Capturing instantaneous neural signal-behavior relationships with concurrent functional mixed models

Reviewer #1 (Public review):

Summary:

The authors aimed to extend a prior fiber photometry analysis process they developed by incorporating the new ability to determine instantaneous, within trial, relationships between the photometry signal and continuously changing variables. They present solid evidence via simulations and example use cases from published datasets highlighting that their approach can capture instantaneous relationships. Overall, while they make a compelling case that this approach is less biased and more insightful, the implementation for many experimentalists remains challenging enough and may limit widespread adoption by the community.

Strengths:

This work builds on prior efforts to analyze photometry signals in a less biased and more statistically sound way. This work incorporates a very important aspect by avoiding the need to summarize individual trials with singular behavioral variables and instead allows for interactions with continuously changing variables to be investigated. The knowledge and expertise of the authors and the presentation provide strong validity and strength to the work. Examples from prior studies in the field are a necessary and important component of the work.

Weaknesses:

While use cases are provided from prior data, a clearer presentation of how common implementations in the field are performed (i.e. GLM) and how one could alternatively use the cFLMM approach would help. Otherwise, most may continue using common approaches of Pearson's correlations and GLMs.

Reviewer #2 (Public review):

The paper presents a regression-based approach for analysing fiber photometry data termed Concurrent Functional Mixed Models (cFLMMs). The approach works by fitting linear mixed effect models separately to each time point in trial aligned data, then applying smoothing to the model coefficients (betas), and computing confidence intervals. The method extends the authors previous work on using FLMMs for photometry data analysis by allowing for the inclusion of predictors whose value changes across timepoints within a trial, rather than just from trial to trial. As fiber photometry is a rapidly expanding field, developing principled methods to analyse photometry data is valuable, particularly as the authors have released an R package that implements their method to facilitate their use by other groups. The basic FLMM approach for using mixed effects models to analyse trial aligned photometry data, detailed by the authors in their previous manuscript (Loewinger et al. 2025, doi: 10.7554/eLife.95802) appears valuable. The aim of incorporating variables that change within trial into this framework is interesting, and the technical implementation appears to be rigorous. However, I have some reservations as to whether the way in which variables that change within trial have been integrated into the analysis framework is likely to be widely useful, and hence how impactful the additional functionality of cFLMM relative to the previously published FLMM will be.

In the original FLMM approach, where predictors change only from trial-to-trial, fitting separate regressions at each timepoint generates a timeseries of betas is for each predictor, indicating when and how the predictor explained variance across the trial. This makes a lot of sense and is widely used in neuroscience data analysis. In extending this approach to incorporate variables that change within trial, the authors have used the same method of fitting separate regression models at each timepoint, to obtain a timeseries of betas for each predictor. It is less clear that this approach makes sense for variables that change within trial. This is because the resulting betas only capture how variation in the predictor across trials at a given timepoint explains variance in the signal, but does not capture effects of variation in the predictor across timepoints within trials. This partitioning of variance in the predictor into a between-trial component whose effect on the signal is modelled, and a within-trial component whose effect on the signal is not, is artificial in many experiment designs, and may yield hard to interpret results.

Consider e.g. the experimental condition considered in Figure 3, taken from Machen et al. 2025 (doi: 10.1101/2025.03.10.642469) in which mice ran down a linear track to collect rewards. In analysing such data, one might want to know how neural activity covaried with the animal's position, but as this variable changes strongly within trial but will have a similar time-course across trials, the cFLMM analysis approach will not work to quantify these effects. This is because variance attributed to position would not capture how neural activity covaried with changes in the animals position within trial, but rather how neural activity covaried with changes in the animals position from trial-to-trial at a given timepoint, which could occur due to e.g. trial-to-trial differences in latency to start moving or running speed. As such, although significant effects of 'position' might be observed, they would not capture covariation between position and neural activity in a straightforwardly interpretable way.

