The Canonical V&S state

The Canonical V&S state#

exp058 · 24 June 2026 · Draft · pdf

Abstract#

Vreeswijk & Sompolinsky predict that a strongly coupled, sparse, inhibition-dominated network falls untuned into a balanced state: large E and I currents cancel, and the residual $𝑂 (1)$ fluctuations drive irregular (CV ≈ 1), asynchronous, rhythmless firing, generated by the deterministic dynamics rather than inherited from the input. We push the conductance-based COBANet into the full four-coupling regime and test three predictions: the balanced signatures at fixed fan-in (Figure 1); their survival as $𝐾 \to \infty$ only under strong coupling, synapse $\propto 𝐽 / \sqrt{𝐾}$ (Figure 2); and that the irregularity is deterministic chaos, by perturbing clones on noise-free input (Figure 3). All three hold: Poisson ISIs, near-zero correlations, a broadband spectrum, a supra-Poisson CV sustained as $𝐾$ grows, and a positive Lyapunov exponent (λ ≈ +28 s⁻¹ vs ≈ 0 for a decoupled control). The lone caveat (Figure 2) is a finite- $𝐾$ rate drift.

Methods#

Network and the $𝐾$ – $𝐽$ mapping. COBANet, $𝑁_{𝐸} = 1024$ , $𝑁_{𝐼} = 256$ , $Δ 𝑡 = 0.25$ ms, $𝑇 = 1000$ ms. Fixed fan-in (exact- $𝐾$ , sparsity $𝑠 = 0.99$ ): every cell draws exactly $𝐾$ inputs. Four recurrent matrices $𝑊^{𝐸 𝐼} = 𝒩︀ (0.6, 0.18)$ , $𝑊^{𝐼 𝐸} = 𝒩︀ (3.0, 0.9)$ , $𝑊^{𝐼 𝐼} = 𝒩︀ (0.4, 0.12)$ , $𝑊^{𝐸 𝐸} = 𝒩︀ (0.4, 0.12)$ μS ( $𝑊^{𝐸 𝐸}$ for completeness: the fixed- $𝐾$ balance sets the rates, not it). Per-cell independent Poisson drive, uncorrelated (E: 8 Hz × 2.10 μS, I: 8 Hz × 0.25 μS). The simulator is set by sparsity $𝑠$ and weight $𝑤$ ; the theory’s fan-in $𝐾$ and coupling $𝐽$ are derived,

𝐾 = (1 - 𝑠) 𝑁_{pre} (1)

𝐽 = \frac{𝑤}{\sqrt{𝐾}} ⟺ 𝑤 = 𝐽 \sqrt{𝐾} (2)

where:

$𝐾$ , the fan-in: how many presynaptic cells each neuron receives input from (the size of its “crowd”);
$𝐽$ , the coupling: the $𝑂 (1)$ synaptic strength held fixed as $𝐾$ grows. The $𝐽 / \sqrt{𝐾}$ scaling is the load-bearing V&S choice, keeping the mean recurrent input $𝑂 (\sqrt{𝐾})$ (large, cancels) while the fluctuation stays $𝑂 (1)$ (survives);
$𝑠$ , connection sparsity (the –ei-sparsity flag); $1 - 𝑠$ is the probability any given pair is wired;
$𝑁_{pre}$ , size of the presynaptic pool a cell draws from ( $𝑁_{𝐸}$ or $𝑁_{𝐼}$ );
$𝑤$ , mean of the recurrent weight matrix in μS (the value passed to –w-ei etc.).

Exact- $𝐾$ divides each synapse by its fan-in, so the total a cell receives, $𝐾 \cdot (𝑤 / 𝐾) = 𝑤$ , is $𝐾$ -independent, giving (2). At $𝑠 = 0.99$ :

matrix	direction	$𝑁_{pre}$	$𝑤$ (μS)	$𝐾$	$𝐽 = 𝑤 / \sqrt{𝐾}$
$𝑊^{𝐸 𝐼}$	E → I	1024	0.6	≈ 10	≈ 0.19
$𝑊^{𝐸 𝐸}$	E → E	1024	0.4	≈ 10	≈ 0.13
$𝑊^{𝐼 𝐸}$	I → E	256	3.0	≈ 3	≈ 1.73
$𝑊^{𝐼 𝐼}$	I → I	256	0.4	≈ 3	≈ 0.23

So $𝐾_{𝐸} \approx 10$ , but the 4× smaller I pool gives only $𝐾_{𝐼} \approx 3$ . V&S needs $𝐾_{𝐸}, 𝐾_{𝐼}$ the same order (4:1 qualifies), but $𝐾_{𝐼} \approx 3$ is marginal: at equal fan-in ( $𝐾_{𝐼} \approx 10$ ) the state stays balanced and asynchronous, the CV merely relaxing from 1.1–1.2 to ≈ 1.0, so the supra-Poisson burstiness is a lumpy- $𝐾_{𝐼}$ shot-noise effect, not a balance failure.

