Logarithmically scaled, gamma distributed neuronal spiking

Abstract Naturally log‐scaled quantities abound in the nervous system. Distributions of these quantities have non‐intuitive properties, which have implications for data analysis and the understanding of neural circuits. Here, we review the log‐scaled statistics of neuronal spiking and the relevant analytical probability distributions. Recent work using log‐scaling revealed that interspike intervals of forebrain neurons segregate into discrete modes reflecting spiking at different timescales and are each well‐approximated by a gamma distribution. Each neuron spends most of the time in an irregular spiking ‘ground state’ with the longest intervals, which determines the mean firing rate of the neuron. Across the entire neuronal population, firing rates are log‐scaled and well approximated by the gamma distribution, with a small number of highly active neurons and an overabundance of low rate neurons (the ‘dark matter’). These results are intricately linked to a heterogeneous balanced operating regime, which confers upon neuronal circuits multiple computational advantages and has evolutionarily ancient origins.


08-Jun-2022 1st Editorial Decision
Dear Michael, Re: JP-TR-2022-282758 "Logarithmically scaled, gamma distributed neuronal spiking" by Daniel Levenstein and Michael Okun Thank you for submitting your Topical Review to The Journal of Physiology.It has been assessed by a Reviewing Editor and by 2 expert referees and I am pleased to tell you that it is considered to be acceptable for publication following satisfactory revision.
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The Journal of Physiology ----------------EDITOR COMMENTS Reviewing Editor: The authors have done an excellent job of writing a topical review on scaling of neuronal spiking.Both reviewers agreed that the review was timely and addressed an important topic in the field..Both reviewers just had extremely minor issues that can be addressed with minor changes to the text.Please rewrite text taking into careful consideration of these points. -----------------

REFEREE COMMENTS
Referee #1: In this manuscript, the authors address the scaling of distributions of neuronal spiking within single neurons as well as across populations.Based on analyses of available experimental data and models they argue that a gamma distribution of firing rates is a better description than lognormal.They discuss "dark matter", irregular/ fluctuation-driven spiking, potential mechanisms of the distribution shape, and other interesting issues.The manuscript is somewhere in between a literature review and novel findings.Given the importance of the topic and the relatively sparse literature on the topic, I see the manuscript as an important contribution.I have a few comments.
Section 2 is important but confusing: "In our example, the log-transform makes both distributions bell-shaped (albeit left-skewed), revealing that the modes of the two log-distributions are similar but the orange distribution has heavier tails." First, there is no orange distribution in figure 2.
What is the difference between 2B and 2C?Some more text in the caption and in the section would be helpful.I recommend writing more text details on the difference and how the measurements were analyzed to give the difference between these two panels.Is the data computer generated?Regarding: Section 5: This section is important for explaining the mechanisms behind the skewed distributions.The argument is that a supra-linear I/O function turns normally distributed input (membrane potential) into firing rates that are not normally distributed (due to the nonlinearity).An argument suggested by Roxin et al, as the authors are well aware.The authors primarily justify this assertion by referring to previous modeling papers.But why not include the experimental work of Petersen and Berg 2016 in these references?Petersen and Berg investigated exactly this in the neuronal network dynamics by measuring membrane potential fluctuation and the supra-linearity across the population and the log-scaling of the spiking.They found that the input distribution is Gaussian and on average 3 sigmas from the threshold, both on single-cell level but also across the population.
Referee #2: In the topical review "Logarithmically scaled, gamma distributed neuronal spiking", the authors discuss findings on the distributions of firing rates and inter-spike intervals (ISI) in, for the most part, cortical neurons.The main thrust of the review is that these distributions are far from Gaussian and are best considered on a logarithmic scale.Overall I believe this is a clear and readable introduction to the topic for the non-expert.This discussion of the differences between Gaussian and logdistributed (log-normal or log-gamma) distributions is well done, as is the discussion of the data.My only concerns would be regarding the details of mechanisms as understood through analysis of single-cell models and networks.I feel some of the details, which I think are of interest, are glossed over.
1 -Relationship between firing rate and CV: I feel Fig. 3 could be explained more clearly.In particular, it would be nice to highlight the difference between the Poisson neuron and the LIF as far as the shape of the CV distribution for difference mean rates.As the authors state, the Poisson process always has CV=1, but this is not the case for the LIF and the ISI distribution narrows for increasing rates, a very general property of IF models and spiking networks in general.This is actually an open challenge in neuroscience since the CV for neurons in-vivo remains high even in the AS, e.g.Compte, Constantinidis et al.J. Neurophysiology 2003.
2 -Dale's principle and the fluctuation-driven regime: In section 5 the authors state that log-scaled ISIs and firing rates are somehow a consequence of Dale's Law.I have to say I don't understand this point.Dale's law state that each neuron is either E or I and cannot have an excitatory effect on some targets and an inhibitory effect on others.From the point of view of the post-synaptic neuron, the requirement for being in the fluctuation-drive regime is that E and I inputs should cancel more or less in the mean.This mechanism is agnostic as to where these inputs are coming from, and hence does not depend on Dale's law.At the network level, to find a self-consistent network state in a recurrent circuit in the fluctuationdriven regime may in some way depend on Dale's law, but it's not clear to me how.Once the network is in the fluctuationdriven regime then log-scaled rates and ISIs follow, yes.
3 -Left-skewed log-rate distributions: The authors note that actual firing rate distributions are not exactly log-normal, but rather they have fatter left tails on a log-scale.They show that this is well fit by a log-gamma.It may also be of interest to show that the firing rate distributions calculated analytically from the theory for LIF neurons as well as the results from simulations of spiking neurons also reproduce this left-skewness and, in fact, can fit in-vivo data quite well, e.g.Roxin et al.J. Neurosci.2011 Fig7.
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09-May-2022
words for one author or 150 words in total for two authors) and a portrait photograph of the two leading authors on the paper.These should be uploaded, clearly labelled, with the manuscript submission.Any standard image format for the photograph is acceptable, but the resolution should be at least 300 dpi and preferably more.A group photograph of all authors is also acceptable, providing the biography for the whole group does not exceed 150 words.

