A PING rhythmicity metric

A PING rhythmicity metric#

exp054 · 15 June 2026 · pdf

Abstract#

A single bounded scalar for how rhythmic a spiking network is: the lobe–trough contrast of its spike-time autocorrelation, $(lobe - trough) / (lobe + trough) \in [0, 1)$ . Driving each excitatory cell with its own private Poisson input makes the metric rate-invariant by construction; across untrained PING networks it reads 0 along the COBA edges and rises smoothly to 0.98 through the PING interior, a rankable gradient that tracks the gamma-gated collapse of the E firing rate from 95 to 3 Hz.

Methods#

The networks here are untrained PING populations driven by external input; the rhythmicity metric is read off the resulting excitatory raster. Write $𝑟 (𝑡)$ for the binned population spike count ( $𝑛$ bins of width $Δ 𝑡$ ) and $ℓ$ for the lag. The metric is read off in a fixed sequence of steps:

Drive each E cell with its own private Poisson channel. Every excitatory cell receives an independent homogeneous Poisson spike train at 100 Hz through a one-to-one identity input weight: there is no shared, dense $𝑊_{in}$ projection, so no two cells share an input channel. Private input removes the input-driven spike coincidence that would otherwise inflate the metric at low firing, which is what makes the contrast rate-invariant by construction (Figures 4–5) with no post-hoc correction.
Bin to a population count. Sum spikes across cells into one count per $Δ 𝑡$ bin, giving the population trace $𝑟 (𝑡)$ of length $𝑛$ .
Raw autocorrelation. Form the lag product $\sum_{𝑡} 𝑟 (𝑡) 𝑟 (𝑡 + ℓ)$ , which counts spike pairs separated by lag $ℓ$ . All lags are computed at once in $𝑂 (𝑛 log 𝑛)$ via the Wiener–Khinchin route (zero-pad $𝑟$ , take its FFT, multiply by the conjugate, inverse-transform) rather than the $𝑂 (𝑛^{2})$ direct sum.
Correct for finite overlap. Only $𝑛 - ℓ$ bin-pairs exist at lag $ℓ$ (the last $ℓ$ samples have no partner), so the raw sum tapers toward zero with lag simply from running out of overlap; dividing by that per-lag overlap $𝑛 - ℓ$ converts it to the average product per available pair and flattens the taper.
Set the chance level. Divide by the mean rate squared ${⟨ 𝑟 ⟩}^{2}$ so rate-matched independent firing sits at $𝐴 = 1$ , and drop the self-paired zero lag. The result is the normalised autocorrelogram
$𝐴 (ℓ) = \frac{1}{{⟨ 𝑟 ⟩}^{2}} \frac{1}{𝑛 - ℓ} \sum_{𝑡} 𝑟 (𝑡) 𝑟 (𝑡 + ℓ) .$
Locate the Mexican hat. A rhythmic $𝐴 (ℓ)$ has a “Mexican-hat” profile: a central lobe above 1 (spikes recur a cycle apart) flanked by a dip below 1 where firing is suppressed between volleys. Scanning out from zero lag, take the trough as the first local minimum of a lightly smoothed $𝐴 (ℓ)$ (it falls near the half-period) and the lobe as the highest point at a shorter lag. Both searches start one bin past zero, so the self-paired zero-lag value dropped in step 5 is excluded from the lobe height: the lobe is read from the first real lag onward, never the trivial self-correlation.
Read the contrast. The metric is the lobe–trough contrast
$contrast = \frac{lobe - trough}{lobe + trough} \in [0, 1),$
zero when lobe equals trough (no structure) and approaching 1 as the trough goes silent. It is bounded by construction, with no trough floor needed.

In words: $𝐴 (ℓ)$ is how much more (or less) likely a spike is to be followed by another one $ℓ$ ms later than under independent firing, with $𝐴 = 1$ the chance floor. A central lobe above 1 says spikes cluster in volleys; a trough below 1 near the half-period says firing is suppressed between them. The contrast is 0 when the spikes carry no such structure (asynchronous) and approaches 1 as sharp volleys separate against near-silence; because it reads the shape of $𝐴 (ℓ)$ , it registers a rhythm whether or not its frequency holds still.