It is therefore not obvious to me that incorporating variables that change within trial into an analysis framework that runs separate regressions at each timepoint in trial aligned data is likely to be widely useful. If scientific questions require understanding how neural activity covaries as a function of variables that change both within and across trials, an alternative approach would be to run a single regression analysis across all timepoints, and capture the extended temporal responses to discrete behavioural events by using temporal basis functions convolved with the event timeseries. This provides a very flexible framework for capturing covariation of neural activity both with variables that change continuously such as position, and discrete behavioural events such as choices or outcomes, while also handling variable event timing from trial-to-trial.

One way that cFLMM is used in the manuscript is to handle variable timing of trial events in trial aligned data. In the Machen et al. data, the time when the animal reaches the reward varies from trial to trial, and this is represented in the cFLMM analysis by a binary variable which changes value at this timepoint. From the resulting beta coefficient timeseries (Figure 3C) it is not straightforward to understand how neural activity changed as the subject approached and then received the reward. A simpler approach to quantify this, which I think would have yielded more interpretable coefficient timeseries would have been to align activity across trials on when the subject obtained the reward, rather than on the start of the trial, allowing e.g. the effect of reward type to be visualised as a function of time relative to reward delivery, and hence to see the differential effects during approach vs consumption. More broadly, handling variable trial timing in analyses like FLMM which use trial aligned data, can be achieved either by separately aligning the data to different trial events of interest or by time warping the signal to align multiple important timepoints across trials. It is not obvious that using cFLMM with binary indicator variables that indicate when task states changed will yield a clearer picture of neural activity than these methods.

It may be that I am missing some key strengths of cFLMM relative to the other approaches I have outlined, or that there are applications where this approach to implementing within-trial variable changes is a natural formalism. However my impression is that while cFLMM represent a technical advance, it is not clear how widely useful the model formalism will be.

Reviewer #3 (Public review):

Summary:

This work is an extension of their previous study (Loewinger et al 2025) describing a statistical framework for the analysis of photometry data using functional linear mixed models with joint confidence intervals, together with an open-source tool implemented in R. The present study extends it by adding the possibility of using 'concurrent' variables (variables that change within a trial) as regressors, for example, capturing the change of speed at each timepoint in the trial. The main claim is that using 'concurrent' regressors can identify associations between signal and behavior that could not be captured by 'non-concurrent' regressors (the value for a regressor on a specific trial is the same for each timepoint), which could lead to misleading conclusions. While the motivation for using time-varying covariates is useful and supported by previous literature (using fixed-effects models, although not cited in this manuscript), the reanalysis of previous studies does not clearly prove the benefit of using concurrent regressors as opposed to non-concurrent, and some of the results are difficult to interpret.

Strengths:

• The motivation for using time-varying covariates is well supported by previous literature using them on fixed-effects models, and here the authors are extending it to mixed-effects models.
• The authors have included this new functionality in their previous open-source R package.

Weaknesses:

• The main weakness of this study is that it is not clear what the conceptual or methodological advance of this work is. As it is written, the manuscript focuses on showing how concurrent regressors offer interpretation advantages over non-concurrent regressors. While the benefit of such time-varying regressors is supported by previous literature (e.g., Engelhard et al., 2020), it is not clear whether the examples provided in the current study clearly support the advantage of one over the other, especially in the reanalysis of Machen et al. (2025), where the choice of regressors is confusing. In this specific example, if the question is about speed and reward type, why variables such as latency to reward or a binary 'reward zone vs corridor' (RZ) regressors are used instead of concurrent velocity (or peak velocity - in the case of the non-concurrent model)? Furthermore, if timing from trial start to reward collection is variable, why not align to reward collection, which would help in the interpretation of the signal and comparison between methods? Furthermore, while for the non-concurrent method, the regressors' coefficients are shown, for the concurrent one, what seems to be plotted are contrasts rather than the coefficients. The authors further acknowledge the interpretational difficulties of their analysis.
• Because the relation between behavioral variables and neuronal signal is not instantaneous, previous literature using fixed effects uses, for example, different temporal lags, splines, and convolutional kernels; however, these are not discussed in the manuscript.
• From the methods, it seems that in the concurrent version of fastFMM, both concurrent and non-concurrent regressors can be included, but this is not discussed in the manuscript.
• The methodological advance is not clearly stated, apart from inputting into fastFMM a 3D matrix of regressors x trial x timepoint, instead of a 2D matrix of regressors x trial.
• This manuscript is neither a clear demonstration of the need for concurrent variables, nor a 'tutorial' of how to use fastFMM with the added extension.

