Beta oscillations and waves in motor cortex can be accounted for by the interplay of spatially structured connectivity and fluctuating inputs
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This manuscript makes a valuable contribution to the field. The authors have developed a compelling network model to study mechanisms for the emergence of oscillations in the beta range in the primary motor cortex during movement preparation, and their propagation as traveling waves across the cortical sheet. The model is able to recapitulate several features of motor cortical activity acquired experimentally. Due to the recent results suggesting a functional role for traveling waves, it is of great interest to discover the mechanisms underlying such phenomena, and this work is an interesting step in that direction. However, the evidence for the reported new insights is incomplete at this stage, due to some weaknesses that remain to be addressed.
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Abstract
The beta rhythm (13–30 Hz) is a prominent brain rhythm. Recordings in primates during instructeddelay reaching tasks have shown that different types of traveling waves of oscillatory activity are associated with episodes of beta oscillations in motor cortex during movement preparation. We propose here a simple model of motor cortex based on local excitatoryinhibitory neuronal populations coupled by longrange excitation, where additionally inputs to the motor cortex from other neural structures are represented by stochastic inputs on the different model populations. We show that the model accurately reproduces the statistics of recording data when these external inputs are correlated on a short time scale (25 ms) and have two different components, one that targets the motor cortex locally and another one that targets it in a global and synchronized way. The model reproduces the distribution of beta burst durations, the proportion of the different observed wave types, and wave speeds, which we show not to be linked to axonal propagation speed. When the longrange connectivity or the local input targets are anisotropic, traveling waves are found to preferentially propagate along the axis where connectivity decays the fastest. Different from previously proposed mechanistic explanations, the model suggests that traveling waves in motor cortex are the reflection of the dephasing by external inputs, putatively of thalamic origin, of an oscillatory activity that would otherwise be spatially synchronized by recurrent connectivity.
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Author Response
Reviewer #1 (Public Review):
This theoretical (computational modelling) study explores a mechanism that may underlie beta (1330Hz) oscillations in the primate motor cortex. The authors conjecture that traveling beta oscillation bursts emerge following dephasing of intracortical dynamics by extracortical inputs. This is a well written and illustrated manuscript that addressed issues that are both of fundamental and translational importance.
We are pleased by the reviewer’s judgement about the importance of the question that we consider and about the presentation of our manuscript.
Unfortunately, existing work in the field is not well considered and related to the present work. The rationale of the model network follows closely the description in Sherman et al (2016). The relation (difference/advance) to this …
Author Response
Reviewer #1 (Public Review):
This theoretical (computational modelling) study explores a mechanism that may underlie beta (1330Hz) oscillations in the primate motor cortex. The authors conjecture that traveling beta oscillation bursts emerge following dephasing of intracortical dynamics by extracortical inputs. This is a well written and illustrated manuscript that addressed issues that are both of fundamental and translational importance.
We are pleased by the reviewer’s judgement about the importance of the question that we consider and about the presentation of our manuscript.
Unfortunately, existing work in the field is not well considered and related to the present work. The rationale of the model network follows closely the description in Sherman et al (2016). The relation (difference/advance) to this published and available model needs to be explicitly made clear. Does the Sherman model lack emerging physiological features that the new proposed model exhibits?
We view the work of Sherman et al (2016) and ours as complementary. Sherman et al propose a model of a single EI module, using the terminology of our manuscript, that is much more detailed than ours since it approximately accounts for the layered structure of the cortex using two layers of multicompartment spiking neurons, each comprising 100 excitatory neurons and 35 inhibitory neurons. This allows a detailed comparison of the model with local MEG signals. We used a much simpler description and only describe the population behavior of local E and I neurons populations in each module. However, contrary to Sherman’s model, this allows us to address the spatial aspect of beta oscillations which is the main target of our work. Our simple description of a local EI module allows us to consider several hundred EI modules with a spatiallystructured connectivity and to analyze the spatiotemporal characteristics of beta activity. We have now described the relation of our work with Sherman et al (2019) in the discussion section (lines 540547).
The authors may also note the stability analysis in: Yaqian Chen et al., “Emergence of Beta Oscillations of a Resonance Model for Parkinson’s Disease”, Neural Plasticity, vol. 2020, https://doi.org/10.1155/2020/8824760
We thank the reviewer for pointing out this paper that had escaped our notice. It presents the stability analysis of a single EI module with propagation delay (and instantaneous synapses). At the mathematical level, the analysis brings little as compared to the much older article of Geisler et al., J Neurophys (2005) that we cite. However, the model specifically proposes to describe beta oscillations in the motor cortex as arising from the interaction between excitatory and inhibitory neurons, as we do. Therefore, we included this reference as well as a reference to the previous work of Pavlides et al., PLoS Comp Biol (2015) where the model was developed.
