Sustained Firing of Model Central Auditory Neurons Yields a Discriminative Spectro-temporal Representation for Natural Sounds
[['It is widely believed that sensory representations are optimized to process the stimuli to which they are exposed in natural environments [1] .', 'Of particular interest is understanding the computational principles that underlie the generation of observed neural firing patterns .', 'A popular hypothesis explored in recent years assumes that neural populations optimize a sparse code .', 'This means that at any given time , only a small subset of a neural population fires to encode a given stimulus [2] .', 'Such a representation is attractive for reasons of coding efficiency ( see , e . g . , [3] ) and conservation of physiological resources [4] .', 'The sparse coding hypothesis has enjoyed particular success in studies of vision ( e . g . , [5] , [6] ) , and has also been supported more recently by both neurophysiological [7] , [8] and computational studies [9]–[11] of the auditory system .', 'However , it has also been observed that some central auditory neurons , when driven by their preferred stimuli , exhibit sustained firing rates .', 'Measuring from auditory thalamus and primary auditory cortex , Wang et al . observed that sustained responses were not simply phase-locked to the fast dynamics of the stimulus , suggesting that this rate-based code represented a meaningful , non-isomorphic transformation of the stimulus [12] , [13] .', 'Indeed , such a code is particularly important for audition since it directly addresses the issue of how to indicate the continued presence of a sound in a complex acoustic environment .', 'Results from Petkov et al . have also illustrated how sustained responses play a role in auditory scene analysis , forming part of the neural basis for the perceptual restoration of foreground sounds against a cluttered background [14] .', 'Moreover , Wang has argued that a rate-based representation is critical for matching fast temporal modulations present in natural sounds to slower rates found in higher cortical areas [15] .', 'Slower dynamics in acoustic signals are believed to be the main carrier of information in speech and music [16]; are commensurate with temporal dynamics of stream formation and auditory grouping [17]; and may play an important role in multi-modal sensory integration [15] .', 'Related computational studies in vision have suggested how this principle may underlie the shapes of simple and complex cell receptive fields in primary visual cortex [18] , [19] .', 'Importantly , a sustained firing rate , i . e . , one that is persistent and therefore slowly changing over time , is related to slow feature analysis , a well-known method for extracting invariances from sensory signals [20] ( see Discussion ) .', 'To the best of our knowledge , however , there are no computational studies that explicitly consider the implications of a sustained firing-based code in central auditory areas .', 'At first glance , the two coding schemes are seemingly at odds: on the one hand a sparse code seeks to minimize the activity of a neural population whereas a sustained firing-based code requires that neural responses persist over time but still form an efficient representation of the stimulus .', "However , it appears that central auditory responses can strike a balance between the two strategies , with a large , transient population response at the onset of a sound , and a sparse subset of preferentially driven neurons exhibiting a strong , sustained response throughout the sound's duration [15] , [21] .", 'This picture suggests a mechanism for detecting and tracking target sounds in noisy acoustic environments and for generating a persistent signal that facilitates a stable perceptual representation .', 'From a computational perspective , a better understanding of these mechanisms can inform models of auditory scene analysis as well as signal processing schemes for hearing prosthetics and automated sound processing systems .', 'A general computational approach for exploring the effects of particular coding strategies in sensory systems is based on optimizing a statistical objective criterion that quantifies the principle governing the transformation between stimulus and internal representation .', 'Upon convergence , one then compares the emergent representation to known properties of the sensory system being studied [1] .', 'Here , we apply this framework to explore how optimizing a sustained firing criterion influences the shapes of model auditory spectro-temporal receptive fields ( STRFs ) when processing natural sounds , and we compare the emergent ensembles to those obtained by optimizing a sparse coding objective .', 'STRFs describe the linear mapping between a spectro-temporal stimulus and an instantaneous firing rate [22] , and have proven useful not only for describing basic processing aspects of auditory neurons [23] , [24] , but also for shedding light on the nature of task-driven plasticity [25] .', 'Figure 1 illustrates how a spectro-temporal stimulus is mapped to a set of instantaneous neural firing rates , whose ensemble response according to a