Neural Tuning Curves Research Articles

Accounting for Bias in the Estimation of r 2 between Two Sets of Noisy Neural Responses.

The Pearson correlation coefficient squared, r 2, is an important tool used in the analysis of neural data to quantify the similarity between neural tuning curves. Yet this metric is biased by trial-to-trial variability; as trial-to-trial variability increases, measured correlation decreases. Major lines of research are confounded by this bias, including those involving the study of invariance of neural tuning across conditions and the analysis of the similarity of tuning across neurons. To address this, we extend an estimator, [Formula: see text], that was recently developed for estimating model-to-neuron correlation, in which a noisy signal is compared with a noise-free prediction, to the case of neuron-to-neuron correlation, in which two noisy signals are compared with each other. We compare the performance of our novel estimator to a prior method developed by Spearman, commonly used in other fields but widely overlooked in neuroscience, and find that our method has less bias. We then apply our estimator to demonstrate how it avoids drastic confounds introduced by trial-to-trial variability using data collected in two prior studies (macaque, both sexes) that examined two different forms of invariance in the neural encoding of visual inputs-translation invariance and fill-outline invariance. Our results quantify for the first time the gradual falloff with spatial offset of translation-invariant shape selectivity within visual cortical neuronal receptive fields and offer a principled method to compare invariance in noisy biological systems to that in noise-free models.SIGNIFICANCE STATEMENT Quantifying the similarity between two sets of averaged neural responses is fundamental to the analysis of neural data. A ubiquitous metric of similarity, the correlation coefficient, is attenuated by trial-to-trial variability that arises from many irrelevant factors. Spearman recognized this problem and proposed corrected methods that have been extended over a century. We show this method has large asymptotic biases that can be overcome using a novel estimator. Despite the frequent use of the correlation coefficient in neuroscience, consensus on how to address this fundamental statistical issue has not been reached. We provide an accurate estimator of the correlation coefficient and apply it to gain insight into visual invariance.

Estimating human neural tuning curves at low-level sound from human psychophysical masking data and the cochlea-frequency maps

We proposed a three-slope method to estimate human frequency tuning curves at low sound levels. As a starting point, cat neural excitation patterns (Allen and Fahey, 1993) are derived from Liberman’s cat cochlea neural tuning curves data (Liberman, 1978) in a mathematical way. A three-slope structure inspired by (Rhode,1978) is proposed to represent neural excitation patterns. The aim of this paper is to estimate the three-slope structure for human neural excitation patterns and then transform human neural excitation patterns into human tuning curves by using a human cochlea-frequency map (Greenwood, 1990). In 1993, Allen and Fahey introduced the concept of the second cochlea map, which was shown to be highly correlated with both neural excitation patterns and distortion product otoacoustic emissions (DPOAE). Thus, three slopes of human neural excitation patterns may be estimated via the second cochlea map and human masking data (Wegel and Lane, 1924). In the present study, slopes are estimated from limited existing psychophysical data, which can be improved with more accurate experimental data in the future.

A Dynamic Efficient Sensory Encoding Approach to Adaptive Tuning in Neural Models of Optic Flow Processing.

This paper introduces a self-tuning mechanism for capturing rapid adaptation to changing visual stimuli by a population of neurons. Building upon the principles of efficient sensory encoding, we show how neural tuning curve parameters can be continually updated to optimally encode a time-varying distribution of recently detected stimulus values. We implemented this mechanism in a neural model that produces human-like estimates of self-motion direction (i.e., heading) based on optic flow. The parameters of speed-sensitive units were dynamically tuned in accordance with efficient sensory encoding such that the network remained sensitive as the distribution of optic flow speeds varied. In two simulation experiments, we found that model performance with dynamic tuning yielded more accurate, shorter latency heading estimates compared to the model with static tuning. We conclude that dynamic efficient sensory encoding offers a plausible approach for capturing adaptation to varying visual environments in biological visual systems and neural models alike.

The barn owls' Minimum Audible Angle.