The modelbased analysis of the traveling nature of the beta frequency bursts appears to be the most original component of the manuscript. Unfortunately, this is also the least worked out component. The phase velocity analysis is limited by the small number (10 x 10) of modeled (and experimentally recorded) sites and this needs to be acknowledged.How were border effects treated in the model and which are they?
We thank the reviewer for these points which gave us the opportunity to clarify them and improve our manuscript. As described in Methods: Simulations (line 847 and seq.) and shown in Fig. S2 (Fig. S10 in the original submission), we actually simulated our model on a 24 × 24 grid and did all our measurements in a central 10×10 grid to take into account that the electrode covers only part of the motor cortex. In addition to minimize border effects, we added on each side of the 24×24 grid two rows of EI modules kept at their (nonoscillating) fixed points of stationary activity, as depicted in Fig. S2. In order to address the concern of the reviewer, and to check that indeed border effects had a minimal impact on our results, we have performed a new set of simulations on a 24×24 grid with periodic boundary conditions. The results are shown in the new supplementary Fig. S9 and are indistinguishable from those reported in the main text and figures. In particular, the proportion of the different wave types and the wave speeds are unaffected by this change of boundary conditions. A paragraph has been added in the revised version (lines 371378) to discuss this point.
How much of the phase velocities are due to unsynchronized random fluctuations? At least an analysis of shuffled LFPs needs to be performed.
The phase velocities are indeed due to unsynchronized random fluctuations (coming from the finite number of neurons in each of our modules as well as, and more importantly, from the uncorrelated local external inputs). In order to check that the spatialstructure of connectivity was important, we followed the suggestion of the reviewer and also performed a new set of simulations to provide a further test. As proposed by the reviewer, after performing the simulations we shuffled in space the signal of the different electrodes and also did a parallel analysis where we shuffled the signal from different electrodes in the recording. We then reclassified the shuffled simulations/recordings in exactly the same way as the original ones. As shown in the new additional Fig. S16, this resulted in the full elimination of time frames classified as “planar waves” both in the model and in the experimental recordings. Additionally, it little modified the proportion of “synchronized” or “random” episodes which is intuitively understandable since shuffling does not change the nature of these states. In order to further assess the impact of connections between modules, we also decided to suppress them, namely to put their range l to zero. In order to avoid modifying the working point of a local module by this manipulation, we focused on the case without propagation delay. Without longrange connection, the local dynamics of each module is little modified. However, as shown in the new Fig. S18a, synchronization between neighboring modules is strongly decreased and the proportion of the different wave types is entirely changed: synchronized states and planar waves disappear and are replaced by random states. These results are described in two new paragraphs (lines 401414 and lines 431435).
Is there a relationship between the localizations of the nonglobal external input and the starting sites of the traveling waves?
This is also an interesting question that parallels some asked by the other reviewers and which we did our best to address. As described in the “Essential revisions” point 5) above, we aligned all “planar wave events” in space and time with the help of the spatiotemporal phase maps of the oscillations. We did find that planar waves were preceded by an increase in the global synchronization index σp, both in simulations and in experiments. In simulations this increase also corresponded to a shift of the global inputs away from their mean, as depicted in the new Fig. 4 in the main manuscript. However, no significant average spatiotemporal profile of the local inputs emerged when we used these temporal alignments. This is presumably due to the large variability of local inputs that can give rise to planar waves. We have described these results in the new section “Properties of planar waves and characteristics of their inputs”.
In summary, this work could benefit from a widening of its scope to eventually inspire new experimental research questions. While the model is constructed well, there is insufficient evidence to conclude that the presented model advances over another published model (e.g. Sherman et al., 2016).
As described in the “Essential revisions” and the discussion section of the manuscript, our work highlights a number of questions that can (and hopefully will) inspire new experimental research. We also hope that we have clarified above that our model complements Sherman et al.’s model and advances it as far as the spatial aspects of beta oscillations in motor cortex are concerned.
