  Cells receive signaling molecules by receptors and relay information via
sensory networks so that they can respond properly depending on the type of
signal. Recent studies have shown that cells can extract multi-dimensional
information from dynamical concentration patterns of signaling molecules. We
herein study how biochemical systems can process multi-dimensional information
embedded in dynamical patterns. We model the decoding networks by linear
response functions, and optimize the functions with the calculus of variations
to maximize the mutual information between patterns and output. We find that,
when the noise intensity is lower, decoders with different linear response
functions, i.e., distinct decoders, can extract much information. However, when
the noise intensity is higher, distinct decoders do not provide the maximum
amount of information. This indicates that, when transmitting information by
dynamical patterns, embedding information in multiple patterns is not optimal
when the noise intensity is very large. Furthermore, we explore the biochemical
implementations of these decoders using control theory and demonstrate that
these decoders can be implemented biochemically through the modification of
cascade-type networks, which are prevalent in actual signaling pathways.