Interaural time differences (ITD) and interaural level differences (ILD) are physical cues that enable the auditory system to pinpoint the position of a sound source in space. This ability is crucial for animal communication and predator-prey interactions. The barn owl has evolved an exceptional sense of hearing and shows abilities of sound localisation that outperform most other species. So far, behavioural studies in the barn owl often used reflexive responses to investigate aspects of sound localisation. Furthermore, they predominately probed the higher frequencies of the owl’s hearing range (> 3 kHz). In the present study we used a Go/NoGo paradigm to measure the barn owl’s behavioural sound localisation acuity (expressed as the Minimum Audible Angle, MAA) as a function of stimulus type (narrow-band noise centred at 500, 1000, 2000, 4000 and 8000 Hz, and broad-band noise) and sound source position. We found significant effects of both stimulus type and sound source position on the barn owls’ MAA. The MAA improved with increasing stimulus frequency, from 14° at 500 Hz to 6° at 8000 Hz. The smallest MAA of 4° was found for broadband noise stimuli. Comparing different sound source positions revealed smaller MAAs for frontal compared to lateral stimulus presentation, irrespective of stimulus type. These results are consistent with both the known variations in physical ITDs and variation in the width of neural ITD tuning curves with azimuth and frequency. Physical and neural characteristics combine to result in better spatial acuity for frontal compared to lateral sounds and reduced localisation acuity at lower frequencies.

Feedback optimizes neural coding and perception of natural stimuli.

Growing evidence suggests that sensory neurons achieve optimal encoding by matching their tuning properties to the natural stimulus statistics. However, the underlying mechanisms remain unclear. Here we demonstrate that feedback pathways from higher brain areas mediate optimized encoding of naturalistic stimuli via temporal whitening in the weakly electric fish Apteronotus leptorhynchus. While one source of direct feedback uniformly enhances neural responses, a separate source of indirect feedback selectively attenuates responses to low frequencies, thus creating a high-pass neural tuning curve that opposes the decaying spectral power of natural stimuli. Additionally, we recorded from two populations of higher brain neurons responsible for the direct and indirect descending inputs. While one population displayed broadband tuning, the other displayed high-pass tuning and thus performed temporal whitening. Hence, our results demonstrate a novel function for descending input in optimizing neural responses to sensory input through temporal whitening that is likely to be conserved across systems and species.

Neural Tuning Curves in Visual Working Memory

Neural Tuning Curves in Visual Working Memory

A random-matrix theory of the number sense.

Number sense, a spontaneous ability to process approximate numbers, has been documented in human adults, infants and newborns, and many other animals. Species as distant as monkeys and crows exhibit very similar neurons tuned to specific numerosities. How number sense can emerge in the absence of learning or fine tuning is currently unknown. We introduce a random-matrix theory of self-organized neural states where numbers are coded by vectors of activation across multiple units, and where the vector codes for successive integers are obtained through multiplication by a fixed but random matrix. This cortical implementation of the 'von Mises' algorithm explains many otherwise disconnected observations ranging from neural tuning curves in monkeys to looking times in neonates and cortical numerotopy in adults. The theory clarifies the origin of Weber-Fechner's Law and yields a novel and empirically validated prediction of multi-peak number neurons. Random matrices constitute a novel mechanism for the emergence of brain states coding for quantity.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Model-based estimation of the frequency tuning of the inner-hair-cell stereocilia from neural tuning curves.

This study proposes that the frequency tuning of the inner-hair-cell (IHC) stereocilia in the intact organ of Corti can be derived from the responses of the auditory fibers (AFs) using computational tools. The frequency-dependent relationship between the AF threshold and the amplitude of the stereocilia vibration is estimated using a model of the IHC-mediated mechanical to neural transduction. Depending on the response properties of the considered AF, the amplitude of stereocilia deflection required to drive the simulated AF above threshold is 1.4 to 9.2 dB smaller at low frequencies (≤500 Hz) than at high frequencies (≥4 kHz). The estimated frequency-dependent relationship between ciliary deflection and neural threshold is employed to derive constant-stereocilia-deflection contours from previously published AF recordings from the chinchilla cochlea. This analysis shows that the transduction process partially accounts for the observed differences between the tuning of the basilar membrane and that of the AFs.