desired coding strategy directly shapes the mapping .', 'In this paper , we show how this framework allows us to not only explore how the timescales of natural sounds are captured by and reflected in an emergent sensory representation , but reveal key similarities between choice of a sustained versus sparse code .', 'Moreover , we demonstrate how a sustained firing-based code suggests a mechanism for an emergent discriminative representation for ensembles of natural stimuli .'], ['We optimized both the sustained objective and sparsity objective using an ensemble of natural stimuli comprising speech , animal vocalizations , and ambient outdoor sounds .', 'Each ensemble of filters was initialized at random using zero-mean , unit variance Gaussian noise , and each STRF covered from 0–250 ms in time and 62 . 5–4000 Hz along the tonotopic axis .', 'For the sustained objective , we considered a wide range of correlation intervals from very brief ( ) to very long ( ) .', 'Examples of emergent STRFs for are shown in Figure 2A .', 'For the spectro-temporal patches shown , red and blue colors indicate that the presence of energy in a particular spectro-temporal region yields excitatory and inhibitory responses , respectively .', 'We observe a variety of STRFs that are highly localized , sensitive to narrowband spectral and temporal events , oriented , and some that are seemingly noise-like and not convergent to any particularly interesting shape .', 'Importantly , such observations about these basic STRF classes align with those made in a number of previous physiological studies ( see , e . g . , [23] , [24] , [27] ) .', 'Moreover , coverage of the STRFs appears to span the full time-frequency space .', 'These results suggest that the sustained firing objective may underlie part of the coding strategy used by central auditory neurons .', 'Shown in Figure 2B are examples of emergent STRFs obtained by optimizing the sparsity objective .', 'Indeed , this particular objective yields STRFs that are highly localized and sparsely distributed , with sensitivity to bandlimited spectral and temporal events .', 'While both objective criteria yield noisy STRFs , it is clear that the sparse ensemble is much more noisy , with a less extensive coverage of the basic sound classes as observed with the sustained ensemble .', 'Since the information-bearing components of natural sounds vary concurrently across multiple timescales , it was expected that the structure of STRFs learned under the sustained objective would vary with the correlation interval .', 'Indeed , inspection of the sustained ensembles for a range of suggested the presence of a number of latent classes whose membership varied smoothly from short to long correlation intervals .', 'To quantify variations in population diversity over ecologically relevant timescales , we performed unsupervised clustering of the emergent STRFs and studied how class membership changed with objective function and correlation interval .', 'We pooled STRFs from the sparse ensemble and from the sustained ensembles for 10 , 25 , 50 , 125 , 250 , 500 , 1000 , and 2000 ms , yielding a total of 3600 STRFs .', 'We then applied normalized spectral clustering to discover latent classes among the pooled STRFs .', 'In general , spectral clustering algorithms require an affinity matrix that specifies pairwise similarities between the objects being clustered .', 'Viewing this affinity matrix as an undirected graph , spectral clustering finds a partition of the graph into groups whose elements have common similarity with one another .', 'A natural measure of similarity between STRFs can be derived from the two-dimensional cross-correlation between pairs of spectro-temporal patches .', 'Such a measure is similar to that considered by Woolley et al . [28] and is desirable since it does not depend on subjective choice of spectro-temporal features to use for clustering .', 'In this work , we defined the measure of similarity between pairs of STRFs as the absolute value of the maximum value of the two-dimensional cross-correlation matrix; we used absolute value since we wished to group similar STRFs regardless of whether they were excitatory or inhibitory .', 'Furthermore , as the STRFs tended to be distributed with a variety of phases in the input space , we considered cross-correlations for arbitrary time-frequency shifts ( see Methods for details ) .', 'Results obtained using normalized spectral clustering of the emergent ensembles into nine classes are shown in Figure 3 .', 'In the center panel of the figure , a stacked bar chart illustrates the the percentage of STRFs at a particular assigned to one of nine classes .', 'Different segment colors correspond to each of the nine classes , and segment width is proportional to the number of STRFs assigned to that class .', 'Surrounding the bar chart are examples from six classes that best illustrate how diversity varies with , namely noisy , localized , spectral , complex , temporal , and directional classes .', 'These labels are qualitative descriptors of each class and not quantitative assessments of the time-frequency characteristics of each category .', 'Inspection of