Across-animal odor decoding by probabilistic manifold alignment
Authors: Pedro Herrero-Vidal
Dmitry Rinberg
Cristina Savin
Venue: NeurIPS 2021 Spotlight (2021)
Keywords: neuroscience
probabilistic models
latent dynamical systems
decoding
neural alignment
across-animal neural analysis

Abstract: Identifying the common structure of neural dynamics across subjects is key for extracting unifying principles of brain computation and for many brain machine interface applications. Here, we propose a novel probabilistic approach for aligning stimulus-evoked responses from multiple animals in a common low dimensional manifold and use hierarchical inference to identify which stimulus drives neural activity in any given trial. Our probabilistic decoder is robust to a range of features of the neural responses and significantly outperforms existing neural alignment procedures. When applied to recordings from the mouse olfactory bulb, our approach reveals low-dimensional population dynamics that are odor specific and have consistent structure across animals. Thus, our decoder can be used for increasing the robustness and scalability of neural-based chemical detection.

Across-animal odor decoding by probabilistic manifold alignment
Pedro Perrero-Vidall Center for Neural Science
Dmitry Rinberg Neuroscience Institute Center for Neural Science NYU Langone Health rinberg@nyu. edu
Cristina Savin Center for Neural Science Center for Data Science New York University cs5360@nyu. edu edu
Neuroscience Institute New York University pmh31 edu
Abstract
Identifying the common structure of neural dynamics across subjects is key for extracting unifying principles of brain computation and for many brain machine interface applications. Here, we propose novel probabilistic approach for aligning stimulus-evoked responses from multiple animals in common low dimensional manifold and use hierarchical inference to identify which stimulus drives neural activity in any given trial. Our probabilistic decoder is robust to a range of features of the neural responses and significantly outperforms existing neural alignment procedures. When applied to recordings from the mouse olfactory bulb, our approach reveals fow-dimensional population dynamics that are odor specific and have consistent structure across animals. Thus, our decoder can be used for increasing the robustness and scalability of neural-based chemical detection.
1 Introduction
Collective network dynamics are the foundation of neural computation, from early sensory encoding []] to working memory [2], decision making (3) motor control [4] Neural population activity often has low-dimensional structure that is qualitatively preserved across sessions [3], behavioral states [5]. and even animals ( Aligning neural datasets from multiple recordings and animal subjects into a common latent space could provide powerful tool for extracting unifying principles of brain computations. Nonetheless, progress is hampered by the lack of robust statistical tools for extracting shared neural population dynamics across datasets.
The alignment of low-dimensional neural manifolds is equally important in brain- wwin-ccompueer interface (BCT) applications, where the assumption of common low dimensional structure neural responses is used to compensate for instability in neural recordings and ensure robust performance over time
9 10 This not only restricted to motor control; in the sensory domain, a new generation of chemical detectors exploit the unsurpassed sensitivity of rodent olfactory receptors, but bypass the limitations training animals to report odor identity by directly decoding it from neural responses
13.33. Unfortunately, the applicability of this idea is limited by the need to learn the mapping between neural activity and chemical identity on an animal- by-nnal-b--nmmaal basis, which involves costly data collection [14). Statistical tools that make use of data from previous animals to quickly calibrate the decoder in a new animal could dramatically increase the practical use of such technologies.
vast number of differ- The basic structure of the rodent olfactory bulb (OB) is well-understood [15]
ent olfactory receptors project to distinct OB subpopulations (glomeruli) whose spatial organization is largely preserved across animals 16 17 Odor identity is encoded in www-dimensional transients of their population dynamics (1 I8 [3] though how the glomerular topography translates into stimulus-dependenn network dynamics is less clear. Nonetheless, given the anatomical and functional similarity of the rodent olfactory bulb, it is likely that the OB response to individual odors is itself
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Figure 1: Overview of the model. A. Different odors elicit distinct trajectories in a shared neural latent dynamical space, with animal-specific measurements resulting distinct observations. The goal is to decode the identity of a test odor, given parameters estimated from dataset that combines measurements from all animals. B. amLDS graphical model with latent dynamics specified by odor-specific & parameters, and animal-spedt specific observation model for measurements X₃.
somewhat stereotyped across animals. With appropriate alignment, one could construct 'universal' odor decoders, applicable across animals.
Here we propose novel probabilistic approach to aligning observations across neural recordings taken from different mice so as to extract commonalities their circuit-level representations of different odors. Our estimation procedure, which uses our 'aligned mixture of latent dynamical systems' (amLDS) graphical model, learns independent mappings of different recordings into common latent manifold In this space, neural trajectories are similar across animals but distinct across odors which allows for robust odor identification. Moreover, our probabilistic decoder provides uncertainty estimates which can be used for optimal decision making. We used extended numerical simulations to evaluate the properties of the decoder in response to a range biologically relevant manipulations, such variations in measurement noise, dimensionality and across-trial variability in the latent dynamics, (mis alignmentt of electrodes across animals, etc. We then tested our procedure recordings from the OB of mice in response several monomolecular odors. When fit this data, our model confirmed that encoding of odors relies on transient dynamics in a shared oww-dimensional manifold and that these dynamics are largely preserved across animals. Taking advantage of this reproducible structure, we were able to obtain remarkable improvements odor identification performance relative alternative approaches.
