  Let $\Lambda$ be a quasi-projective variety and assume that, either $\Lambda$
is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational
maps, or $\Lambda$ parametrizes an algebraic family
$(f_\lambda)_{\lambda\in\Lambda}$ of degree $d$ rational maps on
$\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct
neutral cycles towards the $p$-th bifurcation current letting the periods of
the cycles go to $\infty$, with an exponential speed of convergence. We deduce
several fundamental consequences of this result on equidistribution and
counting of hyperbolic components. A key step of the proof is a locally uniform
version of the quantitative approximation of the Lyapunov exponent of a
rational map by the $\log^+$ of the modulus of the multipliers of periodic
points.
