Pinglab Rythmicity Metric

#exp054 · Gamma-gated sparsity · data v36.0.0
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Abstract

We asked where rhythmic activity appears as reciprocal PING coupling activates across the coupling map. We swept coupling directions in a network and compared autocorrelation structure with private- and shared-input controls.

Rhythmic contrast was weak on uncoupled edges and stronger inside the coupled map; shared input imitated contrast without a recurrent loop. This supports the private-input control, but does 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).

Maps of E rate, I rate and lobe–trough contrast over the coupling grid, above three example E/I rasters showing asynchronous firing and increasingly separated volleys.
Figure 1: (A–C) Per-neuron E rate, I rate and lobe–trough contrast, respectively, across the 11×11 coupling grid; 𝑊𝐸𝐼 and 𝑊𝐼𝐸 denote E-to-I and I-to-E coupling strengths. The high I rate on one zero edge is colour-clipped. (D–F) Rasters at the diagonal conditions marked D, E and F in panel C, respectively; E spikes are black and I spikes red. Each raster shows 200 ms from the first 160 E and 48 I neurons. One seed per condition; no uncertainty estimate.

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).

E/I rasters at every other coupling-grid coordinate: dense edge activity and separated volleys in much of the coupled interior.
Figure 2: (A–AJ) E/I rasters at a 6×6 subset of coupling coordinates, ordered row-major from high to low 𝑊𝐼𝐸 and low to high 𝑊𝐸𝐼; the maps above use all 121 conditions. Display windows and neuron subsets match Figure 1; measurements use all neurons over the full post-burn recording.

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).

E-population autocorrelograms at a 6×6 subset of coupling coordinates, with lobe and trough markers and a chance reference.
Figure 3: (A–AJ) E-population autocorrelograms at the same 6×6 coordinates and in the same row-major coupling order, shown over 0–50 ms. Dotted lines mark the chance reference 𝐴corr=1; markers locate the selected lobe (▲) and trough (▼) of the smoothed curve. 𝑅contrast scores the first lobe and trough, not the later peak’s frequency.

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).

Contrast versus measured E firing rate: private-input null values remain small, while the sparsest shared-input nulls show elevated contrast.
Figure 4: Contrast against measured E firing rate. Black and grey points are the private- and shared-input nulls; red points are the PING coupling grid, including its near-zero edges.

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).

Low-rate shared- and private-input null autocorrelograms selected by approximate rate matching, with each actual rate labelled.
Figure 5: Null autocorrelograms nearest to target E rates of 1, 2.5 and 5 Hz. Shared-input examples (A–C) fired at 1.39/1.39/3.94 Hz; private-input examples (D–F) at 2.00/2.00/3.96 Hz. Within each input type, the first two targets selected the same recording; these are repeated displays, not independent observations.

Methods

Untrained PING populations tested coupling-dependent temporal structure; uncoupled controls tested the influence of input sharing. We reused the spiking recordings and replaced the separate mean-field comparison with newly computed conductance-model results at matching refractory durations.

  1. Sweep coupling. We simulated 1,024 E and 256 I neurons at all 11×11 combinations of 𝑊𝐸𝐼=0–3 µS and 𝑊𝐼𝐸=0–6 µS. Each E neuron received a private 100 Hz Poisson channel with identity weight 0.5. We used a 6 ms GABA decay constant, seed 42, one trial, 0.1 ms steps and 1,000 ms recordings; we discarded the first 100 ms. E/I reset holds were 1.2/0.6 ms. These 136 recordings were reused unchanged for the refreshed theory comparison.
  2. Construct uncoupled controls. We set both coupling strengths to 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 zero connections. The 100 Hz private origin was shared with the coupling grid, giving 136 unique probes.

  1. Measure rates and autocorrelation. We divided each population’s post-burn spike count by neuron count and 0.9 s. We binned E spikes at Δ𝑡bin=1 ms, obtaining counts 𝑛𝐸[𝑘] in 𝑁bin=900 bins. For integer lag ℓ=1,…,100, we calculated

    𝐴corr(ℓ)=1⟨𝑛𝐸⟩2(𝑁bin−ℓ)∑𝑘=0𝑁bin−ℓ−1𝑛𝐸[𝑘]𝑛𝐸[𝑘+ℓ].
    (1)

    Here 𝑘 indexes bins and ⟨𝑛𝐸⟩ is their mean count; physical lag is ℓΔ𝑡bin. Zero-padded FFT correlation, divided by available overlap and mean count squared, gives chance reference 1. We excluded zero lag.

  2. Locate the lobe and trough. We smoothed with weights (0.25,0.5,0.25), filling the excluded zero-lag entry from the first lag. We found the first local minimum from lag 2, and the preceding maximum from lag 1. Their smoothed heights define

    𝑅contrast=𝐴lobe−𝐴trough𝐴lobe+𝐴trough.
    (2)

    For 0≤trough≤lobe and a positive denominator, this lies in [0,1]. A missing trough or invalid denominator leaves the score undefined; no trough floor is imposed on contrast.

  3. Compare mean-field onset. We used the conductance model described in exp033 — Gamma Emerges at a Hopf Bifurcation at 4 mV effective noise, 6 ms GABA decay and 1.2/0.6-ms E/I refractory periods. The refreshed calculation used the cancellation-resistant gain integral specified there. In that separate calculation, we continued fixed points over 401 drives from 0–4 nA and refined the leading-eigenvalue crossing with Brent’s method. We swept 25 drives from 0.1 nA below to 0.55 nA above the crossing in both directions, carrying endpoint states. We integrated 2 s per drive with LSODA, recorded the trajectories and measured peak-to-peak E-rate amplitude (ms−1) over the final 500 ms. The resulting onset was 0.594 nA at 27.6 Hz. The drive sweep was separate from the spiking coupling sweep; we reused its completed numerical results here without another solve.
  1. Compare frequencies. We repeated the theoretical crossing search at inhibitory decays 4.5/6/9/12/18/27 ms. We reused the median frequency across three seeded spiking classifiers from exp041 — Firing Rate Tracks Gamma Frequency at each decay; we did not refit the mean-field noise scale.

Dataset

Appendix: reading the autocorrelogram

The raw lag product ∑𝑘𝑛𝐸[𝑘]𝑛𝐸[𝑘+ℓ] counts spike pairs separated by ℓ bins. Only 𝑁bin−ℓ bin pairs exist at that lag: the last ℓ bins have no partner. Dividing by this overlap converts the raw sum into an average product per available pair, removing the taper caused by finite recording length. Dividing again by ⟨𝑛𝐸⟩2 places independent firing near 𝐴corr=1; this is a reference level, not a lower bound.

The Wiener–Khinchin route computes the lag products by zero-padding the count trace, taking its fast Fourier transform (FFT), multiplying by the complex conjugate and inverse-transforming. This costs 𝑂(𝑛log𝑛) for all lags rather than a full 𝑂(𝑛2) direct correlation, where 𝑂 denotes asymptotic computational scaling.

A central lobe above chance describes short-lag spike clustering. A later trough can reflect suppression between volleys; a subsequent peak near the period is separate evidence of recurrence. Contrast is zero when its lobe and trough are equal, and reaches one if a positive lobe is paired with a zero trough. It summarizes their relative heights without measuring a stable oscillation frequency. Finite recordings, missing extrema and shared-input coincidence therefore matter when interpreting this scalar alongside rasters and longer-lag structure.