Rigorous derivation of the nonlocal reaction-diffusion FitzHugh-Nagumo system
Filbet, Francis
We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigourous link between kinetic and reaction-diffusion models.
 Thank you for the organization of this workshop and for the invitation. So today I would like to reserve some results obtained with Joachim Creva, who is a PhD student in Toulouse and also a colleague of Gregory Faye who works at CNRS. So during this talk I would like first to start with a discussion on the modeling and start with a different description to introduce this Fitts-Hugh-Nagumo model for neuroscience applications. So I will start with a microscopic description in order to motivate the study. That is the derivation of a kinetic model and then some hydrodynamic limits starting from the kinetic equation. So my main result will concern the asymptotic analysis starting from the solution of a kinetic equation in order to obtain a macroscopic model called the Fitts-Hugh-Nagumo. So I will briefly present the solution for this macroscopic model and then give you the result on the asymptotic analysis and then present the main argument of the proof. Okay, so let me start with the modeling issues. So let us start with the simplest model we can do for these applications is the single-neurant model. So this starts with the work of Hodgkin and Uxley even before 52 where they performed some experiment on the neuron cell of a squid and observed some oscillatory behavior of a potential, like exactly like this. Here you have the time evolution of a potential. And what we observe is a periodic behavior in time but with a non-sinusoidal shape for this signal. So what they wanted to do is to derive a simple model in order to reproduce this phenomenon. And then they derived the Hodgkin-Uxley model and later in 61 and 62 Fitts-Hugh and Nagumo Alimoto and Yoshizawa obtained a simplification of this model. This is the one I present here where the two end nodes that we will consider is the membrane potential V of T and another variable called WT which satisfies this ODE system. And here end of V in this talk will be equal to V times 1 minus V squared. And all other terms are linear terms and there is an external current. We see the response on the potential according to this current and what is expected is that if this external current is large enough then you have this behavior on the potential of the membrane. And here this is the phase plane where we have V and W and you can guess the trajectory of your solution on this plane here. So A, B and T are positive constants. Coefficient T is given, T, A and B are positive constants. And I depends on T? Sorry? The external current depends on T? If you want, I will read it here. It can depend on time but even if you have just, it can be used just to initiate the behavior of your solution. But I want to remove it and to replace this part because now I'm not only interested by the behavior of what? A single neuron, so we are interested now in the network of neurons. So I will add a new unknown X which will represent the spatial variable and we assume that these neurons do not move. So X prime of I is equal to 0. I is an index which represents one neuron and you assume that you have N neurons. So we have exactly the same body system for each neuron but now it depends on I. And the current now is replaced by this operator here which modeled the interactions between neurons. Okay, so what is important to emphasize here is that this current depends linearly, I would say, on the V variable. And in order to model the spatial interactions between the neurons, we choose a potential of interactions which depend on two parameters, epsilon and R. So in order to be more precise, we write that the epsilon R is the sum of two terms psi here which will represent long range interactions plus the sum of one over epsilon which describes the intensity of the current and psi of R is a given function. So for instance we have these kind of shapes here in pink, this represents psi of R, one over epsilon psi of R. So with a large intensity and a small support whereas psi describes long range interactions. And here in this picture, this is an experiment made on a monkey, I think. And just to, if we initiate a pulse, we see how it propagates into the network. Okay, and now formally, if we are interested for a large number of neurons and if R goes to zero, we can replace this interaction function, phi, as the sum of psi plus a Dirac times one over epsilon. And formally, we get this kinetic equation where now we are interested in the distribution function of neurons, epsilon, and it is solution of these transport equations. So here it's important to emphasize that this is a pure transport equation in the variable V and W. And on this equation, the X variable only enter in the gain in this operator k psi which is written like this. So you just observe that k psi, psi of f, can be written as the convolution of psi with rho, I will determine, times V minus psi of rho V, where rho and rho V are moments of the distribution function of f with respect to one, to rho, the density, if I'm here, and the first order moment in V called rho V. And I can add epsilon if you want. Another remark is that since here is a transport equation in V and W, if we integrate with respect to V and W, we get exactly the conservation of mass. That is, the density of a neuron, rho epsilon t of x, is actually constant with respect to time, and is equal to the initial density. So this quantity does not evolve with respect to time. So we define rho epsilon V, so now we replace the rho by rho 0. And we can define rho 0 V like this and rho 0 W. And of course, when rho 0 is equal to 0, we set that V and W are also equal to 0. We are not interested by the propagation of the signal when there is no neurons. Okay. And of course, now if we start with this model, we are now interested in the limit epsilon goes to 0. Just before to do that, I would like just to give some references on the derivation of this kinetic equation. So I can refer to some work on the mean field limit starting from Hocke-Keyne-Huxley and Fitzwick-Nagumo systems, and adding noise in the OD, so it's a stochastic differential equation for the microscopic equation. We can derive also a transport equation plus a focal length term between. We can get the kinetic equation for that. There are similar work by Rousseau and Stana recently. I would like also to refer to the work of Micheler, Pignot and Touboul who studied exactly the kinetic equation when now the distribution function does not depend on space. And they prove existence of solution and stability of a stationary state. So in our study here, we will neglect the noise. So it means that at the kinetic level, we