Reviewing Editor:
The authors have done an excellent job of writing a topical review on scaling of neuronal spiking.Both reviewers agreed that the review was timely and addressed an important topic in the field.Both reviewers just had extremely minor issues that can be addressed with minor changes to the text.Please rewrite text taking into careful consideration of these points.
We were delighted to receive this positive evaluation of our submission.In the revised version we have addressed the points raised by the reviewers, as described below.The line number pointers refer to the revised manuscript.

Referee #1:
In this manuscript, the authors address the scaling of distributions of neuronal spiking within single neurons as well as across populations.Based on analyses of available experimental data and models they argue that a gamma distribution of firing rates is a better description than lognormal.They discuss "dark matter", irregular/ fluctuation-driven spiking, potential mechanisms of the distribution shape, and other interesting issues.The manuscript is somewhere in between a literature review and novel findings.Given the importance of the topic and the relatively sparse literature on the topic, I see the manuscript as an important contribution.I have a few comments.
Thank you for this highly favourable evaluation of our manuscript.Section 2 is important but confusing: "In our example, the log-transform makes both distributions bell-shaped (albeit left-skewed), revealing that the modes of the two log-distributions are similar but the orange distribution has heavier tails." First, there is no orange distribution in figure 2.
We agree that the colour is closer to brown than orange.To avoid such confusion, we edited the sentence (L74-75) and added to the figure a legend that labels each distribution.
What is the difference between 2B and 2C?Some more text in the caption and in the section would be helpful.I recommend writing more text details on the difference and how the measurements were analyzed to give the difference between these two panels.Is the data computer generated?
We have updated the text to explain the intuition behind the log-transform (L78-84).We now also include a statistical summary document which provides an example using histograms of datapoints drawn from the PDF, with the (MATLAB) code made publicly available on github.
Section 5: This section is important for explaining the mechanisms behind the skewed distributions.The argument is that a supra-linear I/O function turns normally distributed input (membrane potential) into firing rates that are not normally distributed (due to the nonlinearity).An argument suggested by Roxin et al, as the authors are well aware.The authors primarily justify this assertion by referring to previous modeling papers.But why not include the experimental work of Petersen and Berg 2016 in these references?Petersen and Berg investigated exactly this in the neuronal network dynamics by measuring membrane potential fluctuation and the supra-linearity across the population and the logscaling of the spiking.They found that the input distribution is Gaussian and on average 3 sigmas from the threshold, both on single-cell level but also across the population.
Thank you for this excellent suggestion.We have updated the text to explicitly state that the theoretical work of Roxin et al. was verified experimentally by Petersen & Berg (L396-397).

