Energy-information trade-off makes the cortical critical power law the optimal coding

Stimulus responses of cortical neurons exhibit the critical power law, where the covariance eigenspectrum follows the power law with the exponent just at the edge of differentiability of the neural manifold. This criticality is conjectured to balance the expressivity and robustness of neural codes, because a non-differential fractal manifold spoils coding reliability. However, contrary to the conjecture, here we prove that the neural coding is not degraded even on the non-differentiable fractal manifold, where the coding is extremely sensitive to perturbations. Rather, we show that the trade-off between energetic cost and information always makes this critical power-law response the optimal neural coding. Direct construction of a maximum likelihood decoder of the power-law coding validates the theoretical prediction. By revealing the non-trivial nature of high-dimensional coding, the theory developed here will contribute to a deeper understanding of criticality and power laws in computation in biological and artificial neural networks.