just solve a transport equation. And so for the microscopic model, we just have a deterministic system. Here, the connectivity side is between the neurons is weighted only with respect to the distance, so we can also consider a more general model. And this function is unbound. So formally, if we start with the kinetic equation that I wrote here, and we just compute the time evolution of the moment, so it means that we multiply this equation by V, and that is we can get the time evolution for cross 0 times Vf side. And we get the same for the quantity W epsilon, we multiply by W and integrate. So since there are many linear terms, we get the following ODE system, where now, so we have this time evolution here, where the operator L cross 0 epsilon is defined exactly like this, and of course the system is not closed due to this function N of V, and then there is an error term where appears of course F epsilon, which is given by the difference between these two quantities here. So some remark here, so if we neglect this error term, this corresponds to the kind of reaction diffusion model where the diffusion is now replaced by this non-local operator. So by diffusion, we mean that it propagates the support at infinite speed. It also introduced some dissipation, but there is no regularization here. So the idea in the limit epsilon goes to 0 is to prove that this error term E of F epsilon goes to 0 when epsilon goes to 0. And like this, we can recover the macroscopic Fitzhugh Nagumo model, which is studied in the community of reaction diffusion model for the propagation, we have the different analysis on the propagation of front or pattern formations. Just a remark, here the density rho 0 can be equal to 0 for some x, and so it means that this equation here is not well defined when rho 0 is equal to 0. So it's important to define exactly what we mean by V here. So we set, as I said before, the epsilon of Tx is the ratio between the moments, what I call j epsilon over rho 0 of x if rho 0 of x is strictly positive and is equal to 0 in the other case. So the main result is the following. So my first result will concern the macroscopic model. So here, as you see here, we consider that the equation will be multiplied by rho. But we want to simplify it and to divide by rho f side rho 0. So of course, we can do that if we consider the macroscopic model. And so we studied this macroscopic model, the reaction diffusion system, for this system under the assumption on the side, and if it is L1, non-negative, symmetric, and if rho 0 is L1 and L infinity, and the initial data, w0 and w0 are both L infinity in space, then we can prove that we get a classical solution in the sense that this equation is satisfied, is c1, and L infinity in space. So it means that we propagate the L infinity regularity, and here in this equation, more or less we can see it as an ODE on the banner space, and so we can apply a classical Cauchy-Lipschitz C1. So from the study of this system of equation, we can construct a solution for the same equation multiplied by rho 0. And to do that, it's important to observe that in the region where rho 0 is strictly positive, then the... Oh, yeah, let's see. So there is a contribution of L rho 0 here, and this is a non-local operator in space, but since in this integral there is the weight rho 0, it means that in the values, the x where rho 0 is strictly positive will only enter in the game. We can do whatever we want in the region where rho 0 is equal to 0. And what is the advantage of doing that, studying this model instead of the one written in the rho 0 formulation, is that we can have the propagation of L infinity solution, w and v. And this will be very important in our asymptotic analysis when we will use relative anthropic techniques, because we need some regularity on the macroscopic equation. Okay, now I can state the main result. So if we assume that the initial data is L1 and has some moments bounded uniformly with respect to epsilon, rho 0 is non-negative, L1 and L infinity, and v0 and w0, which are the initial data for the macroscopic system, are both L infinity. If we have well prepared initial data, then we can prove that if we denote by vw, the solution of the macroscopic reaction diffusion system, and w epsilon, the solution computed from the distribution function f epsilon of the kinetic equation, we have this convergence of the two solutions with an explicit rate of convergence with respect to epsilon. So how to prove this theorem? So mainly the proof will be based on relative anthropic techniques, proposed by D'Perna and Lafermose for hyperbolic conservation laws. And here I just give, of course, these techniques have been applied to many systems in kinetic theory and hyperbolic equations, but here I just refer to some recent result on the Vlasov equation on the Pücker-Smale system, which shares many properties with the microscopic model we have here. The main difference is that here in our case there is no transport in x, so which is good and not good. It's good in the sense that we don't have to control space derivative, but what is not good is that we have an equation which is nonlinear in the kinetic equation. We have a nonlinear kinetic equation and we have to control nonlinear term f epsilon and j epsilon, for instance, and to construct solutions, for instance, when we have a classical kinetic equation, we can use some averaging lemma to get some strong compactness on j epsilon. But here, since x, in our case, is only a parameter, there is no hope to do that. Okay. Okay. And the other difficulty comes from this function, n of v, where we have to control when we compute time evolution of the relative entropy, we have to control terms like this. Okay. So to do that, how much time do I have? Five minutes, okay. I think it's enough to give you the main argument of the proof. So the proof is first based on the propagation of moments of the distribution function. So for instance, if I just consider the second order moment in v and w, we can prove that they propagate and we also get from the steepest term, dissipation, which is with a coefficient one over epsilon times d, where d is given by this formula, which can be rewritten here like exactly the square. So it allows us, okay, we get an estimate like this, and we can also propagate higher order moments, for instance, force order moments. And an important point is that we get here two moments in velocity. So it's good for us when we have a term like this. So this is the first argument, and the second two, yeah, but there is a problem here is that the dissipation is written still with the weight rho zero. Okay, but in our term that we want to control, this term does not appear. Okay, so the idea is to remove this term, this weight