David Holcman

Pr Roland Netz

Pr David

Etienne Sandier

Didier Smets

Gershon Wolanski

Asymptotic of Poisson-Nernst-Planck equations and application to the voltage distribution in cellular micro-domains

Keywords: Action Potential AMPA: α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic Excitatory Post-Synaptic Potential Local Electro-Neutrality Principle Ordinary differential equation NMDA: N-methyl-D-aspartate PB: Poisson-Boltzmann PDE: Partial differential equation PNP: Poisson-Nernst-Planck PSD: Post-synaptic density Electro-diffusion, Cusp-shaped Funnel, Poisson-Nernst-Planck, Electroneutrality, Möbius conformal map, Asymptotics, Nonlinear, Partial differential equation, Biophysics. AMS subject classification. 35J66, 35B44, 35B25, 92C05, 92C37, 92C05 Electro-diffusion, Cusp-shaped Funnel, Poisson-Nernst-Planck, Electroneutrality, Schwarz-Christoffel conformal map, Asymptotics, Nonlinear, Partial differential equation AMS subject classification. 35J66, 35B44, 35B25, 92C05, 92C37, 92C05 Electro-diffusion, Poisson-Nernst-Planck equations, Non linear PDE, Conformal mapping, Non electroneutrality Electro-diffusion, Dendritic Spine, Deconvolution, Genetically encoded voltage sensor, Arclight, Concentration polarization Electro-diffusion, Dendritic Spine, Deconvolution, Genetically encoded voltage sensor, Arclight, Concentration polarization

J'ai pu, à son contact, avoir une approche concrète de nombreux domaines des mathématiques appliquées tels que les équations aux dérivées partielles, l'analyse complexe et l'analyse asymptotique. Il m'a également initié à la modélisation mathématique en biologie, et m'a fait découvrir un domaine d'une grande richesse scientifique. Enfin, il m'a permis de développer et d'approfondir une pensée pluridisciplinaire.

Je remercie également les membres de l'équipe que j'ai côtoyée tous les jours au laboratoire pour leurs discussions et leurs conseils. Entre autre, Juergen Reingruber pour son aide sur des problèmes ayant trait à de questions de physique, ainsi que pour ses remarques constructives. Je remercie également Ofir Shukron avec qui j'ai partagé les vicissitudes du quotidien de la recherche. Je remercie tout particulièrement Claire Guerrier et Pierre Parutto pour nos nombreux débats passionnés et extrêmement enrichissants, sans parler du soutien moral indispensable qu'ils m'ont apporté. Je tiens également à remercier Olivier Rodot pour sa bienveillance ainsi que ses précieux conseils. Je remercie Gladys Caré pour ses corrections et ses remarques sur le manuscrit.

Durant ma thèse j'ai eu la chance de rencontrer des chercheurs issus des sciences de la biologie qui m'ont aidè à mieux comprendre des questions importantes de biologie et qui m'ont permis d'aiguiser mon discernement. Je remercie particulièrement Nathalie Rouach, Rafael Yuste ainsi que Taekyung Kwon pour leurs conseils, leurs questions qui ont toujours débouchées sur des problèmes passionnants. Je remercie aussi les professeurs Gershon Wolanski et Roland Netz pour avoir accepté de relire ma thèse. Je remercie également Pierre Sens, Francesco Salvarani, Etienne Sandier, John Sherwood et Didier Smets pour m'avoir fait l'honneur de faire partie de mon jury.

Enfin, je remercie toute ma famille pour leur présence et leur soutien. En particulier, ma grand-mère Ketty Cartailler qui été exceptionnelle dans son écoute et sa bienveillance permanente. Je pense bien sûr à mes parents, dont l'absence est grande. Ils ont été les deux piliers qui m'ont toujours soutenu et m'ont permis de m'engager pleinement dans ma passion pour les sciences, je leur dois tellement.

Et surtout, merci à Marie Michon, qui m'a supporté tout au long de mon cursus universitaire, qui a relu dix fois mes projets de recherche et qui a été là à chaque étape difficile. Merci pour son soutien inestimable et sans faille durant ces dix dernières années.

Résumé

Dans cette thèse nous étudions comment la forme des micro et nano-domaines biologiques impacte l'électrodiffusion dans ces derniers. L'électrodiffusion est décrite à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck (PNP) dans des domaines non-triviaux ayant une forme cuspide ou elliptique. Notre objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles.

Dans la première partie nous calculons la distribution de charges électriques ainsi que le voltage à l'équilibre dans un électrolyte confiné dans un domaine borné, où l'on suppose un fort excès de charge positive. Le problème PNP se réduit à une équation de Liouville dont les conditions de bord de Neumann résultent de la condition de compatibilité de l'équation de Poisson. D'abord, nous étudions les solutions pour des domaines possédant une symétrie radiale. Nous estimons ensuite le premier temps de sortie dans une boule chargée et montrons qu'il dépend de la surface et non du volume de la boule. Nous étudions ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Nous construisons une solution asymptotique dans les cas 2D et 3D en faisant appel à la transformation conforme de Möbius. Celle-ci sépare la partie radiale et la partie angulaire du premier ordre du flot de la solution. Nous montrons que les potentiels en 2D et 3D sont donnés au premier ordre par la même expression. Nous étudions ensuite le cas où la condition de Neumann n'est pas uniforme sur la surface. Nous développons une méthode basée sur la théorie des couches limites afin de construire une approximation dans ce cas. Enfin, nous étudions une géométrie formée d'une ellipse. Nous utilisons une transformation de Schwartz-Christoffel à paramètre dans le but est de déterminer la forme optimale d'un nouveau domaine où le flot longitudinal du champ de vecteur est linéarisé, donnant lieu, au premier ordre, à une solution harmonique pour le potentiel. Pour ces deux types de géométrie nous mettons en évidence l'impact de la courbure du domaine sur les différences de potentiel dans le domaine.

La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. La géométrie de celles-ci est approximée par une tête sphérique connectée à la dendrite par un cou fin cylindrique. On modélise l'électrodiffusion dans l'épine lorsqu'un courant électrique est injecté au sommet et que la dendrite se comporte comme un réservoir électro-neutre et isopotentiel. Á partir de simulations numériques, nous mettons en évidence la polarisation de concentration dans l'épine ainsi que le rôle électrique de chacun des compartiments. Nous comparons ensuite notre modèle à des données de microscopie à encre fluorescente sensible au voltage. Nous développons une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir de la dynamique lente des encres fluorescentes. Enfin nous estimons la résistance du cou et montrons que celle-ci ne suit pas la loi d'Ohm.

Mot-clés: Poisson-Nernst-Planck; Electrodiffusion; Transformation conformes; Polarisation de concentration; Microdomaines; Liouville; Non-electroneutralité; Asymptotique; EDP non linéaires.

Abstract

In this PhD we study how electro-diffusion within biological micro and nano-domains is affected by their shapes. We use the Poisson-Nernst-Planck (PNP) partial differential equations to describe the electro-diffusion in non-trivial shapes such as domains with cusp and ellipse. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales.

In the first part we estimate for several geometries the steady-state distribution of ions and the electrical potential inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. The PNP equations then reduces into a Liouville equation whose Neumann boundary conditions result from the Poisson equation's compatibility condition. Then we start describing the solution for domains having a radial symmetry. To do that, we estimate the mean first passage time (MFPT) in a charged ball. We found it mostly depending on the surface and not on the volume. We further study a geometry composed of a ball with an attached cusp-shaped domain. We construct an asymptotic solution for the voltage in 2D and 3D using the Möbius conformal map. This transformation uncouples the angular and radial components of the leading order Liouville solution flow. We show that to leading order expressions for the voltage in 2D and 3D are identical. We extend our analysis of the cusp-shaped domain for non-uniform Neumann boundary conditions. We develop a method based on boundary-layer theory to approximate the solution. Finally, we study the voltage in an elliptical-shaped domain. We use a parameter dependent Schwartz-Christoffel conformal map to find the optimal shape that linearizes the longitudinal flow of the Liouville solution. The voltage is then computed solving a Laplace equation. For both cusp-shaped and elliptical domains, we find that the curvature affects the electrical landscape.

