 So first of all a general slide. So what we are doing in TVB, what we are interested in are the brain dynamics and basically you want to study the behavior in the functioning but also disfunctioning brain and there especially we are interested in the underlying mechanism why because we can then try to find ways to change the state of the brain like let's say from disfunctioning to functioning or we can also Use this machinery to explain. Let's say task-related brain signals. Okay, so the concept in general is that we consider the brain as a network the network of interacting population and remasters and the basis for that so what we are using is a structure measurement, so that's what goes in TVB in the platform and then the user has to decide on on locker dynamics, so we will go into that later and the idea behind is the working hypothesis so to say is that the brain functioning is constrained by the structure so that there is a relation between structure and function And of course with the modeling we can go very complex. There's a kind of let's say trade-off between the Mathematical site something that we can then really explain and the detail so we try to explain Yeah, the brain dynamics the behavior With models that are simple as possible as complex as necessary Okay Questions so far. It's okay, so you can interrupt any time Okay So how do we describe Brain dynamics or what what it is basically we are looking at time series like what you can see an EG for example, so here's something very generic And we see that's a potential over time and we see that there's Like a damped oscillation, okay, and so how do we explain here that the change over time so what's basically in dynamical systems what we do is to To calculate the derivative of that so this is like the speed the rate of change So if it's positive and then we have a rise if it's negative so below zero then We have its falling so to say Okay, so now I've Introduced here the potential as one variable state variable and then here's another one, which is the change so how it does this time series look like When we combine these two Spaces Variables and then spend a plane which is called the face space Oh, sorry, there's another thing here. So dynamics in general is a study of how things change over time Or space so here I give only examples over time And now we're interested in how this basically was over time looking at these variables. Okay So again Your potential as a function of time then the change of this potential with respect to time and Then here we have in this other plot, which is the state space or face space. It's equivalent then the potential versus the change These are our initial conditions Yeah, that's Before so we start here at zero t zero and here and That's basically these points here the crosses and in if you take both variables together, then we are here, but you have minus point three for the potential and plus point three for the change and Then over time we see so the potentials rising So it's still positive now. It's changing. So it's zero goes negative and Now interesting. So that's how it looks then in this state space what we see over a time and that is spiraling in emotion is with an aspiring in fashion is appearing Okay It's this clear Okay, fine and I'm happy so Again, these are just for the terms these are the initial points and If we let the system so to say relax Which is called in the trajectory. So that's then so release the system at this point. It's doing its motion over time and Which is then called in the state space a trajectory and Then we end up at this point. Yeah, so it gets Then zero in yant Okay, which is exactly this point over here in the in the state space and this is called a equilibrium in that case It's attracting. That's what Petra mentioned. So this is this point is attracting and in other terms what we say and We can also say that it's stable. So to say it's globally attracting If you have questions interrupt Okay Another time series would be in oscillation. Yeah, then here again, we have the potential. So it's oscillating like a Let's say in alpha rhythm. So here we have the change of this oscillation or with respect to time and Then again initial conditions and at the time series potential change Of the potential with respect to time and then here in the state space and then you let it run and then You see it looks similar at the at the beginning. So there is something kind of in Relaxing behavior, but now it's the opposite instead of going inside it goes outside. Okay, it's it's unstable. So to say it's repelling Okay, but now it's basically a force unto such An trajectory that is closed which is called an orbit and if it's attracting then it's called a limit cycle. So that's important. So this is how Oscillation self-sustained oscillations are modeled Okay, so limit cycle is a closed trajectory Okay, I mentioned stability Before and there I will show Yes, I'm the example simple examples what Be mean in general with stability So here we see the three different scenarios. We have a valley. Okay here this U-shape. We have a flat part and then that's like a mountain and Here you will then see the time series for the three Cases and now we start Throwing so to say a ball in it and then see how this ball is moving. Okay So that's the red one on that side or the black one on that side and both are basically Falling down into the valley and to the to the minimum here And we see at the beginning. It's very fast, you know, it's Accelerating and then at decelerating the closer it gets to this and to the valley And and that's what you see also in the time series at the beginning. It's very it's rather steep and then gets flatter and flatter And this is stable. Okay The trajectory here would be this one Yes, this is more let's say One could also and I will show it later than how to I mean, that's let's say an Everyday example, I mean you can do it at home in the kitchen. Let's save it a bowl and it's a sketch Yes, exactly. Yes. Yeah, you could Yeah, so the amplitude would be okay here zero. That's this point and the amplitude here is where we start The back and and this would be then the other one. So we need to turn the round. So, okay This is okay. Okay If you know do the same with the flat so then basically it doesn't matter where we drop the ball It will stay it will not move and then basically we get the flat line Over time. Okay, depending on where we drop the ball Then for the for the mountain it's then I mean if you're good you can balance it But then the status perturbation will bring this ball here either to the to the left or to the right Okay, so this means so and then it goes down to here infinity Which means and in time we have at the beginning. It's very slow So and then with the status perturbation then we have it goes away basically to plus infinity or minus infinity in space, okay Then again the face portraits. So these are now Abstract versions of these trajectories. We see that this is a stable point a stable note We call it and then this is attracting the arrows are basically Summarizing the trajectories That indicating the change the time That's attracting that's a stable note and for the unstable one Here then we see it basically the the arrows are going outwards. Okay, so these are basically condensed Pictures of these these examples Okay So now to make up That's a relation between singles and face portraits Then yeah summarizing the different states that are possible Okay I Will give you here this table. We have the notes just as shown before so this is these are stable states We have notes here That's a stable note. So we see this Exponential behavior and then it's it's a constant in the end. So for this we need Only one variable to describe You know, it's like then a movement like that if you see Like before shown before in a damped oscillation, then we need to interacting Variables so for this It's it goes it increases in one direction. Let's say in y and then X has to pull it so to say in the other direction and then back So we have an interaction a feedback interaction between two variables Which can be then potential and the derivative of the potential And then something can circulate. Okay, so here in the end, it's a stable focus, which means then after a certain time we The system will relax to to zero to baseline Taking the same scheme the same complexity we can also create oscillations like shown before and then the The feed forward and feedback is basically strong enough so that Self-sustained oscillation emerges Okay, and that's Then summarized in the face portray by by these limits like this. Okay, that are attractive attracting from all sides and If you go for a higher dimensions, and it's just so I'm not going to show any Examples that send from dimension free on theoretically they can you can see irregular behavior, which is called in deterministic chaos So you have and let's say and like a limit cycle on one plane And then there's an accumulation of activity in another dimension that Can give rise to very interesting we call it also strange attractors and Geometries and then in the time series, so there's no noise. It's really intrinsic deterministic. You see it an irregular time series and this occurs then from Yeah, the variables for higher than than free Okay, so state space just to go back to that here is one example of states that's Actually the a state diagram of water We have here the state of water as a function of the pressure and the temperature and we see basically that water can take can be Yeah, water itself fluid then then we have ice and then we have vapor yeah, and so on A state space basically represents all the possible states and So these states have unique values, so if you see these these