rho zero epsilon here, and to do that, we just perform a decomposition. So we evaluate an integral like this, and we split the rd into this space. So it means that in a epsilon, there is no contribution, so it's okay. For this guy, it means that we can use the anthropidicipation estimate to control the rate of convergence epsilon. And in this region, we will use some higher order moment estimate to control for any eta, and the idea is to adjust the best eta in order to get this kind of estimate here on this quantity, where now there is no more the weight rho zero. So we lose some rate of convergence, but we remove the weight. And after, the relative entropy technique follows. So we define an entropy by setting eta of z, where z is this variable here, rho zero epsilon rho zero b and rho zero w. We set eta of z equal to this quantity, and then we compute the time we want to prove that the relative entropy, given by this definition here, where you see once again the weight rho zero epsilon appear. And so we want to prove that this quantity goes to zero when epsilon goes to zero. And we have to control the entropy to prove the convergence of the theorem. To do that, we just rewrite our system in an abstract form, where the macroscopic system is written in this form, whereas if we compute the time evolution of the macroscopic quantities derived from the kinetic equation, we have exactly the same equation, plus the error term, given by this quantity. And on the other hand, we have the propagation of the entropy, given by this. It is not exactly dissipated, but it is under control, everything. So then we compute the time evolution of the relative entropy. We split in that way, and from the previous calculation, we get exactly the time evolution of this quantity. So finally, everything can be written like this. So in this equality, only relative terms appear that is different between z epsilon and z, or z and z epsilon. Using classical arguments, we can bound this quantity by some quantity only depending on the initial data, and plus the integral of the relative entropy. And finally, so that is we control this term, and the last step is to control the error term epsilon. And to do that, we will just use the previous result on the dissipation of entropy. We just apply a holder inequality of Cauchy-Schwarz here, and we bound this coefficient alpha t by higher order moments which are controlled with respect to v. And finally, this error term can be controlled with this height and a grand van Lema allowed to conclude the proof. Conclusion. So we have here derived rigorously the macroscopic Fitsu-Nagomo model, starting from a kinetic equation. Some comments. So here we have chosen a more, a kinetic on the potential of interaction which localizes a lot of the interactions. This is the Dirac that we have in the equation. And just a remark, we can start without this very local operator, but by rescaling again the interaction potential psi like this, in that case we can at least formally derive the classical reaction diffusion equation when we replace the non-local operator by a classical Laplacian. And from this model, we know that we have propagation of front during these stabilities, and what we would like to do is to derive or to obtain such property, not for the macroscopic equations, but for the kinetic equation. Thank you. Very good. So you have removed the noise, right? Yes. So question, natural question, can you do with something with the noise? So yes. It's even better because we, if we have the noise for instance in V, so we will get a Fokker-Planck operator in V and it will help for the analysis, I think. But we, what is not clear is that maybe, sorry, yeah, it depends after the rescaling that we have when we, do you see the kinetic equation? What is it? Yeah. If we, if we, if we, it depends after the scaling that we consider for the kinetic equation, maybe if we put one over at silent for instance in front of all these terms, we can have several steady states for the, for the kinetic equation, it may be a problem. Question? This is... So it has some focusing property, so it helps a lot to gain some moments. So we can rewrite this, if we want a general result, indeed we can replace it by some focusing... And this allows us to gain some moments in velocity, it is important. Our at least propagated. So if there are other questions, thank you very much again. Thank you.
 Thank you for the organization of this workshop and for the invitation.  So today I would like to reserve some results obtained with Joachim Creva, who is a PhD student  in Toulouse and also a colleague of Gregory Faye who works at CNRS.  So during this talk I would like first to start with a discussion on the modeling  and start with a different description to introduce this Fitts-Hugh-Nagumo model  for neuroscience applications.  So I will start with a microscopic description in order to motivate the study.  That is the derivation of a kinetic model and then some hydrodynamic limits starting from the kinetic equation.  So my main result will concern the asymptotic analysis starting from the solution of a kinetic equation  in order to obtain a macroscopic model called the Fitts-Hugh-Nagumo.  So I will briefly present the solution for this macroscopic model and then give you the result on the asymptotic analysis  and then present the main argument of the proof.  Okay, so let me start with the modeling issues.  So let us start with the simplest model we can do for these applications is the single-neurant model.  So this starts with the work of Hodgkin and Uxley even before 52  where they performed some experiment on the neuron cell of a squid  and observed some oscillatory behavior of a potential, like exactly like this.  Here you have the time evolution of a potential.  And what we observe is a periodic behavior in time but with a non-sinusoidal shape for this signal.  So what they wanted to do is to derive a simple model in order to reproduce this phenomenon.  And then they derived the Hodgkin-Uxley model and later in 61 and 62  Fitts-Hugh and Nagumo Alimoto and Yoshizawa obtained a simplification of this model.  This is the one I present here where the two end nodes that we will consider is the membrane potential V of T  and another variable called WT which satisfies this ODE system.  And here end of V in this talk will be equal to V times 1 minus V squared.  And all other terms are linear terms and there is an external current.  We see the response on the potential according to this current and what is expected is that if this external current is large enough  then you have this behavior on the potential of the membrane.  And here this is the phase plane where we have V and W and you can guess the trajectory of your solution on this plane here.  