Fisher and shannon information in finite neural populations

While the relationship between information theoretic and Fisher-based measures in the limit of infinite populations is well understood, how these measures compare in finite populations has not yet been systematically explored. We are particularly interested in understanding which stimuli are best encoded by a given neuron within a population and how this depends on the chosen measure. We find that the predictions of Fisher information and of a stimulus-specific decomposition of Shannon information (the SSI) agree very well, even for relatively small population sizes. According to both measures, the stimuli that are best encoded are those falling at the flanks of the neuron’s tuning curve. Background Both Fisher information and measures based on Shannon’s information theory are used to study neural coding. Typically information theory is used only for single neurons and pairs due to tractability constraints, while Fisher information is used for larger populations. Shannon mutual information:- from information theory- a measure of statistical dependency between stimulus and response ensembles- unit: bits Fisher information:- from estimation theory- an upper bound (Cramér-Rao bound) on the variance of an optimal decoder- unit: (stimulus unit)-2 In the limit of large populations, the mutual information can be calculated from the Fisher information1, meaning that the two types of measure are essentially equivalent. For small populations Fisher information overestimates the coding precision. When is it safe to use Fisher information; what size of population is required? What influence does nois