2 Shared odor encoding dynamics across animals
We start by mathematically formalizing the idea of shared stimulus-speciffe neural dynamics with animal-specific measurements (Fig. [A]. The amLDS graphical model assumes d-dimensional latent variables z,, which evolve according to stochastic linear dynamics with odor-specific parameters (Fig. 3):
2, A4 + bR. w.,
where / = (2...7) indexes time within trial; A₁ and by denote parameters specifying the dynamics in response to odor k, with independent noise, W, ~ (0 Qx). The prior for the initial conditions 21 is normal, with zero mean and covariance Qo-
The shared odor-speciiic dynamics are mapped into measured neural responses via an animal-specific linear Gaussian observation model, shared across all odor conditions
(m)
C. Zz Vt
with i..dd noise V. ~ ~O Rm)); C. and R., observation model parameters specific to animal m. We will use N,, to denote the size of observations for animal m. Note that the animal-specific observation model means that it is not strictly required for the neural measurements to be tightly matched. For instance, different animals may have different numbers of measurement channels (e.g. due some electrodes becoming damaged). Moreover, the model accounts for variability in the SNR of different electrodes (across channels and devices), Assuming that the measurement process
"Code available: github.ewm/peeoookrrrrovdv/aaaas

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is consistent in terms of the properties of the electrodes and their alignment on the olfactory bulb surface, one can additionally define a prior that encourages Cs be similar across animals. To keep the procedure general, we only consider the unconstrained version here.
Parameter learning
The dataset D combines measurements from M animals, jointly covering all K stimulus conditions. For each animal-stimulus pair, (m, k). we have m_, trials 2 The goal of the parameter estimation procedure is to derive maximum likelihood estimates of all model parameters, jointly denoted by {Ai:k. b1-k: Qu:K, C1.M, R1.M }. We use expectation maximization (EM) to optimize the parameters, by introducing the variational distribution ((z) and using to lower-bound the marginal likelihood [91
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Due to the Markovian structure of the dynamics, the joint probability of the latent and observations has relatively simple joint Gaussian structure. The E-step optimizes L w.r.t by setting g(z) = P(z|x, x, 0); this computed by traditional Kalman smoothing using the appropriate trial-specific parameters (Suppl. S1).
The M-step optimizes L w.r.t. to A. The cost funetion decomposes into the of terms that only depend on the subject-specific observation parameters (Cms R.,) and the condition-specifi latent dynamic parameters (Aa, Qu: Qa, bu,r), respectively, and can be optimized separately. Taking the derivative of the loss and setting it to zero results in parameter updates:
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Note that not necessary all stimuli are presented to all animals, nor that the number stimulus presentations identical across animals and conditions.
Too keep notation simple, drop the explicit indexing time, trial, stimulus type, and animal of and
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The updates for the observation model are the standard Kalman ones, with the distinction that the index iterates across all trials and stimulus conditions for particular animal (total number Im =x Imnk):
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All the required posterior moments are obtained during the E step via Kalman smoothing as [[ Maj,, R[|,,, ' 2gr PMTM/4 (see Suppl. S1 for further details).
As an important practical side note, the quality of the estimated parameter depends critically on good initialization 0. We achieve this by fitting a simpler factor analysis model (FA), which ignores all temporal dependencies. This directly yields initial conditions for C. and R. The inputs b are initialized the average of the corresponding inferred latent variable 2,, using all trials from stimulus condition k. while parameters A& and Qu:k are initialized heuristically by linear regression using the FA Fn-inferred posterior mean estimates of the latents.
We used Bayesian model comparison to determine the dimensionality of the latent space from data (evaluated separate validation set). As more intuitive measure of goodness-of-fit estimated the reconstruction error when predicting unobserved measurement dimensions (also known leave-neuron-out error), commonly used for assessing the quality of models of neural population dynamics 20 21. More specifically, we use trials from the validation dataset, infer the posterior over the latent trajectory given data measurements in all electrodes except the j-th by Kalman smoothing (Z1.r x j, 0), which is used to predict the values of the response of the jth neuron, x = C (-1)
also
where C is the jth row of C.. We report reconstruction error as the squared distance between the reconstructed and measured activity, {[x] X (l)) where || ||2 is the L2 norm and (+),, is the mean over trials and choices for
Odor decoding as hierarchical inference
At test time, needs to infer the identity of the stimulus given the observations X1.7 in animal M, and the estimated parameters 0, obtained by Bayes rule, P(k[x1.7: 0) x P(xir ( (k).. This requires estimating the marginal likelihood for each possible stimulus condition, which can be computed iteratively during the Kalman filtering procedure. More specifically, using the chain rule and then the Markov structure of the stimulus-specific LDS, have
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Taking the logarithm, rearranging the terms and taking advantage the fact that the poste- rior marginals normal with parameters obtained by Kalman filtering (see Suppl. Info. S1), P P(2//01-11 k, 0) = (N E1jr-1), yields:
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Lastly, here we assume a uniform prior over possible stimuli, to match the statistics of our OB dataset. Nonetheless, for real world applications the prior should reflect natural stimulus statistics.
