Pool-size invariance requires inverse synaptic scaling

Pool-size invariance requires inverse synaptic scaling#

exp047 · 14 July 2026 · pdf

Abstract#

The apparent invariance of pyramidal–interneuron network gamma (PING) firing rate to inhibitory-pool size is a consequence of the simulator’s fan-in normalization, not independence from interneuron count. Excitatory cells are denoted E and inhibitory cells I. The nominal I→E parameter $𝐺_{𝐼 𝐸}$ is divided by the presynaptic pool size, so the realised mean synaptic conductance is $𝑗_{𝐼 𝐸} = \frac{𝐺_{𝐼 𝐸}}{𝑁_{𝐼}}$ . We vary $𝑁_{𝐼}$ under paired controls. Rates remain flat when $𝐺_{𝐼 𝐸}$ is fixed and $𝑗_{𝐼 𝐸} \propto \frac{1}{𝑁_{𝐼}}$ ; when the realised synapse $𝑗_{𝐼 𝐸}$ is fixed, rates change strongly with $𝑁_{𝐼}$ . Pool-size invariance therefore requires compensatory inverse scaling of individual synapses.

Methods#

1. Weight convention. For a dense I→E matrix with shape $𝑁_{𝐼} \times 𝑁_{𝐸}$ , the simulator draws a non-negative weight with nominal mean $𝐺_{𝐼 𝐸}$ and then divides every entry by its presynaptic fan-in $𝑁_{𝐼}$ . Thus

ℰ︀ [𝑊_{𝑘 𝑗}^{𝐼 𝐸}] \approx 𝑗_{𝐼 𝐸} = \frac{𝐺_{𝐼 𝐸}}{𝑁_{𝐼}}, ℰ︀ [\sum_{𝑘 = 1}^{𝑁_{𝐼}} 𝑊_{𝑘 𝑗}^{𝐼 𝐸}] \approx 𝐺_{𝐼 𝐸} . (1)

$𝐺_{𝐼 𝐸}$ is consequently an expected summed coupling, not the conductance of one synapse. An inhibitory volley with active set $𝒜︀$ gives E cell $𝑗$ the conductance increment

Δ 𝑔_{𝑗}^{𝐼} = \sum_{𝑘 \in 𝒜︀} 𝑊_{𝑘 𝑗}^{𝐼 𝐸}, (2)

where:

$ℰ︀$ denotes expectation over random weight initialization;
$𝑊_{𝑘 𝑗}^{𝐼 𝐸}$ is the realised conductance from inhibitory cell $𝑘$ to excitatory cell $𝑗$ ;
$𝑗_{𝐼 𝐸}$ is the mean realised conductance of one I→E synapse;
$𝐺_{𝐼 𝐸}$ is the expected sum of all I→E weights arriving at one E cell;
$𝑁_{𝐼}$ and $𝑁_{𝐸}$ are the inhibitory and excitatory pool sizes;
$𝒜︀$ is the set of inhibitory cells active in a volley; and
$Δ 𝑔_{𝑗}^{𝐼}$ is the resulting inhibitory-conductance increment at excitatory cell $𝑗$ .

Equation 2 therefore depends on both the realised synapses and the number of participating inhibitory cells.

2. Paired controls. We sweep $𝑁_{𝐼} \in {16, 64, 256}$ under two conventions. In the fixed-summed-coupling arm, $𝐺_{𝐼 𝐸} \in {1, 2, 4}$ μS is held fixed, forcing $𝑗_{𝐼 𝐸} \propto \frac{1}{𝑁_{𝐼}}$ . In the fixed-synapse arm, $𝑗_{𝐼 𝐸} \in {3.91, 7.81, 15.63}$ nS is held fixed and the nominal parameter is set to $𝐺_{𝐼 𝐸} = 𝑁_{𝐼} 𝑗_{𝐼 𝐸}$ . The two arms coincide at the reference pool $𝑁_{𝐼} = 256$ .

3. Simulation. Untrained dense PING network, $𝑁_{𝐸} = 1024$ , $Δ 𝑡 = 0.1$ ms, $𝑇 = 500$ ms, 8 independent Poisson-input trials per network and 3 network/input seeds per condition. E→I summed coupling is fixed at 1 μS; input comprises 784 independent 25 Hz Poisson channels. Markers show seed means and error bars ±1 standard deviation (SD).

Results#

Fixed summed coupling produces overlapping rate curves. At $𝐺_{𝐼 𝐸} = 2$ μS, the E rate is 6.01 Hz for $𝑁_{𝐼} = 16$ and 6 Hz for $𝑁_{𝐼} = 256$ . This is the expected consequence of holding the summed I→E coupling fixed while shrinking each realised synapse as $\frac{1}{𝑁_{𝐼}}$ .

Fixed realised synaptic strength gives the opposite result. At $𝑗_{𝐼 𝐸} = 7.81$ nS, increasing $𝑁_{𝐼}$ from 16 to 256 changes the E rate from 16.62 to 6 Hz and the I rate from 120.41 to 34.41 Hz. Every tested synaptic strength shows the same pool-size dependence. The direction is mechanistically consistent: more inhibitory cells at unchanged individual strength produce greater population-level inhibition, while the recurrent feedback reduces both E activity and the I activity it recruits.

Four panels comparing E and I firing rates across inhibitory pool sizes. Rates are flat when summed coupling is fixed, but fall strongly with pool size when realised synaptic strength is fixed. — Figure 1: **Excitatory and inhibitory firing rates across inhibitory-pool size under two I→E scaling controls.** Top: excitatory-cell rate in Hz per cell. Bottom: inhibitory-cell rate in Hz per cell. **(a)** Holding the expected summed I→E coupling $𝐺_{𝐼 𝐸}$ fixed makes the realised synapse $𝑗_{𝐼 𝐸} = \frac{𝐺_{𝐼 𝐸}}{𝑁_{𝐼}}$ shrink as the pool grows. **(b)** Holding $𝑗_{𝐼 𝐸}$ fixed makes summed coupling grow with $𝑁_{𝐼}$ . Each curve is one coupling level; markers are means and error bars ±1 standard deviation (SD) over 3 seeds, with 8 trials per seed. The controls coincide at the reference pool $𝑁_{𝐼} = 256$ . Rates are invariant to pool size only under inverse scaling of individual synapses.

Conclusion#

The network is not intrinsically insensitive to inhibitory-pool size. It is insensitive only along a specific normalization path: individual I→E synapses must scale approximately as $\frac{1}{𝑁_{𝐼}}$ so that expected summed coupling stays fixed. The operative control variable in this dense model is population-level inhibitory drive, jointly determined by pool size, realised synaptic strength, and volley participation. This experiment does not establish that the same inverse scaling holds biologically; it identifies the compensation required for invariance in the model.