Why it is irregular. Each cell sums ≈ $𝐾$ synapses of strength $\propto 𝐽 / \sqrt{𝐾}$ , so the mean recurrent input is $𝑂 (\sqrt{𝐾})$ but its fluctuation is $𝑂 (1)$ . A moderate rate makes the large opposing E and I means cancel to leading order, parking the drive near threshold; the residual $𝑂 (1)$ fluctuations carry each cell over threshold at random times (Poisson, CV → 1), and since each cell hears its own uncorrelated input, the population is asynchronous. These fluctuations are network-generated: they survive $𝐾 \to \infty$ only under $𝐽 / \sqrt{𝐾}$ scaling (Step 4) and persist under noise-free input (Step 5).
Diagnostics. From post-burn-in spikes: per-neuron ISI CV, the Welch spectrum of the population-mean E trace, and the mean pairwise cross-correlogram over 100 random E pairs.
The $𝐽 / \sqrt{𝐾}$ coupling test. Strong coupling means synapse $\propto 𝐽 / \sqrt{𝐾}$ , keeping an $𝑂 (1)$ recurrent fluctuation as $𝐾 \to \infty$ ; holding $𝑤$ fixed instead gives synapse $\propto 1 / 𝐾$ , the weak (mean-field) limit, fluctuation $\propto 1 / \sqrt{𝐾} \to 0$ . At fixed $𝑁_{𝐸} = 1024$ we sweep $𝐾 = 10 \to 160$ under both: strong (weights $\propto \sqrt{𝐾}$ , drive folded into the balance, so rate $\propto 𝐾$ and $𝑔 \propto 1 / \sqrt{𝐾}$ ) and weak (all fixed), reading off whether the irregularity survives. (3 s trials, since the strong rate drifts low at high $𝐾$ and the CV needs enough ISIs.)
Direct Lyapunov exponent (frozen input). Replace the Poisson drive with a quenched DC conductance (a frozen per-cell offset, no per-timestep fluctuation), run two clones on identical input, kick every voltage by $𝜀$ at $𝑡 = 0$ , and track the voltage distance between the copies,
$‖ Δ 𝑉 (𝑡) ‖ = \sqrt{\sum_{𝑖 = 1}^{𝑁_{𝐸}} {(𝑉_{𝑖}^{clean} (𝑡) - 𝑉_{𝑖}^{pert} (𝑡))}^{2}} (3)$
the Euclidean distance between the two copies’ $𝑁_{𝐸}$ E-cell membrane-voltage vectors ( $𝑉_{𝑖}^{clean}$ unperturbed, $𝑉_{𝑖}^{pert}$ kicked). With noise-free input any divergence is network-generated. We compare the balanced four-coupling net against a decoupled control: the same COBA E and I cells under the same DC drive, but with all four recurrent matrices set to ≈ 0, so no cell receives synaptic input from any other. Each decoupled cell is then an independent DC-driven integrator (a clock), and a perturbation has nowhere to spread, fixing the $𝜆 \approx 0$ baseline. (It is not a feedforward network: there is no input-layer pathway, just the same architecture with every recurrent wire cut.) Spiking dynamics contract between spikes and expand only at spike-flips, so $𝜀$ must flip spikes ( $𝜀 = 0.1$ mV; smaller just contracts → spurious $𝜆 < 0$ ). The slope of $log ‖ Δ 𝑉 ‖$ in the post-flip, pre-saturation window is the largest Lyapunov exponent (five seeds). It is an initial-growth estimate ( $‖ Δ 𝑉 ‖$ saturates at the attractor size), so the sign, not the magnitude, is the result.

Results#

Four stacked panels for the canonical four-coupling state: a combined E (black, below) and I (red, above) spike raster with no visible bands, a broadband Welch PSD with no peak, a per-neuron ISI-CV histogram centred near 1, and a flat pairwise cross-correlogram. — Figure 1: Four diagnostics of the four-coupling state (E black, I red above the divider; ⟨r_E⟩ ≈ 17.5, ⟨r_I⟩ ≈ 25.4 Hz). **What we expect.** Irregular (ISI CV ≈ 1; a constant-current cell sits at 0), asynchronous (near-zero pairwise correlation), rhythmless (broadband, no peak). **What we see.** *Raster:* scattered, no bands. *PSD:* broadband, no sustained rhythm; the single-trial spectral max wanders seed to seed (5–90 Hz across six seeds), so it is not a peak, but a *weak* gamma-band E–I resonance does survive seed-averaging (≈ 2× the floor near 40 Hz, the same E↔I loop that sharpens into PING gamma, here heavily damped). *ISI CV:* median 1.11 (E), 1.20 (I). *Cross-correlogram:* flat, peak |C(τ)| ≈ 0.01 despite heavy shared input, the active decorrelation of Renart et al. (2010). **Do they align?** Yes on all three: the COBANet enters the balanced state untuned.

