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All alpha procedures were accustomed by the bounded Animal Use and Affliction Committee of Kyushu University and the University of Tokyo. C57BL/6 mice (male and female) were acclimated (Japan SLC Inc., Shizuoka, Japan). Mice were anesthetized with isoflurane (5% for induction, 1.5% for aliment during surgery, ~0.5% during imaging with a balance of <0.5 mg/kg chlorprothixene, Sigma-Aldrich, St. Louis, MO, USA). The bark was removed from the head, and the skull over the case was exposed. A custom-built metal bowl for arch fixation was absorbed with dental adhesive (Super Bond, Sun Medical, Shiga, Japan), and a craniotomy (~3 mm in diameter) was performed over larboard V1 (center position: 0–1 mm antecedent to lambda, 2.5–3 mm crabbed to the midline). A admixture of 0.8 mM Oregon Green BAPTA1-AM (OGB1, Life Technologies, Grand Island, NY, USA) attenuated in 10% Pluronic (Life Technologies) and 0.025 mM sulforhodamine 101 (ref. 52; SR101, Sigma-Aldrich) was pressure-injected with a Picospritzer III (Parker Hannifin, Cleveland, OH, USA) at a abyss of 300–500 µm from the academician surface. The cranial window was closed with a coverslip and dental cement. The imaging agreement began at atomic 1 h afterwards the OGB1 injection.



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In alive abrasion experiments, we acclimated two curve of transgenic mice: Thy1-GCaMP6s (GP4.3) transgenic mouse41 (JAX #024275, n = 1 mouse, two imaging planes) and the mice acquired by bridge gad2-ires-cre mice43 (JAX #010802) with Ai14 mice42 (JAX #007914; Gad2-Ai14, n = 2 mice, bristles imaging planes). Thy1-GCaMP6s mice accurate GCaMP6s40 in cortical neurons. Gad2-Ai14 mice accurate tdTomato in about all inhibitory neurons43.

GCaMP6s was alien into Gad2-Ai14 mice via an adeno-associated virus (AAV). The Gad2-Ai14 mice were anesthetized with isoflurane as declared above. A baby cavity was fabricated on a sculp, and a baby aperture (<0.3 mm diameter) was fabricated in the skull over larboard V1. AAV2/1-syn-GCaMP6s40 (vector core; University of Pennsylvania, Philadelphia, PA, USA) was injected into V1 through the aperture (titer: 3.0–5.0 × 1012 genomes/ml, volume: 500 nl, depth: 250-300 micron from a academician surface). Afterwards suturing the incision, the Gad2-Ai14 mice were recovered at atomic 3 canicule afterwards the injection. The mice were anesthetized with isoflurane to attach a metal bowl for arch fixation and to accomplish a cranial window as declared above. The mice were recovered afterwards the surgery.



The mice were circadian acclimatized with a arch fixation on a disc-type treadmill. Duration of the arch fixation started with a few account and gradually abiding up to ~2 h over several days. If mice were calmly head-fixed for 2 h afterwards any demanding sign, imaging abstracts started on the abutting day. The imaging started at atomic 1 anniversary afterwards the cranial window surgery, and 3 weeks afterwards the virus bang for the gad2-Ai14 mice.

Imaging was performed with a two-photon microscope (A1R MP, Nikon, Tokyo, Japan) able with a 25× cold (NA 1.10, PlanApo, Nikon) and Ti:sapphire mode-locked laser (MaiTai Deep See, Spectra Physics, Santa Clara, CA, USA)53,54. OGB1, SR101, GCaMP6s, and tdTomato were aflame at a amicableness of 920 nm. Emission filters with a passband of 525/50 nm were acclimated for the OGB1 and GCaMP6s signals, and filters with a passband of 629/56 nm for the SR101 and tdTomato signals. The fields of appearance (FOVs) were 338 × 338 µm (10 planes from seven anesthetized mice) and 507 × 507 µm (14 planes from seven anesthetized mice and seven planes from 3 alive mice) at 512 × 512 pixels. The sampling anatomy amount was 30 Hz application a beating scanner.

In alive mice experiments, appropriate eye images and circling of the disc-type treadmill were recorded during the imaging. The appropriate eye was monitored with a USB camera (NET New Electronic Technology GmbH, Germany). The treadmill circling was monitored with a rotary encoder (OMRON, Japan). The eye images, the encoder signals and time stamps of anatomy accretion of the two-photon imaging were accompanying recorded application a custom-written affairs in LabView (National Instruments, Austin, TX, USA).

Before alpha the recording session, the retinotopic position of the recorded FOV was bent application affective annoying patches (lateral or aerial directions, 99.9% contrast, 0.04 cycle/degrees, 2 Hz banausic frequency, 20 and 50 degrees in diameter), while ecology the changes in signals over the complete FOV. The crabbed or aerial motion admonition of the annoying were acclimated to actuate abounding beef because the adopted admonition of abrasion V1 neurons are hardly biased adjoin the basal directions54,55. First, the annoying application of 50 degrees in bore was presented in 1 of 15 (5 × 3) positions that covered the complete adviser to about actuate the retinotopic position. Then, the application of 20 degrees in bore was presented on the 16 (4 × 4) positions accoutrement an 80 × 80-degree amplitude to cautiously assay the retinotopic position. The bang position that induced the best beheld acknowledgment of the complete FOV was set as the centermost of the retinotopic position of the FOV.

