The authors have declared that no competing interests exist.
Cortical spreading depression (SD) is a spreading disruption of ionic homeostasis in the brain during which neurons experience complete and prolonged depolarizations. SD is the basis of migraine aura and is increasingly associated with many other brain pathologies. Here, we study the role of glutamate and NMDA receptor dynamics in the context of an ionic electrodiffusion model. We perform simulations in one (1D) and two (2D) spatial dimension. Our 1D simulations reproduce the “inverted saddle” shape of the extracellular voltage signal for the first time. Our simulations suggest that SD propagation depends on two overlapping mechanisms; one dependent on extracellular glutamate diffusion and NMDA receptors and the other dependent on extracellular potassium diffusion and persistent sodium channel conductance. In 2D simulations, we study the dynamics of spiral waves. We study the properties of the spiral waves in relation to the planar 1D wave, and also compute the energy expenditure associated with the recurrent SD spirals.
Cortical spreading depression is a wave of neuronal silencing and ion concentration changes that sweeps slowly through the brain. It is the basis of migraine aura, in which a migraine patient sees a defect move through ones visual field 30 minutes to an hour prior to the headache attack. In this paper, we study the mechanisms by which cortical spreading depression travels as a wave through the brain. We construct a detailed mathematical model based on the physics of ion movement as well as what is known about the molecular players of cortical spreading depression. We build a simulation program that successfully solves the resulting set of highly coupled equations. We find that cortical spreading depression can propagate as a wave by distinct but overlapping mechanisms. We also simulate spiral cortical spreading depression waves and study their properties.
Cortical Spreading Depression (SD) is a pathophysiological phenomenon in the central nervous system characterized by a local breakdown in ionic homeostasis resulting in a temporary silencing of neuronal electrical activity. This local ionic disruption propagates at speeds of 27 mm/min [
Classical models of HodgkinHuxley type, suitable for the description of normal electrophysiological activity, cannot be used for SD, which has a much slower time scale and features very large ionic concentration deflections. Many past theoretical models of SD are of reactiondiffusion type [
Glutamate has long been suggested to play an important role in SD. Glutamate was one of the earliest suggested agents for SD initiation and propagation [
Our computational results suggest that the there are two mechanisms that allow SD propagation. One relies on NaP activation and extracellular potassium (K) diffusion and the other relies on NMDAR activation and glutamate diffusion. These two mechanisms, however, cannot be cleanly separated. Even in the absence of NaP currents, in which case SD activation is solely due to NMDAR activation, K diffusion does play a role. When NaP and NMDAR currents coexist, these two mechanisms operate in parallel, and in this sense, our results can be seen as supporting Van Harreveld’s dual hypothesis [
The above study on the NaP and NMDAR currents were conducted using 1D simulations. We further extend our model to 2D. While several detailed models have investigated recurrent SD in either 1D or 0D [
Our model constitutes a nonlinear and highly coupled partial differential algebraic system, which we call the
The model we use here is based on work in [
Let
Next, we need an equation for the electrostatic potential. We use the following chargecapacitance equations:
The transmembrane ion fluxes
All of these fluxes have the form of:
Glutamate dynamics and NMDAR have long been known to play an important role in SD [
Glutamate, after released by neurons into the extracellular space via synaptic release, is taken up by both neurons and glia. Glial glutamate is converted into glutamine and transported back to neurons via glutamine transporters on both the glial and neuronal membranes [
The rates
Parameter  Description  Value 

Reabsorbtion Rate Percent  0.1 [ 

Release Rate  50 

Decay Rate  (42 

Cycle Rate  (84 

Glial Fraction  10^{−3} (see text)  
Extracellular Fraction  10^{−3} (see text)  
Saturation Constant  22.99 

Glutamate Diffusion  7.6 × 10^{−6} 
Our NMDAR model is based on [
Parameter  Description  Value 

NMDAR Permeability  0 − 6 × 10^{−5} cm/sec  
[Mg^{2+}]_{e}  Magnesium Concentration  2mM 
3.94 