3 Numerical experiments on simulated data
To validate the parameter estimation and decoding procedure, we constructed artificial datasets with the same statistical regularities assumed in the graphical model. More specifically, defined

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Figure 2: Parameter estimation and decoding simulated data. A. Example latent dynamics for different odor stimuli; solid lines indicate the average latent trajectory and dashed lines are five random trials. B. Measurements of an example trial in two animals. C. Latents recovered with amLDS: dashed lines shows posterior mean of individual trials, solid lines show across- -tralss average. D. Reconstruction етгог function of assumed latent dimensionality, d (ground truth d 3). E. amLDS decoding performance measured using MAP estimates of the stimulus; K 50. F. Decoding accuracy function of the magnitude of noise in the observations (top). and the latent dynamics (bottom). G. Decoding accuracy for parametric changes model hyperparameters: alignment of axes across animals, C (left), the number of stimulus classes, K (middle) and the dimensionality of the latent space, d (right). H. Decoding accuracy as a function of the amount of data available for a target animal (M of the = 50 stimuli (green), compared to using the same probabilistic model in only the target animal (blue), alignment by canonical correlation analysis (gray), mCCA (dark gray) and Proerustes alignment combined with factor analysis dimensionality reduction (pink). Solid lines show mean accuracy, shaded areas s.c.m. estimated across independent experiments (random seeds).
for amLDS when the target data covers all stimuli (red) only one
shared low dimensional manifold 3) and embedded K 50 latent trajectories evolving over =' 41 time steps (Fig2AA). The stimulus-dependent inputs b2 were constructed using common template with stimulus-specife amplitude (individual dimensions scaled by a value drawn from (1; 0.0004)) and rotation (evenly spaced over 170 degrees). The other latent dynamics parameters were randomly: matrices A₁ have diagonal entries drawn from (0.4;; 0.01) and off-diagonal drawn from (0; 0.04) and the noise covariances Q2 and Qo are diagonal with variances drawn observation model we started from
from (0.555 0.0025). model across-animal variability prototype projection matrix C which we then morph into animal specific parameters Cm by corrupting the individual latent axes with independent additive noise N [0 a²), and rescaling their norm to noisy value around 1.0 (gaussian noise with variance 0.03). Parameter controls the degree of alignment of measurements across animals, with default value 0.1. Finally, the observation noise covariance is set to be diagonal with variances set to the absolute value of independent draws from (O; 0.25) (Fig 3). This setup results in latent dynamics qualitatively similar to odor responses, and has the advantage of explicitly parametrizing the similarity across animals and stimuli.
As a first test of the estimator we simulated 5 animals, with Im.k 50 trials for each stimulus condition, which is on the same order of magnitude to the amount of data one may be able to collect in experiments. amL.D has low computational and memory demands; on a 2.9GHz CPU it takes 58 minutes for parameter learning (with 12500 trials; average over 5 runs) and 4 seconds to infer
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stimulus class in given trial (averaged over 2500 trials). In this setup, model comparison based on reconstruction error was able to correctly discover the dimensionality of the latent dynamics (Fig2D) and reconstruct the underlying trajectories (Fig 2C), up to the expected degeneracies in the latent space scale and alignment [19]. Moreover, the decoding of stimulus identity of unseen test trials (20 per condition) revealed that hierarchical inference is able to correctly identify the true stimulus out of the 50 possible classes 25), where we measure decoding accuracy as the fraction of test trials for which the MAP estimate of the stimulus class recovers the ground truth stimulus identity.
Qualitatively similar results can be obtained for a wide range of model choices. Performance is consistent across a wide range of noise levels in the observations (Fig2F, top), which account for differences in recording quality. The same robustness can be measured for noise in the latent dynamics, due to potential changes in encoding owed to masking odors, variability in sniffing, etc Q ~ (j,, 0.0025), l E 0.9): Fig :: bottom). The estimation is also relatively robust to different degrees of variability of the projections C.m. across animals (Fig2G, left), which may arise for example due to variability dimensionality of the latent dynamics (F middle) and the number of stimulus classes (Fig 2 right) and found that our decoding procedure shows robust performance for a wide range of settings. Given the relatively modest data requirements (50 trials per subject, and condition), these observations suggest that our approach is robust enough to be applicable to experimentally relevant data regimes.
