The dominant neutral constituent in Earth's upper exosphere, atomic hydrogen (H), resonantly scatters solar Lyman-alpha (121.567 nm) radiation, observed as the geocorona. We report here observations of an exospheric response to geomagnetic storms obtained using measurements of the geocorona by Lyman-alpha detectors on the Two Wide-angle Imaging Neutral-atom Spectrometers mission. We introduce a new parameter, NH, the number of H atoms in the spherical shell from a geocentric distance of 3 to 8 Earth radii, to quantitatively characterize in a simplified way global exospheric conditions. Five geomagnetic storms observed during three months in the second half of 2011 are accompanied by abrupt temporary increases, spikes, of NH from 6% to 17%, lasting not longer than a day. These increases seem to show some correlation with the minimum Dst index reached during the peak of each storm.