Neurons in the brain are able to detect and discriminate salient spatio-temporal patterns in the firing activity of presynaptic neurons. It is open how they can learn to achieve this, especially without the help of a supervisor. We show that a well-known unsupervised learning algorithm for linear neurons, Slow Feature Analysis (SFA), is able to acquire the discrimination capability of one of the best algorithms for supervised linear discrimination learning, the Fisher Linear Discriminant (FLD), given suitable input statistics. We demonstrate the power of this principle by showing that it enables readout neurons from simulated cortical microcircuits to learn without any supervision to discriminate between spoken digits, and to detect repeated firing patterns that are embedded into a stream of noise spike trains with the same firing statistics. Both these computer simulations and our theoretical analysis show that slow feature extraction enables neurons to extract and collect information that is spread out over a trajectory of firing states that lasts several hundred ms. In addition, it enables neurons to learn without supervision to keep track of time (relative to a stimulus onset, or the initiation of a motor response). Hence these results elucidate how the brain could compute with trajectories of firing states, rather than only with fixed point attractors. It also provides a theoretical basis for understanding recent experimental results on the emergence of view- and position-invariant classification of visual objects in inferior temporal cortex.