1.94 

0.0213 

0.00277 
We also point out that the nature of the NMDAR desensitization may be due to Zn block [
We note that, in place of
We simulate our equations via a mixed implicitexplicit finite volume routine, based on [
The time course of all variables with
As described above, each time step consists of the following substeps:
Update
Update
Update gating variables.
The updates of
On the other hand, the update of
We use Newton’s method to solve the nonlinear algebraic system. The Jacobian matrix in the Newton method is nonsymmetric. We use preconditioned Krylov subspace solvers for the linear solvers [
Grid sizes 2^{k},
Nonlinear: Newton Line Search
KSP: deflated GMRES
Preconditioner: incomplete LU
Power of 2 bigger than 32.
Nonlinear: Newton Line Search
KSP: Flexible GMRES
Preconditioner: WCycle Multigrid:
SubKSP: Richardson
SubPreconditoner: SOR.
We have also performed convergence studies for certain test problems, which exhibited approximate 2nd order accuracy in space and 1st order accuracy in time. We refer the reader to [
Here, we examine the relative importance of NMDAR and NaP in SD propagation. We first study the behavior of the velocity and duration of SD as we vary the expression level of NaP and NMDAR, with or without extracellular glutamate diffusion. In the presence of glutamate diffusion, we see that increased expression of both channels leads to increased velocity (
Varied over a range of
Varied over a range of
The above results suggest that there are two modes of SD propagation in our model: NMDAR mediated propagation which is dependent on extracellular glutamate diffusion (see
Summary of CSD dynamics of our model. Initiation due to NMDA receptor with propagation caused by the combination of interstitial glutamate and potassium diffusion. Neuronal swelling causes prolonged activation of NMDA receptors.
Summary of Initiation and Propagation due to the persistent sodium channel activation and interstitial potassium diffusion.
An important difference between the NMDAR and NaP mediated modes of propagation is the duration of the SD wave (
Increased NMDAR expression leads to a difference in the time course of the SD wave. As seen in
Each panel shows time courses from four different levels of NMDAR and NaP. For neuronal membrane voltage, as NMDAR permeability increases a secondary bump appears. It appears for even smaller levels of NMDAR, barely visible on the dashed line. The volume graph shows the large reduction in extracellular space.
The figure on the top shows voltage traces for a sampling of NMDAR and NaP permeability. The figure on the bottom is the contour plot for the time course for extracellular voltage for different values of NMDAR permeability. Note that the overshoot is prominent for intermediate values of NMDAR permeability.
The presence of two valleys in the extracellular DC shift (“inverted saddle”) [
The SD time course and the presence of the second valley in the DC shift is strongly influenced by cellular volume changes. We varied the hydraulic permeability coefficient of the neuronal and glial membranes; a high permeability leads to greater volume changes. Without NMDAR, this has almost no effect besides reducing the expansion of neurons and glia. But, with NMDAR there is a prominent effect (see
We vary NMDAR permeability between 4.5 − 6 × 10^{−5}cm/s along the yaxis. Each panel has a different value for hydraulic permeability (water flux). The top and bottom panel have a minimum extracellular space of 2.5% and 10% respectively. For small enough hydraulic permeability, the wave looks no different than a NaP driven wave with no/little NMDA receptor activity.
This shows the difference between the amount of glutamate(concentration times volume) and just concentration.
SD is a 3D phenomenon, and the 1D simulations conducted above and in previous studies cannot fully capture this phenomenon. Here, we consider the spatial patterns formed by SD waves in a 2D crosssection parallel to the cortical surface. 2D spiral and target patterns have been observed in experimental systems [
To create a spiral, we first create an electrophysiologically refractory region in the center of the computational domain by transiently setting the inactivation gating variable of NaP and NMDAR permeability to 0. This prevents the SD wave from penetrating into this region. An SD wave is initiated at the lower half of the left side of the square computational domain, in the same way as the 1D wave. Once this region recovers, we are left with a self sustaining spiral. The behavior of the biophysical variables in a spiral is shown in
All spiral simulations done with Δ
To the best of our knowledge, this is the first computational demonstration of a spiral in a biophysically realistic SD model (see [
We first compute the velocity of the spiral at each point in the computational domain. (for details on how we calculated the 2D velocity see
Speed of wave is calculated at each point in the domain (edges excluded due to edge effects, details provided in
The decrease in speed in comparison to the planar case is consistent with the experimental results in [