Results#

Scalar maps of E rate, I rate, and lobe–trough contrast over the W_EI × W_IE grid, with three example E/I rasters beneath showing asynchronous, emerging, and sharp gamma volleys. — Figure 1: The anchor result: gamma switches on smoothly across the recurrent-weight plane, read as scalar maps over the $𝑊_{𝐸 𝐼} \times 𝑊_{𝐼 𝐸}$ grid (untrained networks, private per-cell Poisson input) with example rasters beneath. **Top**: three per-cell summaries, every cell labelled. **E rate** is high along both zero edges (the loop is broken, E fires at the input-driven ≈95 Hz) and gated down through the interior; **I rate** is silent where $𝑊_{𝐸 𝐼} = 0$ , runs away along $𝑊_{𝐼 𝐸} = 0$ (clipped so the interior is legible), and is controlled once the loop closes; **lobe–trough contrast** (the rhythm scored 0–1) reads exactly 0 along both COBA edges and rises smoothly toward strong coupling, with three points marked along the $𝑊_{𝐼 𝐸} = 2 𝑊_{𝐸 𝐼}$ diagonal. **Bottom**: E/I rasters (E black, I red above) at those points: A the fully-off origin ( $𝑊_{𝐸 𝐼} = 𝑊_{𝐼 𝐸} = 0$ ), asynchronous with no I; B weak coupling, emerging volleys (contrast 0.27, below the half-way mark); C strong coupling, sharp volleys (contrast 0.98). E rate falling, I rate rising, and contrast rising are three readings of the same loop engaging. The full per-cell detail (every raster and autocorrelogram) is in the figures below; the rate-fairness behind the contrast metric is in Figures 4–5.

A 6×6 grid of E/I rasters across the coupling plane: the two zero edges are asynchronous, the interior shows increasingly sharp gamma volleys. — Figure 2: E/I rasters for a 6×6 subset of the grid (every other cell; the heatmaps below use all 121). The two zero edges are the controls: $𝑊_{𝐸 𝐼} = 0$ (left column) leaves I silent and E asynchronous; $𝑊_{𝐼 𝐸} = 0$ (bottom row) lets I fire but not inhibit E, so both stay dense; neither is rhythmic. The interior shows clear gamma volleys (E black, I red) that sharpen as either weight grows, the rhythm the contrast (Figure 1) scores.

A 6×6 grid of E-population autocorrelograms: flat at chance along the zero edges, developing a Mexican-hat lobe-and-trough through the interior. — Figure 3: The E-population autocorrelogram $𝐴 (ℓ)$ for a 6×6 subset of the grid (lag 0–50 ms; dotted line = chance, $𝐴 = 1$ ), with the located lobe (▲) and trough (▼) marked. The two zero edges are flat at 1: asynchronous firing, no structure. Through the interior the Mexican hat emerges: a sharp central lobe at ≈1 ms over a trough near the half-period, with a secondary peak at the full period further out. The contrast in Figure 1 is exactly this lobe-versus-trough read off as one number.

Contrast versus input rate for null networks: the private-input null stays flat near zero across all rates while the shared-input null climbs as firing thins. — Figure 4: The rate-invariance test that justifies the private-input choice. Each line is a **non-rhythmic null network** (no inhibitory loop, no rhythm at any drive) scanned over input rate; a rate-invariant metric should read ≈0 everywhere. With **private input** (black) it does: flat at ≤0.07 across all firing rates. With **shared input** (grey dashed) it instead **climbs to ≈0.50 as firing thins**: cells sharing input channels fire coincidentally, and the metric reads that as rhythm. The real PING cells (red) sit well above the private-input null, so their contrast is genuine, with no correction needed.

Autocorrelograms at matched low firing: the shared-input null shows a spurious central peak, the private-input null is flat around chance. — Figure 5: Why shared input fails and private input does not, at matched low firing rates. **Top (shared input):** coincident spikes from shared channels leave a central peak over a shallow dip (a spurious hat with no inhibitory loop behind it), and the metric marks a lobe (▲) and trough (▼) and reports a non-zero contrast. **Bottom (private input):** the same firing rates, but with one channel per cell the central peak is gone; $𝐴 (ℓ)$ is flat shot-noise around chance and the contrast collapses to ≈0. Same rate, same spike counts; the only difference is whether cells share input.

The onset over its mean-field bifurcation#

The turn-on above is what the network does; the exp033 4D conductance mean-field is why. This final section stacks the two into one manuscript figure: the empirical maps and example rasters directly over the mean-field bifurcation that predicts them. It recomputes the exp033 numerics (Hopf crossing, hysteresis sweep, gamma-vs- $𝜏_{GABA}$ ) and reuses this notebook’s own map and raster rendering, so restyling Figure 1 propagates here automatically, and no figures are copied. (This is the anchor that was formerly its own entry.)