Mathematical modeling and multiscale simulations for vesicular release at neuronal synapses
5 views
Formal Metadata
Title 
Mathematical modeling and multiscale simulations for vesicular release at neuronal synapses

Title of Series  
Part Number 
9

Number of Parts 
17

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

Content Metadata
Subject Area  
Abstract 
Claire Guerrier  Mathematical modeling and multiscale simulations for vesicular release at neuronal synapses Synaptic microdomains are underlying fundamental and yet not completely understood functions, such as learning and memory, breathing, sleeping, and many more. Motivated by understanding and analyzing these neuronal structures, we built a model to study vesicular release at synapses. As a first step, we computed the mean time for a Brownian particle to arrive at a narrow opening defined as the small cylinder joining two tangent spheres. The method relies on Möbius conformal transformation applied to the Laplace equation. We also estimated, when the particle starts inside a boundary layer near the hole, the splitting probability to reach the hole before leaving the boundary layer, which is also expressed using a mixed boundaryvalue Laplace equation. Using these results, we developed model equations and their corresponding stochastic simulations to study vesicular release at neuronal synapses, taking into account their specific geometry.

00:00
Logical constant
Computer programming
Complex (psychology)
Musical ensemble
Dot product
Building
Momentum
Consistency
Projective plane
Algebraic structure
Mathematical model
Functional (mathematics)
Open set
Radical (chemistry)
Arithmetic mean
Goodness of fit
Right angle
Data transmission
Chisquared distribution
02:58
Classical physics
Surface
Computer programming
Statistics
Sequel
Observational study
Multiplication sign
Time zone
Sphere
Infinity
Tangent
Grothendieck topology
Number
Time domain
Independence (probability theory)
Cylinder (geometry)
Plane (geometry)
Root
Arithmetic mean
Oval
Square number
Equation
Length
Field (mathematics)
Rule of inference
Reflection (mathematics)
Projective plane
Complex (psychology)
Mathematical analysis
Independence (probability theory)
Division (mathematics)
Prediction
Cartesian coordinate system
Sphere
Time domain
Dean number
Computer animation
Thermal radiation
Network topology
Equation
Right angle
Boundary value problem
Coefficient
Reduction of order
Arc (geometry)
07:21
Area
Point (geometry)
Process (computing)
Multiplication sign
Mathematical analysis
Similarity (geometry)
Line (geometry)
Mathematical model
Time domain
Sphere
Independence (probability theory)
Computer animation
Green's function
Mathematical singularity
Differential equation
Circle
Absolute value
Boundary value problem
09:26
Geometry
Computer programming
Reflection (mathematics)
Thermal expansion
Power (physics)
Time domain
Arithmetic mean
Regular graph
Computer animation
Equation
Differential equation
Equation
Boundary value problem
10:17
Scaling (geometry)
Thermal expansion
Average
Term (mathematics)
Power (physics)
Time domain
Category of being
Regular graph
Computer animation
Root
Equation
Equation
Boundary value problem
Differential equation
10:36
Decision theory
Multiplication sign
Thermal expansion
Parameter (computer programming)
Mereology
Time domain
Goodness of fit
Regular graph
Arithmetic mean
Term (mathematics)
Equation
Differential equation
Amenable group
Position operator
Condition number
Scaling (geometry)
Reflection (mathematics)
Point at infinity
Division (mathematics)
Volume (thermodynamics)
Thermal expansion
Set (mathematics)
Incidence algebra
Term (mathematics)
Time domain
Bilinear form
Power (physics)
Approximation
Position operator
Field extension
Arithmetic mean
Computer animation
Lattice (order)
Equation
Condition number
Game theory
Boundary value problem
Randbedingung <Mathematik>
14:55
Logical constant
Computer programming
Observational study
Process (computing)
Multiplication sign
Time zone
Mathematical model
Ordinary differential equation
Frequency
Chain
Arithmetic mean
Position operator
Stability theory
Area
Time zone
Theory of relativity
Process (computing)
Building
Forcing (mathematics)
Projective plane
Exponentiation
Mathematical analysis
Independence (probability theory)
Mathematical model
Term (mathematics)
Markov chain
Partial differential equation
Pi
Boundary value problem
Physical system
Resultant
00:02
as if the 10 and you have to tip the what and present to you today is the 1st of the big projects which consist of building model and the numerical simulations to study music loudly that sentences so the 1st step is about as undertaken any easier than in the past such kind of wouldn't want to go to small OK so all this is the 1st president of short overview of synaptic transmission that Atkin consonants is conditional on the program and then managed UBS undertaken uses Alaska so the scientists maybe you and knows what it is Is a structure not very well known to be in there is a functioning that permit to transmit analytical signal from the 1 the presented to you on to another person takes a women electrical signal rights mentioned potential rent here and it triggers the opening of the Baltic state of constant and which of the blue China's on understood the picture and this will prevent the entry of consumer finance in the terminal the orange dots and discussed can go to bind to specific receptors the organized on Dr. musicals about physical although 1 oneshot just opposed to the right and it may find those about there's a goal and the specific receptors on the physical it would trigger the meanings of which was begun in the Senate declared and then it could the combined to the pink recent also specific was it often with leaders and the chemical cascade and everything will prevent the I mean creates a miniature gold signal in the percentage of so it's a very complex phenomenon and just to very interested in what's happening in the press and the ball went out which comes from my and also the consumption and have to find the very small receptor located at the doctor visit became so just because you on the spot says about physical are attached to the membrane from not press which is a complex of potatoes and onions not complex you have receptors for the customer to the judgment of the Bloomington and the views and beliefs if you got some like fall 8 finds the receptors then the same thing as a good will fuse with the membrane momentum that is equal to the rent and then everything will start for the prosecution