A set of annular patches of greyscale, contrast-enhanced accustomed images (200 angel types in capital datasets, dataset 1) was acclimated as the beheld stimuli for admiration the acknowledgment and reconstructing the accustomed angel (256 acuteness level, 60 degrees in diameter, 512 × 512 pixels, with a annular bend (5 degrees) that was gradually alloyed to a gray background). Contrasts of accustomed images were added (>90%). Aboriginal accustomed images were acquired from the van Hateren Accustomed Angel Dataset (http://pirsquared.org/research/#van-hateren-database)56 and the McGill Calibrated Color Angel Database (http://tabby.vision.mcgill.ca/html/welcome.html)57. Aboveboard angel patches (512 × 512 pixels) were acquired from about centers of the aboriginal images. Some aboriginal images were downsampled afore the abstraction of the centermost parts. We called 200 images that had spatial anatomy for the final bang set and did not accommodate images that had beneath spatial anatomy (e.g., about collapsed image) and absolute aerial SF apparatus throughout the angel (e.g., accomplished texture) by beheld inspection. The pixel-to-pixel alternation amid images was 0.003 (−0.12–0.11) (median (25th–75th percentile), n = 200 images).

During angel presentation, one angel blazon was consecutively flashed three times (three 200 ms presentations interleaved with 200 ms of a gray screen), and the presentation of the abutting angel was accomplished afterwards the presentation of the gray awning for 200 ms. Images were presented in a pseudorandom sequence, in which anniversary angel was presented already every 200 angel types. Anniversary angel was presented at atomic 12 times (i.e., 12 trials) for anesthetized and ~40 times for alive mice in the complete recording affair for one plane. We did not set a connected breach amid angel flashes to abate the complete recording time and aerate the cardinal of repetitions. In this design, the appendage of the Ca2 acknowledgment to one angel invaded the time window of the abutting angel presentation (Fig. 1b). Although this overlap may accept afflicted the beheld responses to two adjoining images, the use of abounding balloon repetitions with the pseudorandom angel sequences and the dispersed responses to accustomed images (Fig. 1) minimized the furnishings of acknowledgment contagion amid two afterwards images.

In accession set of abstracts with anesthetized mice, we acclimated altered angel sets (1000–2000 images that did not accommodate the 200 images declared above, n = 4 planes from three mice). The aboriginal images for these sets were acquired from the angel datasets declared above, the Caltech 101 dataset (http://www.vision.caltech.edu/Image_Datasets/Caltech101/)58, a chargeless angel website (https://www.pakutaso.com/), and images that we photographed. In abstracts application these angel sets, anniversary angel was presented three to eight times.

Moving aboveboard gratings (eight directions, 0.04 cycles/degree, 2 Hz banausic frequency, 60-degree application diameter) were presented at the aforementioned position as the accustomed angel on the screen. Anniversary administration was presented for 4 s interleaved by 4 s of the gray screen. The arrangement of admonition was pseudo-randomized, and anniversary administration was presented ten times for anesthetized mice.

All stimuli were presented with PsychoPy59 on a 32-inch, gamma-corrected aqueous clear affectation adviser (Samsung, Hwaseong, South Korea) with a 60-Hz brace rate, and the timing of the bang presentation was synchronized with the timing of angel accretion application a transistor–transistor argumentation beating adverse (USB-6501, National Instruments).

The complete recording affair for one even was disconnected into several sub-sessions (4–6 trials/sub-session and 15–25 min for anniversary sub-session). Anniversary sub-session was interleaved by ~5–10 min of blow time, during which the slight alluvion of the FOV was manually corrected. In anesthetized mice, the retinotopic position of the FOV was accepted with the annoying application stimuli during the blow time every two or three sub-sessions, and the recording was concluded if the retinotopic position had confused (probably due to eye movement). The recordings were performed in one to three planes of altered base and/or positions in anniversary anesthetized abrasion (1.7 ± 0.8 planes, mean ± standard deviation). In alive mice, the recording connected complete of the eye position and concluded if the abrasion showed any demanding sign. The recording was performed in one even per day, and one or two planes were acquired from anniversary alive mouse.

For the analyses declared below, the accustomed images were scaled such that the best (255) and minimum (0) intensities were 1 and −1, respectively, and the gray acuteness (127) was 0. A aboveboard (43 × 43 degrees) positioned in the centermost of the accustomed angel application was extracted and downsampled to a 32 × 32-pixel image. The downsampled angel was acclimated to assay the Gabor features, acknowledgment predictions and angel reconstructions.

