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• W2518631572 abstract "•Correlated trial-by-trial variability can impair the reliability of neural codes•Multidimensional codes of stimulus orientation are highly stable over weeks•Multidimensional correlations restrict variability to non-coding directions•Up to 50% of single-trial, single-neuron “noise” is predictable with correlations Sensory neurons are often tuned to particular stimulus features, but their responses to repeated presentation of the same stimulus can vary over subsequent trials. This presents a problem for understanding the functioning of the brain, because downstream neuronal populations ought to construct accurate stimulus representations, even upon singular exposure. To study how trial-by-trial fluctuations (i.e., noise) in activity influence cortical representations of sensory input, we performed chronic calcium imaging of GCaMP6-expressing populations in mouse V1. We observed that high-dimensional response correlations, i.e., dependencies in activation strength among multiple neurons, can be used to predict single-trial, single-neuron noise. These multidimensional correlations are structured such that variability in the response of single neurons is relatively harmless to population representations of visual stimuli. We propose that multidimensional coding may represent a canonical principle of cortical circuits, explaining why the apparent noisiness of neuronal responses is compatible with accurate neural representations of stimulus features. Sensory neurons are often tuned to particular stimulus features, but their responses to repeated presentation of the same stimulus can vary over subsequent trials. This presents a problem for understanding the functioning of the brain, because downstream neuronal populations ought to construct accurate stimulus representations, even upon singular exposure. To study how trial-by-trial fluctuations (i.e., noise) in activity influence cortical representations of sensory input, we performed chronic calcium imaging of GCaMP6-expressing populations in mouse V1. We observed that high-dimensional response correlations, i.e., dependencies in activation strength among multiple neurons, can be used to predict single-trial, single-neuron noise. These multidimensional correlations are structured such that variability in the response of single neurons is relatively harmless to population representations of visual stimuli. We propose that multidimensional coding may represent a canonical principle of cortical circuits, explaining why the apparent noisiness of neuronal responses is compatible with accurate neural representations of stimulus features. The presentation of stimulus features modulates the responses of single neurons in sensory cortex such that the outside world is represented in the activation pattern of neuronal populations. However, the activity of single neurons shows substantial variability in spike rate and timing across repeated presentations of the same stimulus (Faisal et al., 2008Faisal A.A. Selen L.P.J. Wolpert D.M. Noise in the nervous system.Nat. Rev. Neurosci. 2008; 9: 292-303Crossref PubMed Scopus (1702) Google Scholar). This variability is often called neural noise and poses a problem: how can animals react quickly and reliably to sensory input when the stimulus representation would already be noisy at the first stage of cortical processing? It has been proposed that neural circuits solve this problem by combining information from multiple neurons into a population code. If the variability of neuronal responses were independent, higher precision of stimulus representation would be achieved by combining the responses of more neurons (Beck et al., 2008Beck J.M. Ma W.J. Kiani R. Hanks T. Churchland A.K. Roitman J. Shadlen M.N. Latham P.E. Pouget A. Probabilistic population codes for Bayesian decision making.Neuron. 2008; 60: 1142-1152Abstract Full Text Full Text PDF PubMed Scopus (438) Google Scholar, Knill and Pouget, 2004Knill D.C. Pouget A. The Bayesian brain: the role of uncertainty in neural coding and computation.Trends Neurosci. 2004; 27: 712-719Abstract Full Text Full Text PDF PubMed Scopus (1392) Google Scholar, Ma et al., 2006Ma W.J. Beck J.M. Latham P.E. Pouget A. Bayesian inference with probabilistic population codes.Nat. Neurosci. 