047 — Per-event shunt magnitude sets rates, not summed inhibition

Abstract

How many inhibitory cells does a PING network need? We sweep $N_I$ from 0% to 25% of the excitatory pool ($N_E = 1024$ fixed) on an untrained network and ask how the per-cell firing rates respond. The answer depends on how you scale the $I \to E$ synapses with $N_I$: under constant per-edge weights — the biologically motivated choice — rates are flat across the sweep, even when the summed inhibitory drive varies by ≈ 250×. Under synaptic ($1/N$) or critical-balance ($1/\sqrt{N}$) scaling the rates reduce slightly at small $N_I$. What the network cares about is the per-event shunt magnitude, not the time-averaged total drive — and only constant-per-edge holds that fixed.

Methods

Simulation. Untrained PING net at $\Delta t = 0.1$ ms, $N_E = 1024$ fixed, $N_I$ swept over 256 (the 25% points, with 0% implemented as $N_I = 1$). Reference per-edge weights $W^{EI} \sim \mathcal{N}(1.0, 0.1)$ μS and $W^{IE} \sim \mathcal{N}(2.0, 0.2)$ μS at the canonical $N_I = 256$. Drive: uniform 25 Hz Poisson input on 784 channels, $T = 200$ ms, 4 trials per cell. Reported rates are per-cell averages over trials; rasters come from trial 0.

Three scaling regimes. When we change $N_I$ we have to decide what happens to the strength of each individual $I \to E$ synapse. There’s no neutral choice — the three standard conventions disagree even on what “the same network at different sizes” means. All three coincide at the canonical $N_I = 256$ anchor and only differ once we shrink $N_I$.

The intuition: the network is all-to-all between layers, so each E cell receives one I→E synapse per I cell — i.e. $N_I$ incoming I→E synapses per E cell, with a per-edge weight $W^{IE}$ on each. When we change $N_I$ we have to decide what happens to that per-edge weight. We can keep it fixed (so total drive into each E cell grows with the pool), rescale it as $1/N_I$ (so total drive stays put), or split the difference. These correspond to:

• Constant per-edge — the per-edge weight is fixed at $W^{IE}_\text{ref}$ no matter how big or small the I pool is. Each I→E connection that exists has the same strength as it would in the canonical 256-cell network. Drop from $N_I = 256$ to $N_I = 1$ and an E cell now has one incoming I→E synapse where it used to have 256 — total inhibitory conductance into each E cell drops by $256\times$.

• Synaptic normalization ($1/N$) — per-edge weight scales as $W^{IE}_\text{ref} \cdot (256/N_I)$, so the time-averaged total drive into each E cell is held fixed by construction: more I cells but weaker individual synapses, or fewer I cells but proportionally stronger synapses. At $N_I = 1$ the lone I→E connection is $256\times$ stronger per edge to compensate for the missing 255 contributions. This is the classic mean-field convention: it says “what the cell receives on average” is the meaningful quantity, and population size is just a knob for sampling resolution.

• Critical balance ($1/\sqrt{N}$) — per-edge weight scales as $W^{IE}_\text{ref} \cdot \sqrt{256/N_I}$, the geometric mean of the other two. Brunel and collaborators argued in the asynchronous-irregular literature that this is the “right” scaling for a balanced-state network: it keeps the variance of summed input invariant while letting the mean grow as $\sqrt{N_I}$, which is what you’d want if E firing is driven by input fluctuations around the inhibitory floor rather than by deterministic threshold crossings. At $N_I = 1$ each I→E edge is $16\times$ stronger than reference; at $N_I = 256$ it’s unchanged.

The three regimes thus encode three different stories about what the network is “really” doing — fixed-anatomy, mean-balanced, or variance-balanced. Running all three on the same $N_I$ sweep is the cleanest way to ask which story actually fits the dynamics.

Results

Constant per-edge

Figure 1. Constant-per-edge regime — firing rates vs N_I at fixed input

Figure 2. Single-trial rasters at each N_I (constant-per-edge regime)

Synaptic normalization ($1/N$)

Figure 3. Synaptic-normalization regime — firing rates vs N_I

Figure 4. Single-trial rasters at each N_I (synaptic-normalization regime)

Critical balance ($1/\sqrt{N}$)

Figure 5. Critical-balance regime — firing rates vs N_I

Figure 6. Single-trial rasters at each N_I (critical-balance regime)

Discussion

Why is constant per-edge flat? Three things have to hold simultaneously for the rates not to move with $N_I$:

• Summed inhibition is the wrong quantity. It grows ∝ $N_I$ (≈ 250× across the sweep) and yet $r_E$ doesn’t budge — so the loop is not in a regime where time-averaged total drive sets the rate.
• Per-I-cell drive is fixed. Each I cell still receives the same E → I input regardless of $N_I$, so $r_I$ is set by single-cell biophysics, not by population size.
• Cycle frequency is fixed. The gamma clock period depends on $\tau_\text{AMPA}$, $\tau_\text{GABA}$, and loop gain $G = \Phi'_E \Phi'_I W^{EI} W^{IE}$ (nb033). $G$ is independent of $N_I$ at fixed per-edge weights, so the per-cycle Bernoulli participation gate sets $r_E = p \cdot f_\gamma$ at a pool-size-invariant rate.

The synaptic and critical-balance regimes are the falsifier. Synaptic normalization holds the time-averaged mean drive exactly constant — and rates still drop at small $N_I$. The thing the network actually responds to is the per-event shunt magnitude: one huge shunt and many small ones aren’t the same, even when their time-averaged drive is identical. Only constant per-edge keeps the per-event magnitude fixed across the sweep.

This matches nb046’s structural-bound reading: rate is set within each cycle (does the cell cross threshold before the next I-shunt?), not by mean-field balance across cycles.

Two caveats:

• Untrained network. After training, $W_\text{in}$ adapts and the per-cell operating point shifts; whether the $N_I$-invariance survives training is a separate question.
• 0% endpoint approximation. $N_I = 1$ stands in for 0%. Literal $N_I = 0$ would disable the I-loop entirely and collapse to COBA — see nb025 Figure 1.

Next steps

• After-training check — does the $N_I$-invariance survive training, or does the readout pick a preferred $r_E$ that depends on pool size?
• $W_\text{in} \times N_I$ map — locate the recruitment cliff in two dimensions.
• $\tau_\text{GABA} \times N_I$ map — test whether the affine slope $p$ in $r_E = p \cdot f_\gamma$ is $N_I$-invariant.
• Direct per-event measurement — measure the post-I-burst E-firing-suppression window under each regime and check it correlates with the rate drop.