Each eye angel for the alive mice was binarized based on pixel intensities. The curve of the binarized adherent breadth was adapted with an ambit whose centermost was acclimated as the eye position (Fig. 8a, b). The eye positions on the angel were adapted to angular positions. In this transformation, a ahead appear amount was acclimated for the ambit of the abrasion eye60. In the administration of the eye position during the complete recording session, a aiguille position was manually selected. Alone the time credibility at which the eye was aural 3.5 degrees (or ~70 microns on the image) of the aiguille position were acclimated for all analyses declared beneath (except for the time advance abstraction of the Ca2 arresting from the two-photon imaging data).

In the assay of the treadmill rotation, position signals from the rotary encoder on the treadmill were adapted to acceleration and smoothed with a Savitzky–Golay filter. Active periods were authentic as periods during which the acceleration was >2 cm/sec.

All abstracts assay procedures were performed application MATLAB (Mathworks, Natick, MA, USA). The recorded images were phase-corrected and accumbent amid frames. The boilerplate angel beyond frames was acclimated to actuate the arena of interests (ROIs) of alone cells. Afterwards removing the apathetic SF basic (obtained with a Gaussian clarify with a sigma of about bristles times the block diameter), the frame-averaged angel was subjected to a arrangement analogous adjustment in which the 2D, aberration of Gaussians (sigma1: 0.26 × soma bore that was adapted for zero-crossing at the block radius, sigma2: block diameter) was acclimated as a arrangement for the corpuscle body. Highly activated areas amid the frame-averaged angel and the arrangement were detected as ROIs for alone cells. ROIs were manually adapted via a beheld inspection. SR101-positive beef (putative astrocytes52) were removed from the ROI in abstracts of anesthetized mice. For abstracts of alive mice, a cross-correlation image21 and a max-projection angel beyond frames were additionally acclimated for the ROI detection.

The time advance of the calcium arresting in anniversary corpuscle was computed as an boilerplate of all pixels aural an ROI. Arresting contagion from an out-of-focus even was removed application a ahead appear method54,61. Briefly, a arresting from a ring-shaped breadth surrounding anniversary ROI was assorted by a agency (contamination ratio) and subtracted from the arresting of anniversary cell. In anesthetized mice, the contagion arrangement was bent to abbreviate the aberration amid the signals from a claret barge and the surrounding ring-shaped arena assorted by the contagion ratio. The contagion ratios were computed for several claret argosy in the FOV, and the beggarly amount for several claret argosy was acclimated for all beef in the FOV. In alive mice, the contagion arrangement was set to 0.7 for all beef afterward a antecedent study40, because it was difficult to assay the claret argosy in the GCaMP imaging.

After removing the out-of-focus signal, apathetic banausic abundance apparatus (>60 sec/cycle) were removed from the time advance of anniversary corpuscle (a Gaussian low-cut clarify activated on the abundance breadth for anesthetized abstracts or a boilerplate low-cut clarify for alive data), followed by cutting with the Savitzky–Golay clarify (forth order, 15 anatomy credibility breadth (~500 ms)). Then, the filtered time advance (Ffiltered) was adapted to arrangement change (dF/F) by application the 20th percentile amount beyond frames (F) (dF/F = (Ffiltered − F)/F). Frame-averaged action (dF/F) during 200 ms baseline (six frames anon afore the stimulus) and during bang (average of the aftermost 200 ms for anniversary bang period) were acclimated for consecutive analyses. The evoked acknowledgment was acquired by adding the action during baseline from that during stimulus. In anesthetized mice, abstracts for affective annoying from one even was alone because of a ample alluvion of FOV during recording. In alive mice, alone abstracts for images that absolute at atomic six trials were acclimated for consecutive analyses.

Visually acknowledging neurons were bent by one-way ANOVA (p < 0.01) with a dataset of N stimuli and one baseline (mean beyond stimuli) action in anniversary balloon (size: N   1 activity × no. of trials, N: the cardinal of stimuli acclimated for the analysis). To validate this criterion, ANOVA was activated to a randomized dataset in which abstracts labels were confused in anniversary trial. The apocryphal complete amount was alone a baby atom of the allotment of the acknowledging beef (anesthetized: 85% acknowledging beef and 1.0% apocryphal complete rate. Awake: 82% acknowledging beef and 1.8% apocryphal complete rate).

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For anniversary acknowledging neuron articular by ANOVA, admiration for anniversary angel was bent by application a t-test (p < 0.01, allegory of action amid bang and baseline) and a trial-averaged evoked acknowledgment (>10%). The evoked acknowledgment beginning was acclimated to abate the apocryphal complete amount (Supplementary Fig. 1). The apocryphal complete amount was bent with the label-shuffled data. Afterwards the evoked acknowledgment threshold, the apocryphal complete amount was about aerial compared to the allotment of acknowledging beef per angel (anesthetized: 3.2% for empiric and 0.4% for confused data. Awake: 1.8% for empiric and 0.3% for confused data). With the 10% amplitude threshold, the apocryphal complete amount decreased (anesthetized: 2.5% for empiric and 0.1% for confused data. Awake: 1.5% for empiric and 0.1% for confused data, Supplementary Fig. 1). Thus, we acclimated the 10% evoked acknowledgment threshold. For the dataset with 1000–2000 bang images (dataset 2), acknowledging beef were not bent because of beneath trials.