We now investigate the change in spiral speed and duration as NaP and NMDAR is varied (
Calculated by finding the average value over the whole domain. The zero sections are regions where the spiral dies off due to a lack of propagation. Beyond the NMDAR level shown in the above graphs, the duration becomes too long preventing the spiral from recurring.
Here, we compute the energy consumption due to ionic pumps as the spiral wave propagates through the computational domain. We note that this calculation is made possible by the fact that our model satisfies a free energy identity (see
Top: Maximum and time average of the work done by ion pumps (note the two different
SD and related phenomena have recently been identified as indicators or poor prognosis for patients suffering from stroke and traumatic brain injury [
In this paper, we introduced an electrodiffusion model of SD that includes glutamate and NMDAR dynamics, and performed 1D and 2D simulations. Our 1D simulations varying NaP and NMDAR expression in particular indicated that there are two modes of propagation, whose biophysical mechanism is summarized in Figs
An important future direction is to improve the models of glutamate and NMDAR dynamics. A more biophysically faithful model will include detailed models of glutamate transporters as well as of the glutamineglutamate conversion [
We have demonstrated that the computational framework we developed for the multidomain electrodiffusion model allows for biophysically detailed studies of SD. In the future, we will use our computational framework to investigate the impact of cortical layer structures on SD. The interplay of seizures and SD, studied at the level of an ordinary differential equation in [
Here, details of the ion channel models as well as the parameters used in the simulations are listed.
(PDF)
A description of the calculation of velocity, duration and energy expenditure is given.
(PDF)
Links to the simulation code are provided.
(PDF)
The authors thank the IMA for hosting a workshop on SD in the February of 2018.
Dear Dr Mori,
Thank you very much for submitting your manuscript 'A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model' for review by PLOS Computational Biology. Your manuscript has been fully evaluated by the PLOS Computational Biology editorial team and in this case also by independent peer reviewers. The reviewers appreciated the attention to an important problem, but raised some concerns about the manuscript as it currently stands. While your manuscript cannot be accepted in its present form, we are willing to consider a revised version in which the issues raised by the reviewers have been adequately addressed. We cannot, of course, promise publication at that time.
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Reviewer's Responses to Questions
Reviewer #1: This paper studied the role of glutamate and NMDA receptor dynamics in an electrodiffusion model of cortical spreading depression (SD) in both 1D and 2D spatial dimensions. They found that SD propagation depends on two distinct but overlapping mechanisms: 1) NaP driven SD propagation relies primarily on extracellular K+ diffusion; 2) NMDAR driven SD propagation depends on glutamate diffusion. For the first time, the “inverted saddle” signature of the extracellular voltage shift during SD can be explained by the coexistence of these two mechanisms: the first valley corresponds primarily to NaP activation and the latter valley to NMDAR activation. They also studied the properties of the spiral waves in 2D models. They found that the higher the NMDAR expression, the higher the overall energy consumption, which indicates the potential neuroprotective effect of NMDAR antagonists in recurring SD.
Overall, the article is well organized and presented. However, the following revisions should be considered:
1. About the description in Table 1 and 2, the authors should keep the first letter of second word either lowercase or uppercase consistently. The figure legends and subsection titles also have similar issues.
2. Page 9 from line 162 to 172, for the readers who are not using PETSc software package, it’s hard to understand these terms. It would be better if the authors explain a little bit more about them.
3. Page 10 from line 201 to 204, too many “where”, and it’s not quite easy to understand this sentence.
4. What is the advantage of using electrodiffusion model rather than HodgkinHuxleytype model?
5. About energy consumption, it is not clear how did the author gets the equation (24) and why dG/dt=IbulkImem. What is the difference of the energy consumption here and the ATP energy consumption due to the pump?
Reviewer #2: Review of the manuscript: «A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model».