the alignment of the electrodes on the OB surface. We also varied the
To better quantify decoding performance as function of the amount of data that needs to be collected in new animal, we fixed the amount of data collected from 4 'source' animals (K = 50, Lm. = 50) and varied the amount of new data collected in a fifth 'target' animal, for which we evaluated decoding accuracy. We found that this setup amLDS yields high performance with a minimal amount of additional data (Fig 2 red; one measurement per stimulus condition). Reaching the same decoding performance with data collected in the target animal alone would require many more trials, because the dynamics of all K stimuli need to be learned de novo (Fig 2 blue). In the case of amLDS however, (effectively) one only needs to align the new measurements to the manifolds estimated based on data from previous animals f In fact, observing a few trials for single stimulus is enough to achieve reasonable performance in the target animal 2 green).
We compared the decoding performance of our method to three state-of-the arr methods: 1) across condition alignment by canonical correlation analysis (CCA) [3]23]]
23) 2) multi-set CCA (mCCA)
24 and 3) Procrustes alignment with FA-based dimensionality reduction (FA+Proorustes) [ 6 In
general, for all these approaches the latent dimensions are determined from data that combines all stimulus conditions, without explicit knowledge of the stimulus identity. Decoders are trained based on the resulting representation using stimulus labels.
CCA takes pairs of measurements from two animals and projects them into a shared low dimensional manifold so as to maximize the correlation between them. Since there is no meaningful way of pairing single trials, CCA requires averaging responses across trials within condition and pairing these average responses by stimulus type. Moreover, unlike our approach, both procedures treat measurements as independent over time, ignoring temporal structure in the neural responses. Since simple CCA can only be used for pairs of animals, we increased the number of trials in the single source animal to match the other methods. Additionally, we also used multi-set CCA, a generalization of CCA for more than two animals.
The alternative PA+Procrssss alignment is similar to CCA in that it requires across-trial averages and only considers similarity between pairs of measures; however, it can be applied multiple times to align animals 2 - M to the prototype axes given by animal 1 (the target animal data was used for FA and Procrustes parameter estimates in this case). More formally, we use orthogonal Procrustes 25 where given a set of prototype . and misaligned Özi.r trajectory pairs finds a rotation between the two, by minimizing the average misaligmment after applying a rotation O argmino En liz Ol?:r under the constraint of orthogonality, 00 = I: where | || is the Frobenius norm. The stmuulus-ppecine latent trajectories zf.r are obtained via a dimensionality reduction step, performed independently in each animal by factor analysis. As there no standard way for establishing the shared latent dimensionality based on data for (m)CCA and FA+Procrustes, we set it to the ground truth in all cases. Having established a linear map between observations and a common latent
*While the estimation procedure still uses EM based on the full data, the fractional contribution of the new measurements to the latent parameter updates is comparatively small, and becomes negligible as the number of sources increases.

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Figure 3: Mixtures and concentration. A. Estimation of odor identity across concentrations (right); example trajectories where decreasing concentration induces compression of the stimulus trajectories (left). Shaded area shows s.c.m. across simulations. B. Example trajectories for two odors (A-orange and B-blue) and 50-50% mixture that linearly interpolates between them (left). Frequency of the MAP estimate detecting odor A, as function of the mixture ratio (middle) and three example posteriors (right); dashed lines show individual simulations, with average in solid line.
space by either procedure, individual trials across conditions were all projected in this space. We used a linear SVM with L2 regularization to decode stimulus identity from the latent trajectories [33 26). All controls were constructed using the library scikit-learn [27]. We found that although all three approaches performed above chance, none came close to matching the across animal decoding performance of amLDS.
Robustness to iluctuations in concentration, odor mixtures
Odor decoding in the wild needs to overcome further challenges that are not explicitly captured in the amLDS graphical model, such as fluctuations in concentration that rescale the amplitude and speed of odor specific responses (28). Moreover, odorants are rarely present in isolation and have to be detected when presented in mixtures [29]. We used further numerical simulations to test how the estimator would respond in such scenarios.
We model across-trial -riill variability in odor concentration as changes in the amplitude of the input drive br, [30]. with all other parameters the same before. The parameter learning used data from the highest concentration and the decoder was applied to test trials with lower concentrations (Fig AA). We found that the decoding accuracy degrades gracefully the amplitude of responses decreases, remaining above chance for a wide range of concentrations.
