Inferring causal connectivity from pairwise recordings and optogenetics
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eLife assessment
This useful study adapts methods from causal inference to develop analytical tools for determining the causal influence of single cells on downstream neurons. The simulation evidence is solid, suggesting that these causal methods produce better estimates under biologically-relevant confounds given enough data, although the practical application of the method and the biophysics it relies on is unclear. Nonetheless, this application of causal methods developed in econometrics and other fields could suggest new ways to think about largely observational datasets in neuroscience.
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Abstract
To understand the neural mechanisms underlying brain function, neuroscientists aim to quantify causal interactions between neurons, for instance by perturbing the activity of neuron A and measuring the effect on neuron B. Recently, manipulating neuron activity using light-sensitive opsins, optogenetics, has increased the specificity of neural perturbation. However, using widefield optogenetic interventions, multiple neurons are usually perturbed, producing a confound—any of the stimulated neurons can have affected the postsynaptic neuron making it challenging to discern which neurons produced the causal effect. Here, we show how such confounds produce large biases in interpretations. We explain how confounding can be reduced by combining instrumental variables (IV) and difference in differences (DiD) techniques from econometrics. Combined, these methods can estimate (causal) effective connectivity by exploiting the weak, approximately random signal resulting from the interaction between stimulation and the absolute refractory period of the neuron. In simulated neural networks, we find that estimates using ideas from IV and DiD outperform naïve techniques suggesting that methods from causal inference can be useful to disentangle neural interactions in the brain.
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Author Response
Reviewer #1 (Public Review):
This manuscript presents an inference technique for estimating causal dependence between pairs of neurons when the population is driven by optogenetic stimulation. The key issue is how to mitigate spurious correlations between unconnected neurons that can arise due to polysynaptic and other network-level effects during stimulation. The authors propose to leverage each neuron's refractory period (which begins at approximately random times, assuming Poisson-distributed spikes and conditional on network state) as an instrumental variable, allowing the authors to tease apart causal dependence by considering how the postsynaptic neuron fires when the presynaptic neuron must be muted (i.e., is in its refractory period). The idea is interesting and novel, and the authors show that their modified …
Author Response
Reviewer #1 (Public Review):
This manuscript presents an inference technique for estimating causal dependence between pairs of neurons when the population is driven by optogenetic stimulation. The key issue is how to mitigate spurious correlations between unconnected neurons that can arise due to polysynaptic and other network-level effects during stimulation. The authors propose to leverage each neuron's refractory period (which begins at approximately random times, assuming Poisson-distributed spikes and conditional on network state) as an instrumental variable, allowing the authors to tease apart causal dependence by considering how the postsynaptic neuron fires when the presynaptic neuron must be muted (i.e., is in its refractory period). The idea is interesting and novel, and the authors show that their modified instrumental variable method outperforms similar approaches.
We wish to thank the reviewer for this positive assessment.
However, the scope of the technique is limited. The authors' results suggest that the proposed technique may not be practical because it requires considerable amounts of data (more than 10^6 trials for just 200 neurons, resulting in stimulation of more than 5000 times per neuron). Even with such data sizes, the method does not appear to converge to the true solution in simulations. The method is also not tested on any experimental data, making it difficult to judge how well the assumptions of the technique would be met in real use-cases. While the manuscript offers a unique solution to inferring causal dependence, its applicability for experimental data has not yet been convincingly demonstrated, and would, therefore primarily be of interest to those looking to build on these theoretical results for further method development.
We thank the reviewer for this assessment and agree that the requirement for this many trials makes the estimators practically unsuitable for identifying causal interactions in large systems. However, in the revised manuscript, we can observe that the IV estimator can be beneficial after even a few thousand trials when introducing a newly improved error measurement (which we discovered thanks to these reviews). Moreover, we agree that this work will be of interest to the more theoretically oriented community for methodological improvements; we believe that the methods and causal inference framework will be interesting and useful for the wider neuroscience community. For example, considering the first (new) example in the introduction, even under two-photon single-neuron stimulation, the IV framework should be used to avoid bias amplification.
Reviewer #2 (Public Review):
Lepperød et al. consider the problem of inferring the causal effect of a single neuron's activity on its downstream population. While modern methods can perturb neuronal activity, the authors focus on the issue of confounding that arises when attempting to infer the causal influence of a single neuron while stimulating many neurons together. The authors adapt two basic methods from econometrics that were developed to address causal inference in purely observational data: instrumental variables and difference-in-differences, both of which help correct for unobserved correlations that confound causal inference. The authors propose an experimental procedure where neurons have spike times measured with millisecond precision and a subset of neurons are optogenetically activated. As an instrumental variable, the authors propose using the refractoriness of a stimulated neuron, resulting in absent or delayed spiking which can be used to infer its causal effect in otherwise matched conditions.