areas basically we can measure them and Then the bifurcations these are these lines here are indicating the change of the states and then You can have just for as an example we go from ice to water by increasing the temperature So we keep the pressure constant and then by increasing it say we start here increasing the temperature then the ice melts and we get water and there's something We can also translate this to neuroscience and say okay here the neurons are At rest and then we increase the firing threshold for example, and then the the neurons start firing questions Okay, okay, we'll go into bifurcations Like said before they are bifurcations are in the indicating qualitative changes in the systems behavior They are due to some parameter changes like the temperature or the pressure so and then for for TVB for it could be The inputs this can be a background activity could be input from other areas or temperature like shown before for water as well Then the functioning then the threshold that we change something in the functioning of this population we can change constants time constants like the The kinetics of Excitation and ambition or we can also of course change the Connectivity meaning that we can manipulate the weights or like the stroke example that Petra showed That we can take out of them. Let's say healthy virtualized brain taking parts out. Okay so Questions interrupt if sign, okay Petra showed that before so we our in the in TVB our models are based on Experimental data structure data, so we use we build up then these network models So here what you see in red are then the the brain areas lumped so that these are Population models here in that case Representing brain regions in green are then the structure connections connecting then these brain areas and What TVB what's it the user what you have to do in TVB is to decide for local dynamics, so that's so TVB is then Equipping each of these brain areas with a local dynamics and Then by having so to say a dynamical model at each of these brain areas the The prey I had this this brain model starts to be alive and then things can can interact so these notes are Interacting and this is similar to That's basically what Petra showed before as well, so we have these flocking birds So these these birds are Interacting and then forming basically in time and space these nights behavior And that's so each of these birds could be a neuropopulation. Yeah, that's then now on a different level and There it's interesting what emerges out of their interaction, so I Mean one bird cannot Form or make these nice movements. It's so appealing to us because they it's a collective behavior Okay, and that's basically what we are or what I'm especially I am interested in Using TVB in this emerging behavior Like what we see here in the birds and as an example this could be synchronization then phenomena phenomena in Amongst brain areas So what are these local models? We can have I have to interrupt it. Okay You can have really detailed models like a single Neuron descriptions. So this is here from the blue brain project. They're really Individual neurons are modeled also including the morphology here. So they took them a lot of effort to Simulate here 30,000 neurons That's what you could do in TVB, but I will focus like shown before on populations and So what they had to describe the behavior of such a piece of cortex and we can be summarized that by using a mean field approach and then we would end up With something like this where we have met the projecting neurons here the pyramidal cells and then interneurons, etc. in any be a inhibitory interneurons How Is this? Has this been done? Basically, we have we are looking at the physical space and Then looking there for different types of neurons. So we try to distinguish between let's say excitation inhibition Excitement or inhibitory neurons may be also of the The numbers of neurons and then we come up with the classification we can Have here, let's say type 1 to 3 to n of these Neurons and then we basically lump All these neurons spatially to a point and summarize the behavior of these neurons Okay, and then You describe the the the basic circuitry of the Deselected Neuromarses Okay What can you do with such models? so such models like The chance rid model. That's the chance rid model that Petra also mentioned before Has been used to describe EGM EG So alpha activity like shown here. So that's then said to say the simulated EG over time but also epileptic Activity like insecta spikes spikes tonic firing. So that's then This would be such a normal behavior and that's then the pathological behavior Okay I am now going into More into death how these neural masses are functioning and and then Describing the states what they can do and and also what they cannot do and then introduce Yeah, the bifurcations and it's a the state space of This this model Okay So what we see here is it's a sketch of of of a neuron and How does this neuron work? So we have here axons Hitting the the dendrites. We have the synapse here. That's the dendrite The soma and then the axon again. Okay, so here in blue are such action potentials hitting the the synapse Then on the post-naptic side, we see then the post a potential occurring a change in the potential and This so this post-naptic potential is then Gets in summed up at the soma Okay, and then there's a threshold function at a let's say axon hillock Where it's then decided if so if the potential is high enough. That's what we see here over time That's at the axon hillock if it's so it's a crossing the threshold then the new action potential is Occuring okay How to describe that so I had this this this behavior what you've seen here. So this this red curve. So for this we use Differential equation so that's Yeah the description of a post-naptic potential of a single neuromass and What we see here is The phi one is the the main brain potential then we have the current over here in green phi two and Here's the input firing rate so the perturbation and Then on the left hand side via Like Sean be before we describing the change the change now of Main brain potential and in addition then of the the current Okay, so this equation describes The evolution of the post-naptic potential with respect to time so now we can ask is there is it is it going to relax to a certain value to a baseline after an input is given or not and so for that we look at the Equilibria and that's happening when basically this change or where nothing is changing over With respect to time anymore. So this DTT is zero. There is so to say no emotion Okay, so what we basically do is to set this to zero Okay, and then we see already here the main brain currents have to be zero so there they can be only then a definition of the potential with this remaining equation here that Basically describes that's this that's the remainder Describing the equilibrium and then if you plot this again. So here it's now our parameter. That's the the input And like the action potential and then we see or that you had a firing rate to be exact and here on the x y axis we have then the The potential and this equilibrium basically is now a line. So now how this behaves Is it so to say attracting or repelling is it stable unstable and There's a mathematical machinery behind I'm not showing that I'm Going back to this Here you're going back to to this analog on Mechanics, so we see basically it doesn't matter where we start that we are approaching this line So over time we will see this exponential behavior does matter where we start we are going to approach the red Line here that's indicated with the green arrows. Okay, these are the trajectories done and This also means that actually a single neuromass can do nothing. Yeah, it's it's just relaxing so What is important in this model? For example is how these neuromasses are interacting. So we see here the pyramidal cells are excitatory then we get positive feedback through the Excitement to re-interneurin and then a negative feedback for the inhibitor intern and hopefully I mean you remember as soon as there's inhibition involved So we have a feedback then we can Expect oscillations to a cure Okay Questions, how are we in time? Okay, okay So now I will go through Yeah, to present some bifurcation so how to change states yeah from An oscillation to something that is more a damped oscillation or just going to a constant and I will introduce The basic concept and then Paul is showing simulations of these So the first bifurcation I want to show our is a set of node bifurcation What we see here, and that's the the bifurcation diagram we have the Parameter to vary on the x-axis. It's alpha on the y-axis. It's our it's x. It's for example our potential, okay what we see here that if our parameters negative then basically Our trajectories here are simply indicating that there's no stable Points that you simply go through there's no See a point available at zero then something is happening here that Yeah, this this point is a curing But this still not stable so it's slightly attracting so it means that we are going to approach it fast Then you go slower slower, but then it will accelerate again and will shoot away So in turn and Paul will show the time series later. Basically both is unstable and you see an increase of the potential with respect to time and for higher Parameters then or a positive then we see actually to equilibrium Occuring one that is stable. So here the errors are indicating that we are approaching this one this point In this quadrant we are going to approach this point, which is then a stable node and Here that's a saddle. So that's repelling and this would go so if you're outside would go to infinity And if you're inside this cone, then we