In the second part of this PhD, we model the electrical compartmentalization in dendritic spines. We first approximate the spine as a spherical head connected the dendrite via a thin cylindrical neck. We further model the electrodiffusion in spine when a current composed of positive charges is injected at the spine top while its base is connected to an isopotential reservoir. We investigate numerically the effect of changing the geometry using the electro-diffusion model in non-cylindrical geometry. We find concentration polarization in the spine and identify electrical properties of each compartment. In a second time, we compare our model to experimental data of voltage-sensitive dye fluorescence from microscopy imaging. We develop a deconvolution method to recover the fast voltage dynamic from the slow dye dynamics. We estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius when the injected current is large.

Keywords: Poisson-Nernst-Planck; Electrodiffusion; Conformal mapping; Concentration polarization; Microdomains; Liouville; Non-electro-neutrality; Asymptotics; Nonlinear PDEs.

List of Symbols and Notations

Contents Introduction

In this PhD thesis I develop mathematical models to study how the cell geometry influences the electrical properties of neuronal microdomains such as synapses. Indeed, it is unclear what defines the electrical properties of biological micro-and nano-domains. To address this question I use the electro-diffusion theory, based on the classical coarse-grained Poisson-Nernst-Planck equations, to model the dynamics of ions in solutions and to compute the distribution of the voltage [START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF].

There are several difficulties in this modeling: first if we consider the scale of individual ion, we are left with studying many interacting particles, which is not possible, even numerically, because we are interested in time scale of milli to hundreds of milliseconds. Thus, a coarse-grained approach is well suited. Another difficulty is the variety of small and large scales that create narrow passages for ions and can lead to very high local electrical fields. For example, in the large scale limit of simple geometries such as cables, biological cells have been approximated as electrical simplified devices where resistances and capacitances are organized in networks [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF].

In the continuum limits, these organization leads to the classical cable theory, which is a one dimensional diffusion type equation for the voltage, developed by Lord Kelvin in the ninteenth century, to explain the flow of electricity in submarine cables. This equation is now used to estimate the propagation of an electrical signal in nerve fibers, as demonstrated by the work of Cole, Goldman, Hodgkin and Katz [START_REF] Bart | on the 150 th Anniversary of the Atlantic Cable[END_REF][START_REF] Goldman | Potential, impedance, and rectification in membranes[END_REF][START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF][START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF]. The cable equation, which considers the geometry as a 1D wire, gave good predictions for propagation in copper wires, squid giant axons or ionic channels, where the changes in concentration of ions are negligible. However, in small compartments that cannot be reduced to a one dimensional cable [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF][START_REF] Yuste | Dendritic Spines[END_REF], the assumptions used for the derivation of the cable equations are not satisfied, in particular, the geometry is not cylindrical and the ionic concentration is not necessarily constant. In that context, I develop modeling, asymptotic analysis, numerical simulations and analysis of electrophysiological time series to study the role of small nano to micro structures on the physiological function.

The conclusion of this thesis is that the great variety of synaptic shapes can modulate the electrical conduction of the synaptic current. This conversion of current into voltage depends crucially on the geometry of the synapses. Thus, we propose that these geometrical structures could participate in encoding local memory in the neuronal cells. 3D reconstruction from electron microscopy adapted from [START_REF] Yuste | Dendritic Spines[END_REF]. The red structure is the Post-Synaptique Density (PSD) [START_REF] Yuste | Dendritic Spines[END_REF].

A biological background: From neurons to dendritic spines

Neurons are biological cells specialized in processing and transmitting cellular signals. Their role as independent cell, organized in network, has first been recognized in 1888 by the anatomist Santiago Ramón y Cajal [START_REF] Cajal | Estructura de los centros nerviosos de las aves[END_REF]. Neurons have various sizes and shapes depending on their location and function. Their typical anatomy consists in a soma, an axon and several dendrites (Fig. 1A). The soma contains the cell nucleus and organelles [START_REF] Kandel | Principles of Neural Science[END_REF], while axons and dendrites are both soma projections. Contrary to the axon, the dendrites collect neuronal information in the form of electrical impulses such that they go from the dendrites to the soma. When the sum of electrical impulse reaches a certain threshold, the soma fires a sharp membrane potential perturbation called an Action Potential (AP). An AP initiated at the soma will then propagate in the axon. Two neurons exchange chemical molecules at a specific region called a chemical synapse. It is composed of three compartments: the presynaptic and the postsynaptic terminals and the narrow 20nm width region in between which is referred as the synaptic cleft (see Fig. 1B). Synapses can be either inhibitory or excitatory: the first one makes the soma less likely to fire an AP, while the second one enhance APs. From now on I will focus on the postsynaptic compartment called a dendritic spine.

Dendritic spines (hereafter referred to as spines) have been described by Ramón y Cajal, yet their role remains unclear until today. Dendrite protrusions are found on the majority of excitatory neuronal connections. Interestingly, there is a clear discrepancy between excitatory and inhibitory synapses: the firsts are mostly made on spines, while the seconds favor a direct connection to the dendritic shaft [START_REF] Colonnier | Synaptic patterns on different cell types in the different laminae of the cat visual cortex. An electron microscope study[END_REF]. The spines mushroom shape is composed in average of a bulby head and a thin neck that connects the head to the dendritic shaft (see Fig. 1A-C). Still, there is a large variability in spines shape as show in Fig. 1D. The head diameter is about 1µm, the neck length is in the range of 0.2 to 1.5µm while its radius is <100nm [START_REF] Bartol | Nanoconnectomic upper bound on the variability of synaptic plasticity[END_REF][START_REF] Yuste | Dendritic Spines[END_REF]. The characteristic morphology of the spines has led to the idea that they are biochemical compartments, in particular for calcium ions [START_REF] Yuste | Dendritic spines as basic functional units ofneuronal integration[END_REF]. Yet, the spine presence is not mandatory for such effect since calcium compartmentalization can be found in neurons devoid of spines [START_REF] Goldberg | Ca2+ imaging of mouse neocortical interneurone dendrites: Contribution of Ca2-permeable AMPA and NMDA receptors to subthreshold Ca2 dynamics[END_REF]. electrical compartmentalization in spines was suggested in [START_REF] Tsay | On the electrical function of spines[END_REF], characterized by the spine head and the dendritic shaft not being isopotential [START_REF] Yuste | Electrical Compartmentalization in Dendritic Spines[END_REF] following a synaptic event when the voltage increase in the spine head. It is called an excitatory post-synaptic potential (EPSP). How this EPSP is then filtered by the spine neck, which is modeled electrically as a resistance remains controversial as we discuss below. Electrical compartmentalization implies a large spine neck resistance, an idea that still divides the community [START_REF] Chang | Cortical neurons with particular reference to the apical dendrite[END_REF][START_REF] Coss | The function of dendritic spines: A review of theoretical issues[END_REF][START_REF] Jack | Electric current flow in excitable cells[END_REF][START_REF] Koch | The function of dendritic spinesDevices subserving biochemical rather than electrical compartmentalization[END_REF][START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF][START_REF] Svoboda | Direct measurement of coupling between dendritic spines and shafts[END_REF]. There is no consensus as illustrated by the large range of resistances reported in the literature. Indeed, the conclusions of different optical studies are mixed: some suggest an attenuation of EPSPs [START_REF] Araya | The spine neck filters membrane potentials[END_REF][START_REF] Araya | Activity-dependent dendritic spine neck changes are correlated with synaptic strength[END_REF][START_REF] Harnett | Synaptic amplification by dendritic spines enhances input cooperativity[END_REF], others suggest instead that spines are isopotential with the dendrites and do not alter EPSPs [START_REF] Svoboda | Direct measurement of coupling between dendritic spines and shafts[END_REF][START_REF] Tonnesen | Spine neck plasticity regulates compartmentalization of synapses[END_REF]. Diffusional studies of small fluorescent molecules through spines have predicted neck resistances of 4-50M Ω [START_REF] Svoboda | Direct measurement of coupling between dendritic spines and shafts[END_REF] or 56M Ω [START_REF] Tonnesen | Spine neck plasticity regulates compartmentalization of synapses[END_REF]. Other studies suggest a lack of voltage filtering of dendritic spine [START_REF] Acker | EPSPs Measured in Proximal Dendritic Spines of Cortical Pyramidal Neurons[END_REF][START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF] with a resistance of 30M Ω. I will present evidences in this thesis of the decay of the EPSPs from the spine head to the parent dendrites, and that the resistance of the spine can be derived from the law of electro-diffusion in the spine geometry as that the classical cable equation is not sufficient.

