{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace a binary recurrent layer's ordinary sign activation with a hysteretic update that retains each unit's previous state when its incoming weighted sum is inconclusive. This creates discrete attractors that are robust to small perturbations and can serve as persistent latent memories, iterative classifiers, or compact associative states.",
 "formulas_latex": [
  "$$x_i(t+1)=h(y_i(t)),\\qquad y_i(t)=\\sum_{j=1}^{N}w_{ij}x_j(t),\\qquad x_i(t)\\in\\{-1,+1\\}.$$",
  "$$h(y_i(t))=\\begin{cases}+1,\u0026y_i(t)\\ge Th,\\\\x_i(t),\u0026|y_i(t)|\u003cTh,\\\\-1,\u0026y_i(t)\\le-Th,\\end{cases}\\qquad Th\u003e0.$$",
  "$$z_i=h\\!\\left(\\sum_{j=1}^{N}w_{ij}z_j\\right),\\quad i=1,\\ldots,N.$$",
  "$$\\tilde h_\\tau(y,x)=\\tanh(y/\\tau)+\\rho(Th-|y|)x,$$"
 ],
 "id": 93,
 "implementation": "(1) Integration point: place the module after an encoder or inside an RNN/SSM latent update. The encoder outputs $u\\in\\mathbb R^N$; initialize $x(0)=\\operatorname{sign}(u)$ and run $K$ recurrent updates with learned matrix $W$. (2) Pseudocode: `x = sign(encoder(input)); for t in 1..K: y = W @ x; x = where(y \u003e= Th, +1, where(y \u003c= -Th, -1, x)); output = readout(x)`. In the forward pass use the exact hysteresis rule. In the backward pass use a straight-through estimator whose surrogate is $\\tilde h_\\tau(y,x)=\\tanh(y/\\tau)+\\rho(Th-|y|)x$; stop gradients through the copied-state term if it causes instability, and anneal $\\tau$ from 1 to 0.1. (3) Compute the recurrent field and threshold comparisons exactly; estimate gradients empirically through the surrogate. Constrain $W$ by spectral normalization or replace it with $W/\\max(1,\\|W\\|_2)$ to reduce oscillations. (4) First experiment: train a 32-unit MNIST recurrent classifier for 10 update steps, comparing hysteretic activation against tanh and sign baselines at equal parameter count and FLOPs. Evaluate clean accuracy, accuracy after additive input noise, accuracy after random latent bit flips, convergence rate defined by $x(t+1)=x(t)$, and the number of state changes in the final three iterations. The hypothesis is improved robustness and more stable late-iteration predictions, with comparable clean accuracy.",
 "math_summary": "The paper defines binary states $x_i(t)\\in\\{-1,+1\\}$, recurrent fields $y_i(t)=\\sum_{j=1}^{N}w_{ij}x_j(t)$, and the update $x_i(t+1)=h(y_i(t))$. For threshold $Th\u003e0$, the hysteresis map is $h(y_i(t))=+1$ if $y_i(t)\\ge Th$, $h(y_i(t))=x_i(t)$ if $|y_i(t)|\u003cTh$, and $h(y_i(t))=-1$ if $y_i(t)\\le -Th$. The interval $(-Th,Th)$ is therefore a deadband in which the previous state is copied. A fixed point $z\\in\\{-1,+1\\}^N$ satisfies $z_i=h(\\sum_jw_{ij}z_j)$ for every $i$; its basin is the set of initial states whose iterates converge to $z$. The paper uses integer weights and half-integer thresholds $Th\\in\\{0.5,1.5,2.5,\\ldots\\}$, avoiding equality ambiguity because the fields are integer. The neural-network adaptation uses the exact discrete map in the forward pass and a differentiable straight-through surrogate in the backward pass.",
 "math_tags": [
  "dynamical-systems",
  "combinatorics"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "memory"
 ],
 "paper": {
  "arxiv_id": "2608.23225",
  "arxiv_url": "https://arxiv.org/abs/2608.23225",
  "summary_what_math_gives_to_ml": "The paper studies binary recurrent dynamics with a hysteresis deadband: units switch only when their weighted input crosses a threshold and otherwise retain their previous state. This creates multiple stable fixed points with distinct basins of attraction, offering a route to robust discrete latent memory and attractor-based classification. The threshold is also a controllable architectural parameter that changes attractor multiplicity and basin balance. The most actionable adaptations are a hysteretic recurrent layer trained with a straight-through surrogate and an empirical basin-entropy procedure for avoiding collapse to a single attractor.",
  "title": "Basins of Attraction to Multiple Fixed Points in Discrete-time Hysteresis Neural Networks",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "memory",
  "accuracy"
 ],
 "title": "Hysteretic Attractor Layer",
 "url": "https://synthcore.org/idea/93/hysteretic-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
   }
  }
 }
}