2006; 9: 1432-1438Crossref PubMed Scopus (961) Google Scholar). However, neurons are often correlated in the variability of their response to the same stimulus, which means that simple averaging is insufficient to achieve maximal precision (Averbeck and Lee, 2006Averbeck B.B. Lee D. Effects of noise correlations on information encoding and decoding.J. Neurophysiol. 2006; 95: 3633-3644Crossref PubMed Scopus (133) Google Scholar, Hansen et al., 2012Hansen B.J. Chelaru M.I. Dragoi V. Correlated variability in laminar cortical circuits.Neuron. 2012; 76: 590-602Abstract Full Text Full Text PDF PubMed Scopus (98) Google Scholar, Lee et al., 1998Lee D. Port N.L. Kruse W. Georgopoulos A.P. Variability and correlated noise in the discharge of neurons in motor and parietal areas of the primate cortex.J. Neurosci. 1998; 18: 1161-1170Crossref PubMed Google Scholar, Vinje and Gallant, 2000Vinje W.E. Gallant J.L. Sparse coding and decorrelation in primary visual cortex during natural vision.Science. 2000; 287: 1273-1276Crossref PubMed Scopus (908) Google Scholar). The interdependency between responses of pairs of neurons (i.e., noise correlations [NCs]) has been proposed to influence the amount of information that can be extracted from population codes in different ways, ranging from being beneficial to being mostly irrelevant or harmful (Averbeck et al., 2006Averbeck B.B. Latham P.E. Pouget A. Neural correlations, population coding and computation.Nat. Rev. Neurosci. 2006; 7: 358-366Crossref PubMed Scopus (1035) Google Scholar, Cafaro and Rieke, 2010Cafaro J. Rieke F. Noise correlations improve response fidelity and stimulus encoding.Nature. 2010; 468: 964-967Crossref PubMed Scopus (94) Google Scholar, Cohen and Kohn, 2011Cohen M.R. Kohn A. Measuring and interpreting neuronal correlations.Nat. Neurosci. 2011; 14: 811-819Crossref PubMed Scopus (621) Google Scholar, Cohen and Maunsell, 2009Cohen M.R. Maunsell J.H.R. Attention improves performance primarily by reducing interneuronal correlations.Nat. Neurosci. 2009; 12: 1594-1600Crossref PubMed Scopus (692) Google Scholar, Ecker et al., 2011Ecker A.S. Berens P. Tolias A.S. Bethge M. The effect of noise correlations in populations of diversely tuned neurons.J. Neurosci. 2011; 31: 14272-14283Crossref PubMed Scopus (156) Google Scholar, Fiscella et al., 2015Fiscella M. Franke F. Farrow K. Müller J. Roska B. da Silveira R.A. Hierlemann A. Visual coding with a population of direction-selective neurons.J. Neurophysiol. 2015; 114: 2485-2499Crossref PubMed Scopus (22) Google Scholar, Herrero et al., 2013Herrero J.L. Gieselmann M.A. Sanayei M. Thiele A. Attention-induced variance and noise correlation reduction in macaque V1 is mediated by NMDA receptors.Neuron. 2013; 78: 729-739Abstract Full Text Full Text PDF PubMed Scopus (105) Google Scholar, Montijn et al., 2014Montijn J.S. Vinck M. Pennartz C.M.A. Population coding in mouse visual cortex: response reliability and dissociability of stimulus tuning and noise correlation.Front. Comput. Neurosci. 2014; 8: 58Crossref PubMed Scopus (31) Google Scholar, Seriès et al., 2004Seriès P. Latham P.E. Pouget A. Tuning curve sharpening for orientation selectivity: coding efficiency and the impact of correlations.Nat. Neurosci. 2004; 7: 1129-1135Crossref PubMed Scopus (169) Google Scholar). Another aspect that complicates the study of NCs is that these correlations can be heterogeneous in their size and effects on population codes, depending on factors such as the nature of the presented stimulus features, correlations between neurons other than the pair being studied, and differences in general arousal state (Chelaru and Dragoi, 2016Chelaru M.I. Dragoi V. Negative correlations in visual cortical networks.Cereb. Cortex. 2016; 26: 246-256Crossref PubMed Scopus (17) Google Scholar, Ince et al., 2013Ince R.A.A. Panzeri S. Kayser C. Neural codes formed by small and temporally precise populations in auditory cortex.J. Neurosci. 2013; 33: 18277-18287Crossref PubMed Scopus (43) Google Scholar, Jazayeri and Movshon, 2006Jazayeri M. Movshon J.A. Optimal representation of sensory information by neural populations.Nat. Neurosci. 