It remains unclear how mixtures encoded in OB responses [3]. but reasonable first approxima- tion to use a linear interpolation of sample trajectories for the individual odors present [2] 33 For simplicity, used K 2 and further tied the parameters A and Q of the latent dynamics. We constructed two component mixtures which interpolate between stimuli labeled A and B (Fig 3 left). Using model trained on individual odors, estimated decoding performance trials while varying the ratio between the two components (Fig 3F. middle). The posterior over the two odors appropriately reflects the inherent ambiguity in the observations (Fig 3, right). Overall, these results suggest that the decoding procedure is robust to naturalistic odor variations.
mixture test
4 Across-animal decoding in the rodent olfactory bulb
We tested our model on neural recordings from a 64-site grid-electrode stereotaxically implanted over the dorsal part of the olfactory bulb in five mice [13]. We simultaneously recorded a pressure readout of the animal's s sniff cycle. The animals were presented with five different monomolccular odorants (methyl valerate, ethyl tiglate, benzaldehyde, hexanal and background air), delivered in a temporally precise window (20- 100 trials each). We extracted glomerular responses time window of 2 210ss from inhalation onset. The neural responses were preprocessed by removing the high frequency component (100Hz low-pass filter) and electrical noise (60Hz notch filter). Single electrode signals with peaks exceeding 2 in a 5 second period were considered damaged and removed from the study. The electrode-specine signals were extracted by subtracting the instantaneous mean across all viable electrode sites.
We split the animals into sources, and one target. All data from the source animals and 10% of the trials from the target animal were used for parameter learning and hyperparameter estimation. The
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remaining data from the target animal was used to assess odor decoding performance. When fitting the model to the neural recordings we found that the inferred latent trajectories were separable across stimuli and similar across animals (Fig. A Some across-animal variability in the responses to the same odor would be expected in the model for a limited number of trials due to the independent instantiation of the noise in the latent space, but it may also reflect unaccounted for sources of across-animal variability. The model comparison revealed a relatively small latent dimensionality in the data (Fig. AB; we used d = 7 for all subsequent analyses). The model accurately captured variability in observed electrodes in the target animals, despite the very limited target data (2-6 trials per stimulus; Fig. 4C). The inferred map from the neural measurements into the latent space was similar across animals (Fig. 4D; see also Suppl. Fig. S1), likely reflecting the stereotypy of OB encoding and precise alignment of the electrodes on the bulb surface.
We evaluated the decoding performance of amL.DS on target animal which revealed high accuracy MAP inference based decoding :), with generally sharp single trial posteriors (Fig. 4+) We measured decoding accuracy for all possible choices of target animal and compared it to alternative estimators (Fig. 4G). The CCA, mCCA and FA+Procrsstes estimation procedures were similar to those described for the simulated data, with the same latent dimensionality inferred for amLDS. We used across trial averages within each stimulus condition from the source animals and the 10% fraction of the target animal data for dimensionality reduction and across aniimll aligmment. We used the same data to train linear SVM classifier based on the latent trajectories, which we then used to predict odor identity for the withheld target data. For CCA [22] we used single source animals and the training target data for the alignment and SVM training. averaging decoding performance test data across all possible sources. Our probabilistic decoder significantly outperformed all three alternative alignment methods. For all possible choices of target, amLDS also systematically outperformed probabilistic decoding based on data from the target animal alone, confirming that pooling together data across animals improves data efficiency and overall decoding performance. While the single animal performance is close to that of amLDS given enough data, 82% + 8% vs. 76% = 3% when training with 50% of the target animal's dataset (Fig. 4G), the difference between the two becomes dramatic in the low data regime used for amLDS (10% of target data used for training). with single animal performance going down to 60% = 8%. This reinforces the idea that across- animal decoding is key to limiting the amount of data that needs to be collected for calibrating any new animal at deployment time.
5 Discussion
Animals are unparalleled in their ability to sense odors, which makes them an invaluable resource for a wide range of applications from security medicine (34 BCI approaches to odor detection attempt to bypass costly behavioral training by reading out odor information directly from the olfactory system, but they remain data limited [14). Here we proposed a novel probabilistic procedure for mapping neural responses to odors into a shared manifold. This representation allows for robust odor decoding in a new animal with minimal amounts of further data collection. Calibrating the decoder to a new animal can be achieved with small subset of stimuli, limiting the demand for rare/expensive chemicals. For instance, in our numerical experiments the decoder could extract odor identity in a new animal with as few as 2 trials for each stimulus, or 50 trials of single stimulus. Moreover, we also found the estimation procedure to be robust to modeled fluctuations in odor concentration and the presence of background odors. These benefits were confirmed in data from mouse OB, where we found that decoding quality improved dramatically compared to prcviously-propose alignment procedures, particularly when using a limited amount of data.
Our animmll probabilistic decoding procedure brings neural chemical detection closer to practical applicability. Not only does it improve scalability and robustness, but it also significantly reduces data collection requirements. Furthermore, most odor detection applications have asymmetric costs and need to negotiate complex trade-offs between sensitivity and accuracy. For instance, when detecting Clostridium difficile [37], the medical costs of missing a positive diagnosis needs to be balanced against the financial costs of unnecessarily quarantining someone due to a false alarm. Our probabilistic decoder provides an explicit report of uncertainty associated with possible outcomes, which is critical for optimal decision making in these contexts.