Reviewer #2 (Public Review):
Kang et. al., model the cortical dynamics, specifically distributions of beta burst durations and proportion of different kind of spatial waves using a firing rate model with local EI connections and long range and distance dependent excitatory connections. The model also predicts that the observed cortical activity may be a result of non stationary external input (correlated at short time scales) and a combination of two sources of input, global and local. Overall, the manuscript is very clear, concise and well written. The modeling work is comprehensive and makes interesting and testable predictions about the mechanism of beta bursts and waves in the cortical activity. There are just a few minor typos and curiosities if they can be addressed by the model. Notwithstanding, the study is a valuable contribution towards developing data driven firing rate.
We really appreciate the positive comments of the reviewer and thank her/him for them. We have done our best to correct the typos and to address the questions raised by the reviewer.
 The model beautifully reproduces the proportion of different kind of waves that can be seen in the data (Fig 3), however the manuscript does not comment on when would a planar/random wave appear for a given set of parameters (eg. fixed v ext, tau ext, c) from the mechanistic point of view. If these spatiotemporal activities are functional in nature, their occurrence is unlikely to be just stochastic and a strong computational model like this one would be a perfect substrate to ask this question. Is it possible to characterize what aspects of the global/local input fluctuations or interaction of input fluctuations with the network lead to a specific kind of spatiotemporal activity, even if just empirically ?
This is an important question that parallels some asked by the other reviewers and which we did our best to address. As described in the “Essential revisions” paragraph above, we aligned all “planar wave events” either in phase or at their starting time points. We did find that planar waves were preceded by an increase in the global synchronization index σp, both in simulations and in experiments. In simulations this increase also corresponded to a shift of the global inputs away from their mean, as depicted in the new Fig. 4 in the main manuscript. When we used the same alignment to average spatiotemporal local inputs, we did not see the emergence of any significant patterns. This presumably reflects the high variability of local inputs able to produce a planar wave.
Do different waves appear in the same trial simulation or does the same wave type persist over the whole trial? If former, are the transition probabilities between the different wave types uniform, i.e probability of a planar wave to transit into a synchronized wave equal to the probability of a random wave into synchronized wave?
In the same trial simulation, different types of waves indeed successively appear. The curiosity of the reviewer led us to investigate this interesting point. Since time frames classified as random or synchronized are much more numerous than the planar (and radial) wave ones, it is much more probable that a planar wave transits into a synchronized or a random pattern than the reverse process (i.e., synchronized and random patterns preferentially transit into each other). Nonetheless, we considered questions related to the one of the reviewer. What are the states preceding a planar wave event? Given that a planar wave episode is preceded by a random (or synchronous) episode, is it more likely to be followed by a random or by a synchronous event? We actually find that the entry state is prominently a synchronized state. Furthermore, when the entry state is synchronized, the exit state is also synchronized much more often than would be expected by chance. This shows that most often, planar waves are created from an underlying synchronized persistent state. This has been described in the revised manuscript (lines 443451).
 Denker et al 2018, also reports a strong relationship between the spatial wave category, beta burst amplitude, the beta burst duration and the velocity (Fig 6E  Denker et. al), eg synchronized waves are fastest with the highest beta amplitude and duration. Was this also observed in the model ?
We had long exchanges with Michael Denker about his analysis since there are some differences between his code and what is described in Denker et al. (2017), possibly because of several typos in the Method section of Denker et al (2017). We have checked that the results of our code agree with his but there are some differences with the results obtained on the available datasets and those reported in Denker et al from other data sets. We have now provided the detailed statistics of the different wave types as obtained by our analysis in the simulation of model SN (Fig. S9) and SN’ (Fig. S11) and in the recordings for monkey L (Fig. S10) and monkey N (Fig. S12). In the recording data, the amplitude and speed of the synchronized and planar waves are comparable and higher than in the radial and random wave types. The duration of synchronized events is longer than the one of planar waves and of the other waves types. Comparable results are obtained in the simulations with nonetheless a few differences: the mean amplitude of planar waves is somewhat larger than those of synchronized states, the hierarchy of duration in the different states is respected but the duration themselves are longer in the simulations than in the recordings (about 40 % for the planar waves and almost two times longer for the synchronized states). We attribute these differences to the fact synchronization is slightly less effective in the recordings than in the model. Long synchronization episodes in the recordings are often cutoff by a few time frames where the synchronization index goes below the threshold value for a synchronized pattern. This happens rarely enough not to affect much the global statistics of the different states but it as a much more visible effect on the measured duration of the synchronized states.
