The precision of the neural code is commonly investigated using two families of statistical measures: Shannon mutual information and derived quantities when investigating very small populations of neurons and Fisher information when studying large populations. These statistical tools are no longer the preserve of theorists and are being applied by experimental research groups in the analysis of empirical data. Although the relationship between information-theoretic and Fisher-based measures in the limit of infinite populations is relatively well understood, how these measures compare in finite-size populations has not yet been systematically explored. We aim to close this gap. 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 use a novel Monte Carlo approach to compute a stimulus-specific decomposition of the mutual information (the SSI) for populations of up to 256 neurons and show that Fisher information can be used to accurately estimate both mutual information and SSI for populations of the order of 100 neurons, even in the presence of biologically realistic variability, noise correlations, and experimentally relevant integration times. According to both measures, the stimuli that are best encoded are those falling at the flanks of the neuron's tuning curve. In populations of fewer than around 50 neurons, however, Fisher information can be misleading.

The auditory cortex is necessary for sound localization. The mechanisms that shape bicoordinate spatial representation in the auditory cortex remain unclear. Here, we addressed this issue by quantifying spatial receptive fields (SRFs) in two functionally distinct cortical regions in the pallid bat. The pallid bat uses echolocation for obstacle avoidance and listens to prey-generated noise to localize prey. Its cortex contains two segregated regions of response selectivity that serve echolocation and localization of prey-generated noise. The main aim of this study was to compare 2D SRFs between neurons in the noise-selective region (NSR) and the echolocation region [frequencymodulated sweep-selective region (FMSR)]. The data reveal the following major differences between these two regions: (1) compared with NSR neurons, SRF properties of FMSR neurons were more strongly dependent on sound level; (2) as a population, NSR neurons represent a broad region of contralateral space, while FMSR selectivity was focused near the midline at sound levels near threshold and expanded considerably with increasing sound levels; and (3) the SRF size and centroid elevation were correlated with the characteristic frequency in the NSR, but not the FMSR. These data suggest different mechanisms of sound localization for two different behaviors. Previously, we reported that azimuth is represented by predictable changes in the extent of activated cortex. The present data indicate how elevation constrains this activity pattern. These data suggest a novel model for bicoordinate spatial representation that is based on the extent of activated cortex resulting from the overlap of binaural and tonotopic maps.

Topographic maps are an often-encountered feature in the brains of many species, yet there are no standard, objective procedures for quantifying topography. Topographic maps are typically identified and described subjectively, but in cases where the scale of the map is close to the resolution limit of the measurement technique, identifying the presence of a topographic map can be a challenging subjective task. In such cases, an objective topography detection test would be advantageous. To address these issues, we assessed seven measures (Pearson distance correlation, Spearman distance correlation, Zrehen's measure, topographic product, topological correlation, path length and wiring length) by quantifying topography in three classes of cortical map model: linear, orientation-like, and clusters. We found that all but one of these measures were effective at detecting statistically significant topography even in weakly-ordered maps, based on simulated noisy measurements of neuronal selectivity and sparse sampling of the maps. We demonstrate the practical applicability of these measures by using them to examine the arrangement of spatial cue selectivity in pallid bat A1. This analysis shows that significantly topographic arrangements of interaural intensity difference and azimuth selectivity exist at the scale of individual binaural clusters.

Tuning curves and receptive fields are widely used to describe the selectivity of sensory neurons, but the relationship between firing rates and information is not always intuitive. Neither high firing rates nor high tuning curve gradients necessarily mean that stimuli at that part of the tuning curve are well represented by a neuron. Recent research has shown that trial-to-trial variability (noise) and population size can strongly affect which stimuli are most precisely represented by a neuron in the context of a population code (the best-encoded stimulus), and that different measures of information can give conflicting indications. Specifically, the Fisher information is greatest where the tuning curve gradient is greatest, such as on the flanks of peaked tuning curves, but the stimulus-specific information (SSI) is greatest at the tuning curve peak for small populations with high trial-to-trial variability. Previous research in this area has focussed upon unimodal (peaked) tuning curves, and in this article we extend these analyses to monotonic tuning curves. In addition, we examine how stimulus spacing in forced choice tasks affects the best-encoded stimulus. Our results show that, regardless of the tuning curve, Fisher information correctly predicts the best-encoded stimulus for large populations and where the stimuli are closely spaced in forced choice tasks. In smaller populations with high variability, or in forced choice tasks with widely-spaced choices, the best-encoded stimulus falls at the peak of unimodal tuning curves, but is more variable for monotonic tuning curves. Task, population size and variability all need to be considered when assessing which stimuli a neuron represents, but the best-encoded stimulus can be estimated on a case-by case basis using commonly available computing facilities.

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