02:58
so the here that we did it got some ideas on how best to bring that use freely in the 10 0 lead we know the particles that customers are charged particles that we will make this and the 1st step and that Mexico's justice fanned tensions with the plan can hear you understand in Green Bay London planes in and the binding on this not complex is where the brawn article will find was arrested in a confined here so this is our mother of these items finding the complex find this month here which is the center of the centenarians and of the Independence High .period between the physical pain get sold we are not interested in the time and in the field stamp of the project the probability that Lions starting from this channel will find this small so I represent not only has undertaken a users you for these numbers could find so that now escape program we have here it is the following evident article is a described by a statistic equation eggs .period sequels square root of 2 D at W . efficient radiation from the party candidate about his white notes and the 1st time entering being the main hero make a bomb which represent the main interested in as few as a small hole in red isn't winning this picture is given by this for the 1st time that the article fine the main again about good is reflected in work on the border of the main except under its nose sold so we are interested in the majors passage on to the city and them which is you and we know it's a very well known it's Dean that Disney investors this time following the submission of them extraordinary mental problems which is here at Indianapolis in Europe coalminers wildly as division coefficient and equal 0 is a classical reflection the bonhomie and you equals 0 it's a classical absorption absorption and the Red Sea sold going to do to try to find asymptotic an analysis of the of the site
05:40
OK so those are the 1st 3rd we can little blue ceiling going this is the man that its invariant irritation around the axis of the time here trees vary easy to see and according to the equation we have uh we know that these 4 men is independent of the time if we we consider the cynical coordinates around them so we can uh just study in agreement on problems in the main only got adjusted to the demands of prediction of the domain of the tables in and so on this menacing blue His the ruling the red sphere is here that the red the arc of this small as soon as there is now just signals repair and the equation becomes the following equation where are you recognize here the bastion in single coordinate the isn't there so I'm now in into D the main with into the program which is equivalent to the problem we have unisys as well but I put the division edition right and that we have reflection of the blue black and green onerous is and absorption of the segment the are interested in gas and that takes off you when it's done which is the length of the thing and a goes very small goes to the reserves 1 from the rest of the month so the
07:23
prominent here is that we have a similarity because when it's in those 2 0 the point here is not regular soap 2 the singular this singularity for read we will have to look at the problem of the similarity that it prevents us to use a classical many which are for example the Green functions under this domain because we love the order of the Green functions in its analysis singularity this depends the main we don't have access to this Delaware which is very classical is matching hasn't thinks so we try to find a solution the other cost me the whole we to find a solution in the domain we try to match and then we can obtain something that's here because of the job which is not possible as well so the idea is to dismantle arrives the similarity and we are using conformal laps so he here because we have a tendency because we need to use investment which comes very easy when you stop to think about it because it was transformed this domain into the maintain looming area here because that they most maddening principle this the agenda and new owing to borrow Strickland and you notice that there small hole here in red is going to restrict line here and another maps putting the spheres circle in 2 straight lines and Strickland to circles so indirectly from there the whole here is not spare time it's not a simple model Co but the readiness of the circle is 1 of 2 air base which is here the In the absence of out of the whole unleash you maybe don't remember
09:19
because I didn't tell you this before hairraising although screwed up it's going it's a simple computations so
09:27
sorry this meaning that the and the 1st order we can consider the default is can 1st orders sold putting setting the year AST calling you walk obviously can now consider the question in the matter Domenech much simpler men because at the 1st of its like will the equation is much more complicated because the capture of the geometry of the program in Omega but still the main now is that and we get this this equation we have the reflection of Indiana and began under blue black and green monitors and the absorption of the rent soul 2
10:17
solve this equation will now use scanning property so it's getting the heart and soul
10:24
we put the zinc I called Best squared to Hardage the square root adopting that's look which means that Will
10:37
which means that we are going not thinking it our interests in the region very close to the hole get very close to the responded so my setting
10:50
