Before revision:  We find that optimal decoders are realized with multiple distinct non-monotonic response functionsand that such optimal decoderscan extract information much more efficiently than typical single-layer linear decoders. We also consider the decorrelation of informationembedded in the dynamical patternsand show that decorrelating decoders converge to the upper bound of the mutual information in the weak noise limit. We explore the biochemical implementations of these decoders using control theory and demonstrate that they can be implemented biochemically through  modification of cascade-type networks, which are prevalent in actual signaling pathways.
After revision:  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 they can be implemented biochemically through modification of cascade-type networks, which are prevalent in actual signaling pathways.