Storage Capacity of Generalized Palimpsests

A simple analytical study of a short term memory model is performed. This model consists of a symmetric p-neuron interaction between $N$ neurons. Learning is achieved by a generalized Hebb rule. Saturation is prevented by the introduction of a bound A to the couplings. At each time step, an input pattern is drawn at random, independently of the previous ones. The determination of the life time $T$ of a memorized pattern viewed as a function of $A$ and $N$ is accomplished by a statistical study of the dynamic of the learning process which has been made possible under the assumption that the couplings evolve independently. This simplification reduces the determination of $T$ to a one-dimensional problem, by considering energies rather than couplings. The choice of the optimal value $A_{\rm opt}$ of $A$ is a compromise between the success of the learning process and the maximization of $T$. The essential results are expressed by the formulae $T\propto A^2$ and $A_{\rm opt}\propto N^{\frac{p-1}{2}}$