Neural spike-timing patterns vary with sound shape and periodicity in three auditory cortical fields

Individualized Closed-Loop Acoustic Stimulation Suggests an Alpha Phase Dependence of Sound Evoked and Induced Brain Activity Measured with EEG Recordings

Following repetitive visual stimulation, post hoc phase analysis finds that visually evoked response magnitudes vary with the cortical alpha oscillation phase that temporally coincides with sensory stimulus. This approach has not successfully revealed an alpha phase dependence for auditory evoked or induced responses. Here, we test the feasibility of tracking alpha with scalp electroencephalogram (EEG) recordings and play sounds phase-locked to individualized alpha phases in real-time using a novel end-point corrected Hilbert transform (ecHT) algorithm implemented on a research device. Based on prior work, we hypothesize that sound-evoked and induced responses vary with the alpha phase at sound onset and the alpha phase that coincides with the early sound-evoked response potential (ERP) measured with EEG. Thus, we use each subject's individualized alpha frequency (IAF) and individual auditory ERP latency to define target trough and peak alpha phases that allow an early component of the auditory ERP to align to the estimated poststimulus peak and trough phases, respectively. With this closed-loop and individualized approach, we find opposing alpha phase-dependent effects on the auditory ERP and alpha oscillations that follow stimulus onset. Trough and peak phase-locked sounds result in distinct evoked and induced post-stimulus alpha level and frequency modulations. Though additional studies are needed to localize the sources underlying these phase-dependent effects, these results suggest a general principle for alpha phase-dependence of sensory processing that includes the auditory system. Moreover, this study demonstrates the feasibility of using individualized neurophysiological indices to deliver automated, closed-loop, phase-locked auditory stimulation.

Dissociable Roles of the Auditory Midbrain and Cortex in Processing the Statistical Features of Natural Sound Textures

Sound texture perception takes advantage of a hierarchy of time-averaged statistical features of acoustic stimuli, but much remains unclear about how these statistical features are processed along the auditory pathway. Here, we compared the neural representation of sound textures in the inferior colliculus (IC) and auditory cortex (AC) of anesthetized female rats. We recorded responses to texture morph stimuli that gradually add statistical features of increasingly higher complexity. For each texture, several different exemplars were synthesized using different random seeds. An analysis of transient and ongoing multiunit responses showed that the IC units were sensitive to every type of statistical feature, albeit to a varying extent. In contrast, only a small proportion of AC units were overtly sensitive to any statistical features. Differences in texture types explained more of the variance of IC neural responses than did differences in exemplars, indicating a degree of "texture type tuning" in the IC, but the same was, perhaps surprisingly, not the case for AC responses. We also evaluated the accuracy of texture type classification from single-trial population activity and found that IC responses became more informative as more summary statistics were included in the texture morphs, while for AC population responses, classification performance remained consistently very low. These results argue against the idea that AC neurons encode sound type via an overt sensitivity in neural firing rate to fine-grain spectral and temporal statistical features.

Retinal input integration in excitatory and inhibitory neurons in the mouse superior colliculus in vivo

The superior colliculus (SC) is a midbrain structure that receives inputs from retinal ganglion cells (RGCs). The SC contains one of the highest densities of inhibitory neurons in the brain but whether excitatory and inhibitory SC neurons differentially integrate retinal activity in vivo is still largely unknown. We recently established a recording approach to measure the activity of RGCs simultaneously with their postsynaptic SC targets in vivo, to study how SC neurons integrate RGC activity. Here, we employ this method to investigate the functional properties that govern retinocollicular signaling in a cell type-specific manner by identifying GABAergic SC neurons using optotagging in VGAT-ChR2 mice. Our results demonstrate that both excitatory and inhibitory SC neurons receive comparably strong RGC inputs and similar wiring rules apply for RGCs innervation of both SC cell types, unlike the cell type-specific connectivity in the thalamocortical system. Moreover, retinal activity contributed more to the spiking activity of postsynaptic excitatory compared to inhibitory SC neurons. This study deepens our understanding of cell type-specific retinocollicular functional connectivity and emphasizes that the two major brain areas for visual processing, the visual cortex and the SC, differently integrate sensory afferent inputs.

