The ion response to relativistic electron bunches in the so called bubble or blowout regime of a laser-plasma accelerator is discussed. In response to the strong fields of the accelerated electrons the ions form a central filament along the laser axis that can be compressed to densities 2 orders of magnitude higher than the initial particle density. A theory of the filament formation and a model of ion self-compression are proposed. It is also shown that in the case of a sharp rear plasma-vacuum interface the ions can be accelerated by a combination of three basic mechanisms. The long time ion evolution that results from the strong electrostatic fields of an electron bunch provides a unique diagnostic of laser-plasma accelerators.