Author response:
The following is the authors’ response to the original reviews.
In the revised manuscript, we have expanded the real-data analyses, clarified the relationship between CMP and prior modulated Poisson models, and added discussion of model limitations and future extensions. In summary, the major changes include:
(1) We revised the Introduction, Results, Methods, and Goris-model appendix to clarify the relationship between CMP and prior modulated Poisson models. In particular, we now emphasize that the key distinction is CMP’s continuous-time stochastic gain process.
(2) We moved the simulation-based recoverability analysis from Appendix 3 into the main Results section (Figure 3 in the revised manuscript), making the validation of the inference procedure more visible to readers.
(3) We added new analyses of the inferred gain process. Specifically, we now show the cross-trial gain mean and cross-trial gain variance in Figure 4A to assess whether gain captures stimulus-locked structure, and we added an analysis of pre- versus post-stimulus cross-trial gain variability in Figure 4C to test for gain-variability quenching during stimulus presentation.
(4) We clarified the definitions and implementation of the Baseline Poisson, Poisson-GP, and Goris-style comparison models, including the role of the smoothness prior on the stimulus drive.
(5) We expanded the Discussion to describe future extensions to population recordings, including a GPFA-inspired extension with low-dimensional shared gain activity across neurons.
(6) We added a Discussion paragraph clarifying that the current CMP model captures Poisson and super-Poisson variability, but not sub-Poisson variability, and outlined possible extensions using spike-history terms, renewal-process likelihoods, or alternative count distributions.
eLife Assessment
This work of fundamental significance introduces a novel statistical model of spiking activity that incorporates continuous−time gain modulation. The authors provide exceptional evidence that the model outperforms earlier approaches and alternative candidates in capturing spiking responses across multiple visual areas in the macaque. Beyond its methodological contribution, the study offers new insights into how stimulus−driven variability and internally generated gain fluctuations evolve over time and between brain areas. The framework is likely to find broad application beyond the datasets examined here.
We sincerely thank the Senior Editor, Reviewing Editor, and both reviewers for their careful evaluation and constructive feedback. We are encouraged by the positive assessment of the work and by the recognition of its methodological and conceptual contributions. We especially appreciate the acknowledgement that the continuous-time formulation provides a useful framework for modeling gain modulation in spiking activity, improves upon earlier approaches in capturing responses across multiple visual areas, and offers new insights into how stimulus-driven variability and internally generated gain fluctuations evolve over time and across brain regions.
In the revised manuscript, we have addressed the reviewers’ comments by clarifying the relationship between CMP and prior modulated Poisson models, strengthening the presentation of the simulation-based recoverability analysis, adding new validation analyses of the inferred gain process, and expanding the Discussion of model scope, limitations, and future directions. In particular, we now more clearly distinguish the continuous-time gain process in CMP from Goris-style models with constant or piecewise-constant gain, move the simulation recoverability analysis into the main Results, examine trial-averaged inferred gain and gain-variability quenching, clarify the definitions of the baseline and comparison models, and discuss extensions to population recordings and sub-Poisson variability.
We believe these revisions improve the clarity, rigour, and scope of the manuscript. Below, we address each reviewer comment in turn and describe the corresponding changes made in the revised manuscript.
Public Reviews:
Reviewer #1 (Public Review):
Summary:
In this manuscript, Rupasinghe and co−authors introduce a new statistical model for spiking neurons. Building on earlier work, they propose to model spikes as arising from a Poisson process whereby the firing rate is the product of stimulus drive and astimulus−independent gain signal. The critical innovation of this work is that the gain signal is modeled in continuous time. Earlier explorations of this statistical construction treated the gain−signal as constant within a trial. This innovation is elegant and important. It makes the model richer, more plausible, and more broadly applicable. The authors show that the model parameters are recoverable from realistic amounts of data and then apply the framework to previously studied datasets. They show that the new model outperforms earlier models and alternative candidates in capturing spiking data across four visual areas of the macaque monkey. Analysis of the model parameters replicates some earlier findings and uncovers several new insights. The model and fitting methods can be broadly applied to partition different types of signals and noise from spiking data and are likely to be widely adopted in the systems neuroscience community.