would go Basically to the stable node. Okay, so yeah, let me introduce myself again. My name is Paul And we will now after the theoretical introduction. We'll have a look at some practical examples So all of you should have received the invitation link to Sure To the HPP Collaboratory if not, please Tell me and I'll send another invitation afterwards and if you accepted this invitation you can log in to the human brain project platform and There is actually where I also put the Jupiter notebook, which I'll be demonstrating now So you can exit it and download it and execute yourself Or just follow along on this big screen or on your own computer as you like. So let me show you how to get there Therefore, I'll just take a seat, but please any time you have have questions. Just feel free to interrupt so this is the login page login with your credentials and Once you're locked in Go to collabs and the upper panel here and then type in TVB Simulator and then there should be only one Collaboratory left which you can choose and So you don't have to follow now, but just introducing so there's some description here You can also see the activity log and also the team members Currently working here and if you go on to the left menu there you can see an option examples and And you click on the arrow all these example notebooks will pop up So on the left side examples and all these are Jupiter notebooks using TVB code to run simulations now for this Talk we are just interest interested it in one Jupiter notebook which can be found in the end the back end Which is called Janssen and Ritt modeled by vocation diagram. So this already gives the goal of this Jupiter notebook We will discuss several by vocations Andreas and will introduce the theory and then I'll show some example simulations within the Jupiter notebook And in the end we try to stick them all together and come up with the by vocation diagram of the Janssen and Ritt model So if you click on it the code shot pop up in front of you and Let me increase the screen now so we can all see Okay, so some of you might be familiar with pysons and also Jupiter notebooks some of you not so I'm just giving Short intro on what we are really seeing here So Jupiter notebooks is a way to combine pyson code or other code as well But mainly pyson and a short documentation of what you are doing and also you can view your results at the same time So what you see and then in these cells in with gray background is pyson code and Then once you hit execute up here. It gets done the play button. It gets executed and the output Is shown below so I will not explain every step or each line of code here because that would take too long We are mainly interested in the results of these this code Yeah, and I tried to put some comments in the code and I'll try to which should explain the Yeah, what the code is doing so in the beginning we install some software packages one which are Necessary for what we are doing here one is called. Yes. What again? There's toolbox you don't have the toolbar Okay, you can set view and then toggle few bar or not. Okay Okay, that's strange. Maybe you just have the notebook viewer then and you cannot execute but let's put these questions maybe aside because where I'm I want to show first what this code book your notebook is about and maybe in the breaks we can I can have a look Okay, I mean we have hands-on session in the afternoons as well where there's more time for interaction as well And we we'll do the stuff together for the moment. Just I wanted to show you where it is and What it's doing and in afternoon. Yeah, so we installed some software packages was in which are necessary, especially this one Pides tool which is Software for numerical continuation. I'll explain what that is later some we installed swigs some that which is Compiler and another version of NumPy here Then we include some Python libraries and then we already get to our first example the settle note by vacation Which was discussed by Andreas. So We use differential equations to simulate the dynamics of whatever we want to try to simulate in our case, it's neural masses neural populations and Therefore we choose. Yeah, different differential equations different dimensionalities. We can observe different dynamics in this case We only focus on a one-dimensional model. Okay, this is when I say model. It's not really now Biophysical meaningful model. It just stands for One application to show what what is possible just giving you an example so X dot or X is our variable and X dot is just another way to know that we this is the derivative of X over time Yeah, the X over the time dt what you saw before So it gives us the speed of this variable or how fast it's changing as we move forward in time Okay, and this is depending on two things one the parameter a Which we are free to vary or set and the other one is X itself You know so the speed of this variable depends also on the position of X of itself You know so wherever you are depends on how far it's like This ball example, right if you are up in the area you drop it you are accelerating So it depends on the position you are and once you land in the in the middle and this stable state You know then the speed is zero because you're not changing anymore Okay, so now let's have a look how to use this to run a simulation There's first some code for the continuation and then I define the functions some functions for plotting not so very interesting, but This is the result in the end. Okay So we have a one-dimensional model So we don't have really a face space or a face plane Where you can see oscillations, but we rather just have a face line, which is indicated here, okay? so X our X variable can be somewhere on this on this face line, okay, and X as you can see here is plotted on the x-axis and X dot which is the speed that's plotted on the y-axis Yeah, in this case, it's possible and higher dimensionality is not possible to plot both things at the same time mmm and This curve what you see here is the speed of X You know so depending on the position as we said before and on our parameter a which is this case a is negative one You have different speed for your very variable X Give me five more minutes and Then you can plot the speed either with such a curve Yeah, this is given by our differential equation and you can also indicate the speed by plotting arrows But who Yeah, I asked really five for five more minutes. Yeah The arrows Indicate also the speed so they show they all of them show to the left. What why is that? Because our X dot is negative anywhere in the face space in order in the face line So anywhere you are with X for this type of parameters your speed is negative meaning your X variable will decrease Therefore these arrows all pointing to the left and now also the length of the arrow is important indicating the Amplit or the the magnitude of the speed Long arrows indicating fast movement to the left small arrows indicating slow movement to the left Now this was now plotted the X the position of X against the speed now We can plot the evolution of X Over time, okay So we now change variables and focus on this plot now here X the position of our variable is plotted on the Y axis and Time is plotted on the X axis and now Andreas already mentioned the point of initial conditions or initial point So we choose this and start somewhere in our face space or face line in this case We choose some initial conditions on this line And then we move forward in time and then observe how is our variable moving? Yeah, how is X changing as we move forward in time. This is what we are interested in Turns out since speed is always negative wherever we are X is always decreasing. Okay, so all these different trajectors Trajectories here are different initial conditions meaning we can start at different positions In our face line, but as you can see we always tend towards negative infinity because speed is always negative Okay Now Yeah, let's let's try to finish this For right and now that the goal of this Particularly a notebook is to introduce the sense of bifurcations. Okay, so this is some dynamics. We can observe in one configuration of our model But what happens if we change a so instead of a equals negative one we now plot a equals zero So what you see basically this curve is shifted upwards and at one point it touches the horizontal line at x dot equals zero So what does that mean that at this point the speed is zero at this point? There won't be any change and as we move forward in time So this has some consequences, right? It's because now it's very crucial where you start in the face line What type of initial conditions of x do you use? So again this curve indicates the speed and it seems to be always negative instead of just one point where it's zero and That's also indicated by these arrows again. So all these arrows point to the left Here they are very Long indicating huge speed and here they're getting smaller and smaller And actually at this point, there's no error. There's zero speed Which it says that this point is actually stable or no, it's a fixed point not stable. Yeah, we we had this Term introduced before but it's a fixed point. So if you start directly on it You wouldn't move anymore Yeah, but if you start somewhere on the right of this line You would get attracted to this point Yeah, so you move left until you reach a point and their speed is zero you stay there If you move if you start slightly just a slight tiny tiny bit to the left on the line You would get repelled. So you move towards negative infinity and that's what we are looking now also in the trajectory plot so again x is now on the y-axis and these trajectories indicate different initial conditions and here the X-axis is