  We introduce a persistent random walk model with finite velocity and
self-reinforcing directionality, which explains how exponentially distributed
runs self-organize into truncated L\'evy walks observed in active intracellular
transport by Chen et. al. [\textit{Nat. mat.}, 2015]. We derive the
non-homogeneous in space and time, hyperbolic PDE for the probability density
function (PDF) of particle position. This PDF exhibits a bimodal density
(aggregation phenomena) in the superdiffusive regime, which is not observed in
classical linear hyperbolic and L\'evy walk models. We find the exact solutions
for the first and second moments and criteria for the transition to
superdiffusion.