Part I: Electrostatics of non-neutral biological micro-domains

Dendritic spines [START_REF] Yuste | Dendritic Spines[END_REF] and astrocytic processes [START_REF] Rusakov | Disentangling calcium-driven astrocyte physiology[END_REF] regulate synaptic transmission. Filopodia are finger-like structures at the nanometer-scale that contribute among other to the connectome development [START_REF] Cooper | The Cell: A Molecular Approach[END_REF][START_REF] Kaiser | Mechanisms of Connectome Development[END_REF][START_REF] Mattila | Filopodia: molecular architecture and cellular functions[END_REF] and the immune response [START_REF] Kress | Filopodia act as phagocytic tentacles and pull with discrete steps and a load-dependent velocity[END_REF].

Although, those structures are engaged in various cognitive tasks such as learning and memory [START_REF] Henneberger | Longterm potentiation depends on release of D-serine from astrocytes[END_REF][START_REF] Sanders | Elimination of dendritic spines with long-term memory is specific to active circuits[END_REF][START_REF] Segal | Dendritic spines and long-term plasticity[END_REF][START_REF] Sibille | The Neuroglial Potassium Cycle during Neurotransmission: Role of Kir4.1 Channels[END_REF], yet, little is known about them. Indeed, if great geometrical details can be obtained from electron microscopy techniques [START_REF] Yuste | Dendritic Spines[END_REF], accessing directly quantities such as the voltage or ionic concentrations in small domains remains, in general, impossible [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Jayant | Targeted intracellular voltage recordings from dendritic spines using quantum-dot-coated nanopipettes[END_REF][START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF]. Consequently, we use a modelin gapproach to investigate these micro-domains, that we shall model as ball connected to a cylinder, elongated ellipses or cusp-shaped domains [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF]. The physical model I used to describe biological electrolyte solutions is the steady-state electro-diffusion in a non-neutral electrolyte.

The Poisson-Nernst-Planck equations

The motion of ions in solutions can driven by two physical forces: diffusion and an electrical force. For such ion, the Smoluchowski limit of the Langevin equation is written as

γ ẋ = ze E(x, t) + 2γk B T m ẇ, (1)

Introduction

where γ is the friction coefficient per unit mass, k B T the thermal energy, m is the mass, zeE(x, t) is the electrostatic force and ẇ represents a Gaussian white noise. The Probability Distribution Function (PDF) p i (x, t) os an ion i whose motion is described by the Smoluchowski equation ( 1) is solution of the Fokker-Planck equation [START_REF] Schuss | Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry, and Biology[END_REF] ∂p i (x, t) ∂t

= D i ∇ ∇p i (x, t) - z i e k B T p i (x, t)E(x, t) , (2)
where D i is the diffusion constant. In a dilute electrolyte, we assume that charges do not interact with each other. Consequently, the density of charges ρ i (x, t) of the i th specie is N i p i (x, t), where N i is the number of charges i [START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF]. The Fokker-Planck equation ( 2) for the density of particle becomes:

∂ρ i (x, t) ∂t = D i ∇ ∇ρ i (x, t) - z i e k B T ρ i (x, t)E(x, t) . (3)
This equation ( 3) represent the Nernst-Planck (NP) equations, that were described at the end of the nineteenth century by Walther Nernst and Max Planck [START_REF] Nernst | Zur Kinetik der in Lsung befindlichen Körper[END_REF][START_REF] Nernst | Die Electromotorische Wirksamkeit der lonen[END_REF][START_REF] Planck | Ueber die Erregung von Electricität und wärme in electrolyten[END_REF]. One can notice that for uncharged particles, the equation mentioned above simply reduces to the classical diffusion equation [START_REF] Schuss | Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry, and Biology[END_REF][START_REF] Evans | Partial Differential Equations[END_REF]. When there are no external field, the electrical field E(x, t) is given by

E(x, t) = -∇V (x, t), (4)
where V (x, t) is the electrical potential. Then, the Maxwell-Gauss formula [START_REF] Feynman | Mainly Electromagnetism and Matter, Basic Books[END_REF]is the classical Poisson equation that connect the charge density to the voltage ∆V (x, t) = e ε r ε 0 n i=1 z i ρ i (x, t), [START_REF] Andelman | Electrostatic Properties of Membranes: The Poisson-Boltzmann Theory[END_REF] where ε r ε 0 is the electrical permitivity. It was noticed that one could also add fixed charges to eq. ( 5) [START_REF] Nadler | Ionic diffusion through confined geometries: from Langevin equations to partial differential equations[END_REF][START_REF] Zheng | Poisson-Boltzmann-NernstPlanck model[END_REF]. The equations (3) and ( 5) form the Poisson-Nernst-Planck (PNP) equations. The PNP equations have been applied successfully in the study of ionic channels [START_REF] Barcilon | Ion Flow through Narrow Membrane Channels: Part II[END_REF][START_REF] Barcilon | Ion Flow Through Narrow Membrane Channels: Part I[END_REF][START_REF] Goldman | Potential, impedance, and rectification in membranes[END_REF][START_REF] Nadler | Ionic diffusion through confined geometries: from Langevin equations to partial differential equations[END_REF][START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF][START_REF] Singer | A Poisson-Nernst-Planck model for biological ion channelsan asymptotic analysis in a three-dimensional narrow funnel[END_REF][START_REF] Taflia | Diffusion of interacting particles in confined domains and applications to biology[END_REF]], yet relying on geometrical simplifications. Indeed, solving PNP is a burdensome task which is often overpassed imposing Local Electro-Neutrality Principle (LENP) [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF][START_REF] Rubinstein | Electro-Diffusion of ions[END_REF].

Local electro-neutrality principle: The LENP consists in replacing the Poisson equation [START_REF] Andelman | Electrostatic Properties of Membranes: The Poisson-Boltzmann Theory[END_REF] with an algebraic relation between the ionic densities ρ i :

i z i ρ i (x, t) = 0, for x ∈ Ω. (6)
In other words, a given region of space must always contain the same number of charges of either sign, thus ensuring its macroscopic neutrality. Although that cannot be deduced from Maxwell's laws, the LENP is routinely used in electro-chemistry 0.1. Part I: Electrostatics of non-neutral biological micro-domains when the characteristic size L c of the domain of interested is much larger (∼ tens of microns) than the Debye length [START_REF] Rubinstein | Voltage against current curves of cation exchange membranes[END_REF]:

λ D = εε 0 k B T eF n i=1 z 2 i c 0 i , (7)
where c 0 i is the bulk concentration [START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF]. Nonetheless, such simplification prevents any effects of the boundary on the voltage to be accounted for. Still, matched asymptotic expansion based methods exist to bypass such problems [START_REF] Bazant | Current-voltage relations for electrochemical then films[END_REF][START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF][START_REF] Olesen | Strongly nonlinear dynamics of electrolytes in large ac voltages[END_REF]]. Yet, these approaches assume λ D L c and an electro-neutral bulk. So, applying the LENP to geometries having characteristic length of dozens of nanometers up to few micrometers may lead to misleading interpretations [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF][START_REF] Rubinstein | Electro-Diffusion of ions[END_REF].