Strengths:
(1) Through clever use of advanced statistical techniques, the authors manage to infer critical information from single−trial single−cell data.
(2) The question of which aspect of a spike train is signal and which is noise is omnipresent in neuroscience. By improving our ability to characterize the distinct factors that shape spiking activity, this work makes a fundamental contribution to the literature.
We sincerely thank the reviewer for the thoughtful and detailed evaluation of our manuscript. We are pleased that the continuous-time formulation and its methodological contributions were viewed as elegant, important, and broadly applicable. We also appreciate the reviewer’s recognition that the framework provides a useful way to separate stimulus-driven and modulatory components of neural variability from single-trial, single-cell data. The reviewer’s comments helped us improve the precision of our framing, clarify the relationship between CMP and prior modulated Poisson models, and strengthen the validation of the inferred gain process. Below, we respond to each point in turn and describe the revisions made in the manuscript.
Weaknesses:
Overall, I find the work impressive and important. I have a couple of questions and suggestions.
(1) The work is entirely focused on single−cell data. While this is a great starting point, expanding the approach to spiking activity in neural populations is an importantfuture goal.
We thank the reviewer for this important suggestion. We agree that extending the CMP framework to population recordings is a natural and important direction for future work. In the present study, we focus on single-neuron responses to establish the continuous-time model, validate the inference, and characterize how stimulus-driven activity and stochastic gain fluctuations can be separated at the level of individual cells. However, the same modeling principles could be extended to simultaneously recorded neural populations by introducing shared latent structure across neurons. For example, one natural direction would be to combine CMP with ideas from Gaussian Process Factor Analysis [Keeley et al., 2020], using low-dimensional shared gain activity to capture population-wide fluctuations, while retaining neuron-specific stimulus-driven components. Such an extension would allow the model to capture correlated variability and shared modulatory dynamics across neural ensembles. In the revised manuscript, we have expanded the Discussion to describe this possible future extension to population recordings.
To address this comment, we expanded the Discussion (Page 14: lines 473-478) to describe a possible GPFA-inspired extension of CMP to population recordings.
(2) Line 49−53: These statements seem incorrect to me. The modulated Poisson model , as introduced in Goris et al (2014), is a process model that can perfectly be used to generate spike trains (within a trial, spiking emerges from a Poisson process, which canbe homogeneous or inhomogeneous). Moreover, the model contains a parameter thatrepresents the duration of the counting window (delta t). The dependency of over− dispersion on the size of the time bins for real neurons is shown in Figure 1b (inset plot) of that paper (and shown to resemble the model prediction). This time− dependency was further explored by the same authors in Goris et al (2018 − Journal ofVision) and also in Henaff et al (2020 − Nature Communications). I suggest that the authors rephrase this argument (here and at some later points in the paper). They could just say that the Goris model makes the simplistic and implausible assumption that, within a given trial, gain does not fluctuate. This is clearly an important limitation and the key difference with the continuous model introduced here.
We sincerely thank the reviewer for identifying this lack of clarity in our original description. We agree that our original description was not sufficiently precise. The modulated Poisson model introduced by Goris et al. (2014) is indeed a generative process model and can be used to generate spike trains, with spiking arising from a Poisson process that may be homogeneous or inhomogeneous within a trial. We apologize for implying otherwise.
Our intended point was that, in the original formulation, the modulatory gain is represented as a scalar random variable associated with a counting window or trial, and therefore does not explicitly model gain as a continuously time-varying process within a trial. Thus, the key limitation addressed by CMP is not the use of a Poisson process, but the assumption that gain is constant or piecewise constant over the relevant interval.