Neural Tuning to Numerosity Relates to Perceptual Tuning in 3-6-Year-Old Children.

Here we test for the existence of neural tuning to numerosity in the developing brain in the youngest sample of children tested with fMRI to date. Although previous research shows evidence of numerical distance effects in the intraparietal sulcus of the developing brain, those effects could be explained by patterns of neural activity that do not represent neural tuning to numerosity. These data provide the first robust evidence that from as early as 3-4 years of age there is developmental continuity in how the intraparietal sulcus represents the values of numerosities. Moreover, the study goes beyond previous research by examining the relation between neural tuning and perceptual tuning in children.

Limited range correlations, when modulated by firing rate, can substantially improve neural population coding

Neural activities are unreliable indicators of features of the external world [1], and an open question is how our nervous systems function robustly in the presence of this noise. One possibility arises from the observation that variability is often correlated between neurons [2], leading to the important theoretical question of when and how noise correlations affect neural population codes. Much work has investigated this issue, leading to impressive insights about how the relationship between the statistical structures of signals vs noise affects neural population coding. Despite this progress, an important issue has been largely overlooked by the field: that of firing-rate-dependent correlations. Notably, the same pair of neurons can display different noise correlations in response to different stimuli; those correlations coefficients typically increase with increases in the neurons' firing rates [2,3]. In this paper, we investigate the role of so-called limited range correlations on population codes, either in the presence, or the absence of rate-modulation of the correlation coefficients. Limited-range correlations are frequently-observed population-wide correlation structures in which cells with similar tuning curves have positive noise correlations, and the correlations decrease with decreasing tuning curve similarity [2,4]. These patterns of noise correlation are typically harmful to population coding [5] (yielding worse population coding performance than would be obtained with the same tuning curves and noise variances for all cells, but no noise correlations); these effects are somewhat dependent on the degree of heterogeneity in the population's tuning curves [6,7]. Experimentally reported noise correlations are usually averaged over stimuli, thereby masking any stimulus dependence. Herein, we will demonstrate that, when correlation coefficients increase with the product of mean neural firing rates (as in [2,3]), the stimulus-averaged correlation coefficients will display limited-range structure. When the rate dependence of these correlations is ignored, those correlations appear to be quite harmful to the population code, in accordance with previous theoretical work [5]. Surprisingly, when the rate dependence is taken into account, the correlations can yield much better population coding than would be obtained in the presence of uncorrelated noise. These effects persist for either homogeneous or heterogeneous sets of neural tuning curves. One prior study [8] also found that rate-dependent correlations can have very different impacts on population codes than can rate-independent ones, but did not make the connection between firing-rate-dependent correlations and the frequently observed limited-range correlation structure. Overall, our results emphasize that, for understanding the impact of limited-range correlations on neural population coding, the firing-rate dependence of those correlations is a potentially important consideration. Thus, it is important to report not only stimulus-averaged correlation coefficients, but also the relationship between those correlation coefficients and the neural firing rates.

Topology-defined units in numerosity perception

What is a number? The number sense hypothesis suggests that numerosity is "a primary visual property" like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.

Optimum neural tuning curves for information efficiency with rate coding and finite-time window.

An important question for neural encoding is what kind of neural systems can convey more information with less energy within a finite time coding window. This paper first proposes a finite-time neural encoding system, where the neurons in the system respond to a stimulus by a sequence of spikes that is assumed to be Poisson process and the external stimuli obey normal distribution. A method for calculating the mutual information of the finite-time neural encoding system is proposed and the definition of information efficiency is introduced. The values of the mutual information and the information efficiency obtained by using Logistic function are compared with those obtained by using other functions and it is found that Logistic function is the best one. It is further found that the parameter representing the steepness of the Logistic function has close relationship with full entropy, and that the parameter representing the translation of the function associates with the energy consumption and noise entropy tightly. The optimum parameter combinations for Logistic function to maximize the information efficiency are calculated when the stimuli and the properties of the encoding system are varied respectively. Some explanations for the results are given. The model and the method we proposed could be useful to study neural encoding system, and the optimum neural tuning curves obtained in this paper might exhibit some characteristics of a real neural system.