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Decoding performance on neural data from mouse olfactory bulb. A. Inferred trajec- tories for two monomolecular odors (colors) in two animals (solid and dashed lines), averaged across 24 trials. B. Model selection based on reconstruction error performance; dashed lines show errors estimated for each target animal, solid line their average. C. Example odor-evoked responses and amLDS reconstructions for 2 stimuli and 2 animals. Shaded areas show 95% confidence intervals based on inferred posterior. D. Similarity of observation model across animals. E. Confusion ma- trix for amLDS decoding. F. Corresponding single trial posterions; colors mark different inferred odors, trials are grouped by presented odor. G. Decoding performance for amI.DS and alternative approaches based on CCA, mCCA, Procrustes alignment and probabilistic decoder trained using data from the target animal alone (50% training, 50% test). Lines correspond to different choices for the target animal. Error bars show s.e.m. across 10 folds of the target data, for CCA s.e.m. across sources.
Figure 4:
Alternative procedures for neural datasets alignment are usually not explicitly probabilistic | Instead, they rely on a sequence of independent steps: first a (usually off-the-shelf) dimensionality reduction procedure. Dimensionality reduction is followed by across-condine condition data alignment within the latent manifold, using approaches such as CCA (and its multivariate generalizations (39] 40)). or
Procrustes alignment 6 9 Lastly, common decoder is trained on the aligned across-condition data [22 10 9.. These procedures are popular as replacement for costly and inconvenient BCI re-calibration prosthetics [42] but their application to across-animal alignment remains rare 7 6 The multi-step procedure, although formally suboptimal, allows for some degree of flexibility in the individual component steps. For instance, the map into the latent manifold could be nonlinear, e.g. using Isomap 7 43 There also exist several nonlinear alignment procedures, such as kernel CCA Distance Covariance Analysis, although such extensions have yet to translate into BCI practice [4]] In contrast, the hierarchical model proposed here is jointly estimated and linear in both the map and the latent dynamics. The linearity assumption here is not just a simplifying mathematical assumption, but it reflects domain-speciff knowledge about the underlying neurophysiology of the system.
Neural activity has complex temporal dynamics and structured across-trial variability 45) Previous approaches to aligning population activity across conditions via CCA or Procrustes alignment fail to capture either of these key statistical regularities. First, they treat the measurements across time as independent. Second, they intrinsically rely paired measures, and since the correspondence of individual moments in time across trials in different animals is essentially arbitrary, they are restricted to across-trial average responses that may obscure important features of the underlying computation is also less clear how to determine the latent dimensionality of the data using these procedures. In contrast, amLDS extracts common low-dimensional dynamics using single trials, exploiting known features of the underlying computation. Moreover, the animal-specific observation model provides principled framework for determining data dimensionality and for handling different sources of experimental variability (e.g. due to grid misalignment or missing data from defective electrodes). The probabilistic formulation also lends itself to variety of generalizations. One could easily convert from Gaussian to Poisson observation noise for modeling spiking data instead of
46. 47).
*Some exceptions exist, such as probabilistic CCA
38 but they are commonly used in practice.

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or replace the linear dynamics with shared nonlinear equivalents [50, [T] It should also be possible to extend the prior over trajectories to allow for some degree of heterogeneity across animals for instance by allowing the parameters of the latent dynamics to slightly vary across animals, or to fall into few distinct 'dynamies classes' (formally, an LDS mixture) [8]. Hence, amLDS can be thought of as yet another building block in the expanding statistical toolkit for understanding neural computation through the lens of dynamics in low-dimensional manifolds.
the raw electric signals in our data [48 49
The representational stereotypy in the rodent olfactory bulb and the diwdemmesiooall structure of its odor-induced dynamics are well documented [16] [3] make this system an ideal test case for out model. Nonetheless, we expect the same statistical tools to prove useful in other systems. The shared latent dynamics are best thought as reflecting a common computational solution, not literal match (in contrast to the hierarchical switching linear dynamical system approach of Ref. [8). Hence, our model assumptions remain reasonable in circuits where representations are learned, rather than genetically encoded, as long as the dynamical system underlying the computation is low dimensional and the task strongly restricts the topology of the solution [52] (e.g. attractor dynamics for evidence integration (3). We hope that our results inspire the search for common dynamic signatures in other brain areas and model systems.
or subpopulation-te -os---ttssoppuaationn
Author contributions. CS and PHV conceived the original idea, developed the model and wrote the manuscript. PHV performed the numerical simulations and the data analysis. CS supervised the project. PHV collected the neural data in DR lab, previously published Shor et al. 2022 [3]
Declaration of competing interest. DR is one of the founders of a company that commercializes
related bio-electronic technology [13
Acknowledgements. PHV is supported by training grant R90DA043849 (NIH). CS is supported by NIMH award IRO1MH12557 1-01, NSF award 1922658 and Google research faculty award. DR
was supported by DARPA BAA 15-35. We thank Caroline Haimerl, Edoardo Balzani and Erez Shor for their constructive feedback.