2006; 9: 690-696Crossref PubMed Scopus (1) Google Scholar, Miller et al., 2014Miller J.E. Ayzenshtat I. Carrillo-Reid L. Yuste R. Visual stimuli recruit intrinsically generated cortical ensembles.Proc. Natl. Acad. Sci. USA. 2014; 111: E4053-E4061Crossref PubMed Scopus (163) Google Scholar, Moreno-Bote et al., 2014Moreno-Bote R. Beck J. Kanitscheider I. Pitkow X. Latham P. Pouget A. Information-limiting correlations.Nat. Neurosci. 2014; 17: 1410-1417Crossref PubMed Scopus (284) Google Scholar, Pitkow et al., 2015Pitkow X. Liu S. Angelaki D.E. DeAngelis G.C. Pouget A. How can single sensory neurons predict behavior?.Neuron. 2015; 87: 411-423Abstract Full Text Full Text PDF PubMed Scopus (83) Google Scholar, Schölvinck et al., 2015Schölvinck M.L. Saleem A.B. Benucci A. Harris K.D. Carandini M. Cortical state determines global variability and correlations in visual cortex.J. Neurosci. 2015; 35: 170-178Crossref PubMed Scopus (140) Google Scholar). Therefore, one of the most relevant challenges in neurophysiology is explaining how accurate sensory representations can be generated by neuronal populations in the face of instantaneous single-neuron response fluctuations. Pairwise dependencies might be important for neuronal populations that represent sensory information, but it has been hypothesized that the underlying structure of neural responses may be multidimensional—i.e., dependent on interactions among more than two neurons (Franke et al., 2016Franke F. Fiscella M. Sevelev M. Roska B. Hierlemann A. da Silveira R.A. Structures of neural correlation and how they favor coding.Neuron. 2016; 89: 409-422Abstract Full Text Full Text PDF PubMed Scopus (70) Google Scholar, Kanitscheider et al., 2015Kanitscheider I. Coen-Cagli R. Pouget A. Origin of information-limiting noise correlations.Proc. Natl. Acad. Sci. USA. 2015; 112: E6973-E6982Crossref PubMed Scopus (73) Google Scholar, Latham et al., 2003Latham P.E. Deneve S. Pouget A. Optimal computation with attractor networks.J. Physiol. Paris. 2003; 97: 683-694Crossref PubMed Scopus (34) Google Scholar, Pasupathy and Connor, 2002Pasupathy A. Connor C.E. Population coding of shape in area V4.Nat. Neurosci. 2002; 5: 1332-1338Crossref PubMed Scopus (299) Google Scholar, Pillow et al., 2008Pillow J.W. Shlens J. Paninski L. Sher A. Litke A.M. Chichilnisky E.J. Simoncelli E.P. Spatio-temporal correlations and visual signalling in a complete neuronal population.Nature. 2008; 454: 995-999Crossref PubMed Scopus (857) Google Scholar, Schneidman et al., 2006Schneidman E. Berry 2nd, M.J. Segev R. Bialek W. Weak pairwise correlations imply strongly correlated network states in a neural population.Nature. 2006; 440: 1007-1012Crossref PubMed Scopus (1121) Google Scholar). A related computational problem in natural vision holds that many features are present simultaneously, and these features are thought to be represented with high fidelity by ensembles of neurons in early sensory areas (Baddeley et al., 1997Baddeley R. Abbott L.F. Booth M.C.A. Sengpiel F. Freeman T. Wakeman E.A. Rolls E.T. Responses of neurons in primary and inferior temporal visual cortices to natural scenes.Proc. Biol. Sci. 1997; 264: 1775-1783Crossref PubMed Scopus (320) Google Scholar, Eichhorn et al., 2009Eichhorn J. Sinz F. Bethge M. Natural image coding in V1: how much use is orientation selectivity?.PLoS Comput. Biol. 2009; 5: e1000336Crossref PubMed Scopus (38) Google Scholar, Froudarakis et al., 2014Froudarakis E. Berens P. Ecker A.S. Cotton R.J. Sinz F.H. Yatsenko D. Saggau P. Bethge M. Tolias A.S. Population code in mouse V1 facilitates readout of natural scenes through increased sparseness.Nat. Neurosci. 2014; 17: 851-857Crossref PubMed Scopus (107) Google Scholar, Kayser et al., 2003Kayser C. Salazar R.F. König P. Responses to natural scenes in cat V1.J. Neurophysiol. 2003; 90: 1910-1920Crossref PubMed Scopus (95) Google Scholar, Vinje and Gallant, 2000Vinje W.E. Gallant J.L. Sparse coding and decorrelation in primary visual cortex during natural vision.Science. 2000; 287: 1273-1276Crossref PubMed Scopus (908) Google Scholar). Such concurrent representation of multiple stimulus features is known to exist in higher visual and non-visual brain areas (DiCarlo et al., 2012DiCarlo J.J. Zoccolan D. Rust N.C. How does the brain solve visual object recognition?.Neuron. 