time again. So how does x evolve now as we move forwards in time and As you can see it's or as I said before it's very crucial now where you start. So I plot it here at x equals zero this Dash dotted line which indicates the saddle point Which was here again x at zero you have the saddle point here Settle meaning that it's Attracting from one direction and it's repelling from another direction if it would be attracting from both sides We'll come to this later It's a stable fixed point and it's if it's repelling from both sides and it's an unstable fixed point But if it's from one direction attracting the other one, it's repelling means it's a saddle Okay, so what that's the the term you call it and this is also indicated here by the trajectories You choose different initial conditions, which when x is positive you will approach the saddle You know from the positive side and if x is negative you will get repelled and as you can see here You tend towards negative infinity and now just quickly the last case It's one one or two more minutes. Sorry Afterwards, we're having a coffee break if we increase a even further to a equals one now what happens to our Dynamics and now this is the interesting case what happens again this curve Which indicates the speed of x is shifted upwards again? And what you can see there are no two intersections with the horizontal line at x dot equals zero meaning Actually at two positions in the face line the speed is zero Giving rise to two fixed points Yeah, but now we have to decide on the stability of these fixed points because before we Called this a saddle because from one direction it's attracting the other one is repelling now What about these two fixed points here one? fixed point is Laying at x equals negative one and the other one is at x equals positive one and you can already see from the From the your scheme here that an empty circle indicates an unstable fixed point and a full circle indicates a Stable fixed point meaning that this one is attracting indicated also by the arrows Yeah, so if you start in the neighborhood of this somewhere here or over there you get attracted to this fixed point so x will tend towards one and If you start here if you start directly on the unstable fixed point It's also stable or you would stay there forever because speed is negative But remember just a slight perturbation to the left or to the right will then repel you to On this side towards negative infinity again Yeah, because there is no other fixed point no other objects you can be attracted to and on the right side you will get repelled and you end up on the stable fixed point Yeah, and so stable fixed point unstable fixed point and this is the saddle How do trajectories look like okay? Dash line here unstable fixed point Solid line will not really visible in between all the trajectories But that's the stable fixed point and as I said before if you start in the neighborhood of this stable fixed point So every anywhere x greater than negative one all these initial conditions You will approach the stable fixed point at x equals one on the other hand if you start in a neighborhood of this unstable fixed point Somewhere between yeah one and or in less than one all these you will get repelled from this unstable fixed point and either here to the stable fixed point or you reach the negative infinity again We're just one more minute, and then we're done with this set an old bifurcation. Really sorry I Introduced this Pides toolbox in the beginning. Yeah, so it is and I said it's for numerical continuation What do you really want to know now or what a bifurcation diagram should tell us is what happens as we what happens to the Dynamics if we move if we change one parameter a you know this is our parameter we changed up here and Here in this example we indicated that we can calculate this pretty easy by hand sometimes it's not so easy analytically then has to be done numerically and We found these different points Yeah, fixed points settle points and we were really interested in those points because they give us a hint on the dynamics of the system So it would be very nice if we can track these points or if we can tell how Do these points move and evolve as we vary one parameter? Okay, and as you can see here for a negative one There's no fixed point or no object on the face line for a equals zero There's one settle point and for a equals one. There's two points here So what the continuation software now does is it varies this parameter a and it looks for these Points if we can find any in there and this is basically what you can see here, okay? So for negative values of a as I said, there's no fixed points or no objects That's why here it's empty as we approach a equals zero. There's a settle point here and This gives rise as we increase a even further to positive value This gives rise to a stable fixed point and an unstable fixed point and as you increase a even further they both grow apart as you can see, you know and Okay, that's it. I think enough for the morning and then we'll continue after the coffee break. Thank you So welcome back, I hope you had a good coffee tea refresh so we would now continue with Introducing some bifurcations that are then important to understand in this Locker dynamic model, so the transmit model Paul just showed Yeah, the calculation so the integration of the settle note bifurcation. They basically Have a switch from no stability to Is the stability there only one point basically is stable and depends on the initial condition There we end up. Okay, so I will now move on and go to oscillations and Yeah deal with the question how they emerge and that's called Hopf bifurcation and trying to hopf bifurcation to be exact and that's in a super critical Form here, which means that you're going to see a stable Oscillation so here now as mentioned before. Hope you remember for an oscillation. We need two Parameters, sorry two variables to describe it. Yeah for line. It's not possible It's easier than using your arms And you see that it's better in the plane on the plane and So this plane is here basically x1 and x2 so this could be like the post-anoptic potential and then the current and Again on the x-axis. We have a parameter. This could be something like this alpha like excitability, okay, so for Negative values there. We see there's actually only this line here so the behavior for negative alpha, so The system is less excited. We see The spiraling and like shown before so we start doesn't matter where we start We have a movement at the beginning as fast and then we will see that over time You're going to approach this point and this is then showing over when we see the time series then later Paul will show it Then it's simply a damped oscillation, so it's fading away with the time So now what happens if you get closer to to zero to this to this point here We're obviously something is happening and then what we see it takes longer and longer To approach this point and that's the bifurcation point, so we see that there's a change in the behavior from relaxation behavior to to this what is the equilibrium or picks point And then for positive alphas we see that it's still attracting but now Locally, it's actually repeling. So here if you would start close to this line, then it goes outside Okay, so from here we go. You have a movement outside outwards and then from outside this Orbit the cycle here then we it's attracting so and this basically for positive alphas if the system is highly excited we see this limit cycle and this limit cycle is attracting Basically, it's here in this bifurcation diagram with respect to Alpha it is Here shown as this cone. Okay, so this face portray is simply now here a slice and then this slide Intersection and then we see that it's from outside this limit cycle it is so to say it the trajectories are attracted to Onto this limit cycle if you're inside this limit cycle locally, it's unstable and then we have a spiraling out motion onto this limit cycle So Paul will show the calculation Questions Okay So we stop here, right the equilibrium carapace at an odd bifurcation and now we continue with the half bifurcation So this is the mass behind it again I don't want to focus too much on the equations here Just I want to say or I just want to point out that now this is a two-dimensional system Yeah, so we're now moving on a plane. It's a face plane before we had the face line We just call them y1 y2. This is not not biological meaningful. It's just Yeah, mathematical framework of mathematical theory behind it to generate such bifurcations and runoff of bifurcation Again y1 or y2 and the dot above it means dy1 Over dt or the speed of this variable or the change over time. Okay okay, some code here to For the continuation We'll continue then the equilibrium as we saw before the stable point or the unstable points and also the limit cycles I'll show you how that means in a second Then some more code to plot vector fields and then basically in the end we loop Yeah, if we look just once very shortly into this equation. There's these State variables one y1 and y2 here and also there's a parameter b Okay, and this will be our bifurcation parameter. So before we call it a We varied that to create a set of node bifurcation now We have this parameter called b which we also will vary to create the runoff of bifurcation and This is already what we want to see or what we are interested in as we said There's now a face plane on the x-axis. We have the state variable y1 and on the y-axis it's y2 and Now our initial point So we saw in one of the beginning slides of this talk is Somewhere on this plane before it was on the line now What's on the plane and again these arrows indicate the speed and the direction