  In this work we discuss possible physical origins for non-trivial exponents
in the athermal rheology of soft materials at low but finite driving rates. A
key ingredient in our scenario is the presence of a self-consistent mechanical
noise that stems from the spatial superposition of long-range elastic responses
to localized plastically deforming regions. We study analytically a mean-field
model, in which this mechanical noise is accounted for by a stress diffusion
term coupled to the plastic activity. Within this description we show how a
dependence of the shear modulus and/or the local relaxation time on the shear
rate introduces corrections to the usual mean-field prediction, concerning the
Herschel-Bulkley-type rheological response of exponent 1/2. This feature of the
mean-field picture is then shown to be robust with respect to structural
disorder and partial relaxation of the local stress. We test this prediction
numerically on a mesoscopic lattice model that implements explicitly the
long-range elastic response to localized shear transformations, and we conclude
on how our scenario might be tested in rheological experiments.
