PING locks E rate ≈10× below COBA

PING locks E rate ≈10× below COBA#

exp025 · 30 May 2026 · pdf

The trained networks this entry uses are produced once in the shared training hub, exp022 (Training), and reused here rather than retrained.

Abstract#

Head-to-head comparison of COBA (recurrent inhibitory loop disabled) against PING (loop active) on MNIST under matched architecture and training recipe. PING locks the hidden E rate to ≈ 12 Hz while COBA runs at ≈ 181 Hz; accuracy is within a few points across the two, so the loop buys better than an order-of-magnitude per-spike economy. The rate-vs-accuracy frontier traced by sweeping the $𝜃_{𝑢}$ rate regulariser shows the floor is structural, not a trade-off the optimiser can navigate away.

Method#

Training recipe (canonical / medium tier):

Parameter	Value
Integration timestep $Δ 𝑡$	0.1 ms
Trial duration $𝑇$	200 ms
MNIST samples (80/20 stratified split of 2000)	1600 train / 400 test (≈ 2.9% of the 70k-sample MNIST corpus)
Epochs	100

Two configurations of the same COBANet architecture, differing only in whether the E→I→E inhibitory loop is active. COBA (–ei-strength 0) disables the loop; excitatory cells drive each other but receive no structured inhibition. PING (–ei-strength 1) enables the loop, producing pyramidal-interneuron gamma (PING) oscillations at a cadence set by $𝜏_{AMPA}$ and $𝜏_{GABA}$ .

Architecture. $𝑁_{𝐸} = 1024$ excitatory cells, $𝑁_{𝐼} = 256$ inhibitory cells, single hidden layer. Input: 784 channels (MNIST pixels), Poisson-encoded at 25 Hz peak rate. Readout: mem-mean (time-averaged E spike vector projected through a trained linear layer $𝑊_{out}$ ; see ar006 for the full readout specification). Dale’s law enforced.

What is trainable. Only the input weights $𝑊_{in}$ ( $784 \times 1024$ , 95% sparse) and the readout $𝑊_{out}$ ( $1024 \times 10$ ). The recurrent weights $𝑊_{𝑒 𝑒}$ (fixed at zero, since Börgers-style PING needs no E→E coupling), $𝑊_{𝑒 𝑖}$ , and $𝑊_{𝑖 𝑒}$ are initialised once and held requires_grad = False. The synaptic time constants $𝜏_{AMPA}$ , $𝜏_{GABA}$ are module-level constants. 813k trainable parameters out of 2.4M total.

Training. Adam, lr = $4 \times 10^{- 4}$ , batch size 256, gradient norm clipped to 1.0. Cross-entropy loss on 10-class MNIST (definition in ar006). Gradient stabiliser: –v-grad-dampen 1000 (uniform scaling of per-step voltage gradients). Three seeds (42, 43, 44) for baselines, one seed (42) for sweep cells.

Recipe difference. The only parameter that differs between COBA and PING besides –ei-strength is the $𝑊_{in}$ initialisation: COBA uses mean 0.3 (std 0.03), PING uses mean 1.2 (std 0.12). PING needs stronger input drive to reliably recruit the I-loop at init.

Spike-budget regulariser. To probe the rate axis, the training loss adds a soft upper bound on per-trial spike count. For each E cell $𝑛$ with mean per-trial spike count ${\bar{𝑧}}_{𝑛}$ :

𝐿_{rate} = 𝜆 \sum_{𝑛} {ReLU ({\bar{𝑧}}_{𝑛} - 𝜃_{𝑢})}^{2},

with $𝜃_{𝑢}$ the per-cell budget (spikes/trial) and $𝜆 = 10^{- 3}$ ; only cells over budget contribute, cells under it are free, and the total loss is cross-entropy plus $𝐿_{rate}$ . We sweep six budgets (off, 5, 2, 1, 0.5, 0.2 spikes/trial; at $𝑇 = 200$ ms this is no penalty, 25, 10, 5, 2.5, 1 Hz), giving twelve $(model, 𝜃_{𝑢})$ cells.

Rate-floor decomposition. The affine law $𝑟_{𝐸} = 𝑝 \cdot 𝑓_{𝛾}$ (exp041, exp046) factors the E rate into per-cycle participation $𝑝$ and gamma frequency $𝑓_{𝛾}$ . Both are measured at every cell: $𝑓_{𝛾}$ from the Welch PSD peak of the E-population trace, $𝑝$ via I-burst peak detection and per-(cell, cycle) spike counting (style of exp046).

Basin and landscape probes. To test basin attractivity, four PING networks are trained with $𝑊_{in}$ initialised at 0.05, 0.1, 0.3, and 1.2 ( $𝜃_{𝑢} = 0.2$ from epoch 0; Figure 3). To map the loss landscape around the operating point, each network trained at the heaviest penalty ( $𝜃_{𝑢} = 0.2$ ) has its $𝑊_{in}$ scaled by a common scalar $𝑠 \in [0.05, 3]$ at inference with all other weights frozen, metrics averaged over the test set at 24 values of $𝑠$ (Figures 4–5).