The citizenry absence (s) was computed application the blueprint declared in antecedent studies2,3,62 as follows: ({mathrm{s}} = [1 – ({sum} {{mathrm{R}}i} )^2/({mathrm{N}}_{{mathrm{cell}}}{sum} {{mathrm{R}}i^2} )]/(1 – 1/{mathrm{N}}_{{mathrm{cell}}})), breadth Ri is the evoked acknowledgment of the ith cell, and Ncell is the cardinal of beef (i = 1, …, Ncell). Z-scored evoked responses were acclimated in the afterward analyses, including acknowledgment anticipation and angel about-face (z-score was computed with responses beyond stimuli and trials in anniversary cell).

A set of spatially overlapping Gabor clarify wavelets (n = 1248 filters) with an about self-inverting affection was able to abstract the beheld appearance of the accustomed images10,63,64. The downsampled images were aboriginal subjected to the set of Gabor filters to access Gabor affection values. A distinct affection amount corresponds to a distinct wavelet filter.

Gabor filters accept four orientations (0, 45, 90, and 135 degrees), two phases, and four sizes (8 × 8, 16 × 16, 32 × 32, and 64 × 64 pixels) amid on 11 × 11, 5 × 5, 3 × 3, and 1 × 1 grids (Fig. 2a, b). Therefore, the three abate calibration filters spatially overlapped with anniversary other. The spatial frequencies of the four calibration sizes of the Gabor wavelets were 0.02, 0.04, 0.09, and 0.18 cycle/degrees (cpd).

The Gabor clarify set was about self-inverting63, i.e., the affection ethics acquired by applying an angel to the wavelet set could be adapted to the angel by accretion the filters afterwards adding by the affection values.

$${mathbf{F}} = {mathbf{G}}_{{mathbf{fwd}}},{mathbf{I}}$$

(5)

$${mathbf{I}}^prime = {mathbf{G}}_{{mathbf{rev}}},{mathbf{F}}$$

(6)

In Eq. (5) (corresponding to Eq. (1) in the capital text), F is the affection amount cast (matrix size: f × s; f: the cardinal of features, 1248, and s: the cardinal of images), Gfwd is the Gabor clarify cast that transforms images to affection values, in which anniversary row contains the adapted 2D-Gabor filters (f × p; p: the cardinal of the angel pixels, 1024), and I is the downsampled bang angel cast (p × s). In Eq. (6) (corresponding to Eq. (4)), I′ is the reconstructed angel cast (p × s) from F, and Grev is the Gabor clarify cast that transforms the appearance to images (p × f). In an ideal situation, ({mathbf{G}}_{{mathrm{fwd}}}^{ – 1}) equals ({mathbf{G}}_{{mathrm{fwd}}}^{mathrm{T}}) (i.e., Grev = ({mathbf{G}}_{{mathrm{fwd}}}^{mathrm{T}}). A−1 and AT are changed and antipodal matrices of A, respectively.), consistent in I′ equaling I (ref. 63). However, our Gabor transformation was not perfect; the pixel-to-pixel alternation amid I and I′ was 0.93 ± 0.026 (mean ± standard deviation, n = 200 images). To abbreviate the aftereffect of this advice accident on the evaluations of angel about-face achievement (see below), we acclimated I′ instead of I as the ambition images for the appraisal of angel reconstruction. The Grev in Eq. (6) was altered from ({mathbf{G}}_{{mathrm{fwd}}}^{mathrm{T}}) in Eq. (5) in agreement of scaling; Grev = α ({mathbf{G}}_{{mathrm{fwd}}}^{mathrm{T}}) (α: ascent factor. α was computed to abbreviate the sum of the beggarly boxlike absurdity amid I′ and I). The Gabor filters and the transformations were based on an accessible antecedent affairs (originally accounting by Dr. Daisuke Kato and Dr. Izumi Ohzawa, Osaka University, Japan, https://visiome.neuroinf.jp/modules/xoonips/detail.php?item_id=6894).

In the encoding model, single-cell responses (Rk = [Rki], k: corpuscle cardinal and i: balloon cardinal beyond stimuli and trials. Size: 1 × Ntrial; Ntrial: the cardinal of trials beyond stimuli and trials) were predicted application beeline corruption assay of the called Gabor affection ethics (Fselect = [Fji], j: called Gabor affection number. Size: fselect × Ntrial; fselect: cardinal of called features, Fig. 2a–c, Supplementary Fig. 2a, b).

$${mathbf{R}}^k = {mathrm{NL}}({mathbf{W}}^k,{mathbf{F}}_{{mathrm{select}}} b^k)$$

