Decision Time and Gamma Cycles

Background

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

Results

Inhibitory decay and network dynamics

We tested one trained network on 500 MNIST test images, 50 per digit, with ten random spike encodings per image. We reused each encoding at 19 inhibitory decay settings, $\tau_{\mathrm{GABA}}$, from 3 to 30 ms. Each presentation lasted 200 ms, with network state reset between presentations. All five figures use this shared sweep of 95,000 presentations. Sampling seeds, encoding rates and simulation settings are listed in the parameter table. No uncertainty estimates are shown.

Longer inhibitory decay slowed the rhythm and reduced firing (Figure 1, A and C). From 3 to 30 ms decay, the spectral peak fell from 100.2 to 17.7 Hz and excitatory firing fell from 26,589 to 5,377 spikes/s. The fraction of excitatory neurons active per accepted cycle stayed between 20.9% and 24.5% (B). Spectral peaks can select harmonics; these results come from one trained network.

Evidence across time

Higher-frequency groups generally developed high correct-digit scores sooner (Figure 2). The pattern varied by image; image 115 (F) kept much lower scores than most examples.

The nine examples are the first selected image indices: 0, 7, 29, 51, 71, 115, 140, 161 and 164. For digit $c$, $z_c(t)$ counts its output spikes by time $t$:

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 applied softmax before averaging the correct digit’s score across ten encodings. All scores start at 0.1. The network was trained using mean voltage; this spike-count readout was not retrained or calibrated.

Evidence per completed volley

Several images reached high scores within a few completed volleys (Figure 3). The traces do not align exactly: early scores vary with decay, image 115 remains distinct, and later counts use different trial subsets. Score saturation can also make traces look aligned.

We detected inhibitory volleys after 1 ms Gaussian smoothing, requiring peaks at least 2 ms apart and prominence of at least one spike or 20% of the maximum, whichever was larger. Accepted sequences needed a complete interior cycle and at least 50% mean inhibitory participation. A volley ended at the trough between its peak and the next; we sampled the accumulated output score there. We did not extend traces beyond the last bounded completion.

Sustained correct decisions

Higher frequencies generally accompanied shorter decision times, while median volley counts stayed near one or two over much of the range (Figure 4). Decisions before the first trough count as one; this does not mean a whole volley was necessary. These plots include only eligible successful trials. Frequency uses the full 200 ms, including activity after the decision, so the association does not show that frequency alone sets classification speed.

A lasting correct decision is the first update after which the correct digit stays the unique output-spike-count winner through 200 ms. Ties break stability. This looks back over the full trial; it is not an online stopping rule.

Of 95,000 trials, 84,284 had a lasting correct decision. Of these, 84,237 also had an accepted frequency and appear in A. Panel B also requires a decision no later than the final bounded completion and at least eight of ten encodings with enough completed volleys at that count, leaving 80,341 trials.

Spike cost at 80% accuracy

Slower settings saved spikes at the cost of time (Figure 5). At 3 ms decay, reaching 80% accuracy took 38 ms and 5,135 mean network spikes. At 30 ms decay, it took 144 ms and 3,706 spikes: 28% fewer spikes, but 3.8 times longer. Mean I-volley frequencies were about 107 and 15 Hz.

For each setting, predictions used the largest accumulated output spike count; the lowest digit won ties. All 19 settings reached 80%. Unlike Figure 4, accuracy could fall again after crossing the target. The 80% target was chosen for this exploratory comparison, and the curve was selected on the same test sample. Costs are averages, not individual stopping costs or physical energy. Other accuracy targets can give different curves.