Linear mixed models (LMM) are a class of methods for testing the association between covariates (‘independent’ variables) and outcomes (‘dependent’ variables) for repeated measures data (see Methods section Linear mixed models for a brief introduction to LMM). They can be used to analyze trial-level summary measures (e.g. AUCs) pooled across all trials and animals, preventing the loss of information from trial-averaging. However, this still requires condensing signals into scalar summary measures, which coarsens within-trial information across time-points. In contrast, functional regression methods can be used to model a photometry time series as a ‘function,’ which makes it possible to test the association between covariates of interest and the signal value at each time-point in the trial (see Methods section Functional linear regression for a brief introduction to functional regression). However, most functional regression methods require trial averaging of the signals prior to analysis, which discards trial-level information.

The photometry analysis framework we are introducing is based on Functional Linear Mixed Models (FLMM), which combines the benefits of LMM and functional regression to extract the information in the signal both across- and within- trials (Cui et al., 2022; Scheipl et al., 2015; Davidson, 2009; Morris and Carroll, 2006). By modeling covariates that can vary between (1) trial (e.g. cue-type, latency-to-press), (2) session (e.g. training stage), and (3) animal (e.g. treatment group), FLMM estimates the effects (termed functional fixed-effects) and statistical significance of those covariates for longitudinal designs. By further including functional random-effects, the framework can also estimate and adjust for between-animal differences in (1) photometry signal dynamics, and (2) effects of a covariate on the signal, within- and across- trials. In essence, this enables one to model between-animal and between-trial variability in both the ‘shape’ of the signal, and the evolution of covariate effects across trial time-points. The result is a single plot of the coefficient estimates for each covariate, which visualizes when (and to what extent) the covariate has a statistically significant (fixed-effect) association with the photometry signal during a trial.

In Figure 2A, we illustrate the steps in FLMM parameter estimation, which are implemented by our software (see Methods section Functional mixed models for details). The first input is a matrix Y, where column s, Y(s), contains the photometry signal value at trial time-point s from every trial in every session and animal, as shown in Figure 2A(1). The other inputs are covariate matrices, X and Z, associated with the fixed-effects and random-effects regression coefficients, respectively. Then, for each time-point, s, we fit a LMM model to Y(s) as the outcome variable, with the conditional mean of animal i modeled as:(1)E[Yi(s)|Xi,Zi,γi(s)]=Xiβ(s)+Ziγi(s).

This yields estimates for fixed-effect regression coefficients (β^(s), common across animals), random-effect coefficients (γi^(s), animal-specific), and β^(s) pointwise 95% confidence intervals (CIs), as shown in the last row of Figure 2A(1). We then smooth the β^k across trial time-points for each covariate k, as illustrated in Figure 2A(2). The smoothed pointwise 95% CI (dark gray) is constructed using a closed form covariance expression when Yi(s)|Xi,Zi,γi(s) is modeled as Gaussian, or through a bootstrap-based procedure for other distributions. To account for the multiple comparisons of inspecting coefficients across the entire trial, we construct a joint 95% CI (light gray), yielding the plots in Figure 2A(3). We detail the joint 95% CI estimation procedure, the subject of our statistical methodology contribution, in section Functional mixed models and the Discussion. By treating the signal as a ‘function,’ FLMM exploits the correlation across trial time-points to construct narrower joint CIs, thereby enabling one to identify more significant effects.

Figure 2B illustrates the FLMM output of an example analysis. The inset in Figure 2B [bottom left] shows how the effect at a given time-point is estimated and interpreted: when correlating (with an LMM) the signal values (pooled across trials and animals) measured at trial time-point s=1.7 sec with the covariate, Latency, the slope of that line is β^1(1.7)=0.82. More generally, the interpretation of the FLMM plot for covariate k at time-point s, β^k(s), is the ‘average change in the photometry signal at trial time-point s associated with a one unit increase in covariate k, holding other covariates constant.’ Figure 2B [bottom right] shows estimated functional random-effects for an example animal and illustrates how these random-effects model individual differences in the Latency–signal association. Each period for which the joint 95% CI does not contain 0 anywhere denotes that the fixed-effect coefficients are statistically significantly different from 0 throughout the entire period. The plot conveys: (1) where effects are statistically significant; (2) the estimated effect magnitudes; and (3) joint 95% CIs, thereby providing a complete, interpretable, and simplified presentation of statistical results.

To demonstrate how our method can be used to answer scientific questions in photometry experiments, we reanalyzed data from a recent article proposing a new model for mesolimbic dopamine’s role in learning (Jeong et al., 2022) (all analyses are on photometry data collected with dLight1.3b in the nucleus accumbens core). Note that this is a different study from the one described in Figure 1. We used this study for two reasons. First, the dataset exhibits many common characteristics of photometry data that can dilute effects, or even invalidate results if left unaccounted for. Second, the dataset contains data from multiple experiments, which allows us to illustrate how FLMM can be used to test hypotheses across a range of experimental designs. In this section, we discuss how FLMM handles those characteristics; in subsequent sections, we show how those hypotheses can be posed in our framework.

One of the most important characteristics of these data is that the dopamine (DA) measurements were collected in within-subject nested longitudinal experiments. For example, in the first behavioral task we discuss, mice were trained across multiple sessions. Each session involved the delivery of a sequence of 100 sucrose rewards. The authors analyzed average dopamine (AUC) during the ‘reward period’ time-window (aligned to the first lick after reward-delivery) as a trial (Figure 3A). Thus trials were nested in session, which were nested in animals (Figure 3B). Photometry experiments often exhibit this type of nested longitudinal structure, which can induce correlation patterns in the data within-animal, and obscure effects if not accounted for statistically. This structure can occur in both within-subjects and between-subjects designs, as illustrated in Figure 3D. For example, these data exhibited high within-animal correlations across sessions (Figure 3E), and across trials within-session (Figure 3F).