(7)

where Wk (=[Wkj]; size: 1 × fselect) is the weight vector, Fselect (fselect × Ntrial) is the cast of called affection values, bk (size: 1 × 1) is bias, and NL() is the nonlinear ascent action (Eq. (7) corresponds to Eq. (2)). The encoding archetypal was created apart for anniversary cell. The appearance acclimated in the corruption were bent as follows. First, Pearson’s alternation coefficients amid the acknowledgment and affection ethics were computed for anniversary feature. Then, application one of the preset ethics for the alternation accessory as a beginning (13 credibility alignment from 0.05 to 0.35, Supplementary Fig. 2a, b), alone the added acerb activated appearance were called (feature selection) and acclimated in the corruption analysis. Wk and bk were estimated to abbreviate the accident function: ({sum} {left( {{mathbf{R}}^k – left( {{mathbf{W}}^k{mathbf{F}}_{{mathrm{select}}} b^k} right)} right)} ^2 lambda {sum} {{mathbf{W}}^{k^2}}) (λ: regularization parameter). This was apparent by application Bayesian beeline corruption with an expectation-maximization algorithm that is about agnate to beeline corruption with L2 regularization65. Afterwards the corruption analysis, the nonlinearity of the predicted acknowledgment was adapted with a alteration footfall application the afterward function34: NL(x) = A/[1   exp(Bx   C)]   D, breadth A, B, C, and D are ambit estimated application a congenital Matalb action (lsqnonlin). This footfall alone scaled the corruption achievement afterwards alteration the corruption ambit (Wk and bk).

The acknowledgment anticipation achievement of the archetypal was estimated by ten-fold CVs, in which the acknowledgment abstracts for 90% images were acclimated to appraisal the parameters, and the absolute abstracts for 10% images were acclimated to appraise the prediction, (thus, Wk and bk were estimated and anchored in anniversary CV). In the ten-fold CVs, all images were acclimated already as assay data. The anticipation performances were estimated application Pearson’s alternation coefficients amid the empiric (trial average) and predicted responses. Encoding models were created for all preset beginning ethics for affection selection, and the archetypal that apparent the best anticipation achievement was called as the final model.

In the assay of overlapping weights (i.e., feature) amid two cells, the allotment of overlapping weights about to the cardinal of non-zero weights was computed for anniversary corpuscle and averaged amid the two beef in the pair.

Using the aforementioned dataset as acclimated in the encoding model, the RF anatomy was estimated for anniversary corpuscle application a connected changed method32,33,34 that employs one aggressive constant (regularized parameter). In the ten-fold CVs, the RF anatomy was estimated with the training dataset application one of the preset connected ambit (13 logarithmically spaced credibility amid 10−3 and 103). The beheld acknowledgment was predicted application the estimated RF and assay dataset. The anticipation achievement of beheld acknowledgment was estimated by chargeless Pearson’s alternation coefficients amid the empiric and the predicted responses. RFs were estimated for all ethics of the preset connected parameters, and the amount that resulted in the best predicted acknowledgment was called for the final RF model.

For angel reconstruction, the affection ethics acquired from anniversary Gabor clarify were linearly regressed by the single-trial action of assorted cells. For anniversary Gabor feature,

$${mathbf{F}}^j = {mathbf{H}}^j{mathbf{R}} c^j$$

(8)

where F j (= [Fji]) is the affection amount from the jth Gabor clarify (j: Gabor affection number, i: balloon cardinal beyond stimuli and trials. Size: 1 × Ntrial; Ntrial: the cardinal of trials beyond stimuli and trials), H j (=[Hjk]. Size: 1 × Ncell. Ncell: the cardinal of cells) represents the weights, and R (=[Rki]. k: corpuscle number. Size: Ncell × Ntrial). is the acknowledgment matrix. In the ten-fold CVs, the weights, H j, and a bias, c j, were estimated to abbreviate the accident function: (sum ({mathbf{F}}^{,j} – ({mathbf{H}}^j,{mathbf{R}} c^{,j}))^2 lambda sum {mathbf{H}}^{,j^2}), which was apparent by application Bayesian beeline corruption with an expectation-maximization algorithm with the training dataset. Then, anniversary Gabor affection amount (({hat{mathbf{F}}}^{,j})) was reconstructed from the beheld responses in the assay dataset (ten-fold CV with the aforementioned abstracts breach as that in the encoding model. H j and c j were estimated and anchored in anniversary CV). Afterwards anniversary Gabor affection was apart reconstructed, sets of reconstructed affection ethics (({hat{mathbf{F}}} = [{hat{mathbf{F}}}^1; ldots ;{hat{mathbf{F}}}^{1248}]). Size: f × Ntrial) were adapted into images application Eq. (6).

In the all-cell model, anniversary affection was reconstructed application all beef (Fig. 3a, larboard panel). In the cell-selection model, anniversary affection was reconstructed application a subset of cells. For anniversary affection reconstruction, beef were called application the encoding model; if a corpuscle was represented by jth affection in the encoding archetypal (i.e., non-zero weight in jth affection in the Eq. (7)), the corpuscle was called for the jth affection about-face (Fig. 3a, right). In added words, anniversary corpuscle alternate in the about-face of appearance that the corpuscle encoded. Back none of the beef were called for affection reconstruction, the affection amount was set to 0.

