text:
March 14 coronavirus news
By Jessie Yeung, Brett McKeehan, Ivana Kottasová and Amir Vera, CNN
Updated 1024 GMT (1824 HKT) March 15, 2020
US emergency: President Donald Trump said he was declaring a "national emergency" to free up federal resources to combat the coronavirus. Meanwhile, new travel restrictions into the US go into effect today.
Talk to us: Do you have a question about the coronavirus or a story about the outbreak? Share them here.
Stay updated: Our coronavirus newsletter. The Coronavirus: Fact vs Fiction podcast. Alerts on the pandemic in the CNN app. All the latest information and updated case numbers here.
Sort by LatestOldest Dropdown arrow
4:16 a.m. ET, March 14, 2020
How to keep coronavirus fears from affecting your mental health
From CNN's AJ Willingham
Coronavirus is a serious situation and deserves your vigilance and attention -- but the deluge of information, precautions and warnings can take a real toll on your mental health.
Here are some tips on maintaining a happy medium:
Pare down your sources of information. Find a few sources you trust, like the CDC or a community authority, and stick with them. Limit the frequency of your updates, be disciplined with your social media use and know when to walk away.
Name your fears. It may help to sit down and really consider what specific threats worry you. If your fears are practical ones, think about a plan: What are other options if you can't telework? Do you have savings or support? Being prepared for your fears will help keep them in scale.
Think outside yourself: Since action can allay our anxieties, you may want to also consider what you can do to help others who may be more affected by the outbreak than you, like service or hourly workers who may have to put themselves in disproportionate danger.
Seek support, but do it wisely. If you want to run to a friend to discuss the latest outbreak cluster or your family's contingency plans, try not to create an echo chamber where overwhelmed people further overwhelm each other.
Look for someone who is handling it differently, or for professional help if it's an available option.
Pay attention to your basic needs. Don't forget the essential, healthy practices that affect your wellbeing every day -- getting enough sleep, proper nutrition, physical activity, and getting outside as much as possible.
Don't chastise yourself for worrying. You are allowed to worry or feel bad, and those feelings are valid in times of crisis. The key is to work toward understanding and contextualizing your fears so they don't keep you from living your healthiest life.
US hospitals aren't prepared for coronavirus pandemic, experts warn
From CNN's Jen Christensen, Elizabeth Cohen and John Bonifield
Nurses at a COVID-19 testing station set up by the University of Washington Medical Center in Seattle on Friday, March 13. (AP Photo/Ted S. Warren)
As the coronavirus spreads throughout the US, some health experts are warning that hospitals are not prepared to manage the anticipated number of patients if there is a spike in severe cases.
It's unclear exactly how many people might need to be hospitalized, but doctors are trying to figure that out as healthcare systems prepare for what could be a large influx of patients.
Here's one estimate by Dr. James Lawler at the University of Nebraska Medical Center:
On March 5, he predicted that over the next two months, 4.8 million patients will be hospitalized in the US because of the coronavirus -- including 1.9 million stays in the intensive care unit.
"This estimation is just that, an estimation," Lawler said in an emailed statement. "However it is based upon the best epidemiological modeling and opinion of experts in pandemics and respiratory viral disease."
The US healthcare system will likely struggle. Lawler's prediction of 4.8 million hospital patients could mean trouble for the US -- the entire country doesn't even have 1 million hospital beds, according to the American Hospital Association.
Of course, this 4.8 million figure wouldn't all happen at once. Rather, it would be spread out over many weeks, with outbreaks peaking at different times in different communities. Resources will still be severely stretched -- but not in all areas at once.
We can still lower these numbers if we act now. Lawler's estimate "assumes no attempt to lessen the outbreak" -- so the numbers can be altered if people follow official guidance like social distancing, avoiding gatherings, and practicing good hygiene, he said.
Test kits are "no longer available" in parts of Australia, health chief says
Medical staff conduct coronavirus tests at a hospital in Adelaide, Australia, on March 11. Tracey Nearmy/Getty Images
Stocks of coronavirus test kits have run out in parts of Australia -- and supplies elsewhere are running low, the government's chief medical officer has warned.
Dr. Brendan Murphy addressed fellow healthcare professionals in an open letter, posted to the Royal Australian College of General Practitioners' website.
"Unfortunately, the extreme pressure on our personal protective equipment (PPE) stocks continues, and the situation regarding pathology test kits, reagents and swabs is deteriorating rapidly, with kits no longer being available in some regions of the country," he wrote.
"Pathology collection centres have also experienced large backlogs in testing appointments in some parts of Australia, and emergency testing facilities have had to be established in some areas to ensure that urgent patients can get access to testing."
To address the issue, he urged medical staff to only refer patients for a coronavirus pathology test if they meet the following criteria:
Traveled internationally in the 14 days before illness onset, OR close or casual contact with a confirmed case in the 14 days before illness onset.
Have symptoms of fever OR acute respiratory infection (shortness of breath, cough, sore throat) with or without fever.
Australia is bracing for a wave of infections. The country has 189 cases, according to the World Health Organization, and the number will likely keep rising.
Health officials in the state of Queensland say they are preparing for the possibility that up to 25% of the its population may be infected in the next six months. According to the Australian Bureau of Statistics, Queensland's population is 5 million -- meaning officials are preparing to deal with 1.25 million infections.
Across the border in New South Wales, officials say they expect 20% of the state's population -- about 1.6 million people -- will be infected.
Apple is closing every store outside greater China until March 27
A closed Apple Store is pictured in a deserted shopping mall in Rome, Italy, on March 12. Andreas Solaro/AFP/Getty Images
Apple will close all its stores worldwide -- except those in greater China -- until March 27, CEO Tim Cook said in a tweet today.
The company has also committed $15 million to help with the global coronavirus outbreak.
All of Apple's stores in greater China have reopened after being closed for much of the past two months, while the region grappled with the coronavirus outbreak.
"As rates of new infections continue to grow in other places, we're taking additional steps to protect our team members and customers," the company said in a statement.
Apple employees worldwide will begin to work remotely where possible, while all sites will undergo deep cleaning and health screenings. Hourly workers will continue to be paid.
Customers can still shop on the Apple website, or get customer support online.
Apple's donations, totaling $15 million, have been directed to medical response for patients, and to help lessen the economic impact of the pandemic, the statement said.
Vietnam suspends entry for travelers coming from 26 European countries, as well as the UK
From Chermaine Lee in Hong Kong
A passenger wearing a face mask walks in the nearly empty departure hall of Noi Bai International Airport in Hanoi on March 12. Manan Vatsyayana/AFP/Getty Images
Vietnam is suspending travel from certain European countries, according to a government statement issued Saturday.
The ban will apply to travelers who have been in 26 European countries, known as the Schengen Area, and the UK in the past 14 days.
Visas upon arrival will also no longer be issued for all foreign nationals, the statement said.
The travel restrictions come into effect at midnight Sunday local time, and will last 30 days.
Vietnam has 32 cases of the coronavirus, 16 of which have been discharged from hospital.
A wave of travel bans: Vietnam's ban echoes similar ones imposed by the US, Hong Kong, Turkey, Cambodia and other countries in the past few days.
The US and Hong Kong both implemented bans for the Schengen Area. Turkey and Cambodia's bans didn't cover as many countries, and singled out several of the hardest-hit European nations, like Spain and Germany.
Colombia closes its border with Venezuela in response to coronavirus outbreak
From CNN's Daniel Silva in Miami
People coming from Venezuela hold their documents on the border at Simon Bolivar International Bridge, in Cucuta, Colombia, on March 12. Colombia declared a "health emergency" due to the coronavirus pandemic. Schneyder Mendoza/AFP/Getty Images
Colombia will close its border with Venezuela in an effort to halt the spread of coronavirus, President Ivan Duque announced late Friday -- the same day Venezuela announced its first two cases.
The closure will take effect early Saturday local time, Duque said on Twitter.
Colombia will also restrict access to foreign nationals and non-residents who have been to Europe or Asia in the last 14 days. Colombian nationals and residents returning to the country will face mandatory quarantine for 14 days.
Colombia's border with Ecuador will remain open with greater control and coordination between both countries, Duque said.
The first two cases in Venezuela are a woman who traveled to the US, Italy and Spain, as well as a man who traveled to Spain. Both are in quarantine.
Colombia confirmed its first case on March 7 -- a 19-year-old-woman who traveled to Italy.
New Zealand wants all arrivals to self-isolate for two weeks
From CNN's Chermaine Lee in Hong Kong
New Zealand Prime Minister Jacinda Ardern speaks to media on March 14 in Auckland. (Dave Rowland/Getty Images)
New Zealand Prime Minister Jacinda Ardern today announced that all travelers entering the country will be required to "self-isolate" for 14 days.
The new measures go into effect at midnight Sunday local time and apply to nearly every traveler, regardless of nationality, including New Zealand citizens and residents.
The only exemptions are the Pacific Islands -- but travelers from there will also need to self-isolate if they show symptoms.
"This decision will mean New Zealand will have the widest ranging and toughest border restrictions of any country in the world," Ardern said.
The measures will be reviewed in 16 days, she added.
New Zealand has six coronavirus cases so far. Ardern said none are the result of community transmission.
Trump doesn't need testing or quarantine, White House physician says
US President Donald Trump announcing a national emergency at the White House in Washington, DC, on Friday, March 13. Photo by Chip Somodevilla/Getty Images
US President Donald Trump does not need to be tested or quarantined for coronavirus despite being in proximity to infected dignitaries at Mar-a-Lago, the White House physician said in a statement today.
Three members of a Brazilian delegation that traveled to the US have tested positive. One of them was seated at the same dinner table as Trump at Mar-a-Lago last Saturday, while another shook the President's hand and took a photograph with him, according to the physician's statement.
"These interactions would be categorized as LOW risk for transmission per CDC guidelines, and as such, there is no indication for home quarantine at this time," the statement said.
"Additionally, given the President himself remains without symptoms, testing for Covid-19 is not currently indicated."
There has been some flip-flopping about this: On Thursday, the White House press secretary said Trump and Vice President Mike Pence "do not require being tested at this time."
Then on Friday, when asked about it by a reporter, Trump said he would "likely" receive a coronavirus test "fairly soon."
The White House was constantly evaluating whether to test the President and take measures to prevent the virus from spreading among Trump's close confidants, according to one senior official, who said aides are keenly aware of the optics of testing the commander-in-chief.
House passes coronavirus relief bill backed by Trump
From CNN's Clare Foran
House TV
The House just passed the coronavirus relief measure negotiated between Speaker Nancy Pelosi and the Trump administration.
H.R. 6201, or the Families First Coronavirus Response Act, passed with bipartisan support and a final tally of 363-40, with 40 Republicans voting against it and Independent Justin Amash voting "present."
President Donald Trump's backing of the legislation cleared the way for a broad, bipartisan vote. The Senate is expected to take up the measure when it returns to session next week.
What's in the bill: The broad legislative package includes provisions for paid emergency leave and free coronavirus testing. Trump tweeted his support for the bill earlier this evening, after Pelosi announced a deal with the administration.
The deal came together after intense back-and-forth negotiation between Pelosi and Treasury Secretary Steven Mnuchin, and prolonged uncertainty over whether and when a consensus deal could be struck -- with House Democrats warning that they would go ahead and pass legislation one way or another.
In a series of tweets, Trump put an end to the uncertainty by saying "I fully support" the legislation.
"This Bill will follow my direction for free Coronavirus tests, and paid sick leave for our impacted American workers. I have directed the Secretary of the Treasury and the Secretary of Labor to issue regulations that will provide flexibility so that in no way will Small Businesses be hurt. I encourage all Republicans and Democrats to come together and VOTE YES!," he said.
, Brett McKeehan, Ivana Kottasová and Amir Vera, CNN"},"speakable":{"@type":"SpeakableSpecification","cssSelector":[".speakable"]}}


