  Perceptrons are the building blocks of many theoretical approaches to a wide
range of complex systems, ranging from neural networks and deep learning
machines, to constraint satisfaction problems, glasses and ecosystems. Despite
their applicability and importance, a detailed study of their Langevin dynamics
has never been performed yet. Here we derive the mean-field dynamical equations
that describe the continuous random perceptron in the thermodynamic limit, in a
very general setting with arbitrary noise and friction kernels, not necessarily
related by equilibrium relations. We derive the equations in two ways: via a
dynamical cavity method, and via a path-integral approach in its supersymmetric
formulation. The end point of both approaches is the reduction of the dynamics
of the system to an effective stochastic process for a representative dynamical
variable. Because the perceptron is formally very close to a system of
interacting particles in a high dimensional space, the methods we develop here
can be transferred to the study of liquid and glasses in high dimensions.
Potentially interesting applications are thus the study of the glass transition
in active matter, the study of the dynamics around the jamming transition, and
the calculation of rheological properties in driven systems.
