. The upper panel shows in￾stantaneous ISIs for both neurons against the cur￾rent delay τ at firing events, and the lower panel shows the corresponding instantaneous phase de￾fined by (55). The coloured curves are the frozen￾delay branches from the two-… view at source ↗
Figure 5
Figure 5. Figure 5: Numerical simulation of the recipro￾cal two-node Haken model with two plastic delays showing (ISI, τ1, τ2). The colour of data points indi￾cates the corresponding instantaneous phase. The simulations show switching between low and high ISIs. Parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic and quantification of node-wise commensurability classes for a randomly connected [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We develop a theory of phase-locked activity in delayed spiking networks using the Haken Lighthouse model as an analytically tractable event-based description of neural dynamics. For networks with fixed delays, we derive self-consistency conditions for phase-locked states and an associated linear stability theory formulated directly in terms of spike-time perturbations. The framework is illustrated for a delayed autapse, a reciprocally coupled two-cell network, and spatially structured rings with distance-dependent coupling and conduction delays, where circulant symmetry allows stability to be decomposed into Fourier modes. We then introduce an activity-dependent white matter plasticity rule in which myelination modulates axonal conduction speed and hence communication delay. This leads naturally to a slow--fast system with state-dependent delays, in which frozen phase-locked branches organise the adaptive dynamics. The plasticity rule selects commensurate delay--period relationships, providing a mechanism for the emergence of synchrony, other frequency-locked states, slow switching between competing phase-locked patterns, and the organisation of heterogeneous delays into discrete delay--period classes. Direct simulations of the event-driven network support the analytical predictions and illustrate how adaptive conduction can reshape the attractor structure of a delayed spiking network and generate long-timescale transitions. These results provide a tractable mathematical framework for studying how activity-dependent myelination may regulate temporal coordination, synchrony, and communication through coherence in spiking neural systems.

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 develops a theory of phase-locked activity in delayed spiking networks using the Haken Lighthouse model. For fixed delays it derives self-consistency conditions and linear stability in spike-time perturbations, illustrated on autapses, two-cell networks, and circulant rings via Fourier-mode decomposition. It then introduces an activity-dependent white matter plasticity rule that modulates conduction delays, producing a slow-fast system whose attractors are claimed to be organized by the frozen phase-locked branches of the fixed-delay subsystem. This rule is asserted to select commensurate delay-period relationships, enabling synchrony, frequency locking, slow switching, and discrete delay classes; direct event-driven simulations are stated to support the predictions.

Significance. If the central claim holds, the work supplies a tractable mechanism linking activity-dependent myelination to the emergence and switching of phase-locked states in spiking networks, with potential relevance to communication-through-coherence hypotheses. The fixed-delay analysis, particularly the circulant symmetry reduction to Fourier modes, is analytically clean and yields falsifiable predictions. The simulations are presented as direct support, though quantitative agreement metrics are not detailed in the abstract.

major comments (2)
  1. [Abstract; section introducing the adaptive rule] Abstract and the section introducing the adaptive rule: the claim that the plasticity rule produces a slow-fast system whose long-term attractors are organized by the frozen phase-locked branches rests on the unshown extension of the event-based map and spike-time perturbation equations to state-dependent delays. The fixed-delay self-consistency conditions and linear stability are derived first, but no re-derivation or continuity argument for time-varying delays is supplied; this extension is load-bearing for the headline result on commensurate delay-period selection and branch organization.
  2. [Simulations section] Simulations section: the statement that 'direct simulations of the event-driven network support the analytical predictions' lacks quantitative error measures (e.g., deviation of realized delay-period ratios from the predicted commensurate values or comparison of observed switching times to the slow timescale). Without these, it is unclear whether the state-dependent case actually follows the frozen branches or requires additional assumptions on timescale separation.
minor comments (2)
  1. [Section introducing the adaptive rule] Notation for the plasticity rule (myelination modulating conduction speed) should be introduced with an explicit equation number when first stated, to allow direct reference in the stability discussion.
  2. [Abstract] The abstract lists 'slow switching between competing phase-locked patterns' as a consequence but does not indicate which network topologies (autapse, two-cell, or rings) exhibit this behavior in the simulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying two points where the manuscript would benefit from greater explicitness. We address each major comment below and will revise the manuscript to strengthen the presentation of the state-dependent case and the simulation validation.