Based on this, they develop a collection of estimators to measure the pairwise causal relationship of one neuron on another. By simulating a variety of small networks, the authors show that, provided enough data is present, the proposed causal methods provide estimates that better match underlying connectivity than methods based on ordinary least squares or naive cross-correlograms (CCHs). However, the methods proposed require extensive data and highly targeted stimulation to converge.
Strengths:
The value of the paper comes from its attempt to find neuroscience applications for methods from fields where causal analysis of observational data is required. Moreover, as the field develops improved methods of measuring anatomical neuronal connectivity using molecular, physiological, and structural approaches, the question of the causal influence of one neuron's spiking on another remains vital. The authors thoughtfully lay out the necessary conditions - and difficulties - required to establish this type of causal functional influence and suggest one potential approach. The collection of models tested highlighted both the strengths and difficulties of the suggested approaches.
We wish to thank the reviewer for the positive feedback, we are delighted to share your view that obtaining methodology for estimating causal influence is vital.
Weaknesses:
- I found the paper's introduction to its analysis techniques to be very confusingly written, particularly as it is designed to bridge fields. It is vital that the ideas are communicated more clearly. Some topics are explained multiple times, even after being used previously, other ideas and notations are introduced and immediately dropped (e.g. the "do operator", the ratio of covariances in the introduction to instrumental variables), and still others are introduced with no clear explanation (e.g. the weight term w, the "|Y->Y-Y*" notation, and the notation in the methods with "Y(Z=0)").
We thank the reviewer to point out this lack of clarity and we extensively rewrote the paper to make it more accessible. The do operator is used in the methods to define Y(Z=0), but is now removed from the introduction to reduce the number of concepts introduced early in the text. The w term is now defined from the generative model. The difference in differences notation is written out fully to be clear and a sketch of the method intuition is added to Figure 1.
- Of particular importance, the introduction of the Z,X, and Y variables in the first full paragraph on page five, it could be made much more clear that this method is pairwise: Z and X reference the spiking of one specific stimulated neuron at two time points and Y references one specific downstream neuron. 2) In the third paragraph of the same page, the authors refer to the "refractoriness of X" and "spiking of X onto Y", but this language confuses the neurons with variables in a way that took considerable time to unpack. 3) This was not helped by Figure 1b, which suggested that Z_i, X_i, and Y_i applied to all neurons and merely reflected time points around stimulation. 4) Similarly, the introduction of the Y* variable in the difference of differences method, which the authors view as one of the main contributions, is given little clear explanation or intuition. I assume "shifted on window-size left" means measuring the presence of spiking at the same time step as X, but I see no clear definition of this. 5) The confusion about variables remains when, in Figure 1d, a "transmission probability" goes below 0 and above 1.
Thank you for pointing out this lack of clarity, the suggested explanation of the variables XYZ is adopted.
The language is clarified such that variables and neurons are separated.
Figure is fixed such that variables refer to the neurons they represent.
We have now improved the explanation of DiD with a figure for intuition.
We have now redefined the “transmission probability” to effective connectivity to reduce confusion.
I also found the network models studied after the first section and the relevant variables difficult to understand with the detail necessary to interpret the results. For example, the cartoon in Figure 2a does not seem to match the text description. I see no explanation for the external "excitatory confounder" and "inhibitory confounder" terms, nor what is done to control the (undefined) \sigma_max/\sigma_min term. I don't see anything in the methods about distinct inhibitory and excitatory neurons either. Further, the violin plots (e.g. Fig 2d) seem quite noisy (e.g. is Br, DiD really bimodal?), and it is not clear what distribution is being covered by them. If this is computational simulations, I would imagine more samples could be generated. The same vagueness issues hold for the networks in section 2.4 and 2.7.
We have now clarified the implementation of the excitatory and inhibitory confounder and how we distinguish between excitatory and inhibitory neurons and defined the condition number. The violin plots were removed in Fig 2 since the large variance represented changes across external drive which produced largely incomparable statistics. To illustrate variance, we now show the standard deviation of the absolute error in line plots 2e and 2g.
- Broadly speaking, the causal estimates appear better in the sense of having smaller errors, but it's not clear to me if they are actually good or not. What does an error of 0.4 mean in terms of your ability to estimate the causal structure, and what exactly does the Error(w{greater than or equal to}0) notion refer to? It would be useful to see actual reconstructions of ground truth versus causally inferred connectivity to better understand the method's strengths and weaknesses.