Parallel treatments of photons, electrons, phonons, and molecules as energy carriers, aiming at fundamental understanding and descriptive tools for energy and heat transport processes from nanoscale continuously to macroscale. Topics include the energy levels, the statistical behavior and internal energy, energy transport in the forms of waves and particles, scattering and heat generation processes, Boltzmann equation and derivation of classical laws, deviation from classical laws at nanoscale and their appropriate descriptions, with applications in nano- and microtechnology.
Gang Chen. 2.57 Nano-to-Macro Transport Processes. Spring 2012. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.


Mrs. Lorraine Smith, age 80, formerly of Olean, New York died early Wednesday morning, May 6, 2015 at The Absolut of Salamanca, New York following a long illness.
She was born January 27, 1935 in Ogdensburg, New Jersey. She was married in Andover, New Jersey to Mr. Raymond Smith who survives.
Mrs. Smith was employed as a driver for Carrier Coach in Olean, New York prior to her retirement. She had previously been employed with Thatcher Glass in Dover, New Jersey.
She was a member of St. Mary's Roman-Catholic Church in Olean and enjoyed cooking.
Surviving besides her husband are: a daughter, Deborah Rogers, Lawrenceville, GA; three grandchildren; seven great-grandchildren; and a brother, Gregory Galiando.
There will be no visitation or local services.
Burial will be in St. Thomas Aquinas Cemetery in Ogdensburg, New Jersey.


text:
Vlingo responds to privacy issues raised about its Android app
The popular Android app Vlingo has come under a bit of fire the past several days, as it seems the application is sending a bit more data than they explain in their privacy agreement. The folks over at Android Pit found some things that raised a few eyebrows, so we reached out to Vlingo to see what was what, and if we need to worry. We spent some time talking to product engineers, and our conclusion is that everything's on the up-and-up, but there were some issues with the way their privacy agreement was written or presented to the user and a software bug or two at work.
Things get a little muddy, partially because there's more than one Android version. One is available in the Market for any device to download, and there's a more customized version offered by OEM's like Samsung on the Galaxy Note. Different versions with different licenses and agreements simply led to the wrong version of the privacy agreement being presented to the user. The developers and staff at Vlingo recognize that there's an issue, and were completely transparent about the entire thing.
They also came across a bug that allowed the service to run even if the user initially canceled the request, and another that sends location data when none is requested. Again, Vlingo was up front about the issue and answered any questions we asked. They even have set up an opt-out process for folks who don't want to use the product with these issues, and they will delete all user data from anyone who requests it.
Yes, it's bad when software bugs force an application to send the wrong data. It's also bad when users aren't presented with the correct use policies -- even though most would never read them. But these types of things happen, and the real test is how the company reacts when presented with issues of this sort. And Vlingo aced it. They were courteous, and seemed genuinely concerned about the issues, without trying to back pedal or lay the blame at someone else's feet. This kind of transparency with the community is exactly what we deserve and expect. Hit the break for the official statement, in its entirety.
More info about the privacy concerns: Android Pit
We take any claims about our customers' privacy and security very, very seriously. We certainly appreciate that we have individuals who are passionate enough about Vlingo's products and about their own privacy rights to conduct this sort of in depth investigation. No question it has raised some real issues, and we have already begun to address the bugs internally.
First, to be clear, Vlingo does make use of information about each device in order to improve the quality of our service. Some examples of this include:
We use the current location of the device to improve search results, for example, to display nearby restaurants when the user does a restaurant search.
We use the device make & model to improve recognition accuracy since microphone characteristics can vary from one type of device to the next.
We use names from the address book on the device to improve speech recognition accuracy (and to spell those names correctly) when users speak those names while performing tasks like voice dialing or SMS dictation.
We use song titles and artist names from music on the device to improve speech recognition accuracy when users speak those names while requesting that specific music be played.
We use the carrier information to work around some issues we've found on some carrier-specific wap gateways
While we transmit and store this information, Vlingo itself does not store any user-identifiable information—meaning we have no way to associate a list of songs or contact names to the user they came from.
Even though we intend to be very transparent about what information we are using in our Privacy Policy, we have discovered that our Privacy Policy is somewhat out of date. For example, we are not mentioning the fact that we are sending song titles and artist names from the device. We will be updating the Privacy Policy to reflect this.
The above information is currently being sent via the HTTP protocol. We are in the process of migrating our communication protocols from HTTP to more secure HTTPS. Already, some of our client applications are using HTTPS, and new versions of our applications will be using HTTPS in the future.
In many cases, we are using the device IMEI as the unique identifier to distinguish communication from different devices to our system, which matches the convention used by many other mobile applications to identify unique devices. The IMEI cannot be used to track the actual identification of the user, but we understand that in some rare cases, the IMEI can be abused. We're investigating switching to other mechanisms like cookies in the future.
We have found several bugs that are causing us to send more data than we intend:
If you start the application for the first time using the widget (and only using the widget) and then cancel out of the initial disclaimer, we are currently sending some device-specific data to the server erroneously.
Our application is currently including the device location information with device information like contact names and song titles. Our intention is to only send location with a speech recognition request, and the location should be omitted from background device-information transmissions.
These issues are errors and are by no means intentional. We plan to fix them as soon as possible and will release updates to the user community as well as through our OEM partners. In addition, we will be improving our processes to ensure that our application behavior with respect to privacy matches our intention.
For users who are happy with the Vlingo application and want to continue using it, we thank you and recommend that you continue using the service. For users who want to stop using Vlingo and are concerned that our servers can still contain data from your device, you can contact support@vlingo.com with your IMEI, and we will remove sensitive information such as contact names from our servers.
Again, we are grateful to the individuals that uncovered these security issues. We are all very proud of the service that Vlingo provides to millions of happy active users, we apologize for these mistakes, and we are doing all we can to ensure that our users can continue accessing our application in full knowledge that their personal information is secure.