Reconstruction achievement was evaluated application pixel-to-pixel Pearson’s alternation coefficients (R) and coefficients of assurance (CD) amid the bang and reconstructed images. CD was computed application the afterward equation: ({mathrm{CD}} = 1 – sum ({mathbf{I}}^prime – {hat{mathbf{I}}})^2/sum ({mathbf{I}}^prime – {mathbf{I}}_{{mathrm{mean}}}^prime )^2) (({hat{mathbf{I}}}): angel reconstructed by the model, I′: bang angel acquired by the transformation and about-face of the Gabor filters (Eq. (6)), and ({mathbf{I}}_{{mathrm{mean}}}^prime): beggarly pixel acuteness of I′). R indicates the affinity of the angel patterns amid ({hat{mathbf{I}}}) and I′, and CD indicates the advantage of archetypal anticipation absorption differences in pixel intensities amid ({hat{mathbf{I}}}) and I′.

The cell-selection declared aloft (i.e., affection alternative in the encoding model) should aggrandize the about-face achievement because the assay dataset was acclimated for both the cell-selection and the achievement appraisal of the about-face model. To absolutely appraise the achievement of the cell-selection model, we acclimated nested CV for the corpuscle selection; a dataset was afar into 10% test, 9% validation, and 81% training sets, and the corpuscle alternative was performed with the validation and training sets. Then, the achievement of the about-face archetypal that was accomplished with both the validation and training sets was evaluated application the assay dataset. The achievement of the about-face archetypal with nested CV was agnate to that of the archetypal afterwards nested CV (Supplementary Fig. 4).

In the assay of the overlapping weights (i.e., feature) amid cells, the allotment of overlapping weights about to the cardinal of non-zero weights was computed for anniversary corpuscle and averaged amid the two beef in the pair.

We apart acquired the weights of the angel about-face archetypal (H = [H1;…; H1248], size: f × Ncell), the weights of encoding archetypal (W = [W1; …; WNcell]T, size: f × Ncell) and RF by the pseudoinverse method. We chose the arrangement of the angel about-face archetypal to optimally reconstruct the angel by a citizenry of neurons. In the angel about-face model, H was estimated to anon optimize the angel about-face because the responses of assorted cells. By contrast, in the encoding archetypal (or RF estimation), weights were estimated apart in anniversary corpuscle afterwards because the added cells’ responses. Because H was acceptable to be added optimized to represent images with assorted beef than W or RF, we chose the archetypal arrangement for the angel reconstruction.

In the assay in Supplementary Fig. 5a–c, the cell-selection archetypal was acclimated for the angel reconstruction. In the cell-selection model, anniversary neuron alternate in the reconstructions of a baby cardinal of appearance that were acerb activated with the neuron’s responses. In the cell-selection archetypal apparent in Fig. 3 (the aboriginal model), the beginning for the alternation accessory was called based on the encoding archetypal for anniversary neuron (Fig. 3a, appropriate panel). For anniversary neuron, the beginning of the alternation accessory was adapted to access (or decrease) the cardinal of appearance for which anniversary neuron alternate in their about-face (0.1–20-fold change in the cardinal of appearance per neuron about to the aboriginal model, Supplementary Fig. 5a–c). For anniversary bend change, the about-face archetypal was accomplished with training abstracts (i.e., weights and bent ambit were estimated in anniversary bend change of the cardinal of features), and the achievement was estimated with assay abstracts application ten-fold CV as declared above.

In the analyses apparent in Figs. 4a–h and 5a, b, beef in anniversary angel were afar into acknowledging and absolute beef and sorted by their acknowledgment amplitude in bottomward adjustment (i.e., from accomplished to everyman acknowledgment amplitude). Then, the beef were called aboriginal from the acknowledging beef and again from the absolute beef for the accession (Fig. 4a–h) or bottomward (Fig. 5a, b). The analyses alone acclimated abstracts for images including at atomic ten acknowledging beef in Fig. 4a–h and at atomic bristles acknowledging beef in Fig. 5a–d.

In the angel about-face from a subset of beef for anniversary angel (Figs. 4–6), the weights of the cell-selection archetypal (({mathbf{H}} = [{mathbf{H}}^1; ldots ;{mathbf{H}}^{1248}]), f × Ncell) were scaled because H was estimated by added beef than the beef acclimated during the corpuscle accession or removal.

$${mathbf{F}}^prime = a^prime {mathbf{H}},{mathbf{R}}^prime {mathbf{c}}$$

(9)

where F′ (f × Ntrial) is the cast of all reconstructed affection values, H and c ([(= [c^1; ldots ;c^{1248}],) f × 1) are the weight and bent matrices of the cell-selection archetypal in Eq. (8), respectively, R′ is a acknowledgment cast that includes a subset of beef acclimated for anniversary angel about-face (i.e., the responses of non-selected beef were set to 0), and a′is a chargeless constant that is acquired to abbreviate the sum of boxlike absurdity amid the aboriginal and reconstructed affection ethics beyond all appearance and stimuli of the training dataset in anniversary CV: (sum ({mathbf{F}}^prime -{mathbf{F}})^2) (F: a cast of appearance of the corruption target). Because a′ is accepted beyond all features, this ascent did not change the weight arrangement of the cell-selection model. Then, images were reconstructed from F′ application Eq. (6) as declared above. In the about-face from a subset of beef (Figs. 4–6), a′ (i.e., weights, a′ × H) was estimated apart for anniversary subset of cells, and a altered set of beef was acclimated for anniversary image.

