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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 diffusion 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-field 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