redpajama_set_name: RedPajamaCommonCrawl


There are some exciting changes in the Integrated Postsecondary Data Education System (IPEDS) data collection. Most notably, IPEDS has started to collect more information on postsecondary distance education. In the 2011-12 data collection year, IPEDS began asking institutions if they were exclusively a distance education institution (i.e. all of their programs are offered via distance education), so we now know that in Fall 2011 there were 30 Title IV institutions (i.e. institutions that participate in the federal financial aid programs) in the U.S. that were exclusively distance education institutions. In the current data collection year (2012-13), we are collecting information on the number of students enrolled in distance education and which programs are offered as completely distance education programs.
Distance education, for the purposes of IPEDS, is "Education that uses one or more technologies to deliver instruction to students who are separated from the instructor and to support regular and substantive interaction between the students and the instructor synchronously or asynchronously." A distance education course is a course in which all the instructional content is delivered via distance education. A distance education program is a program in which all the courses, or instructional content, are distance education courses. Courses that have some of the instructional content delivered while the students and instructors are in the same room are not distance education courses and programs in which even one course is not a distance education course are not distance education programs. For this purpose, examinations, orientation, and practical experience components of courses or programs are not considered instructional content.
How Did NCES Decide to Add Distance Education Items to IPEDS?
As distance education has become more prevalent in U.S. higher education, we at the National Center for Education Statistics (NCES) receive many requests for data on this topic. To respond both to the changing nature of higher education and to the interests of our data providers and users, a meeting of the IPEDS Technical Review Panel (TRP) was held to discuss possible changes to IPEDS to include items on distance education. Panelists include data providers from all sectors, state and system offices, national associations, data users, and policy experts. The distance education additions to IPEDS are the result of recommendations based on the 23rd IPEDS TRP. All TRP recommendations can be accessed at the TRP website.
What Will We Know About Distance Education?
We will know which institutions offer distance education and to what extent.
Which institutions are exclusively distance education (this already available in the IPEDS Data Center and as a search criteria on College Navigator).
For those institutions that are not exclusively distance education, we will have information on whether or not those institutions offer any distance education opportunity and for what student level (undergraduate and/or graduate).
We will have information about the number of undergraduate and graduate students who are enrolled in distance education courses in the fall. The data will include the number of students enrolled exclusively in distance education courses, in some distance education courses, and in no distance education courses. For those students enrolled exclusively in distance education courses, institutions will be asked to report where the student is located while enrolled.
We will have information about which programs are able to be completed through exclusively distance education courses. This information will be available by field of study (Classification of Instructional Program (CIP) code) and award level. These data will appear on College Navigator and will be especially helpful for prospective students interested in distance education opportunities.
When Will This New Distance Education Data Be Available?
Some of these new data, namely whether institutions are exclusively distance education institutions, are already available through the IPEDS Data Center.
Data on which programs at an institution are available to be completed entirely through distance education courses were collected in the Fall 2012 data collection and preliminary data will be available Spring 2013.
The data on enrollment in distance education courses are part of the upcoming Spring 2013 data collection. We expect the preliminary data to be available in Fall 2013.
We invite you to visit the IPEDS website for more information about the IPEDS program. To view the survey materials for the 2012-13 data collection (blank forms, instructions, faqs and other resources), visit the IPEDS data provider center.


Welcome to the LineageOS wiki!
You can find various information, how-tos, build instructions and much more here in our wiki.
You can check if your device is on our list of supported devices.
You can look at our information on requesting a device.
To grab information needed for a bug report, you can setup ADB and then capture a log.
Then, learn where to report your bug by reading our how-to on reporting bugs.
This wiki is powered by Jekyll.
You can look at the source on GitHub.
And of course, we have a comprehensive guide on contributing.
Every officially-supported device has a comprehensive set of build instructions. Simply choose your device from our list of supported devices to get started.
We even have a nice list of how-tos, for inquisitive souls.
To contribute, you'll need to be able to produce builds for your device. Pick your device from our list of supported devices to get started.
Once you're successfully running your own build, you can begin to make your changes. All the apps included in LineageOS can be found in packages/apps, and core parts of the system can be found in frameworks/base.
Once you've finished making your change, simply follow our guide on submitting to Gerrit.
If you unofficially maintain a device, and think it's good enough to go live, follow our instructions on submitting a port. One of our developer relations staff will reach out to you, and we'll work to get the ball rolling on official builds for your device.
Enhance your apps with new Lineage-specific APIs using the LineageOS platform SDK.
You can view all current pages here.


text:
\section*{Author Summary}

Cortical spreading depression (CSD) is a traveling wave of depolarization that is associated with migraine and brain injury. Massive changes in cerebral blood flow have been associated with CSD, but thus far no mathematical modeling has incorporated these physiologically important changes. In this manuscript, we present a mathematical model for CSD that incorporates blood flow and oxygen consumption in order to examine the effects of the brain vasculature on CSD and vice versa.

\section{Introduction}\label{sec:intro}

Cortical spreading depression (CSD) is a self-propagated depolarization that occurs in the gray matter of many species~\cite{bures1974mechanism}. In humans, it is known to occur during brain injury, stroke, and subarachnoid hemorrhage \cite{kraig2002spreading}. There is also strong evidence that CSD is responsible for the migraine aura \cite{tfelt-hansen2010migraine-csd,hadjikhani2001mechanisms,lashley1941patterns}, a sensory hallucination associated with the migraine attack.

Although CSD was discovered in 1944 by Le\~ao \cite{le1944spreading}, we still do not have a detailed understanding of how CSD is manifest.  In particular, CSD has been associated with massive changes in cortical perfusion. The magnitudes of these changes vary by animal species, but significant decreases and increases in blood flow volume occur in all species tested~\cite{brennan2010update,charles2009cortical,busija2008mechanisms}. Also common to all species tested is a mismatch in the delivery of substrates to meet metabolic demands, resulting in a derangement of neurovascular coupling~\cite{chang2010biphasic,piilgaard2009persistent}.

Blood delivery is known to play a significant role in CSD in several ways.  Changes in perfusion can induce peri-infarct depolarizations (PID), which are electrophysiologically identical to CSD. Conditions that mimic the effects of hypoperfusion, such as oxygen glucose deprivation (OGD) and exposure to ouabain (an inhibitor of the ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase), also generate spreading depolarizations~\cite{somjen2001mechanisms,lauritzen2010clinical,dreier2011role}.

Perfusion changes can also modulate the signature characteristics of CSD. Depending
on the levels of the underlying oxygenation or blood pressure, the amplitude and duration of
depolarization and the velocity of propagation of CSD can be altered~\cite{dreier2011role,sukhotinsky2008hypoxia}.
Clearly, CSD in vivo cannot be understood without reference to the vascular
changes that condition -- and are conditioned by -- the phenomenon.  There is a need to further explore the implications of the effects of perfusion and metabolism on various aspects of CSD.

Such explorations naturally lead to the development of mathematical models in which many mechanisms can be studied independently and/or simultaneously.  Previous mathematical models of CSD have accounted for ionic diffusion~\cite{tuckwell1978mathematical}, cellular membrane ionic currents~\cite{tuckwell1978mathematical,kager2000simulated}, the ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase and other membrane pumps~\cite{tuckwell1978mathematical,kager2000simulated},  and extra- and intracellular volume changes~\cite{shapiro2001osmotic,yao2010continuum}.

However, mathematical models of CSD have not looked at the dynamical implications of neurovascular coupling and metabolism.  Here we formulate a five-compartment continuum model for CSD  that uses known physiological data relating effective blood vessel diameter and extracellular potassium concentration to model oxygen delivery in the brain.  The compartments in our model are:  a somatic  neuronal compartment, a dendritic neuronal compartment, an extracellular space compartment, a vascular tree compartment, and a glial compartment.

We show that the oxygen deprivation that results from both metabolic demand and vasoconstriction modifies the characteristics of CSD waves. Our results predict faster, longer duration CSD \emph{in vivo} than \emph{in vitro}, due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal.   In addition, the model explains differences between CSD \emph{in vivo} and CSD in brain slices due to variant arterial constriction and dilation during the CSD event.

\section{Methods\label{sec:model}}

\begin{figure}[t]
\includegraphics[width=\textwidth]{cellmodel}
\caption[{\textbf{Diagram of the model.} Neurons (mustard) consist of various sodium and potassium channels as well as the {${\textrm{Na}^+}/{\textrm{K}^+}$--}ATPase. Dendrites additionally consist of NMDA channels. Glia (teal) are incorporated as potassium buffers. Blood vessels (pink) bring in oxygen to supply the {${\textrm{Na}^+}/{\textrm{K}^+}$--}ATPase. The cellular-level model is taken into the continuum limit (upper right) to yield a model with 5 compartments: neural cell bodies, neural dendrites, glial, vascular, and extracellular space. }]{}
\label{fig:diagram}
\end{figure}

To study the important new elements that affect and are affected by CSD, we formulate a five-compartment continuum mathematical model, see Fig~\ref{fig:diagram}. Neurons comprise two of the five compartments: a compartment representing the dendritic processes ($d$), and a compartment representing the cell bodies (somatic compartment $s$). The {ECS} ($e$), the vascular bed ($v$), and glia {(g)} comprise the remaining compartments. Though cell swelling has been shown to occur during CSD~\cite{takano2007cortical,andrew2007physiological}, we make the simplifying assumptions that the {ICS and ECS} volume fractions remain fixed, based on the findings of Yao et al.~\cite{yao2010continuum} who found that osmotic effects do not significantly affect the propagation of CSD waves.

The {ICS and ECS} compartments include only the {most relevant} ions (sodium, potassium, chloride) and channels that have been shown to be responsible for the instigation and spread of CSD~\cite{yao2010continuum,kager2000simulated,kager2002conditions}. In the somatic membranes, we include T-type sodium channels, P-type sodium channels, delayed-rectifier potassium channels,  A-type potassium channels, and the ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase. In the dendritic membranes, we additionally include NMDA channels.

Our model assumes that the vascular compartment does not exchange fluid with the extracellular space. The effective diameters of proximal arterioles {control} the blood flow rate.  In turn, the vascular diameters are coupled to neuronal activity through {ECS} potassium concentrations proximal to dendritic processes, which is also buffered by astrocytes.