When does recurrent connectivity improve neural population coding?

Neural systems contain many cells, and an important problem is to understand if and how those neurons work together to form a functioning system. In sensory neuroscience -- which is our focus -- this function is to encode information about a stimulus so that it can be transmitted to other brain areas. An experimentally accessible, and hence popular, way to assess collective behavior is to measure the trial-to-trial covariability in the responses of multiple neurons over repeats of the same stimulus, called noise correlations. Repeat presentations of the same stimulus yield different responses on each trial, and that variability is typically correlated across different neurons [1,2] (although, see [3] for one counterexample). In the past two decades, the impact of these noise correlations on population coding has generated great interest. While nothing can add more information about the stimulus than was contained in the inputs (the data processing inequality), noise correlations determine the extent to which noise in the neural system degrades the amount of information that a neural population conveys about a stimulus (for example, see [4-6]). There are two main ways in which these noise correlations can be generated: the cells may receive common (noisy) input from (some of) the same upstream source(s), or the cells may be (recurrently) coupled to one another. At the same time, most work on noise correlations and population coding (with a few notable exceptions, including [7,8]) ignores their mechanistic origins. Interestingly, the few studies that have considered the mechanistic origins of noise correlations [7,8] have concluded that recurrent connectivity tends to hinder population coding, or has little overall effect: even though recurrent coupling can “sharpen” neural tuning curves, this advantage is more than offset by the fact that it generates harmful noise correlations. This begs the question of when -- if ever -- recurrent coupling can improve population coding, and of whether noise correlations with different mechanistic origins have different impacts on coding performance? To address these issues, we are investigating models in which groups of cells with, and without, recurrent coupling are driven by noisy inputs. The cells themselves are then noisy spike generators, and we are varying the fraction of shared inputs to the cells (which modifies the noise correlations due to common input), and the inter-neuronal connectivity, and computing the coding capacity of the resultant networks. Preliminary results indicate that, in some cases – similar to those discussed in [9] – recurrent coupling can combat the cell-intrinsic variabililty enough that the population’s coding performance is improved even though the noise correlations the are generated by that coupling do themselves hinder that coding performance. We are in the process of expanding and generalizing on this work. Support for this work came from NSF grants DMS-1122106, and CRCNS DMS-1208027 and a Burroughs- Wellcome Fund Career Award at the Scientific Interface to ESB.

A digital hardware design for real-time simulation of large neural-system models in physical settings

The organization of neural systems reflects the specific complexities of the physical environments in which they operate. In order to address this relationship more directly, there is increasing interest in testing real-time neural simulations that interface with the physical world. We describe a new simulation approach that allows us to run large, sophisticated neural models on low-power embedded commodity hardware such as field-programmable gate arrays (FPGAs). A custom digital circuit was designed to approximate the collective outputs of populations of neurons that have correlated activity. These populations are taken to represent physical quantities in their spike rates. Information processing (e.g. function approximation) is taken to be determined by synaptic weights. This design is based on the Neural Engineering Framework (NEF), which bridges the gap between neural activity and higher-level behaviour [1,2]. Populations are grouped together on hardware execution components, which we call “population units”, that perform time-multiplexing in order to simulate 1024 populations per timestep. The population unit represents each population as a weighted sum of principal components of the neural tuning curves summed with a model of the associated high-frequency spike-related fluctuations. These principal components span the functions that weighted sums of spikes can approximate without being dominated by spike-related noise. Populations running on the same population unit use the same principal components, which saves memory and improves the speed of the simulation. Clustering is performed prior to simulation, which groups together populations which can be accurately represented by shared principal components. The hardware does not need to be customized or regenerated in order to simulate different networks. It can be programmed with a network description generated by a compiler that operates as a backend to the Nengo simulator. (Nengo was used to run the Spaun model [2].) The design was implemented on an FPGA and was able to run simulations of up to 45 thousand populations of neurons (a realistic surrogate model of about 1-5 million point neurons) in real-time at 12-bit accuracy on a 1 millisecond timestep. Input and output to the hardware is over Gigabit Ethernet and can be collected from a PC running Nengo for simulation control and visualization. This implementation allows real-time approximate simulation of about the same scale as the largest real-time GPU simulations in Nengo, but using much less power. Furthermore, the FPGA chip is suitable for embedded applications such as mobile robots, cameras, etc. This work greatly facilitates simulation of an essential feature of neural systems, their embodiment and interaction with the physical world.