Hearing as adaptive cascaded envelope interpolation

The human auditory system is designed to capture and encode sounds from our surroundings and conspecifics. However, the precise mechanisms by which it adaptively extracts the most important spectro-temporal information from sounds are still not fully understood. Previous auditory models have explained sound encoding at the cochlear level using static filter banks, but this vision is incompatible with the nonlinear and adaptive properties of the auditory system. Here we propose an approach that considers the cochlear processes as envelope interpolations inspired by cochlear physiology. It unifies linear and nonlinear adaptive behaviors into a single comprehensive framework that provides a data-driven understanding of auditory coding. It allows simulating a broad range of psychophysical phenomena from virtual pitches and combination tones to consonance and dissonance of harmonic sounds. It further predicts the properties of the cochlear filters such as frequency selectivity. Here we propose a possible link between the parameters of the model and the density of hair cells on the basilar membrane. Cascaded Envelope Interpolation may lead to improvements in sound processing for hearing aids by providing a non-linear, data-driven, way to preprocessing of acoustic signals consistent with peripheral processes.

Context Sensitivity across Multiple Time scales with a Flexible Frequency Bandwidth

Everyday auditory streams are complex, including spectro-temporal content that varies at multiple timescales. Using EEG, we investigated the sensitivity of human auditory cortex to the content of past stimulation in unattended sequences of equiprobable tones. In 3 experiments including 82 participants overall, we found that neural responses measured at different latencies after stimulus onset were sensitive to frequency intervals computed over distinct timescales. Importantly, early responses were sensitive to a longer history of stimulation than later responses. To account for these results, we tested a model consisting of neural populations with frequency-specific but broad tuning that undergo adaptation with exponential recovery. We found that the coexistence of neural populations with distinct recovery rates can explain our results. Furthermore, the adaptation bandwidth of these populations depended on spectral context-it was wider when the stimulation sequence had a wider frequency range. Our results provide electrophysiological evidence as well as a possible mechanistic explanation for dynamic and multiscale context-dependent auditory processing in the human cortex.

Plasticity of Multidimensional Receptive Fields in Core Rat Auditory Cortex Directed by Sound Statistics

Sensory cortical neurons can nonlinearly integrate a wide range of inputs. The outcome of this nonlinear process can be approximated by more than one receptive field component or filter to characterize the ensuing stimulus preference. The functional properties of multidimensional filters are, however, not well understood. Here we estimated two spectrotemporal receptive fields (STRFs) per neuron using maximally informative dimension analysis. We compared their temporal and spectral modulation properties and determined the stimulus information captured by the two STRFs in core rat auditory cortical fields, primary auditory cortex (A1) and ventral auditory field (VAF). The first STRF is the dominant filter and acts as a sound feature detector in both fields. The second STRF is less feature specific, preferred lower modulations, and had less spike information compared to the first STRF. The information jointly captured by the two STRFs was larger than that captured by the sum of the individual STRFs, reflecting nonlinear interactions of two filters. This information gain was larger in A1. We next determined how the acoustic environment affects the structure and relationship of these two STRFs. Rats were exposed to moderate levels of spectrotemporally modulated noise during development. Noise exposure strongly altered the spectrotemporal preference of the first STRF in both cortical fields. The interaction between the two STRFs was reduced by noise exposure in A1 but not in VAF. The results reveal new functional distinctions between A1 and VAF indicating that (i) A1 has stronger interactions of the two STRFs than VAF, (ii) noise exposure diminishes modulation parameter representation contained in the noise more strongly for the first STRF in both fields, and (iii) plasticity induced by noise exposure can affect the strength of filter interactions in A1. Taken together, ascertaining two STRFs per neuron enhances the understanding of cortical information processing and plasticity effects in core auditory cortex.

Amplitude modulation encoding in the auditory cortex: comparisons between the primary and middle lateral belt regions