the cluster groupings reveal rich structural variations over a wide range of correlation intervals .', 'In particular , the STRFs labeled according to the noisy class are found to dominate the sparse ensemble , with a large presence in the sustained ensemble for .', 'Membership in this class drops for between 10 and 125 ms , and begins to increase at 125 ms . We also observe that short correlation intervals ( 10 , 25 , and 50 ms ) have a large concentration of localized STRFs , with membership dropping with increasing .', 'While the temporal class holds relatively steady across the sustained ensembles , we find that membership in the directional , complex , and spectral classes varied smoothly across .', 'In general , we find that ensemble diversity is maximized for ( max . entropy of 3 . 08 bits ) , but the overall trends suggest rich ensemble structure between 10 and 250 ms , which is notably in the range of the timescales of natural sounds [29] , .', 'This is further supported by the increasing presence of noisy STRFs for large correlation intervals ( 1000 and 2000 ms ) .', 'In addition to studying structural variations in the shapes of the emergent STRFs , it is also of interest to examine the structure of the STRF outputs in response to natural sounds .', 'In particular , we sought to address the extent to which enforcing sustained responses does indeed yield responses that persist over time .', 'We defined the neuron to be significantly “active” when its firing rate exceeded 1 standard deviation over time .', "While this is not meant to be a precise measure of a neuron's activation ( since , for instance , the firing rate is not used to modulate a Poisson spike generation process ) , such a measure nevertheless quantifies and characterizes a strong versus weak ensemble response to natural stimuli .", 'Shown in Figure 4A are the distribution of activation times for individual neurons for ensembles of 10 and 125 ms in response to a held-out set of natural stimuli .', 'The neurons are shown sorted according to decreasing median activation time , and the interquartile ranges of activation time are indicated by the shaded regions .', 'We observed that the most diversity in median activation times across ensembles occurred in approximately the top 10% of the most persistent neurons .', 'To summarize these observations , we considered the distribution of median activation times of the top 10% of neurons with most persistent responses ( i . e . , the top 40 neurons ) ; these distributions are illustrated as boxplots in Figure 4B .', 'As noted previously with the clustering results , shorter values favor mostly localized and noisy STRFs and consequently it was expected that activations would be brief .', 'Interestingly , however , we observe that with increasing , median activations peak between 50 and 500 ms and fall off for large despite the STRFs being optimized to promote sustained responses over long intervals .', 'This overall trend aligns with the previous clustering results that demonstrate how population diversity is maximized over intervals corresponding to timescales that predominate natural stimuli .', 'The STRFs corresponding to the top 10% most persistent responses for are shown in Supplementary Figure 1 , and we find that they generally have a spectral tuning , but are fairly narrowband and localized .', 'Additionally , we considered the responses of the top 40 most persistent responses obtained using the sparsity objective function; the distribution of median activations is in the first column of Figure 4B .', 'We find that the sparse ensemble yields responses most similar to those for short .', 'How do the emergent STRFs learned under the sustained firing objective compare to those observed in physiological studies ?', 'Broadly speaking , we find that the emergent STRFs share many of the trends with biological receptive fields typically observed in animal models .', 'We explored this issue by comparing our model ensembles with a set of 1586 STRFs recorded from awake , non-behaving ferret primary auditory cortex using TORC [31] and speech stimuli [27] , [32] ( see Methods for more details ) .', 'Where applicable , we also compared our results with reported results from anesthetized ferrets by Depireux et al . [23] and cats by Miller et al . [24] in the literature .', 'Illustrative examples of the types of STRFs found in the neural data are shown in Figure 5 .', 'In particular , we find neural STRFs that are qualitatively similar those found in the localized , complex , noisy , and directional clusters shown earlier in Figure 3 .', 'Because the temporal and spectral sampling rates used in our model are higher than those used in the physiological data , we did not find good matches with the temporal and spectral classes .', 'To visualize the overlap between the spectro-temporal modulation coverage of the neural and model STRFs , we used the ensemble modulation transfer function ( eMTF ) .', 'The eMTF is derived by averaging the magnitude of the 2D Fourier Transform of each neuron in a given ensemble , and jointly characterizes modulations in time ( rate , in Hz ) and in frequency ( scale , in cyc/oct ) .', 