Inhibition, Accuracy and Firing

Abstract

We compared MNIST accuracy and excitatory firing in trained COBA and PING classifiers across activity ceilings, with three independent training seeds per frontier condition. Without a rate penalty in the reduced-pool study, PING reached 89.8% test accuracy at 16.6 Hz, versus 90.9% at 88.2 Hz for COBA. Additional single-seed probes measured cycle participation and input-coupling sensitivity. These comparisons neither establish a structural firing-rate floor nor isolate a benefit of gamma timing or measure energy use.

Results

Accuracy-rate activity ceilings

Full-MNIST reference classifiers reached mean final validation accuracies of 91.4% for PING and 95.3% for COBA (Figure 1A–C). In the separate reduced-training-pool comparison, at the unpenalised operating points, PING reached 89.8% at 16.6 Hz and COBA reached 90.9% at 88.2 Hz. These measurements do not establish a structural lower firing-rate limit (Figure 1D).

PING participation and frequency

PING participation varied from 0.16 to 0.26 and oscillation frequency from approximately 60 to 18 Hz. The $p_{\mathrm{part}} f_\gamma$ approximation differed from measured E rate by up to 27.5%, so participation was not constant. PING accuracy spanned 81–90%; COBA fell from 89% to 62% as the ceiling tightened (Figure 2).

Input-coupling learning curves

Final validation accuracies were 76.5% / 76.6% / 76.8% / 76.6%, while final I rates were 5.6 / 5.8 / 5.6 / 5.6 Hz. The similar endpoints across these initializations do not prove basin attractivity (Figure 3).

Inference input-scaling response

COBA’s penalty reached approximately 2 at $s = 3$. The empirical inhibitory-rate crossing marks a sampled transition, not a fitted bifurcation (Figure 4).

Accuracy-rate input scaling

PING’s highest sampled accuracy was approximately 83%. At input scale $s = 3$, COBA reached approximately 63% at 8 Hz. This single direction of weight scaling does not map the full loss landscape (Figure 5).

Methods

We reused networks and learning histories from the exp022 — MNIST and reanalysed recorded inference measurements; no new training or simulation was performed.

Compute

1. Prepare digit inputs. Full-MNIST reference classifiers in Figure 1 A–C used all 60,000 official-training images, split into 54,000 optimization and 6,000 validation images. The activity-ceiling and input-coupling studies used a separate 7,000-image pool, split into 6,300 optimization and 700 validation images. Sampling and class-stratified splitting used seed 42. Pixels drove 784 Poisson channels at a 25 Hz maximum, for 200 ms with 0.1 ms steps.

2. Compare network configurations. Both conductance-based networks had 1,024 excitatory (E), 256 inhibitory (I), and 10 output leaky-integrate-and-fire neurons. Pyramidal-interneuron gamma (PING) enabled fixed E↔I coupling; COBA disabled it; E→E and I→I coupling were zero throughout. Only input and readout weights trained; class scores were mean pre-reset output membrane voltages. Voltage-gradient damping was 1,000 for both COBA and PING.

3. Train with activity ceilings. Networks trained for 50 epochs with AdamW (zero weight decay), learning rate $4 \times 10^{-4}$, batch size 256, and gradient-norm clipping at 1. Cross-entropy was supplemented by:

   $$r_b = 1 / (N_E T_{\mathrm{present}}) \sum_{n \in E} n_{\mathrm{spike}}(b,n), \quad L_{\mathrm{rate}} = \lambda_{\mathrm{rate}} / B \sum_b \max(r_b - r_{E,\mathrm{ceil}}, 0)^2. \tag{1}$$

   Here $n_{\mathrm{spike}}(b,n)$ counts excitatory neuron $n$‘s spikes in presentation $b$, $E$ is the excitatory population, $N_E$ its size, $T_{\mathrm{present}}$ its duration in seconds, and $B$ minibatch size. Rates $r_b$ and ceilings $r_{E,\mathrm{ceil}}$ are in hertz; $\lambda_{\mathrm{rate}} = 0.041$ $\mathrm{Hz}^{-2}$ weights the dimensionless penalty $L_{\mathrm{rate}}$. Six ceiling conditions (Training settings) and three seeds yielded 36 reduced-pool networks. Six separate full-MNIST reference networks trained without an activity penalty; their final-checkpoint digit-0 recordings lasted 200 ms, and their learning curves averaged the three validation accuracies at each epoch.

Analyse

4. Evaluate training endpoints. Final-epoch weights, rather than validation-selected weights, supplied the endpoint comparisons. Each network was evaluated on the same 1,000 official-test images; frontier points show means and standard errors across seeds 42–44.

5. Measure cycle participation. For seed 42, oscillation frequency $f_\gamma$ was the 5–150 Hz peak of trial-averaged Welch spectra of E activity 1. Participation $p_{\mathrm{part}}$ was the fraction of E-neuron/cycle pairs containing at least one spike, with cycles delimited by I-burst midpoints (Measurement details).

   $$r_E \approx p_{\mathrm{part}} f_\gamma. \tag{2}$$

   This diagnostic approximation relates mean E rate $r_E$ (Hz), dimensionless $p_{\mathrm{part}}$, and $f_\gamma$ (Hz); repeated spikes and differing cycle/frequency aggregation prevent treating it as an identity.

6. Vary input coupling. Twelve PING networks used four initial input-coupling means and three seeds, with a 1 Hz ceiling throughout; validation histories measured recruitment during training. Separately, seed-42 PING and COBA networks trained at 1 Hz were evaluated after multiplying all input weights by dimensionless $s \in [0.05, 3]$, holding other weights fixed, on the same 1,000 test images at 24 scales.

Present

7. Expose retained comparisons. We displayed endpoint comparisons, cycle-participation probes and input-coupling sensitivity with their recorded seed roles and aggregation.