A neural measure of interaural correlation based on variance in spike count

Interaural Correlation (IAC) is related to variance in Interaural Time Difference (ITD) and Interaural Level Difference (ILD). While normalized IAC can account for behavioral performance in discrimination tasks, so can models directly employing this variance as a cue. Attempts at identifying a neural correlate of IAC discrimination have focused on changes in mean spike count, typically at the peaks of ITD tuning curves. We propose that IAC discrimination relies on variance in spike rates on the slope of neural tuning curves (for ITD). We developed a physiologically based hemispheric-balance model of IAC, where fluctuations in the ratio of activity between left- and right-brain serve as the detection cue to a reduction in IAC from unity, a ratio that is stimulus power invariant. Adjusting model parameters, we find that two orders of magnitude less activity is required in the variance-based, compared with mean spike-rate based, model, in order to achieve the same performance. This adjustment also revealed a necessary neural time integration of 10 ms, which is comparable with physiological estimates. The model was tested by recording neural responses from the midbrain of anesthetized guinea pigs to noise stimuli of various IAC.

The Sign Rule and Beyond: Boundary Effects, Flexibility, and Noise Correlations in Neural Population Codes

Over repeat presentations of the same stimulus, sensory neurons show variable responses. This “noise” is typically correlated between pairs of cells, and a question with rich history in neuroscience is how these noise correlations impact the population's ability to encode the stimulus. Here, we consider a very general setting for population coding, investigating how information varies as a function of noise correlations, with all other aspects of the problem – neural tuning curves, etc. – held fixed. This work yields unifying insights into the role of noise correlations. These are summarized in the form of theorems, and illustrated with numerical examples involving neurons with diverse tuning curves. Our main contributions are as follows. (1) We generalize previous results to prove a sign rule (SR) — if noise correlations between pairs of neurons have opposite signs vs. their signal correlations, then coding performance will improve compared to the independent case. This holds for three different metrics of coding performance, and for arbitrary tuning curves and levels of heterogeneity. This generality is true for our other results as well. (2) As also pointed out in the literature, the SR does not provide a necessary condition for good coding. We show that a diverse set of correlation structures can improve coding. Many of these violate the SR, as do experimentally observed correlations. There is structure to this diversity: we prove that the optimal correlation structures must lie on boundaries of the possible set of noise correlations. (3) We provide a novel set of necessary and sufficient conditions, under which the coding performance (in the presence of noise) will be as good as it would be if there were no noise present at all.

Neural computation with efficient population codes

Brain function depends on populations of neurons that perform computations on perceptually and behaviorally relevant variables. One of the main goals of neuroscience is to understand how the responses of populations of neurons and the connectivity patterns between groups of neurons allow brains to perform a wide range of neural computations. The Neural Engineering Framework (NEF) is a promising approach to designing neural models that perform many neural computations [1,2]. The central thesis behind the NEF is that populations of neurons represent, and perform computations on, low-dimensional time-dependent variables. By characterizing how neurons in a population encode a variable, and how the variable can be decoded from the distributed representation, functioning neural circuits are constructed that allow a comparison with experimental data at a range of levels from single neuron responses to connectivity patterns to perceptual and behavioral performance. The particular models that result from this approach depend on how the neural encoding and decoding processes are characterized. This is where emerging principles of neural computation constrain models of neural encoding and decoding. We describe how efficient codes are used to design neural circuit models that perform a wide variety of computations. The fundamental characteristic of the efficient code is that the neural representation is adapted to the statistics of the environment. Here, we take an efficient code to be one where the preferred stimuli are drawn from the prior distribution and the neural tuning curves are proportional to the likelihood function [3-6]. We show that in an efficient code, a general center-of-mass decoder can extract Bayesian estimates of encoded variables or functions of encoded variables, which allows for the construction of networks that perform many computations [3,4]. Networks constructed using the method we describe have several nice functional properties and match many experimental observations. First, neural tuning properties match the statistics of the variables they process. Second, in this framework, normalization is an essential computation at each stage of processing. This is consistent with normalization being described as a canonical neural computation [7]. Also, computation in networks of neurons using efficient coding is robust to neuronal loss and uses local connection rules. Finally, the networks we describe are flexible and can incorporate changes to environmental statistics or goals using gain modulation or changes in tuning curve widths. The overall result is to show the importance of several emerging principles of neural computation in an already successful modeling framework.

