{
 "artifacts": null,
 "category": "architecture",
 "description": "Add a bank of temporal branches whose synapses apply heterogeneous delays before coincidence pooling. A branch selectively responds to an ordered pair or short sequence when the delay difference compensates the sequence interval, while low delay dispersion makes it primarily an order-invariant event detector. The branch threshold creates sparse, interpretable activations that can replace part of a recurrent or attention-based temporal module.",
 "formulas_latex": [
  "$$\\mathrm{Ca}_{j}=\\sum_f w_{jf}\\,\\mathbf{1}_{\\{|a_{jf}-t_c|\u003c\\tau_{\\mathrm{w}}\\}}$$",
  "$$I_{j}=\\eta_{j}\\,g\\!\\left(\\max(0,\\mathrm{Ca}_{j}-\\theta)\\right)$$",
  "$$a_{ji}=t_i+d_{ji},\\qquad |(t+d_{j2})-(t+\\Delta+d_{j1})|\u003c\\tau_{\\mathrm{w}}\\iff |d_{j2}-d_{j1}-\\Delta|\u003c\\tau_{\\mathrm{w}}$$",
  "$$\\widetilde{\\mathrm{Ca}}_j(t_c)=\\sum_f w_{jf}\\,\\sigma\\!\\left(\\beta\\left[\\tau_{\\mathrm{w}}-|a_{jf}-t_c|\\right]\\right),\\qquad \\widetilde I_j=\\eta_j\\min\\!\\left(\\max(0,\\widetilde{\\mathrm{Ca}}_j-\\theta),I_{\\max}\\right)$$"
 ],
 "id": 3168,
 "implementation": "Integrate the module immediately before a temporal encoder or classifier. Inputs are timestamped token or event embeddings $x_i\\in\\mathbb{R}^D$ with times $t_i$. Instantiate $B$ branches and $F$ delayed synapses per branch. Each synapse has a source projection, a nonnegative delay $d_{jf}$, and weight $w_{jf}$. For each candidate coincidence time $t_c$ on a discretized temporal grid, calculate arrivals $a_{jf}=t_i+d_{jf}$ and accumulate their soft coincidence contributions. Pseudocode: `for branch j: Ca=0; for synapse f: a=t[source[f]]+d[j,f]; Ca += w[j,f]*sigmoid(beta*(tau_w-abs(a-tc))); I[j]=eta[j]*clamp(relu(Ca-theta),0,Imax)`. Use the soft equation during training, anneal `beta` from 10 to 100, and use hard indicators for evaluation. Initialize delays from narrow and broad dispersions, for example $\\sigma_d\\in\\{0,\\Delta/2,\\Delta,2\\Delta\\}$, with $\\Delta$ covering the task's interval range. Keep delays fixed initially, then compare with projected nonnegative learnable delays. Present isolated events, simultaneous pairs, and ordered pairs at intervals $\\Delta$; measure activation sparsity and an order-selectivity index such as $(A_{AB}-A_{BA})/(A_{AB}+A_{BA}+\\epsilon)$. The first cheap test should use a small GRU or Transformer on synthetic two-event sequences plus permuted-MNIST or Speech Commands. Compare equal-parameter baselines. Success means higher ordered-sequence accuracy at equal FLOPs, sparse branch activations, and a monotonic increase in order selectivity as $\\sigma_d$ grows.",
 "math_summary": "The paper defines branch calcium accumulation as $$\\mathrm{Ca}_{j}=\\sum_f w_{jf}\\mathbf{1}_{\\{|a_{jf}-t_c|\u003c\\tau_{\\mathrm{w}}\\}}$$, where branch $j$ receives synapses indexed by $f$, $w_{jf}$ is the synaptic weight, $a_{jf}$ is the arrival time after axonal delay, $t_c$ is the branch coincidence reference time, and $\\tau_{\\mathrm{w}}$ is the integration-window half-width. Its output is $$I_j=\\eta_j g(\\max(0,\\mathrm{Ca}_j-\\theta))$$, where $\\eta_j$ is branch-to-soma efficiency, $\\theta$ is the calcium threshold, and $g$ is monotone and saturating. For an input at time $t_i$ sent through a synapse with delay $d_{ji}$, use $a_{ji}=t_i+d_{ji}$. For two inputs at times $t$ and $t+\\Delta$, coincidence occurs when $|(t+d_{j2})-(t+\\Delta+d_{j1})|\u003c\\tau_{\\mathrm{w}}$, equivalently $|d_{j2}-d_{j1}-\\Delta|\u003c\\tau_{\\mathrm{w}}$. Therefore a branch detects interval $\\Delta$ when its delay-difference signature matches that interval. Sample delays from a distribution with dispersion $\\sigma_d$, such as truncated $\\mathcal{N}(\\mu_i,\\sigma_d^2)$, to control the range of detectable intervals. During gradient training replace the indicator by $s_\\beta(x)=\\operatorname{sigmoid}(\\beta x)$ and anneal $\\beta$ upward; at inference use the hard indicator. The threshold $\\theta$ controls sparse all-or-none activation, and $g(z)=\\min(z,I_{\\max})$ gives bounded branch amplitude.",
 "math_tags": [
  "dynamical-systems",
  "probability",
  "approximation-theory"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "attention"
 ],
 "paper": {
  "arxiv_id": "2609.04195",
  "arxiv_url": "https://arxiv.org/abs/2609.04195",
  "summary_what_math_gives_to_ml": "The paper supplies a concrete temporal-computation primitive: synaptic delays form a physical matching key, and a branch fires only when delayed arrivals fall inside a narrow coincidence window and exceed a calcium-like threshold. The transferable asset is the algebraic condition that a delay difference compensates an input sequence interval, allowing the same module to distinguish unordered events from ordered sequences. A practical neural implementation is a bank of branches with fixed or learnable delays, soft coincidence pooling during training, and thresholded branch outputs before a recurrent, state-space, or attention-based temporal encoder. The main falsifiable prediction is that increasing delay dispersion should improve order selectivity while reducing performance on order-invariant event detection.",
  "title": "Axonal delay dispersion decides whether a neuron detects an event or a sequence, and predicts cortical column diameter",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "scalability"
 ],
 "title": "Delay-signature coincidence branches",
 "url": "https://synthcore.org/idea/3168/delay-signature-coincidence-branches",
 "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
   }
  }
 }
}