This paper studies the capability of a recurrent neural network model to\nmemorize random dynamical firing patterns by a simple local learning rule. Two\nmodes of learning/memorization are considered: The first mode is strictly\nonline, with a single pass through the data, while the second mode uses\nmultiple passes through the data. In both modes, the learning is strictly local\n(quasi-Hebbian): At any given time step, only the weights between the neurons\nfiring (or supposed to be firing) at the previous time step and those firing\n(or supposed to be firing) at the present time step are modified. The main\nresult of the paper is an upper bound on the probability that the single-pass\nmemorization is not perfect. It follows that the memorization capacity in this\nmode asymptotically scales like that of the classical Hopfield model (which, in\ncontrast, memorizes static patterns). However, multiple-rounds memorization is\nshown to achieve a higher capacity (with a nonvanishing number of bits per\nconnection/synapse). These mathematical findings may be helpful for\nunderstanding the functions of short-term memory and long-term memory in\nneuroscience.\n