FLMM can model effect heterogeneity, as well as correlation patterns within- and across- trials, through the inclusion of functional random-effects. These allow one to estimate what is common across all animals, and what is unique to each. This is critical because photometry datasets often exhibit between-animal variability in covariate-signal associations. For example, in the task described above, the time between rewards, (IRI: ‘inter-reward interval’) varied between trials unpredictably (Figure 3C), and the authors reported that ‘reward period’ AUC was correlated with IRI. Figure 3G shows that the magnitude of this correlation varied considerably across animals and sessions. FLMM can model this variability in IRI–DA correlation magnitude through the inclusion of animal- and session-specific random effects. More broadly, by varying which covariates and random-effects are included in the model, FLMM can analyze data from experimental designs that include within- and between-subject contrasts. For example, one can implement functional versions of methods like correlations, or repeated measures ANOVA, and, more generally, model a wide range of dependence structures. Finally, FLMM can accommodate missing data and subjects with different sample sizes, which are often unavoidable characteristics of photometry experiments. In Figure 3H we show how this dataset included photometry recordings of behavioral sessions at various stages of training that differed across animals. If not accounted for statistically, this can obscure associations between the signal and covariates. For example, average reward period AUC levels were reported to increase across sessions. Thus animals with data collected only on early sessions may appear to have lower AUC levels than other animals, which can increase uncertainty in estimates of covariate effects.

We first recreate an analysis of the experiment described in the previous section. Jeong et al., 2022 reported, in section ‘Tests 1 and 2’, that the trial-level ‘reward period’ AUC was positively correlated with IRI. The authors fit separate Pearson correlations to data from each animal. To illustrate how to test this question in our framework, we show a recreation of their analysis in Figure 4A, B, and the FLMM analysis estimates of the IRI–DA association in Figure 4C, D. While Jeong et al., 2022 assessed the IRI–AUC correlation in each animal separately, FLMM can test the IRI–DA association at each trial time-point and in all animals jointly, thereby improving our ability to detect effects. To implement a test most similar to the Pearson correlation, we fit an FLMM model with IRI as the (fixed-effect) covariate. Given the between-animal and between-session heterogeneity in signal profiles highlighted in Figure 3, we included nested random-effects to account for the variability across both subjects and sessions. The relationship is significantly positive throughout the time-window [–0.5, 1.5] sec.

These results corroborate the paper’s findings, and provide finer-grained details inaccessible with standard analyses. For example, the temporal dynamics, revealed by FLMM, suggest that the neural signal associated with IRI may be composed of two distinct components. The first component rises rapidly starting at around –0.75 sec and decreases quickly after lick-onset (the first lick after reward-delivery). Since the association reaches its peak before lick-onset, the signal may reflect motivation, movement, or reward detection. The second component begins after lick-onset and rises and falls slower than the first component. The timing suggests it may track sucrose consumption. Importantly, the reward period AUC in the paper averages across these putative components. In contrast, FLMM is able to partially disentangle them through their distinct temporal dynamics, and offers insight into the role the components play in behavior.

FLMM can also identify results completely obscured by standard methods. We show this by recreating an analysis in which Jeong et al., 2022 report that reward period AUC was positively correlated with that trial’s Reward Number. Figure 3B illustrates the definition of Reward Number, the cumulative trials/rewards received across sessions. The finding that the Reward Number–AUC correlation was positive was controversial because it conflicted with the prediction of the Reward Prediction Error (‘RPE’) model, a prevailing hypothesis for the role of DA in reward learning. The authors argued that RPE predicts a negative Reward Number–DA association, while their model (‘ANCCR’) predicts a positive association. Using FLMM on the same data, we estimate that the Reward Number–DA association is in fact negative within-session. The discrepancy by findings arises because the analysis in Jeong et al., 2022 pools trials across sessions in a manner that does not consider the session number that a reward was delivered on. FLMM finds the effect above by accounting for the nested session/trial task structure, and suggests why it occurs by estimating changes at each trial time-point.

The source of the conflicting results above is most clearly illustrated by fitting Reward Number–AUC correlations in each animal and session separately. In Figure 5A and Appendix 4—figure 1 we show session-specific linear regression fits overlaid on the fits from the session-pooled analysis conducted by Jeong et al., 2022. We parameterized session-specific models to yield intercepts (highlighted as large black circles) with the interpretation ‘the expected AUC value on the first trial of the session.’ These intercepts tend to increase across sessions, ultimately resulting in a positive overall Reward Number–AUC correlation in the session-pooled analysis (see Appendix 4—figure 1). However, the majority of the within-session slopes are actually negative, indicating that the AUC decreases across trials within-session. The apparent disagreement between the positive Reward Number–AUC correlation estimated in the session-pooled analysis, and the negative Trial Number–AUC association identified in the within-session analysis, is an example of Simpson’s paradox. This is caused by pooling data without taking into account its hierarchical structure.

To resolve this ‘paradox,’ we fit an FLMM that models the nested design by including both between-session and within-session fixed-effects, as well as nested random-effects. Estimating the Session Number and Trial Number functional coefficients reveals that within-session and between-session effects have distinct temporal dynamics. For example, Figure 5D, E shows that the FLMM estimates that the mean signal decreases significantly within-session beginning immediately post-lick, but increases across-sessions for a brief interval around the lick. Importantly, the Session Number–DA association peaks and begins falling before the lick, while the Trial Number–DA correlation becomes significant only after the lick. Together these results suggest that DA increases across sessions in response to reward anticipation, but decreases within-session during reward consumption.

An alternative interpretation of the opposing within- and between-session effects is that DA rises within-session, but appears to decrease due to photobleaching. We assess this in Appendix 4.3. We repeat the above analyses on the signal aligned to reward-delivery, and analyze two other behavioral experiments, collected on the same animals. These show that, within-session, DA responses increase to reward-predictive cues, but decrease to reward-delivery (predicted and unpredicted). That we can detect both within-session increases and decreases to separate events (cue-onset and reward-delivery) occurring on the same trials, suggests that photobleaching is not hiding within-session DA increases. In personal communications, Jeong et al., 2022 agree that while photobleaching cannot be definitively ruled out, a parsimonious interpretation of these findings is that there is a DA reduction to unpredicted rewards within-session, independent of photobleaching.

