Historical overview of macroscopic and kinetic models in neuroscience

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A microscopic spiking neuronal network for the age-structured model

Regarding the chaos propagation method, in a series of outstanding papers, Pakdaman, Perthame and Salort (PPS) [Pakdaman et al., 2010, Pakdaman et al., 2013, Pakdaman et al., 2014] introduced a very versatile model for the large-scale dynamics of neuronal networks. These equations describe the probability distribution of the time elapsed since the last spike red as an age-structured nonlinear PDE.
Inspired by the dynamics of these macroscopic equations, we work on a microscopic model describing the dynamics of a nite number of neurons, and that provides a realistic neural network model consistent with the PPS model, in the sense that in the thermodynamic limit, propagation of chaos and convergence to the PPS equation is proved.
In this model, the state of each neuron i is described by a R+-valued variable Xi;N t corresponding to the time elapsed since last discharge. Of course, this approach is quite dierent from classical literature, where the key variable is the voltage: this is an important originality of the PPS model. The spiking interaction between neurons is considered as a the global activity M at the network level. Specically, a neuron with age x (duration since it red its last spike) res a spike with an instantaneous intensity a(x;M) where M is the global activity of the network. Subsequently to the spike emission, two things happen: the age of the spiking neuron is reset to 0, and the global variable M increases its value by an extra value of J=N. The coecient J represents the mean connectivity of the network. For each N 2 N, let us consider a family (N1 t ; : : : ;NN t )t0 of i.i.d. standard Poisson processes.
Let us also consider a family (1; : : : ; N) of i.i.d. real valued random variables with probability law b.
These coecients represent delays in the transmission of information from the cell to whole network. Furthermore, we assume that the family of delays is independent of the Poisson processes and the initial conditions.

On the statistical description of neuron networks: the weak connectivity conjecture

Most large-scale neuronal networks can described by a density function f = f(t; ) 0 describing the probability density of nding neurons in some state 2 (typically stands for a intern neuron time, the membrane voltage or the couple voltage-conductance of the neuron in the FhN model) at time t 0. The density f evolves according to an integral and/or partial dierential equation @tf = LM(t)f; f(0; ) = f0; (1.5.12).
where the operator f 7! Lmf is linear for any given network state m 2 R, and the evolution of M(t) is
also given by some constraints, dierential or delay equation M(t) = M[f] = M[(f(s))js2[0;t]].

Annealed convergence and propagation of chaos in the general case

We now turn our attention to the case of non-translation invariant networks where the law of delays
and synaptic weights depend on the index of neuron i in population . In this case we will see that the propagation of chaos property remains valid as well as convergence to the mean-eld equations (2.3.1), no more for almost all realization of the disorder, but in average across all possible congurations. Denoting Ei the expectation over all possible distributions i , we have: Theorem 2.3.4 (Annealed convergence in the general case). We assume that (H1)-(H4) are valid and that network initial conditions are chaotic in M2(C), and that the interaction does not depend on the postsynaptic neuron state (i.e., b(w; x; y) = `(w; y)). Let us x i 2 N, then the law of process (Xi;N t ; 􀀀 t T) solution to the network equations (2.2.1) averaged over all the possibles realizations of the disorder, converge almost surely towards the process ( X i t ; 􀀀 t T) solution to the mean eld equations (2.3.1). This implies in particular the convergence in law of (Ei[Xi;N t ]; 􀀀 t T) towards ( X t ; 􀀀 t T) solution of the mean eld equations (2.3.1). Extensions to the spatially extended neural eld case are discussed in Appendix 2.7.

Application: dynamics of the ring-rate model with random connectivity

In the previous section, we derived limit equations for networks with random connectivities and synaptic weights. The motivation of these mathematical developments is to understand the role of specic connectivity and delays patterns arising in plausible neuronal networks. More precisely, it is known that anatomical properties of neuronal networks aect both connectivities and delays, and we will specically consider the two following facts:
• Neurons connect preferentially to those anatomically close.
• Delays are proportional to the distance between cells.
At the level of generality of the previous sections, we obtained very complex equations, from which it is very hard to uncover the role of random architectures. However, as we already showed in previous works [Touboul et al., 2011], a particularly suitable framework to solve these questions is provided by the classical ring-rate model. In that case, we showed in dierent contexts that the solution to the mean-eld equations is Gaussian, whose mean and standard deviation are solution of simpler dynamical system.