In macaques, the middle lateral auditory cortex (ML) is a belt region adjacent to the primary auditory cortex (A1) and believed to be at a hierarchically higher level. Although ML single-unit responses have been studied for several auditory stimuli, the ability of ML cells to encode amplitude modulation (AM)-an ability that has been widely studied in A1-has not yet been characterized. Here, we compared the responses of A1 and ML neurons to amplitude-modulated (AM) noise in awake macaques. Although several of the basic properties of A1 and ML responses to AM noise were similar, we found several key differences. ML neurons were less likely to phase lock, did not phase lock as strongly, and were more likely to respond in a nonsynchronized fashion than A1 cells, consistent with a temporal-to-rate transformation as information ascends the auditory hierarchy. ML neurons tended to have lower temporally (phase-locking) based best modulation frequencies than A1 neurons. Neurons that decreased their firing rate in response to AM noise relative to their firing rate in response to unmodulated noise became more common at the level of ML than they were in A1. In both A1 and ML, we found a prevalent class of neurons that usually have enhanced rate responses relative to responses to the unmodulated noise at lower modulation frequencies and suppressed rate responses relative to responses to the unmodulated noise at middle modulation frequencies.NEW & NOTEWORTHY ML neurons synchronized less than A1 neurons, consistent with a hierarchical temporal-to-rate transformation. Both A1 and ML had a class of modulation transfer functions previously unreported in the cortex with a low-modulation-frequency (MF) peak, a middle-MF trough, and responses similar to unmodulated noise responses at high MFs. The results support a hierarchical shift toward a two-pool opponent code, where subtraction of neural activity between two populations of oppositely tuned neurons encodes AM.

Sleep Differentially Affects Early and Late Neuronal Responses to Sounds in Auditory and Perirhinal Cortices

A fundamental feature of sleep is reduced behavioral responsiveness to external events, but the extent of processing along sensory pathways remains poorly understood. While responses are comparable across wakefulness and sleep in auditory cortex (AC), neuronal activity in downstream regions remains unknown. Here we recorded spiking activity in 435 neuronal clusters evoked by acoustic stimuli in the perirhinal cortex (PRC) and in AC of freely behaving male rats across wakefulness and sleep. Neuronal responses in AC showed modest (∼10%) differences in response gain across vigilance states, replicating previous studies. By contrast, PRC neuronal responses were robustly attenuated by 47% and 36% during NREM sleep and REM sleep, respectively. Beyond the separation according to cortical region, response latency in each neuronal cluster was correlated with the degree of NREM sleep attenuation, such that late (>40 ms) responses in all monitored regions diminished during NREM sleep. The robust attenuation of late responses prevalent in PRC represents a novel neural correlate of sensory disconnection during sleep, opening new avenues for investigating the mediating mechanisms.SIGNIFICANCE STATEMENT Reduced behavioral responsiveness to sensory stimulation is at the core of sleep's definition, but it is still unclear how the sleeping brain responds differently to sensory stimuli. In the current study, we recorded neuronal spiking responses to sounds along the cortical processing hierarchy of rats during wakefulness and natural sleep. Responses in auditory cortex only showed modest changes during sleep, whereas sleep robustly attenuated the responses of neurons in high-level perirhinal cortex. We also found that, during NREM sleep, the response latency predicts the degree of sleep attenuation in individual neurons above and beyond their anatomical location. These results provide anatomical and temporal signatures of sensory disconnection during sleep and pave the way to understanding the underlying mechanisms.

Modeling stimulus-dependent variability improves decoding of population neural responses

OBJECTIVE

Neural responses to repeated presentations of an identical stimulus often show substantial trial-to-trial variability. How the mean firing rate varies in response to different stimuli or during different movements (tuning curves) has been extensively modeled in a wide variety of neural systems. However, the variability of neural responses can also have clear tuning independent of the tuning in the mean firing rate. This suggests that the variability could contain information regarding the stimulus/movement beyond what is encoded in the mean firing rate. Here we demonstrate how taking variability into account can improve neural decoding.

APPROACH

In a typical neural coding model spike counts are assumed to be Poisson with the mean response depending on an external variable, such as a stimulus or movement. Bayesian decoding methods then use the probabilities under these Poisson tuning models (the likelihood) to estimate the probability of each stimulus given the spikes on a given trial (the posterior). However, under the Poisson model, spike count variability is always exactly equal to the mean (Fano factor  =  1). Here we use two alternative models-the Conway-Maxwell-Poisson (CMP) model and negative binomial (NB) model-to more flexibly characterize how neural variability depends on external stimuli. These models both contain the Poisson distribution as a special case but have an additional parameter that allows the variance to be greater than the mean (Fano factor  >  1) or, for the CMP model, less than the mean (Fano factor  <  1).

MAIN RESULTS

We find that neural responses in primary motor (M1), visual (V1), and auditory (A1) cortices have diverse tuning in both their mean firing rates and response variability. Across cortical areas, we find that Bayesian decoders using the CMP or NB models improve stimulus/movement estimation accuracy by 4%-12% compared to the Poisson model.

