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Why Connection Balance Can Either Calm a Network or Stir It Up ✛ ✎

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 · 1 min
Neurons and Cognition Disordered Systems Chaotic Dynamics
When incoming signals to each neuron nearly cancel each other out, the network can freeze or, conversely, start pulsing chaotically, depending on how the neuron itself is built.

Each neuron in the brain receives many signals from its neighbors: some push it forward, others hold it back. When these signals nearly cancel each other out, the neuron operates in a balanced regime. For a long time, it was thought that such balance only served to prevent the network from becoming overloaded. But it turns out that balance governs the entire "weather" within the network.

Balance acts like a volume knob for the self-generated signal: it removes the constant hum that each neuron hears from itself via feedback.

If the neuron is built like a simple amplifier, balance makes the network calmer. If the neuron saturates, like an overloaded microphone, balance can plunge the network into chaos. And if the neuron responds equally to plus and minus, balance has no effect at all. This discovery helps explain why real brain networks teeter on the edge of order and chaos: the anatomical balance of synapses can predict which regime the network operates in.

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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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Albert Einstein
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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