Karl Friston's Free Energy Principle posits that biological systems maintain their integrity by minimizing variational free energy, which acts as an upper bound on sensory surprise. Mathematically, free energy $F$ is defined as: $$F = D_{KL}[q(\vartheta)||p(\vartheta)] - \langle \ln p(y|\vartheta) \rangle_q$$ where $D_{KL}$ is the Kullback-Leibler divergence between the recognition density $q$ and the prior $p$, and the second term is expected log-likelihood.

In predictive coding, the brain is organized hierarchically. Higher levels continuously generate top-down predictions about sensory inputs. Lower levels compute bottom-up prediction errors (the difference between expected and actual input). Internal models are updated solely to minimize these prediction errors.