2012; 73: 415-434Abstract Full Text Full Text PDF PubMed Scopus (918) Google Scholar, Sigala et al., 2008Sigala N. Kusunoki M. Nimmo-Smith I. Gaffan D. Duncan J. Hierarchical coding for sequential task events in the monkey prefrontal cortex.Proc. Natl. Acad. Sci. USA. 2008; 105: 11969-11974Crossref PubMed Scopus (100) Google Scholar, Stokes et al., 2013Stokes M.G. Kusunoki M. Sigala N. Nili H. Gaffan D. Duncan J. Dynamic coding for cognitive control in prefrontal cortex.Neuron. 2013; 78: 364-375Abstract Full Text Full Text PDF PubMed Scopus (419) Google Scholar), but it is unknown whether multidimensional coding is used in primary visual cortex (V1) to enable efficient representations of natural scenes. Moreover, long-term stability of these codes and their correlation structure is required for many aforementioned models of the cortex to be neurophysiologically plausible, and the extent of this temporal stability is as yet unknown. We therefore set out to investigate two important factors in the interaction of correlated variability with population coding: (1) the stationarity of correlations over time and (2) the presence and potential use of higher-dimensional correlations in population coding. For instance, can responses of neuronal triplets be described well by the pairwise interactions of its members? If not, what might be the use of higher-dimensional interactions? We recorded long-term (>4 weeks) neuronal responses to drifting gratings and natural movies from the same populations of L2/3 (cortical layer 2/3) V1 neurons in awake mice using GCaMP6 calcium imaging (Chen et al., 2013Chen T.-W. Wardill T.J. Sun Y. Pulver S.R. Renninger S.L. Baohan A. Schreiter E.R. Kerr R.A. Orger M.B. Jayaraman V. et al.Ultrasensitive fluorescent proteins for imaging neuronal activity.Nature. 2013; 499: 295-300Crossref PubMed Scopus (3647) Google Scholar). Our data show that neuronal responses are variable across trials but relatively stable across days. We observed that multidimensional correlations are of critical importance for the efficacy of population codes by restricting variability to those directions in neural space that are perpendicular to the axes coding for stimulus features. Moreover, these correlations can be used to predict up to half of the instantaneous noise in single-neuron activity. We conclude that much of the trial-by-trial fluctuation shown by individual neurons is not noise but might be functionally important for the neuronal population code of sensory input. We performed chronic GCaMP6m imaging in V1 L2/3 of awake mice to study the variability in responses of neuronal populations to drifting gratings (n = 9 mice recorded long term, over 2–5 weeks) and natural movies (n = 4 mice recorded long term; n = 5 mice recorded short term, on a single day; Figure 1A). We found that the orientation tuning and responses to natural movies of many neurons were reliable over this period (Figures 1B–1F). However, some neurons showed weak or non-orientation-tuned responses across trials and recording sessions. Such neurons might be located at the edge of the focal plane in some recording sessions or might be non-responsive to visual stimulation by our drifting gratings. We therefore excluded these from further analyses and included only neurons that showed a non-random orientation preference across days (for long- and short-term recordings, respectively, on average 48% of 43–158 neurons and 55% of 130–181 neurons per mouse were consistently tuned to orientation across recording sessions; Figures S1 and S2). Although tuning to the orientation of drifting gratings was stable over days for many neurons, the responses of these stable neurons for subsequent trials of the same orientation were still variable (Figure 1D). We hypothesized that some of this variability might be related to the behavioral state of the animal. Because pupil size is positively correlated with arousal (Bradshaw, 1967Bradshaw J. Pupil size as a measure of arousal during information processing.Nature. 1967; 216: 515-516Crossref PubMed Scopus (117) Google Scholar, Coull et al., 2004Coull J.T. Jones M.E.P. Egan T.D. Frith C.D. Maze M. Attentional effects of noradrenaline vary with arousal level: selective activation of thalamic pulvinar in humans.Neuroimage. 2004; 22: 315-322Crossref PubMed Scopus (126) Google Scholar, Vinck et al., 2015Vinck M. Batista-Brito R. Knoblich U. Cardin J.A. Arousal and locomotion make distinct contributions to cortical activity patterns and visual encoding.Neuron. 