\subsection{Membrane potential and ion transport using a neuronal model}\label{sec:membraneion}
The membrane potentials  of the neuronal compartments, $E_{m,\ast}$ ($\ast$ is either $s$ for somatic or $d$ for dendritic), are governed by the coupled partial differential equations
\begin{align}
C_m\frac{\partial E_{m,s}}{\partial t}&=-I_{s,\textrm{tot}} +\overbrace{\frac{1}{2R_a\delta_d^2}\left(E_{m,d}-E_{m,s}\right)}^{\textrm{dendrite coupling}}{,} \\
C_m\frac{\partial E_{m,d}}{\partial t}&=-I_{d,\textrm{tot}} +\overbrace{\frac{1}{2R_a\delta_d^2}\left(E_{m,s}-E_{m,d}\right) }^{\textrm{soma coupling}}
\end{align}
where $C_m$ is the membrane capacitance per unit surface area ($\mu$farad/cm$^2$) for both the somatic and dendritic membranes, $R_a$ is the input resistance of the effective dendritic tree (ohms), $\delta_d$ is the half length of the effective dendritic tree (cm), and $\displaystyle I_{\ast,\textrm{tot}}=\sum_{\textrm{ions}} I_{\ast,\textrm{ion,tot}}=\sum_{\textrm{ions}} \sum_{\textrm{channels}}I_{\ast,\textrm{ion,channel}}$ are the spatially-dependent total cross-membrane ionic currents per unit surface area (mA/cm$^2$) for each neuronal compartment. Following Kager et al.~\cite{kager2000simulated,kager2002conditions} and Yao et al.~\cite{yao2010continuum}, the total cross-membrane currents, $I_{\ast,\textrm{tot}}$, are given for the three major ions (sodium, potassium, and chloride) and are the sum of the active and passive (leak) sodium and potassium currents, the chloride (leak) current, and the sodium-potassium exchange pump current (see Appendix \ref{appendix}).  Here we model the ion exchange between the somatic and dendritic portions of the neuron by a flux proportional to the difference between the ion concentrations. The exchange coefficient $D_{ion}/2\delta^2_d$ is estimated using the molecular diffusion coefficient and mean length of the dendrites, adjusted by the volume ratio of soma and dendrites.

The local rates of change of the ECS ions (${\textrm{Na}^+}, {\textrm{K}^+}$, and ${\textrm{Cl}^-}$) are due to membrane ionic currents, diffusion of extracellular ions, and the buffering of ECS potassium by glial cells. Note that all of the model differential equations have only time derivatives in them with the exception of the ECS diffusion equations for the ions. However, all of these equations depend implicitly on the spatial coordinate as a result of the spatial distribution of the ions.
\begin{align}
\frac{\partial([\textrm{ion}]_{e})}{\partial t}&=\overbrace{\frac{1}{f_eF}\left\{\frac{{A}_sI_{s,\textrm{ion,tot}}}{ V_s}+\frac{{A}_dI_{d,\textrm{ion,tot}}}{ V_d} \right\}}^{\textrm{through channels}}
+\overbrace{\frac{\partial}{\partial x}\left(D_{\textrm{ion}}\frac{\partial [\textrm{ion}]_{e}} {\partial x}\right)}^{\textrm{extracellular diffusion}}.  \label{eq:ecs}
\end{align}
The notation, $D_\textrm{ion}$, corresponds to the ion diffusion coefficient in aqueous solution taking into account tortuosity and volume fraction, see~\cite{nicholson1981ion,nicholson2001diffusion}, and $F$ is the Faraday constant. The quantities
 ${A}_{(\ast)}$ are the surface areas of the neuronal compartments in the total fixed volume given by the sum of the fixed somatic volume $V_s$, dendritic volume $V_d$, and extracellular volume, $V_e$.   The ECS volume fraction is given by $f_e=V_e/(V_s+V_d)$.  The equation for {ECS} potassium is modified by adding the buffering flux term, $v_\textrm{buffer}$, given in Section~\ref{sec:buffer}.  The equations for the rates of change of {ICS} ions are
\begin{align}
\frac{\partial([\textrm{ion}]_{i,s})}{\partial t}&=\overbrace{-\frac{{A_s}}{FV_s}I_{s,\textrm{ion,tot}}}^{\textrm{through channels}} + \overbrace{\frac{D_{\textrm{ion}}(V_d+V_s)}{2\delta_d^2V_s}\left([\textrm{ion}]_{i,d}-[\textrm{ion}]_{i,s} \right) }^{\textrm{exchange between soma and dendrites}}\label{eq:ics1} \\
\frac{\partial([\textrm{ion}]_{i,d})}{\partial t}&=-\frac{{A_d}}{FV_d}I_{d,\textrm{ion,tot}}+\frac{D_{\textrm{ion}}(V_s+V_d)}{2\delta_d^2V_d}\left([\textrm{ion}]_{i,s}-[\textrm{ion}]_{i,d} \right) . \label{eq:ics2}
\end{align}
Note that we have not included the diffusion of {ICS} ions because any appreciable diffusion (relative to cell size) would require {an ion} to first become an ECS ion, diffuse extracellularly, and then becomes an ICS ion.

As Cl$^-$ is the only ECS anion in our simulations, to ensure electroneutrality, its initial extracellular concentration ($[{\textrm{Cl}^-}]_{e}$) was determined by the sum of the concentrations of the major cations: $[{\textrm{Cl}^-}]_{e}=[{\textrm{Na}^+}]_{e}+[{\textrm{K}^+}]_{e}$.  We chose the initial intracellular concentrations of chloride ($[{\textrm{Cl}^-}]_{i,\ast}$) so that its Nernst potentials matched the membrane potentials. To achieve intracellular electroneutrality  in the soma and dendrites, we assume the existence of immobile anions.  Although calcium can have major effects on neurotransmitter dynamics and other secondary messenger effects, we ignored the calcium concentration because of its relatively small values.

\subsection{Exchange pumps}

During CSD, the ionic concentrations in the {ECS} and {ICS} are considerably displaced from steady-state. This displacement occurs primarily because of fluxes through voltage-gated sodium and potassium channels.  Here, we include the sodium-potassium exchange pumps ({${\textrm{Na}^+}/{\textrm{K}^+}$}--ATPase) in the neuronal membranes, whose primary role is to restore the ionic concentrations back to their homeostatic state.  {The ionic pumps are active and consume energy.}  When local oxygen levels are depleted, ATP is in short supply.  Therefore, the function of the ionic pumps {in our model} is related to oxygen consumption and vascular flows.

These pumps are involved in the movements of  {ICS} sodium and {ECS} potassium against their electrochemical gradients {and} require active ionic pumps that consume {energy.  The} pumps are fueled by the dephosphorylation of ATP in the cell~\cite{keener2009mathematical} given by
$${\textrm{ATP}} + 3{{\textrm{Na}^+}}_i+2{{\textrm{K}^+}}_e \xrightarrow{\textrm{pump}} {\textrm{ADP}} + P_i+3{{\textrm{Na}^+}}_e+2{{\textrm{K}^+}}_i.$$
ATP is replenished by the reattachment of a phosphate ion to ADP {and} is powered by cellular respiration through both aerobic and anaerobic processes. When local oxygen in the tissue is very low, the normal ATP dynamics are perturbed.

The ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase is a trans-membrane protein with two extracellular binding sites for potassium, three intracellular binding sites for sodium, and a single intracellular binding site for ATP.  In each neuronal  compartment, its potassium and sodium currents are given by $I_{\ast,\textrm{K,pump}}=-2I_{\ast,\textrm{pump}}$ and $I_{\ast,\textrm{Na,pump}}=3I_{\ast,\textrm{pump}}$  (as noted in the beginning of Section~\ref{sec:membraneion}, $\ast$ is either $s$ for somatic or $d$ for dendritic), respectively, with \begin{equation}
\displaystyle I_{\ast,\textrm{pump}} = I_\textrm{max}\gamma_{\ast,\textrm{pump},1}{\gamma_{\textrm{pump},2}}\label{eq:pump}\nonumber
\end{equation}
where
\begin{align}
\gamma_{\ast,\textrm{pump},1}([{\textrm{K}^+}]_{e},[{\textrm{Na}^+}]_{i,\ast})
=\left(1+\frac{[{\textrm{K}^+}]_{e,0}}{[{\textrm{K}^+}]_{e}}\right)^{-2}\left(1+\frac{[{\textrm{Na}^+}]_{i,0}}{[{\textrm{Na}^+}]_{i,\ast}}\right)^{-3}.\label{eqn:pumpnak}
\end{align}
Equation~\ref{eqn:pumpnak} is given in~\cite{kager2000simulated} and the expressions on the right are dependent on {ECS} potassium and {ICS} sodium concentrations. These equations implicitly assume that ATP is plentiful, which is not the case when oxygen is limited or when metabolic needs are high.

In the oxygen-limited regime, where ATP is limited, we modify this expression with an
additional oxygen {dependent} term
\begin{equation}
\gamma_{\textrm{pump},2}([{\textrm{O}_2}])=2\left(1+\frac{[{\textrm{O}_2}]_0}{(1-\alpha)[{\textrm{O}_2}]+\alpha[{\textrm{O}_2}]_0}\right)^{-1}
\label{eqn:gammaO}
\end{equation}
where $\alpha$ is the percentage (about 5\%) of ATP production that is independent of oxygen~\cite{klein2003principles,chih2003energy},  the subscript $0$ denotes the equilibrium values, and $[{\textrm{O}_2}]$ is the tissue oxygen concentration (see Fig~\ref{fig:gammaO}). This expression indicates that the pumping rate will be reduced whenever there is a decrease of the oxygen level in the tissue ({see} Appendix~\ref{sec:oxygenmodel} for details).

\begin{figure}
\begin{center}
\includegraphics[width=19.62pc]{figure1.eps}
\caption[{\textbf{Oxygen availability affects the ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase.} Shown is the relationship between tissue oxygen concentration and the oxygen-dependent portion of the pump rate (Eq~\ref{eqn:gammaO}). Since some ATP is generated even in the absence of oxygen, the pump rate does not completely go to zero as the oxygen concentration approaches zero. The steady state oxygen concentration is $0.2 {\textrm{mM}}$.}]{}
\label{fig:gammaO}
\end{center}
\end{figure}

\subsection{Potassium buffering by glial cells}
\label{sec:buffer}
Astrocytes, a type of glial cell, play important roles in the instigation and propagation of CSD as well as in neurovascular coupling through neurotransmitter-mediated signaling pathways~\cite{attwell2010glial}.  A principal role of astrocytes is the clearance of local increases {of ECS} potassium~\cite{somjen1975electrophysiology}. This buffering is achieved through a variety of inward rectifying potassium channels in the glial membrane and is bolstered by the extreme polarity of glial cell membranes with membrane potential near the Nernst potential for potassium~\cite{walz2000role}. For this study, we are not interested in the exact mechanisms of glial potassium buffering, but are interested in reproducing accurate potassium dynamics for our model.  Thus, we incorporate astrocyte effects through empirical potassium buffering.