To improve clarity, we have added a paragraph in the text before figure 2 explaining a new error measure. Since the estimators give the transmission probability and not the inferred connection strength directly, we previously computed a regressed error as in Das & Fiete 2020. This error measure is equivalent to the sine of the angle between $W$ and $\hat{W}$. This error measure is not ideal and gives an indirect population measure with deviations scaled during the error regression. Upon further reflection, we realized that we could define the error directly using our definition of effective connectivity on the generative model to obtain a much cleaner and more interpretable measure. This further led us to remove one of the proposed methods (brew) as it did not perform well under this new error measure. All error measurements are updated in all figures. Error(w{greater than or equal to}0) means that we only look at positive weights; now clarified in the text
- I found the section on optogenetic modeling to be unsatisfying in its realism. The general result that 1 photon excitation hits a wide collection of neurons is undisputed, but the simulation does not account for a number of key factors - optogenetic receptor expression is distributed across the axons and dendrites of a cell, not only soma, scattering in tissue greatly affects transmission, etc. Moreover, experiments that attempt to do highly targeted activation have other methods for exactly this reason, such as multiphoton activation or electrophysiology. The message of decreasing performance as a function of stimulus size is important, but I struggle with the idea of the model being "realistic".
We thank the reviewer for pointing out this unsatisfactory comparison with realistic scenarios. To mitigate we have changed the wording, but kept the simulation as is. As the reviewer pointed out optogenetic receptor expression is distributed, and here we have assumed an expression that only affects soma (experimentally plausible according to Grødem et al 2023 (10.1038/s41467-023-36324-3)), scattering in tissue is included according to the Kubelka-Munk model.
- The authors spend a great deal of analysis of stimulation, but little time on measurement. It seems like this approach demands a highly precise measure of spike time to know if a neuron is firing or not at a given millisecond due specifically being in a refractory state. A stimulated but refractory neuron will still likely spike as soon as it can after the momentary delay, and given the noise in the network this difference might not be easily detectable in the delay-to-spike of the downstream neuron, even assuming one spike in the presynaptic neuron is likely to cause a spike in the downstream. It would be useful to see this aspect considered with the same detail as the rest of the study.
We thank the reviewer for pointing out this. We have now added a paragraph discussing this: “As outlined in \citep{ozturk2000ill}, ill-conditioning can affect statistical analysis in three ways and therefore similarly in inverse connectivity estimates from measured activity. First, measurement errors such as a temporal shift in spike time estimate e.g. due to low sampling frequency, inaccurate spike sorting, or general noisy measurement due to animal movement etc. In the presence of ill-conditioning the outputs will be sensitive (unstable) to small input changes. If errors are included in some variables, the inference procedures will require information about the distributional properties of these errors. Second, optimized inference can give misleading results in the presence of ill-conditioning, caused by bad design or sampling.
There will always exist a natural variability in the observations which necessitates the assessment of ill-conditioning before performing statistical analysis. Third, rounding errors can lead to small changes in input under ill-conditioning. This numerical problem is often not considered in neuroscience but will become evermore relevant when large-scale recordings require large-scale inferences.”
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eLife assessment
This useful study adapts methods from causal inference to develop analytical tools for determining the causal influence of single cells on downstream neurons. The simulation evidence is solid, suggesting that these causal methods produce better estimates under biologically-relevant confounds given enough data, although the practical application of the method and the biophysics it relies on is unclear. Nonetheless, this application of causal methods developed in econometrics and other fields could suggest new ways to think about largely observational datasets in neuroscience.
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Reviewer #1 (Public Review):
This manuscript presents an inference technique for estimating causal dependence between pairs of neurons when the population is driven by optogenetic stimulation. The key issue is how to mitigate spurious correlations between unconnected neurons that can arise due to polysynaptic and other network-level effects during stimulation. The authors propose to leverage each neuron's refractory period (which begins at approximately random times, assuming Poisson-distributed spikes and conditional on network state) as an instrumental variable, allowing the authors to tease apart causal dependence by considering how the postsynaptic neuron fires when the presynaptic neuron must be muted (i.e., is in its refractory period). The idea is interesting and novel, and the authors show that their modified instrumental …
Reviewer #1 (Public Review):
This manuscript presents an inference technique for estimating causal dependence between pairs of neurons when the population is driven by optogenetic stimulation. The key issue is how to mitigate spurious correlations between unconnected neurons that can arise due to polysynaptic and other network-level effects during stimulation. The authors propose to leverage each neuron's refractory period (which begins at approximately random times, assuming Poisson-distributed spikes and conditional on network state) as an instrumental variable, allowing the authors to tease apart causal dependence by considering how the postsynaptic neuron fires when the presynaptic neuron must be muted (i.e., is in its refractory period). The idea is interesting and novel, and the authors show that their modified instrumental variable method outperforms similar approaches.
However, the scope of the technique is limited. The authors' results suggest that the proposed technique may not be practical because it requires considerable amounts of data (more than 10^6 trials for just 200 neurons, resulting in stimulation of more than 5000 times per neuron). Even with such data sizes, the method does not appear to converge to the true solution in simulations. The method is also not tested on any experimental data, making it difficult to judge how well the assumptions of the technique would be met in real use-cases. While the manuscript offers a unique solution to inferring causal dependence, its applicability for experimental data has not yet been convincingly demonstrated, and would therefore primarily be of interest to those looking to build on these theoretical results for further method development.
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