We next describe an example comparing the signal ‘temporal dynamics’ across conditions, from section ‘Tests 3–7’ of Jeong et al., 2022. In this experiment, each trial consisted of a presentation of a 2 sec cue, followed by a reward 3 sec after the cue-onset. After many training sessions, the delay was lengthened to 9 sec (Figure 6A–B). On the first 9 sec delay session, the authors report that “dopaminergic cue responses [(Cue Period AUC)] showed no significant change,” relative to trials from the last short-delay session. Jeong et al., 2022 argue that their finding conflicts with what an RPE hypothesis of dopamine coding would anticipate (Kobayashi and Schultz, 2008). Like the authors, we compared the average signals between the last short-delay and the first long-delay sessions. However, we sought to test whether the signal ‘dynamics’ over trial time-points changed after lengthening the delay. To directly compare the difference in signal magnitudes between short- and long-delay sessions, at each time-point, we fit a FLMM model with a single binary covariate representing short/long-delays (similar to a functional paired t-test). The FLMM estimated mean DA was significantly lower in the latter parts of the initial cue-elicited DA response (∼1–2.5 sec after cue-onset) on long- compared to short-delay trials (shown in Figure 6H). This effect may have been occluded because the AUC analyzed in Jeong et al., 2022 was constructed by averaging signal values in a time window that contains opposing effects: the significant (relative) reduction in mean DA ∼0.5–2.5 sec after cue-onset (marked as interval (2) in Figure 6H), identified by FLMM, is potentially diluted in the AUC by the (non-significant) increase in the cue response in the first ∼0.5 sec (marked as interval (1) in Figure 6H). In our reanalyses we identified other experiments where effects were obscured by the use of AUCs averaging over opposite effects (see Appendix 4.2).

This example shows how FLMM can detect subtle effects that are difficult to identify by eye, and thus hard to construct appropriate summary measures for. For example, the delay-change effect is significant only during the falling-portion of the cue-elicited transient (0.5–2.5 sec after cue-onset). This small relative reduction (shown in Figure 6H) during a later portion of the cue-response is overshadowed by the much larger overall DA response immediately following (0–0.5 sec) cue-onset (shown in Figure 6G). Finally, we show animal-specific functional random-effect estimates in Figure 6E, F. These provide intuition about how FLMM adjusts for between-animal differences in the ‘temporal dynamics’ of covariates effects.

We further analyze this dataset in Appendix 2, to compare FLMM with the approach applied in Lee et al., 2019 of fitting pointwise LMMs (without any smoothing) and applying a Benjamini–Hochberg (BH) correction. Our hypothesis was that the Lee et al., 2019 approach would yield substantially different analysis results, depending on the sampling rate of the signal data (since the number of tests being corrected for is determined by the sampling rate). The proportion of time-points at which effects are deemed statistically significant by FLMM joint 95% CIs is fairly stable across sampling rates. In contrast, that proportion is both inconsistent and often low (i.e. highly conservative) across sampling rates with the Lee et al., 2019 approach. These results illustrate the advantages of modeling a trial signal as a function, and conducting estimation and inference in a manner that uses information across the entire trial.

We conducted experiments to assess how FLMM performs on synthetic data based on the Delay Length experiment data from Jeong et al., 2022 introduced in Using FLMM to compare signal ‘temporal dynamics’ across conditions. We simulated data in the small sample sizes typical in photometry experiments. We incorporated the key characteristics discussed earlier, namely trial-to-trial and session-to-session correlation, as well as animal-to-animal variability in photometry signal (i) magnitudes and (ii) levels of association with the covariates. We did this by treating the parameter estimates from a FLMM model, fit to the Delay Length dataset, as the ‘true’ parameter values. This ensured values fell within a realistic range. As the simulated data contained a single binary covariate, this allowed comparison with other statistical approaches discussed in the Introduction, namely the permutation method (Jean-Richard-Dit-Bressel et al., 2020), a paired samples t-test, and a (non-functional) LMM. The latter two only work with summary measures, so we applied them to the reward period AUC summary measure used in Jeong et al., 2022. Figure 7A shows average traces of the real data that the simulations are based on. Figure 7B presents simulated trials from one ‘animal,’ and Figure 7C presents simulated session averages from seven ‘animals’.

The quantitative measures for comparison are pointwise and joint 95% CI coverage, and statistical power. Figure 7D shows that the FLMM achieves joint 95% CI coverage at roughly the nominal level for all sample sizes tested. The permutation method, Perm, (Jean-Richard-Dit-Bressel et al., 2020) provides only pointwise CIs and thus yields low joint coverage. We next analyzed the cue period to allow comparison between the average performance of FLMM, evaluated at time-points in that time-window, with the performance of standard methods applied to the AUC summary measure. FLMM achieved roughly the nominal pointwise coverage (Figure 7E), while Perm does not, likely because the cluster-bootstrap often performs poorly in small sample settings (Ju, 2015). FLMM yields substantially better statistical power than the other methods (Figure 7F). The LMM and t-test exhibit lower power, likely because effects are diluted by analyzing summary measures. In Appendix 5.2, we present additional simulation results that demonstrate that the performance of summary measure methods is highly sensitive to minor changes in the length of the time-interval that the AUC summarizes. These results show how FLMM improves pointwise coverage and statistical power compared to standard methods and provides reasonable joint coverage.

We introduced a Functional Linear Mixed Modeling framework for photometry data analysis that: (1) enables hypothesis testing at every trial time-point; (2) accounts for individual differences and trial-to-trial variability; and (3) allows modeling of longitudinal experimental designs. We extend the FLMM framework in Cui et al., 2022 to enable parameter estimation with pointwise and joint 95% CIs for general random-effect specifications. Previous work provided FLMM implementations for either joint CIs for simple random-effects models (Cui et al., 2022; Sergazinov et al., 2023; Crainiceanu et al., 2024), or pointwise CIs for nested models (Scheipl et al., 2016). However, they do not allow computation of joint 95% CIs in the presence of general random-effects specifications, which is helpful for modeling many characteristics of photometry datasets. When conducting the analyses in this paper, we found that nested random-effects specifications, enabled by our method, substantially improved model fits compared to models without nested random-effects. We describe FLMM further in the Methods section Functional mixed models and derive our methodology in Appendix 3.

