  Criticality is deeply related to optimal computational capacity. The lack of
a renormalized theory of critical brain dynamics, however, so far limits
insights into this form of biological information processing to mean-field
results. These methods neglect a key feature of critical systems: the
interaction between degrees of freedom across all length scales, which allows
for complex nonlinear computation. We present a renormalized theory of a
prototypical neural field theory, the stochastic Wilson-Cowan equation. We
compute the flow of couplings, which parameterize interactions on increasing
length scales. Despite similarities with the Kardar-Parisi-Zhang model, the
theory is of a Gell-Mann-Low type, the archetypal form of a renormalizable
quantum field theory. Here, nonlinear couplings vanish, flowing towards the
Gaussian fixed point, but logarithmically slowly, thus remaining effective on
most scales. We show this critical structure of interactions to implement a
desirable trade-off between linearity, optimal for information storage, and
nonlinearity, required for computation.