Non electro-neutrality: Modeling an unbalance between the number of positive and the negatives charges cannot be done with the LENP, since equation ( 6) is no longer true. For example, studying the electrical potential near a charged surface that results from a particular organization of the ions into non-electro-neutral layers cannot be done with the LENP [START_REF] Andelman | Electrostatic Properties of Membranes: The Poisson-Boltzmann Theory[END_REF]. Nevertheless, non electro-neutrality is particularly relevant in life sciences where the cellular membrane is rich in ion channels that regulate the transport of ions and molecules between the interior and the exterior of the cell. Indeed, most of the channels, such as AMPA and NMDA [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF][START_REF] Kandel | Principles of Neural Science[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF][START_REF] Yuste | Dendritic Spines[END_REF], are ion selective. After activation triggered by a specific chemical cascade, they induce a current composed exclusively of Na + and Ca 2+ cations leading to a charge unbalance. In this context, how a small compartment geometry impacts the distribution of the voltage in the presence of an excess of positive charges is still unclear. Additionally, the literature shows a strong discrepancy between the concentration of mobile cations (K + (155mM) and Na + (12mM)) and anions (Cl -(4.6mM)) [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF]. This large excess of positive charge is probably balanced by negatively charged molecules of various size, but their diffusion coefficients are certainly smaller than those of the mobile ions and this difference could even be amplified in tortuous domains [START_REF] Rusakov | Geometric and viscous components of the tortuosity of the extracellular space in the brain[END_REF][START_REF] Syková | Diffusion in Brain Extracellular Space[END_REF].

Stationary PNP for non-neutral electrolytes

Background: I investigated the consequences of an excess of positive charge on the voltage and on the charges distribution in a dimensionless confined domain Ω. For this purpose, I introduced a (1, 1) electrolyte (z p = z m = 1) where the number of positive charges N p is in large excess compare to the negative ones N m . I defined the ratio ζ = N m /N p 1 such that, in this limit, the leading order term U 0 (x) of the adimensionalized potential U (x) is solution of the Liouville equation

-∆U 0 (x) = µ(N p )e -U 0 (x) in Ω (8) ∂U 0 (x) ∂n = - l B N p L c |∂Ω| on ∂Ω,

Introduction

where L c is the domain characteristic length and

µ(N p ) = l B N p L c Ω e -U 0 (s) ds > 0. (9)
The Liouville equation ( 8) has been widely used in physics and biology to describe the voltage in electrolytes (µ > 0) [START_REF] Cervera | Ionic conduction, rectification, and selectivity in single conical nanopores[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Rubinstein | Electro-Diffusion of ions[END_REF][START_REF] Wei | Current Rectification at Quartz Nanopipet Electrodes[END_REF], in the representation of gravitationaly induced self-aggregation (µ < 0) [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particle, III[END_REF][START_REF] Chipot | On the solutions of Liouville systems[END_REF][START_REF] Wolansky | On steady distributions of self-attracting clusters under friction and fluctuations[END_REF], and much more [START_REF] Caglioti | A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description[END_REF][START_REF] Chandrasekhar | An Introduction to the Study of Stellar Structure[END_REF][START_REF] Gelfand | Some Problems in the Theory of Quasi-Linear Equations[END_REF]. The well-possessedness of solutions for [START_REF] Araya | Sodium channels amplify spine potentials[END_REF] has been studied extensively for bounded domains [START_REF] Bandle | Asymptotic behaviour of large solutions of quasilinear elliptic problems[END_REF][START_REF] Gogny | Sur les états déquilibre pour les densités électroniques dans les plasmas[END_REF][START_REF] Keller | On solutions of ∆u = f (u)[END_REF][START_REF] Krzywicki | Some results concerning the Poisson-Boltzmann equation[END_REF][START_REF] Krzywicki | A nonstationary problem in the theory of electrolytes[END_REF][START_REF] Krzywicki | A note on the PoissonBoltzmann equation[END_REF] as well as its monotonicity [START_REF] Chanillo | Conformally invariant systems of nonlinear pde of Liouville type[END_REF][START_REF] Gidas | Symmetry and related properties via the maximum principle[END_REF][START_REF] Lions | Two geometric properties of solutions of semilinear problems[END_REF]. Nevertheless, for some cases an analytical solution can be explicitly computed. Due to the radial symmetry, the problem [START_REF] Araya | Sodium channels amplify spine potentials[END_REF] reduces to an ODE. For d = {1 , 2}, the problem ( 8) can be integrated, and in addition, it exists µ * > 0 such that for µ ∈ [0 , µ * ] the solution of ( 8) is not singular for r ∈ [START_REF][END_REF][START_REF] Ablowitz | Method for Solving the Sine-Gordon Equation[END_REF]. It follows

         d = 1 : U 1D 0 (r) = -ln cos 2 µ 2 r , µ * = π 2 2 d = 2 : U 2D 0 (r) = ln 1 - µ 8 r 2 2 , µ * = 8. ( 10
)
Goal: My goal is to study the solution of equation ( 8) in a 3D ball, in particular when the number of charge N p is large.

Results: (Chap. 1) Albeit µ has to be bounded, by expressing µ(N p ) as a function of N p I obtain that: as N p → ∞, it follows lim λ→∞ µ(λ) = µ * from below.

Consequently, for d = 1, 2, the condition 0 < µ(N p ) < µ * holds for N p ∈ R + , and the Liouville problem (8) admits a non singular solution for r ∈ [0, 1] for which a log-singularity develops at the boundary r = 1 when N p → ∞.

For d = 3, the solution of the initial value problem (8) cannot be directly computed. Contrary to d = 1 and 2, the critical value µ * can only be estimated numerically and is approximatively ≈ 10.7. Nonetheless, it is possible to estimate the solution in the two following regimes:

U 3D 0 (r) =        - l B N p L c r 2 8π + O(N 2 p ) for N p 1 2 ln(1 -r 2 + O N -1 p ) for N p 1. ( 11
)
The potential U 3D 0 affects the distribution of charged particles inside the domain. To investigate this distribution, I computed the time to escape this domain through a small hole. Such analysis is motivated from biology, where the flux of ion through a small window will characterize the electrical current. This problem falls into the category of the narrow escape problems that I introduce below.

The narrow escape problem

Background: The narrow escape problem in diffusion theory aims to compute the narrow escape time (NET) of a Brownian particle to a small absorbing window 0.1. Part I: Electrostatics of non-neutral biological micro-domains while the rest of the domain is characterized by the reflecting boundary of a bounded domain (Fig. 2). It was first introduced in the context of the theory of sound [START_REF] Helmholtz Von | Theorie der luftschwingungen in röhren mit offenen enden[END_REF][START_REF] Rayleigh | The Theory of Sound[END_REF], and a lot of applications were found for biology such as modeling the early steps of viral infection [START_REF] Holcman | Time scale of diffusion in molecular and cellular biology[END_REF][START_REF] Lagache | Quantitative analysis of virus and plasmid trafficking in cells[END_REF], or for stochastic models of chemical reactions [START_REF] Dao Duc | Threshold activation for stochastic chemical reactions in microdomains[END_REF][START_REF] Holcman | Stochastic chemical reactions in microdomains[END_REF], and many other models [START_REF] Gehlen | Nuclear geometry and rapid mitosis ensure asymmetric episome segregation in yeast[END_REF][START_REF] Greber | A superhighway to virus infection[END_REF][START_REF] Holcman | The narrow escape problem[END_REF][START_REF] Holcman | Control of flux by narrow passages and hidden targets in cellular biology[END_REF]. My goal is to develop asymptotic methods to compute the NET in the limit of the small size of the window, for the various geometries of cellular structures. For instance let's consider a Brownian particle whose trajectory is characterized by the stochastic equation

Brownian trajectory

Ẋ = √ 2D ẇ, ( 12
)
where D is the diffusion coefficient and ẇ is the white noise. The mean first passage time (MFPT) τ (x) for a Brownian particle, starting at position x, to reach at small absorbing domain ∂Ω a located on the boundary ∂Ω (Fig. 2) is the solution of the mixed boundary value problem for the Pontryagin-Andronov-Vitt (PAV) equation [START_REF] Holcman | The narrow escape problem[END_REF] D∆ τ (x) = -1 for x ∈ Ω (13) ∂ τ ∂n (x) = 0 for x ∈ ∂Ω \ ∂Ω a τ (x) = 0 for x ∈ ∂Ω a .

