Output Evidence Against Time and Gamma Cycles

Background

We imported the trained PING network from exp022 — Training Runs and kept its weights fixed. The parameter table lists the network and simulation settings.

Evidence across time

We encoded each of the first nine MNIST test images into 100 random spike trains and reused each train at $\tau_{\mathrm{GABA}} = 3, 6, 12$ ms, where $\tau_{\mathrm{GABA}}$ is the inhibitory decay time constant. Brighter pixels had higher average spike rates, with 25 spikes per second for a fully bright pixel. Each presentation lasted 200 ms, with 0.1 ms simulation steps. For each trial, the logit for digit $c$ was its accumulated output spike count:

Here, $t$ is elapsed time, $\Delta t_{\mathrm{sim}}$ is the timestep, $k$ indexes simulation steps, and $s_c[k]$ is one when output neuron $c$ spikes, otherwise zero. We converted these logits into scores using softmax:

Here, $p_c(t)$ is the score for digit $c$, and $j$ ranges over all ten digits. We averaged the correct digit’s score after softmax across 100 trials, separately for each decay setting. These scores are uncalibrated.

Evidence per completed volley

We detected inhibitory peaks after 1 ms Gaussian smoothing, requiring ≥2 ms spacing and prominence ≥max(1 spike, 20% of the maximum). Accepted sequences required a complete interior cycle and mean inhibitory participation ≥50%.

We defined each volley’s completion as the minimum of smoothed inhibitory activity between its peak and the next. At that boundary, we sampled the correct digit’s softmax score from all output spikes accumulated since onset.

We averaged these samples across 100 trials for each decay setting, using completed-volley counts available in every trial and setting. The final unbounded volley is excluded; count zero shows the initial score of 0.1 at time zero, not a detected volley. Clustering and score saturation can mimic alignment.

Sustained correct decisions

For the same nine images and 100 trials per decay setting, we found the earliest update at which the correct digit had the largest cumulative output spike count and remained the unique winner through 200 ms. Ties interrupt stability; trials without a stable correct decision were excluded.

We used the first completed-volley trough boundary at or after each stable decision. This is the first sampled point at which the decision is already stable, numbered from one. Counts required accepted volley sequences, a decision no later than the final completion boundary, and the shared completed-volley count range. We summarised median decision time and median completed-volley count against inhibitory decay. The two summaries can include different trial subsets. Trial counts are integers; medians can be fractional. Stability was assessed retrospectively, not by a stopping rule.

Accuracy at matched coordinates

We compared accuracy against time, cycle position and cumulative input spikes, alongside paired-trial coverage. The sample comprised 500 images, 50 per digit, with three paired encodings and inhibitory decay times of 3, 6 and 12 ms. Uncertainty was summarised with image-bootstrap 95% intervals. Accuracy is omitted below 80% coverage. Later coordinates select subsets. Equal spike counts do not guarantee equal information.

Accuracy and spike cost

We compared accuracy versus deadline, mean network spikes versus deadline, and accuracy versus spike cost at encoding rates of 12.5, 25 and 50 Hz and inhibitory decay times of 3, 6 and 12 ms. We evaluated five deadlines within each setting, using 500 images and three paired encodings. Accuracy uncertainty was summarised with image-bootstrap 95% intervals. Cost includes excitatory, inhibitory and output spikes, excluding input. Spike cost is not physical energy; encoding rate also changes drive.

Pareto candidates and frequency

We compared Pareto candidates using deadline, accuracy and mean network spikes at encoding rates of 12.5, 25 and 50 Hz, for each inhibitory decay setting. Candidates were classified as empirically nondominated or dominated. Frontier membership is sample-dependent.

We also measured peaks of mean excitatory-population power spectral densities at the same encoding rates. Peaks can reflect harmonics or heterogeneous rhythms. The sample and cost definition were the same as in the accuracy and spike-cost comparison.