Author response:

Common responses:

We thank the editors for considering our paper and the reviewers for their thoughtful and detailed feedback. Based on the comments, we will revise our manuscript to better describe how our approach differs from modeling strategies that are common in the field. We also aim to elaborate on the advantages of fastFMM and what scientific questions it is designed to answer. Finally, we will provide more background on our example analyses and the interpretation of the results.

Within this response, “within-trial timepoints”, “time-varying predictors/behaviors”, and “signal magnitude” are used as specific examples of the general concepts of functional domain”, “functional co-variates”, and “functional outcome”, respectively. To make statements or examples more concrete, we may use the former neuroscience-specific terms when making general claims about functional models.

- ncFLMM, cFLMM: non-concurrent or concurrent functional linear mixed models.

- FUI: fast univariate inference. An approximation strategy to perform FLMM Cui et al. (2022).

- fastFMM the R package that implements FUI.

- CI confidence interval.

Before specific line-by-line responses, we provide a brief comparison between cFLMM and fixed effects encoding models. All three reviewers suggested that fixed effects models could be an existing alternative to cFLMM (Reviewer 1 (1B), Reviewer 2 (2C), Reviewer 3 (3A)). Their shared comments highlight that our revision should articulate the advantages and applications of cFLMM relative to existing analysis strategies.

Functional regression methods like cFLMM produce functional coefficient estimates that quantify how the magnitude of predictor-signal associations evolve across an ordered functional domain such as within-trial timepoints. Standard scalar outcome regression methods, like the GLMs specified in Engelhard et al. (2019), model these associations and their corresponding coefficients as fixed across the functional domain. While GLM encoding models may include time-varying predictors, these analysis strategies do not model the predictor–signal association as changing over the functional domain.

Moreover, encoding models are less suited to hypothesis testing in clustered or longitudinal settings (e.g., repeated-measures datasets) and yield regression coefficient estimates that are only interpretable with respect to the units of the basis functions. In contrast, cFLMM provides time-varying coefficient estimates that are interpretable as statistical contrasts in terms of the original variables and produces hypothesis tests in clustered settings. cFLMM can be applied to datasets that define covariates in terms of the same flexible representations of covariates used in encoding models; this is a modeling choice rather than a methodological characteristic.

The remainder of this provisional author response will respond to reviewers’ concerns line-by-line, approximately in the order they appear.

Reviewer #1 (Public review):

We thank Reviewer 1 for their comments, especially their efforts to provide first-hand experience with loading and applying fastFMM. We hope that recent improvements to fastFMM’s public release and vignettes address Reviewer 1’s concerns about ease-of-use.

(1A) Overall, while they make a compelling case that this approach is less biased and more insightful, the implementation for many experimentalists remains challenging enough and may limit widespread adoption by the community.

We believe the reviewer may have experimented with an old version of fastFMM, so their experience may not reflect recent rewrites and improvements. fastFMM v1.0.0+ is now stable, validated on CRAN, and contains new example data and step-by-step tutorials. We designed fastFMM’s model-fitting code to be similar to common GLM packages in R to reduce the learning curve for new users.

(1B) …a clearer presentation of how common implementations in the field are performed (i.e. GLM) and how one could alternatively use the cFLMM approach would help.