So A, B and T are positive constants.  Coefficient T is given, T, A and B are positive constants.  And I depends on T?  Sorry?  The external current depends on T?  If you want, I will read it here.  It can depend on time but even if you have just, it can be used just to initiate the behavior of your solution.  But I want to remove it and to replace this part because now I'm not only interested by the behavior of what?  A single neuron, so we are interested now in the network of neurons.  So I will add a new unknown X which will represent the spatial variable and we assume that these neurons do not move.  So X prime of I is equal to 0.  I is an index which represents one neuron and you assume that you have N neurons.  So we have exactly the same body system for each neuron but now it depends on I.  And the current now is replaced by this operator here which modeled the interactions between neurons.  Okay, so what is important to emphasize here is that this current depends linearly, I would say, on the V variable.  And in order to model the spatial interactions between the neurons, we choose a potential of interactions which depend on two parameters, epsilon and R.  So in order to be more precise, we write that the epsilon R is the sum of two terms psi here which will represent long range interactions  plus the sum of one over epsilon which describes the intensity of the current and psi of R is a given function.  So for instance we have these kind of shapes here in pink, this represents psi of R, one over epsilon psi of R.  So with a large intensity and a small support whereas psi describes long range interactions.  And here in this picture, this is an experiment made on a monkey, I think.  And just to, if we initiate a pulse, we see how it propagates into the network.  Okay, and now formally, if we are interested for a large number of neurons and if R goes to zero, we can replace this interaction function, phi, as the sum of psi plus a Dirac times one over epsilon.  And formally, we get this kinetic equation where now we are interested in the distribution function of neurons, epsilon, and it is solution of these transport equations.  So here it's important to emphasize that this is a pure transport equation in the variable V and W.  And on this equation, the X variable only enter in the gain in this operator k psi which is written like this.  So you just observe that k psi, psi of f, can be written as the convolution of psi with rho, I will determine, times V minus psi of rho V,  where rho and rho V are moments of the distribution function of f with respect to one, to rho, the density, if I'm here, and the first order moment in V called rho V.  And I can add epsilon if you want.  Another remark is that since here is a transport equation in V and W, if we integrate with respect to V and W, we get exactly the conservation of mass.  That is, the density of a neuron, rho epsilon t of x, is actually constant with respect to time, and is equal to the initial density.  So this quantity does not evolve with respect to time.  So we define rho epsilon V, so now we replace the rho by rho 0.  And we can define rho 0 V like this and rho 0 W.  And of course, when rho 0 is equal to 0, we set that V and W are also equal to 0.  We are not interested by the propagation of the signal when there is no neurons.  Okay.  And of course, now if we start with this model, we are now interested in the limit epsilon goes to 0.  Just before to do that, I would like just to give some references on the derivation of this kinetic equation.  So I can refer to some work on the mean field limit starting from Hocke-Keyne-Huxley and Fitzwick-Nagumo systems,  and adding noise in the OD, so it's a stochastic differential equation for the microscopic equation.  We can derive also a transport equation plus a focal length term between.  We can get the kinetic equation for that.  There are similar work by Rousseau and Stana recently.  I would like also to refer to the work of Micheler, Pignot and Touboul who studied exactly the kinetic equation  when now the distribution function does not depend on space.  And they prove existence of solution and stability of a stationary state.  So in our study here, we will neglect the noise.  So it means that at the kinetic level, we just solve a transport equation.  And so for the microscopic model, we just have a deterministic system.  Here, the connectivity side is between the neurons is weighted only with respect to the distance,  so we can also consider a more general model.  And this function is unbound.  So formally, if we start with the kinetic equation that I wrote here,  and we just compute the time evolution of the moment,  so it means that we multiply this equation by V,  and that is we can get the time evolution for cross 0 times Vf side.  And we get the same for the quantity W epsilon,  we multiply by W and integrate.  So since there are many linear terms,  we get the following ODE system,  where now, so we have this time evolution here,  where the operator L cross 0 epsilon is defined exactly like this,  and of course the system is not closed due to this function N of V,  and then there is an error term where appears of course F epsilon,  which is given by the difference between these two quantities here.  So some remark here, so if we neglect this error term,  this corresponds to the kind of reaction diffusion model  where the diffusion is now replaced by this non-local operator.  So by diffusion, we mean that it propagates the support at infinite speed.  It also introduced some dissipation, but there is no regularization here.  So the idea in the limit epsilon goes to 0 is to prove that this error term  E of F epsilon goes to 0 when epsilon goes to 0.  And like this, we can recover the macroscopic Fitzhugh Nagumo model,  which is studied in the community of reaction diffusion model  for the propagation, we have the different analysis on the propagation of front  or pattern formations.  Just a remark, here the density rho 0 can be equal to 0 for some x,  and so it means that this equation here is not well defined when rho 0 is equal to 0.  So it's important to define exactly what we mean by V here.  So we set, as I said before, the epsilon of Tx  is the ratio between the moments, what I call j epsilon over rho 0 of x  if rho 0 of x is strictly positive and is equal to 0 in the other case.  So the main result is the following. So my first result will concern the macroscopic model.  So here, as you see here, we consider that the equation will be multiplied by rho.  But we want to simplify it and to divide by rho f side rho 0.  So of course, we can do that if we consider the macroscopic model.  