E rate is affine in gamma frequency

E rate is affine in gamma frequency#

exp041 · 2 June 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#

exp037 varied $𝜏_{GABA}$ at inference and the per-cell E rate tracked the gamma cycle. Does that survive re-training at each $𝜏_{GABA}$ ? Yes. Across $𝜏_{GABA} \in {4.5, 6, 9, 12, 18, 27}$ ms × 3 seeds, $𝑟_{𝐸} = 1.15 + 0.183 \cdot 𝑓_{𝛾}$ with $𝑅^{2} = 0.994$ . The slope is per-cycle E-cell participation; the intercept is a non-rhythmic baseline; accuracy stays at ≈ 86–92% across the sweep. The cycle clock constrains what the network can become.

Method#

Sweep. Six $𝜏_{GABA}$ values ${4.5, 6, 9, 12, 18, 27}$ ms × three seeds = 18 networks, trained in the shared hub to the gamma standard (50 epochs on MNIST, Adam at $4 \times 10^{- 4}$ , batch 256, mem-mean readout, no spike budget, $Δ 𝑡 = 0.1$ ms, $𝑇 = 200$ ms); only $𝜏_{GABA}$ varies.

Measuring $𝑓_{𝛾}$ . For each cell, $𝑓_{𝛾}$ is the parabolic-interpolated peak of the Welch PSD on the per-trial population E trace; the fit (Figure 4) uses per-trial peak medians, which avoid the centroid bias of trial-mean PSD peaks. Parabolic interpolation with peak-bin values $(𝑦_{0}, 𝑦_{1}, 𝑦_{2})$ :

𝑓_{𝛾} = freq [peak] + \frac{1}{2} \frac{𝑦_{0} - 𝑦_{2}}{𝑦_{0} - 2 𝑦_{1} + 𝑦_{2}} \cdot Δ 𝑓, Δ 𝑓 = 5 Hz .

It is needed because the bare 5 Hz bin quantisation would coarsen $𝑓_{𝛾}$ across the six conditions; on a well-isolated peak the interpolation error is $𝑂 ({(Δ 𝑓)}^{3})$ .

The predicted law. The shape $𝑟_{𝐸} = 𝑎 + 𝑝 \cdot 𝑓_{𝛾}$ is predicted by cycle dynamics, not curve-fitted. Within one cycle of duration $1 / 𝑓_{𝛾}$ , a fraction $𝑝$ of E cells emits exactly one spike (those nearest threshold when the I shunt drops); the rest are still recovering, so the cyclic per-cell rate is $𝑝 \cdot 𝑓_{𝛾}$ . At long $𝜏_{GABA}$ the I conductance never fully decays and the cycle dissolves into a tonic bath, leaving a feedforward baseline $𝑎$ independent of $𝑓_{𝛾}$ :

𝑟_{𝐸} = \underset{feedforward baseline}{\underset{⏟}{𝑎}} + \underset{cyclic contribution}{\underset{⏟}{𝑝 \cdot 𝑓_{𝛾}}} .

We fit this across the 18 cells.

Convergence. Accuracy plateaus by ≈ epoch 15 while the E rate keeps climbing through training (Figure 1), so the fit uses the final-epoch rates. Fitting $𝑟_{𝐸} = 𝑎 + 𝑝 \cdot 𝑓_{𝛾}$ across the 18 cells is tight, and forcing the intercept through zero barely loosens it, so the law is not an artefact of the free intercept:

fit	$𝑎$ (Hz)	$𝑝$ (Hz/Hz)	$𝑅^{2}$
affine	1.15	0.183	0.994
through origin	0	0.213	0.966

Results#

Per-cell accuracy and E-rate over training epochs across the τ_GABA sweep. — Figure 1: Per-cell accuracy (top) and E rate (bottom) over training, one line per cell. Accuracy plateaus by ≈ epoch 15; the E rate keeps climbing through the 50 epochs, so the final-epoch rates are the ones fit.

Population-E power spectra by τ_GABA, gamma peak shifting with the inhibitory time constant. — Figure 2: Trial-mean Welch PSDs by $𝜏_{GABA}$ ; dots mark the parabolic-interpolated peak. The peak shifts cleanly from ≈ 20 Hz at $𝜏_{GABA} = 27$ ms to ≈ 51 Hz at $𝜏_{GABA} = 4.5$ ms, with no overlap between adjacent conditions.

One MNIST trial through each τ_GABA network; the gamma cycle period lengthens with τ_GABA. — Figure 3: One MNIST trial through each network. The cycle period stretches from ≈ 20 ms ( $𝑓_{𝛾} \approx 51$ Hz) at short $𝜏_{GABA}$ to ≈ 50 ms ( $𝑓_{𝛾} \approx 20$ Hz) at long, the eye and the spectrum agreeing.

Post-training E rate against gamma frequency; points lie on the affine fit line. — Figure 4: The law itself. Top: mean post-training E rate vs $𝑓_{𝛾}$ , six clusters × three seeds, error bars from seed variance; the affine fit passes through every error bar. Bottom: per-cluster accuracy is flat, so the rate change is not paid in classification.