2015; 86: 740-754Abstract Full Text Full Text PDF PubMed Scopus (378) Google Scholar), we performed eye-tracking during neurophysiological recordings (Figure S3) and found that during trials in which the mouse’s pupil size was large, neuronal responses showed a higher variability (SD) across repetitions to the preferred orientation, as well as increased NCs, despite similar levels of mean activity (Figure 1E; Figure S3). We aimed to better quantify the observation that variability can be high across trials mere seconds apart, whereas tuning similarity is stable across days (Figure S4). We therefore calculated NC and signal correlation (SC) matrices for each recording session. This splits the neuronal responses into a signal component that encodes grating orientation (Figures 2A and 2B ) and a noise component that reflects trial-by-trial fluctuations (Figures 2C and 2D). The stability of the population response can be approximated by calculating the Pearson correlation between pairs of SC matrices recorded during different sessions. Analysis of SC versus inter-recording time in days (Figure 2B) yielded three main results. First, when two sessions were recorded on the same day (inter-recording time was 0 days), the correlation between these recordings was relatively high (r = 0.52 ± 0.088, mean ± SD) but clearly far from identical (r = 1.0). This means that pairwise neuronal responses to the same stimuli across repetitions are largely defined by short-term fluctuations that occur on the order of minutes to hours. Second, longer intervals, on the order of days, decrease correlations at a slower pace. Even after an interval of a month, the correlations were well above zero (one-sample t test across four recording pairs with largest time intervals, mean interval 29.8 days, p < 0.01; Figure 2B). After fitting the data with exponential decay functions, we found that SC half-lives were similar across animals (mean ± SEM half-life across mice, 37.3 ± 10.8 days). Third, when repeating these analyses for across-recording similarity in NCs, the effects were comparable in form and magnitude. We found that r was 0.48 ± 0.097 (mean ± SD) for recordings made on the same day and that the mean ± SEM of half-life across mice was 41.1 ± 6.0 days (Figures 2D and 2E). These results suggest that correlation structures are relatively stable over time, showing a slow decay that is in line with previous reports of multiday stability of orientation tuning (Chen et al., 2013Chen T.-W. Wardill T.J. Sun Y. Pulver S.R. Renninger S.L. Baohan A. Schreiter E.R. Kerr R.A. Orger M.B. Jayaraman V. et al.Ultrasensitive fluorescent proteins for imaging neuronal activity.Nature. 2013; 499: 295-300Crossref PubMed Scopus (3647) Google Scholar, Lütcke et al., 2013Lütcke H. Margolis D.J. Helmchen F. Steady or changing? Long-term monitoring of neuronal population activity.Trends Neurosci. 2013; 36: 375-384Abstract Full Text Full Text PDF PubMed Scopus (69) Google Scholar, Mank et al., 2008Mank M. Santos A.F. Direnberger S. Mrsic-Flogel T.D. Hofer S.B. Stein V. Hendel T. Reiff D.F. Levelt C. Borst A. et al.A genetically encoded calcium indicator for chronic in vivo two-photon imaging.Nat. Methods. 2008; 5: 805-811Crossref PubMed Scopus (380) Google Scholar). However, these results also indicate that there are large fluctuations in pairwise NCs and SCs on the order of minutes to hours (cf. Figures S4 and S5). Next, we asked whether higher-dimensional representations would be similarly stable over days. We calculated a multidimensional population code similarity, based on the distance between neural representations of the same orientation on two different recording days (the number of dimensions in this approach equals the number of neurons; see Experimental Procedures). The similarity has a maximum of 1.0 and is normalized by the average trial-by-trial variability in multidimensional representations of the same orientation. A value of 0.0 indicates that representations of the same orientation are as distant as the mean trial-by-trial variability within those recordings. First, we performed this procedure for pairs of neurons, each time averaging across 100 randomly selected groups. After fitting the data with an exponential decay function, we found that this metric’s similarity was 0.51 ± 0.04 (mean ± SD) for recordings on the same day and that the mean ± SEM of the half-life across mice was 62.0 ± 9.2 days (Figure 3A). Next, this procedure was repeated with different group sizes (triplets, quadruplets, etc.), yielding a half-life for each dimensionality. Analysis of half-lives as a function of dimensionality showed that high-dimensional representations of visual stimulus orientation are more temporally stable than pairwise representations by the same populations (Figure 3B). We observed a consistent within-animal effect, in which high-dimensional codes were more stable than pairwise codes (t test of high-dimensional half-lives normalized to pairwise; p < 0.001; Figure 3C), although raw half-lives were quite variable across animals (mean ± SEM of half-life decay times; at pairwise: 62.0 ± 9.2 days; at maximum dimensionality: 74.4 ± 10.3 days). Earlier work has demonstrated that non-zero spike-count correlations between pairs of neurons can lead to higher-dimensional network states that are hard to predict from pairwise correlations (Schneidman et al., 2006Schneidman E. Berry 2nd, M.J. Segev R. Bialek W. Weak pairwise correlations imply strongly correlated network states in a neural population.Nature. 2006; 440: 1007-1012Crossref PubMed Scopus (1121) Google Scholar). Therefore, the multidimensional correlation structure of a neuronal population could in principle contain more information than might be apparent from the responses of pairs of neurons. It is unknown whether neurons in mouse visual cortex encode stimuli in a lower-dimensional (e.g., pairwise) way or whether stimuli are represented in a multidimensional response space that cannot be inferred from lower-dimensional statistical interdependencies. To investigate the potential of multidimensional population coding, we created a Mahalanobis-distance-based decoder that assumes multivariate Gaussian responses and can be used for any number of dimensions (i.e., a variant of a quadratic discriminant analysis; Figure 4A; see Figure S6 for analysis of this assumption). We used Mahalanobis space because this normalizes all variability, even across multiple dimensions and regardless of its direction (see Experimental Procedures). This means that Mahalanobis space automatically incorporates multidimensional correlations. The variability normalization also means that decision boundaries between stimulus classes are always linear in Mahalanobis space (but not necessarily in non-normalized response space; e.g., cf. Figures 4A and 4E; De Maesschalck et al., 2000De Maesschalck R. Jouan-Rimbaud D. Massart D.L. The Mahalanobis distance.Chemom. Intell. Lab. Syst. 2000; 50: 1-18Crossref Scopus (1531) Google Scholar). This simplifies the neural computations necessary to optimally extract information from a population code, assuming that neural circuits can perform response normalization (Carandini et al., 1997Carandini M. Heeger D.J. Movshon J.A. Linearity and normalization in simple cells of the macaque primary visual cortex.J. Neurosci. 1997; 17: 8621-8644Crossref PubMed Google Scholar, Lee and Maunsell, 2009Lee J. Maunsell J.H.R. A normalization model of attentional modulation of single unit responses.PLoS ONE. 2009; 4: e4651Crossref PubMed Scopus (180) Google Scholar, Montijn et al., 2012Montijn J.S. Klink P.C. van Wezel R.J. Divisive normalization and neuronal oscillations in a single hierarchical framework of selective visual attention.Front. Neural Circuits. 2012; 6: 22Crossref PubMed Scopus (13) Google Scholar, Reynolds and Heeger, 2009Reynolds J.H. Heeger D.J. The normalization model of attention.Neuron. 2009; 61: 168-185Abstract Full Text Full Text PDF PubMed Scopus (875) Google Scholar, Ringach, 2010Ringach D.L. Population coding under normalization.Vision Res. 2010; 50: 2223-2232Crossref PubMed Scopus (25) Google Scholar) and decorrelation (Wiechert et al., 2010Wiechert M.T. Judkewitz B. Riecke H. Friedrich R.W. Mechanisms of pattern decorrelation by recurrent neuronal circuits.Nat. Neurosci. 