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Connection Balance as the Volume Knob of Chaos ✛ ✎

Original: "Local connectivity balance shapes population dynamics in random recurrent networks"
· Shotaro Takasu, Richard Gast, Ann Kennedy
arXiv:2608.30008v1 · 2026-08-30 · arXiv non-exclusive · independent analysis · 2 min
Neurons and Cognition Disordered Systems Chaotic Dynamics
Local balance of excitatory and inhibitory connections can either calm a network or plunge it into chaos, depending on how each neuron is wired.

Imagine a large orchestra without a conductor. Each musician hears everyone else and adjusts to the collective hum. If every instrument amplifies its neighbors, the sound builds to a screech; if some occasionally dampen it, the orchestra may fall into silence or break into cacophony. Neural networks behave much the same way: connections between cells act as volume knobs, some boosting the signal, others turning it down.

In the new model, each connection gets its own volume knob, but with a common rule: the sum of all "pluses" and "minuses" at each neuron is nearly zero. That is local balance.

For a long time, such balance was thought to be needed only to keep the network from "exploding" due to random outliers. But it turns out to change the very music. If neurons act as simple amplifiers, balance acts as a muffler: the network stays calm even with strong connections. If neurons saturate, like an overloaded microphone, balance instead cranks the volume up to chaos.

The reason is that balance removes the constant hum each neuron hears from itself through feedback. If that hum is already absent (as with odd activation functions, where the average signal is zero), the volume knob changes nothing.

Experiments on virtual networks of a thousand neurons showed that for linear amplifiers, the critical connection strength at which activity explodes increased from 1.4 to 1.8 under full balance. For saturating neurons, balance turned a quiet network into a chaotic one, and the chaos dimension (the number of independent "voices" in the orchestra) first rose and then fell. Such a nonmonotonic response means balance is not just a switch but a fine-tuning knob.

Previously, researchers often assumed neurons have an odd activation function, meaning they respond symmetrically to plus and minus. That simplification masked the effect of balance. Now it is clear that in the real brain, where neurons saturate, the balance of excitation and inhibition can directly decide whether the network stays silent or generates complex activity. The orchestra without a conductor turns out to be sensitive to how exactly the musicians turn their volume knobs.

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Under full balance (when the sum of incoming weights is zero), the time-averaged activity of each neuron in the network is strictly zero, even if the network is in a chaotic regime. This is analytically proven in the appendix of the paper.

(1+\partial_t)x_i = \sum_{j=1}^N J_{ij}\phi(x_j)
The activity of each node changes under the sum of inputs from other nodes, transformed by the activation function.
J_{ij} = \bar{J}_{ij} - \frac{\mu}{N}\sum_{k=1}^N \bar{J}_{ik}
From the random weight, the row average multiplied by the balance parameter μ is subtracted. At μ=1, the sum of incoming weights is zero.
q = g^2 \left\{ \langle\langle \phi,\phi \rangle\rangle_{x\sim N(0,q)} + (1-\mu)^2 \langle \phi \rangle^2_{x\sim N(0,q)} \right\}
The variance of activity at the fixed point is determined by the sum of fluctuations and the square of the mean, with the mean contribution suppressed by balance.
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collective behavior random network brain connectivity network weight brain rhythmic transport redundancy Recurrent Neural Networks emergent collective learning neural activity stochastic self organization ion channel dynamics nonlocality complex system dynamics power grid stabilization bottleneck effect feedback loop information storage log-normal distribution Braess's paradox in biology