We demonstrated the capabilities of FLMM by reanalyzing datasets from a recently published study Jeong et al., 2022. First, we showed that the main scientific questions asked by the authors can easily be answered by specifying covariates of interest, fitting an FLMM model, and referring to the coefficient estimate plots for results. Furthermore, we showed that FLMM allows direct comparison – and testing – of differences in signal time-courses between conditions, something that is difficult with standard methods relying on summary measure tests. Finally, we showed that FLMM can reveal significant effects that were tested by the authors, but obscured by summary measure analyses.

The specific effects identified with FLMM bear on one of the key questions in Jeong et al., 2022, namely which of two competing models – RPE and ANCCR – better describe changes in DA responses to unpredicted rewards over trials. For example, in personal communications, Jeong et al., 2022 still argue that the observed DA increases across-sessions in the random IRI task may be inconsistent with RPE. However, as a result of our reanalyses, they now argue that the within-session decreases, identified by FLMM, conflict with the predictions of the initial ANCCR model presented in Jeong et al., 2022. The predictive/explanatory power of RPE and ANCCR as models for measured DA responses depend, however, on their respective model specifications (e.g. RPE reward function) and is thus beyond the scope of the present work. The difficulty in comparing the two models is further compounded by a number of alternative explanations for our findings that cannot be definitively ruled out. For example, satiation or habituation may contribute to within-session DA decreases. An initial exploration suggests that this may not be the sole cause, as the temporal dynamics of within-session reductions vary between-session (Appendix 4—figure 2), despite equal levels of food-restriction. The between-session variability in dynamics may, however, reflect learning-related changes in consummatory behavior, and thus may not rule out satiation as an explanation for the within-session DA reductions. Moreover, the authors suggested that the reward-delivery mechanism produces stimuli that could act as a reward-predictive cue. Some of the DA changes may, therefore, reflect learning about this cue. Finally, between-session DA increases could be due to elevations in indicator expression levels over training. This was, however, explored by Jeong et al., 2022 and they reported that transients unrelated to reward did not increase in magnitude across sessions (see Figure S8F-H in Jeong et al., 2022). As the authors note, if reward-related DA increases across sessions were caused by higher indicator levels, one would expect non-reward-related DA activity to also rise.

In personal correspondence, the authors of Jeong et al., 2022 offered the following possible explanations for the within-session DA decreases within the ANCCR framework. Specifically, while the ANCCR simulations described in Jeong et al., 2022 did not consider the session a trial occurred on, there may be session boundary-effects in animal learning. For instance, animals may estimate the rate of reward to be lower at the start of a session, due to a degraded memory from prior sessions, or to generalization between the high reward rate within the head-fixed context and the low rate outside it. Then the systematic increase in animals’ estimate of reward rate during the session would result in a corresponding decrease in the learning rate of the reward–reward predecessor representation (Burke et al., 2023). If so, the magnitude of the reward-reward predecessor representation increase at reward-delivery would reduce across sessions, and result in a negative Session Number–DA correlation.

To further demonstrate the utility of FLMM in answering common neuroscience questions, we reanalyze data in which the authors tested whether, across several sessions of Pavlovian learning, DA activity ‘backpropagates’ from reward delivery (3 sec after cue-onset) to the presentation of reward-predictive cues (see Appendix 4.4). We argue that FLMM is well-suited to assess questions like these: the method tests how the signal timing evolves both within- and across-sessions by providing a null hypothesis test (i.e. to assess statistical significance) at each time-point. Our results align with those reported by Jeong et al., 2022. In contrast, Amo et al., 2022 reported fiber photometry data that they argue supports the backpropagation hypothesis, despite being collected from a similar brain region and behavioral task. If researchers apply FLMM to test the backpropagation hypothesis in the future, we recommend fitting models that account for between-animal heterogeneity. Our initial exploration suggested that if the backpropagation hypothesis holds, the timing of when that ‘backpropagating hump’ begins to transition may differ between animals. In order to thoroughly test that hypothesis, it may, therefore, be necessary to specify an FLMM mean model that explicitly accounts for this potential variability in timing.

Adjudicating between ANCCR and RPE is beyond the scope of this paper, but our findings suggest that future experiments may be needed to test the explanations proposed in light of our results. Even in the Delay Length change experiment, in which FLMM identified significant changes obscured by AUC analyses, it is not clear to us whether proposed versions of RPE and ANCCR would predict the specific change in dynamics observed. As such, modeling trial-level mesolimbic DA as a vector (e.g. see Wärnberg and Kumar, 2023, Lee et al., 2024) or function may improve theories for its role in reward learning. By estimating trial-level temporal dynamics, we hope that FLMM will make it possible to further refine these theories.

For further evaluation of FLMM, we reanalyzed a second study (Coddington et al., 2023, see Appendix 6) that measures calcium dynamics, showing the applicability of the method to different photometry sensors. There, we demonstrate additional capabilities to: (1) model how signal ’dynamics’ early in training predict behavior later in training; and (2) estimate covariate interactions across both trial time-points and sessions. Finally, we carried out simulation experiments on realistic synthetic data to verify that, in small sample settings, the FLMM framework yields joint and pointwise CIs that achieve approximately 95% coverage, and improve statistical power versus the alternative approaches introduced. As FLMM commits type I errors at roughly nominal levels and outperforms methods that analyze trial-averaged data, we verify that FLMM is not overly susceptible to trial-level noise or spurious correlations.

