According to a popular hypothesis, short-term memories are stored as persistent neural activity maintained by synaptic feedback loops. This hypothesis has been formulated mathematically in a number of recurrent network models. Here we study an abstraction of these models, a single neuron with a synapse onto itself, or autapse. This abstraction cannot simulate the way in which persistent activity patterns are distributed over neural populations in the brain. However, with proper tuning of parameters, it does reproduce the continuously graded, or analog, nature of many examples of persistent activity. The conditions for tuning are derived for the dynamics of a conductance-based model neuron with a slow excitatory autapse. The derivation uses the method of averaging to approximate the spiking model with a nonspiking, reduced model. Short-term analog memory storage is possible if the reduced model is approximately linear and if its feedforward bias and autapse strength are precisely tuned.