In the study of the behaviour under iteration of a transcendental entire map, both its singular set and its forward orbit, called its postsingular set, play a key role. In fact, all previous results providing a model for the topological dynamics of transcendental maps, concerned functions with bounded postsingular set. In this talk, I will introduce a new class of functions with escaping critical values for which I have constructed a model for their dynamics. In particular, this will allow us to conclude that their Julia sets consist of a collection of dynamic rays or hairs, that is, injective curves that escape to infinity uniformly, that split at (preimages of) critical points, together with their corresponding landing points. Time permitting, I will also comment on further results obtained as part of my PhD.