{
 "artifacts": null,
 "category": "dynamics",
 "description": "Build a recurrent associative-memory layer whose hidden variables are phases rather than unconstrained activations, and whose symmetric couplings may be positive or negative. Positive edges attract two units to the same phase, while negative edges attract them to phase difference \\(\\pi\\), allowing memories containing both correlation and anti-correlation constraints to remain stable after the external input is removed.",
 "formulas_latex": [
  "$$\\dot{\\phi}_{i}=\\omega_{i}-\\sum_{j\\neq i}K_{ij}(x_{ij})\\sin(\\phi_{i}-\\phi_{j})$$",
  "$$F(\\phi\\mid x)=-{1\\over 2}\\sum_{i\u003cj}K_{ij}(x_{ij})\\cos(\\phi_{i}-\\phi_{j})$$",
  "$$R(x)=R_{\\mathrm{off}}(1-x)+R_{\\mathrm{on}}x,\\qquad G(x)=R(x)^{-1}$$",
  "$$K_{ij}=\\frac{\\kappa}{N}A_{ij},\\qquad A_{ij}=A_{ji}=\\sum_{\\mu=1}^{P}\\cos\\!\\left(\\theta_i^\\mu-\\theta_j^\\mu\\right),\\qquad \\phi_i^{t+1}=\\phi_i^t-\\Delta t\\sum_{j\\ne i}K_{ij}\\sin(\\phi_i^t-\\phi_j^t)$$"
 ],
 "id": 306,
 "implementation": "(1) Exact integration point: implement this as a recurrent refinement layer between an encoder and a classifier or retrieval head. The encoder maps an input to an initial phase vector \\(\\phi^0\\in[0,2\\pi)^N\\), for example \\(\\phi_i^0=2\\pi\\,\\sigma(z_i)\\). The recurrent layer performs 10–50 relaxation steps. Store a symmetric coupling matrix \\(K\\) as a fixed Hebbian buffer for the first experiment, or make it trainable with the exact symmetry parameterization \\(K=(W+W^T)/2\\). (2) Pseudocode: inputs are \\(\\phi^0\\), symmetric \\(K\\), step size \\(\\Delta t\\), and number of steps \\(T\\). For each \\(t=0,\\ldots,T-1\\), compute \\(D_{ij}=\\phi_i^t-\\phi_j^t\\), \\(g_i=\\sum_{j\\ne i}K_{ij}\\sin(D_{ij})\\), and update \\(\\phi^{t+1}=\\operatorname{wrap}_{2\\pi}(\\phi^t-\\Delta t g)\\). Decode using \\([\\cos\\phi,\\sin\\phi]\\), or retrieve pattern \\(\\mu\\) by maximizing \\(\\sum_i\\cos(\\phi_i-\\theta_i^\\mu)\\). The paper's continuous-time equation supplies the vector field and its \\(F\\) supplies an energy diagnostic. (3) Compute \\(K\\) from the phase-Hebbian formula, symmetrize it exactly, and normalize by \\(N\\). Estimate basin size, energy decrease, and retrieval accuracy empirically. Check \\(F(\\phi^{t+1})\\le F(\\phi^t)\\); reduce \\(\\Delta t\\) if energy rises. For learned couplings, use \\(K_{ij}=\\kappa\\tanh(W_{ij})/N\\) to bound interaction strength. (4) First cheap experiment: use binary MNIST or CIFAR-10 class prototypes converted to phases, corrupt 20–50% of phases with additive noise or flips by \\(\\pi\\), and compare against a standard Hopfield network with the same parameter count and recurrent-step budget. Include memories with deliberately anti-correlated phase relations, where unsigned couplings cannot encode the target constraints. Measure retrieval accuracy versus corruption level, energy monotonicity, relaxation steps to convergence, and robustness after removing the input clamp. Success means higher anti-correlated-pattern retrieval at equal recurrent FLOPs, a larger basin of attraction, or faster convergence; failure is no improvement over the unsigned control.",
 "math_summary": "For \\(N\\) oscillators, \\(\\phi_i(t)\\in\\mathbb{R}\\) is the phase of unit \\(i\\), \\(\\omega_i\\) is its intrinsic frequency, and \\(K_{ij}=K_{ji}\\) is a signed symmetric coupling. The paper uses \\(\\dot{\\phi}_i=\\omega_i-\\sum_{j\\ne i}K_{ij}(x_{ij})\\sin(\\phi_i-\\phi_j)\\), where \\(x_{ij}\\in[0,1]\\) is a memristor state and \\(K_{ij}(x_{ij})\\) is the effective edge weight. In the homogeneous rotating frame \\(\\omega_i=0\\), the dynamics are gradient descent on \\(F(\\phi\\mid x)=-\\frac12\\sum_{i\u003cj}K_{ij}(x_{ij})\\cos(\\phi_i-\\phi_j)\\). Therefore \\(K_{ij}\u003e0\\) favors equal phase and \\(K_{ij}\u003c0\\) favors phase difference \\(\\pi\\). For a software layer, use normalized signed couplings \\(K_{ij}=\\kappa A_{ij}/N\\), with \\(A=A^T\\), and interpret the final phase vector modulo \\(2\\pi\\). A phase-memory coupling can be constructed as \\(A_{ij}=\\sum_{\\mu=1}^{P}\\cos(\\theta_i^\\mu-\\theta_j^\\mu)\\), where \\(\\theta^\\mu\\in[0,2\\pi)^N\\) is stored pattern \\(\\mu\\). The hardware resistance model is \\(R(x)=R_{\\mathrm{off}}(1-x)+R_{\\mathrm{on}}x\\) and \\(G(x)=R(x)^{-1}\\), with \\(R_{\\mathrm{on}},R_{\\mathrm{off}}\u003e0\\); signed effective weights require differential excitatory/inhibitory branches rather than a single nonnegative conductance.",
 "math_tags": [
  "dynamical-systems",
  "geometry",
  "optimization",
  "algebra"
 ],
 "ml_areas": [
  "rnn",
  "retrieval",
  "graph-nn"
 ],
 "paper": {
  "arxiv_id": "2607.00286",
  "arxiv_url": "https://arxiv.org/abs/2607.00286",
  "summary_what_math_gives_to_ml": "The paper provides a directly implementable signed phase-coupling primitive whose dynamics descend an explicit energy, while negative couplings create persistent anti-phase attractors rather than merely transient anti-synchronization. The transferable asset is the combination of bounded phase states, symmetric signed interactions, and an energy landscape in which coupling signs determine whether stored relations are equality or opposition. A practical neural-network experiment is to replace a conventional binary associative-memory layer or recurrent attention block with a differentiable phase-relaxation layer using signed couplings, then test whether it denoises phase-corrupted patterns with fewer update steps and better robustness to anti-correlated structure.",
  "title": "Self-Organized Learning in Oscillatory Neural Networks with Memristive Signed Couplings",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "stability",
  "sample-efficiency"
 ],
 "title": "Signed Phase-Attractor Layer",
 "url": "https://synthcore.org/idea/306/signed-phase-attractor-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}