SIGNIFICANCE

Moreover, the uncertainty of the non-Poisson decoders more accurately reflects the magnitude of estimation errors. In addition to tuning curves that reflect average neural responses, stimulus-dependent response variability may be an important aspect of the neural code. Modeling this structure could, potentially, lead to improvements in brain machine interfaces.

Neural timing of stimulus events with microsecond precision

Temporal analysis of sound is fundamental to auditory processing throughout the animal kingdom. Echolocating bats are powerful models for investigating the underlying mechanisms of auditory temporal processing, as they show microsecond precision in discriminating the timing of acoustic events. However, the neural basis for microsecond auditory discrimination in bats has eluded researchers for decades. Combining extracellular recordings in the midbrain inferior colliculus (IC) and mathematical modeling, we show that microsecond precision in registering stimulus events emerges from synchronous neural firing, revealed through low-latency variability of stimulus-evoked extracellular field potentials (EFPs, 200–600 Hz). The temporal precision of the EFP increases with the number of neurons firing in synchrony. Moreover, there is a functional relationship between the temporal precision of the EFP and the spectrotemporal features of the echolocation calls. In addition, EFP can measure the time difference of simulated echolocation call–echo pairs with microsecond precision. We propose that synchronous firing of populations of neurons operates in diverse species to support temporal analysis for auditory localization and complex sound processing.

We routinely rely on a stopwatch to precisely measure the time it takes for an athlete to reach the finish line. Without the assistance of such a timing device, our measurement of elapsed time becomes imprecise. By contrast, some animals, such as echolocating bats, naturally perform timing tasks with remarkable precision. Behavioral research has shown that echolocating bats can estimate the elapsed time between sonar cries and echo returns with a precision in the range of microseconds. However, the neural basis for such microsecond precision has remained a puzzle to scientists. Combining extracellular recordings in the bat’s inferior colliculus (IC)—a midbrain nucleus of the auditory pathway—and mathematical modeling, we show that microsecond precision in registering stimulus events emerges from synchronous neural firing. Our recordings revealed a low-latency variability of stimulus-evoked extracellular field potentials (EFPs), which, according to our mathematical modeling, was determined by the number of firing neurons and their synchrony. Moreover, the acoustic features of echolocation calls, such as signal duration and bandwidth, which the bat dynamically modulates during prey capture, also modulate the precision of EFPs. These findings have broad implications for understanding temporal analysis of acoustic signals in a wide range of auditory behaviors across the animal kingdom.

A Hierarchy of Time Scales for Discriminating and Classifying the Temporal Shape of Sound in Three Auditory Cortical Fields

Auditory cortex is essential for mammals, including rodents, to detect temporal "shape" cues in the sound envelope but it remains unclear how different cortical fields may contribute to this ability (Lomber and Malhotra, 2008; Threlkeld et al., 2008). Previously, we found that precise spiking patterns provide a potential neural code for temporal shape cues in the sound envelope in the primary auditory (A1), and ventral auditory field (VAF) and caudal suprarhinal auditory field (cSRAF) of the rat (Lee et al., 2016). Here, we extend these findings and characterize the time course of the temporally precise output of auditory cortical neurons in male rats. A pairwise sound discrimination index and a Naive Bayesian classifier are used to determine how these spiking patterns could provide brain signals for behavioral discrimination and classification of sounds. We find response durations and optimal time constants for discriminating sound envelope shape increase in rank order with: A1 < VAF < cSRAF. Accordingly, sustained spiking is more prominent and results in more robust sound discrimination in non-primary cortex versus A1. Spike-timing patterns classify 10 different sound envelope shape sequences and there is a twofold increase in maximal performance when pooling output across the neuron population indicating a robust distributed neural code in all three cortical fields. Together, these results support the idea that temporally precise spiking patterns from primary and non-primary auditory cortical fields provide the necessary signals for animals to discriminate and classify a large range of temporal shapes in the sound envelope.SIGNIFICANCE STATEMENT Functional hierarchies in the visual cortices support the concept that classification of visual objects requires successive cortical stages of processing including a progressive increase in classical receptive field size. The present study is significant as it supports the idea that a similar progression exists in auditory cortices in the time domain. We demonstrate for the first time that three cortices provide temporal spiking patterns for robust temporal envelope shape discrimination but only the ventral non-primary cortices do so on long time scales. This study raises the possibility that primary and non-primary cortices provide unique temporal spiking patterns and time scales for perception of sound envelope shape.