In the revised manuscript, we have rephrased the Introduction to clarify this distinction. We now describe the Goris model more accurately as a modulated Poisson framework in which gain is constant over the counting window, and we emphasize that CMP extends this framework by replacing this assumption with a continuous-time stochastic gain process. We have also added discussion of related time-dependent analyses and extensions [Goris et al., 2018, H´enaff et al., 2020], as thoughtfully suggested by the reviewer.
In addition, we revised the Results and Methods to clarify how the Goris-style baselines were implemented in our comparisons. Specifically, all Goris-style results reported in the main model comparisons use versions with a smoothness prior on the stimulus drive, where the stimulus-dependent firing rates are set to the smooth firing-rate estimates obtained from the Poisson-GP model. This ensures that the comparisons focus on different assumptions about the temporal structure of the gain process, rather than differences in stimulus-drive estimation. We also clarified the comparison to Goris-style variants without this smoothness prior, in which the stimulus-drive parameters are estimated directly under the corresponding Goris-style likelihood (Figure 5 - figure supplement 2). These results show that the smoothness prior on the stimulus drive substantially improves model performance. Finally, we revised the Figure 1 caption and the Goris-model appendix to make these distinctions explicit.
To address this comment, we revised the Introduction (Pages 2-3: Lines 49-77), Results (Page 9: Lines 263-266, 273-277, Page 11: Lines 319-326), Methods (Page 21), Figure 1 caption, and Goris-model appendix to clarify that CMP extends the Goris framework by modeling gain as a continuously time-varying process within trials.
(3) Line 54−55: I think the first part of the claim is a bit misleading. There is nothing in the Goris model that would inherently limit it to homogeneous Poisson processes, as seems to be implied by this description. The model is built on the assumption thatspike generation within a trial arises from a Poisson process. This may very well be an inhomogeneous Poisson process (i.e., a stimulus−dependent time−varying firing rate). Homogeneous and inhomogeneous Poisson processes both give rise to Poisson distributed spike counts (and thus a mixture of Poisson distributions across trials in the Goris model). I suggest the authors clarify this description a bit. Note that the two model variants illustrated in Figure 1b and c were also explored in Henaff et al (2020 − Nature Communications).
We thank the reviewer for this helpful clarification. We agree that the Goris model is not limited to homogeneous Poisson spiking and can incorporate a stimulus-dependent, time-varying firing rate within trials. We did not intend to imply otherwise, and we have revised the relevant text to avoid this misunderstanding.
Our intended point was that, in formulating continuous-time extensions of the modulated Poisson framework, we explicitly model the time-varying stimulus drive using a smoothness prior, as in the CMP framework, and then consider different assumptions about the temporal structure of the gain process, including constant gain and independently resampled gain across time bins. This highlights the distinction between piecewise-constant gain assumptions and the fully continuous gain process introduced in CMP.
In the revised manuscript, we have clarified this distinction in the Introduction, Results, and Methods. We now state that the Goris-style variants use stimulus-dependent, time-varying Poisson firing rates, and that the main difference between these variants and CMP lies in the temporal structure assumed for the gain process. We have also acknowledged related variants explored in Goris et al. [2018] and H´enaff et al. [2020], and clarified that our continuous-time formulations of the Goris model differs by imposing a smoothness prior on the stimulus drive. This allows us to estimate a regularized time-varying stimulus component while comparing different assumptions about gain dynamics, ensuring that the comparison focuses on the temporal structure of the gain process rather than differences in stimulus-drive estimation. We also highlight in Figure 5 - figure supplement 2 that even for the Goris-style models, versions that use a smoothness prior on the stimulus drive outperform versions that do not, which are closer to the original modulated Poisson formulation.
To address this comment, we revised the Introduction (Pages 2-3: Lines 49-77), Results (Page 9: Lines 263-266, 273-277, Page 11: Lines 319-326), Methods (Page 21) to clarify that the Goris-style variants allow stimulus-dependent time-varying firing rates and differ from CMP primarily in their assumptions about gain dynamics. We also added citations to related time-dependent extensions of the modulated Poisson framework.
