Gell-Mann–Low Criticality in Neural Networks

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, required 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 parametrize 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.

Figure 1

(a) Model schematics. (b) Variance. Variance as a function of $N$ for ${\overline{g}}_3=0.1$ and ${\overline{g}}_2^2=0.01$ (${\overline{g}}_2^2\ll {\overline{g}}_3$ regime); blue diamonds: simulation; dashed dark blue line: linear system; gray solid line: renormalized theory. (c) Flow of couplings. Direction of flow (arrows) theoretically predicted by Eqs. (5) and (6). The transition line (black, dashed) separates the converging region (green) from the diverging region (red). Region of nonphysical bistable regime in darker red. Simulated flow of couplings at $\ell =(N/2)$, with $N\in \{32,48,64,96,128,192\}$ (dark to light shading of blue markers; squares include $N=256$, 384) for different initial conditions (yellow edge). Predicted flow by Eqs. (5) and (6) (gray solid curves, thicker within the range of simulated $\ell$). (d) Memory. Reconstruction accuracy (0: complete forgetting; 1: perfect reconstruction) over time of a Gaussian input for different strengths of the nonlinearities (legend). (e) Classification. Share of correctly assigned parity to 3-bit strings (blue curve) and standard deviation across repeated trainings (shaded area), for ${\overline{g}}_2={\overline{g}}_3=0.3$. Decay timescales of the slowest (red), middle (orange), and fastest (green) modes available at readout. Inset: spatial visualization of all 3-bit strings, colored by their parity. In (d) and (e): Gray dots or curve show performance away from criticality with a nonvanishing mass $m_1$, corresponding to a correlation length of a single lattice spacing.