  We introduce a formalism to deal with the microscopic modeling of vehicular
traffic on a road network. Traffic on each road is uni-directional, and the
dynamics of each vehicle is described by a Follow-the-Leader model. From a
mathematical point of view, this amounts to define a system of ordinary
differential equations on an arbitrary network. A general existence and
uniqueness result is provided, while priorities at junctions are shown to
hinder the stability of solutions. We investigate the occurrence of the Braess
paradox in a time-dependent setting within this model. The emergence of Nash
equilibria in a non-stationary situation results in the appearance of Braess
type paradoxes, and this is supported by numerical simulations.