The authors present a threedomain continuum model of neural tissue susceptible to spreadingdepression (SD). The model includes a neuronal, glial and extracellular domain, and accounts for electrodiffusive ion transport within the domains, as well as for ion exchange between the domains via a set of ion channels, ion pumps and NMDA synapses. The simulations explore in a convincing way the differences and interplay between the propagation of the SD wave predicted by (1) persistent sodium activation + K+ diffusion, and (2) NMDA activation + Glutamatediffusion, in a way which can explain various experimental observations. This is done both for a 1D planar wave and for 2Dspiralling waves (the latter of which this model is the first ever to simulate). A model like this should be welcomed by the neuroscience community as a tool for exploring and testing hypothesis regarding the mechanisms underlying the pathophysiology of SD.
The modelling work seems well conducted, and I would much like to see it published in PLoS CB. There are, however, some improvements that I think should be made to the manuscript beforehand, mainly in terms of how the model is introduced, how the modelling assumptions are discussed, and how the simulations are related to the biological system/phenomenon at hand. I think this would amount to a minor to moderate revision.
####### Moderately Major points:
##### 1 Relating to biology
1.1. I miss in the introduction a brief overview of the neurophysiology of SD. One thing that confuses me a bit is the role of Nap in this and previous models of the authors, and also the previous (cited) model by Kager et al. This may reflect my lack of knowledge, but I thought that the common view of SD in relation to the K+ diffusion hypothesis was that the membrane depolarization came from APfiring (e.g., via Nat and Kdr) and the following increase in the K+ reversal potential due to a gradual increase in extracellular K+ (see e.g., Ayata & Lauritzen 2015). There are probably good reasons to exclude APs in a coarsegrained model like this. However, should Nap be understood as a playing the role as a standin mechanism for Nat, e.g. does Nap sort of work as a temporally averaged version of Nat on a long time scale, or is it in itself the key mechanism? That is, is the depolarization (in biology and in the model) explained mainly by a persistent sodium current or by K+ reversal potential changes which could follow from AP firing?
1.2. I will not demand it, but I think a schematic figure 1 that illustrates what goes into the model would help the reader to get into the material.
1.3. Fig. 1 is introduced with (line 143144): “Since S is linear in s we can directly solve the above equations. Fig. 1 shows time profiles of an example 1D simulation”. I think the simulation should be explained more carefully to help the reader get into the material. How was SD triggered in the system? Was there an input signal? What was the recording position?
1.4. Likewise, the spiral simulations (line 242243) were introduced with: “In two spatial dimensions, we can obtain more interesting dynamics. In the following, we focus on spirals. To create a spiral, we first create an electrophysiologically refractory region in the center of the computational domain.” The way this is written makes it appear like the spirals were created “just for fun”. I would suggest that this subsection is introduced in a more biologyoriented fashion, i.e. by briefly explaining that such spirals have been observed in biological systems, and as such motivating why one would like to simulate them.
1.5. Relating to the above quite, it is unclear to me how “an electrophysiological refractory region” was created in the center of the computational domain, and what it represents biologically. This must be more carefully explained.
1.6. Line 279281 read: “We also note that the range of parameter values of Nap and NMDAR for which a spiral does not form is much larger than the corresponding range of propagation failure for 1D planar wave. Given the recurring nature of the SD spiral, a higher expression level of the active currents are needed for its generation.” This is a nice model prediction. Can it be tied to some experimental observations, or discussed in terms of where/when we see spirals or planar waves?
1.7. Generally (like in my points 1.31.5), the simulated results could be described in some more detail, and could be related more strongly to the biological scenario that is being simulated. Such improvements are likely to increase the impact that this work can have on the community.
1.8. In Fig. 13, for small distances, the Average work is larger than the Maximum work. This seems wrong. Is it something here that I don´t understand?
#### 2 Eletrodiffusive continuum model
By necessity, a coarse grained model like this rests on a series of assumptions regarding how the detailed structures and morphologies of neuropil can be collapsed into a treedomain continuum in a meaningful manner. Whereas I believe that most of the assumptions made in the proposed model are sound and motivated, I feel like several of them could be stated and discussed in some further detail.
2.1. As suggested by eq. 5 continuous electrodiffusion occurs internally in all three domains. While it has previously been motivated that transport through the extracellular space and through the gapjunction coupled astrocytic syncytium can be represented as a continuums (Chen & Nicholson 2000), I have not seen a corresponding motivation for such continuous transport in neurons. I think this at least should be commented on when introduced, and perhaps given a brief discussion.
