Storage capacity diverges with synaptic efficiency in an associative memory model with synaptic delay and pruning

It is known that storage capacity per synapse increases by synaptic pruning in the case of a correlation-type associative memory model. However, the storage capacity of the entire network then decreases. To overcome this difficulty, we propose decreasing the connectivity while keeping the total number of synapses constant by introducing delayed synapses. In this paper, a discrete synchronous-type model with both delayed synapses and their prunings is discussed as a concrete example of the proposal. First, we explain the Yanai-Kim theory by employing statistical neurodynamics. This theory involves macrodynamical equations for the dynamics of a network with serial delay elements. Next, considering the translational symmetry of the explained equations, we rederive macroscopic steady-state equations of the model by using the discrete Fourier transformation. The storage capacities are analyzed quantitatively. Furthermore, two types of synaptic prunings are treated analytically: random pruning and systematic pruning. As a result, it becomes clear that in both prunings, the storage capacity increases as the length of delay increases and the connectivity of the synapses decreases when the total number of synapses is constant. Moreover, an interesting fact becomes clear: the storage capacity asymptotically approaches 2/π due to random pruning. In contrast, the storage capacity diverges in proportion to the logarithm of the length of delay by systematic pruning and the proportion constant is 4/π. These results theoretically support the significance of pruning following an overgrowth of synapses in the brain and may suggest that the brain prefers to store dynamic attractors such as sequences and limit cycles rather than equilibrium states.