And so we studied this macroscopic model, the reaction diffusion system,  for this system under the assumption on the side, and if it is L1, non-negative, symmetric,  and if rho 0 is L1 and L infinity, and the initial data, w0 and w0 are both L infinity in space,  then we can prove that we get a classical solution in the sense that this equation is satisfied,  is c1, and L infinity in space.  So it means that we propagate the L infinity regularity, and here in this equation,  more or less we can see it as an ODE on the banner space,  and so we can apply a classical Cauchy-Lipschitz C1.  So from the study of this system of equation, we can construct a solution for the same equation  multiplied by rho 0.  And to do that, it's important to observe that in the region where rho 0 is strictly positive,  then the...  Oh, yeah, let's see.  So there is a contribution of L rho 0 here, and this is a non-local operator in space,  but since in this integral there is the weight rho 0, it means that in the values,  the x where rho 0 is strictly positive will only enter in the game.  We can do whatever we want in the region where rho 0 is equal to 0.  And what is the advantage of doing that, studying this model instead of the one written in the rho 0 formulation,  is that we can have the propagation of L infinity solution, w and v.  And this will be very important in our asymptotic analysis when we will use relative anthropic techniques,  because we need some regularity on the macroscopic equation.  Okay, now I can state the main result.  So if we assume that the initial data is L1 and has some moments bounded uniformly with respect to epsilon,  rho 0 is non-negative, L1 and L infinity, and v0 and w0, which are the initial data for the macroscopic system,  are both L infinity. If we have well prepared initial data, then we can prove that if we denote by vw,  the solution of the macroscopic reaction diffusion system, and w epsilon, the solution computed from the distribution function f epsilon  of the kinetic equation, we have this convergence of the two solutions with an explicit rate of convergence  with respect to epsilon.  So how to prove this theorem?  So mainly the proof will be based on relative anthropic techniques,  proposed by D'Perna and Lafermose for hyperbolic conservation laws.  And here I just give, of course, these techniques have been applied to many systems in kinetic theory and hyperbolic equations,  but here I just refer to some recent result on the Vlasov equation on the Pücker-Smale system,  which shares many properties with the microscopic model we have here.  The main difference is that here in our case there is no transport in x, so which is good and not good.  It's good in the sense that we don't have to control space derivative,  but what is not good is that we have an equation which is nonlinear in the kinetic equation.  We have a nonlinear kinetic equation and we have to control nonlinear term f epsilon and j epsilon, for instance,  and to construct solutions, for instance, when we have a classical kinetic equation,  we can use some averaging lemma to get some strong compactness on j epsilon.  But here, since x, in our case, is only a parameter, there is no hope to do that.  Okay.  Okay.  And the other difficulty comes from this function, n of v,  where we have to control when we compute time evolution of the relative entropy,  we have to control terms like this.  Okay.  So to do that, how much time do I have? Five minutes, okay. I think it's enough to give you the main argument of the proof.  So the proof is first based on the propagation of moments of the distribution function.  So for instance, if I just consider the second order moment in v and w,  we can prove that they propagate and we also get from the steepest term,  dissipation, which is with a coefficient one over epsilon times d,  where d is given by this formula, which can be rewritten here like exactly the square.  So it allows us, okay, we get an estimate like this,  and we can also propagate higher order moments, for instance, force order moments.  And an important point is that we get here two moments in velocity.  So it's good for us when we have a term like this.  So this is the first argument, and the second two, yeah,  but there is a problem here is that the dissipation is written still with the weight rho zero.  Okay, but in our term that we want to control, this term does not appear.  Okay, so the idea is to remove this term, this weight rho zero epsilon here,  and to do that, we just perform a decomposition.  So we evaluate an integral like this, and we split the rd into this space.  So it means that in a epsilon, there is no contribution, so it's okay.  For this guy, it means that we can use the anthropidicipation estimate  to control the rate of convergence epsilon.  And in this region, we will use some higher order moment estimate  to control for any eta, and the idea is to adjust the best eta  in order to get this kind of estimate here on this quantity,  where now there is no more the weight rho zero.  So we lose some rate of convergence, but we remove the weight.  And after, the relative entropy technique follows.  So we define an entropy by setting eta of z,  where z is this variable here, rho zero epsilon rho zero b and rho zero w.  We set eta of z equal to this quantity, and then we compute the time  we want to prove that the relative entropy, given by this definition here,  where you see once again the weight rho zero epsilon appear.  And so we want to prove that this quantity goes to zero when epsilon goes to zero.  And we have to control the entropy to prove the convergence of the theorem.  To do that, we just rewrite our system in an abstract form,  where the macroscopic system is written in this form,  whereas if we compute the time evolution of the macroscopic quantities derived  from the kinetic equation, we have exactly the same equation,  plus the error term, given by this quantity.  And on the other hand, we have the propagation of the entropy,  given by this.  It is not exactly dissipated, but it is under control, everything.  So then we compute the time evolution of the relative entropy.  We split in that way, and from the previous calculation,  we get exactly the time evolution of this quantity.  So finally, everything can be written like this.  So in this equality, only relative terms appear  that is different between z epsilon and z, or z and z epsilon.  Using classical arguments, we can bound this quantity  by some quantity only depending on the initial data,  and plus the integral of the relative entropy.  And finally, so that is we control this term,  and the last step is to control the error term epsilon.  And to do that, we will just use the previous result  on the dissipation of entropy.  We just apply a holder inequality of Cauchy-Schwarz here,  and we bound this coefficient alpha t by higher order moments  which are controlled with respect to v.  