Visualization of an example concurrent functional linear mixed model on synthetic data.

A) A-i) Functional outcomes (photometry signal) and inputs (animal velocity) over one trial, A-ii) labeled model equation for the time point marked with cross-hairs in (i) and (iii), A-iii) output functional fixed coefficients β0(s) and β1(s). B) Example experimental data and interpretation of effects, including the B-i) heat maps of the photometry dataset and speed dataset (groups are subjects i, rows are trials j, and columns are time points s), and B-ii) fixed effects at a given timepoint (left) and random effects at the same point (right).

Fitted coefficients for concurrent and non-concurrent models of the expected signal with respect to licking behavior in mice randomly delivered sucrose solution.

A) Features of the experimental data. A-i) The distribution of licking behaviors, averaged within mice and smoothed for readability, with one curve per mouse. A-ii) Photometry signal, averaged within mice. B) Estimates of the functional fixed coefficient β1(s) corresponding to different choices of FLMM and behavioral covariate. B-i) Concurrent model with an instantaneous functional covariate tracking licking behavior. B-ii-iv) Non-concurrent models with a trial-specific scalar covariate for the lick rate ii) 0.5 seconds after reward delivery, iii) 1.0 seconds after reward delivery, and iv) 2.0 seconds after reward delivery. Fixed effect estimates for trial and session can be found in Appendix 3.

Comparisons of coefficient estimates for an experiment where mice receive different rewards at variable times.

A Fixed coefficients in a non-functional linear mixed model fit to A-i the average of the photometry signal in the reward period or A-ii the sum, with ∗∗∗∗: p ≤ 0.0001. B Functional fixed coefficients in a non-concurrent functional linear mixed model, corresponding to B-i) the intercept, i.e., the expected signal when latency is average and the mouse is given water; B-ii) the expected signal when latency is average and the mouse is given SMS; B-iii) the slope of mean signal change with respect to (w.r.t.) change in latency; and B-iv) the slope of mean signal change w.r.t. change in latency when the mouse is rewarded with SMS. C Functional fixed coefficients in a concurrent functional linear mixed model, corresponding to C-i) the intercept, i.e., the expected change in signal when a mouse in the baseline water condition has not yet received a reward; C-ii) unrewarded SMS, i.e., the expected change in signal when a mouse in the SMS condition has not yet received a reward; C-iii) baseline reward change, i.e., the expected difference in signal between a mouse receiving water and not receiving water; and C-iv) SMS reward change minus baseline reward change, i.e., the expected difference between receiving SMS and not receiving SMS and the value described in A-iii.

Nominal coverage achieved in simulations, reported as A) joint coverage and B) pointwise coverage for different values of measurement noise (meas. noise) and between subject variability (b/w subj. var.).

Simulations were performed on 200 replicates of each condition. The horizontal red dotted line indicates the theoretical 95% coverage mark.

Functions corresponding to the fixed effects and a sample of 10 i.i.d. draws of subject-specific random effects.

The fixed intercept β0(s) is the true relationship between signal and time when velocity is zero at time s, and the fixed slope β1(s) is the true relationship between signal and velocity at time s. The random intercept γi,0(s) is individual perturbation of the fixed intercept for mouse i at time s, and the random slope γi,1(s) is the individual perturbation of the fixed slope for mouse i at time s.

Data points in the insets correspond to the true values generated for the demonstration.

Joint (A) and pointwise (B) coverage in simulations when incorporating a random slope.

Traces of synthetic photometry signals for five subjects with one observation each.

Point estimates and confidence intervals for all fixed effects included in the models compared in Modeling behaviors that change over the trial.

Axes are omitted because the scale of the concurrent and non-concurrent models are different based on the covariate of lick or lick rate.