Dynamic self-loops in networks of passive and active binary elements

Models of coupled binary elements capture memory effects in complex dissipative materials, such as transient responses or sequential computing, when their interactions are chosen appropriately. However, for random interactions, self-loops - cyclic transition sequences incompatible with dissipative dynamics - dominate the response and undermine statistical approaches. Here we reveal that self-loops originate from energy injection and limit cycles in the underlying physical system. We furthermore introduce interaction ensembles that strongly suppress or completely eliminate self-loops, allowing statistical studies of memory in large dissipative systems. Our work opens a route towards a unified description of passive and active multistable materials using hysteron models.