(4) The extension to the continuous case is very elegant!
We thank the reviewer for the positive comment and are pleased that the continuous-time formulation was viewed as elegant.
(5) I find the result shown in Appendix 3 critically important. The recoverability of the model for realistic amounts of data is foundational for the rest of the paper. I wouldconsider including this analysis in the main results section. Not all readers may check Appendix 3, but they should know about this result.
We thank the reviewer for emphasizing the importance of this result. We agree that demonstrating parameter recoverability is foundational to the paper and should be visible to readers in the main Results section. In the revised manuscript, we have moved the simulation-based validation from Appendix 3 into the main Results. This section now describes the synthetic CMP dataset, the inference procedure used to estimate the latent stimulus-drive and gain processes, and the comparison between true and inferred GP hyperparameters. These results show that the proposed inference framework can accurately recover the ground-truth stimulus drives, gain processes, and hyperparameters from realistic amounts of simulated data.
To address this comment, we moved the simulation-based recoverability analysis from Appendix 3 into the main Results section (Page 6: Lines 200-211 and Figure 3).
(6) Figure 3: I am wondering whether the inferred gain is capturing some response fluctuations that originate from the cell’s phase−selectivity. Could the authors compute the trial−averaged inferred gain (ideally, aligned to stimulus−phase at the start of the trial if this experimental parameter varied across repeats)? If they have successfully partitioned the response variance, the trial−averaged gain should have no systematic temporal structure. If it has a sinusoidal modulation, it may partially capture stimulus−drive. This could be an interesting test to run on all model fits to further validate that the partitioning into a signal and noise component succeeded as intended.
We thank the reviewer for this insightful suggestion. We agree that verifying that the inferred gain does not capture stimulus-driven structure is an important validation of the model. In the revised manuscript, we have added the trial-averaged inferred gain to Figure 4A for the example neuron. This analysis shows that the trial-averaged inferred gain is relatively flat and neither resembles the inferred stimulus drive nor exhibits clear stimulus-locked temporal structure. This suggests that trial-specific gain fluctuations largely average out across repeats, consistent with the interpretation that the gain process captures random trial-to-trial variability rather than stimulus-driven activity.
We also note that a direct comparison of this inferred gain trace across methods is not possible for the Goris-style baselines, because these models do not infer a continuous trial-specific gain process. Instead, they marginalize over scalar or time-bin-independent gain variables when computing likelihoods and Fano factor curves. Thus, the trial-averaged gain diagnostic is specific to the CMP model, where the posterior over the continuous-time gain process is explicitly inferred.
To address this comment, we added the trial-averaged inferred gain to Figure 4A and clarified that it does not show a clear stimulus-locked temporal structure (Page 7: Lines 229-236).
(7) One common observation that is currently not explored is the quenching of neuronal response variability following stimulus onset (Churchland et al 2010 − NatureNeuroscience), which was suggested to reflect a quenching of gain variability in Goris et al (2024 − Nature Reviews Neuroscience). Building on the previous suggestion, the authors could compute the temporal evolution of cross−trial gain variability from the inferred gain traces. Do they recognize a reduction in gain variability following stimulus onset? If so, it would be worthwhile to show this.
We sincerely thank the reviewer for this valuable suggestion. We agree that examining whether gain variability decreases following stimulus onset provides an important test of the inferred gain process. In the revised manuscript, we have added an analysis of the temporal evolution of cross-trial gain variability before and after stimulus onset.
First, in Figure 4A, we now show the cross-trial variance of the inferred gain for the example neuron. This trace shows larger gain variability during the stimulus-off period and a reduction following stimulus onset, suggesting that the inferred gain captures a stimulus-related quenching of trial-to-trial variability. To quantify this effect across the population, we also added a pre- versus post-stimulus comparison in Figure 4C. Following the approach of Churchland et al. [2010], we compared gain variability in two matched 400-ms windows: a pre-stimulus window ending at stimulus onset and a stimulus-period window beginning 100 ms after stimulus onset. For each neuron and stimulus condition, we computed the cross-trial variance of the inferred gain at each time bin, averaged this quantity within each window, and then compared the pre- and post-stimulus values across neuron-stimulus pairs.