Reviewer #3 (Public Review):
In this manuscript, the authors consider a rate model with recurrently connections excitatoryinhibitory (EI) modules coupled by distancedependent excitatory connections. The ratebased formulation with adaptive threshold has been previously shown to agree well with simulations of spiking neurons, and simplifies both analytical analysis and simulations of the model. The cycles of beta oscillations are driven by fluctuating external inputs, and traveling waves emerge from the dephasing by external inputs. The authors constrain the parameters of external inputs so that the model reproduces the power spectral density of LFPs, the correlation of LFPs from different channels and the velocity of propagation of traveling waves. They propose that external inputs are a combination of spatially homogeneous inputs and more localized ones. A very interesting finding is that wave propagation speed is on the order of 30 cm/s in their model which is consistent with the data but does not depend on propagation delays across EI modules which may suggest that propagation speed is not a consequence of unmylenated axons as has been suggested by others. Overall, the analysis looks solid, and we found no inconsistency in their mathematical analysis.
We thank the reviewer for his comments and for his expert review.
However, we think that the authors should discuss more thoroughly how their modeling assumptions affect their result, especially because they use a simple ratebased model for both theory and simulations, and a very simplified proxy for the LFPs.
In the revised manuscript, we have performed additional simulations to test different modeling assumptions as suggested by the reviewer and discussed further below.
The authors introduce anisotropy in the connectivity to explain the findings of Rubino et al. (2006), showing that motor cortical traveling waves propagate preferentially along a specific axis. They introduce anisotropy in the connectivity by imposing that the long range excitatory connections be twice as long along a given axis, and they observe waves propagating along the orthogonal axis, where the connectivity is shorter range. Referring specifically to the direction of propagation found by Rubino et al, could the authors argue why we should expect longer range connections along the orthogonal axis? In fact, Gatter and Powell (1978, Brain) documented a preponderance of horizontal axons in layers 2/3 and 5 of motor cortex in nonhuman primates that were more spatially extensive along the rostrocaudal dimension as compared with the mediolateral dimension, and Rubino et al. (2006) showed the dominant propagation direction was along the rostrocaudal axis. This is inconsistent with the modeling work presented in the current manuscript.
This is an important comment and we thank the reviewer for pointing out these data in Gatter and Powell (1978). Since the experimental data show that planar wave propagation directions are anisotropically distributed, we have tried and investigated what the underlying mechanism of this anisotropy could be in the framework of our model. Anisotropy in connectivity is an obvious possibility. Given our result, and the data of Gatter and Powell, it appears however that it is not the underlying cause of the observed anisotropy direction in the motor cortex (in the framework of our model). We have thus investigated another possibility, namely that the local external inputs are anisotropically targeting the motor cortex, being more spread out along a given axis (lines 510529 and new Fig. 5gl). We find that planar waves propagate preferentially along the orthogonal axis. This leads us to conclude that the observed propagation anisotropy could be of consequence of the external input being more spread out along the mediolateral axis. Data addressing this issue could be obtained using retroviral tracing techniques.
The clarity and significance of the work would greatly improve if the authors discussed more thoroughly how their modeling assumptions affect their result. In particular, the prediction that external inputs are a combination of local and global ones relies on fitting the model to the correlation between LFPs at distant channels. The authors note that when the model parameter c=1, LFPs from distant channels are much more correlated than in the data, and thus have to include the presence of local inputs. We wonder whether the strong correlation between distant LFPs would be lower in a more biologically realistic model, for example a spiking model with sparse connectivity and a spiking external population, where all connections are distant dependent. While the analysis of such a model is beyond the scope of the present work, it would be helpful if the authors discussed if their prediction on the structure of external inputs would still hold in a more realistic model.
This is a legitimate question that we indeed asked ourselves. In a previous work with a simpler chain model, we only considered finite size fluctuations. We found good agreement between our simplified description of finite size fluctuations and simulations of a spiking network with fully connected modules and sparse distancedependent connectivity. This leads us to believe that our description of finitesize fluctuations is reliable in this setting. Assuming that it is the case, we find that with 104 neurons or more per module finite size noise is not strong enough to replace our local external inputs. Even with 2000 neurons per modules the intrinsic fluctuations the network is very synchronized (new Fig. S15eg). With 200 neurons per module, the intrinsic fluctuations are strong enough to replace the fluctuating local inputs (Fig. S15ad) but this is quite a low number. Our description of local noise would have to underestimate the fluctuation in a more sparsely connected network by a significant amount for agreement with the data to be obtained without local inputs. Moreover, it seems to us quite plausible that different regions of motor cortex receive different inputs but, of course, this can only settled by further experiments. Together with the new Fig. S15, we have added a paragraph to address this question in the manuscript (lines 379400).