2.2. The tortouosity given in Table 4 suggests that the porous medium approximation was used, but the tortouosity does not appear in eq. 5. Was or wasn´t it part of the model?
2.3. Eq. 4 suggests that the extracellular space interacted with an omnipresent bath solution. This assumption appears somewhat bold, and should be discussed when introduced. Is it a technicality, or does the magnitude of the bath interaction play an important role for the simulation outcome? Was it tuned? What could it represent? Blood vessels, which are not part of the model, or leakage to deeper layers or into white matter?
2.4. The code for the model should be made available online. I think it has the potential to be a much used tool for exploring SD by many labs, at least if it is moderately easy to download and run the code.
#### 3. Language
Some parts of the manuscript seems hastily written, and the authors should do a revision before the final version is submitted. Especially the figure texts are often very sparse, sometimes should provide more information.
#### 4. Minor comments
Fig 6: “Each graph is 4 different NMDAR” should read: “Each panel shows four different NMDAR”.
Fig 10: “have very wide arcs that the main drivers do not have”: I was not able to understand this from looking at the figure. I think the arcs should be explained more carefully in the main text, since these results are not easy to wrap ones head around.
Fig 11: “Period of each point, time between each depolarization for each
point in the domain.” This should be made into a sentence and related explicitly to something that we see in the figure.
Fig. 13: “by averaging around the spiral center”. Should this be averaging over a circle around the spiral center? Also: “with a smaller increasing” should be “with a smaller increase”.
Line 46: “in either 1D or 0D”. Whereas I understand what 0D means, it seemed a bit odd. Perhaps explain that this is about point models with no spatial extension?
Eq. 35. Is it necessary to use two notations (1,2,3 or n,g,e) for index k? It would be tidier to use n,g,e consistently.
Line 116117: Typo: “Bg greater than Bg.”
Line 198: “An important difference between the two modes of propagation”. I think you should define here what the two modes are.
Line 303: “NMDAR driven propagation depends on glutamate diffusion and is strongly influenced by volume changes.” Is this not the case for K+ diffusion? Please comment!
Line 337: “Fast Na+ currents will require very fine timestepping, which will lead to further challenges in the numerical method.” Again, I wonder here if Nap could essentially have the same long term effect as a (temporally smeared) seizure. If appropriate, this could be discussed here. Also, I suppose that the inclusion of fast Nacurrents in a continuummodel is also poses some conceptual problems? For example, in 1D, does it not effectively correspond to the assumption that all neurons (at a given x) fire completely synchronous APs?
Ayata, C., & Lauritzen, M. (2015). Spreading Depression, Spreading Depolarizations, and the Cerebral Vasculature. Physiological Reviews, 95(3), 953–993.
Chen, K. C., & Nicholson, C. (2000). Spatial buffering of potassium ions in brain extracellular space. Biophysical Journal, 78(6), 2776–2797.
**********
Largescale datasets should be made available via a public repository as described in the
Reviewer #1: Yes
Reviewer #2: No: Model code should be made available through an online depository
**********
PLOS authors have the option to publish the peer review history of their article (
If you choose “no”, your identity will remain anonymous but your review may still be made public.
Reviewer #1: No
Reviewer #2: Yes: Geir Halnes
Submitted filename:
Dear Dr Mori,
We are pleased to inform you that your manuscript 'A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model' has been provisionally accepted for publication in PLOS Computational Biology.
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Reviewer's Responses to Questions
Reviewer #1: The authors significantly improved the manuscript and addressed each of my comments. I only have one more minor issue, the figure resolution is too low and figure size is too large. It looks like it is larger than A4 size. The authors should address the issue before publication.
Reviewer #2: The authors have appropriately addressed all the concerns that I had with the original submission, and I suggest that the paper is accepted for publication in PLoS CB.
**********
Largescale datasets should be made available via a public repository as described in the
Reviewer #1: None
Reviewer #2: Yes
**********
PLOS authors have the option to publish the peer review history of their article (
If you choose “no”, your identity will remain anonymous but your review may still be made public.
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Reviewer #2: Yes: Geir Halnes
PCOMPBIOLD1900996R1
A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model
Dear Dr Mori,
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