And finally, this error term can be controlled with this height  and a grand van Lema allowed to conclude the proof.  Conclusion.  So we have here derived rigorously the macroscopic Fitsu-Nagomo model,  starting from a kinetic equation.  Some comments.  So here we have chosen a more,  a kinetic on the potential of interaction  which localizes a lot of the interactions.  This is the Dirac that we have in the equation.  And just a remark, we can start without this very local operator,  but by rescaling again the interaction potential psi like this,  in that case we can at least formally derive the classical  reaction diffusion equation when we replace the non-local operator  by a classical Laplacian.  And from this model, we know that we have propagation of front  during these stabilities, and what we would like to do  is to derive or to obtain such property,  not for the macroscopic equations,  but for the kinetic equation.  Thank you.  Very good.  So you have removed the noise, right?  Yes.  So question, natural question, can you do with something with the noise?  So yes.  It's even better because we,  if we have the noise for instance in V,  so we will get a Fokker-Planck operator in V  and it will help for the analysis, I think.  But we, what is not clear is that maybe,  sorry, yeah, it depends after the rescaling that we have  when we, do you see the kinetic equation?  What is it?  Yeah.  If we, if we, if we, it depends after the scaling that we consider  for the kinetic equation, maybe if we put one over at silent  for instance in front of all these terms,  we can have several steady states for the,  for the kinetic equation, it may be a problem.  Question?  This is...  So it has some focusing property,  so it helps a lot to gain some moments.  So we can rewrite this, if we want a general result,  indeed we can replace it by some focusing...  And this allows us to gain some moments in velocity,  it is important.  Our at least propagated.  So if there are other questions, thank you very much again.  Thank you.
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Thank you for the organization of this workshop and for the invitation. So today I would like to reserve some results obtained with Joachim Creva, who is a PhD student in Toulouse and also a colleague of Gregory Faye who works at CNRS. So during this talk I would like first to start with a discussion on the modeling and start with a different description to introduce this Fitts-Hugh-Nagumo model for neuroscience applications. So I will start with a microscopic description in order to motivate the study. That is the derivation of a kinetic model and then some hydrodynamic limits starting from the kinetic equation. So my main result will concern the asymptotic analysis starting from the solution of a kinetic equation in order to obtain a macroscopic model called the Fitts-Hugh-Nagumo. So I will briefly present the solution for this macroscopic model and then give you the result on the asymptotic analysis and then present the main argument of the proof. Okay, so let me start with the modeling issues. So let us start with the simplest model we can do for these applications is the single-neurant model. So this starts with the work of Hodgkin and Uxley even before 52 where they performed some experiment on the neuron cell of a squid and observed some oscillatory behavior of a potential, like exactly like this. Here you have the time evolution of a potential. And what we observe is a periodic behavior in time but with a non-sinusoidal shape for this signal. So what they wanted to do is to derive a simple model in order to reproduce this phenomenon. And then they derived the Hodgkin-Uxley model and later in 61 and 62 Fitts-Hugh and Nagumo Alimoto and Yoshizawa obtained a simplification of this model. This is the one I present here where the two end nodes that we will consider is the membrane potential V of T and another variable called WT which satisfies this ODE system. And here end of V in this talk will be equal to V times 1 minus V squared. And all other terms are linear terms and there is an external current. We see the response on the potential according to this current and what is expected is that if this external current is large enough then you have this behavior on the potential of the membrane. And here this is the phase plane where we have V and W and you can guess the trajectory of your solution on this plane here. So A, B and T are positive constants. Coefficient T is given, T, A and B are positive constants. And I depends on T? Sorry? The external current depends on T? If you want, I will read it here. It can depend on time but even if you have just, it can be used just to initiate the behavior of your solution. But I want to remove it and to replace this part because now I'm not only interested by the behavior of what? A single neuron, so we are interested now in the network of neurons. So I will add a new unknown X which will represent the spatial variable and we assume that these neurons do not move. So X prime of I is equal to 0. I is an index which represents one neuron and you assume that you have N neurons. So we have exactly the same body system for each neuron but now it depends on I. And the current now is replaced by this operator here which modeled the interactions between neurons. Okay, so what is important to emphasize here is that this current depends linearly, I would say, on the V variable. And in order to model the spatial interactions between the neurons, we choose a potential of interactions which depend on two parameters, epsilon and R. So in order to be more precise, we write that the epsilon R is the sum of two terms psi here which will represent long range interactions plus the sum of one over epsilon which describes the intensity of the current and psi of R is a given function. So for instance we have these kind of shapes here in pink, this represents psi of R, one over epsilon psi of R. So with a large intensity and a small support whereas psi describes long range interactions. And here in this picture, this is an experiment made on a monkey, I think. And just to, if we initiate a pulse, we see how it propagates into the network. Okay, and now formally, if we are interested for a large number of neurons and if R goes to zero, we can replace this interaction function, phi, as the sum of psi plus a Dirac times one over epsilon. And formally, we get this kinetic equation where now we are interested in the distribution function of neurons, epsilon, and it is solution of these transport equations. So here it's important to emphasize that this is a pure transport equation in the variable V and W. And on this equation, the X variable only