eLife assessment
This manuscript makes a valuable contribution to the field. The authors have developed a compelling network model to study mechanisms for the emergence of oscillations in the beta range in the primary motor cortex during movement preparation, and their propagation as traveling waves across the cortical sheet. The model is able to recapitulate several features of motor cortical activity acquired experimentally. Due to the recent results suggesting a functional role for traveling waves, it is of great interest to discover the mechanisms underlying such phenomena, and this work is an interesting step in that direction. However, the evidence for the reported new insights is incomplete at this stage, due to some weaknesses that remain to be addressed.

Reviewer #1 (Public Review):
This theoretical (computational modelling) study explores a mechanism that may underlie beta (1330Hz) oscillations in the primate motor cortex. The authors conjecture that traveling beta oscillation bursts emerge following dephasing of intracortical dynamics by extracortical inputs. This is a well written and illustrated manuscript that addressed issues that are both of fundamental and translational importance. Unfortunately, existing work in the field is not well considered and related to the present work. The rationale of the model network follows closely the description in Sherman et al (2016). The relation (difference/advance) to this published and available model needs to be explicitly made clear. Does the Sherman model lack emerging physiological features that the new proposed model exhibits? The …
Reviewer #1 (Public Review):
This theoretical (computational modelling) study explores a mechanism that may underlie beta (1330Hz) oscillations in the primate motor cortex. The authors conjecture that traveling beta oscillation bursts emerge following dephasing of intracortical dynamics by extracortical inputs. This is a well written and illustrated manuscript that addressed issues that are both of fundamental and translational importance. Unfortunately, existing work in the field is not well considered and related to the present work. The rationale of the model network follows closely the description in Sherman et al (2016). The relation (difference/advance) to this published and available model needs to be explicitly made clear. Does the Sherman model lack emerging physiological features that the new proposed model exhibits? The authors may also note the stability analysis in: Yaqian Chenet et al., "Emergence of Beta Oscillations of a Resonance Model for Parkinson's Disease", Neural Plasticity, vol. 2020, https://doi.org/10.1155/2020/8824760
The modelbased analysis of the traveling nature of the beta frequency bursts appears to be the most original component of the manuscript. Unfortunately, this is also the least worked out component. The phase velocity analysis is limited by the small number (10 x 10) of modeled (and experimentally recorded) sites and this needs to be acknowledged. How much of the phase velocities are due to unsynchronized random fluctuations? At least an analysis of shuffled LFPs needs to be performed. How were border effects treated in the model and which are they? Is there a relationship between the localizations of the nonglobal external input and the starting sites of the traveling waves?
In summary, this work could benefit from a widening of its scope to eventually inspire new experimental research questions. While the model is constructed well, there is insufficient evidence to conclude that the presented model advances over another published model (e.g. Sherman et al., 2016).

Reviewer #2 (Public Review):
Kang et. al., model the cortical dynamics, specifically distributions of beta burst durations and proportion of different kind of spatial waves using a firing rate model with local EI connections and long range and distance dependent excitatory connections. The model also predicts that the observed cortical activity may be a result of non stationary external input (correlated at short time scales) and a combination of two sources of input, global and local.
Overall, the manuscript is very clear, concise and well written. The modeling work is comprehensive and makes interesting and testable predictions about the mechanism of beta bursts and waves in the cortical activity. There are just a few minor typos and curiosities if they can be addressed by the model. Notwithstanding, the study is a valuable …
Reviewer #2 (Public Review):
Kang et. al., model the cortical dynamics, specifically distributions of beta burst durations and proportion of different kind of spatial waves using a firing rate model with local EI connections and long range and distance dependent excitatory connections. The model also predicts that the observed cortical activity may be a result of non stationary external input (correlated at short time scales) and a combination of two sources of input, global and local.
Overall, the manuscript is very clear, concise and well written. The modeling work is comprehensive and makes interesting and testable predictions about the mechanism of beta bursts and waves in the cortical activity. There are just a few minor typos and curiosities if they can be addressed by the model. Notwithstanding, the study is a valuable contribution towards developing data driven firing rate.