read point-by-point responses

  1. Referee: Abstract and the section introducing the adaptive rule: the claim that the plasticity rule produces a slow-fast system whose long-term attractors are organized by the frozen phase-locked branches rests on the unshown extension of the event-based map and spike-time perturbation equations to state-dependent delays. The fixed-delay self-consistency conditions and linear stability are derived first, but no re-derivation or continuity argument for time-varying delays is supplied; this extension is load-bearing for the headline result on commensurate delay-period selection and branch organization.

    Authors: We agree that an explicit continuity argument or brief re-derivation for slowly varying delays is needed to make the organization of attractors by the frozen branches fully rigorous. Because the plasticity rule evolves on a timescale much slower than the spiking dynamics, the instantaneous event map at any fixed time is identical to the fixed-delay map evaluated at the current delay value; the linear stability analysis then carries over directly to the quasi-static case. We will add a short subsection (or appendix) that states this continuity explicitly and shows how the spike-time perturbation equations remain valid under adiabatic variation of the delays. This will be incorporated in the revised manuscript. revision: yes

  2. Referee: Simulations section: the statement that 'direct simulations of the event-driven network support the analytical predictions' lacks quantitative error measures (e.g., deviation of realized delay-period ratios from the predicted commensurate values or comparison of observed switching times to the slow timescale). Without these, it is unclear whether the state-dependent case actually follows the frozen branches or requires additional assumptions on timescale separation.

    Authors: We accept that quantitative agreement metrics would make the support for the analytical predictions more convincing. In the revised version we will report (i) the mean and standard deviation of the realized delay-to-period ratios relative to the nearest commensurate values predicted by the fixed-delay theory, and (ii) the ratio of observed switching times to the slow plasticity timescale, together with a brief discussion of how close the separation remains throughout the trajectories. These additions will be placed in the simulations section and its figure captions. revision: yes

Circularity Check

0 steps flagged

No circularity; fixed-delay derivations and plasticity rule are independent modeling steps

full rationale

The paper first derives self-consistency conditions and spike-time linear stability directly from the Haken Lighthouse event-based equations for fixed delays (autapse, two-cell, circulant rings). The activity-dependent myelination rule is then introduced as an independent modeling choice that induces a slow-fast system with state-dependent delays, with the claim that frozen phase-locked branches organize the attractors. No equation reduces a prediction to a fitted input by construction, no self-citation is load-bearing for the central result, and no ansatz or uniqueness theorem is smuggled in. The derivation chain remains self-contained against the model equations and the stated plasticity rule.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim depends on the Haken Lighthouse model being an analytically tractable event-based description, on the validity of linear stability formulated in spike-time perturbations, and on the specific functional form of the new plasticity rule; none of these receive independent external validation in the abstract.

axioms (2)
  • domain assumption Self-consistency conditions for phase-locked states can be written directly from the event-based delay equations
    Invoked for fixed-delay networks in the first part of the abstract.
  • standard math Linear stability analysis can be performed via small perturbations of spike times
    Formulated directly in terms of spike-time perturbations for the stability theory.
invented entities (1)
  • Activity-dependent white matter plasticity rule no independent evidence
    purpose: Modulates axonal conduction speed (hence delay) according to recent neural activity, producing state-dependent delays
    New rule introduced to create the slow-fast adaptive system; no independent biological evidence supplied in the abstract.

pith-pipeline@v0.9.1-grok · 5776 in / 1572 out tokens · 32245 ms · 2026-06-26T12:38:43.089374+00:00 · methodology

discussion (0)

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