enter in the gain in this operator k psi which is written like this. So you just observe that k psi, psi of f, can be written as the convolution of psi with rho, I will determine, times V minus psi of rho V, where rho and rho V are moments of the distribution function of f with respect to one, to rho, the density, if I'm here, and the first order moment in V called rho V. And I can add epsilon if you want. Another remark is that since here is a transport equation in V and W, if we integrate with respect to V and W, we get exactly the conservation of mass. That is, the density of a neuron, rho epsilon t of x, is actually constant with respect to time, and is equal to the initial density. So this quantity does not evolve with respect to time. So we define rho epsilon V, so now we replace the rho by rho 0. And we can define rho 0 V like this and rho 0 W. And of course, when rho 0 is equal to 0, we set that V and W are also equal to 0. We are not interested by the propagation of the signal when there is no neurons. Okay. And of course, now if we start with this model, we are now interested in the limit epsilon goes to 0. Just before to do that, I would like just to give some references on the derivation of this kinetic equation. So I can refer to some work on the mean field limit starting from Hocke-Keyne-Huxley and Fitzwick-Nagumo systems, and adding noise in the OD, so it's a stochastic differential equation for the microscopic equation. We can derive also a transport equation plus a focal length term between. We can get the kinetic equation for that. There are similar work by Rousseau and Stana recently. I would like also to refer to the work of Micheler, Pignot and Touboul who studied exactly the kinetic equation when now the distribution function does not depend on space. And they prove existence of solution and stability of a stationary state. So in our study here, we will neglect the noise. So it means that at the kinetic level, we just solve a transport equation. And so for the microscopic model, we just have a deterministic system. Here, the connectivity side is between the neurons is weighted only with respect to the distance, so we can also consider a more general model. And this function is unbound. So formally, if we start with the kinetic equation that I wrote here, and we just compute the time evolution of the moment, so it means that we multiply this equation by V, and that is we can get the time evolution for cross 0 times Vf side. And we get the same for the quantity W epsilon, we multiply by W and integrate. So since there are many linear terms, we get the following ODE system, where now, so we have this time evolution here, where the operator L cross 0 epsilon is defined exactly like this, and of course the system is not closed due to this function N of V, and then there is an error term where appears of course F epsilon, which is given by the difference between these two quantities here. So some remark here, so if we neglect this error term, this corresponds to the kind of reaction diffusion model where the diffusion is now replaced by this non-local operator. So by diffusion, we mean that it propagates the support at infinite speed. It also introduced some dissipation, but there is no regularization here. So the idea in the limit epsilon goes to 0 is to prove that this error term E of F epsilon goes to 0 when epsilon goes to 0. And like this, we can recover the macroscopic Fitzhugh Nagumo model, which is studied in the community of reaction diffusion model for the propagation, we have the different analysis on the propagation of front or pattern formations. Just a remark, here the density rho 0 can be equal to 0 for some x, and so it means that this equation here is not well defined when rho 0 is equal to 0. So it's important to define exactly what we mean by V here. So we set, as I said before, the epsilon of Tx is the ratio between the moments, what I call j epsilon over rho 0 of x if rho 0 of x is strictly positive and is equal to 0 in the other case. So the main result is the following. So my first result will concern the macroscopic model. So here, as you see here, we consider that the equation will be multiplied by rho. But we want to simplify it and to divide by rho f side rho 0. So of course, we can do that if we consider the macroscopic model. And so we studied this macroscopic model, the reaction diffusion system, for this system under the assumption on the side, and if it is L1, non-negative, symmetric, and if rho 0 is L1 and L infinity, and the initial data, w0 and w0 are both L infinity in space, then we can prove that we get a classical solution in the sense that this equation is satisfied, is c1, and L infinity in space. So it means that we propagate the L infinity regularity, and here in this equation, more or less we can see it as an ODE on the banner space, and so we can apply a classical Cauchy-Lipschitz C1. So from the study of this system of equation, we can construct a solution for the same equation multiplied by rho 0. And to do that, it's important to observe that in the region where rho 0 is strictly positive, then the... Oh, yeah, let's see. So there is a contribution of L rho 0 here, and this is a non-local operator in space, but since in this integral there is the weight rho 0, it means that in the values, the x where rho 0 is strictly positive will only enter in the game. We can do whatever we want in the region where rho 0 is equal to 0. And what is the advantage of doing that, studying this model instead of the one written in the rho 0 formulation, is that we can have the propagation of L infinity solution, w and v. And this will be very important in our asymptotic analysis when we will use relative anthropic techniques, because we need some regularity on the macroscopic equation. Okay, now I can state the main result. So if we assume that the initial data is L1 and has some moments bounded uniformly with respect to epsilon, rho 0 is non-negative, L1 and L infinity, and v0 and w0, which are the initial data for the macroscopic system, are both L infinity. If we have well prepared initial data, then we can prove that if we denote by vw, the solution of the macroscopic reaction diffusion system, and w epsilon, the solution computed from the distribution function f epsilon of the kinetic equation, we have this convergence of the two solutions with an explicit rate of convergence with respect to epsilon. So how to prove this theorem? So mainly the proof will be based on relative anthropic techniques, proposed by D'Perna and Lafermose for hyperbolic conservation laws. And here I just give, of course, these techniques have been applied to many systems in kinetic theory and hyperbolic equations, but here I just refer to some recent result on the Vlasov equation