  We investigate the dynamics of large stochastic networks with different
timescales and nonlinear mean-field interactions. After deriving the limit
equations for a general class of network models, we apply our results to the
celebrated Wilson-Cowan system with two populations with or without slow
adaptation, paradigmatic example of nonlinear mean-field network. This system
has the property that the dynamics of the mean of the solution exactly
satisfies an ODE. This reduction allows to show that in the mean-field limit
and in multiple populations with multiple timescales, noise induces canard
explosions and Mixed-Mode Oscillations on the mean of the solution. This sheds
new light on the qualitative effects of noise and sensitivity to precise noise
values in large stochastic networks. We further investigate finite-sized
networks and show that systematic differences with the mean-field limits arise
in bistable regimes (where random switches between different attractors occur)
or in mixed-mode oscillations, were the finite-size effects induce early jumps
due to the sensitivity of the attractor.