The difficulty here is to construct an asymptotic solution τ (x), when the ratio ε = |∂Ωa| |∂Ω| tends to zero. The NET diverges as the absorbing part of the boundary shrinks, thus rendering the computation a singular perturbation problem. The computation is related to the calculation of the principal eigenvalue of the mixed Dirichlet-Neumann problem for the PAV equation [START_REF] Holcman | The narrow escape problem[END_REF][START_REF] Holcman | Control of flux by narrow passages and hidden targets in cellular biology[END_REF][START_REF] Schuss | Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry, and Biology[END_REF]. To solve [START_REF] Bandle | Asymptotic behaviour of large solutions of quasilinear elliptic problems[END_REF], one approach consists in using the Neumann function N (x, ξ) [START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF][START_REF] Schuss | Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry, and Biology[END_REF]. Indeed, the integration of the Green's identity of N (x, ξ) and τ (x) leads to an integral form of the NET τ (ξ). Then, once averaged over the domain boundary, the NET is the solution of the Helmholtz integral equation:

Introduction

For ξ outside of the boundary layer located near the absorbing hole ∂Ω a , this representation allows computing the leading order approximation of the MFPT. For example, the MFPT in a three-dimensional domain Ω, when the target is a circular absorbing window ∂Ω a of radius a centered at 0 on the boundary ∂Ω (Fig. 2), is given by [START_REF] Holcman | The narrow escape problem[END_REF]

τ (x) = |Ω| 4aD 1 - L(0) + N (0) 2π a ln a + O(a) , (15)
where L(0) and N (0) are the principal curvatures of the surface at the center of the absorbing boundary ∂Ω a . The MFPT of a Brownian particle moving on the surface of a sphere of radius R is [START_REF] Cheviakov | Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps[END_REF][START_REF] Holcman | Stochatic Narrow escape. Theory and Applications[END_REF][START_REF] Singer | Narrow escape, part III: Non-Smooth Domains and Riemann Surfaces[END_REF]

τ (x) = 2R 2 D ln 1 δ + O(1) , (16)
where a and δ are linked such that a = R sin (δ/2). This method has also been extended to the NET on Riemannian manifolds [START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF][START_REF] Singer | Narrow escape, part II: The circular disk[END_REF][START_REF] Singer | Narrow escape, part I[END_REF][START_REF] Singer | Narrow escape, part III: Nonsmooth domains and riemann surfaces[END_REF].

In [START_REF] Ward | Summing logarithmic expansions for singularly perturbed eigenvalue problems[END_REF][START_REF] Ward | Strong localized perturbations of eigenvalue problems[END_REF] an asymptotic framework, based on combining matched asymptotic expansions and potential theory, was developed to analyze PDE problems with strong local changes in the boundary conditions. In this approach a boundary layer solution is constructed near the absorbing boundary of small size ε [START_REF] Ward | Summing logarithmic expansions for singularly perturbed eigenvalue problems[END_REF][START_REF] Ward | Strong localized perturbations of eigenvalue problems[END_REF]. This method was developed mainly in [START_REF] Cheviakov | An asymptotic analysis of the mean first passage time for narrow escape problems: Part II: The sphere[END_REF][START_REF] Kolokolnikov | Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps[END_REF][START_REF] Ward | Summing logarithmic expansions for singularly perturbed eigenvalue problems[END_REF][START_REF] Ward | Strong localized perturbations of eigenvalue problems[END_REF][START_REF] Ward | The onset of thermal runaway in partially insulated or cooled reactors[END_REF].

Narrow escape theory in a highly charged ball

Goal: The PNP equations serve here to compute the electric field for a single specie, where I do not assume local electro-neutrality. To derive a relationship between the outward current and the voltage, I compute the escape rate of moving ions from the steady-state density in a ball to a small absorbing window in its boundary. I studied how the potential U 3D 0 , found in the charged ball [START_REF] Ashrafuzzaman | Membrane Biophysics, Biological and Medical Physics[END_REF], affects the MFPT in the large charge regime N p 1. The MFPT τ (x) is the solution of the PAV boundary value problem [START_REF] Schuss | Theory and Applications of Stochastic Processes[END_REF]

D ∆ τ (x) -∇ τ (x) • ∇U 3D 0 (x) = -1 for x ∈ Ω (17) ∂ τ (x) ∂n + τ (x) ∂U 3D 0 (x) ∂n = 0 for x ∈ ∂Ω r τ (x) = 0 for x ∈ ∂Ω a ,
where ∂Ω a represents the small absorbing window and ∂Ω r = ∂Ω\∂Ω a (see Fig. 2).

Results: (Chap. 1) I found that the current of particles is controlled by the small absorbing window in the boundary of the ball, as predicted by the Narrow Escape Theory [START_REF] Holcman | Time scale of diffusion in molecular and cellular biology[END_REF][START_REF] Schuss | The narrow escape problem for diffusion in cellular microdomains[END_REF], while the voltage is independently regulated by the coupled PNP equations.

For N p 1 equation [START_REF] Ashrafuzzaman | Membrane Biophysics, Biological and Medical Physics[END_REF] gives that the electric field near the boundary is large:

|∇U 3D 0 (x)| 1.
In this limit, the study of the PAV problem [START_REF] Bart | on the 150 th Anniversary of the Atlantic Cable[END_REF] reveals that the MFPT from x ∈ Ω to ∂Ω a is the sum of the MFPT from x to ∂Ω and of the MFPT from ∂Ω to ∂Ω a on the surface ∂Ω. This approximation means that to reach ∂Ω a in 0.1. Part I: Electrostatics of non-neutral biological micro-domains a highly charged ball, a charge is first transported by the field to the reflecting part ∂Ω r of the sphere with overwhelming probability and it then finds ∂Ω a by diffusing on the surface. A second consequence of the charge distribution is the control of the flux of particles through a small hole at the surface (see Fig. 2). Using the MFPT for N independent charges, the electrical current J through the small window is given by

J = ze τ N = DN e 2R 2 ln R 2a + O(1)
for a R. ( 18
)
Once a current flows into a dielectric ball such as a spine head, the excess of charges (N e) is first pushed toward the boundary before moving by Brownian motion to ∂Ω a (small disk of size a). This result shows that the current J is governed by the geometry [START_REF] Bartol | Nanoconnectomic upper bound on the variability of synaptic plasticity[END_REF], and that a key parameter is the radius a.

0.1.5 The PNP equations in a domain with a cusp-shaped funnel Background: The local curvature is a key geometrical element for controlling charge distribution in various media, such as in the air as illustrated by the lightning rod [START_REF] Courant | Methods of Mathematical Physics[END_REF] or the corona discharge [START_REF] Gallot-Lavallée | Dielectric Materials and Electrostatics[END_REF]. However, these effects for electrolyte are not very well understood and exhibit very non-linear phenomena like current rectification [START_REF] Lan | Voltage-Rectified Current and Fluid Flow in Conical Nanopores Acc[END_REF][START_REF] Perry | Characterization of Nanopipettes[END_REF] or shock wavees [START_REF] Mani | Deionization shocks in microstructures[END_REF][START_REF] Nam | Experimental Verification of Overlimiting Current by Surface Conduction and Electro-Osmotic Flow in Microchannels[END_REF]. I investigate the curvature effect, extending the precedent results for radial operators to a geometry composed of a convex domain composed of a ball with an attached cusp-shaped funnel on its surface as shown in Fig. 3A. Such geometry was studied in [START_REF] Holcman | Brownian motion in dire straits[END_REF] for the narrow escape theory of a Brownian particle. However, the problem here is radically different since the PDE is a non-linear equation and boundary conditions are not homogeneous. In fact, those two constraints call for two different methods in order to construct an asymptotic solution of the non-dimensionalized potential u(x), which is solution of

-∆u(x) = e -u(x) in Ω (19) ∂u(x) ∂n = -σ < 0 on ∂Ω,
where σ represents the normalized electric field at the boundary ∂Ω.

Both cusp-shaped geometries in R 2 and R 3 can be reduced to a dimensionless planar cusp-shaped funnel Ω formed by two bounding circles A and B of dimensionless radii 1 (see Fig. 3B). The goal is to construct an asymptotic solution in the limit of the small ε that represents the opening of the funnel. As in [START_REF] Holcman | Brownian motion in dire straits[END_REF], I first map the domain Ω conformally with a Möbius transformation w as illustrated

Introduction

details in [START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF][START_REF] Holcman | Brownian motion in dire straits[END_REF]) and represents the solution inside the red disk in Fig. 3C, away from the cusp. However, the boundary condition at

θ = π is - σ √ ε 2
, which I consider large in the limit σ 1. Goal: My goal is to show that an asymptotic approximation to the Liouville problem [START_REF] Bazant | Conformal mapping of some non-harmonic functions, in transport theory[END_REF] can be constructed in the limits mentioned above.

