Functional hierarchies in brain dynamics characterized by signal reversibility in ferret cortex

Brain signal irreversibility has been shown to be a promising approach to study neural dynamics. Nevertheless, the relation with cortical hierarchy and the influence of different electrophysiological features is not completely understood. In this study, we recorded local field potentials (LFPs) during spontaneous behavior, including awake and sleep periods, using custom micro-electrocorticographic (μECoG) arrays implanted in ferrets. In contrast to humans, ferrets remain less time in each state across the sleep-wake cycle. We deployed a diverse set of metrics in order to measure the levels of complexity of the different behavioral states. In particular, brain irreversibility, which is a signature of non-equilibrium dynamics, captured by the arrow of time of the signal, revealed the hierarchical organization of the ferret’s cortex. We found different signatures of irreversibility and functional hierarchy of large-scale dynamics in three different brain states (active awake, quiet awake, and deep sleep), showing a lower level of irreversibility in the deep sleep stage, compared to the other. Irreversibility also allowed us to disentangle the influence of different cortical areas and frequency bands in this process, showing a predominance of the parietal cortex and the theta band. Furthermore, when inspecting the embedded dynamic through a Hidden Markov Model, the deep sleep stage was revealed to have a lower switching rate and lower entropy production. These results suggest functional hierarchies in organization that can be revealed through thermodynamic features and information theory metrics.


Major
• In the introduction you mention "One promising analysis framework considers the description of brain states in terms of brain dynamics operating out of equilibrium, based on time series reversibility features (Deco et al., 2022;Lynn et al., 2021)."The list is much bigger and interesting, you should mention the recent articles of de la Fuente et al., 2023, Sanz-Perl et al., 2021, Gilson 2023, Cruzat 2022, etc..
• It is not clear how the irreversibility level was calculated.You mention overlapping sliding windows (of 1 second), but you do not mention the overlap.Both quantities must be specified.For example, for a Markov chain, it is not that the probability of observing a transition between any given two states in the configuration space is the same as the probability of observing the reverse transition.The detailed balance condition is a series of equations that require that that the transition probability matrix P for the Markov process that possess a stationary distribution π satisfies that for all i, j. • You mention "if the system follows detailed balance" I would say satisfies the detailed balance conditions...the transition probabilities between any two states are equal to the reverse probabilities...again this is wrong... and then the entropy production will be zero.. for this you have to assume that 0 log 0 = 0, which is not explicitly stated in the equations.
• In the section results, you mention "We inspected the temporal asymmetry of the dynamical system", which dynamical system?seems to me that you only look at the experimental time series without referring to any dynamical system.
• Then you mention that you measure non-reversibility and consequently the level of nonequilibrium of the system.While for some non-equilibrium steady states found in statistical mechanics, this relationship holds, it is not clear to me what you mean here.What is the equilibrium state from which you measure the level of non-equilibrium of the system?
• In the results you mention "If the relation between the network states was the same in both directions (symmetrical), then the entropy level would be 0." Again you are confusing entropy with entropy production.
• Fig 3f, you show transition probabilities but the values that you show reveal that these are not transition probabilities, as the sum of each row is not one.
• In the discussion you mention "We show how this measure of irreversibility can be used as a proxy of non-equilibrium", Actually you don't show that, you only vaguely mention it.
• In the discussion you mention "Previous studies have inspected non-equilibrium of signals in macaque monkeys (Deco et al. 2022)" You forgot one relevant article by one of the authors of this manuscript (Sanz Perl 2021).
• In the discussion you mention that the SWS behavioral state exhibited a lower entropy production, with respect to the awake state.This result is in agreement with the recent result obtained using human fMRI data (Gilson 2023), a relevant relationship may appear here, you should compare and comment on this.
• In the discussion you mention " In conclusion, we demonstrated how to direct metrics derived from thermodynamics such as signal reversibility".Your custom measure of signal reversibility (equation 3) seems not to be derived from thermodynamics.

Minor
• In the definition of determinism you mention "The average determinism of a directed network X with N .. with N what?
• In the definition of degeneracy, it is not clear to me how you compute the entropy of an average.Is the distribution of averages?Also, seems that a parenthesis is not necessary and you define a network as X and then refer to it as x.
• In the definition of mutual information, the definition itself is missing, you only mention properties and mention "strictly independent" while being independent is a property that a pair or random variables holds or not.Statistical interdependence is different.
• In the results you mention "The absolute difference between the irreversibility matrices (forward minus backward)" what do you mean by irreversibility matrices, seems to me that you compute the difference between the functional connectivity matrices.
• Fig 1c what is the time scale?After you mention that you compute irreversibility values (difference between forward and backward matrix) at each time point.It is not clear at all how this quantity is computed following your definition in Equation 3. Is an average over all pairs?
• How do you explain the differences between irreversibility and entropy production between panels C and I?
• In the discussion you mention "the extensive recording time (>3 hours per animal)".This should appear before not in the conclusion.Before you only mentioned resting activity during the light cycle between 12 a.m. and 6 p.m., but is not clear if you used all the data.
• If you want to illustrate a transition probability a color bar with the values is missing.
• In figure 1 is not clear what HMM-MAR brings to the irreversibility framework.
• In the caption of panel I) you mention "The highest entropy" do you mean entropy production?
• • In my opinion, the computational code used to obtain the results should always be available, readable, and usable without even needing to request it.
Figure 2B is badly explained.Took me a while to understand what you did and I'm not sure I understood.You should be more explicit on what each column means.Only the color of the rectangle guides what you mean.Also, you should be more explicit in the x-axis of the histograms.It is not clear how you measure irreversibility between and within regions