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13


Reviews:
[{"review_id": "dsFujLedIz3", "reviewer": "Reviewer_FTaj", "summary": "The authors develop a method for aligning the neural activity of multiple animals. Specifically, they address the challenge of odor detection, where neural activity from multiple animals is recorded in response to many stimuli (odors), and then the odor needs to be decoded from a given animal’s neural activity. They use a hierarchical and probabilistic approach, in which there is a shared low-dimensional space of neural activity that is shared across animals, with different low->high dimensional mappings for each animal. The authors demonstrate the utility of the approach in simulations and on experimental neuroscience data.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Quality:\n\nThe quality of this paper is high. The submission is technically sound. The method was tested out rigorously in simulations and also shown to work well on experimental neuroscience data.\n\n\nOriginality:\n\nThis is a new method that builds upon linear dynamical systems / graphical modeling to create an effective model for the given problem.\n\nThe relevant literature was cited well.\nOne additional piece of relevant literature is a hierarchical switching linear dynamical system (which is similar to your approach but has an unknown k during training), as in\nhttps://www.biorxiv.org/content/10.1101/621540v1.full\n\n\nSignificance:\n\nI think the broad topic that the authors are trying to address - aligning neural activity across subjects has broad interest across the neuro/AI communities. However, the specific application of odor detection seems limited in terms of its broad appeal. I think this paper would be much stronger (and in my opinion, a clear accept rather than a borderline accept), if you were able to demonstrate its generality to other applications as well.\n\n\nClarity:\n\nWhile much of the paper was written clearly, I found several aspects difficult to understand. In particular, I think it would be helpful to be more explicit in the beginning about what aspects of the model are learned during the training phase, and then what you are trying to predict during the testing/decoding phase (both for your model and the comparison models). Here are a couple specific points along those lines: \n\nDuring training, is it assumed that the odor is always known? It seems like yes, but please clarify up front.\n\nLines 171-196: I initially found this section confusing since you describe decoding after describing the state-space alignment, so I was confused how you were actually doing decoding w/ CCA and FA+Procrustes. Having a sentence upfront about the overall approach would be helpful for clarity.\n\nOn your target animal, are you updating any of the parameters of the model besides learning the C and R for that new animal?\n\nWhen decoding in the comparison methods, are you only using the target animal labels (odors) for the purpose of aligning the datasets, or are they also used when training the decoder?\n\n\n\nOther:\n\nI think multi-set CCA (which is an extension of CCA for >2 subjects) would be another natural comparison.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations are addressed\n\nPotential negative societal impacts are not mentioned in the text", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "7"}, "extra_text": {}}, {"review_id": "XhFRVfCdrHw", "reviewer": "Reviewer_tRLv", "summary": "Recent advances in recording techniques allow us to record from a large population of neurons - this high dimensional activity can typically be described by low dimensional latent factors. The authors leverage this low dimensional structure to capture the shared dynamical structure across animals in neural recordings from the olfactory bulb as animals are exposed to various odor stimuli. They fit a linear dynamical system (LDS) model with latent parameters and trajectories estimated for each odor that are shared across animals. These latent trajectories are subsequently mapped onto the neural recordings by employing animal specific observation parameters. The LDS model parameters are optimized using expectation maximization (EM). In order to decode the odor identities, the marginal likelihood for all stimulus conditions is computed. They validate the model on simulated data as well as neural recordings from the olfactory bulb. ", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths: Aligning low dimensional structure in neural population recordings across animals is an open problem in neuroscience. The paper provides a novel approach to recover these shared latent dynamics across animals. They demonstrate that their model can recover previously observed low dimensional transient dynamics for odor stimuli in data recorded from the olfactory bulb. They are also able to decode the odor identity across animals by training the model on source animals and a small subset of trials from a target animal. Moreover, the model outperforms CCA and Factor analysis followed by Procrustes alignment on across animal decoding.\n\nWeaknesses: The authors make a strong simplifying assumption of identical dynamics across animals that has not been adequately justified in the paper. \n\nNeural recordings during each session and across animals might be obtained from different locations in the same area or from a vastly different subset of neurons. It’s highly likely that the recorded neurons are part of different circuits and coding for different dynamical features. \n\nAnimals also have different sniffing cycles that could introduce an added level of complexity in the temporal dynamics that will not be accurately captured by this model. It is not necessary that the dynamics will be exactly temporally aligned. \n\nOther factors such as varying levels of attention and arousal could further impact the dynamics of odor coding. \n\nDue to the underlying assumption, this method is limited in scope and its applicability to other tasks and areas remains unclear. Maybe a weaker assumption with constraints on the dynamics parameters; for example, by employing a hierarchical model (Linderman et al., 2019), could make this model more flexible and as a result, more widely applicable.\n\nOriginality: To the best of my knowledge, this is the first work that aims to align neural dynamics across animals by having a common dynamics model.