B

Results: (Chap. 2-3) Unfortunately, the classical matched-asymptotic techniques [START_REF] Bender | Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory[END_REF][START_REF] Schuss | Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry, and Biology[END_REF] fails because of the exponential term in the Liouville equation. Nonetheless, I first construct the outer-solution in the form of a serie in powers of ε, which is a valid approximation away from the boundary θ = c √ ε. Then, at θ = π since the outer solution cannot satisfy all boundary conditions, I use a boundarylayer correction. An approximation of the solution can be obtained by freezing the power-law term and neglecting the first order derivatives, for which the equation reduces to a 1D parameter-dependant Liouville equation. Then I construct a uniform approximation v unif (θ) putting the outer and the boundary-layer solution together. As the result, in the small ε limit, the potential in the mapped domain Ω w , for the planar cusp Ω ⊂ R 2 is to the leading order

v unif (θ) = ln 1 √ ε -ln 1 π 2 ln π 6 |∂Ω ε | 2 3 ε 3/2 (20) -2θ 1 π 2 ln π 6 |∂Ω ε | 2 3 ε 3/2 arctan θ √ ε + ln cos 2 1 -4 π 1 σ √ ε 2 θ , 0.1. Part I: Electrostatics of non-neutral biological micro-domains
where |∂Ω ε | is the length of the cusp opening. For a three dimensional funnel Ω ⊂ R 3 , the solution reads as

v unif (θ) = 1 π ln 8ε π 4 |∂Ω ε | (1) -β (θ -sin(θ)) (21)
+ ln cos 2 1 2 - 2 π 1 σ √ ε θ + β,
where β is a constant that depend on the solution in the bulk (red circle Fig. 3C). I then use the uniform solution v unif (θ) to expose the consequences of a cusp on the potential. Indeed, the narrow region S in the cusp has a potential that is lower than the potenial at the north pole N . It contrasts with the potential in the disk or in the ball for which the curvature is constant and the boundary is isopotential. Interestingly, numerical simulations show that the difference of potential between the two poles N and S (see Fig. 3A) is maximal for a certain σ * .

PNP equations in an charged cusp-shaped funnel

Goal: The goal consists now to deal with non-zero Neumann boundary conditions at the cusp. However, non-homogeneous Neumann boundary conditions under the Möbius transformation lead to unwieldy terms. Indeed, conformal maps are a very efficient approach once applied to harmonic functions [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF], or more generally to conformally invariant systems of equation [START_REF] Bazant | Conformal mapping of some non-harmonic functions, in transport theory[END_REF], but unfortunately the Liouville problem does not fall into this category. Nevertheless, in the previous section the cumbersome terms that resulted from the Laplace operator under the Möbius map were dealt with by reducing the geometry into a 1D arc. Sadly, the non-zero Neumann boundary conditions on the cusp prevent such simplification. One is now forced to handle a two dimensional problem. Nonetheless, insight can be gained observing how the field lines transform in the domain Ω w , as shown in Fig. 4A-B.

Results: (Chap. 2-3-5) I show that in the limits of σ 1 and ε → 0 these lines are parallel to the radius vector, except in a small region near the funnel. I thus distinguish two regions A and B as illustrated in Fig. 4C. In A, the angular derivatives are negligible relative to the radial ones. Hence, after a regular expansion of the solution, it follows that the θ derivatives can be neglected relative to the r derivatives. The equation is then solved along the rays θ = const for r ∈ [1 -√ ε, 1] simply by solving a 1D Liouville problem.

In region B, the boundary conditions show that the θ-derivatives are no longer small, moreover they dominate the radial ones, so a boundary layer correction is needed. In the small ε limit, the solution in region B can be approximated once again by a the Liouville problem in 1D, where the space variable is replaced by the angular coordinate θ. As this solution is defined up to a constant, I use it to match the solution in B to the solution in the region A. I consequently obtain the uniform The field lines inside the original domain Ω (A) and its image domain Ω w (B), are computed numerically from equation [START_REF] Bazant | Conformal mapping of some non-harmonic functions, in transport theory[END_REF]. The blue lines originate from the bulk, while the orange ones start in the cusp. The domain Ω w is subdivided into three regions: the bulk Ω 1 w , the region Ω 2 w connecting the bulk to the region Ω 3 w inside the funnel. (C) Representation of the two subregions A (blue) and B (green

) of Ω w . solution v unif (ρ, θ) =                ln 8 2(1 -cos(θ)) + σε πσ|e iθ -1| 2 2 , θ ∈ [0, π - √ ε] ln cos 2 π 2 (θ -(π - √ ε)) 2 ε 1 - 4 σε + ln (4 + σε) 2 2(πσ) 2 , θ ∈ [π - √ ε, π]. (22)
Interestingly, the leading order approximation in a solid funnel (Ω ⊂ R 3 ) is identical to the solution computed in the planar cusp. Furthermore, using ( 22), I estimate the potential difference between the center of mass C and the south pole S as well as C and the north pole S. In both cases the differences grow as ln (σ 2 ) to the leading order. The impact of the cusp geometry is revealed by computing the potential difference between the two poles N and S. Indeed, in the large σ limit, the leading order terms ln (σ 2 ) cancel each other and the difference converges to a constant that depends on the geometry of the domain.

Finally, an important remark is that the Möbius transformation in the problem described above does not entirely simplify the geometry. Indeed, the transformed geometry cannot be reduced to a 1D circular arc because of the non-homogeneous boundary conditions. However, the map does simplify the flow of the solution, and this allows to compute an asymptotic solution of the problem.

The electro-diffusion in a thin ellipse

Background: Thin elliptically-shaped domains are ubiquitous in biology and are involved in critical tasks such as synaptic regulation, immune efficiency and much more. For example, astrocytes -star-shaped cells that mostly support neuronshave nanoscopic protrusions that can be as thin as 30-50nm and that help regulate the synaptic transmission [START_REF] Heller | Probing nanoorganization of astroglia with multi-color super-resolution microscopy[END_REF][START_REF] Rusakov | Disentangling calcium-driven astrocyte physiology[END_REF]. An other example is, the neural growth cone, a protrusion neurons use to operate in their environment. They are mostly composed of thin 100-300nm finger-like structures called filopodia and whose length is about 10µm [START_REF] Mattila | Filopodia: molecular architecture and cellular functions[END_REF]. Yet, measuring key electrophysiological quantities such as voltage or ionic concentrations in such small compartments is impossible due to experimental limitations Electrical models for elongated structures, such as the linear cable equation [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF] are based on assumptions like local electro-neutrality or fixed concentration ∇c i (x) = 0. Those principles have been applied successfully for bigger compartments such as axons whose diameter lies between 1 to 20µm, and up to 0.5mm for the famous Hodgkin-Huxley squid giant axon. Nonetheless, those approximations don't hold in nano-domains [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF][START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF][START_REF] Sparreboom | Eijkel Principles and applications of nanofluidic transport[END_REF][START_REF] Wei | Current Rectification at Quartz Nanopipet Electrodes[END_REF][START_REF] Woermann | Language lateralization by Wada test and fMRI in 100 patients with epilepsy[END_REF].

Goal: I investigate the effects of electro-diffusion in thin nano-domains represented by an elongated ellipse. My goal is to construct an asymptotic solution of the stationary PNP in such domain and study the impact of the geometry on the voltage. Results: (Chap. 4-5) I find that for an excess of positive charge, the ellipse curvature induces a steady-state longitudinal potential gradient. I study the PNP equation ( 19) inside an elongated ellipse in R 2 whose major and minor axises are a = 1 and b 1 respectively. I build an asymptotic solution in the limit b 1 to

Introduction

the Liouville problem (8) using a similar approach as the one I used for the cuspshaped domain. My approach relies on distinguishing two subregions A and B of Ω as shown in Fig. 5A.