of the State change. Okay, so long arrows again mean high speed small errors low speed or very slow change and the Arrow of the direction is also the direction of the change and Now here are just some sample trajectories plotted into the face plane So this would be some different initial conditions Okay, the blue line starts here The red line starts down here at y1 negative 1 and y2 also negative 2 and here are some Yeah more inner initial points as well And basically what you can see is all of them spiral down towards the origin, right? y1 equals 0 here and y2 equals 0 as well So this seems to be attracting anywhere we start and a face plane We'll get attracted by these stable focus before we had a stable fixed point where we directly approach it Now this is called a stable focus. We have where you first draw some rounds and spiral into it So what happens if we now plot y1 over time? so Here we plot y1 against y2 and now we plot the state variables over time So this starting point here equals t t1 or t0. Yeah, there's Time point one now somewhere then here is time point two ten points three time at four and so on so forth and If you go into infinity you reach this fixed point you stay there forever So now this we take all these values for y1 and y2 and plot them on new plot. Okay This one is actually wrong We have two state variables here one in yellow and one in blue y1 and y2 and we plot them across time and What you can see here? Yeah, so basically this is one very very strong damped oscillation You know it makes one turn maybe and then the rest of the oscillation are here very very small And you directly or more or less directly approach this fixed point in the middle a stable focus, okay? now We set B Equal negative one in this case Now this is our bifurcation parameter So we vary B and we see what changes in the dynamics of the system for B equals zero This is what you get here So you still spiral down you still can choose different initial conditions as you see here and up there But you still tend towards the spiraling behavior And you will try to approach this fixed point or that's stable focus in the origin of this face plane Again, we plot now the variables y1 and y2 over time And this is what we get down here Very now this over here was very strongly damped, but as we Approach the bifurcation point the damping is reduced more and more. So this is why we have these sustained oscillations As I move forward in time Now what happens if we cross? The spifurcation point what happens if we increase B even further to B equals one? This is depicted here. We actually create or we give birth to a stable limit cycle You know, it's a limit cycle Which you can see here and it's globally attracting Meaning anywhere you start in a face plane whether here or in here Over there you will always be attracted onto this limit cycle and basically this is an orbit which continues forever So you will cycle on it as you go forward some time and the interesting fact is before and the the origin was a stable focus And now it changed its stability and it got unstable. So if we If we start directly at zero and zero Directly on the origin. This is a fixed point again. So you wouldn't move because speed is zero But as soon as you slightly perturb just like this perturbation from there You will then spiral outwards and approach the stable limit cycle and this is what you we see here So we have sustained oscillations with fixed amplitude now I said we were using this continuation software again and and Okay, is it big? Yeah so it tracks the the points of The stable fixed points unstable fixed points in the face plane or in the face line Okay, I think I hope you can still hear me. I just shut off So for B equal having negative values you can see this on the left side here We have a stable on our stable fixed point or stable focus. Okay, so there's always you will always be approaching this point wherever you start for a different different places in the state space and At B equals zero we said we have this hopf bifurcation, which is indicated here by the blue point H equals one and After this point The origin at y equals y1 equals zero is getting unstable Okay, so you start somewhere near to this point You will spiral outwards and what you will approach is this limit cycle So we have to think as this like a like a section through the three-dimensional plot Andreas showed you before Yeah, so we are just focusing here on the parameter B and y1, but this thing here Actually just plots the maximum and minimum values of the limit cycle. So if you go back to the face plane This is the limit cycle, right and the maximum value for y1 is some here around one and they're at negative one And these two values are plotted and this gives Ryan rise to this cone and actually in three I mentioned will be a real cone and here you cut through the cone the particular section Okay, and as you can see the amplitude of the limit cycle increases as we Increase B and this is the supercritical hopf bifurcation. Yeah You have any questions on this yet? Yes Yeah, so Actually, we have two state variables, right? Giving rise to a cycle which you can see here. However, we are just is it working on? However, we in this bifurcation diagram. We just plot one variable Okay, and so we have to make some reduction. We cannot show the whole circle. It would be a line, right? To make things in inside the cycle still visible We just choose to plot the minimum value of y1 or of this state variable We are interested in and the maximum value as we move a very parameter B So why one is the maximum value of why one or you can also look here the maximum value of where one is Positive one and minimum value is negative one now this Changes, yeah, because the limit cycle increases as we increase B and that's what's depicted here You know for B equals zero or before there was no limit cycle, but then there's a minimum and Maximum value up here and the minimum value for why one down there Okay, so this is all still just mathematical theory and no neurons simulated on neuromasses But still we want to show What difference one and two dimensional systems can have regarding their dynamics and especially what happens if we simulate a stimulus So we run the system. We integrate it with some noise So it will fluctuate around for example around a stable focus or stable fixed point And then at one time point we hear we hit it Yeah, we move one of the state variables outwards Far far away and then we observe what happens as it comes back or is it is it gone or what happens to system as we stimulate it You can think of maybe evoked potentials in EEG or MEG So therefore we use both systems the system we used to Simulate the set a note hope but the set a note bifurcation and now the two-dimensional system for the and run off of bifurcation So here the equations our integrator some setup for the noise and basically we will implement a stimulus at after a quarter of simulation time half of the simulation time and three quarters of simulation time and This is depicted down here. So this is our two-dimensional system. You can already see the oscillations So if we would run it without noise Like we will spiral downwards. Oh, sorry. I forgot. It's to say that we use this system for Be somewhere here. So before the hop-by-furcation, you know So basically what we have is a stable focus. We would stay we would spiral down towards this point here at the origin and the other system we Initiate in this configuration where there is an unstable in a stable fixed point Okay, otherwise, we would have the problem that we would tend towards negative infinity pretty fast Regardless of any stimulus So what happens? We in we give some noise. So each time we look at the speed We also had some noise that that pushes us around this stable focus So we get these small oscillations and after a quarter of simulation time as I said we Throw a stimulus to one of the state variables with which kicks it far away from the stable center This gives rise to this huge oscillations here. And what you can see there is also some transient Yeah, so it takes some time to approach back the stable focus or at least to Reduce the size of the amplitude below the size of the noisy oscillations around it. Okay So after a stimulus, there's a certain time period where you can still see like after effects And this happens after the second and also after the third stimulation You know, so this is the noisy part where you see some small oscillations And then you see big oscillations as we stimulate the system Now what happens for the one-dimensional case? We have a stable focus here. We Right and that x equals one and we had some noise to it giving fluctuation around a stable point and then as soon as we hit As soon as we hit it or we give it a stimulus we push it far away from the fixed point here in the positive direction and As soon as stimulus is off. We fall back onto it. So there's no transient So we directly approach back the limits as the stable fixed point and we directly Get lost in this noisy fluctuation around the stable fixed points. So this is the difference Me what does it mean like if you want to model evoke potentials any oscillations you again You know, that's the point you need to a two-dimensional system at least to observe Patterns like this instead of this. Okay Then you do you have any questions on this part? Otherwise, we go back to theory and then see the next way for occasion so questions, okay, so you've seen basically it does supercritical hop-by-focation and Have it a stable focus and then on the other side we have unstable one with Stable limit cycle that's wrong here. It's a stable limit cycle is attractive Of course Like it's already indicated here. There also