Firing Rate Tracks Gamma Frequency

Results

Inhibitory-timescale training response

Inhibitory-timescale spectra

Per-network interpolated frequencies spanned approximately 8–68 Hz across the inhibitory-decay sweep (Fig. 2).

Inhibitory-timescale rasters

Rate and accuracy versus frequency

Individual network rates spanned 2.3–18.4 Hz and accuracies 81–92% across the six inhibitory-decay conditions (Fig. 4).

Alternative rate-frequency fits

Table 1: Both least-squares fits used the same six condition means and centred total sum of squares. The origin-constrained fit tests how much the association depends on a free intercept.

Methods

We compared final-epoch dynamics across matched networks trained at different inhibitory decay times, keeping the evaluation data fixed.

1. Reuse matched trained networks. Six inhibitory GABA decay constants, ${𝜏}_{\text{GABA}}\in \left\{4.5,6,9,12,18,27\right\}$ ms, and seeds 42–44 defined 18 PING networks. Training used 6,300 MNIST training images and 700 held-out validation images, 50 epochs of AdamW with zero weight decay, learning rate $4\times {10}^{-4}$, batches of 256, a time-averaged membrane-potential readout and no spike budget. Only inhibitory decay varied between conditions; simulation used 0.1 ms timesteps and 200 ms trials. We reused final-epoch weights to measure endpoint dynamics, without retraining or selecting weights by test performance.

2. Measure fixed-trial responses. Each network received the same fixed subset of 1000 images from the official MNIST test partition. We measured classification accuracy, mean excitatory spikes per neuron per second, and each trial’s population-E trace over the full 200 ms. The illustrative raster used image index 0 and seed 42; a fixed random seed of 0 selected displayed neurons without replacement.

3. Estimate rhythm frequency. We demeaned each trial’s trace and used a Welch density estimate with one full-trial Hann window [1], then averaged PSDs across trials. The largest peak between 5 and 150 Hz defined the candidate gamma frequency; its neighbouring linear-power values gave

   $${𝑓}_{\text{peak}}={𝑓}_{\text{peak,bin}}+\frac{1}{2}\frac{{𝑦}_0-{𝑦}_2}{{𝑦}_0-2{𝑦}_1+{𝑦}_2}\cdot \mathrm{\Delta }{𝑓}_{\text{bin}}.$$

   (1)

   Here ${𝑓}_{\text{peak}}$ is the interpolated spectral-peak frequency, ${𝑓}_{\text{peak,bin}}$ the peak-bin frequency, and $\mathrm{\Delta }{𝑓}_{\text{bin}}=5$ Hz the bin spacing; ${𝑦}_0$, ${𝑦}_1$ and ${𝑦}_2$ are PSD values immediately below, at and above that bin. Because the peak search is restricted to the defined gamma band, the reported estimator is ${𝑓}_{𝛾}\equiv {𝑓}_{\text{peak}}$. We clamped the correction to half a bin, using zero offset for zero curvature or a spectrum endpoint. Interpolation reduces bin quantisation but can remain biased [2]. Per-trial peak distributions were diagnostics; their medians did not enter the fit.

4. Fit the rate–frequency relation. We averaged each network’s frequency and excitatory rate over the three seeds, then fitted the six condition points with equal weight by least squares:

   $${𝑟}_{𝐸}=𝑎+{𝛽}_{𝑟𝑓}\cdot {𝑓}_{𝛾},{𝑟}_{𝐸}={𝛽}_{𝑟𝑓,0}\cdot {𝑓}_{𝛾}.$$

   (2)

   Here ${𝑟}_{𝐸}$ is mean excitatory firing rate in hertz, $𝑎$ is the affine intercept in hertz, and ${𝛽}_{𝑟𝑓}$ and ${𝛽}_{𝑟𝑓,0}$ are dimensionless fitted slopes. Both fits report ${𝑅}_{\text{fit}}^2$, the coefficient of determination using centred total sum of squares; error bars are sample standard deviations divided by $\sqrt{3}$.

5. Display matched endpoint evidence. We displayed the retained frequency, rate and accuracy comparisons across inhibitory decay times, preserving seed-level values and across-seed summaries.