We investigate the energy efficiency of signaling mechanisms 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 find optimum numbers of channels that maximize energy efficiency. The optima depend on several factors: the relative magnitudes of the signaling cost (current flow through channels), the fixed 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 influence energy efficiency. We find that energy-efficient 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 significant constraint, trade-offs between information transfer and energy can strongly influence the number of signaling molecules and synapses used by neurons and the manner in which these mechanisms represent information
