044 — Rate floor stable across a 20× Δt sweep — physical Hz, not artefact (trains)

Abstract

Training recipe (canonical / medium tier):

The Δt audit asks whether the nb025 headline E rate is a physical (Hz) property of the trained network or an artefact of the integration timestep. The rate stays in a 9.5–14 Hz band across a 20× Δt sweep, accuracy holds at 88.5–90.7%, and the gamma cycle period in physical ms is invariant — but the rate dependence on Δt is non-monotonic, with Δt = 0.5 ms producing a lower mean rate than Δt = 0.25 ms. The architectural separation from COBA (≈ 96 Hz at the same recipe, from nb025) holds at every Δt.

Methods

Train one PING from scratch per $\Delta t \in \{0.05, 0.1, 0.25, 0.5, 1.0\}$ ms × seed $\in \{42, 43, 44\}$ = 15 cells. Total physical time $T = 200$ ms held constant (step count varies 4000 → 200). Batch size 64 throughout — smaller than nb025’s 256, but matched across the sweep so per-step compute and memory stay comparable, and the Δt = 0.05 cells (4000 timesteps × $N_E$ × $N_I$) fit in a single A100. All other PING recipe parameters held to nb025.

After training, run inference on the test set; report mean E rate (Hz), accuracy, and a single-trial raster from seed 42 per Δt for visual cycle-period inspection.

Results

Figure 1. PING Δt audit — post-training E rate and accuracy vs integration timestep

The E rate stays in a 9.5–14 Hz band across the 20× Δt sweep — a 4.6 Hz span (1.48× ratio). The relationship is not monotonic: Δt = 0.5 ms gives a lower mean rate than Δt = 0.25 ms. Accuracy and the qualitative gamma dynamics are unaffected (2.2 pp variation in accuracy; I rate moves in a similar 40–55 Hz band).

Figure 2. Trained-PING rasters at each Δt (seed 42, MNIST digit 0 sample 0) — x-axis is physical time in ms

The raster strip is the load-bearing diagnostic. Across the 20× Δt sweep, the gamma cycle period in milliseconds is invariant — bursts at the same physical cadence in every panel. The cycle physics is Δt-invariant.

Why the rate varies with Δt

In the discretised LIF integrator with forward Euler, $V_{t+1} = V_t + \Delta t \cdot \dot V$, a single timestep advances the membrane farther in mV when Δt is larger. The effect on the rate is non-monotonic across this sweep — Δt = 0.5 ms produces a lower mean rate than Δt = 0.25 ms — which suggests the interaction with refractory windows and per-burst dynamics is not a simple “coarser Δt → higher rate” relation. Two effects act in opposite directions:

• Threshold crossing per step is more permissive at coarse Δt. A subthreshold $V_t$ with positive drift may not cross $V_\text{th}$ in 0.05 ms but does in 1 ms. This biases per-burst participation upward at coarse Δt.
• Refractory granularity changes with Δt. Refractory is held constant in physical ms ($\text{ref\_ms\_E} = 3$ ms → $3/\Delta t$ steps); but the exact refractory exit moment is quantised to the integration grid, which shifts which cycle a cell rejoins.

The net effect across this sweep is a 4.6 Hz band (9.5–14 Hz) with no monotone ordering. The cycle frequency (cycle period set by $\tau_\text{AMPA}$ and $\tau_\text{GABA}$) is Δt-invariant; the per-cycle participation probability $p$ in nb041’s law $r_E = a + p \cdot f_\gamma$ has some Δt-dependence that this sweep does not fully disentangle.

Convergence

Figure 3. Per-cell training curves — convergence check across Δt sweep

Two convergence observations:

1. Accuracy converges within ~ 15 epochs in every cell. All Δt values reach 86–91% accuracy by epoch 15 and plateau through epoch 100.
2. E rate is still drifting at epoch 100 in some cells. The reported rate-vs-Δt values are snapshots at epoch 100; with longer training individual rates would move further but the band (9.5–14 Hz) and the non-monotone ordering are unlikely to collapse.

The gamma cycle observation from Figure 2 is independent of training convergence — it’s a property of the underlying dynamics and would hold at any training duration.

Discussion

This audit confirms the structural claims are preserved across Δt:

1. The gamma cycle period is Δt-invariant — the cycle is genuine network dynamics, not a step-counting artefact.
2. The qualitative architectural separation survives. At every Δt the PING network fires in the 9.5–14 Hz band; the COBA equivalent from nb025 sits at ≈ 96 Hz. The order-of-magnitude rate gap that motivates the forbidding claim does not collapse at coarse or fine Δt.
3. The numerical headline rate has a Δt-dependent residual. Reporting ”≈ 10 Hz” without qualifying Δt is acceptable; the right precise statement is “9.5–14 Hz across $\Delta t \in [0.05, 1]$ ms with non-monotone ordering, accuracy invariant at 88.5–90.7%.”

Next steps

The ar010 framing should anchor the rate claim at $\Delta t = 0.1$ ms and explicitly cite this audit’s Figure 2 (the Δt-invariant gamma cycle) as the structural-bound argument’s load-bearing observation. The Figure 1 rate-vs-Δt scaling is itself worth one paragraph of methods, noting that the per-cycle integration of the LIF dynamics has a residual Δt effect on participation probability.

A natural follow-up is a retrained version of nb041 at multiple Δt values — six $\tau_\text{GABA}$ × three Δt would test whether the affine law’s slope $p$ scales predictably with Δt or wobbles. That experiment is not on the ar010 critical path but is the cleanest way to establish the Δt → $p$ functional form.