  Energy efficient information transmission may be relevant to biological
sensory signal processing as well as to low power electronic devices. We
explore its consequences in two different regimes. In an ``immediate'' regime,
we argue that the information rate should be maximized subject to a power
constraint, while in an ``exploratory'' regime, the transmission rate per power
cost should be maximized. In the absence of noise, discrete inputs are
optimally encoded into Boltzmann distributed output symbols. In the exploratory
regime, the partition function of this distribution is numerically equal to 1.
The structure of the optimal code is strongly affected by noise in the
transmission channel. The Arimoto-Blahut algorithm, generalized for cost
constraints, can be used to derive and interpret the distribution of symbols
for optimal energy efficient coding in the presence of noise. We outline the
possibilities and problems in extending our results to information coding and
transmission in neurobiological systems.