\n\nQuality: The paper is well motivated and the writing and figures is easy to follow. \n\nClarity:\nThere are some minor parts that can be improved or might need further explanation.\nLine 102: (typo) evaluated on on.\nLine 77: It is not necessary to introduce a variational distribution since the authors are using regular EM. \nFig 4: I’m not sure I understand the source of variation between the latent trajectories for the same stimulus and different animals in panel A. Some further explanation here would be helpful.\n\nSignificance:\nAligning low dimensional dynamics across sessions and animals is elemental for identifying common encoding principles in the brain. However, as mentioned above, the strong assumptions in the model put stringent constraints on the dynamics and limit its applicability to other brain areas or contexts. \n\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "M7TWRcmEH0", "reviewer": "Reviewer_Qtbh", "summary": "The report provides a method for decoding stimulus identity (odor) using data from multiple animals. The framework is straightforward and the performance appears respectable. I anticipate that this approach could be of use in neuroscience labs and could help with developing models and BCI. Given its general applicability, it would strengthen the impact if its use was demonstrated on spiking data and/or other types of stimuli or behaviors. This is a well written paper with helpful illustrations. I expect it would support interesting discussions.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Neural activity is variable, even when sensory stimuli or motor performance are repeated. However, the underlying perceptions or intentions could be quite similar. When analyzing neural activity, it can be good to isolate the conserved components, despite this variability. A straighforward example is averaging over trials, but this only works when the dynamics are relatively time-locked. When the dynamics can play out over a range of speeds, the problem is less trivial. In olfaction, sync'ing to the sniff cycle is one way to register across trials, and in that case, trials are time warped to match a landmarks in a sniff cycle. For this paper, the author(s) developed an approach for solving this problem, and it is more general than timewarping to match the sniff cycle. It can be applied to a range of applications, sensory modalities, and behaviors. The framework allows decoding across animals. The approach is based on the idea that each animal has a set of neural activity manifolds that are each specific for a stimulus or behavior. Manifolds can be combined across animals for decoding. A graphical model, an aligned mixture of latent dynamical systems (amLDS), is computed from the dataset with this idea. They compare performance to CCA and FA+Procrustes. On the one hand, these are standard tools and so it is a reasonable comparison. On the other hand, they are very general tools that are used without taking advantage of the structure of the data, and so it is not so surprising that they provide less accurate results. The performance they present in Fig. 4 looks respectable.\n\nSince they present olfaction data, I am curious to compare this approach to simple sync'ing to a sniff cycle (and they have that data, line 219). But I do not consider this a major weakness. I wish they had presented an application to spiking data. I will not mark that as bad weakness, but I do think it is something that others in the field would like to see. Relatedly, the impact could be greater if the technique were applied to a range of data types (e.g., visual stimuli, motor activity in a reaching task, parietal activity in a two-alternative forced-choice task). Again, it is not a weakness, but I would appreciate seeing it, and I bet I am not alone.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "No concerns.", "review_text": "", "extra_scores": {"ethical_concerns": "I have no ethical concerns with this report.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "I_2indSscLm", "reviewer": "Reviewer_HVvC", "summary": "This submission is a very well done application of the probabilistic inference to decoding of odour identity from neural recordings in mice. Interestingly, the latent code is assumed to have temporal dynamics shared across animals. The main benefits (high accuracy, data savings,  robustness to incomplete data) are shown in simulation and neural recordings. The method outperforms the existing (non-probabilistic) neural alignment procedures. \n", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "To the best of my knowledge, this is a novel method for this important (BMI) application. It should have a large impact on the olfactory field, as it enables an effective combination of neural recordings from many animals.\nThe relevant work is adequately cited and discussed with two methods implemented for comparison.\n\nThe paper is clearly written and comprehensive. \n\nMinor comments:\n- L59: Shouldn't $x_1$ be $\\mathbf{z}_1$?\n- L119: Equation, first line is missing an integral over $\\mathbf{z}_t$.\n- Update Figure 4 B caption (\"dashed lines...\")", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I assume the odours used in the experiment were very different. I'd be curious to learn about the temporal latent characteristics over more odours. What are the limits to discriminability? Is the detection of Clostridium difficile even feasible? I'm looking forward to updates into the \"space of odours\" following up this work. \n\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {"ethical_concerns": "In terms of potential applications - is anyone considering using BMI for odour detection? Perhaps mentioning non-invasive recordings, or BMI improving life of human patients would be more convincing to a general audience?"}}]

Author rebuttal:

Meta-review: This paper presents a novel probabilistic alignment of stimulus driven neural activity for decoding stimulus identity. Reviewers agree that this is an important contribution that can contribute to advancing neuroscientific understanding. At the same time, reviewers also agree on the assumptions that may limit its applicability outside the olfactory system. Please make sure to add the discussions through the rebuttal in the final version.
Decision: Accept (Spotlight)