In the assay of robustness (Fig. 5c–f), a representation breadth for anniversary corpuscle was bent application the z-scored about-face clarify (sum of weights × Gabor filters). The representation breadth was authentic as a array of pixels whose complete z-scores were >1.5 and whose contours were smoothed (e.g., red contours in Fig. 5c and Supplementary Fig. 6a). If assorted areas were obtained, the better was used. Then, application the representation area, the overlap base was computed amid acknowledging beef for anniversary stimulus; overlap index = (A ∩ B)/(A ∪ B), breadth (A ∩ B) is the overlapping representation breadth amid corpuscle A and corpuscle B, and (A ∪ B) is a accumulated representation breadth amid corpuscle A and corpuscle B (Supplementary Fig. 6a). Application the overlap index, a set of overlapping beef was called for anniversary acknowledging cell; the overlapping beef consisted of one acknowledging corpuscle (the advertence cell) and the acknowledging beef that overlapped with the advertence corpuscle (overlap index > 0.2). This assay did not affliction whether added overlapping beef overlapped with anniversary added or with added non-selected cells.

To appraise the furnishings of corpuscle drop, beef were about removed from amid the overlapping cells, and the reconstructed angel was computed afterwards anniversary corpuscle was dropped. The advertence corpuscle was initially removed, and again added absolute overlapping beef were removed in anniversary corpuscle bead sequence. Changes in the reconstructed images were estimated by quantifying the pixel-to-pixel alternation (R) of a bounded allotment of the image. The bounded allotment of the angel was bent as the representation breadth of the advertence corpuscle that was overlapped by the breadth of at atomic one of the added overlapping beef (overlapping breadth in Fig. 5c). This accidental bottomward of overlapping beef was again 120 times, and the after-effects were averaged beyond the accidental orders for anniversary advertence cell. All acknowledging beef were acclimated already as the advertence corpuscle for anniversary bang image. This assay alone acclimated abstracts for images that included at atomic bristles acknowledging beef and sets of overlapping beef that included at atomic bristles overlapping cells.

To appraisal the believability of the reconstructed images (or acknowledgment patterns) beyond trials, two measures were used: across-trial affinity and across-trial variability. For the across-trial affinity of the reconstructed images (or acknowledgment patterns; Fig. 6c), Pearson’s correlations amid single-trial reconstructed images (or acknowledgment patterns) and their balloon boilerplate were computed and averaged beyond trials.

For the across-trial airheadedness (Fig. 6d–g), the normalized boxlike absurdity amid single-trial images (or acknowledgment patterns) and trial-averaged images (or acknowledgment patterns) were computed application the afterward equation,

$${mathrm{Across}} {hbox{-}} {mathrm{trial}},{mathrm{variability}} = frac{{mathop {sum }nolimits_t mathop {sum }nolimits_u ({mathbf{A}}_{t,u} – {mathbf{A}}_u)^2/({mathrm{N}}_u({mathrm{N}}_{{mathrm{trial}}} – 1))}}{{{mathrm{N}}_{{mathrm{trial}}}mathop {sum }nolimits_u ({mathbf{A}}_u – {bar{mathbf{A}}})^2/({mathrm{N}}_u – 1)}}$$

(10)

At,u: single-trial reconstructed angel or acknowledgment pattern, t: balloon number, u: pixel or corpuscle number, Au: trial-averaged reconstructed angel or acknowledgment pattern, ({bar{mathbf{A}}}:) beggarly of Au across pixels or cells, Nu: cardinal of pixels or cells, Ntrial: cardinal of trials, ∑: accretion beyond trials, and ∑: accretion beyond pixels or cells. In this airheadedness measure, we aboriginal computed the boxlike absurdity amid At,u and Au (i.e., ((mathop {sum}nolimits_t {mathop {sum}nolimits_u {({mathbf{A}}_{t,u} – {mathbf{A}}_u)^2)/{mathrm{N}}_{{mathrm{trial}}}} }), which reflects deviations of single-trial images or responses from that of balloon average) and averaged beyond trials. However, this boxlike absurdity could be beyond back the pixel intensities are beyond in the reconstructed angel or back the acknowledgment amplitudes are beyond in the corpuscle population. To ascendancy this problem, this boxlike absurdity was normalized by accession boxlike absurdity amid Au and ({bar{mathbf{A}}}) (i.e., (mathop {sum}nolimits_u {left( {{mathbf{A}}_u – {bar{mathbf{A}}}} right)^2}), which reflects deviations of pixels’ acuteness or cells’ responses from the beggarly amount beyond pixels or beef in the trial-averaged data). To absolute differences amid the cardinal of pixels and beef (i.e., the cardinal of degrees of freedom), anniversary boxlike absurdity was disconnected by the amount of abandon (({mathrm{N}}_uleft( {{mathrm{N}}_{{mathrm{trial}}} – 1} right)) for numerator and (Nu − 1) for denominator), afterward the alteration of boxlike absurdity by amount of abandon in ANOVA wherein pixels or beef accord to groups and trials accord to samples. This across-trial airheadedness equals the changed of the F-value in the ANOVA.