Scaling of Perceptual Errors Can Predict the Shape of Neural Tuning Curves

Weber's law, first characterized in the 19th century, states that errors estimating the magnitude of perceptual stimuli scale linearly with stimulus intensity. This linear relationship is found in most sensory modalities, generalizes to temporal interval estimation, and even applies to some abstract variables. Despite its generality and long experimental history, the neural basis of Weber's law remains unknown. This work presents a simple theory explaining the conditions under which Weber's law can result from neural variability and predicts that the tuning curves of neural populations which adhere to Weber's law will have a log-power form with parameters that depend on spike-count statistics. The prevalence of Weber's law suggests that it might be optimal in some sense. We examine this possibility, using variational calculus, and show that Weber's law is optimal only when observed real-world variables exhibit power-law statistics with a specific exponent. Our theory explains how physiology gives rise to the behaviorally characterized Weber's law and may represent a general governing principle relating perception to neural activity.

Top-Down Modulation of Lateral Interactions in Visual Cortex

The primary visual cortex (V1) changes its computation according to the perceptual task being performed. We propose that this cognitive modulation results from gating of V1 intrinsic connections. To test this idea, using behavioral paradigms that engage top-down modulation of V1 contextual interactions, we recorded from chronically implanted electrode arrays in macaques. We observed task-dependent changes in interactions between V1 sites measured both by correlation between spike trains and by coherence between local field potentials (LFP-LFP coherence). The direction of the changes in aggregate activity, as measured by LFPs, depended on perceptual strategy: perceptual grouping increased LFP coherence between sites crucial for the task, whereas perceptual segregation lowered the LFP coherence. Using spiking activity as a measure, we found that the behaviorally driven changes in correlation structure between neurons dramatically increased the stimulus-related information that they convey; this additional increase in encoded information at the level of neuronal ensembles equals that obtained from task-driven reconfigurations of neural tuning curves. The improvements in information encoding were strongest for stimuli with greatest discrimination difficulty.

Binaural displays based on interaural phase compensation.

The asymmetry of many neural tuning curves to the interaural time delay (ITD) of wideband stimuli around their maximum implies that such external temporal differences are compensated by some internal phase‐delay mechanism rather than a system of axonal time delays. An internal phase delay explains why the ITD that generates maximum neural activity (best delay) is generally less than a half period of the tuning curve’s modulation (“π‐limit”). The interaural phase circle (IPD circle) and the smoothed interaural phase spectrum (SIPS) are new binaural displays, which are based on these physiological findings. The IPD‐circle displays interaural phase difference (IPD) and interaural phase coherence (IPC) of single frequency channels in a polar diagram as vector angle and magnitude, respectively. Its two‐dimensionality facilitates the study of IPD and IPC fluctuations. The SIPS shows an activation matrix of binaural coincidence detectors across their best frequencies and internal phase delays. It serves similarly to the traditional cross‐correlogram as a display of interaural temporal differences across frequency. Its computation is based on the interaural phase spectrum, which is smoothed in frequency‐dimension by the auditory filter shapes. These displays describe neural ITD tuning curves with their asymmetry, and could provide a basis for future physiologically orientated models.