Beyond the technical capabilities of our method, we believe there are a number of ways in which it can enhance scientific practice. First, FLMM may reduce biases arising from summary statistic analyses. Photometry measures aggregate activity from a collection of neurons, which may contain sub-populations with different functions. By coarsening the temporal resolution, summary measures may obscure the heterogeneity of the target neural population, and bias the study of systems towards sub-populations that exhibit larger signals. Through analyzing each time-point, FLMM may help disentangle separate signal components arising from different neural sub-populations. Second, we found that FLMM made it easier to identify model misspecification, as estimate plots often provided obvious visual indications (e.g. CI magnitude varying enormously across sub-intervals of the trial). We recommend fitting a few models and selecting one based on fit criteria (e.g. AIC/BIC), without considering statistical significance. Finally, we hope FLMM will improve reproducibility by removing the need for some signal pre-processing steps. Any data pre-processing will have down-stream consequences on the analysis results and interpretations from most statistical methods. In addition to possibly drowning out effects, some pre-processing steps (e.g. signal smoothing) can reduce reproducibility if the approaches differ across labs. FLMM can be used to eliminate some common pre-processing steps by including them as part of the model, or make their effect on the analysis results clearer, as we summarize in the next paragraph.

FLMM can help model signal components unrelated to the scientific question of interest, and provides a systematic framework to quantify the additional uncertainty from those modeling choices. For example, analysts sometimes normalize data with trial-specific baselines because longitudinal experiments can induce correlation patterns across trials that standard techniques (e.g. repeated measures ANOVA) may not adequately account for. Even without many standard data pre-processing steps, FLMM provides smooth estimation results across trial time-points (the ‘functional domain’), has the ability to adjust for between-trial and -animal heterogeneity, and provides a valid statistical inference approach that quantifies the resulting uncertainty. For instance, session-to-session variability in signal magnitudes or dynamics (e.g. a decreasing baseline within-session from bleaching or satiation) could be accounted for, at least in part, through the inclusion of trial-level fixed or random effects. Similarly, signal heterogeneity due to subject characteristics (e.g. sex, CS+ cue identity) could be incorporated into a model through the inclusion of animal-specific random effects. Inclusion of these effects would then influence the width of the confidence intervals. By expressing one’s ‘beliefs’ in an FLMM model specification, one can compare models (e.g. with AIC). Even the level of smoothing in FLMM is largely selected as a function of the data, and is accounted for directly in the equations used to construct confidence intervals. This stands in contrast to ‘trying to clean up the data’ with a pre-processing step that may have an unknown impact on the final statistical inferences.

FLMM is applicable in a wide range of experimental settings. We provide an analysis guide and open source package, fastFMM (our package is available on CRAN and also at https://github.com/gloewing/fastFMM, which has links to our analysis guide and examples of using the package in both R and python), that is the first, to our knowledge, to provide joint CIs for functional mixed models with nested random-effects. For demonstration, we selected a dataset collected on a behavioral paradigm, photometry sensor, and signaling pathway that are well characterized, making it easier to evaluate the framework. Although selecting summary measures and time-windows to quantify is relatively easy here, this may not be the case when collecting data under new conditions. In such cases, FLMM can characterize the association between neural activity and covariates without strong assumptions about how to summarize signals. Importantly, the package is fast and maintains a low memory footprint even for complex models and relatively large datasets (see Functional mixed models package for an example). The FLMM framework may also be applicable to techniques like electrophysiology and calcium imaging. For example, our package can fit functional generalized LMMs with a count distribution (e.g. Poisson). Additionally, our method can be extended to model time-varying covariates. This would enable one to estimate how the level of association between signals, simultaneously recorded from different brain regions, fluctuates across trial time-points. This would also enable modeling of trials that differ in length due to, for example, variable behavioral response times (e.g. latency-to-press). In this paper, we specified FLMM models with linear covariate–signal relationships at a fixed trial time-point across trials/sessions, to compare the FLMM analogue of the analyses conducted in Jeong et al., 2022. However, our package allows the modeling of covariate–signal relationships with non-linear functions of covariates, using splines or other basis functions. One must consider, however, the tradeoff between flexibility and interpretability when specifying potentially complex models, especially since FLMM is designed for statistical inference.

To conclude, we believe research on statistical methods for fiber photometry is necessary, given the widespread use of the technique and the variability of analysis procedures across labs. We hope that our proposed FLMM framework and software further enable investigators to extract the rich information contained in photometry signals.

Linear mixed modeling (LMM, also known as multilevel or hierarchical modeling) is a method for testing the association between covariates (i.e. ‘predictor’ or ‘independent’ variables) and an outcome (‘dependent’) variable for repeated measures data (or multilevel/longitudinal). We provide a brief description of mixed models below, but see Aarts et al., 2014; Yu et al., 2022; Magezi, 2015; Barr et al., 2013; Barr, 2013; Baayen et al., 2008 for detailed descriptions of their use in neuroscience and psychology.

LMM enables hypothesis testing in repeated measures designs, and can account for and characterize individual differences through the inclusion of ‘random-effects.’ Conceptually, random-effects serve as animal-specific regression coefficients (one may include more than just animal-specific random-effects (e.g. session-level random-effects)). This allows the relationship between a covariate of interest (e.g. locomotion) and the outcome (e.g. lever pressing) to vary between-animals in the model, while still pooling data from all animals to estimate parameters. For example, they can be used to model between-animal differences in photometry signal magnitudes on baseline conditions (e.g. on a control condition). This can allow the model to capture how variables of interest (e.g. behavior) are associated with the signal after ‘adjusting’ for the fact that some animals may, on average, exhibit lower signal magnitudes on all conditions. While LMMs may not feel intuitive initially, they actually share connections to many familiar hypothesis testing procedures. Just as ANOVAs, t-tests, and correlations can be cast as special cases of linear models, repeated measures versions of these tests (e.g. paired sample t-tests, repeated measures ANOVAs, and MANOVAs) have similar connections to linear mixed models. We recommend sections 1.2–1.3 of Fitzmaurice, 2008 for more explanation of these connections. This reference also includes descriptions of the capacity of LMM (and thus FLMM) to accommodate: (1) unbalanced designs, (2) varying sample sizes across individuals, and (3) more complicated correlation structures than methods like repeated measures ANOVA.