This analysis revealed a significant reduction in inferred gain variability following stimulus onset (one-sided paired Wilcoxon signed-rank test, p≤ 10−15), consistent with gain variability quenching [Churchland et al., 2010, Goris et al., 2024]. We now report this result in the main text and illustrate it in Figure 4A and Figure 4C. This provides additional evidence that the inferred CMP gain captures meaningful trial-to-trial variability and its temporal modulation around stimulus presentation.
To address this comment, we added the cross-trial gain variance trace to Figure 4A and a population-level pre- versus post-stimulus gain-variability quenching analysis (Page 7 and 8: Lines 239-248) in Figure 4C.
(8) Line 543−565: I want to make sure I understand the Baseline Poisson model and Poisson−GP correctly. For the baseline model, I had imagined that the authors would simply use the stimulus−conditioned PSTH as an estimate of the time−dependent firing rate, coupled with an inhomogeneous Poisson process assumption. But they additionally assume a Gamma prior on the firing rate to compensate for the sparsenessof the data (sometimes only 5 repeats per condition). The Poisson−GP includesexactly the same model components, but now the time−dependent firing rate is modeled by a Gaussian process. Doing this massively improves the goodness−of−fit (Fig 4A). Do I understand this correctly?
We thank the reviewer for this careful reading. Yes, this understanding is broadly correct, and we have revised the manuscript to clarify the relationships among the Baseline Poisson, Poisson-GP, and Goris-style models. The Baseline Poisson model estimates a stimulus- and time-dependent firing rate independently for each stimulus condition and time bin, using a Gamma-Poisson formulation to regularize the estimate when the number of repeats is limited. The Poisson-GP model uses the same conditionally Poisson observation model, but replaces these independent time-bin-wise rate estimates with a smooth stimulus-specific Gaussian process model for the log firing rate.
We have also clarified how the Goris-style models were implemented. All Goris-style results reported in the main model comparisons use versions with a GP prior on the stimulus drive. In these versions, the stimulus-dependent firing rates are set to the smooth firing-rate estimates obtained from the PoissonGP model, and the gain parameters are then fit under either the independent-gain or constant-gain assumptions. We used these GP-smoothed versions as stronger baselines. In Figure 5, Figure Supplement 2, we additionally compare these models to Goris-style variants without the GP prior on the stimulus drive, in which the stimulus-drive parameters are estimated directly under the corresponding Goris-style likelihood. This comparison shows that adding a GP smoothness prior to the stimulus drive substantially improves held-out model fit. Together, these analyses clarify that the GP-smoothed stimulus drive improves the Goris-style baselines, while the continuous-time gain process in CMP provides an additional improvement by capturing temporally structured trial-to-trial variability.
To address this comment, we clarified the definitions of the Baseline Poisson, Poisson-GP, and Goris-style models (Pages 8-9: Lines 255-259, 263-266, 273-277), and revised the text (Page 11: Lines 319-326) describing Figure 4 - figure Supplement 2 to make explicit how this existing comparison isolates the effect of the GP prior on the stimulus drive.
Reviewer #2 (Public Review):
Summary:
Neurons have varied responses to external stimuli that cannot be explained by naive Poisson models. Previous work has quantified and partitioned higher−than−Poisson variability in the brain into different components. The authors improve on these methods to infer how both the stimulus drive and internal gain dynamics impact neuronal variability continuously in time. The clean and well−reasoned model is rigorously developed and then applied to neural data across the visual hierarchy. This lends new insights into how variability is partitioned, agreeing with and extending previous work on how that variability changes from early visual areas (LGN, V1) through to higher, motion−sensitive areas (area MT). Another key contribution is that this partitioning can be fully addressed as a continuous−time process, which allows for the dissection of how the timescale of fluctuations in these two components changesacross the brain’s processing arc.