this using the scaling we can put a big white walls of society paucity of Estes there is simple and then we can go over the extension of BY in until to and injecting women expansion into the equation and expanding the equation now we obtained here hasn't adequately questionable Ypsilanti and we we will now take care of the terms of feeding of much interest in modern times of the equation the first one in 1 of the 2 lives and where the 2nd 1 is 1 of its Exeter so using this extensions now we are looking because we are looking for a summit the candidacies ,comma king of the meaning of the term looking for the owner of potentially we 1st consider the 1st of which is the taking the 2nd American of glazing over team he calls so you think equation and the monetary conditions we have proved we have on the blue 1 reminder as we can show there isn't that what is involved is not independent of the game this is what we get from the 1st of the 2nd you go to the cinema about gives us the equation fears and integrating over again we can sure that why 1 is also an independent of the it's simple computation and you this will take complete immigration for why always the incident and you think the American mission now and the whole which they in the world in the title of while solution this year we going the form all the all the time of the solution being Weisel near the whole going back into our 1st and this apparent as we had the hasty these forms this will it's the was 0 in on the whole the solution is an approximation is not in a position as the leader of the town Office region near the because of the amenities we have 200 filling during years the rest of the parameters but this approximation of decision is not looked far from the whole but we cannot use the reflection that the last last bomb Commission so diluted we will use the divisions which there he did well all above of liberation movements equal to the middle of the board of the classical equations so this good annual your walls minus of the volume of phenomenology just comes from the equation dealer new requirements 1 and the other parts because we of reflection everywhere except that the whole said that these into work on the border is exactly and involves a solution of the local it's good because we have a solution at the so we can then you think our 1st all the time of the solutions but the whole give these equations and then fine a and we now and that the 1st all the time thinking about the decision in its following the whole domain so To get a toll to Indonesia's passage chancery we just put the value of aid in view of St and then consider whether S is equal to 0 when not far from the old it's supposed to want to hold its 1st meeting of the role and we get an interest as a chance followed and logical because as an indelible memory layout we also have the meaning of the didn't and we know as well underway it the case OK
14:56
so we concluded that it is not among the main force pastime to a smaller rival located between 2 attendance for and we could show that these investors Chinese Constance outside of the monetary area as an adjustment to its and he's wearing these those that these investors stand well approximated by a person process so this is the result of the world was already known so it's meaning that we can we can consider that the removal of these particles to the small hole is personal and those the 1st of the mission that will offer the arrival is an exponential this is very interesting because you think this analysis in further development of the project that I don't have time to present here we could compute as well the probabilities where particles stops inside the here because periods we have the investors to go to the hold when you are far from the whole because we said it's because usually means we are far what's happening when your clothes the problem is that still the program so using the same kind of analysis what we have here we could compute the probability to reach this year and that escaping the and it's the same kind of computation and using this we could build a model of the active zone which is a place where physical are located and the constant channel located as well to investigate the influence of the position of the channels on the clothing on physical not planning of this included strikes at under stability which it poverty to have diffusion the British have I am about coming to this physical on arts and using this week could built and 1 of them the Senate determine how much bigger wizard the whole of stuff we have over there but the fact that these investors Chinese person and is really what is important here in our study because if you want to monitor the presenting coming now and you have a Brenham particle in Uganda were enhancing relations if you aren't very simple not that simple because of the men is not that firm that OK you can do it but you would wait for hours because of the very small hole you here and because of that there very long time you have to find the small holes the fact that it's a process allows you to use standing is terrorism to say OK it's personnel and I can at least use for what's happening because it's constant in doesn't of the positions of independent of back and that's what we did in the following of this project and I wanted to show you this because it's a very said about anything is there he said going after that because we could show that the 1st Boston is independent of the position of the studies pursuant to this work was published in the scientific togetherness relations I want to send just might proviso that development and my mates and organizer of this conference for the opportunity to present my work to thinking if the land and some of across the board president of the community the 1st Standing approximately 2 because of this because it doesn't have the position they didn't happen when you start so it's fair to say that you have that meaning the violence is the same as it's well approximated any other questions probably not in the sense that he was