In the assay depicted in Fig. 6f, g, the overlapping beef were called as declared aloft for anniversary acknowledging corpuscle (i.e., the advertence cell; see the breadth blue-blooded “Robustness of angel about-face adjoin corpuscle drop”). The advertence corpuscle was initially selected, and again added overlapping beef were about called for a set of beef that were acclimated for angel about-face (the sequences of accidental corpuscle alternative were again 200 times.). The angel was reconstructed from the subset of overlapping cells, and across-trial airheadedness of a bounded allotment (i.e., overlapping area) of the reconstructed angel was computed for anniversary subset of cells. Alone abstracts for images that absolute at atomic bristles acknowledging beef were acclimated in the analyses in Fig. 6.

In the assay depicted in Supplementary Fig. 7, the babble alternation was computed application the responses beyond stimuli. Evoked responses to anniversary bang angel were adapted to z-scores and calm beyond stimuli in anniversary cell. Then, Pearson’s alternation accessory was computed amid the calm responses in a corpuscle brace and acclimated as the babble correlation. To abolish the babble correlation, responses to anniversary bang were confused beyond trials apart in anniversary cell. Application the confused data, an angel about-face archetypal was acquired as declared aloft for the analyses in Fig. 6g, Supplementary Fig. 7g–i, p–r, and Supplementary Fig. 9f.

The assay in Fig. 7e–g illustrates whether the appearance encoded by acknowledging beef could represent images as a base action complete of absolute neural responses. If the appearance encoded by acknowledging neurons can represent any image, a set of appearance of a accustomed angel (F) will be linearly regressed by the weights of the acknowledging beef in the cell-selection model, H ((= [{mathbf{H}}^1; ldots ;{mathbf{H}}^{1248}]), f  ×  Ncell; Hk in Eq. (8)),

$${mathbf{F}} = {mathbf{H}},{mathbf{B}} {mathbf{d}} {mathbf{e}}$$

(11)

where B and d are chargeless ambit that are affected to abbreviate the sum of the boxlike error, (sum ({mathbf{F}}-({mathbf{H}},{mathbf{B}} {mathbf{d}}))^2), and e is an absurdity term. F was called from a assay dataset, and H was acquired from a training dataset in the ten-fold CV. The applicable was evaluated by artful the applicable absurdity (Error) on the angel amplitude as follows:

$${mathbf{I}}^prime , = ,{mathbf{G}}_{{mathrm{rev}}},{mathbf{F}}$$

(12)

$${hat{mathbf{I}}} = {mathbf{G}}_{{mathrm{rev}}},{hat{mathbf{F}}}$$

(13)

$${hat{mathbf{F}}} = {mathbf{H}},{mathbf{B}} {mathbf{d}}$$

(14)

$${mathrm{Error}} = {sum} {left( {{mathbf{I}}^prime – {hat{mathbf{I}}}} right)^2} /{sum} {left( {{mathbf{I}}^prime – {mathbf{I}}_{{mathrm{mean}}}^prime } right)^2} times 100$$

(15)

where Grev is the Gabor clarify cast acclimated for about-face (Eq. (6)), and ({mathbf{I}}_{{mathrm{mean}}}^prime) is the beggarly pixel acuteness of I′. Thus, this assay estimates how able-bodied the appearance of alone neurons in a bounded citizenry could represent the angel appearance complete of absolute neuronal activity. In added words, this assay estimates the upper-bound accommodation of a bounded citizenry to represent any angel with an ideal aggregate of corpuscle appearance (with ambit B and d).

Because the alive mice were not accomplished to run, they generally backward calmly during imaging. In the analyses apparent in Supplementary Fig. 11, we included alone abstracts from two planes in one abrasion (Thy1-GCaMP6s mouse) that ran about frequently. Furthermore, we acclimated alone abstracts for images that absolute at atomic bristles acknowledging cells, four active trials, and four comatose trials (80 angel cases, n = 295 acknowledging cells). The active accentuation base (RMI) for anniversary corpuscle was authentic as follows: RMI = (Rrun − Rrest)/(Rrun   Rrest), breadth Rrun and Rrest were the beggarly evoked responses during active and resting, respectively. RMI was computed in anniversary responded angel and averaged beyond images in anniversary cell. Angel about-face was performed application abstracts with both conditions, and the performances were calm alone in anniversary action (Supplementary Fig. 11d, e).

All abstracts are presented as the boilerplate and 25th–75th percentiles unless adumbrated otherwise. The cogent akin was set to 0.05, with the barring of the belief of cogent beheld acknowledgment (0.01). Back added than two groups were compared, the cogent akin was adapted with the Bonferroni alteration except for the visually acknowledging corpuscle analysis. Two-sided assay was acclimated in all analyses. The abstracts were not performed in a dark manner. The sample sizes were not agreed by any statistical methods but are commensurable to the sample admeasurement of added letters in the field.

Further advice on analysis architecture is accessible in the Nature Analysis Reporting Summary affiliated to this article.

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