 Following Kager et al.~\cite{kager2000simulated}, we modeled the potassium-buffering flux, $v_{\text{buffer}}$, by the following  differential equation,
 \begin{eqnarray}
 v_{\text{buffer}}(x,t) &=& -\frac{\partial B(x,t)}{\partial t} = \mu_+{[{\textrm{K}^+}]_eB(x,t)}\exp\left(\frac{[{\textrm{K}^+}]_e-5.5}{-1.09} \right) - \mu_- {(B_0 - B(x,t))},
 \end{eqnarray}
where $B\ ({\textrm{mM}})$ is the free buffer concentration, the rate constants $\mu_+=\mu_-=8.0\times10^{-6}\textrm{ms}^{-1}$ determine the speed at which potassium is buffered, and $B_0=200 \ {\textrm{mM}}$ is the effective total buffer concentration. This equation describes
strong buffering of extracellular potassium for concentrations above $5.5 \ {\textrm{mM}}$, but is limited by saturation of the finite buffer. As the amount of buffered potassium increases, re-release of potassium into the extracellular space becomes more favorable. The initial value of the free buffer concentration is set to maintain steady state when the extracellular potassium concentration is at its rest value ($3.5 \ {\textrm{mM}}$).

\subsection{Neurovascular coupling and oxygen supply}

We now describe how the tissue oxygen level is affected by and influences CSD. First, we assume that there exists an effective blood vessel radius $r$ in the tissue, and that the cerebral blood flow rate (CBF) is given by
\begin{equation}
\textrm{CBF}=\textrm{CBF}_0\frac{r^4}{r^4_0} \label{effectivebloodradius}
\end{equation}
in which the equilibrium values are again denoted by the subscript 0. This expression is based on the empirical observation that blood flow through the small vessels{,} where nutrient exchange primarily occurs{, can be} modeled as Poiseuille flow where the volume flow rate is proportional to $r^4$~\cite{zamirphysics}.

We use an empirical model for the effective vessel radius $r$, based on replicating the activity observed in many experimental studies on the subject~\cite{knot1996extracellular,mccarron1990potassium}.  Extracellular potassium is known to dilate vessels at lower concentration elevations (less than $\sim17 \ {\textrm{mM}}$), and constrict vessels at higher concentration levels.  To mimic this vascular response, Farr and David~\cite{farr2011models} constructed a plot of the radius of cerebral arterioles versus extracellular potassium based upon currents through potassium channels in the membranes of vascular cells.  To reproduce this plot, we  assume that the effective vessel radius is given by
\begin{align}
r([{\textrm{K}^+}]_e)&=r_0{\overbrace{\exp\left\{-\left(\frac{[{\textrm{K}^+}]_e-3.5}{a}\right)^2\right\}}^{\textrm{constriction}}}
\times \frac{\overbrace{\left(1+ be^{-[([{\textrm{K}^+}]_e-10)/c]^2} \right)}^{\textrm{dilation}}}{1+be^{-\left(6.5/c\right)^2}},
\label{eqn:dilation}
\end{align}
which is a product of constricting and dilating terms. Since dendrites are long neuronal processes that occupy a greater volume than cell bodies, we take $[{\textrm{K}^+}]_e$ to be the concentration in the dendritic {ECS}.  The parameter  $a\ ({\textrm{mM}})$ controls the constriction response, $b$ controls the amount of maximal dilation in the vessel, and $c\ ({\textrm{mM}})$ controls the dilation response.  We found values of $a=50 \ {\textrm{mM}}$, $b=0.18$, $c=3 {\textrm{mM}}$ in (\ref{eqn:dilation}) to mimic the plot of Farr and David given by Fig~\ref{fig:radius}.  We can use this expression in (\ref{effectivebloodradius}) to give us a simple relationship for the CBF in terms of the extracellular potassium concentration.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=19.62pc]{figure2}
\end{center}
\caption[{\textbf{Relationship between vascular caliber and $[{\textrm{K}^+}]_e$} Effective vascular radius $r$ as a function of  {ECS} potassium concentration $[{\textrm{K}^+}]_e$.}]{}
\label{fig:radius}
\end{figure}

We model the temporal evolution of the tissue oxygen concentration $[{\textrm{O}_2}]$, using a reaction-diffusion equation
\begin{equation}
\frac{\partial [{\textrm{O}_2}]}{\partial t}=D_{{\textrm{O}_2}}\frac{\partial^2 [{\textrm{O}_2}]}{\partial x^2}+S
\end{equation}
with nonlinear source term given by
\begin{align}
\lefteqn{S=\overbrace{\textrm{CBF}\frac{[{\textrm{O}_2}]_b-[{\textrm{O}_2}]}{[{\textrm{O}_2}]_b-[{\textrm{O}_2}]_0}}^{\textrm{vascular supply}}- \overbrace{\textrm{CBF}_0\times P([{\textrm{O}_2}])\times (1-\gamma)}^{\textrm{background oxygen consumption}}  } \nonumber \\
&\quad - \underbrace{\textrm{CBF}_0\times P([{\textrm{O}_2}])\times\gamma \Bigg(\frac{\gamma_{s,\textrm{pump},1}([{\textrm{K}^+}]_{e},[{\textrm{Na}^+}]_{i,s})+\gamma_{d,\textrm{pump},1}([{\textrm{K}^+}]_{e},[{\textrm{Na}^+}]_{i,d})}{\gamma_{\textrm{s,pump},1}([{\textrm{K}^+}]_{e,0},[{\textrm{Na}^+}]_{i,0})+\gamma_{\textrm{d,pump},1}([{\textrm{K}^+}]_{e,0},[{\textrm{Na}^+}]_{i,0}} \Bigg)}_{\textrm{tissue oxygen consumption due to }{\textrm{Na}^+}/{\textrm{K}^+}\textrm{--ATPase}}
\label{eqn:source}
\end{align}
where
\[ P([{\textrm{O}_2}]) =  \frac{\gamma_{\textrm{pump},2}([{\textrm{O}_2}])-\gamma_{\textrm{pump},2}(0)}{\gamma_{\textrm{pump},2}([{\textrm{O}_2}]_0)-\gamma_{\textrm{pump},2}(0)}.
\]
The first term on the right in the source is the amount of oxygen transferred from the blood stream to the tissue and is given by the product of CBF and the normalized concentration difference in oxygen tension between the blood and the tissue. The second term on the right in the source represents the consumption of oxygen by the sodium-potassium exchange pump and other cellular processes that are assumed to remain steady during CSD. The pump consumption is given by the product of the equilibrium CBF and pump rate normalized by the steady-state pump rate. A fraction $0<\gamma<1$ of the total oxygen consumption at steady-state is due to the pump. Experimental estimates of this parameter have ranged from as low as $0.10$~\cite{laughlin1998metabolic} to as high as $0.70$~\cite{aperia2001regulation}. The final term in the source term is the consumption of tissue oxygen beyond their steady state value by ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase. We include simulations over the full spectrum for thoroughness.  Note that we have $S=0$ at steady-state.

\section{Results\label{sec:simulations}}

We performed one-dimensional simulations by breaking up our model  into a system of ODEs solved using Matlab routine \texttt{ode15s} with reflecting boundary conditions. The computational domain was set to a length of $5.52$ cm and discretized into $46$ grid points. This domain was sufficiently long for the propagating wave to become stable and for boundary effects to remain insignificant.   We used a minimal amount of potassium to induce CSD, finding that for injections of Gaussian boluses with $120$ micron width, a ${\textrm{K}^+}$ concentration of $15 {\textrm{mM}}$ was sufficient to induce CSD. This concentration is {the} near previously-reported ceiling for potassium concentration in a non-CSD brain~\cite{heinemann1977ceiling,hansen1988brain} and confirmed to us that our phenomenological potassium buffer was behaving in a physiologically realistic manner.

Throughout these results, we report the duration of the CSD event. We measured the duration of potassium elevation by taking the total amount of time that $[{\textrm{K}^+}]_{e}$ is above $6\ {\textrm{mM}}$. The transient sodium channel had negligible effect on the ionic currents during CSD (data not shown), so we chose to omit it from these simulations.