We will provide a clearer description of existing methods in the revised manuscript. Briefly, inference with fastFMM can accommodate large datasets that contain clustered data, repeated measures, or complex hierarchical effects, e.g., experiments with multiple animals and multiple trials per animal. When encoding models are fit to each cluster (e.g., animal, neuron) separately, we are not aware of a principled method to pool these cluster-specific models together to quantify uncertainty or yield an appropriate global hypothesis test.

Reviewer #2 (Public review):

Reviewer 2’s thoughtful feedback helped structure our points in the common response above, which we will refer to when applicable. In our response, we aim to clarify the problems that cFLMM solves and characterize the advantages in interpretability.

(2A) The aim of incorporating variables that change within trial into this framework is interesting, and the technical implementation appears to be rigorous. However, I have some reservations as to whether the way in which variables that change within trial have been integrated into the analysis framework is likely to be widely useful, and hence how impactful the additional functionality of cFLMM relative to the previously published FLMM will be.

We hope that the common response addresses these concerns. We were motivated to provide a concurrent extension of fastFMM based on our experience with statistical consulting in neuroscience research. Questions that benefit from a functional approach are common and often not adequately modeled with a non-concurrent approach, such as the variable trial length analysis we describe below.

(2B) It is less clear that this approach makes sense for variables that change within trial…This partitioning of variance in the predictor into a between-trial component whose effect on the signal is modeled, and a within-trial component whose effect on the signal is not, is artificial in many experiment designs, and may yield hard to interpret results.

We thank Reviewer 2 for highlighting a point that we did not adequately explain and that we will address further in the revision. The pointwise and joint CIs estimated by fastFMM account for uncertainty in the coefficient estimates due to variation in the predictors across within-trial timepoints. cFLMM targets a statistical quantity, or estimand, that is defined by trial timepoint specific effects, so the first step of our estimation strategy fits separate pointwise mixed models. However, models from every within-trial timepoint are then combined to calculate uncertainty and smooth the coefficient estimates. Thus, the widths of the pointwise and joint CIs depend on the estimated between-timepoint covariance and a smoothing penalty. Loewinger et al. (2025a) provides further details in Appendices 2 and 3, describing the covariance structure and detailing the power improvements of FUI compared to multiple-comparisons corrections.

Other functional regression estimation strategies jointly fit the entire model with a single regression, e.g., functional generalized estimating equations Loewinger et al (2025b). However, these methods use basis expansions of the coefficients. In contrast, the encoding models mentioned in 2C below and Reviewer 3 (3A) apply basis-expansions of the covariates, and the resulting model does not capture how signal–covariate associations evolve across some functional domain. Although the first stage in the fastFMM approach fits pointwise linear models, this is only one of three steps in the estimation strategy. fastFMM yields coefficient estimates comparable to those that would be obtained from functional regression estimation strategies that jointly estimate the functional coefficients in a single regression. We mention this to distinguish between the target statistical quantity (functional coefficients) and the estimation strategy (pointwise vs. joint).

(2C) …an alternative approach would be to run a single regression analysis across all timepoints, and capture the extended temporal responses to discrete behavioural events by using temporal basis functions convolved with the event timeseries. This provides a very flexible framework for capturing covariation of neural activity both with variables that change continuously such as position, and discrete behavioural events such as choices or outcomes, while also handling variable event timing from trial-to-trial.

Our understanding is that the suggested approach aims to quantify the association between the outcome and within-trial patterns in covariates. This is a great question and we will incorporate a discussion of this into the revision. However, temporal basis functions convolved with the covariate time series cannot directly characterize these relationships. Encoding models can detect the contribution of predictors to neural signals while remaining agnostic to the precise relationship, but this flexibility can come at the cost of interpretability. The coefficients of the convolutions may not be translatable into a clear statistical contrast in terms of the original covariates.

In our paper, we provide examples of cFLMM models with simple signal-covariate relationships. The coefficient estimates quantify the expected change in signal given a one unit change in the original predictors. Let 𝑌(𝑠) be the outcome and 𝑋(𝑠) be some covariate at within-trial timepoint 𝑠. For brevity, we will suppress subject/trial indices and random effects in the following notation. The coefficient at time point 𝑠 can be captured by the generic mean model

𝔼[𝑌(𝑠) ∣ 𝑋(𝑠) = 1] − 𝔼[𝑌 (𝑥)|𝑋(𝑠) = 0].