The model beautifully reproduces the proportion of different kind of waves that can be seen in the data (Fig 3), however the manuscript does not comment on when would a planar/random wave appear for a given set of parameters (eg. fixed v_ext, tau_ext, c) from the mechanistic point of view. If these spatiotemporal activities are functional in nature, their occurrence is unlikely to be just stochastic and a strong computational model like this one would be a perfect substrate to ask this question. Is it possible to characterize what aspects of the global/local input fluctuations or interaction of input fluctuations with the network lead to a specific kind of spatiotemporal activity, even if just empirically ? Do different waves appear in the same trial simulation or does the same wave type persist over the whole trial? If former, are the transition probabilities between the different wave types uniform, i.e probability of a planar wave to transit into a synchronized wave equal to the probability of a random wave into synchronized wave?
Denker et al 2018, also reports a strong relationship between the spatial wave category, beta burst amplitude, the beta burst duration and the velocity (Fig 6E  Denker et. al), eg synchronized waves are fastest with the highest beta amplitude and duration. Was this also observed in the model ?

Reviewer #3 (Public Review):
In this manuscript, the authors consider a rate model with recurrently connections excitatoryinhibitory (EI) modules coupled by distancedependent excitatory connections. The ratebased formulation with adaptive threshold has been previously shown to agree well with simulations of spiking neurons, and simplifies both analytical analysis and simulations of the model. The cycles of beta oscillations are driven by fluctuating external inputs, and traveling waves emerge from the dephasing by external inputs. The authors constrain the parameters of external inputs so that the model reproduces the power spectral density of LFPs, the correlation of LFPs from different channels and the velocity of propagation of traveling waves. They propose that external inputs are a combination of spatially homogeneous inputs …
Reviewer #3 (Public Review):
In this manuscript, the authors consider a rate model with recurrently connections excitatoryinhibitory (EI) modules coupled by distancedependent excitatory connections. The ratebased formulation with adaptive threshold has been previously shown to agree well with simulations of spiking neurons, and simplifies both analytical analysis and simulations of the model. The cycles of beta oscillations are driven by fluctuating external inputs, and traveling waves emerge from the dephasing by external inputs. The authors constrain the parameters of external inputs so that the model reproduces the power spectral density of LFPs, the correlation of LFPs from different channels and the velocity of propagation of traveling waves. They propose that external inputs are a combination of spatially homogeneous inputs and more localized ones. A very interesting finding is that wave propagation speed is on the order of 30 cm/s in their model which is consistent with the data but does not depend on propagation delays across EI modules which may suggest that propagation speed is not a consequence of unmylenated axons as has been suggested by others. Overall, the analysis looks solid, and we found no inconsistency in their mathematical analysis. However, we think that the authors should discuss more thoroughly how their modeling assumptions affect their result, especially because they use a simple ratebased model for both theory and simulations, and a very simplified proxy for the LFPs.
The authors introduce anisotropy in the connectivity to explain the findings of Rubino et al. (2006), showing that motor cortical traveling waves propagate preferentially along a specific axis. They introduce anisotropy in the connectivity by imposing that the long range excitatory connections be twice as long along a given axis, and they observe waves propagating along the orthogonal axis, where the connectivity is shorter range. Referring specifically to the direction of propagation found by Rubino et al, could the authors argue why we should expect longer range connections along the orthogonal axis? In fact, Gatter and Powell (1978, Brain) documented a preponderance of horizontal axons in layers 2/3 and 5 of motor cortex in nonhuman primates that were more spatially extensive along the rostrocaudal dimension as compared with the mediolateral dimension, and Rubino et al. (2006) showed the dominant propagation direction was along the rostrocaudal axis. This is inconsistent with the modeling work presented in the current manuscript.
The clarity and significance of the work would greatly improve if the authors discussed more thoroughly how their modeling assumptions affect their result. In particular, the prediction that external inputs are a combination of local and global ones relies on fitting the model to the correlation between LFPs at distant channels. The authors note that when the model parameter c=1, LFPs from distant channels are much more correlated than in the data, and thus have to include the presence of local inputs. We wonder whether the strong correlation between distant LFPs would be lower in a more biologically realistic model, for example a spiking model with sparse connectivity and a spiking external population, where all connections are distant dependent. While the analysis of such a model is beyond the scope of the present work, it would be helpful if the authors discussed if their prediction on the structure of external inputs would still hold in a more realistic model.
