adaptive-conduction-delays-haken-lighthouse

name	adaptive-conduction-delays-haken-lighthouse
description	Theory of phase-locked activity in delayed spiking networks using the Haken Lighthouse model — an analytically tractable event-based framework bridging integrate-and-fire networks and coupled phase oscillators. Derives self-consistency conditions for phase-locked states with multiple fixed delays, linear stability theory formulated in spike-time perturbations, and activity-dependent white matter plasticity (myelination-modulated conduction speed) creating slow-fast state-dependent delay systems that self-organize toward commensurate delay-period relationships. Activation: Haken Lighthouse model, phase locking, conduction delays, white matter plasticity, spike-time perturbations, circulant ring networks, slow-fast adaptive delays, myelination, commensurate timing, temporal coordination.
metadata	{"arxiv_id":"2606.21508","published":"2026-06-19","authors":"Stephen Coombes, Rüdiger Thul, Stefan Ruschel, Rachel Nicks"}

Adaptive Conduction Delays and Phase Locking in Spiking Haken Lighthouse Networks

arXiv: 2606.21508 | Published: 2026-06-19

Haken Lighthouse Model

The model describes N pulse-coupled nodes on a directed weighted graph:

dθ_i/dt = S(ψ_i(t))                              (1)
ψ_i(t) = Σ_j w_ij (η_ij * s_j)(t)                (2)
s_j(t) = Σ_m δ(t - T_j^m)                        (3)

• θ_i ∈ S: phase variable of node i
• S(x): sigmoidal firing-rate function (monotonically increasing, positive)
• ψ_i: synaptic input from delayed spike trains
• η_ij(t) = η(t - τ_ij): causal kernel with propagation delay τ_ij
• α-function kernel: η(t) = α²·t·e^{-αt}·H(t), Q = (1 + d/dt/α)²
• Firing times T_j^m defined by threshold: θ_j(T_j^m) = 2π (phase wraps)

Key insight: This is an event-driven spiking system where continuous phase evolution couples to discrete spike times, making it amenable to analytical treatment while preserving spike-level temporal precision.

Phase-Locked State Theory

Self-Consistency Equation

For phase-locked states T_i^m = mT + φ_i·T with φ_i ∈ [0,1):

2π = ∫₀ᵀ dt S( Σ_j w_ij P(t + (φ_i - φ_j)T - τ_ij) )           (7)

where P(t) is the T-periodic synaptic drive function, equivalently:

• Fourier series: P(t) = Σ_n P_n e^{iω_n t} with ω_n = 2πn/T
• P_n = η̂(ω_n)/T where η̂ is Fourier transform of η
• Closed form for α-function: P(t) = α²e^{-αt}/(1-e^{-αT}) · [t + T·e^{-αT}/(1-e^{-αT})]

Stability Analysis (Spike-Time Perturbations)

Linear stability is formulated directly in terms of spike-time perturbations:

• Perturb firing times T_i^m → T_i^m + ε_i^m
• Derive characteristic equation for ε evolution
• For ring networks with circulant symmetry: stability decomposes into Fourier modes (twisted states)
• Eigenvalue problem: det[I - M(λ)] = 0 where M encodes delayed spike interactions

Network Architectures Analyzed

1. Single-Node Autapse: fold bifurcations of regular-spiking branches, dynamic instabilities of inter-spike intervals
2. Reciprocal Two-Node: synchronous, anti-synchronous, and asymmetric phase-locked states; two-delay → one-delay reduction
3. Ring Networks with Distance-Dependent Delays: circulant symmetry allows stability diagonalization via Fourier modes; traveling waves analyzed systematically

White Matter Plasticity (Slow-Fast System)

Plasticity Rule

Activity-dependent myelination modulates conduction speed and hence delay:

dτ_ij/dt_slow = ε · F(activity_ij)               (plasticity)
dθ_i/dt_fast = S(ψ_i(t))                          (spiking)

• τ_ij evolves on timescale 1/ε >> 1 relative to spiking
• Creates a state-dependent delay problem (delays depend on network state)
• Frozen phase-locked branches organize the adaptive dynamics

Key Results from Adaptive Dynamics

1. Synchrony emergence: Plasticity drives networks toward synchronized states even from asynchronous initial conditions
2. Slow switching: Networks exhibit long-timescale transitions between competing phase-locked patterns
3. Commensurate delay classes: Heterogeneous delays self-organize into discrete delay-period classes (τ ≈ nT/k for integers n,k)
4. Attractor reshaping: Adaptive conduction reshapes the attractor landscape of the delayed spiking network

Slow-Fast Interpretation

• Fast subsystem: Spiking dynamics with frozen delays (phase-locked branches)
• Slow subsystem: Delay evolution guided by activity statistics
• Critical manifold: Set of phase-locked states parameterized by delay values
• Plasticity selects points on the critical manifold that satisfy commensurate timing

Methodology Summary

1. Model specification: Define Haken Lighthouse network with delays τ_ij on graph
2. Phase-locking analysis: Solve self-consistency equations (7) for φ_i and T
3. Stability computation: Linearize spike-time perturbations, solve characteristic equation
4. Symmetry exploitation: For circulant networks, diagonalize via Fourier modes
5. Slow-fast decomposition: Separate fast spiking from slow delay evolution
6. Critical manifold analysis: Track how frozen branches organize adaptive dynamics
7. Event-driven simulation: Validate analytical predictions with direct spike-time simulation

Activation Keywords

adaptive conduction delays, phase locking, Haken Lighthouse model, white matter plasticity, activity-dependent myelination, delayed spiking networks, spike-time perturbations, slow-fast systems, commensurate timing, communication through coherence, event-driven SNN, circulant ring networks, twisted states, temporal coordination, synchrony emergence, delay-period relationships

Common Pitfalls

• Delays as parameters vs. dynamical variables: Traditional models treat τ as fixed; this work shows τ should be treated as a slow dynamical variable when myelination is considered
• α-function vs. other kernels: Results use α-function for analytical tractability; other kernels (exponential, double-exponential) require numerical treatment
• Circulant assumption: Ring network stability diagonalization requires distance-dependent coupling symmetry; arbitrary topologies need full eigenvalue computation
• Timescale separation validity: Slow-fast analysis requires ε << 1; if plasticity is too fast, the frozen-branch approximation breaks down
• Spike-time vs. phase perturbations: Stability must be analyzed in spike-time domain (not phase domain) for event-driven models — phase perturbations alone miss timing-dependent effects