We present a new formulation of the hyperdynamics method in which the biasing effect is local, making it suitable for large systems. In standard hyperdynamics, the requirement that the bias potential be zero everywhere on the dividing surface bounding the state has the consequence that as the system size increases the boost factor decays to unity, regardless of the form of the bias potential. In the new method, the bias force on each atom is obtained by differentiating a local bias energy that depends only on the coordinates of atoms within a finite range of this atom. This bias force is thus independent of the bias force in distant parts of the system, providing a method that gives a constant boost factor, independent of the system size. We demonstrate for some realistic atomistic systems that the method gives escape rates in excellent agreement with direct molecular dynamics simulations.