Origins of scale invariance in vocalization sequences and speech

To communicate effectively animals need to detect temporal vocalization cues that vary over several orders of magnitude in their amplitude and frequency content. This large range of temporal cues is evident in the power-law scale-invariant relationship between the power of temporal fluctuations in sounds and the sound modulation frequency (f). Though various forms of scale invariance have been described for natural sounds, the origins and implications of scale invariant phenomenon remain unknown. Using animal vocalization sequences, including continuous human speech, and a stochastic model of temporal amplitude fluctuations we demonstrate that temporal acoustic edges are the primary acoustic cue accounting for the scale invariant phenomenon. The modulation spectrum of vocalization sequences and the model both exhibit a dual regime lowpass structure with a flat region at low modulation frequencies and scale invariant 1/f² trend for high modulation frequencies. Moreover, we find a time-frequency tradeoff between the average vocalization duration of each vocalization sequence and the cutoff frequency beyond which scale invariant behavior is observed. These results indicate that temporal edges are universal features responsible for scale invariance in vocalized sounds. This is significant since temporal acoustic edges are salient perceptually and the auditory system could exploit such statistical regularities to minimize redundancies and generate compact neural representations of vocalized sounds.

The efficient coding hypothesis posits that the brain encodes sensory signals efficiently in order to reduce metabolic cost and preserve behaviorally relevant environment information. In audition, recognition and coding depends on the brain’s ability to accurately and efficiently encode statistical regularities that are prevalent in natural sounds. Similarly, efficient audio coding and compression schemes attempt to preserve salient sound qualities while minimizing data bandwidth. A widely observed statistical regularity in nearly all natural sounds is the presence of scale invariance where the power of amplitude fluctuations is inversely related to the sound amplitude modulation frequency. In this study, we explore the physical sound cues responsible for the scale invariant phenomenon previously observed. We demonstrate that for animal vocalizations, including human speech, the scale invariant behavior is fully accounted by the presence of temporal acoustic edges that are largely created by opening and closing of the oral cavity and which mark the beginning and end of isolated vocalizations. The findings thus identify a single physical cue responsible for the universal scale invariant phenomenon that the brain can exploit to optimize coding and perception of vocalized sounds.

Interpretation of correlated neural variability from models of feed-forward and recurrent circuits

Neural populations respond to the repeated presentations of a sensory stimulus with correlated variability. These correlations have been studied in detail, with respect to their mechanistic origin, as well as their influence on stimulus discrimination and on the performance of population codes. A number of theoretical studies have endeavored to link network architecture to the nature of the correlations in neural activity. Here, we contribute to this effort: in models of circuits of stochastic neurons, we elucidate the implications of various network architectures—recurrent connections, shared feed-forward projections, and shared gain fluctuations—on the stimulus dependence in correlations. Specifically, we derive mathematical relations that specify the dependence of population-averaged covariances on firing rates, for different network architectures. In turn, these relations can be used to analyze data on population activity. We examine recordings from neural populations in mouse auditory cortex. We find that a recurrent network model with random effective connections captures the observed statistics. Furthermore, using our circuit model, we investigate the relation between network parameters, correlations, and how well different stimuli can be discriminated from one another based on the population activity. As such, our approach allows us to relate properties of the neural circuit to information processing.

The response of neurons to a stimulus is variable across trials. A natural solution for reliable coding in the face of noise is the averaging across a neural population. The nature of this averaging depends on the structure of noise correlations in the neural population. In turn, the correlation structure depends on the way noise and correlations are generated in neural circuits. It is in general difficult to identify the origin of correlations from the observed population activity alone. In this article, we explore different theoretical scenarios of the way in which correlations can be generated, and we relate these to the architecture of feed-forward and recurrent neural circuits. Analyzing population recordings of the activity in mouse auditory cortex in response to sound stimuli, we find that population statistics are consistent with those generated in a recurrent network model. Using this model, we can then quantify the effects of network properties on average population responses, noise correlations, and the representation of sensory information.