In order to provide better intuition about LMM in the photometry context, we illustrate the role of random-effects in Figure 8. In panel A, we plot data from five animals (photometry data from Coddington et al., 2023, with behavioral data simulated for illustrative purposes). On multiple trials, to examine the association between the signal and behavior on a test session. Averaging the signal across trials and correlating its peak amplitude with average session behavior results in only five observations, shown in the inset. In Figure 8B, we show the alternative approach that correlates a summary of each trial (e.g. peak amplitude) with the trial-level behavioral variable. The linear regression is fit to many more observations than in the session average strategy, but the standard regression hypothesis testing procedure may now be invalid, as we violated the independence assumption by pooling repeated observations within animals. Moreover, it ignores that the number of samples (trials) differ between animals. We magnify the inset from Figure 8B in Figure 8C to emphasize how within-subject correlation arises, at least partially, from animal-to-animal variability. Indeed, the linear regressions (Ordinary Least Squares or ‘OLS’) fit to each animal separately differs from the fit obtained if we first pool all animals’ trial-level data (‘All Animals OLS Fit’). This is expected since (1) each animal may have a different ‘true’ regression line (i.e. individual differences), and (2) the estimated lines may differ ‘by chance’ alone due to sampling error. An LMM allows us to pool across animals and trials but still account for individual differences through ‘random-effects,’ thereby striking a balance between (i) the line fit to the trial-level data pooled across animals, and (ii) the animal-level fits. In Figure 8D we show the fit estimated from the LMM (determined by β^(LMM)), as well as the subject-specific fits that include the random-effects (defined by the sum: β^(LMM)+γ^i for subject i). In Figure 8C the All Animals OLS Fit is ‘pulled’ towards Animal 1’s data because we failed to account for the fact that more trials were recorded from Animal 1. In Figure 8D, the LMM fit accounts for differing sample sizes across animals and is no longer unduly influenced by Animal 1: it is more similar to the majority of the other animals’ subject-specific LMM fits than the OLS fit. Moreover, we see that Animal 3 in Figure 8C has an OLS fit that differed considerably from all other animals. The LMM subject-specific fit is, however, ‘shrunk’ towards the population fit, determined by β^(LMM).

The LMM used to estimate the fits presented in Figure 8D is(2)Yi,j=β0(LMM)+γ0,i+Xi,j(β1(LMM)+γ1,i)+ϵi,j

where Yi,j and Xi,j are the outcome variable (e.g. trial behavior) and covariate (e.g. photometry signal peak amplitude), respectively, associated with subject i on trial j. For short hand, we concatenate the ‘fixed’ effects, random-effects and covariates into vectors, β(LMM)=[β0β1]T, γi=[γ0,iγ1,i]T and Xi,j=[1Xi,j]T. β(LMM) is the population regression coefficient estimate that is shared across all subjects, and γi is subject i’s random-effect, its deviation from the shared population slope β(LMM). Thus the slope that determines the fit specific to subject i is β(LMM)+γi. That is, the LMM estimates: (i) what is common among all animals; and (ii) what is unique to each animal. This imbues LMM (and its generalizations) with a flexibility that enables one to account for individual differences and to model correlated data arising from a wide range of complex experimental designs. LMM still requires, however, one to analyze summary measures (e.g. AUCs), and thus their application in photometry analyses leads to discarding of within-trial signal information across time-points the downsides of summary measure analyses are well-characterized in the statistics community (e.g. see section 1.2 of Fitzmaurice, 2008).

We now describe function-on-scalar regression, a functional data analysis method where the outcome variable is a function, rather than a scalar value. We treat the photometry time series from one trial as a ‘function’, which allows us to test the association between any trial-level experimental variables of interest (e.g. cue type, latency-to-press) and the photometry signal value at each time-point in the trial. For example, if photometry data for two groups is collected on one trial, one could fit a separate t-test at every time-point in the trial. The t-test at time s then assesses the mean difference between the signal magnitude of group 1 at time s and the signal magnitude of group 2 at time s. Plotting the test statistics vs. time yields a curve representing how differences between groups change over time. This provides the intuition behind the functional version of a t-test, and is a special case of the functional linear model. This functional version of the t-test differs from the common approach of comparing the groups’ signals using a single summary measure (e.g. peak amplitude in the trial).

In essence, the functional linear model (here, we describe the function-on-scalar linear model but use the functional linear model as short hand) is a linear regression with a functional coefficient β(s) across time-points s:(3)Yi(s)=Xiβ(s)+ϵi(s)

where Yi(s)∈R and ϵi(s)∈R are the photometry value and error term of animal i at time-point s in the signal. The p×1 vector, β(s), is the regression coefficient at time-point s, applied to the covariates, Xi∈Rp. This functional linear regression framework allows for testing the effects of (scalar) variables at every time-point, adjusting for multiple covariates (continuous or discrete) and their interactions. Functional regression thus exploits within-trial signal information. However, it cannot be directly used to analyze multiple trials per animal, since it assumes independence across observations of the signal between-trials (see Chapter 5 of Crainiceanu et al., 2024).

Our framework is based on Functional Linear Mixed Models (FLMM), which is a form of function-on-scalar regression that combines the benefits of linear mixed effects modeling and functional linear regression. We provided an informal presentation of the approach in the Results section Functional Linear Mixed Models (FLMM), with the goal of enabling researchers with a wide range of statistical backgrounds to understand its application and results. Here, we provide a high-level overview of the methodology, with further technical details provided in Appendix 3. FLMM models each trial’s signal as a function that varies smoothly across trial time-points (i.e. along the ‘functional domain’). It is thus a type of non-linear modeling technique over the functional domain, since we do not assume a linear model (straight line). FLMM and other functional data analysis methods model data as functions, when there is a natural ordering (e.g. time-series data are ordered by time, imaging data are ordered by x-y coordinates), and are assumed to vary smoothly along the functional domain (e.g. one assumes values of a photometry signal at close time-points in a trial have similar values). Functional data analysis approaches exploit this smoothness and natural ordering to capture additional information during estimation and inference. Our FLMM approach is based on an estimation procedure that was first proposed in Cui et al., 2022. We selected this method because (1) it allows for calculation of joint 95% CIs, (2) it is readily scalable for the dataset sizes and random-effect specifications needed for neuroscience experiments, and (3) it uses common syntax for model specification from the well-known lme4 package.