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Table of contents :

Acknowledgments
1 General Introduction 
1.1 Presentation
1.1.1 Part I: Neuronal networks
1.1.2 Part II: The role of homeoprotein diusion in morphogenesis
1.1.3 Part III: On a subcritical Keller-Segel equation
1.1.4 Plan of the Thesis
1.2 Mathematical toolbox
1.2.1 Mean-eld macroscopic equations: propagation of chaos property
1.2.2 Uniqueness of stationary solutions and nonlinear convergence
1.3 Biomathematical background
1.4 Main results
1.4.1 Randomly connected neuronal networks with delays
1.4.2 On a kinetic FitzHugh-Nagumo equation
1.4.3 Competition and boundary formation in heterogeneous media
1.4.4 On a subcritical Keller-Segel equation
1.5 Perspectives and open problems
1.5.1 A microscopic spiking neuronal network for the age-structured model
1.5.2 On the statistical description of neuron networks
I Neuronal networks 
2 Limits on randomly connected neuronal networks 
2.1 Introduction
2.2 Setting of the problem
2.3 Main results
2.3.1 Randomly connected neural mass models
2.3.2 Quenched convergence and propagation of chaos in the translation invariant case .
2.3.3 Annealed convergence and propagation of chaos in the general case
2.4 Application: dynamics of the ring-rate model with random connectivity
2.4.1 Reduction to distributed delays dierential equations
2.4.2 Small-world type model and correlated delays
2.5 Proofs
2.6 Discussion
2.6.1 Relationship with pathological rhythmic brain activity
2.6.2 Cluster size and synchronization in primary visual area
2.6.3 Macroscopic vs Mesoscopic models
2.6.4 Perspectives
2.7 Appendix A: Randomly connected neural elds
3 On a kinetic FitzHugh-Nagumo equation 
3.1 Introduction
3.1.1 Historical overview of macroscopic and kinetic models in neuroscience
3.1.2 Organization of the paper
3.2 Summary of the main results
3.2.1 Functional spaces and norms
3.2.2 Main results
3.2.3 Other notations and denitions.
3.3 Analysis of the nonlinear evolution equation
3.3.1 A priori bounds.
3.3.2 Entropy estimates and uniqueness of the solution
3.4 The linearized equation
3.4.1 Properties of A and B »
3.4.2 Spectral analysis on the linear operator in the disconnected case
3.5 Stability of the stationary solution in the small connectivity regime
3.5.1 Uniqueness of the stationary solution in the weak connectivity regime
3.5.2 Study of the Spectrum and Semigroup for the Linear Problem
3.5.3 Exponential stability of the non linear equation
3.6 Open problems beyond the weak coupling regime
3.7 Appendix A: Mean-Field limit for Fitzhugh-Nagumo neurons
3.8 Appendix B: Strong maximum principle for the linearized operator
II The role of homeoprotein diusion in morphogenesis 
4 Local HP diusion and neurodevelopment 
4.1 Introduction
4.2 Model
4.2.1 Theoretical Description
4.3 Results
4.3.1 Ambiguous boundary in the absence of non cell-autonomous processes
4.3.2 Unpredictable patterns in the absence of morphogen gradients
4.3.3 Precise patterning for competitive systems with spatial cues and HP diusion
4.3.4 Stability of the front
4.4 Discussion
4.5 Appendix A: Supplementary material
4.5.1 Mathematical Model
4.5.2 Stationary solutions in the cell autonomous case
4.5.3 Uniqueness of the front in the presence of HP diusion
4.5.4 Movies
5 Competition and boundary formation in heterogeneous media 
5.1 Introduction
5.1.1 Biological motivation
5.1.2 General model and main result
5.2 Analysis of the parabolic problem
5.2.1 Uniform bounds
5.2.2 Monotonicity in time
5.2.3 Monotonicity in space
5.2.4 Uniform positivity of the solutions
5.3 Asymptotic analysis as  » vanishes and front position
5.3.1 The limit as  » vanishes
5.3.2 WKB change of unknown
5.4 Characterization of the front and transition layer
5.5 Application
5.5.1 Model
III On a sub-critical model of chemotaxis 
6 On a subcritical Keller-Segel equation 
6.1 Introduction and main results
6.1.1 The subcritical Keller-Segel Equation
6.1.2 The particle system
6.1.3 Weak solution for the P.D.E
6.1.4 Notation and propagation of chaos
6.1.5 Main results
6.1.6 Comments
6.1.7 Plan of the paper
6.2 Preliminaries
6.3 Well-posedness for the system of particles
6.4 Convergence of the particle system
6.5 Well-posedness and propagation of chaos
6.6 Renormalization and entropic chaos
Bibliography 

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