Strengths:
(1) The model is cleanly derived and thoroughly documented, including usable code shared in a GitHub repo. This makes the method immediately portable to other neural systems.
(2) This is a clear and well−presented piece of work. The figures and writing are clear and understandable, and all pieces of the derivations are included in the main text and supplementary information.
(3) Comparisons to other models, particularly the one from Goris et al., 2014 shows how this Continuous Modulated Poisson (CMP) model outperforms previous work.
(4) New insights about how variability partitioning changes across the visual stream from LGN to MT are revealed, including how the gain fluctuates on longer timescales in higher visual areas. Another key result about the anticorrelation between the variance in stimulus drive and gain fluctuations comports with theories about how neurons maintain efficient, reliable encoding.
(5) In addition to the results reported here, this work will serve as an excellent tutorial for students and postdocs first delving into the sources of variability in the brain.
We sincerely thank the reviewer for the thoughtful and positive assessment of our work. We are pleased that the model development, empirical analyses, and presentation were viewed as clear, rigorous, and useful for the broader neuroscience community. We also appreciate the reviewer’s recognition that the continuous-time formulation meaningfully extends prior variability-partitioning approaches by allowing stimulus drive and internal gain dynamics to be characterized across temporal scales. The reviewer’s comments helped us further clarify the positioning of the work, expand the Discussion of model scope and limitations, and better articulate future extensions. Below, we address the specific suggestions raised by the reviewer and describe the revisions made in the manuscript.
Weaknesses:
The work is somewhat incremental, building on previous studies of the partitioning of variability in the brain, but it provides important new extensions, as noted above.
Regarding the comment on incremental contribution, we agree that our framework builds directly on previous variability-partitioning approaches, especially the modulated Poisson framework of Goris et al. However, the main goal of this work is to move this class of models from a count-based formulation to a continuous-time spike-train framework. This extension is important because it allows us to model gain as a temporally structured latent process, characterize how variability depends on the timescale over which spikes are counted, and infer the temporal covariance structure of stimulus-independent fluctuations. In addition, the CMP framework provides analytic expressions for the Fano factor as a function of bin size, introduces the EPL covariance function for slowly decaying gain dynamics, and enables direct comparisons of gain amplitude and timescale across visual areas. In the revised manuscript, we have clarified this positioning and emphasized how CMP extends prior variability-partitioning models while preserving their interpretability.
To address this comment, we revised the Introduction (Pages 3-4: Lines 108-111 and Lines 118-121) and Discussion (Page 13: Lines 425-429) to clarify better how CMP builds on prior variability-partitioning models while extending them to continuous-time spike-train data.
The only major gap I would suggest addressing in the Discussion is the observation of sub−Poisson variability in the brain. It seems clear that this model can extend to sub− Poisson variability and its partitioning and perhaps even show how that varies in real time, with an animal’s attentional state. That is, of course, beyond the scope of the current work, but could be mentioned in the Discussion.
We thank the reviewer for this suggestion. We agree that sub-Poisson variability is an important phenomenon observed in neural data. Because the CMP model uses a conditionally Poisson observation model with stochastic gain modulation, it naturally captures Poisson and super-Poisson variability but does not generate sub-Poisson spike count statistics in its current form. In the revised manuscript, we have clarified this limitation in the Discussion and outlined possible extensions that could address sub-Poisson variability, including spike-history terms, renewal-process likelihoods, and alternative count distributions [Truccolo et al., 2005, Paninski et al., 2007, Aghamohammadi et al., 2024]. We also note that such extensions could allow future models to examine how sub-Poisson and super-Poisson components vary with behavioral state, attention, or arousal.
To address this comment, we added a Discussion paragraph describing the current model’s limitation for sub-Poisson variability and possible extensions to capture it (Page 14: Lines 460-471).
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