'We first applied normalized spectral clustering to the neural STRFs to obtain nine clusters .', 'Next , we computed the eMTF for each cluster , extracted isoline contours at the 65% level , and overlaid these curves on the eMTF of the model STRFs for .', 'These results are shown in Figure 6 and illustrate the overlap between the model and neural data , particularly at the “edges” of the neural STRF modulations .', 'While the overlap is not complete , it is clear that the modulation spectra of each ensemble are not disjoint .', 'Moreover , the model eMTF suggests a general ensemble sensitivity to relatively fast modulations; this point is explored further in a later section ( “Emergent STRFs capture spectro-temporal modulation statistics of stimulus” ) .', 'To better characterize the relationship between the neural and model data , we employed a statistical comparison of the distribution of the two datasets .', 'If the models truly generated STRFs similar to those in physiological studies , then one might expect a nearest-neighbor ( NN ) similarity distribution akin to one derived from the neural ensemble we considered .', 'We computed the symmetric KL-divergence between each of the model and within-physiology NN similarity distributions ( shown in Supplemental Figure 2 ) .', 'We found that the sustained-response ( presented here ) and sustained-shape ( presented later in this paper ) distributions had KL divergences of 0 . 80 and 0 . 85 , respectively , whereas the sparse distribution had a KL distance of 1 . 05 .', 'KL typically measures the expected number of bits required to code samples from one distribution using codes from the other .', 'While these numbers are difficult to assess in absolute terms , they give a sense of how the different model optimizations and constraints compare to each other .', 'These numbers reveal that the sustained ensembles are similarly comparable to the physiology , whereas the sparse ensemble has a somewhat worse match .', 'Of course , caution must be taken with these numbers because the set of neural STRFs we analyzed represent only a subset of mappings that likely exist in central auditory areas .', 'Next , we measured a variety of parameters from the neural and model STRFs ( for ) that more fully characterized the extent of spectro-temporal coverage and modulation sensitivity of the ensembles ( see Methods ) , the results of which are summarized in Figure 7 .', 'Based on the distribution of directionality indices , shown in panel ( A ) , we observe that the model STRFs are largely symmetric , with the majority of neurons having no preference for upward or downward moving input stimuli ( mean0 ) .', 'As indicated by the tails of this distribution , however , a subset of neurons have a strong directional preference .', 'This agrees with the neural STRFs , and similar observations have been made in MGB and primary auditory cortex of cats by Miller et al . , as well as in measurements by Depireux et al . from primary auditory cortex of ferrets .', 'Furthermore , panel ( B ) illustrates that a large number of model STRFs are fairly separable , with a peak in the separability index ( SPI ) distribution around 0 . 10 and an average value of 0 . 26 .', 'This trend aligns with values reported in the literature by Depireux et al . in measurements from ferret auditory cortex ( mean of approx . 0 . 25 ) .', 'However , it is worth noting that this low level of separability is not uniformly reported across physiological studies of receptive field of mammalian auditory cortex .', 'For instance , the physiological data analyzed in the current study ( examples of which are shown in Figure 5 ) do yield a higher average SPI ( mean\u200a=\u200a0 . 37 ) .', 'The temporal modulation statistics of the model STRFs , as quantified by best rate ( BR ) , also align generally with results reported from mammalian thalamus and cortex .', 'In panel ( C ) we observe a broad , bandpass distribution of best rates , with an average of 23 . 9 Hz .', 'Reported physiological results from Miller et al . show similarly broad ranges of temporal tuning with preferences around 16 Hz and 30 Hz range for cortex and thalamus , respectively .', 'The neural STRFs we analyzed show a somewhat slower tuning , with an average BR of 9 . 5 Hz .', 'Furthermore , in panel ( D ) , we computed the normalized average rate profile from the model STRFs .', 'We observe a peak at 7 . 8 Hz , with an upper 6-dB cutoff of 34 . 4 Hz .', 'Here we find a close overlap with the rate profile computed from the neural STRFs as well as with average profile results as reported by Miller et al . ( peak at 12 . 8 Hz; upper 6-dB cutoff at 37 . 4 Hz ) .', 'The spectral modulation statistics of the model STRFs , as quantified by best scale , are generally faster than those reported from studies of thalamic and cortical nuclei .', 'The distribution of best scales shown in panel ( E ) is bandpass with a wide range of slow to fast spectral coverage , with an average tuning of 1 . 40 cyc/oct .', 'The neural STRFs , in contrast , are tuned to much slower scales ( mean\u200a=\u200a0 . 47 cyc/oct ) .', 'Similarly , results from Miller et al . in MGB indicate a generally slower tuning ( 0 . 