The construction of joint CIs in the context of functional data analysis is an important research question; see Cui et al., 2022 and references therein. Each point at which the pointwise 95% CI does not contain 0 indicates that the coefficient is statistically significantly different from 0 at that point. Compared with pointwise CIs, joint CIs take into account the autocorrelation of signal values across trial time-points (the functional domain). Therefore, instead of interpreting results at a specific time-point, joint CIs enable joint interpretations at multiple locations along the functional domain. This aligns with interpreting covariate effects on the photometry signals across time-intervals (e.g. a cue period) as opposed to at a single trial time-point. Previous methodological work has provided functional mixed model implementations for either joint 95% CIs for simple random-effects models (Cui et al., 2022), or pointwise 95% CIs for nested models (Scheipl et al., 2016), but to our knowledge, do not provide explicit formulas or software for computing joint 95% CIs in the presence of general random-effects specifications. We found nested random-effects specifications substantially improved model fits in the neuroscience experiments we reanalyzed, and helped model sophisticated behavioral designs (see Appendix 3.3 for more details). We, therefore, derived an extension of the estimator for the covariance of the random-effects (Cui et al., 2022), based on a Method of Moments approach described by Greven et al., 2010.

Mixed effects modeling (and thus FLMM) requires one to specify the random-effects in the modeling process. In essence, this amounts to selecting which covariate effects (i.e. regression coefficients) might vary across, for example, animals or sessions (see Appendix 3.3 for more details on specification and interpretation of random-effects). While this may seem unfamiliar at first, in practice, most experimental designs (and the appropriate model formulations) fall under a few categories that can be modeled with extensions of familiar methods (e.g. FLMM analogues of repeated measures ANOVA, correlations, etc.). Our method is fully implemented by our package and uses the same syntax as the common mixed effects package lme4, thereby allowing one to easily fit the models and plot the results. Thus the rich literature on linear mixed model selection and model syntax can be used for FLMM, as many of the same principles apply (e.g. see reviews of LMM and applications in neuroscience and psychology in Aarts et al., 2014; Yu et al., 2022; Magezi, 2015; Barr et al., 2013; Barr, 2013; Baayen et al., 2008). Users may improve their analyses by evaluating a few candidate models and selecting the one with the best model fit criteria. To that effect, AIC, BIC, and cAIC (Säfken et al., 2018) are already provided automatically in our software implementation. We provide the code to fit models with our package below and we include all code used for this paper (e.g. data pre-processing, model-fitting) on the GitHub page: https://github.com/gloewing/photometry_FLMM (copy archived at Loewinger, 2024). Finally, since fiber photometry sums photon counts over many neurons (the activity of which may be modeled as only weakly dependent given model parameters), we appeal to the central limit theorem to motivate the adoption of a Gaussian-likelihood for the conditional distribution of Yi,j(s)|Xi,j,γi,j(s). However, when photon counts are low, our package can be used to fit functional generalized LMMs with a count distribution like a Poisson (see Cui et al., 2022).

Figure 1 was generated from data (Coddington et al., 2023) available at Dudman, 2023. All signals are measurements of calcium dynamics in mesolimbic dopamine cells (DAT-Cre::ai32 transgenic mice were injected with a Cre-dependent jRCaMP1b virus across the ventral midbrain) recorded from fibers in the nucleus accumbens. In this experiment, head-fixed mice were exposed to a 0.5 sec stimulus, followed by a reward 1 sec after cue-offset. Signals are aligned to cue-onset.

Figure 1A was generated from trial-averaging Lick+ trials from session 8 for control animals that received no optogenetic stimulation. To ensure the animals were sufficiently trained we selected session 8, because it is the latest session in which all control animals have recorded data available in this dataset. Figure 1B was generated from trial-averaging Lick+ and Lick- trials and then plotting the pointwise difference. Figure 1C was generated arbitrarily from the first control animal in the dataset (i.e. animal ID 1), which we selected without inspecting other animals’ data to avoid biases. The data plotted are a subset of Lick+ trials (a sequence that starts from the first and ends on the last trial and takes every 10th trial) on the final training session for that animal (session 11). We selected this session to ensure the animal was as well trained as possible. Figure 1D shows session averages on Lick+ trials from sessions 8–11.

We made Figure 1E and F for explanatory purposes, and the specific choices we made in creating these figures are not meant to reflect any specific photometry analysis procedures. We nevertheless include a description of how we generated them for completeness. Figure 1E was the trial-averaged trace from session 8 for Lick- trials for Animal 1. Figure 1F was the trial-averaged trace from session 8 for Lick- and Lick+ trials separately for Animal 1. The inset contains an individual point for each animal from session 8. A given point was calculated by trial-averaging Lick+ and Lick- for each animal separately, and then finding the maximum difference.

Our primary experimental results in the Results section come from a reanalysis of the data from Jeong et al., 2022. Here, we provide a high-level description of the methods used, with additional details in Appendix 4. We describe the models we implemented and where possible, we explain the authors’ hypotheses, data pre-processing, and analysis procedures. We often center all covariates to be mean zero to allow an interpretation of the intercept as the mean signal for a trial when all covariates are at their average values. For each of the following analyses, we fit a collection of models and selected the final one based on which exhibited the best AIC and BIC. Finally, the R scripts used to replicate the analyses or derive information from data (e.g. extracting consummatory lick bouts) can be found on the GitHub page: https://github.com/gloewing/photometry_FLMM (copy archived at Loewinger, 2024).

We analyzed the data described in the section ‘Experiment 3: Tests 4–5’ (Jeong et al., 2022). The authors noted that “[W]hen a learned delay between cue onset and reward (3 seconds) is extended permanently to a new, longer delay (9 seconds), [Reward Prediction Error] predicts that as animals learn the longer delay…there will be a concomitant reduction in the dopamine cue response due to temporal discounting (46). By contrast, [their model] predicts little to no change in the dopamine cue response as the structure of the task is largely unchanged…Experimentally, [they] observed that although animals learned the new delay rapidly, dopaminergic cue responses showed no significant change” (Jeong et al., 2022).

