  Cyclically sheared jammed packings form memories of the shear amplitude at
which they were trained by falling into periodic orbits where each particle
returns to the identical position in subsequent cycles. While simple models
that treat clusters of rearranging particles as isolated two-state systems
offer insight into this memory formation, they fail to account for the long
training times and multi-period orbits observed in simulated sheared packings.
We show that adding interactions between rearranging clusters overcomes these
deficiencies. In addition, interactions allow simultaneous encoding of multiple
memories which would not have been possible otherwise. These memories are
different in an essential way from those found in other systems, such as
multiple transient memories observed in sheared suspensions, and contain
information about the strength of the interactions.