In contrast, the change in signal associated with patterns in within-trial covariates can be written as

𝔼[𝑌 (𝑠₁) ∣ 𝑋(𝑠₂) = 1] − 𝔼[𝑌 (𝑠₁) ∣ 𝑋(𝑠₂) = 0]

for all pairs of timepoints 𝑠₁, 𝑠₂. While simple lagged or offset outcome-predictor associations can be incorporated as covariates in cFLMM, the approach does not capture all within-trial timepoints 𝑠₁, 𝑠₂. Encoding models also do not target the above estimand. Instead, a full function-on-function regression could estimate the above. This topic can be incorporated into our revision and may be a future line of inquiry.

(2D) In the Machen et al. data…From the resulting beta coefficient timeseries (Figure 3C) it is not straightforward to understand how neural activity changed as the subject approached and then received the reward. A simpler approach to quantify this, which I think would have yielded more interpretable coefficient timeseries would have been to align activity across trials on when the subject obtained the reward. More broadly, handling variable trial timing in analyses like FLMM which use trial aligned data, can be achieved either by separately aligning the data to different trial events of interest or by time warping the signal to align multiple important timepoints across trials.

In this experiment, mice waited in a trigger zone, ran through a linear corridor, then received a food reward in the reward delivery zone of either water or strawberry milkshake Machen et al. (2026). Mice received different rewards between sessions but the same reward within all trials of a given session. This design complicated the analysis, as the reward type produced prominent differences in average latency (water: 3.3 seconds, milkshake: 2.0 seconds). The authors wanted to disentangle whether mean differences in the signal across reward types reflected differences in motivation to obtain the reward or differences in reaction to reward receipt.

We agree that performing a reward-aligned analysis would be an intuitive approach to visualize the differences in average signal for mice that received milkshake compared to water. In fact, we provide a ncFLMM reward-aligned analysis in Figure S1 of Machen et al. (2025). We will add this analysis to the revision and thank the reviewer for the suggestion. We emphasize, however, that this method answers a different question. It does not identify how the signal change associated with receiving the milkshake evolves with respect to latency, especially if the relationship is non-linear. Time warping faces similar obstacles in this setting, especially since sufficiently flexible curve registration can induce similarity due purely to noise. Generally, time warping does not lend itself to hypothesis testing as it is unclear how to propagate uncertainty from the time warping model into final hypothesis tests.

We believe cFLMM is an appropriate choice for the specific question, and we will revise the manuscript to better reflect its advantages. The functional coefficient estimates in Figures 3C-iii and 3C-iv provide insights that are not possible to derive from the proposed alternatives. For example, we can infer that for short latencies, we do not see a significant difference in signal magnitude for mice receiving water and mice receiving the milkshake. However, for latencies longer than around 2 seconds, receiving the milkshake is associated with an additional positive change in signal. We agree that we should make Figure 3C and the accompanying discussion more clear and thank Reviewer 2 for their feedback on interpretation.

Reviewer 3 (Public review):

(3A) …it is not clear what the conceptual or methodological advance of this work is. As it is written, the manuscript focuses on showing how concurrent regressors offer interpretation advantages over non-concurrent regressors. While the benefit of such time-varying regressors is supported by previous literature (e.g., Engelhard et al., 2020), it is not clear whether the examples provided in the current study clearly support the advantage of one over the other…

We assume Reviewer 3 is referencing “Specialized coding of sensory, motor and cognitive variables in VTA dopamine neurons Engelhard et al. (2019). We hope that the Common response sufficiently contrasts the settings where each approach can be applied. Because these models have different goals and assumptions, they are appropriate for answering different questions.