Their analysis of DA cue-response was based on a baseline subtracted AUC (AUC [0,2 sec] - [–1,0 sec] AUC relative to cue onset). They write in their supplement (p. 4) that “Experiments 2–5: [T]o analyze learning-dependent dynamics of dopamine response, AUC of ΔF/F during 2 [sec] from CS +onset was normalized by AUC during baseline period.”

Our final model included a random slope for the long-delay (9 sec) indicator (Delayi,l), where Delayi,l=1 indicates long-delay trials:E[Yi,j,l(s)|Xi,l,Zi,l,γi(s)]=β0(s)+γ0,i(s)+Delayi,l[β1(s)+γ1,i(s)].

We analyzed all trials on the last session of the short-delay (2 sec) and all trials on the first session of the long-delay (8 sec). Because the experimental design included the delay length switch between sessions, the covariate Delayi,l does not include a trial index j (i.e. all trials on session l have the same delay length). This model assumes γi(s)∼iidN(0,Σγ(s)), where γi(s)=[γ0,i(s)γ1,i(s)]T. This can be fit with the code:

model_fit = fui(photometry ~ delay + (delay | id), data = photometry_data)

We provide the fastFMM package that implements our proposed methods. The package can be downloaded on CRAN, or on the GitHub page: https://github.com/gloewing/fastFMM. This GitHub also includes our analysis guide, and examples of calling the package from python. We also recommend lme4’s thorough package resources, since our package is based on the lme4 software and model syntax (e.g. see Table 2 in Bates, 2014, Table 1 in Barr et al., 2013 and Bates, 2010). Appendix 3.4 contains more information about functional data analysis modeling approaches.

Our fastFMM package scales to the dataset sizes and model specifications common in photometry. The majority of the analyses presented in the Results included fairly simple functional fixed and random effect model specifications because we were implementing the FLMM versions of the summary measure analyses presented in Jeong et al., 2022. However, we fit the following FLMM to demonstrate the scalability of our method with more complex model specifications:E[Yi,j,l(s)|Xi,j,l,Zi,j,l,γi(s)]=β0(s)+γ0,i(s)+SNi,l[β1(s)+γ1,i(s)]+TNi,j,l[β2(s)+γ2,i(s)]+IRIi,j,l[β3(s)+γ3,i(s)]+Licki,j,l[β4(s)+γ4,i(s)]+TLi,j,l[β5(s)+γ5,i(s)].

We use the same notation as the Reward Number model in Reanalysis methods: Using FLMM to test associations between signal and covariates throughout the trial, with the additional variable TLi,j,l denoting the Total Licks on trial j of session l for animal i. In a dataset with over 3200 total trials (pooled across animals), this model took ∼1.2 min to fit on a MacBook Pro with an Apple M1 Max chip with 64 GB of RAM. Model fitting had a low memory footprint. This can be fit with the code:

model_fit = fui(photometry ~ session + trial +iri + lick_time + licks +            (session + trial + iri + lick_time + licks | id),            parallel = TRUE,            data = photometry_data)

For the experiments reported in Using FLMM to compare signal ‘temporal dynamics’ across conditions, we generated synthetic photometry signals based on the Delay Length data and model described in Reanalysis methods: Using FLMM to compare signal ‘temporal dynamics’ across conditions. We simulated the photometry signals using the functional linear mixed effects model.Yi,j,l(s)=β0(s)+γ0,i(s)+Delayi,j,l[β1(s)+γ1,i(s)]+ϵi,j,l(s)

where β0(s) is the (functional) intercept at time-point s on all trials, γ0,i(s) is the random (functional) intercept of subject i at time-point s on all trials. The (functional) coefficient β1(s) represents the average change in the mean signal at trial time-point s on long-delay trials (compared to short-delay trials). ϵi,j,l(s) is the error term and γk,i(s)⊥⊥ϵi,j,l(s) for all i∈{1,2,...,n}, j∈{1,2,...,J}, k∈{0,1}, l∈{1,2} and s∈{1,2,...,S}. All other notation is described in Notation and Reanalysis methods: Using FLMM to compare signal ‘temporal dynamics’ across conditions.

For each simulation replicate, we drew γk,i∼iidNT(0,Σ^kγ) where Σ^kγ is the S×S covariance matrix of random-effect k across trial time-points, estimated in the analysis described in Reanalysis Methods: Using FLMM to compare signal ‘temporal dynamics’ across conditions. For each s, we set β(s)=β^(s), where β^(s) was the estimated coefficient vector, at s, from the model described in Reanalysis methods: Using FLMM to compare signal ‘temporal dynamics’ across conditions. We drew ϵi,j,l∼iidNT(0,Σiϵ) where ϵi,j,l represents the S×1 vector of errors across trial time-points. The errors were drawn independently across different trials and subjects. To induce additional correlation across time-points within a trial, we set Σiϵ=5∗Σ^mϵ, where Σ^mϵ is the estimated covariance matrix of the model residuals fit to the observed photometry data of animal m. For each simulated subject, i, the index m was drawn uniformly from the observed animal indices {1,...,7} without replacement when the simulated sample size n≤7. Otherwise, we set the first 7 to the indices {1,...,7}, and the remaining n−7 were drawn without replacement from the indices {1,...,7}. This induced some correlation between signals from different subjects when n>7, which had a small but predictably deleterious effect on method performances.

For comparisons in the cue period window, we assessed the average performance of FLMM evaluated at time-points in that window, against the performance of standard methods that analyzed a summary measure of the window (cue period average-signal, which we denote as AUC) as the outcome variable. The LMM was applied to trial-level AUCs, the paired t-test was applied to subject- and condition-level AUCs. The performances of the Perm and FLMM were quantified according to the average pointwise performance during the cue period. We present the full consecutive threshold criteria of Perm in Figure 7 and show the performance of the 1/2 consecutive threshold criteria in Appendix 5.2 for completeness. We implemented the consecutive threshold criteria method, adapted to our simulated sampling rate of 15 Hz.