58 cyc/oct ) , whereas measurements from cortical neurons , while having a similarly wide range of tunings as with the model , indicate a slower average value of 0 . 46 cyc/oct and an upper cutoff of approx .', '2 cyc/oct .', 'Finally , the ensemble average scale profile , shown in panel ( F ) , is bandpass and exhibits a peak at 0 . 7 cyc/oct with an upper 6-dB cutoff of 2 . 9 cyc/oct .', 'The neural STRFs , however , are much slower with peak at 0 . 2 cyc/oct and an upper cutoff of 1 . 9 cyc/oct .', 'This is similar to observations from MGB by Miller et al . , where they reported that the ensemble average scale profile is generally low-pass , with average scale profile peaks and upper 6-dB cutoffs at 0 cyc/oct and 1 . 3 cyc/oct , respectively , with similar observations in cortex .', 'In summary , while we cannot map the emergent STRFs to any exact synapse , they nevertheless reflect the general processing characteristics of various stations along in the central auditory pathway .', 'There is good alignment with the neural STRFs and reported results in mammalian MGB and primary auditory cortex with respect to directional sensitivity and spectro-temporal separability .', 'The temporal modulation statistics of the emergent sustained STRFs appear to be most similar to those measured from thalamus and cortex .', 'Furthermore , the model STRFs are generally faster with regard to spectral modulations than those measured from thalamus and cortex .', 'To explore the relationship between STRFs optimized to promote sustained responses and those that explicitly maximize population sparsity , we compared the average responses of the sustained ensemble for with the sparse ensemble .', 'Specifically , we used the converged STRFs to analyze a held-out set of natural stimuli , computed a histogram of the population responses at each time , and computed the average histogram across the entire test input ( see Methods ) .', 'Since the sparse ensemble was optimized to yield a highly kurtotic firing rate distribution , it was of interest to examine the shape of the distribution when promoting sustained responses .', 'Results comparing the average histograms of sustained versus sparse responses is shown in Figure 8 , with log-probabilities shown on the vertical axis to emphasize differences between the tails of the distributions .', 'The main observation is that both the sustained and sparse ensembles have distributions that have long tails and are are highly peaked around a firing rate of zero .', 'For reference , we show the average histograms obtained by filtering the stimulus through the first 400 principal components of the stimulus ( see Supplemental Figure 3 ) as well as through a set of 400 random STRFs; a zero-mean , unit variance Gaussian distribution is also shown .', 'Therefore , despite promoting temporally persistent responses , the sustained responses yield a population response that is not altogether different from an ensemble that explicitly maximizes kurtosis .', 'Interestingly , this observation was also made by Berkes and Wiscott in the context of complex cell processing in primary visual cortex ( see Sec . 6 of [33] ) .', 'Finally , we sought to explore the consequences of relaxing the constraint that the responses be mutually uncorrelated .', 'Rather than directly constrain the responses , we considered constraints to the shapes of the model STRFs .', 'This was achieved by solvingi . e .', ', we require the STRFs to form an orthonormal basis .', 'So long as the stimuli are bounded , this set of constraints meets our requirements that ( 1 ) the output of the STRFs be bounded and ( 2 ) we minimize redundancy in the learned ensemble .', 'As before , the optimization is described in the Methods .', 'We consider an ensemble size of STRFs initialized at random .', 'Examples of shape-constrained STRFs that optimize the sustained objective function for are shown in Figure 9 .', 'Again , we observe STRFs that are bandpass , localized , oriented , and sensitive to a variety of spectral and temporal input .', 'However , there was an apparent difference between the speed of the spectro-temporal modulations and those from STRFs learned subject to the response constraints .', 'It is well known that natural sound ensembles are composed largely of slow spectro-temporal modulations [29] , [30] , [34] .', 'However , the emergent STRFs learned subject to response constraints appear to be tuned to relatively fast spectral and temporal modulations , whereas the STRFs learned subject to shape constraints appear to have a broader tuning .', 'To further examine how both sets of constraints jointly capture and are related to the spectro-temporal modulations observed in stimulus , we compared the average 2D modulation profile of the stimulus to the eMTFs derived from both sets of constraints .', 'An interesting view of how the emergent STRFs capture the spectro-temporal modulations of the stimulus is illustrated in Figure 10 for .', 'Shown is the average 2D modulation profile of the stimulus overlaid with a single isoline contour ( at the 65% level ) of the eMTFs learned subject to response ( thick red lines ) and shape constraints ( thick black lines ) .', 'We also show the constellation of BR versus BS for each ensemble ( indicated by ‘’ and ‘’ for response and shape constraints , respectively ) .', 'As implied by the contours , the response constraints yield STRFs that follow the spectro-temporal “edge” of the stimulus , while the shape constraints explicitly capture most of the “slowness” of the stimulus .', 'As mentioned previously , the response constraints effectively force the temporal response of the sustained ensemble to be sparse , which consequently results in highly selective STRFs that tend to be tuned to fast modulations .', 'Nevertheless , they implicitly capture the spectro-temporal extent of the stimulus .', 'Moreover , since the shape constraints effectively force the STRFs to form a basis that spans the input space , this results in neurons that explicitly capture the slow modulations of the stimulus .', 'Similar observations were made across the range of , and for each case it was clear that the spectro-temporal modulations of the stimulus are fully captured by the combination of both sets of constraints .'], ['The combination of shape and response constraints on the sustained objective function yield STRF ensembles that appear to jointly capture the full range of spectro-temporal modulations in the stimulus .', 'However , the distinct differences in MTF coverage illustrate the tradeoff between redundancy and efficiency in sensory representations .', 'In particular , the shape constraints yield STRFs that are somewhat akin to the first few principal components of the stimulus ( see Supplemental Figure 3 ) .', 'This is not surprising given that the objective function defines a notion of variance of linear projections , the component vectors of which are constrained to form an orthonormal basis .', 'However , since the responses are not strictly enforced to be uncorrelated , orthonormality imposed on the filter shapes does not necessarily reduce redundancy in the resulting neural responses .', 'In contrast , the response constraints yield STRFs that are highly selective to the input and are thus comparatively “fast” in the modulation domain .', 'This representation can be thought of as more efficient since at any given time only a few neurons have a large response .', 'However , while the shapes of individual STRFs fail to explicitly capture the slow spectro-temporal modulations predominant in natural sounds , it instead appears that the ensemble MTF of the response-constrained STRFs collectively forms a contour around the high-energy modulations of the stimulus that implicitly capture its spectro-temporal extent .', 'Is this contouring of the average modulation spectrum of natural sounds something performed by the auditory system ?', 'The neural STRFs we considered certainly had an eMTF that reflects a tuning to slower modulations near the MTF origin .', 'However , there is some evidence that the auditory system uses an “edge”-sensitive , discriminative modulation profile for analyzing sound .', 'Woolley et al . [36] , in an avian study , showed that the eMTF of neurons from Field L ( the avian A1 analog ) has a bandpass temporal modulation profile ( at low scales ) that facilitates a discriminative tuning of temporal modulations among classes of natural sounds .', 'Nagel and Doupe [37] have also shown examples of avian Field L STRFs that orient themselves near the spectro-temporal “edge” of the stimulus space .', 'Moreover , Rodriguez et al . [38] , in a study of mammalian IC neurons , showed that neural bandwidths can scale to better capture fast , but less frequently occurring , modulations .', 'In light of these observations , the modulation profiles observed from the sustained STRFs for both response and shape constraints are consistent with the notion that the auditory system makes an explicit effort to capture all modulations present in natural sounds: fast , feature-selective , and consequently discriminative modulations , as well as frequently occurring slow modulations .', 'The notion that sustained neural firings form part of the neural representation of sensory systems is not limited exclusively to the auditory modality .', 'In fact , the sustained firing objective considered in this paper is related to a broad class of sensory coding strategies referred to collectively under the temporal slowness hypothesis .', 'This concept proposes that the responses of sensory neurons reflect the time-course of the information-bearing components of the stimulus—which are often much slower with respect to the fast variations observed in the stimulus—and may therefore reflect invariant aspects of the sensory objects in the environment .', 'Examples of early neural network models exploring slowness as a learning principle were considered by Földiák [39] , Mitchison [40] , and Becker [41] .', 'More recently , a number of computational studies , particularly in vision , have established slowness as a general sensory coding strategy and have revealed relationships with a number of general machine learning techniques .', 'Here we outline the connections between the sustained firing criterion considered in