Identified GABAergic and Glutamatergic Neurons in the Mouse Inferior Colliculus Share Similar Response Properties

GABAergic neurons in the inferior colliculus (IC) play a critical role in auditory information processing, yet their responses to sound are unknown. Here, we used optogenetic methods to characterize the response properties of GABAergic and presumed glutamatergic neurons to sound in the IC. We found that responses to pure tones of both inhibitory and excitatory classes of neurons were similar in their thresholds, response latencies, rate-level functions, and frequency tuning, but GABAergic neurons may have higher spontaneous firing rates. In contrast to their responses to pure tones, the inhibitory and excitatory neurons differed in their ability to follow amplitude modulations. The responses of both cell classes were affected by their location regardless of the cell type, especially in terms of their frequency tuning. These results show that the synaptic domain, a unique organization of local neural circuits in the IC, may interact with all types of neurons to produce their ultimate response to sound.SIGNIFICANCE STATEMENT Although the inferior colliculus (IC) in the auditory midbrain is composed of different types of neurons, little is known about how these specific types of neurons respond to sound. Here, for the first time, we characterized the response properties of GABAergic and glutamatergic neurons in the IC. Both classes of neurons had diverse response properties to tones but were overall similar, except for the spontaneous activity and their ability to follow amplitude-modulated sound. Both classes of neurons may compose a basic local circuit that is replicated throughout the IC. Within each local circuit, the inputs to the local circuit may have a greater influence in determining the response properties to sound than the specific neuron types.

Population responses in primary auditory cortex simultaneously represent the temporal envelope and periodicity features in natural speech

Speech perception relies on a listener's ability to simultaneously resolve multiple temporal features in the speech signal. Little is known regarding neural mechanisms that enable the simultaneous coding of concurrent temporal features in speech. Here we show that two categories of temporal features in speech, the low-frequency speech envelope and periodicity cues, are processed by distinct neural mechanisms within the same population of cortical neurons. We measured population activity in primary auditory cortex of anesthetized guinea pig in response to three variants of a naturally produced sentence. Results show that the envelope of population responses closely tracks the speech envelope, and this cortical activity more closely reflects wider bandwidths of the speech envelope compared to narrow bands. Additionally, neuronal populations represent the fundamental frequency of speech robustly with phase-locked responses. Importantly, these two temporal features of speech are simultaneously observed within neuronal ensembles in auditory cortex in response to clear, conversation, and compressed speech exemplars. Results show that auditory cortical neurons are adept at simultaneously resolving multiple temporal features in extended speech sentences using discrete coding mechanisms.

Onset-Duration Matching of Acoustic Stimuli Revisited: Conventional Arithmetic vs. Proposed Geometric Measures of Accuracy and Precision

Onsets of acoustic stimuli are salient transients and are relevant in humans for the perception of music and speech. Previous studies of onset-duration discrimination and matching focused on whether onsets are perceived categorically. In this study, we address two issues. First, we revisit onset-duration matching and measure, for 79 conditions, how accurately and precisely human listeners can adjust the onset duration of a comparison stimulus to subjectively match that of a standard stimulus. Second, we explore measures for quantifying performance in this and other matching tasks. The conventional measures of accuracy and precision are defined by arithmetic descriptive statistics and the Euclidean distance function on the real numbers. We propose novel measures based on geometric descriptive statistics and the log-ratio distance function, the Euclidean distance function on the positive-real numbers. Only these properly account for the fact that the magnitude of onset durations, like the magnitudes of most physical quantities, can attain only positive real values. The conventional (arithmetic) measures possess a convexity bias that yields errors that grow with the width of the distribution of matches. This convexity bias leads to misrepresentations of the constant error and could even imply the existence of perceptual illusions where none exist. This is not so for the proposed (geometric) measures. We collected up to 68 matches from a given listener for each condition (about 34,000 matches in total) and examined inter-listener variability and the effects of onset duration, plateau duration, sound level, carrier, and restriction of the range of adjustable comparison stimuli on measures of accuracy and precision. Results obtained with the conventional measures generally agree with those reported in the literature. The variance across listeners is highly heterogeneous for the conventional measures but is homogeneous for the proposed measures. Furthermore, the proposed measures show that listeners tend to under- rather than to overestimate the onset duration of the comparison stimuli. They further reveal effects of the stimulus carrier on accuracy and precision which are missed by the conventional measures. Our results have broad implications for psychophysical studies that use arithmetic measures to quantify performance when geometric measures should instead be used.