  The study of adaptive dynamics, involving many degrees of freedom on two
separated timescales, one for fast changes of state variables and another for
the slow adaptation of parameters controlling the former's dynamics is crucial
for understanding feedback mechanisms underlying evolution and learning. We
present a path-integral approach \`a la Martin-Siggia-Rose-De Dominicis-Janssen
(MSRDJ) to analyse nonequilibrium phase transitions in such dynamical systems.
As an illustration, we apply our framework to the adaptation of gene-regulatory
networks under a dynamic genotype-phenotype map: phenotypic variations are
shaped by the fast stochastic gene-expression dynamics and are coupled to the
slowly evolving distribution of genotypes, each encoded by a network structure.
We establish that under this map, genotypes corresponding to reciprocal
networks of coherent feedback loops are selected within an intermediate range
of environmental noise, leading to phenotypic robustness.