\subsection{Oxygen-clamped simulations}

First, we simulated our model with the oxygen coupling completely turned off, i.e., $\gamma=0$~(see Eq~\ref{eqn:source}).  Thus, the {${\textrm{Na}^+}/{\textrm{K}^+}$--}ATPase does not consume any oxygen, and we assume that the tissue oxygenation is held at its steady state value. In this situation, we obtain CSD waves with a velocity of $3.2$ millimeters per minute and with {ECS} potassium concentration increasing to a maximum value of $45.7 \ {\textrm{mM}}$. This result is similar to {ECS} elevations reported elsewhere in the literature~\cite{hansen2008extracellular}. The total duration of the elevated {ECS} potassium concentration is $65.4$ seconds (Fig~\ref{fig:uncoupled}).
\begin{figure}[!h]
\begin{center}
\includegraphics[width=19pc]{figure3}
\end{center}
\caption[{\textbf{CSD in absence of oxygen consumption.} Shown are the time-courses of the propagating waves of  {ECS} potassium concentration, free buffer capacity, membrane potential, and {${\textrm{Na}^+}/{\textrm{K}^+}$}--ATPase pump rate at positions $120$ microns apart for simulations performed in the absence of oxygen consumption by the ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase. The ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase operates at rates determined solely by $[{\textrm{K}^+}]_e$ and $[{\textrm{Na}^+}]_i$.}]{}
\label{fig:uncoupled}
\end{figure}

The potassium buffer acts similar in that in Kager et al.~\cite{kager2000simulated}. It saturates rapidly, after which it is responsible for a net re-release of potassium into the extracellular environment. The buffer acts this way in all simulations as it is independent of the oxygen level in our model. For this reason, we omit further mention of the potassium buffer.

\subsection{Blood vessel-clamped simulations}

Next, we simulated our theoretical CSD where the effective blood vessel diameter remained fixed, thereby fixing the maximal oxygen flux rate into the tissue.  We varied the oxygen coupling parameter $\gamma$ between $0$ (no oxygen consumption by pump) and $1$ (maximal oxygen coupling) in increments of $0.025$.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=19.62pc]{figure4}
\end{center}
\caption[{\textbf{Oxygen coupled to ${\textrm{Na}^+}/{\textrm{K}^+}$--ATPase. } Fixing the effective vascular radius while coupling oxygen to the {${\textrm{Na}^+}/{\textrm{K}^+}$}--ATPase gives a purely consumption-based view of oxygen during CSD. Defining coupling as the fraction of oxygen consumed by the pump at steady-state, these simulations show that all of the CSD characteristics -- speed, duration, maximum $[{\textrm{K}^+}]$ -- increase with an increase in the coupling constant $\gamma$. The duration of the wave increases linearly from $\gamma=0$ to $\gamma\approx0.4$ before increasing linearly at a different rate when $\gamma>0.4$}]{}
\label{fig:fixedvessel}
\end{figure}

The results shown in Fig~\ref{fig:fixedvessel} illustrate the response of CSD to oxygen coupling. By increasing $\gamma$, we increase the duration and the amplitude of the CSD waveform. To get a deeper understanding of these results, Fig~\ref{fig:fixedvessel2} shows the macroscopic observables.  One sees that increases in $\gamma$ lead to greater drops in oxygenation. This deoxygenation results in decreases in the magnitude of the sodium-potassium--ATPase pump current, resulting in longer tails of recovery for both membrane potential and extracellular potassium concentration.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=19pc]{figure5}
\end{center}
\caption[{\textbf{Fixed vascular caliber.} Time courses at a fixed position, $780 \mu\textrm{m}$  downstream of the original stimulus. These simulations show the effects of oxygen consumption on CSD for a range of the oxygen coupling parameters, $\gamma$. Increasing $\gamma$ prolongs the duration and magnifies the amplitude of the CSD because it implies that the pump consumes more of the available oxygen, thereby resulting in larger oxygen depletions.}]{}
\label{fig:fixedvessel2}
\end{figure}

\subsection{Coupled vasculature}

From the previous results, one sees that oxygenation has a definite impact on susceptibility of the tissue to CSD, as defined by its wave propagation speed, and on its recovery.  The next question is whether vasoconstriction has an impact on CSD.
We performed a series of simulations where we activated the vascular coupling that we defined in Eqs.~\ref{eqn:dilation} and \ref{eqn:source}.  We simulated an idealized vascular response, using parameters mimicking the vascular behavior of Farr and David~\cite{farr2011models}. In these simulations, we varied the oxygen coupling parameter, $\gamma$, in order to see the variability in behavior one might expect to see in an idealized CSD experiment.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=19.62pc]{figure6}
\end{center}
\caption[{\textbf{``Typical" vascular coupling.}  Time courses at a fixed position, $780 \mu\textrm{m}$ downstream of the original stimulus. Simulations performed using vessel of Farr and David~\cite{farr2011models}. After a short dilation period, the vessels constrict significantly to about 40\% of their rest radii, before recovering.}]{}
\label{fig:farrdavidcsd}
\end{figure}

The results, shown in Fig~\ref{fig:farrdavidcsd}, illustrate the effects of vascular coupling. One can see that the effective blood vessel radius drops rapidly to approximately 40\% of its original value. {This} drop leads to steep sustained reductions in the oxygen concentration during the metabolic challenge {from} the CSD wave. The deoxygenation is reflected in the pump rate, where a large increase in inward potassium current at the beginning of CSD is rapidly diminished as oxygen is depleted. The implications of this chain of events are visible in the {ECS} potassium profile.

As we noted earlier, {the} vascular response is variable {and depends} on factors such {as} species, {individuals}, and metabolic states. In Fig~\ref{fig:unfixedduration}, the effects of this variability are seen in simulations across a grid of values for the parameters $a,b,\gamma$~(Eq.~\ref{eqn:dilation}).  The duration of the CSD event decreases as the parameter $a$ (ECS ${\textrm{K}^+}$ concentration at 63\% constriction) and the parameter $b$ (percent maximal dilation) {are} increased.  In Fig~\ref{fig:unfixedspd}, we have plotted the speed of CSD waves across {a} spectrum of possible vascular responses. These vascular responses are parameterized mainly by {the} two parameters, {$a$ and $b$}.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=41pc]{figure8}
\end{center}
\caption[{\textbf{Duration of CSD for different vascular responses.} Shown are CSD durations plotted against percent dilation $b$ (the maximum vasodilation), constriction parameter $a$ (this parameter has the units of mM and is the width of a Gaussian curve that controls how fast $r$ drops as $[{\textrm{K}^+}]$ increases~(Eq.~\ref{eqn:dilation})), and oxygen coupling constant $\gamma$. Duration is defined by the length of time that potassium concentration is elevated to a level greater than $6 \ {\textrm{mM}}$. Increasing $a$, (shown from $30-80 \ {\textrm{mM}}$) decreases the amount of constriction, resulting in quicker recovery from CSD. Likewise, increasing $b$, which increases the maximum dilation of the vessels, also reduces the duration of CSD.}]{}
\label{fig:unfixedduration}
\end{figure*}
\begin{figure*}[!h]
\begin{center}
\includegraphics[width=41pc]{figure9}
\end{center}
\caption[{\textbf{Speed of CSD for different vascular responses.} Increasing the constriction parameter $a$ makes the vessel less sensitive to potassium, weakening the resulting constriction. The speed decreases as $a$ increases from $30 \ {\textrm{mM}}$ up to $80 \ {\textrm{mM}}$, but increasing the amount of dilation by increasing $b$ seems to have a greater effect.}]{}
\label{fig:unfixedspd}
\end{figure*}

\section{Discussion\label{sec:discussion}}

 Since its discovery, CSD has been known to involve massive changes in vascular caliber, and hence, perfusion~\cite{le1944spreading}. Up to now, these {vascular} changes, which can have profound effects on cortical function and thus on CSD itself, have not been incorporated into CSD models. Using our model, we were able to examine the effects of vascular activity on CSD.

\subsection{Relationship between vascular activity and CSD}

 Our numerical experiments confirmed that oxygen delivery plays a significant role in the dynamics of CSD. Even in the absence of constriction, oxygen depletion is seen. Additionally, vasoconstriction and vasodilation were seen to further modify the characteristics of the CSD wave, particularly in the low-$\gamma$ regime. By varying the relationship between vasoconstriction and {ECS} potassium, we achieved a continuum of CSD responses that could explain real physiological variability across species, within members of the same species, and {for}  the same animal.

\subsubsection{CSD uncoupled from vascular activity}

Our modeling of CSD uncoupled from oxygen concentration resembles current CSD models, which rely on similar assumptions and conductances. Such models are relevant to CSD in {{\it ex vivo}} brain slices, where the perfusing medium has contact with an essentially unlimited source of oxygen. In this case, the pump is always able to perform at a rate determined by $[{\textrm{K}^+}]_e$ and $[{\textrm{Na}^+}]_i$, and the total pump current is constrained only by the total number of available pumps present, provided that there is sufficient nutrient available. Our model predicts that CSD waves {propagate} more slowly and recover more quickly in slices than {{\it in vivo}}.

\subsubsection{CSD coupled with vascular activity}

Coupling an inert vasculature {with} CSD, one obtains a sense of the metabolic demands that CSD places on the cortex. The large displacements of sodium and potassium require the use of energy to {return the cortex to homeostasis.}  {The} physical constraints placed on blood delivery by the physiologically reasonable assumption of finite volume-flow-rate were seen to result in depletion of oxygen and reduction in potassium flux through the pumps. This finding is consistent with {{\it in vivo}} measurements of tissue oxygen and mitochondrial redox state -- even with mild dilation. Several studies have found that both variables moved into a more-reduced oxidative state~\cite{takano2007cortical,sakadvzic2009simultaneous}. In our model, the depletion of oxygen is due to increased metabolic demand driven by the {${\textrm{Na}^+}/{\textrm{K}^+}$}--ATPase.

\subsubsection{The effects of vasoconstriction/vasodilation}

Furthermore, we show that vasoconstriction can both decrease the tissue's ability to slow down the CSD wave, and impair its ability to recover. These effects are due to the reductions in blood flow causing a significant additional drop in cortical oxygenation levels. In Fig~\ref{fig:farrdavidcsd}, the vessel is seen to constrict to approximately 40\% {of} its original radius. This constriction is within the experimental range reported in Chang et al.~\cite{chang2010biphasic}. Due to the {power law} relationship between blood flow and effective blood vessel radius, even a small reduction in effective radius has large blood flow implications. A 60\% constriction results in blood flow dropping to 2.6\% of its original value. This effect is visible in the oxygenation level, which undergoes further decreases. Our simulations show that both wave speed and recovery time increase when {vascular caliber is reduced}.