2010; 13: 1003-1010Crossref PubMed Scopus (77) Google Scholar). Using this decoder, we performed a bootstrapping procedure of orientation decoding for all dimensionalities, in which the dimensionality is defined as the number of cells within one randomly drawn group of neurons. We integrated information ranging from 1 to 100 groups (i.e., samples) of neurons of different sizes (i.e., dimensionalities), ranging from 2 to 14 neurons per group (see Experimental Procedures; Figure 4B). This analysis showed that the decoder’s performance saturated ∼60–80 randomly drawn groups of neurons for all dimensionalities and all mice, so we performed all further analyses with the integration of 100 random groups for each dimensionality. To quantify the effect of experimentally measured correlations on neuronal population codes, we compared the decoder’s performance on the real data to its performance on shuffled datasets, in which trials were randomized across repetitions of the same stimulus type. This procedure preserves stimulus tuning but destroys NC structures and therefore allows the identification of effects that are only due to NCs of a certain dimensionality. For independent decoding (i.e., dimensionality 1) there was no difference between shuffled and non-shuffled datasets (orientation decoding accuracy, mean ± SEM across animals: 49.5% ± 4.1% for shuffled, 50.5% ± 3.6% for non-shuffled; errors corrected over shuffled: 1.2% ± 2.0%; paired t test, p = 0.336), but the decoding of stimulus orientation gradually improved based on the higher-dimensional structure present in the recorded data (orientation decoding accuracy at dimensionality 15: 44.7% ± 4.2% for shuffled, 53.2% ± 4.0% for non-shuffled; errors corrected over shuffled: 15.8% ± 2.0%; paired t test, p < 0.001; Figure 4C). We fitted a half-logistic growth function to the observed performance across dimensionalities 2–15 and calculated the asymptotic performance that would hypothetically be reached (see Experimental Procedures). Asymptotic performance showed an even larger difference between shuffled and non-shuffled performance than independent and pairwise performance, suggesting that V1 neuronal populations encode unique information in high-dimensional space that cannot be inferred using lower-dimensional representations (Figure 4D). This effect is not due to simply taking information from more neurons; if this were the case, decoding performance for any dimensionality (especially the low-dimensional ones) would not saturate ∼60–80 random groups (Figure 4B); more importantly, there would be no difference between shuffled and non-shuffled decoding performance (Figure 4C). This within-dimensionality saturation effect, combined with the across-dimensionality increase in performance, shows that additional information on stimulus orientation is encoded in higher-dimensional neuronal response space that is not present in a lower-dimensional space. Moreover, the effect was present when using a range of time windows, showing that multidimensional coding is not dependent on particular epochs during stimulus processing (Figure S6L). As with any classification problem, variability orthogonal to a decision boundary increases the likelihood that a stimulus is misclassified, while variability parallel to decision boundaries is irrelevant for the classification of a stimulus pair (Figure 4E). We therefore tested whether neuronal circuits might be more variable along dimensions that are not relevant for coding primary stimulus features (i.e., orientation for V1; Figure 4F). In a higher-dimensional space with N neurons, all instances of a stimulus feature (i.e., orientation) can be captured within a single curve. In this case, a line tangential to the orientation coding curve exists for each orientation. Movement along this line corresponds to a shift in the encoded stimulus orientation, while all N − 1 other possible directions are irrelevant for encoding orientation of that stimulus. These other dimensions may then be used to represent other stimulus features, such as contrast and spatial frequency, or for modulatory effects, such as attention, arousal, or other factors, without interfering with the encoding of orientation (Figure 4G). We tested the non-uniformity of variability in higher-di" @default.
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• W2518631572 date "2016-08-01" @default.
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• W2518631572 title "Population-Level Neural Codes Are Robust to Single-Neuron Variability from a Multidimensional Coding Perspective" @default.
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