We identify a time-dependent class of metrics with potential applications to cosmology, which emerge from negative-tension branes. The cosmology is based on a general class of solutions to Einstein-dilaton-Maxwell theory, presented in hep-th/0106120. We argue that solutions with hyperbolic or planar symmetry describe the gravitational interactions of a pair of negative-tension q-branes. These spacetimes are static near each brane, but become time-dependent and expanding at late epoch — in some cases asymptotically approaching flat space. We interpret this expansion as being the spacetime's response to the branes' presence. The time-dependent regions provide explicit examples of cosmological spacetimes with past horizons and no past naked singularities. The past horizons can be interpreted as S-branes. We prove that the singularities in the static regions are repulsive to time-like geodesics, extract a cosmological ``bounce'' interpretation, compute the explicit charge and tension of the branes, analyse the classical stability of the solution (in particular of the horizons) and study particle production, deriving a general expression for Hawking's temperature as well as the associated entropy.