Our simulations also show that vasodilation plays a role in CSD. Predictably, vasodilation appears to precondition the tissue so that it is  {better} able to withstand the increase of  {ECS} ${\textrm{K}^+}$ that accompanies the CSD wave. Due to the power law relationship, an increase of 14\% in the vessel radius results in a 70\% increase in blood volume flow rate.  The result shown in Fig~\ref{fig:unfixedspd} is a tissue that is {better} able to withstand potassium elevations, thereby causing slow-downs in the CSD wave. Vasodilation (beyond the effective resting radius) seems to play a significant role in the recovery from CSD, causing decreases in the recovery time (Fig~\ref{fig:unfixedduration}).

\subsection{Clinical and translational implications}

If a major goal of CSD modeling is to understand its role in human disease, incorporation of perfusion and metabolism is an essential step. CSD is a near-complete leveling of ionic gradients which challenges the ability of homeostatic mechanisms to compensate. CSD is also triggered by changes in perfusion {and} metabolism. Finally, during CSD, the relationships between brain activity, {perfusion,} and metabolism - neurovascular coupling - are altered. These complex relationships can be expected to lead to a variety of responses, depending on the state of the brain tissue surrounding the depolarization. Biological data from animals and humans bears out this complexity and variability. Spreading depolarizations can be relatively innocuous  - repetitive CSD in mouse over several days appears to cause no overt injury~\cite{sukhotinsky2011chronic}. However, they can also be quite harmful, enlarging infarct and contusion areas in both animals and humans~\cite{dreier2009cortical,dreier2011role}. These deleterious effects are almost certainly due to alterations in the vascular response to tissue depolarization, and thus cannot be understood from a modeling standpoint without explicit incorporation of perfusion and metabolism.

Our model more realistically represents conditions observed in the brain during experimental CSD and the spreading depolarizations of migraine and brain injury. Though it simplifies a complex vascular/metabolic response, it has the distinct advantage of making specific, quantifiable predictions which can be used to generate hypotheses for further experimentation. A particular advantage is the ability to explore the whole ``CSD/metabolic parameter space," which is not possible in biological experiments. This modeling could have important implications for study of {the role of} CSD in migraine, as the conditions which could generate such a massive depolarization in an awake behaving person remain obscure.

\subsection{Assumptions, limitations, and future directions}
In this study, in order to make our model as widely applicable as possible, we have not considered the geometry of any particular vascular network. Our lumped model is assumed to be a good approximation of oxygen delivery dynamics in the brain, in general, for gray-matter.  Further insight into the system may be gained by targeting specific tissue types for simulation.

Our simplified model relating arterial effective diameter to ${\textrm{K}^+}$ is likely applicable during CSD but does not mechanistically explain all the subtleties of neurovascular coupling under more normal conditions. Though ${\textrm{K}^+}$ is involved in the coupling of neural and astrocytic activity to blood flow~\cite{filosa2004calcium}, other mediators including arachidonic acid derivatives, purines, nitric oxide, and possibly neurotransmitters are involved as well~\cite{brennan2010update,attwell2010glial,gordon2008brain}. For the purposes of CSD, however,   {the} supra-physiological swings in  {ECS} potassium likely {provide} a good leading-order approximation of vascular behavior. Many of these other mediators also work via their effects on potassium channels~\cite{farr2011models}.

Swelling of individual cells~\cite{mazel2002changes,kager2000simulated,shapiro2001osmotic} and the tissue as a whole~\cite{chang2010biphasic} appears to occur during CSD. Whether cell-swelling plays an important  role in CSD is a controversial topic, as recent computational studies have cast some doubt on its impact~\cite{yao2010continuum,bennett2008quantitative}. For this reason, we have omitted cell swelling from our model.

Finally, this model is not equipped to account for the complexity of the post-CSD state. Chang et al.~\cite{chang2010biphasic} found that extracellular ${\textrm{K}^+}$ levels remained constant during a long--lasting hypoperfusion and depolarization that was found to follow CSD. As this phase may be clinically relevant, future modeling will focus on trying to understand the ${\textrm{K}^+}$--independent mechanisms involved in the etiology of this period.

\section*{Acknowledgements}
The authors would like to thank Tom Chou (UCLA Biomathematics and Mathematics)  and Tim David (University of Canterbury) for their insightful comments.

The work presented here is based on work and discussions held by the authors before, during, and after the American Institute of Mathematics (AIM) SQuaRE workshops on Modeling Cortical Spreading Depression held in Palo Alto, California in May 2010, August 2011, and August 2012.   We thank AIM for their financial support of the SQuaRE {workshops} and special thanks go to Estelle Basor for her warm hospitality and ensuring a convenient and stimulating work environment.

We also thank the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) who supported a Focussed Research Group on Cerebral Blood Flow, Neurovascular Coupling, and Cortical Spreading Depression in June 2013.

The authors thank  grant number T32GM008185 from
the National Institute of General Medical Sciences and grant  number 58386MA from the Army Research Office (JCC); the National Institutes of
Health (NINDS NS059072 and NS070084) and the Department of Defense (CDMRP PR100060)  (KCB); the Natural Sciences and Engineering Research Council of Canada and the Mathematics in Technology and Complex Systems (HH), the National Science
Foundation through grant DMS-1022848 (RMM, HH, JJW), and agreement No. 0635561 (JCC), and the Research Grants Council of the Hong Kong Special Administrative Region, China, CityU 104211 (JJW).

\onecolumn

redpajama_set_name: RedPajamaArXiv


Kitchen Remodel & Exterior Renovation – The Duncan/Potter Home Home – Arlington, MA – Custom Contracting, Inc.
Perched on a hill in the Jason Heights neighborhood of Arlington, the home of Sharon Duncan and Thomas Potter has all the charms – and challenges – of a structure built in 1904.
Since the couple moved there in 2000, the house has undergone extensive renovations. Sharon and Thomas have done some work themselves, from plugging up leaky basement windows to knocking down walls to removing plenty of paint. Some work, however, has called for outside expertise.
A few years ago, one of our clients referred us to Sharon and Thomas. After speaking with general manager Scott Blain (and several other contractors), they chose to go with us.
The kitchen now features custom, white-painted wood cabinetry with recessed doors. In the pantry, we rebuilt the beautiful glass-door cabinets that stand in front of white-tile walls (the same combination that Sharon recalled from her visit to the famed Gamble House in California).
The new stone counters are also a nod to the past and a break from the shiny granite counters that are so popular now. On a similar note, rather than installing recessed lighting, Sharon and Thomas chose traditional lighting fixtures. In fact, the main fixture – a schoolhouse fixture – looks quite similar to the one in the original kitchen.
Improved venting for the oven and a new dishwasher helped bring the kitchen into the 21st century. But the crown jewel of the room is an old oven once owned by Sharon's grandmother. Formerly an all-gas appliance, it has been refurbished to run on a combination of gas and electric. It fits right in with the classic look of the renovated kitchen.
In addition to the kitchen, we renovated much of the rest of the home's interior. This was heavy-duty work that entailed tearing down walls to the studs, removing trim, upgrading electrical, installing insulation and plaster walls, introducing central air conditioning and more. All of the spaces – from the living room to the music room to the master bedroom – mimic the kitchen in that they are clearly updated without clashing with the home's basic character.
One of Sharon's favorite aspects of our work involved the living room's original stained glass windows, which previously had been removed and stored in the attic. Project manager and lead carpenter Dave Surrette was able to gently reinstall these gorgeous windows, which now sit safely behind weather-proof exterior windows.
This past spring, we returned to renovate the flat-roof garage, which was in rough shape. Among other things, we replaced the roof and floor, installed new windows and doors, and built a new storage area on top. Just one more way we've helped to make the home more functional.


Manmohan Acharya is a poet and lyricist from India. He is also a researcher and published author. His contribution appears significant by inspiring the classical dance, Odissi, for the first time to enter into Bollywood with his lucid Sanskrit lyrics.
His Gita-Milindam contains 15 gunjans (songs) in different melodious rhythms. Manomohan Acharya speaks for the common mass. His creative technique suggests his attitude to life. Looking at his off-beat themes, none can deny that he is projecting the voice of the mass and hence is a postmodern poet. A devotional song from his Gitamohanam  is included in the 2009 movie, The Desire. He authored many Sanskrit poetry to his credit. Some worth-mentioning are Gita-bhaaratam, Palli-panchaasika (Khandakavya), subhasa-charitam (Mahakavya), Sri Sivananda-Laharika (Philosophical Kavya) and Yati-giti-satakam (Sataka-kavya). As a playwright, Manomohan Acharya has written many dance dramas, including Arjuna-Pratijnaa, Shrita-kamalam, Pada-pallavam, Divya-Jayadevam, Pingalaa, Mrtyu, Sthitaprajnah, Tantram, Purva-sakuntalam, Uttara-sakuntalam and Raavanah.
Manmohan Acharya was born in the village Lathanga in Jagatsinghpur district in Orissa, India in 1967. He was born in a Sanskrit family of Lathanga, Orissa of parents Pandit Mayadhar Acharya and Parvati Devi.
Manomohan Acharya has contributed a lot to modern Sanskrit literature both as Poet and Play-wright.
Gitamohanam. 7 of the songs were adapted for the Bhakti Sangeeta C.D. of International Prajnan Mission; a devotional song is included in the 2009 movie The Desire.
Gita-bhaaratam(lyrics). Compilation of patriotic songs.
Lyrical Sanskrit poetry 'Gita-bharatam' is very much amazing. This literary document is meant to express the glory of the great India. This kavya contains 10 numbers of sweet patriotic songs. In this poetry. somewhere the literary style of Jayadeva is traced and somewhere the Raagas and Taalas of Oriya heritage have become the tune of this kavya. Above all the literary style of Manmohan is unique. He posed Sanskrit language as a world glory through his literature. In his melodious song, Madhura-bhaaratam, Sweet India the poet comprehend each particle of India a sweet one.
Palli-panchasika is a Khandakavya written by Acharya Manmohan published in the year 1987. Dr. Acharya successfully added a new kavya of Panchasika type to the treasury of Sanskrit literature. This work basically describes the external nature of pallis, the villages and the internal natural expression of a son towards his mother, who lives abroad.