this study and previous work .', 'Our definition of the sustained firing objective , , was adapted from a notion of temporal stability proposed by Hurri and Hyvärinen termed temporal response strength correlation ( TRSC ) [18] .', 'This study considered modeling of simple cells in primary visual cortex , and their objective function was defined as ( 3 ) for a single fixed .', 'By maximizing subject to the decorrelation constraints , they showed the emergence of spatial receptive fields similar to those observed in simple cells in primary visual cortex .', 'It is clear that the objective functions and are equivalent for a single time step , but the main difference between the two is that we sought to enforce temporal stability over a time interval , rather than between two distinct times and .', 'Interestingly , optimization of the TRSC objective was shown by Hyvärinen to yield a solution to the blind source separation problem [42] , suggesting perhaps that in the auditory domain , such a criterion may underlie separation of overlapping acoustic sources .', 'The sustained firing objective is also related to a well-known model of temporal slowness known as slow feature analysis ( SFA ) [20] .', 'The computational goal of SFA is to find a mapping of an input that extracts the slow , and presumably more invariant , information in the stimulus .', 'Briefly , for an input , linear SFA finds mappings that minimize ( 4 ) subject to , , and .', 'Note that the input is not necessarily the raw stimulus but could represent a non-linear expansion of the input , akin to applying a kernel function in a supp
['Introduction', 'Results', 'Discussion', 'Methods']
['The processing characteristics of neurons in the central auditory system are directly shaped by and reflect the statistics of natural acoustic environments , but the principles that govern the relationship between natural sound ensembles and observed responses in neurophysiological studies remain unclear .', 'In particular , accumulating evidence suggests the presence of a code based on sustained neural firing rates , where central auditory neurons exhibit strong , persistent responses to their preferred stimuli .', 'Such a strategy can indicate the presence of ongoing sounds , is involved in parsing complex auditory scenes , and may play a role in matching neural dynamics to varying time scales in acoustic signals .', 'In this paper , we describe a computational framework for exploring the influence of a code based on sustained firing rates on the shape of the spectro-temporal receptive field ( STRF ) , a linear kernel that maps a spectro-temporal acoustic stimulus to the instantaneous firing rate of a central auditory neuron .', 'We demonstrate the emergence of richly structured STRFs that capture the structure of natural sounds over a wide range of timescales , and show how the emergent ensembles resemble those commonly reported in physiological studies .', 'Furthermore , we compare ensembles that optimize a sustained firing code with one that optimizes a sparse code , another widely considered coding strategy , and suggest how the resulting population responses are not mutually exclusive .', 'Finally , we demonstrate how the emergent ensembles contour the high-energy spectro-temporal modulations of natural sounds , forming a discriminative representation that captures the full range of modulation statistics that characterize natural sound ensembles .', 'These findings have direct implications for our understanding of how sensory systems encode the informative components of natural stimuli and potentially facilitate multi-sensory integration .']
['We explore a fundamental question with regard to the representation of sound in the auditory system , namely: what are the coding strategies that underlie observed neurophysiological responses in central auditory areas ?', 'There has been debate in recent years as to whether neural ensembles explicitly minimize their propensity to fire ( the so-called sparse coding hypothesis ) or whether neurons exhibit strong , sustained firing rates when processing their preferred stimuli .', 'Using computational modeling , we directly confront issues raised in this debate , and our results suggest that not only does a sustained firing strategy yield a sparse representation of sound , but the principle yields emergent neural ensembles that capture the rich structural variations present in natural stimuli .', 'In particular , spectro-temporal receptive fields ( STRFs ) have been widely used to characterize the processing mechanisms of central auditory neurons and have revealed much about the nature of sound processing in central auditory areas .', 'In our paper , we demonstrate how neurons that maximize a sustained firing objective yield STRFs akin to those commonly measured in physiological studies , capturing a wide range of aspects of natural sounds over a variety of timescales , suggesting that such a coding strategy underlies observed neural responses .']
['computational neuroscience', 'audio signal processing', 'signal processing', 'biology', 'computational biology', 'sensory systems', 'neuroscience', 'engineering', 'coding mechanisms']