LETTER Communicated by Andreas Andreou Energy-Efcient Coding with Discrete Stochastic Events

We investigate the energy efciency of signalingmechanisms that transfer information by means of discrete stochastic events, such as the opening or closing of an ion channel. Using a simple model for the generation of graded electrical signals by sodium and potassium channels, we nd optimum numbers of channels that maximize energy efciency. The op-tima depend on several factors: the relative magnitudes of the signaling cost (current ow through channels), the xed cost of maintaining the system, the reliability of the input, additional sources of noise, and the relative costs of upstream and downstream mechanisms. We also analyze how the statistics of input signals inuence energy efciency. We nd that energy-efcient signal ensembles favor a bimodal distribution of channel activations and contain only a very small fraction of large inputs when energy is scarce. We conclude that when energy use is a signicant con-straint, trade-offs between information transfer and energy can strongly inuence the number of signaling molecules and synapses used by neu-rons and the manner in which these mechanisms represent information.