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Cochlear implants, also known as bionic ears, are surgically implanted biomedical devices that can provide hearing to some deaf people by direct electrical stimulation of the auditory nerve. A crucial question for the design of future cochlear implants is that of how many electrodes might achieve optimal hearing performance in patients. It is efficient to avoid using more electrodes if this does not provide a performance improvement. Whether an improvement can be gained by inclusion of more electrodes depends crucially on physical properties such as distance of the electrode array from the auditory nerve and current spread. The response of individual fibers in the auditory nerve to electrical stimulation is stochastic, and it is proposed that the interface between an array of electrodes and the auditory nerve can be thought of as a communication channel in which only uncoded transmission can be used. A discrete memoryless channel model for this interface is defined and used as the basis for obtaining numerical estimates of the optimal number of electrodes in the array as a function of array-to-nerve distance. While the only true indicator of improved hearing through cochlear implants is via empirical audiological measurements, the discrete memoryless channel model allows maximization of mutual information as a proxy measure, under the hypothesis that there exists a monotonic relationship between mutual information and perceptibility.