(3B) In this specific example, if the question is about speed and reward type, why variables such as latency to reward or a binary “reward zone vs corridor” (RZ) regressors are used instead of concurrent velocity (or peak velocity - in the case of the non-concurrent model)? Furthermore, if timing from trial start to reward collection is variable, why not align to reward collection, which would help in the interpretation of the signal and comparison between methods? Furthermore, while for the non-concurrent method, the regressors' coefficients are shown, for the concurrent one, what seems to be plotted are contrasts rather than the coefficients. The authors further acknowledge the interpretational difficulties of their analysis.

Thank you for pointing out that we were not clear. This was mentioned by multiple reviewers and highlights the need to elaborate on our motivation in the revision. In this example, we wanted to investigate the change in signal-reward association as a function of within-trial timepoints, not the association between instantaneous velocity and the signal. “Slow” or “fast” means “mouse with below or above average latency”. We ask you to please refer to Reviewer 2 (2C) where we discuss why event alignment is an insufficient correction.

The functional coefficient estimates in Figure 3C are interpreted as contrasts because the fixed effect coefficients capture the difference in expected signal between strawberry milkshake and water along the functional domain. An advantage of cFLMM is that it is easy to specify models in which the coefficients correspond to interpretable contrasts of the signal across conditions. The coefficient estimate shown in Figure 3B-ii also corresponds to a contrast because the estimates capture the difference in mean signal from strawberry milkshake and water. Equations (7) and (8) in the section “Materials and methods” and sub-section “Variable trial length analysis” provide additional details on the fixed effect coefficients. Based on this confusion, we will convert the two 1 x 4 sub-plots of 3B and 3C into two 2 x 2 sub-plots to avoid unintended direct comparisons.

To contextualize how we “acknowledge the interpretational difficulties of [our] analysis”, we stated that a non-concurrent FLMM attempting to control for a time-based covariate is difficult to interpret. The concurrent FLMM provides a straightforward interpretation directly related to the question of interest, which we discuss above in Reviewer 2 (2D).

(3C) Because the relation between behavioral variables and neuronal signal is not instantaneous, previous literature using fixed effects uses, for example, different temporal lags, splines, and convolutional kernels; however, these are not discussed in the manuscript.

Thank you for this suggestion. All three reviewers raised this topic (see Reviewer 1 (1B), Reviewer 2 (2C), and the Common responses), and we will incorporate our response in the revision.

(3D) From the methods, it seems that in the concurrent version of fastFMM, both concurrent and non-concurrent regressors can be included, but this is not discussed in the manuscript.

This is an important point that we mentioned implicitly. In our cFLMM specification of the Jeong et al. (2022) model, “we incorporated trial-specific covariates for trial number and session, modeling these as increasing numerical values rather than identical categorical variables”, which are also plotted in Appendix 3. In Box 1, “if the functional covariate of interest is a scalar constant across the domain, the models fit by the concurrent and non-concurrent procedure are identical”. We will explicitly point out that cFLMM can perform inference on combinations of functional and constant covariates.

(3E) The methodological advance is not clearly stated, apart from inputting into fastFMM a 3D matrix of regressors x trial x timepoint, instead of a 2D matrix of regressors x trial.

Prior to our work described in this Research Advance, it was not obvious that the existing approximation approach in fastFMM could be generalized to cFLMM. During the writing of the article, a fastFMM user reached out for help with producing pseudo-concurrent FLMMs by duplicating rows in a nonconcurrent model, which both underscores the unmet need for cFLMMs and the difficulty in fitting them with available tools.

The “under-the-hood” differences are described in Appendix 4. Concurrent FLMM with fast univariate inference was theoretically possible as early as Cui et al. (2022). The univariate step was straightforward, but guaranteeing “fast” and “inference” was not. We needed to verify, for example, that the method-of-moments estimation of the random effects covariance matrix generalized to cFLMM, which is not a trivial step. Characterizing whether the method achieved asymptotic coverage required extensive simulation studies (Figure 4, Appendix 2). Future work may focus on fully characterizing the asymptotic convergence in high noise or high complexity regimes.