exists a subcritical one which means that an unstable limit cycle can exist as well So here everything is inverted. Okay from the science You have basically the same space No for positive alpha. We have an unstable focus. So we have this spiral going out Then it's zero. It's again. It looks identical But the arrows are pointing outwards so it's unstable and then For negative values locally it's stable. So which means if you're inside the limit cycle then We get attracted to this line here to zero to the zero line if you're outside then The system will go away to infinity. Okay, so there's a this Limits like this unstable limit cycle can act as a separate tricks to kind of separating the outside from the entire Side, but still you can there's something stable inside, but globally So to say it's unstable Okay I would like to go back and to I mean these are all local bifurcations. So which means that basically You can predict already like Paul showed in the simulation with the noise if you the change Due to the bifurcation. So you don't have to cross the bifurcation to see what comes after so For example here. That's what Paul showed this supercritical bifurcation where you have the stable focus on for negative Alphas then if you have noise you're going to see always like a permanent Transient like a damped oscillation occurring. Okay, so you actually know already What is coming after the bifurcation point so on why I'm saying that that there are also bifurcations that or changes that you cannot predict So and for that I would like to go back to two demonstrators I would like to go back to the saddle node bifurcation There there was no solution for here negative alpha. So this is basically doesn't matter where you stimulate There's a flow this trajectory is showing that you simply flow through the the states of which means that the x the potential for example is increasing and this also happens here in this part so if even if there is Stability then the question is actually where is this going? Okay, so these arrows where they are pointing to So this is here one dimension is example and What happens if we increase the dimension so this can happen in a more or let's say realistic models they go to the chance in red and there are other objects so to say in the repertoire of the dynamics and It can look like that so basically That's X now here X and this Representation is face portray is basically this line here. So we see what we see here is basically just this part for Positive alphas so we have here a saddle which is this one Over here. I hope you see that and we have a stable node Which is that one so here this indicates that they are connected Okay, so if you're close here, then you're going you're close to this unstable one to this unstable node then you're going to to be To have a movement towards the stable node over here. Okay But now what can also be what happens if we are on the other side of this? Unstable node so it's here. It's attracting there. It's repelling That's this part. So what happens if we go in this direction? So we are initializing somewhere here the system and then what under certain circumstances what can happen is that basically this Motion the trajectory falls back then to itself so that it's connected and then basically it goes in a different path then back To the to the stable node over here So that's what we call heteroclinic Channel heteroclinic because two different objects are connected. Okay Now what happens if these this unstable one and Stable one are colliding. That's what happening at zero. Okay So now they they move towards each other as we approach from positive alphas zero and then at one point here we see that basically the The saddle and the node are colliding building this a saddle node But still we have this this big cycle. Okay, which is now a homoclinic one because there's only one stable point at this And then for negative Perimeter alpha and then there's no Stable point available anymore and then what can happen that? still this cycle exists and that's another Mechanism to create Oscillation, so that's a limit cycle what we see here But something that we could not predict by just looking at the local behavior, which is that a note Okay, that there is a limit cycle occurring. So this that's an emergent property out of the interaction of the elements in models Okay, so Paul will show some simulations Okay, so back to the notebook The slides where we left off was this one here the stimulation and Now we are coming to this shilnikov set of nodes homoclinic bifurcation and we'll introduce us now with a biological Biological meaningful model before we just set these example toy examples normal forms now. We're using this smaller slikar model, which is Model for the membrane potentials are they derived it from a crap So they use muscle muscle tissue from a crap to drive these Potentials nevertheless they can give right. Yeah, they give rise to Different patterns of oscillation are used for different for different explanations of excitability and neurons as well So again here the mass it's getting a little bit more complicated as you can see But actually what I again what I want to focus on here is V dot and W dot It's a two-dimensional system. So we're still acting on a on a plane. Okay These are just helper functions, which appear up here again now our bifurcation parameter is this one here I applied it. So this is like an applicator Current to or you give some input to this model from the outside V is the membrane potential of this cell and W is the current through potassium channels and Now let's see. Okay. There's again some code for the continuation Little bit more this case some utility functions to plot the face plane and Yeah, you see it's due to all these heteroclinic homoclinic orbits, which I'll show in a second It's getting more and more code already So this where This is what we want to focus on here We are again on the face plane. We know already. This is one variable. This is the other state variable and On this face plane you have trajectories. You have these arrows indicating where to go and at what speed and you have these points for fixed points Yeah, they can be Unstable focusing like here. They can be stable fixed points or even settle points You know so and the circle means unstable black circle means stable and half full half empty means a saddle point So there's from one direction again. It's attracting and through other directions. It is repelling Okay, so what you see here is also these lines the black one here and the green one and on the top These are called null clients. Yeah, so for one variable on this line The speed is negative. So it just the other variable can move. Okay, so these arrows for example up here They're very small, but at this point they are completely horizontally. So meaning there's no change At on this line, there's no change for variable w. Yeah, and the same Holds true for the black curve and the state variable v Meaning that if at one point on this line One the speed for the change of one variable is zero Meaning that we're both of these lines intersect the speed of both has to be zero Okay, and this gives rise to fixed points whether they are unstable or Stable and that's what you can see here. So these lines intersect at one two three positions And this is where we find our fixed points now the colored trajectories here the the colored curves are trajectories and One is starting here from the unstable focus the blue one it starts really close to it Where you can see it spirals out and approaches the stable fixed point over here the other two orange Trajectories one starts very close to the right of This settle point and it does all this loop To approach the stable fixed point and the other trajectory starts very close to the left of the settle point and we'll approach as well the stable Fixed point so this is what we call hetero clinic orbits You start from one object in the face plane and you approach another one like these two are hetero clinic orbits Now we said I Applicated is the current we apply to the model and this is our bifurcation parameter So in this case I equaled 20 what happens now if we increase I to well this 39.9 something Is it what you can see basically here if let's me show them both next to each other these two Fixed points these two objects they approach each other and in this point they merge actually and What you see is you lose the stable fixed point and only this saddle point remains Okay, so what does it mean as I said the settle point has one unstable and one stable direction If you start very close to the right you will do this loop And it approaches itself again Yeah, so you do this loop and you come back to it where you started this is called a homo clinic orbit Yeah before it was called hetero clinic orbits where you start from one object in the face plane approaching a separate one Now this is a homo clinic orbit you started from one point you approach yourself again And again from this unstable focus you spiral outward and you approach this Settle point over here. Okay, so this is why it's also called a homo clinic orbit and set a note bifurcation Because we have a stable node and a settle here colliding with each other and After they collided so if we increase I even further to I equals 60 they annihilate so they're finally gone So you can also see this through the null clients these null clients now here the black one and the green one They only intersect in one point. So the other two intersections are gone and this point which remains is this Unstable focus and what you can see is you spiral outwards and you approach the stable limit cycle Okay, so there's a stable limit cycle born through this homo clinic bifurcation of the