Tutorial 39: Learnable Delays

March 25, 2026 · View on GitHub

SC-NeuroCore supports trainable per-synapse delays in PyTorch surrogate gradient training. Each connection has a weight AND a delay — both optimized by backpropagation. Delays are differentiable via linear interpolation between integer delay bins.

Why Learnable Delays?

Standard SNNs have fixed delays (or none). Adding trainable delays lets the network learn temporal structure: which input spikes should arrive simultaneously (for coincidence detection) and which should be staggered (for sequence recognition). Research shows delays can replace entire layers — same accuracy with fewer parameters.

    Standard SNN (no delays)         With trainable delays
    ─────────────────────────         ───────────────────────
    input A ──┐                      input A ──[d=0]──┐
              ├──→ neuron                              ├──→ neuron
    input B ──┘                      input B ──[d=3]──┘

    A and B arrive simultaneously    B arrives 3 timesteps later
    → only rate matters              → timing encodes information

flowchart LR
    subgraph Standard["Fixed-delay SNN"]
        SA["spike A"] --> N1["neuron"]
        SB["spike B"] --> N1
    end
    subgraph Delayed["Delay-trained SNN"]
        DA["spike A"] -->|"d=0"| N2["neuron<br/>coincidence<br/>detector"]
        DB["spike B"] -->|"d=3 (learned)"| N2
    end

    style Standard fill:#fff3e0
    style Delayed fill:#e8f5e9

1. DelayLinear Module

import torch
from sc_neurocore.training.delay_linear import DelayLinear

layer = DelayLinear(
    in_features=64,
    out_features=32,
    max_delay=16,      # delays in [0, 16) timesteps
    learn_delay=True,  # make delays trainable
    init_delay=2.0,    # initial delay for all synapses
)

print(f"Parameters: {sum(p.numel() for p in layer.parameters())}")
# Weight: 64*32 = 2048, Delay: 64*32 = 2048 → 4096 total

2. Training Loop

import torch.nn as nn
from sc_neurocore.training.surrogate import atan_surrogate

# Simple delayed SNN: input → DelayLinear → LIF → output
delay_layer = DelayLinear(10, 5, max_delay=8, init_delay=1.5)
optimizer = torch.optim.Adam(delay_layer.parameters(), lr=0.01)

for epoch in range(100):
    delay_layer.reset()  # clear spike history between sequences
    v = torch.zeros(5)

    total_spikes = torch.zeros(5)
    for t in range(20):
        x = torch.randn(10) * (0.5 if t < 5 else 0.0)  # input burst
        current = delay_layer.step(x)
        v = 0.9 * v + current
        spike = atan_surrogate(v - 1.0)
        v = v - spike.detach()
        total_spikes = total_spikes + spike

    loss = -total_spikes[0]  # maximize first neuron's spikes
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

3. Export Trained Delays to Hardware

import numpy as np

# Integer delays for Verilog deployment
int_delays = delay_layer.delays_int
print(f"Learned delays: {int_delays}")

# Flat array for Projection(delay=array)
delay_array = delay_layer.to_nir_delay_array()

# Use in network engine
from sc_neurocore.network.projection import Projection
proj = Projection(src_pop, tgt_pop, weight=0.1, delay=delay_array)

4. How Differentiable Delays Work

For each synapse connecting input i to output j with delay d:

delayed_input[j] = sum_i W[j,i] * interp(history, t - D[j,i])

Where interp linearly interpolates between integer delay bins:

interp(history, t - 2.3) = 0.7 * history[t-2] + 0.3 * history[t-3]

This makes the delay parameter D differentiable: the gradient tells the optimizer whether to increase or decrease the delay.

5. Per-Synapse Delays in the Network Engine

The trained delays integrate with the simulation engine:

from sc_neurocore.network.population import Population
from sc_neurocore.network.projection import Projection

pop_a = Population(HodgkinHuxleyNeuron, n=10, label="a")
pop_b = Population(HodgkinHuxleyNeuron, n=5, label="b")

# Per-synapse delays from training
delays = delay_layer.to_nir_delay_array()
proj = Projection(pop_a, pop_b, weight=0.1, delay=delays)

print(f"Delay mode: {proj.delay_mode}")   # "per_synapse"
print(f"Max delay: {proj.max_delay}")     # max of learned delays