Referee #2:
In the topical review "Logarithmically scaled, gamma distributed neuronal spiking", the authors discuss findings on the distributions of firing rates and inter-spike intervals (ISI) in, for the most part, cortical neurons.The main thrust of the review is that these distributions are far from Gaussian and are best considered on a logarithmic scale.Overall I believe this is a clear and readable introduction to the topic for the non-expert.This discussion of the differences between Gaussian and log-distributed (log-normal or log-gamma) distributions is well done, as is the discussion of the data.My only concerns would be regarding the details of mechanisms as understood through analysis of single-cell models and networks.I feel some of the details, which I think are of interest, are glossed over.
Thank you for this overall positive evaluation of the manuscript and the suggestion on the ways to improve it.
1 -Relationship between firing rate and CV: I feel Fig. 3 could be explained more clearly.In particular, it would be nice to highlight the difference between the Poisson neuron and the LIF as far as the shape of the CV distribution for difference mean rates.As the authors state, the Poisson process always has CV=1, but this is not the case for the LIF and the ISI distribution narrows for increasing rates, a very general property of IF models and spiking networks in general.This is actually an open challenge in neuroscience since the CV for neurons in-vivo remains high even in the AS, e.g.Compte, Constantinidis et al.J. Neurophysiology 2003.
Thank you for raising this point.We revised the manuscript to explicitly point to the fact that the shape of log ISI distribution of IF neuron is not constant, unlike the Poisson neuron (L158-160).
We also considered adding two additional panels (reproduced below) to the figure.We decided that Fig. 3C provides a sufficient illustration but would be happy to get further feedback on this.
2 -Dale's principle and the fluctuation-driven regime: In section 5 the authors state that log-scaled ISIs and firing rates are somehow a consequence of Dale's Law.I have to say I don't understand this point.Dale's law state that each neuron is either E or I and cannot have an excitatory effect on some targets and an inhibitory effect on others.From the point of view of the post-synaptic neuron, the requirement for being in the fluctuation-drive regime is that E and I inputs should cancel more or less in the mean.This mechanism is agnostic as to where these inputs are coming from, and hence does not depend on Dale's law.At the network level, to find a self-consistent network state in a recurrent circuit in the fluctuation-driven regime may in some way depend on Dale's law, but it's not clear to me how.Once the network is in the fluctuation-driven regime then log-scaled rates and ISIs follow, yes.
We completely agree that the title of this section was misleading.We therefore changed it to "Logscaled ISIs and mean firing rates from balanced input" and would like to thank the reviewer for pointing this out.In fact, it was not our intention to say that log-scaled ISIs and firing rates are the consequence of Dale's principle.Rather we want to say that we can presume Dale's principle and proceed based on this fundamental assumption, simply because these are the networks that one encounters.
What would have happened if Dale's principle did not hold, i.e., if there was a single population of neurons that have both E & I synapses, or perhaps even individual synapses that are both E & I? Clearly in this case simple formalisms like Wilson-Cowan do not hold.It is possible that E-I balanced regimes would exist in such networks, but they probably would require some structural conditions on the number and strength of E & I synapses.This is an interesting theoretical question, which however is well outside the scope of the manuscript (because Dale's principle does hold after all).
3 -Left-skewed log-rate distributions: The authors note that actual firing rate distributions are not exactly log-normal, but rather they have fatter left tails on a log-scale.They show that this is well fit by a log-gamma.It may also be of interest to show that the firing rate distributions calculated analytically from the theory for LIF neurons as well as the results from simulations of spiking neurons also reproduce this left-skewness and, in fact, can fit in-vivo data quite well, e.g.Roxin et al.J. Neurosci.2011 Fig7.
Thank you for this suggestion.We fully agree that this is highly relevant to the points that the review is making, and have edited the text to explicitly make this point (L396-397).

28-Jul-2022 1st Revision -Editorial Decision
Dear Michael, Re: JP-TR-2022-282758R1 "Logarithmically scaled, gamma distributed neuronal spiking" by Daniel Levenstein and Michael Okun I am pleased to tell you that your Topical Review article has been accepted for publication in The Journal of Physiology, subject to any modifications to the text that may be required by the Journal Office to conform to House rules.
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