  Encoding temporal information from the recent past as spatially distributed
activations is essential in order for the entire recent past to be
simultaneously accessible. Any biological or synthetic agent that relies on the
past to predict/plan the future, would be endowed with such a spatially
distributed temporal memory. Simplistically, we would expect that resource
limitations would demand the memory system to store only the most useful
information for future prediction. For natural signals in real world which show
scale free temporal fluctuations, the predictive information encoded in memory
is maximal if the past information is scale invariantly coarse grained. Here we
examine the general mechanism to construct a scale invariantly coarse grained
memory system. Remarkably, the generic construction is equivalent to encoding
the linear combinations of Laplace transform of the past information and their
approximated inverses. This reveals a fundamental construction constraint on
memory networks that attempt to maximize predictive information storage
relevant to the natural world.