Most Internet users, or smartphone users are savvy enough to know they cannot upload huge videos. By making the UGC videos available on your site you hope to attract clients that shoot short videos often and understand the impact on the web surfing public.
This would additionally help you by helping to create a network of "followers" and "Likes" for the content on the site.


Parents have already been cautioned about the health issues surrounding too much television time but now, researchers also say that certain shows – those that are fast paced – can also ruin concentration.
The warning came after psychologists evaluated 60 four-year-old children. Children from the research group were randomly assigned to watch fast-paced children's programming, slow-paced children's programming or color and draw, for a total of nine minutes.
After the nine minutes were over, each group of children were given a series of tests. The first test was a mental function test that determined their problem-solving abilities and ability to remember what they were told. Children were then tested on their ability to delay self-gratification and maintain self-control.
Children in the fast-paced television program scored significantly lower than the children that watched the slow-paced programming in both mental function and delayed gratification. There was no difference, however, noted between the children that watched the slow-paced programming and the children that spent the time drawing.
While it was stated that more research is needed, lead author and University of Virginia psychology believes that parents should really consider not just the amount of time children watch television but also the quality of those programs. She says that the data indicates that young children's ability to learn and practice self-control can be significantly altered, even just after a short time of viewing the wrong kind of programming.
Lillard offered a couple of different theories to explain why the fast-programming had such an effect on the children. First, she believes that the constant change in scenery and the characters constantly in motion may be responsible for the inability to concentrate. The differences between 'fantasy' and 'real-life' characters may also be a factor. Lillard also mentioned that children may even adopt the frantic and unfocused behaviors of the characters in fast-paced shows, without knowing it.
Child developmental specialist, Dr. Dimitri Christakis at Seattle Children's Hospital pointed out that there are some "notable weaknesses" in the research. He stated that holes in the research failed to indicate if older children are affected in the same manner as younger children. He also pointed out that the study did not include a large enough group of children nor did it indicate how long the programming has an effect on young children.
Despite the flaws, however, Christakis states that there is some significant messages for parents. "This is a small study but it is [an] important one because if it is true that there is a link between TV viewing and intellectual capacity in children, it is concerning," Christakis told CTV News.
Christakis also indicated that he agrees with the theory that, what children watch is just as important as how much they watch.
"It was the type of programming that actually made a difference. The fast-paced cartoon was associated with deficits in attention, the slow-paced was not and this, I think, I critical for parents and, for that matter, educators and researchers to realize," Christakis stated.


haha wow! tell him he can always come over to make some cakes here too!
Looks very delicious, way to go Aubrey!
I'm going to have to make that! My husband is a big chocolate mousse lover – though I don't think I'll be lucky like you, to have him make it. Thanks for sharing!
what a mighty mighty good man.
J'adore tes assiettes! OU est-ce que tu les as trouvée?
I love your plates…. Where did you find it?
And I love also your blog. Merci pour ces belles choses.
Oh, my husband must see this! Yum!
Next time add a splash of your favourite rum or liquor!


Published 04/24/2019 08:19:50 am at 04/24/2019 08:19:50 am in St George Discount Carpet Cleaning.
st george discount carpet cleaning our system also removes more moisture from carpets and other surfaces as well which helps them dry thoroughly and stay cleaner longer st george discount carpet cleaning.
st george carpet cleaning upholstery,st george carpet cleaning qld,st george utah carpet cleaning,st george carpet cleaning qld time,st george carpet cleaning qld school,st george carpet cleaning qld stock,st george discount carpet cleaning,saint george utah carpet cleaning,st george carpet cleaning qld transport,st george carpet cleaning qld police,saint george carpet cleaning.


North East Calgary real estate & homes for sale with North East Calgary real estate plus condos in sought after communities. With new North East Calgary homes in plus just listed homes for sale you're sure to find a easy place to call home. Looking to settle down in Calgary but wondering where to start? Even if you think you know sometimes it can be hard to find the perfect East Calgary home.
Northeast Calgary real estate statistics: As of April 22, 2019 there are currently 620 properties for sale in Northeast Calgary listed active, with an average asking price of $382,307. The highest-priced property is $4,600,000 the lowest priced property can be purchased for around $99,000.
Find Northeast Calgary real estate & Community Reviews Information Locate Northeast Calgary real estate reviews, NE Calgary Community REVIEWS, statistics, NE Calgary community maps . As residents in NE Calgary will tell you this is a GREAT place to live. Wonderful restaurants, shops, parks and of course, people! Every day we see improvements to the neighborhoods with community events attract more people each year.


text:
Russian cosmonauts execute 8-hour spacewalk to investigate possible sabotage aboard the ISS
by Alex Hollings Dec 14, 2018
Back in August, the crew aboard the International Space Station noticed a slight drop in air pressure within the station. The drop wasn't life threatening, but any change in the status of the their small habitable cocoon of technology hung hundreds of miles above the earth's surface warrants investigation, so the team set out to find the cause of the drop.
It didn't take long before they found the culprit, a small hole in the side of a Soyuz habitation module currently docked with the ISS. Initial speculation assume the hole must have been caused by a micro-meteorite strike, but subsequent investigation of the approximately 2mm breach of the spacecraft's exterior wall seemed to suggest that the hole may have been man-made instead.
This image, which was uploaded to Twitter by NASA and then deleted, does seem to show tool marks.
"It was done by a human hand. There are traces of a drill sliding along the surface. We don't reject any theories," Dmitry Rogozin, the head of Russian space agency Roscosmos, told Russian state media at the time. The subsequent media fervor about the possibility of space-sabotage soon dissipated, thanks in part to the outreach efforts of the astronauts and cosmonauts on board who made a concerted effort via social media to show that there were no tensions among American, European, and Russian inhabitants of the space station — despite rapidly escalating tensions between their parent nations back on earth and even in other orbital endeavors.
The entire six-member Expedition 56 crew gathers in the Cupola, the International Space Station's "window to the world," for a team portrait. In the front row, from left, are NASA astronauts Serena Auñon-Chancellor, Commander Drew Fuestel and Ricky Arnold. Behind them, from left, are Roscosmos cosmonauts Sergey Prokopyev and Oleg Artemyev. At the top of the group is astronaut Alexander Gerst of ESA (European Space Agency).
This week, cosmonauts Oleg Kononenko and Sergey Prokopyev left the safety of the space station to execute what was to be a six-hour investigation into the hole. The space-walk proved difficult, as the Soyuz capsules are not designed for repair work in orbit and have no railings or hand-holds like can be found throughout the Space Station itself, which was purpose-built to allow orbital service work to be done by spacewalking astronauts.
Cosmonauts on this week's spacewalk cut a section from the damaged Soyuz capsule. (NASA)
After around seven hours and 45 minutes, the two men returned from their spacewalk carrying a 10-inch slab of the thermal insulation and debris shield from the hull of the Soyuz. That sample will be sent back to earth, where Russian investigators will pour over it in hopes of unlocking more clues into the cause of the hole in the ship.
The spacewalk was broadcast live, which served as a comical juxtaposition against Moscow's claims regarding the "unprecedented complexity" of the cosmonaut's task. Of course, there's no question that the work was difficult, dangerous, and complex… but at some points, it also looked a bit like a frustrated dad beating on a lawnmower that wouldn't start.
The Russian's are performing this spacewalk with surgical precision. pic.twitter.com/RZvqpjpuhQ
— Michael Baylor (@nextspaceflight) December 11, 2018
Sabotage seems unlikely, as the hole wasn't significant enough to cause any real problems and surfaced after the capsule had already been in orbit for some time. It seems far more likely that the hole was caused by an error in manufacturing and then patched using materials that ultimately wore away, either through the intense heat and friction from launch or as a result of the massive temperature fluctuations that can be found in orbit.
The 10-inch hole was then sealed with a thermal blanket and tape… which is troubling in itself, as that same spacecraft is still expected to serve as the boat home for crew members Sergey Prokopyev, Alexander Gerst, and Serena Auñón-Chancellor.
Feature image courtesy of NASA
About Alex Hollings
Alex Hollings writes on a breadth of subjects with an emphasis on defense technology, foreign policy, and information warfare. He holds a master's degree in communications from Southern New Hampshire University, as well as a bachelor's degree in Corporate and Organizational Communications from Framingham State University.
Watch: Take a 'Fly By' Tour of the International Space Station! You Have to See This!
NASA is planning a space station that orbits the moon - Here's what you need to know
SpaceX nods to the movie 'Alien' in successful launch of Crew Dragon capsule
The true story behind NASA's 1973 Skylab 'mutiny'
Soyuz vs F-14: 6 Questions with Astronaut Reid Wiseman
Dutch: MH17 Downed by Russian-built SAM