Three panels versus fan-in K on a log axis: median ISI CV, pairwise correlation peak, and population rates, each comparing weights scaled proportional to sqrt(K) (black) against weights held fixed (grey), with the I rate in red. — Figure 2: Fan-in $𝐾 = 10 \to 160$ at fixed $𝑁_{𝐸} = 1024$ , equivalently sparsity $𝑠 = 1 - 𝐾 / 𝑁_{𝐸}$ from 0.99 to 0.84 (top axis), with recurrent weights scaled $\propto \sqrt{𝐾}$ (black, the V&S $𝐽 / \sqrt{𝐾}$ rule) versus held fixed (grey); $𝑟_{𝐼}$ shown in red. ±1 SD over three seeds, 3 s trials. **What we expect.** Strong coupling keeps the $𝑂 (1)$ fluctuations, so irregularity *survives* as $𝐾$ grows; weak coupling loses them ( $\propto 1 / \sqrt{𝐾}$ ), so cells *regularise*. Asynchrony holds in both. **What we see.** *Irregularity:* strong stays supra-Poisson (CV 1.2–1.3, easing to 1.11 at $𝐾 = 160$ ); weak decays to ≈ 1.0, the Poisson floor. *Asynchrony:* both ≈ 0.006 throughout. *Rates:* strong $𝑟_{𝐸}$ drifts 17.5 → 4.9 Hz (the $𝑂 (1 / \sqrt{𝐾})$ finite- $𝐾$ correction); weak holds ≈ 17 Hz. **Do they align?** Yes: only $𝐽 / \sqrt{𝐾}$ keeps the recurrent irregularity alive as $𝐾$ grows, and the gap is the signature. The one mismatch is the strong-coupling rate drift (Next steps); whether the survival is *self-generated* is settled by Figure 3.

Voltage distance between two clones versus time since the kick: the balanced (black) trace spikes then collapses to near zero and stays flat, while the decoupled (grey) trace stays elevated and noisy around 12 mV throughout. — Figure 3: Two clones on *identical* quenched-DC input, the second kicked by ε = 0.1 mV at $𝑡 = 0$ ; voltage distance $‖ Δ 𝑉 (𝑡) ‖$ , seed-mean (bold) with a ±1 SD band over five seeds, lightly smoothed. **What we expect.** If chaotic, the kick is *amplified* ( $‖ Δ 𝑉 ‖$ grows exponentially, λ > 0), and with noise-free input that can only be network-generated. The decoupled control should not grow (λ ≤ 0). **What we see.** *Balanced (black):* $‖ Δ 𝑉 ‖$ rises steeply (fitted rate λ ≈ +28 s⁻¹) then saturates at the attractor size (≈ 250 mV). *Decoupled (grey):* neither amplified nor forgotten, a fixed phase offset, λ ≈ 0. **Do they align?** Yes: λ > 0 for the balanced net, ≈ 0 for the control, deterministic chaos, self-generated. Caveat: an *initial-growth* estimate (saturation caps the window), so the sign, not the magnitude, is the result; a renormalised Benettin scheme would pin exact λ (Next steps).

Next steps#

Figures 1–3 confirm the balanced signatures, their $𝐽 / \sqrt{𝐾}$ -only survival, and a positive Lyapunov exponent: the state is self-generated and deterministically chaotic. Two refinements remain.

Hold the rate as $𝐾$ grows. Strong-coupling $𝑟_{𝐸}$ drifts 17.5 → 4.9 Hz (the $𝑂 (1 / \sqrt{𝐾})$ finite- $𝐾$ correction). A DC offset to E alone fails: it pushes E off the balanced point and inflates CV to 1.4–1.7. Honest pinning must co-adjust the E and I drives along the balanced manifold (solve the 2D balance for a target $(𝑟_{𝐸}, 𝑟_{𝐼})$ ), giving a clean constant-rate CV comparison.
Pin down the asymptotic $𝜆$ . Figure 3 gives the sign; the magnitude is an initial-growth estimate (ε-dependent, saturation-capped). The exact exponent needs a renormalised Benettin scheme (lockstep clones, periodic rescaling of $Δ 𝑉$ , averaging $log$ -growth over a long trajectory), which also yields the full spectrum and the attractor dimension.