In region A, in the small b limit, the leading order term of the non-homogeneous Neumann boundary condition simplifies as

∂u ∂n ≈ ∂u(x, y) ∂x x∈∂Ω . (23)
The expression above indicates that the derivatives ∂u/∂y are negligible relative to the derivatives ∂u/∂x. A regular expansion of the solution suggests that the equation can be solved along the lines y = const using the 1D Liouville problem. I construct a solution in the region B, using conformal transformations. Indeed, a transformation mapping the interior of an ellipse into the unit disk [START_REF] Kanas | On conformal representation of the interior of an ellipse[END_REF][START_REF] Szego | Conformal Mapping of the Interior of an Ellipse onto a Circle[END_REF] would simplify efficiently conformally invariant problems. Nevertheless, it fails to simplify the stationary PNP because of the exponential term. To rather simplify the vector field, I used a parameter-dependent Schwarz-Christoffel map [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF][START_REF] Nehari | Conformal mapping[END_REF]. My idea was to find the optimal shape under the SC Schwarz-Christoffel map.

It maps Ω into the bell-bottom shaped domain Ω w = w(Ω) and the region B into B w = w(B) as illustrated in Fig. 5B-C. The goal is to find the optimal shape under the SC map such that the leading order solution becomes linear on B w = w(B), in other words transforming the Liouville problem into a simple Laplace equation. The last step consists in matching the two solutions from the regions A and B into an uniform solution:

u unif (x, y) =                ln   2 2 + bσ 1 -y 2 πσ 2 cos 2   1 2 πσx 2 + bσ 1 -y 2     for 0 ≤ y ≤ y 0 u A (1 -y 0 ) + νy 0 arcsinh σ(1 -y 0 ) ν -arcsinh σ(1 -y) ν for y 0 ≤ y ≤ 1, (24)
where ν depends on the solution constructed in region A.

From the uniform approximation u unif [START_REF] Biess | Diffusion in a dendritic spine: the role of geometry[END_REF], I compute three characteristic differences of potential inside the ellipse that I summarize in the following table (Tab. 1) for y 0 = 1 -b. Interestingly, I find that the difference of potential between the major and the minor axis tip ∆a-b is maximal for a given value σ.

The analysis of the thin ellipse reveals that there is a gradient of potential in the direction of the major axis. There is no such gradient if the ellipse is approximated by a cylinder even in the non-neutral regime. Indeed, the cylinder null curvature would prevent longitudinal changes in the potential, except a the cylinder tip, assuming it is sealed. With the advent of new imaging methods in biology, it is now possible to record voltage responses in small neuronal compartments during their electrical activity [START_REF] Brinks | Two-Photon Lifetime Imaging of Voltage Indicating Proteins as a Probe of Absolute Membrane Voltage[END_REF][START_REF] Jin | Single action potentials and subthreshold electrical events imaged in neurons with a fluorescent protein voltage probe[END_REF][START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF][START_REF] Xu | Voltage imaging with genetically encoded indicators[END_REF][START_REF] Zhang | Optogenetic Approaches to Drug Discovery in Neuroscience and Beyond[END_REF]. This opens new avenues for studying how changes in voltage propagate in the nanoscale neuronal compartments, and it also means that better models will be required to describe such changes. On the theoretical part, there are plenty of frameworks one can use to study the electrical properties of nanocompartments. Monte-Carlo simulations are employed to explore the dynamics of the concentration of diffusing ions such as calcium [START_REF] Franks | A Monte Carlo model reveals independent signaling at central glutamatergic synapses[END_REF][START_REF] Franks | Complexity of calcium signaling in synaptic spines[END_REF]; however, there are no simple theoretical frameworks that capture the physical correlation between ion concentration and the electricfields. So far, the description of electrical properties was mostly based on the linear cable theory [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. This theory describes the propagation of an electrical potential along a conducting cable coated in a dielectric membrane. The linear cable theory is a good approximation as long as the thickness of the cable is large enough (> 200µm) so the longitudinal fluctuations of the charge concentration remain negligible. Otherwise the cable theory fails to predict the voltage [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. Consequently, it is mandatory to develop a model that accounts for the complexity of the geometry, and at the same time, for the coupling between the electrical species and the electrical field.

u(0, 0) -u(0, 1) -2 ln bσ 1 -(1 -b) 2 + 2 bσ + 2 + arcsinh(bσ(1 + b/2)) 1 + b/2 u(0, 0) -u(b, 0) -2 ln cos πbσ 4 + 2bσ u(0, 1) -u(b, 0) 2 ln (bσ + 2) cos πbσ 4+2bσ bσ 1 -(1 -b) 2 + 2 + arcsinh(bσ(1 + b/2)) 1 + b/2
As suggested above, the PNP equations are well suited for this role.

Deconvolution of Arclight voltage sensor fluorescence

Background: There are various methods used in experimental biology to measure certain physical quantities -in particular the voltage -in micro and nano compartments. Contrary to the neuron soma, axons or dendrites, the small size of dendritic spines prevents direct voltage measurement. The solution is to use indirect methods such as the GEVI based on Genetically Engineered Voltage Indicators, that I will describe below. The voltage sensor used by my collaborators is the sensor Arclight Goal: My goal is to develop a deconvolution procedure to recover the dynamic of the voltage from the fluorescence.

Results: (Chap. 6-7) I assume that the Arclight fluorescence response A rc (t) results from a convolution of the voltage V (t), such that A rc (t) = (V * K)(t), where K(t) is a convolution kernel. I construct the kernel K(t) by comparing the voltage and the fluorescence in a region where they are both measurable, i.e. the soma (Fig. 6B). The kernel K(t) describes the time delay of the fluorescence activation compared to the voltage dynamics. Based on the deconvolution kernel K(t), I deconvolve the fluorescent responses in the spine head and the adjacent portion of the dendritic shaft represented, in Fig. 6C, by regions of interest (ROIs) R1 and R2 respectively. One can observe that although the region R1 contains the spine neck, because of its thickness, its fluorescence is negligible. The fluorescence is approximated by a family of function

f γ 1 ,γ 2 ,α,β 1 ,β 2 (t) = t α (β 1 exp(-γ 1 t) + β 2 exp(-γ 2 t)) . ( 25
)
This approximation allows the calculation of an analytical solution for the voltage,

V (t) = β 1 e -γ 1 t A t α-1 (t(1 -2τ γ 1 ) + 2τ α) + τ 2 t α-2 (tγ 1 ) 2 + α 2 -α -2γ 1 αt + β 2 e -γ 2 t A t α-1 (t(1 -2τ γ 2 ) + 2τ α) + τ 2 t α-2 (tγ 2 ) 2 + α 2 -α -2γ 2 αt .( 26
∂ 2 V ∂x 2 (x, t) = - F εε 0 (c p (x, t) -c m (x, t)) ( 27
)
∂c i ∂t (x, t) = D i ∂ ∂x ∂c i ∂x (x, t) + z i e k B T c i (x, t) ∂V ∂x (x, t) , i = {p, m}.
Here V (t) represents the voltage drop V head (t) -V dend (t).

Goal: I aim to estimate numerically, based on the deconvolved fluorescence, the electrical current I(t) that enter the neck (see Fig. 7A). Then I use this current to estimate the neck resistance. Finally, I study the concentration profile in the spine neck.

Results: (Chap. 6-7) I first reconstruct the electrical current I(t) that arrives at the neck entrance and results from the stimulation current I stim (t) received by the head. I model it as the sum of a resistive and a capacitive terms:

I(t) = G V head (t) + C dV head (t) dt . ( 28
)
Using the expression above for the current as a flux boundary condition for the positive charges in PNP, I develop an algorithm to extract the conductance G and

Introduction

the capacitance C from the data. I show that the PNP based electro-diffusion model reproduces the voltage drop observed experimentally between the two region R1 and R2. The voltage drop and the current I(t) estimated numerically are then used to characterize the spine neck resistance R neck = V (0, t) / I(t) for several neck lengths L that have been measured experimentally. I find that the resistance behaves linearly relatively to the neck length L.

Especially, electro-diffusion simulations predict a significant change in the local concentration of positive charges along the neck. Indeed, the difference in concentration is 33mM (the imposed concentration on the other end is 163mM ), leading to a chemical gradient between the dendrite and the spine head. Although the injected current I stim (t) is composed of positive charges, the concentration gradient in the neck is driven by positive and negative charges. Such phenomenon is called concentration polarization [START_REF] Strathmann | Ion-Exchange Membrane Separation Processes, Membrane science and technology series[END_REF].

