Continuous partitioning of neuronal variability

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 a stimulus-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.

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 important future goal.

(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 can be homogeneous or inhomogeneous). Moreover, the model contains a parameter that represents 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 of Vision) and also in Hénaff 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.

(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 that spike 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 Hénaff et al (2020 - Nature Communications).

(4) The extension to the continuous case is very 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 would consider including this analysis in the main results section. Not all readers may check Appendix 3, but they should know about this result.

(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.

(7) One common observation that is currently not explored is the quenching of neuronal response variability following stimulus onset (Churchland et al 2010 - Nature Neuroscience), 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.

(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 sparseness of the data (sometimes only 5 repeats per condition). The Poisson-GP includes exactly 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?

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 changes across 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.

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.

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.

Author response:

Reviewer #1 (Public review):

We 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 and broadly applicable, and that the empirical analyses provide meaningful new insights into neural variability across the visual hierarchy. We appreciate the reviewer’s constructive suggestions and clarifications, which will help us improve the precision, clarity, and scope of the manuscript. Below we respond to each point in turn and outline the revisions we will make.

(1) Extension to neural populations: We thank the reviewer for this important suggestion. We agree that extending the framework to population recordings is a natural next step. In this work, we focus on single-cell data to establish the model and validate inference. In the revised manuscript, we will expand the Discussion to outline how the framework could be generalized to population activity, for example by incorporating shared latent-variable structure.

(2) Clarification regarding the Modulated Poisson model: We thank the reviewer for pointing this out. We agree that our description was not sufficiently precise and may have been unclear. The modulated Poisson model introduced in Goris et al. (2014) is indeed a generative process model that can be used to generate spike trains, and we apologize for the inaccurate characterization of this framework. Our intended point was that the original formulation assumes gain is constant within a trial (or counting window) and does not provide a principled mechanism for modeling continuously time-varying gain fluctuations within trials. In the revised manuscript, we will clarify this distinction and revise the relevant passages accordingly. We will also cite and discuss related extensions and analyses in Goris et al. (2018) and Hénaff et al. (2020) to provide a more accurate and complete characterization of prior work.

(3) Continuous extensions of the Goris model: 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 will revise the relevant text to avoid this misunderstanding. Our intended point was that, in formulating continuous-time extensions, we explicitly model the time-varying stimulus drive using a GP prior, as in the CMP framework, and then consider different assumptions about the temporal structure of the gain process, including constant and finely sampled gain. This highlights the distinction between piecewise-constant gain assumptions and the fully continuous gain process introduced in our model. We will clarify this distinction in the revised manuscript. We will also acknowledge related variants explored in Hénaff et al. (2020) and more clearly describe how our formulation differs, including the role of smoothness priors on the stimulus drive and gain processes.

(4) Continuous-time extension: We thank the reviewer for the positive comment and are pleased that the continuous-time formulation was viewed as elegant.

(5) Parameter recovery analysis: We thank the reviewer for emphasizing the importance of this result. We agree that demonstrating parameter recoverability is foundational to the paper. In the revised manuscript, we will move the Appendix 3 analysis into the main Results section and clearly illustrate how our inference procedure faithfully recovers the generative parameters in simulation studies.

(6) Validation of gain–stimulus separation: 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 will compute the trial-averaged inferred gain, to assess whether it exhibits systematic temporal structure. This analysis will provide an additional check that the partitioning between stimulus drive and gain fluctuations operates as intended.

(7) Temporal evolution of gain variability: We thank the reviewer for this valuable suggestion. We agree that examining whether gain variability decreases following stimulus onset is an important and relevant analysis. In the revised manuscript, we will compute the temporal evolution of cross-trial gain variability from the inferred gain traces and assess whether a quenching effect is observed after stimulus onset. If present, we will report and illustrate this result.

(8) Clarification of Baseline Poisson and Poisson-GP models: We thank the reviewer for this careful reading. Yes, this understanding is correct. The Baseline Poisson model uses a stimulus-conditioned PSTH as an estimate of the time-dependent firing rate and includes a Gamma prior to regularize rate estimates in conditions with sparse repeats. The Poisson-GP model retains the same structure but models the time-dependent firing rate using a stimulus-specific Gaussian process prior, which substantially improves goodness-of-fit. In the revised manuscript, we will clarify this description. We will also highlight that Figure 4 – figure supplement 2 illustrates how introducing a GP smoothness prior on the stimulus drive markedly improves model fit, even within the Goris-style model.

Reviewer 2 (Public review):

We thank the reviewer for the thoughtful and positive assessment of our work. We are pleased that the model development, empirical analyses, and presentation were found to be clear and rigorous. We appreciate the recognition that the continuous-time formulation meaningfully extends prior variability-partitioning approaches and enables a more precise characterization of how stimulus drive and internal gain dynamics evolve across temporal scales. We are also encouraged that the cross-area analyses and model comparisons were viewed as providing new insights and clear empirical improvements. Below, we address the specific suggestions raised by the reviewer.

Positioning relative to prior work: Regarding the comment on incremental contribution, we agree that our framework builds directly on earlier variability-partitioning approaches. Our goal was to extend these models to continuous time and to develop a principled inference framework capable of characterizing how gain dynamics evolve across temporal scales. We will further clarify this positioning in the revised manuscript.

Extension to sub-Poisson variability: 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 builds on a Poisson observation model with stochastic gain modulation, it naturally captures Poisson and super-Poisson variability but cannot generate sub-Poisson spike count statistics in its existing form. We will clarify this limitation in the revised manuscript and expand the Discussion to outline potential extensions that could address sub-Poisson variability, such as incorporating spike-history effects, renewal-process models, or alternative count distributions.