on the Pücker-Smale system, which shares many properties with the microscopic model we have here. The main difference is that here in our case there is no transport in x, so which is good and not good. It's good in the sense that we don't have to control space derivative, but what is not good is that we have an equation which is nonlinear in the kinetic equation. We have a nonlinear kinetic equation and we have to control nonlinear term f epsilon and j epsilon, for instance, and to construct solutions, for instance, when we have a classical kinetic equation, we can use some averaging lemma to get some strong compactness on j epsilon. But here, since x, in our case, is only a parameter, there is no hope to do that. Okay. Okay. And the other difficulty comes from this function, n of v, where we have to control when we compute time evolution of the relative entropy, we have to control terms like this. Okay. So to do that, how much time do I have? Five minutes, okay. I think it's enough to give you the main argument of the proof. So the proof is first based on the propagation of moments of the distribution function. So for instance, if I just consider the second order moment in v and w, we can prove that they propagate and we also get from the steepest term, dissipation, which is with a coefficient one over epsilon times d, where d is given by this formula, which can be rewritten here like exactly the square. So it allows us, okay, we get an estimate like this, and we can also propagate higher order moments, for instance, force order moments. And an important point is that we get here two moments in velocity. So it's good for us when we have a term like this. So this is the first argument, and the second two, yeah, but there is a problem here is that the dissipation is written still with the weight rho zero. Okay, but in our term that we want to control, this term does not appear. Okay, so the idea is to remove this term, this weight rho zero epsilon here, and to do that, we just perform a decomposition. So we evaluate an integral like this, and we split the rd into this space. So it means that in a epsilon, there is no contribution, so it's okay. For this guy, it means that we can use the anthropidicipation estimate to control the rate of convergence epsilon. And in this region, we will use some higher order moment estimate to control for any eta, and the idea is to adjust the best eta in order to get this kind of estimate here on this quantity, where now there is no more the weight rho zero. So we lose some rate of convergence, but we remove the weight. And after, the relative entropy technique follows. So we define an entropy by setting eta of z, where z is this variable here, rho zero epsilon rho zero b and rho zero w. We set eta of z equal to this quantity, and then we compute the time we want to prove that the relative entropy, given by this definition here, where you see once again the weight rho zero epsilon appear. And so we want to prove that this quantity goes to zero when epsilon goes to zero. And we have to control the entropy to prove the convergence of the theorem. To do that, we just rewrite our system in an abstract form, where the macroscopic system is written in this form, whereas if we compute the time evolution of the macroscopic quantities derived from the kinetic equation, we have exactly the same equation, plus the error term, given by this quantity. And on the other hand, we have the propagation of the entropy, given by this. It is not exactly dissipated, but it is under control, everything. So then we compute the time evolution of the relative entropy. We split in that way, and from the previous calculation, we get exactly the time evolution of this quantity. So finally, everything can be written like this. So in this equality, only relative terms appear that is different between z epsilon and z, or z and z epsilon. Using classical arguments, we can bound this quantity by some quantity only depending on the initial data, and plus the integral of the relative entropy. And finally, so that is we control this term, and the last step is to control the error term epsilon. And to do that, we will just use the previous result on the dissipation of entropy. We just apply a holder inequality of Cauchy-Schwarz here, and we bound this coefficient alpha t by higher order moments which are controlled with respect to v. And finally, this error term can be controlled with this height and a grand van Lema allowed to conclude the proof. Conclusion. So we have here derived rigorously the macroscopic Fitsu-Nagomo model, starting from a kinetic equation. Some comments. So here we have chosen a more, a kinetic on the potential of interaction which localizes a lot of the interactions. This is the Dirac that we have in the equation. And just a remark, we can start without this very local operator, but by rescaling again the interaction potential psi like this, in that case we can at least formally derive the classical reaction diffusion equation when we replace the non-local operator by a classical Laplacian. And from this model, we know that we have propagation of front during these stabilities, and what we would like to do is to derive or to obtain such property, not for the macroscopic equations, but for the kinetic equation. Thank you. Very good. So you have removed the noise, right? Yes. So question, natural question, can you do with something with the noise? So yes. It's even better because we, if we have the noise for instance in V, so we will get a Fokker-Planck operator in V and it will help for the analysis, I think. But we, what is not clear is that maybe, sorry, yeah, it depends after the rescaling that we have when we, do you see the kinetic equation? What is it? Yeah. If we, if we, if we, it depends after the scaling that we consider for the kinetic equation, maybe if we put one over at silent for instance in front of all these terms, we can have several steady states for the, for the kinetic equation, it may be a problem. Question? This is... So it has some focusing property, so it helps a lot to gain some moments. So we can rewrite this, if we want a general result, indeed we can replace it by some focusing... And this allows us to gain some moments in velocity, it is important. Our at least propagated. So if there are other questions, thank you very much again. Thank you.
model the spatial interactions between the with the kinetic equation that I wrote here, and do that, it's important to observe that in the prove that if we denote by vw, the solution of proof. So the proof is first based on the abstract form, where the macroscopic system is time evolution of the macroscopic quantities time evolution of the relative entropy. We split non-local operator by a classical Laplacian. And