shenikov type Is this clear more or less? Does it make sense? If not, please ask questions anytime Yeah, so these two collide here they vanish and you remain with a stable limit cycle now What implications does this specific bifurcation diagram have for us if we want to simulate action potentials in neurons? Okay, and this is what this model is for basically so We will again try to simulate it at a little bit of noise and to see what happens to the trajectory as we start close To the stable fixed point. Okay Maybe not I equals 20 a little further approaching 39, but I think I used 38 38 yeah, and we are interested in this variable V which is the membrane potential So if we record membrane potentials, we can then track down the spikes All action potentials if this neuron or its muscle has it and that's what you see here so V measured in millivolt and time in milliseconds and The threat at the time series of our model now as you can see this time series is getting wiggly again Because we had some noise in each integration step. Well, as you can see there are several spikes Across the time series. How does it look like in the face plane? Okay, this is depicted down here as I said I equals 38 So we're still before the bifurcation the subtle note Homo clinic bifurcation, which means we have this unstable focus or the unstable spiral We have the saddle point here and the fixed point over there And as you can see now in blue that this is the trajectory and it's wiggly because we had noise And most of the time we have these sub threshold fluctuations. Now. This is these periods here We're below below a spike, but as soon as we cross What we call a separatrix because it separates the dynamics as soon as we cross the saddle point if noise drives us More to the right on this face plane You have to do all this loop to come back to your attractor the stable attractor over here So if we start here, you give some noise on the system. You might fluctuate around this area That's why it's so dark blue over here and at some points maybe noise is enough to push you across the separation border and then you have to do this loop here which describes the action potential and Yeah, so this is basically the case before the bifurcation happens and this is again a bifurcation diagram Where we track down the solution with the software we mentioned before so now it's getting a little more complicated as you can see We have the solid line here, which is the stable point this one We have this dash dotted line, which is the saddle point and they both touch each other and the annihilate in Settle note bifurcation here and then this third line up here. This is the unstable focus Okay, where you spiral outwards and this up here and down there again are the minimum and maximum values of your stable limit cycle Which is created through in the end this green line here is representing the homo clinic orbit now the point where we have This bifurcation, it's homo clean orbit here and both of these fixed point touch So right at the bifurcation, we have a homo clinic orbit We have the unstable spiral and we have the saddle note depicted here by the green line We have an unstable focus here. There's this point. We have the saddle note here and We have the homo clinic orbit the green line and from there on to the right We only have this limit cycle and the unstable focus left Okay, then I think we have all parts together to stick them now Into the Janssen and Ritt bifurcation diagram All right any questions further questions on to the specific bifurcation for now See Okay, so now we are going to the Janssen Ritt model which is Six dimensional meaning that we have six variables So post-naptic potential to describe post-naptic potential I showed to okay, and then We have three neural masses. So this Pyramid cell Mass of pyramids as interneurons, et cetera inhibitory. So we in total we have six and What we've seen so far is then I should say looking for the equilibrium then stability so we are stable or not and then how Stability is changing and what the different states Are so we have now all the ingredients to like post that to explain so to say the Johnson or to understand the Janssen Ritt model and We start So to say with the oops Yeah, that I showed at the beginning I can show it again Let's see if this Try it again It is this model that I'm talking about okay, so we have here the pyramidal cells and then the interneurons Acceleratory and inhibitory Okay, I will So now first we look for the stability now because at six dimensional you cannot display it It's so to say a projection and What we see from the state is only the post-naptic potential of pyramidal cells as a function of the input To the to the pyramidal stars by pyramidal stars because they are meant to generate EG and MEG and because of their Mophology how they look like Okay, and here that's the equilibrium curve that we see it's not a line before it was usually just a straight line because it's non-linear the system is non-linearly coupled and This is basically causing this this falling what we see here, and then that's the equilibrium curve and What is already suggested for a certain input here? Yeah, let's say if the input is higher than than four then there's only one equilibrium If it's lower than I don't know minus one then it's also only one equilibrium is Available and there's a certain range here between minus one and four. Let's say that's where they are free so we we are talking here already about Multistability that's There's not just one solution possible. They're dependent on where we are so What depending on the history we can take different states. Okay, so if they are stable or not there we use this mathematical Machinery I'm not we haven't talked about that to decide if something is stable or not But basically this is here indicated by the dotted lines Okay, and then this is also indicating that if stability is changing on this equilibrium curve That we have a we see a local bifurcation. So locally meaning on this Equilibrium curve something is changing and Then voila, we see the different local bifurcation. We see hopf bifurcations Over here that sub supercritical. So here we should see a limit cycle somewhere oscillation should appear Here it's a supercritical. So a separatrix should appear And then we have in the folding of the equilibrium curve. We also have bifurcation points Settle note and a special one that I would like to skip Okay, and if we look then at the limit cycle, so that's a special question to address that with Contination software that Paul is going to show You can basically show the the maximum and minimum of limb of these Limit cycles and we see like expected here in this upper branch that we have this In turn of half supercritical that they are connected and forming this at this branch of Limit cycles, so if you we are up here, then we see oscillations there another Range of limit cycles which is over here So and settle note like I said before that's a local one We do not see the the bill at this you cannot see locally the the birth of the limit cycle so we have to take into account that there is something up here and Are going on here and then what we see basically then if you look at this point for limit cycles that there is a huge Limit cycle appearing then I mean huge meaning that The amplitude is bigger than the other one over here the red one Okay, so Just to summarize because see I just bifurcation diagram is not complete Because you do not see the pay a face portraits for different input values and That's basically done here. It's the same representation So we have here again the the post-optic potential of the premises as a function of the inputs And then here would be sort of is the Folded equilibrium curve and then now we see that this here is a note Then this here is a stable focus with the separatrix around to the unstable limit cycle and If you have two limit cycles that are coexisting then for a small range of input One with a large amplitude in clue Which is basically showing then in the eG simulated eG or local field potential like a spike wave behavior, so this has been used to model epilepsy and Then here in red we have the other limit cycle which is more harmonic And this has been used to describe alpha activity Okay Yeah, and that's what Paul is then going to simulate Questions So now about the Janssen-Ritt equations again, I showed the math behind it Where as you can see we we came from one over two dimensional systems and now we skip three four five We directly approach six dimensions Really just don't focus too much on this It's why one f y zero two by five And our bifurcation parameter here is this meal over there It's just the input External input from other brain areas through stimulation or whatever onto the pyramidal cells in this Janssen-Ritt model. Okay, so What happens to the dynamics of these three neural masses once we change the input? Where we are running out of time, so I really Focus on the main results for very low mu you can see we approach Stable, so this is the post synaptic potential of the pyramidal cells over time Okay, so what happens is that we move forward in time to the post synaptic potential for low input mu equals 0.05 You approach a stable fixed point as you saw in the diagram and there's there's where you stay for the rest of the time Okay, there's nothing interesting or nothing more happening over there first certainly higher values of mu 0.13 You get these spike spike and wave penance actually as Andreas showed you In the pyramidal cells and this is what made it so interesting for modeling epileptic seizures for example now because this is what you also see in human EEG people are in the crisis and If you increase mu even further then you