(3F) This manuscript is neither a clear demonstration of the need for concurrent variables, nor a 'tutorial' of how to use fastFMM with the added extension.

We hope that the Common responses clarifies how cFLMM compares to existing approaches and fills a gap in the data analysis landscape for neuroscience. The fastFMM R package vignettes contain example analyses, and we intend for these files to be work in tandem with the manuscript. To provide more guidance for interested analysts, we can explicitly reference these tutorials within the revision.

Planned revisions

The following summary is not exhaustive.

Writing additions:

Per 1B, 2C and 3A, the Common responses will be incorporated in the revision.

Per 2B, we will discuss function-on-function regression and explore how to estimate statistical contrasts for complex within-trial relationships. Relatedly, we will clarify that the CIs in fastFMM are constructed using an estimate of the within-trial covariance of the predictors, and clarify the definition of pointwise and joint CIs.

Per 3D, we will explicitly state that concurrent FLMMs can include covariates that are constant over within-trial timepoints.

Though we cannot prescribe a universally correct model selection procedure, we will mention that AIC, BIC, and other summary statistics can inform the specification of the random effects.

Analysis modifications:

Parts of Appendix 3 may be included in Figure 2 to directly address the question investigated by Jeong et al. (2022) and Loewinger et al (2024).

When discussing Machen et al. (2025) data, the supplementary analysis with reward-aligned ncFLMM models might be added to clarify the ncFLMM/cFLMM difference.

Per \ref{rvw2:encoding}, the additional analysis aimed at disentangling latency and reward in Machen et al.’s variable trial length data may be incorporated as an additional sub-figure in Figure 3.

Aesthetic changes:

Figure 3 will be reorganized to avoid unintended direct comparisons between the coefficients of the non-concurrent and concurrent model.

Citations for Machen et al. (2026) will be updated to reflect publication of the preprint.

The version number for fastFMM will be updated.

References

Cui E, Leroux A, Smirnova E, Crainiceanu CM. Fast Univariate Inference for Longitudinal Functional Models. Journal of Computational and Graphical Statistics. 2022; 31(1):219–230. https://doi.org/10.1080/10618600.2021.1950006, doi: 10.1080/10618600.2021.1950006, pMID: 35712524.

Engelhard B, Finkelstein J, Cox J, Fleming W, Jang HJ, Ornelas S, Koay SA, Thiberge SY, Daw ND, Tank DW, Witten IB. Specialized coding of sensory, motor and cognitive variables in VTA dopamine neurons. Nature. 2019 Jun; 570(7762):509–513. https://www.nature.com/articles/s41586-019-1261-9, doi: 10.1038/s41586-019-1261-9.

Jeong H, Taylor A, Floeder JR, Lohmann M, Mihalas S, Wu B, Zhou M, Burke DA, Namboodiri VMK. Mesolimbic dopamine release conveys causal associations. Science. 2022; 378(6626):eabq6740. https://www.science.org/doi/abs/10.1126/science.abq6740, doi: 10.1126/science.abq6740.

Loewinger G, Cui E, Lovinger D, Pereira F. A statistical framework for analysis of trial-level temporal dynamics in fiber photometry experiments. eLife. 2025 Mar; 13:RP95802. doi: 10.7554/eLife.95802.

Loewinger G, Levis AW, Cui E, Pereira F. Fast Penalized Generalized Estimating Equations for Large Longitudinal Functional Datasets. ArXiv. 2025 Jun; p. arXiv:2506.20437v1. https://pmc.ncbi.nlm.nih.gov/articles/PMC12306803/.

Machen B, Miller SN, Xin A, Lampert C, Assaf L, Tucker J, Herrell S, Pereira F, Loewinger G, Beas S. The encoding of interoceptive-based predictions by the paraventricular nucleus of the thalamus D2R+ neurons. iScience. 2026 Jan; 29(1):114390. doi: 10.1016/j.isci.2025.114390.