Grey-box modeling and hypothesis testing of functional near-infrared spectroscopy-based cerebrovascular reactivity to anodal high-definition tDCS in healthy humans

Transcranial direct current stimulation (tDCS) has been shown to evoke hemodynamics response; however, the mechanisms have not been investigated systematically using systems biology approaches. Our study presents a grey-box linear model that was developed from a physiologically detailed multi-compartmental neurovascular unit model consisting of the vascular smooth muscle, perivascular space, synaptic space, and astrocyte glial cell. Then, model linearization was performed on the physiologically detailed nonlinear model to find appropriate complexity (Akaike information criterion) to fit functional near-infrared spectroscopy (fNIRS) based measure of blood volume changes, called cerebrovascular reactivity (CVR), to high-definition (HD) tDCS. The grey-box linear model was applied on the fNIRS-based CVR during the first 150 seconds of anodal HD-tDCS in eleven healthy humans. The grey-box linear models for each of the four nested pathways starting from tDCS scalp current density that perturbed synaptic potassium released from active neurons for Pathway 1, astrocytic transmembrane current for Pathway 2, perivascular potassium concentration for Pathway 3, and voltage-gated ion channel current on the smooth muscle cell for Pathway 4 were fitted to the total hemoglobin concentration (tHb) changes from optodes in the vicinity of 4x1 HD-tDCS electrodes as well as on the contralateral sensorimotor cortex. We found that the tDCS perturbation Pathway 3 presented the least mean square error (MSE, median <2.5%) and the lowest Akaike information criterion (AIC, median -1.726) from the individual grey-box linear model fitting at the targeted-region. Then, minimal realization transfer function with reduced-order approximations of the grey-box model pathways was fitted to the ensemble average tHb time series. Again, Pathway 3 with nine poles and two zeros (all free parameters), provided the best Goodness of Fit of 0.0078 for Chi-Square difference test of nested pathways. Therefore, our study provided a systems biology approach to investigate the initial transient hemodynamic response to tDCS based on fNIRS tHb data. Future studies need to investigate the steady-state responses, including steady-state oscillations found to be driven by calcium dynamics, where transcranial alternating current stimulation may provide frequency-dependent physiological entrainment for system identification. We postulate that such a mechanistic understanding from system identification of the hemodynamics response to transcranial electrical stimulation can facilitate adequate delivery of the current density to the neurovascular tissue under simultaneous portable imaging in various cerebrovascular diseases.

The article describes a complete study about the effect of electrical stimulation on hemodynamics, more precisely vessel circumference and total hemoglobin, by means of a model -data comparison of 11 human participants. The original published physiological model include neuronal, astrocyte, perivascular and smooth muscle cells compartments tightly coupled.
The innovation of the study (apart from new data compared to the published pilot study) lies in the methodological part: The interesting biological point is that the authors highlight the four pathways by which electrical stimulation affects the cerebral blood flow via temporal dynamics of the vessel circumference. In order to better identify and understand these contributions separately, the authors choose to linearize and reduce the physiological model to transfer functions to obtain a grey-box linear model, and then compare it to individual participant's data and to the average.
In my opinion the definition of these four pathways and above all the model linearization and reduction are crucial points that make the strength of the idea and thus an interesting study to publish. Nonetheless, these methodological points are difficult to understand under the present form of the article: on one hand because of the organization of the document and on the other hand because of a lack precision about how are implemented the pathways (equations) and how is realized the post -processing of the model (linearization and reduction ( Fig. 9B and 10), as indicated in the three first sections of this review.
I specify here that my review concerns essentially the model computing part of the article as I'm an expert neither in brain electrical stimulation nor experimental protocols.

Major issue #1: Reorganization of the manuscript
I know that guidelines of the journal encourage to write the results part before the methods one, but in this article, doing it the classical way would greatly improve the direct understanding of the framework imagined by par the authors. At the moment we begin to understand the approach only with Fig. 10 at first reading of the document.
In a new organization, I would keep content of Fig. 1 in the beginning with methods as I understood were acquired in a preceding pilot study and are thus here inputs to the questioning. Apart from the first §, sub-section A in methods section is the acquisition protocol and the most important part are the data used for the estimation/identification ( Fig.1) thus the details of this section may be placed in the Suppl. Mat. to only keep the essentials parts in the main text, except details of processing concerning frequency matters. Also sub-section C « physiologically detailed model linearization for grey-box analysis of experimental data » in results section contains in my opinion a majority of methodological content except the ultimate sentence so that its position in the manuscript should be reviewed.
If the authors want to keep the present organization, I strongly recommend to add a scheme of the whole framework in order to clarify the different parts of the manuscript.

Major issue #2: How is obtained Fig. 2B
Reading the 17 equations of the physiological model, I found that the effects contribute to vessel circumference x via the I highly recommend to specify these points because it's a reproduction issue. Also Fig. 2B should be improved in terms of temporal resolution.
The question arising then if the four contributions lead to the same dimension, why making the reduction of the model and not just compare the MSE values to conclude about pathway 3 as the more important contribution ? This issue is also link to the improvement of methodological figures, as suggested in the next section. Figures 9B and 10, where a first order transfer function is mentioned both in each pathway without specifying if it's the same one or four different ones in Fig. 9B and one vasoactive signal applied to all pathways in Fig. 10. In the objective to help the reader to catch directly their idea, an illustrative figure would be a combination of these two figures which would reduce discrepancies of series and parallel arrows, by adding the four intermediate variables names. Adding the variables names on the arrows and specify name or equations of function in the squares which would get rid of, among other, the discrepancy between the parallel aspect of Fig. 9B, the parallel one of Fig. 10 and the serial of the physiological model in Fig. 10.

Major issue #3: Clarification of
The accompanied text should contain these equations i.e. Sub-Section B « pathways equations or models » of the Suppl. Mat. is important to better understand the approach because they highlight the link between "stimulation" variables and studied pathways variables. Thus, this section should be incorporated in the corresponding method section.

Major issue #4: section (D) participant-specific grey-box linear model dynamics
The use of the system identification toolbox of Matlab must be better explained in this section by specifying the variables obtained in the grey-box model ; And in the following section the variables 'tfest' obtained by the reduction named 'balred'.

Minor issues: Overall review
The authors could improve the overall aspect and coherence of the manuscript by  Reviewing typos : at the end of section titles, spaces after commas and full stops ;  Choosing meaningful variable names as, for instance,  Fig. 4 would benefit spatially and therefore gain sharpness to be organized (A and B) vertically instead of horizontally

Minor issue #2: Text improvement -example of sub-section C of the Methods section
A certain number of sentences could be rethink to be more pedagogical in the direction of organization of ideas inside a sub-section which also will avoid repetition. An example is the sentence "the TDCS current density in the brain's neurovascular tissue was assumed to be proportional…" and the same one "transcranial electrical stimulation induced current density … arbitrary gain" (around Eq. 1).
 Improvement of the vocabulary precision For remarkable example, the different occurrences of the term model (physiologically detailed model or physiological model, pathway models or grey-box model, …) to discriminate the steps of the study's framework should be chosen once and kept throughout the manuscript to strengthen the understanding of this crucial point of hypothesis testing). One occurrence in " Figure 10 shows …"  Explanation of the Biological criteria in "We identified four nested ways …"  Structure of sentences: be sure that sentences contain a verb (example "In pathway 1, the vasoactive …")