jump out of this limit cycle Explaining the spike wave pattern on to something more like an alpha-rhythm like alpha oscillations in what you also can see in human EEG Okay, so varying this parameter mu seems to be pretty interesting Yeah, you can observe different kind of kind of dynamics as you move this parameter. So Let me skip this. This is the continuation and this is what you already saw from Andreas's plots, you know the equilibrium curve and you find different local bifurcations on it and then from these local bifurcation especially half bifurcations you stand start the continuation of the limit cycles and The scheme is always the same. Yeah, so solid lines mean stable dash dotted lines means unstable and Limit cycles are depicted here again with the maximum and the minimum value of this one state Variable we are looking at why is here on this case? so you can see there's stable limit cycles and Unstable limit cycles and again another one another stable limit cycle up there Down here. Yeah, the minimum and maximum value again Okay, so We said if we have very low mu There's a stable point down here And if we choose our initial condition somewhere in this range, you will approach the stable point Okay, and you stay there for the rest of time as we increase mu some value around here point 13 as I said You land on the big limit cycle which gives the rise to the spike wave patterns if you move even further if you increase mu to 0.2 for example, you land on this still limit cycle Yeah, this limit cycle is created at this point and then it Yeah, it vanishes at this point again. So this white looks like a bubble or something but It's it's the section again what we're interested in so you end up here giving rise to alpha oscillates what you saw before Now what happens if we? Dynamically vary this parameter mu so we simulate the system meaning we integrate it over time We see what happens what kind of dynamics can we explore and then in each time step? We vary mu so we want what we want to do here is we start down here Then we increase view to some value over here, and then we move back To lower values of mu and what we all in see then Is something interesting or we call hysteresis or hysteresis effects? And this code basically down here does it? Okay, so this is what the result basic in the end is up here. This is the post synaptic potentials of the pyramidal cells and This is the time series. Okay, so over time. We actually simulated 60 cells milliseconds or one minute and sorry and This is where we start in the beginning. So this is what we Expected right we see some noisy fluctuations around a stable fixed point Corresponding to this part down here. Yeah, there's a stable fixed point. No oscillation. No nothing But as we increase mu you see these spike and wave patterns Becoming more and more frequent and until they are here systemic or a rhythmic Spike and waving spike wave patterns and this is due to the big limit cycle Which we just discussed and as we increase mu even further than this we leave this Spike wave patterns and we approach the alpha oscillations Okay, and this bit depicted here because it's so compressed and time you you can't really see the cycle each cycle But you see this vexing and veining of the amplitude. This is pretty you can see this pretty clearly here Now we increased mu. This is down here. Yeah, so we increase mu from point was it point Point one or five we increase it to Point one four and then we decrease it again. So until 30 seconds we increase mu and then we Change the sign and we decrease it. Yeah, this is depicted here from zero to 30 seconds. We increase and then we decrease and The interesting thing now is we don't add and we don't end up at the state where we started from Yeah, so this is what you see here We started at a fixed point and noisy fluctuations and then we passed through this spike and wave patterns But what actually is interesting as we increase mu we stay up here and actually the alpha oscillation doesn't change So why is that under us? The reason for that basically is that the by stability and that the Equilibrium curve is folded I got back here maybe Yeah, so it's folded and Then if you start somewhere here, so that's the folding of the equilibrium curve It's now it's simpler to show it here. So we start somewhere here We move and in a positive direction. So there's a jump up something that we could not predict I mean, it's simply losing stability and fine-stand stability up here So where we have then these oscillations Okay, and then if we decrease again to the original value to the initial input value Then basically we stay up here. So to basically full With close to sight I go back and to make a cycle we need to decrease further the inputs and then to this point where we then but so to say Jump down and then you can increase again and go back to the original value. Okay, so This state basically depends on the history and that's what's a poor shot in the simulation Is there do you have questions? Actually, you wouldn't see the spiking No, you see the alpha oscillation still So here you you see at the beginning was just the fluctuations. Okay, then you had to spikes. So like Spike wave complexes and then if you increase further then you go through this bifurcation here and you jump up in this branch in this harmonic Limit cycle the red one basically for alpha. So and then if you go back to the initial point Which is still so to say in the red Range so in this regime where you have this harmonic oscillations Then you stay there and you a qualitatively you you don't see a change Yeah Yeah, that's it Okay, so any further questions if not That was the end of our talk and we would then continue with the architecture of the virtual brain Sure Again, what's that? Let's see. Ah, there it is. Okay. Just give it some time Okay, what maybe you can Yeah Yeah Yeah, yeah, okay. Yeah, it's it's right at the bottom Right Sometimes are these are these adapters Maybe can let me show I think it's this adapter. Sometimes they fail. Can I show you so What we're doing is okay. So here's in another unit, but it's basically the same parameter. Yeah, so we start here yeah, and We start with our PSP down here. Oh, yeah, sorry So we start here This is right and we start with our post synaptic potentials We start also somewhere below here and it is egg-tracked as you can see this is stable Stable so it anything in this range here and PSP is it attracted so you will approach this point. Okay now, however For values mu around 50 to 100. There's also a second attractor Yeah, there's not only one attractor. So there's a possibility also to be attracted by this one here Which is a stable spiral. Yeah They're the only way they are separated through these lines here, which are unstable limit cycles. Okay, right? So let's say you start here directly on the line and now you increase mu. What happens You want you walk along this line until you reach this point now suddenly this line loses its stability If you increase mu even further this line is gone. However, there's a new attractor now Which is the large limit cycle this one here in blue They had a minimum at a maximum and the minimum values of this large limit cycle Okay, and you are here and you are closest to this limit cycle. So you get attracted There and you end up on this big spike wave oscillation. That is what you see here the blue blue limit cycle is this one and If you move even further with mu here suddenly also this big limit cycle is gone Yeah, and you have up here somewhere in then in nowhere. So if you are here and suddenly increase mu you end up here Now, what is the attractor in at this point? Well, it's this other the second limit cycle in red Yeah, so if you are here, you will get attracted up here to this stable limit cycle in red And you stay on there now we move mu from 150 back towards 50 So what happens? You are here. You are always on this limit cycle just on the red one always on the red one always on the red one Always on the red one always on the red one until this point where suddenly this stable limit cycle is gone It's the super critical hop-fly vacation here what remain remains. However is another attractor at this point Which is this stable focus. So now you're up here And you have to so before you we started here at 50 We did this like this loop up here, and it's not a loop. It's not closed. Yeah, so this is what his raises You start here you jump up there you move back here now you end up here to come back down here Yeah, you would you would have to move further along this axis Until this point here this one the black curve until this one loses its stability Yeah, and then the only tractor is the yellow curve down here So if you move across this point, you will fall back down here So to close the loop you have to jump up here go back down here What again? Yeah, when the when the stable point disappears or when the stable limit cycle disappears, and There's another attractor in the face pain place Yeah Face space may maybe can I show the next slide as well? There's a reason for that if you can go back. I mean this transition is that this unstable limit cycle which is here is Colliding with a stable limit cycle so in the face portrayal in blue we have the big limit cycle in red inside the Small limit cycle, and then there's another one. You may not see it in dashed lines here. It's the unstable limit cycle acting as a separate trick so separating between inside the Harmonic red small limit cycle and then the blue limit cycle so here at this point At this separation is gone. So these the blue one and unstable one are colliding Vanish and then only the red one is left. So that's that's a global another global by vocation that I haven't introduced but yeah