Fig. 2: Transitions among different memorized attractors in an indexed memory RC.

A, B By changing the index value p as in panel (A), one can toggle among different states shown in panel (B) that have been memorized as attractors in the indexed memory RC (Dimensions v_x and v_z are not shown). C An instance of a failed switch, where the memory RC evolves to an untrained state after changing p. D Two typical transition matrices where six different states [as the ones in Fig. 1(B)] were memorized. E The average of transition matrices from 25 memory RCs with different random seeds and the same settings as in panel (D). F Variances in the success rate with respect to the starting and destination attractors, revealing a stronger dependence on the latter. The result is from the ensemble of 25 RC networks, each trained with ten different chaotic attractors. G A deliberately chosen memory RC with 16 memory states has relatively low success rates among many states for visualizing the basin structures shown below. H Attracting regions of two different destination states on the manifold of a starting state. The memory RC originally operates on this shell-shaped starting attractor, and the index p is toggled at different time steps when the memory RC is at different positions on this attractor. Such a position can determine whether the switching is successful, where the successful positions are marked blue and failed positions are marked lighter orange. I Local two-dimensional slices of the basin structures of the two destination states from panel (H) in the high-dimensional RC hidden space. Again, the blue and lighter orange regions mark the basins of the memorized state and of some untrained states leading to failed switches, respectively. The quantities ϵ₁ and ϵ₂ define the scales of perturbations to the RC hidden space in two randomly chosen perpendicular directions. The ϵ₁ = ϵ₂ = 0 origin points are the RC hidden states taken from random time steps while the memory RC operates at the corresponding destination states.