Pinglab Rythmicity Metric

Abstract

We swept an 11×11 reciprocal-coupling grid in an untrained PING circuit and compared autocorrelation contrast with private- and shared-input null controls. At seed 42, selected diagonal conditions had lobe–trough contrasts 0.000301, 0.268 and 0.989. Coupled conditions generally showed stronger temporal structure, but shared input could also produce elevated contrast without feedback. These single-seed observations do not establish rate invariance or the spiking transition’s bifurcation type.

Results

Coupling-map structure

On either zero-coupling edge the loop was broken: E fired at 168.91 Hz and contrast was 0.000301, while I was silent when E-to-I coupling was zero. The diagonal examples in panels D/E/F had contrasts 0.000301/0.268/0.989. Coupled conditions generally showed stronger temporal structure, although the map was not strictly monotonic (Fig. 1).

Coupled-grid rasters

Many coupled-interior conditions showed separated volleys, whereas the zero-E-to-I column had silent I neurons and the zero-I-to-E row lacked inhibitory feedback to E (Fig. 2).

Coupled-grid autocorrelograms

Edge autocorrelograms remained near chance. In the coupled interior, short-lag clustering, suppression between volleys and later recurrence peaks supplied complementary temporal evidence (Fig. 3).

Input-sharing controls

Without an inhibitory loop, the private-input null had a maximum contrast of 0.0392 over 2.00–168.91 Hz. The shared-input null reached 1 at 0.0109 Hz, but that estimate contained only 10 E spikes across the population. At 1.39 Hz its contrast was 0.127. Shared afferents can create short-lag coincidence detected by this score; the sparse extreme also illustrates its finite-sample sensitivity. These single-seed controls do not prove rate invariance or establish rhythmicity from contrast alone (Fig. 4).

Rate-matched null structure

Shared-input examples showed central coincidence without inhibitory feedback; private-input examples showed smaller contrast and finite-sample fluctuations around chance. Their approximate rate matching did not make rates or spike counts equal (Fig. 5).

Methods

1. Sweep coupling. We simulated untrained PING networks (1,024 E/256 I neurons) at 11×11 combinations of $W_{E I}=0$–3 µS and $W_{I E}=0$–6 µS. Each E neuron received private 100 Hz Poisson input with identity weight 0.5. Settings were: GABA decay 6 ms; E/I reset holds 1.2/0.6 ms; seed 42; one trial; 0.1 ms steps; 1,000 ms duration, discarding the first 100 ms.

2. Test input sharing. With both couplings zero, we scanned private input at 1/2/5/10/20/40/70/100 Hz and shared input at 8/12/16/20/28/40/60/100 Hz. Shared input used 200 channels, weight 0.2 and 95% initial zeros. Reusing the grid’s 100 Hz private origin gave 136 unique probes.

3. Measure rates and autocorrelation. Per-neuron rates were post-burn spike counts divided by population size and 0.9 s. For E counts $n_{E[k]}$ in $N_{\text{bin}}=900$ bins of width $\Delta t_{\text{bin}}=1$ ms, we calculated

   $$A_{\text{corr} ( \ell )} = \frac{1}{{\langle n_{E} \rangle}^{2} ( N_{\text{bin}} - \ell )} \sum_{k = 0}^{N_{\text{bin}} - \ell - 1} n_{E [ k ]} n_{E [ k + \ell ]} . \tag{1}$$

   Here $A_{\text{corr}(\ell)}$ is normalized autocorrelation, $k$ indexes bins, $\langle n_E\rangle$ is their mean count, and $\ell=1,\ldots,100$ gives lag $\ell\Delta t_{\text{bin}}$. We used zero-padded FFT correlation with overlap and squared-mean normalization: chance reference 1, zero lag excluded.

4. Calculate lobe–trough contrast. After filling zero lag from lag 1, we smoothed with weights $(0.25,0.5,0.25)$. We selected the first local minimum from lag 2 and the preceding maximum from lag 1. Their heights $A_{\text{trough}}$ and $A_{\text{lobe}}$ define

   $$R_{\text{contrast}} = \frac{A_{\text{lobe}} - A_{\text{trough}}}{A_{\text{lobe}} + A_{\text{trough}}} . \tag{2}$$

   Contrast $R_{\text{contrast}}$ lies in $[0,1]$ when $0\le A_{\text{trough}}\le A_{\text{lobe}}$ and the denominator is positive. Missing troughs or invalid denominators give undefined scores; no trough floor is applied.