Sigmoid-rate source and polyglot fidelity evidence¶

This page records the scientific scope, finite-step equation, five-runtime parity, failure atomicity, and controlled benchmark used to promote SigmoidRateNeuron to the polyglot-complete catalogue.

Scientific scope¶

Wilson and Cowan (1972), doi:10.1016/S0006-3495(72)86068-5, derive coupled nonlinear differential equations for interacting excitatory and inhibitory populations. SigmoidRateNeuron does not claim to reproduce that complete system. It declares a reduced scalar motif:

[ \tau\dot r=-r+\sigma(\beta(I-\theta)). ]

The citation therefore supports the population-rate and sigmoid inspiration; fidelity is evaluated against the explicitly declared scalar equation. The coupled family remains a separate WilsonCowanUnit surface.

Exact finite-step contract¶

With current held constant over one step, the logistic target is constant and the scalar ODE has the exact update

[ r_{n+1}=d r_n+(1-d)\sigma(\beta(I-\theta)), \qquad d=e^{-\Delta t/\tau}. ]

All five runtimes implement that update, branch-stable logistic evaluation, finite validation, a bounded-rate postcondition, and candidate-before-mutation failure atomicity. Reset changes only the dynamic rate.

Executable parity matrix¶

Runtime	Executed surface	Enrolled result
Python	public scalar and atomic batch	reference
Rust engine	PyO3 modular batch, zero crate-root delta	byte-identical to Python
Rust safety	independently compiled module	byte-identical to Python; 8/8 tests pass
Julia	`simulate_trace`	byte-identical to Python
Go	service plus generated C-shared ABI	byte-identical to Python
Mojo	exported shared-library C ABI	maximum absolute difference `3.08e-14`

The configured parity case is r=0.25, tau=10, beta=2, theta=1, dt=0.5, and I=3. The first six Python values are:

Text Only

0.2857007338135623
0.3196603222932904
0.3519636820991432
0.38269158845670403
0.41192087713731845
0.43972463658754457

tests/test_sigmoid_rate_backends.py also executes empty batches, a timestep fifty times the time constant, explicit-unavailable backends, malformed native results, and invalid Go/Mojo contracts whose output buffers must remain unchanged.

Generated fixed-point co-simulation¶

The paired TOML and JSON schemas preserve the hand exact-relaxation trajectory within 5e-12 over a 256-step sign-changing input. The production equation compiler lowers the same schema to a Q32.32 Verilog module. Icarus Verilog co-simulation reads the public r_out and spike_out ports and establishes:

maximum absolute rate difference 0.014879114367180313, below 0.016;
every emitted rate remains in [0, 1];
spike_out remains zero on every step, so positive rates are not recast as binary events.

The state bound includes the 0.125-argument sigmoid and exponential-relative lookup-table quantisation. It is an H1 generated-RTL trajectory result, not bit identity for transcendental functions.

Controlled benchmark¶

benchmarks/bench_model_sigmoid_rate.py measures the full 200,000-step trace through each public dispatcher, five times after warm-up. It fails if any backend is absent, the standalone Rust-safety test binary fails, or a trace or final rate differs by more than 5e-12.

The run is pinned to one logical CPU but is not exclusively isolated. The artifact records raw samples, source hashes, exact Rust/Go/Mojo binary hashes, runtime versions, affinity, and load. It explicitly rejects production-speed, cross-host, hardware, and universal-ranking interpretations.

Backend	Median call	Median ns/step	Maximum error
Python	73.788 ms	368.938	0
Rust	46.985 ms	234.926	0
Julia	17.493 ms	87.467	0
Go	97.270 ms	486.350	0
Mojo	14.610 ms	73.048	`3.08e-14`

The Python/Rust/Julia/Go canonical trace SHA-256 is 5241be414683ce92ba9886c13c0a9f5ef84886d5d48ddda05fc892b72274e07d. Mojo has a distinct binary trace hash because the tolerated libm-level difference is real and disclosed.

Boundaries¶

This is continuous rate output, not a spike train. Positive rates are not counted as events.
The paper citation does not turn the scalar unit into the full coupled Wilson-Cowan model.
The benchmark is local regression evidence, not a deployment claim.
The generated Q32.32 claim is bounded co-simulation only. No formal equivalence, synthesis, timing, device, or PPA result is claimed.

Reproduction¶

Bash

go test ./services ./neurons/sigmoid_rate -run SigmoidRate -count=1
cargo test --manifest-path src/sc_neurocore/accel/rust/Cargo.toml \
  sigmoid_rate --lib -j 4

PYTHONPATH=bridge:src:. .venv/bin/python -m pytest -q \
  tests/test_model_sigmoid_rate.py \
  tests/test_cosim_sigmoid_rate.py \
  tests/test_sigmoid_rate_backend_loading.py \
  tests/test_sigmoid_rate_backends.py \
  tests/test_bench_sigmoid_rate.py

taskset -c 4 env PYTHONPATH=bridge:src:. .venv/bin/python \
  benchmarks/bench_model_sigmoid_rate.py \
  --json benchmarks/results/local_python_2026-07-14_sigmoid_rate.json