Results#

Two-by-two panel: COBA and PING single-trial rasters, per-epoch learning curves, and the accuracy–rate frontier across the spike-budget penalty. — Figure 1: The headline comparison in one 2×2 frame (rasters, learning, and the accuracy–rate frontier together). **Top:** trained-baseline single-trial rasters on the same MNIST digit 0 input. COBA (*–ei-strength 0*) fires densely and asynchronously at ≈ 181 Hz (I silent, loop off); PING (*–ei-strength 1*) fires in gamma bands at ≈ 22 ms cadence (≈ 12 Hz per E cell) with synchronous I bursts (red) above E (black), on the same architecture, parameter count, and recipe, with only PING’s recurrent E↔I matrices non-zero. **Bottom left:** test accuracy per epoch (mean over three seeds); both reach ≈ 91%, so both learn the task. **Bottom right:** the accuracy–rate frontier across the spike-budget penalty $𝜃_{𝑢}$ . PING sits up-and-left of COBA, the same accuracy at a fraction of the hidden-E rate, with the $𝜃_{𝑢}$ -off operating points starred and labelled (PING 91% at ≈ 12 Hz, COBA 91% at ≈ 181 Hz). COBA red, PING black; E black and I red in the rasters. *The headline of this entry: gamma gating buys an order-of-magnitude per-spike economy at matched accuracy.*

PING participation fraction p and gamma frequency f_gamma across the spike-budget sweep, with the p·f_gamma product overlaid on the measured E rate. — Figure 2: Five $(PING, 𝜃_{𝑢})$ sweep cells, 256 test trials each. **$𝑝$ stays in 0.18–0.23** across the entire sweep (the architecture protects the participation gate), while **$𝑓_{𝛾}$ slides from ≈ 46 Hz to ≈ 17 Hz** as the penalty tightens. The grey dashed $𝑝 \cdot 𝑓_{𝛾}$ curve overlays the measured E rate within 4%; the rate change is entirely in $𝑓_{𝛾}$ . PING accuracy holds at 92–98% the whole way; COBA’s collapses from 91% to 68%.

Per-epoch test accuracy and E/I firing rates for four PING networks initialised across the recruitment cliff, one column per W_in value. — Figure 3: Per-epoch training traces from four PING networks (seed 42, $𝜃_{𝑢} = 0.2$ from epoch 0), one per column. Top: test accuracy. Bottom: test-set E (black) and I (red) firing rates. Recruitment is $𝑊_{in}$ -ordered: at $𝑊_{in} = 0.05$ and $0.1$ the I population stays silent for the first ≈ 8 epochs and engages only weakly in the final one or two; at $𝑊_{in} = 0.3$ the loop recruits by epoch 7; at $𝑊_{in} = 1.2$ it is active from epoch 1. The lower the input drive, the later the loop crosses the recruitment threshold. Final accuracies: 76% / 81% / 79% / 82%; final I rates: 6.9 / 2.9 / 10 / 20 Hz.

Inference-time W_in scale sweep: CE loss, spike-budget penalty, total objective, test accuracy, and E/I rates versus scalar s for PING and COBA. — Figure 4: Inference-time $𝑊_{in}$ scale sweep on the two networks trained under the heaviest penalty ( $𝜃_{𝑢} = 0.2$ ); every $𝑊_{in}$ weight multiplied by a common scalar $𝑠$ , all other weights frozen, 24 values of $𝑠 \in [0.05, 3]$ . Top row: CE loss, spike-budget penalty $𝐿_{rate}$ , total objective CE + $𝐿_{rate}$ . Bottom row: test accuracy (chance dotted), E rate, I rate. PING black, COBA red. Vertical dashed line at $𝑠 = 1$ marks the trained operating point; dotted line marks ≈ $𝑓^{*}$ , PING’s recruitment cliff. Loss panels clipped at 4; COBA’s penalty reaches ≈ 44 at $𝑠 = 3$ ( ${rate}^{2}$ scaling).

The W_in scale sweep re-projected with hidden E rate on the x-axis, trained operating points starred for PING and COBA. — Figure 5: The same 24-point sweep as Figure 4, re-projected with hidden E rate on the x-axis. Filled stars mark each cell’s trained operating point ( $𝑠 = 1$ ): PING at $𝐸$ ≈ 3.8 Hz, COBA at $𝐸$ ≈ 1.3 Hz. PING’s accuracy reaches its plateau (≈ 88%) just past its trained operating point; COBA climbs slowly and only reaches ≈ 71% even at $𝐸$ ≈ 33 Hz.