Simulation of PNP in a spine-shaped domain

Goal: I now present the steady-state PNP equations, that describe the concentration of positive c p (x) and negative c m (x) charges and the voltage V (x) inside a three dimensional bounded domain composed of a sphere connected to a cylindrical neck as shown in Fig. 7B. The equations are given by

∆V (x) = -F εε 0 (c p (x) -c m (x)) (29) 0
= D i ∇ ∇c i (x) + z i e k B T c i ∇V (x) ,
Where i = {p, m}. In that model, only positive charges can enter the spine domain.

Results: (Chap. 6-7) Applying the electro-diffusion model above to a spinelike geometry (Fig. 7A) reveals that most of the voltage drop is carried by the spine neck. The spine head resistance is thus negligible such that the entire spine resistance is carried by the neck: R spine ≈ R neck . Using PNP I estimate the effect of the spine head radius on the averaged voltage and I find that it is almost constant ( V ball ∈ [1.52 , 1.56]mV ). Consequently, the neck radius is one of the most critical parameters in defining the conversion of current into voltage. I then estimate the I-V relation for various neck radii r 0 , showing a saturation for large currents that contrasts with the linear behavior expected from the Ohm's law. Then, to clarify the dependency of the spine neck resistance on its radius, I simulate the PNP equations when a steady-state current is injected inside the spine head. Surprisingly, the resulting curve could not be fitted with a power laws 1/r α 0 having the same exponent α. On the contrary, albeit a power law approximation matched well the numerical simulations, the exponent α depends in the injected current. This is a clear deviation from the classical Ohm's law for a resistance where α = 2 does not depend on the current.

Organization of the thesis

This thesis is organized as follows:

Part I, Chapter 1: I present stationary solutions of Poisson-Nernst-Planck on a finite segment, a disk and a ball. I compute an asymptotic solution of the stationary PNP in a ball for two regimes: for a small and for a large number of confined charges. I then show that in a charged ball, the mean-first-passage time of a charged particle to exit the domain through a small circular opening is almost identical to the MFPT of a Brownian particle on a sphere to an narrow opening located on the surface. I finally compute the outflux current exiting through the small opening window.

Chapter 2, 3, 4 and 5: I construct asymptotic approximations of the stationary PNP for domains with a cusp-shaped funnel or ellipses attached in R 2 and R 3 . I present the method I have developed based on conformal transformations. I then show that the local mean curvature of a domain impacts the voltage profile, and thus the electrical properties of the electrolyte.

Part II, Chapter 6 and 7: I present an application of PNP to biology, showing how the spine geometry affects its electrical properties. I first analyze and process data from fluorescence microscopy to study how the voltage is transduced by dendritic spines. I develop a deconvolution method to recover the voltage from the fluorescence of the voltage sensor Arclight. I model the electro-diffusion in the spine using PNP. Then I estimate the electrical current in the spine neck. I discuss the concentration polarization in the spine neck. Finally, I present results on the spine resistance showing that it is mostly carried by the spine neck. We use the Poisson Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion of ions in neuronal micro-compartments.

Part

It is an open question to determine the relaxation and the steady state distribution of voltage when an initial charge of ions is injected into a bounded domain. We start here by computing the steady state PNP equation in a ball, which is also the Liouville-Gelfand-Bratú equation, with the difference that the boundary condition is Neumann, not Dirichlet and there is a minus sign in the exponent of the exponential term. The entire boundary is impermeable to particles (ions) and the electric field satisfies the compatibility condition of Poisson's equation. We construct a steady state radial solution and find that the voltage is maximal in the center and decreases toward the boundary. We study the limit of large charge in dimension 1, 2 and 3, and find that the difference of potential between the center and the surface increases with the log of the total number of charge and not linearly (as the classical capacitance theory of electrostatic does). This log-singularity in dimension three is obtained from an asymptotic argument and cannot be derived from the analysis of the phase portrait. Finally, we apply the present results to derive the relation between the outward current and the voltage in a dendritic spine, a fundamental microdomain involved in communication between neurons. We compute the escape rate of an ion from the steady density in a ball (which is a model of the spine head) to a small absorbing window in a sphere. In conclusion, we predict here that the current is defined by the small absorbing window, as suggested by the Narrow Escape Theory, while voltage is independently controlled by the PNP equation.

Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains

Introduction

The non-linear system of Poisson-Nernst-Planck (PNP) equations has been widely used to study properties of the electric field in local nanodomains such as ionic channels [START_REF] Barcilon | Ion Flow through Narrow Membrane Channels: Part II[END_REF][START_REF] Barcilon | Ion Flow Through Narrow Membrane Channels: Part I[END_REF][START_REF] Goldman | Potential, impedance, and rectification in membranes[END_REF][START_REF] Nadler | Ionic diffusion through confined geometries: from Langevin equations to partial differential equations[END_REF][START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF][START_REF] Singer | A Poisson-Nernst-Planck model for biological ion channelsan asymptotic analysis in a three-dimensional narrow funnel[END_REF]. It was also used to simulate the equilibration of ions between large reservoirs through narrow necks [START_REF] Graf | A Dynamic Lattice Monte Carlo Model of Ion Transport in Inhomogeneous Dielectric Environments: Method and Implementation[END_REF][START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF] and to study the effect of interacting ions in ionic channels [START_REF] Taflia | Diffusion of interacting particles in confined domains and applications to biology[END_REF].

Using the PNP model, we compute here the distribution of a single specie of unscreened positive charges in a ball for different regimes, including the limit of large charges. The stationary PNP equation with Neumann and no-flux conditions on the boundary of a bounded domain Ω is the classical Poisson equation with an exponential term, that model the electrical potential and density of charges in Ω. This equation is also known as the Liouville-Gelfand-Bratú-type equation for the electric potential with however two major differences: first, the boundary condition on ∂Ω is Neumann and not Dirichlet and second, there is a minus sign in the exponent, normalized over the domain Ω. This question is thus different from the Newtonian potential of a cluster of self-gravitating mass distribution [START_REF] Chipot | On the solutions of Liouville systems[END_REF][START_REF] Wolansky | On steady distributions of self-attracting clusters under friction and fluctuations[END_REF]. In addition, this equation should not be confused with the Poisson-Boltzmann, which is a model for two populations of negative and positive ions with the same valence charge. We compute the solution of the stationary PNP equation in spherical symmetry in dimensions ≤ 3 with respect to the (dimensionless) total charge λ. We construct asymptotic approximations of the solutions for small and large λ. The onedimensional case is solved explicitly and it is characterized by a log-singularity at the boundary that develops in the large λ limit. The explicit solution in two-dimension has also a singularity on the boundary. We also obtain a similar asymptotic behavior in three-dimension, although the solution cannot be computed explicitly and we provide an asymptotic and numerical argument for large λ showing again a logsingularity at the boundary. We note that this log-singularity in dimension three cannot be derived from the analysis of the phase portrait because it occurs at the initial point of the dynamics.

The present modeling and analysis is motivated by the need to compute voltage changes in bounded domains and how it develops a boundary layer for large λ. We find that the drop of the solution from the center to the boundary converges to a finite value as λ increases to infinity. Finally, we apply this analysis to predict the voltage change (solution of the PNP equation) for idealized neuronal microdomains, such as dendritic spines. These structures (see Fig. 1.1) are approximated as a spherical dielectric membrane filled with an ionic solution, connected to the dendrite by a cylindrical narrow neck. A large effort was dedicated to study the mathematics of diffusion in such structures [START_REF] Holcman | Stochatic Narrow escape. Theory and Applications[END_REF][START_REF] Holcman | Diffusion laws in dendritic spines[END_REF], but very little is known about their electrodiffusion properties, even experimentally where almost no data are available at the nanometer resolution (see [START_REF] Araya | Sodium channels amplify spine potentials[END_REF][START_REF] Araya | Dendritic spines linearize the summation of excitatory potentials[END_REF][START_REF] Araya | The spine neck filters membrane potentials[END_REF]). This high resolution is necessary to evaluate the change of voltage and whether or not electro-neutrality is satisfied. Despite converging experimental efforts, the electrical properties of these structures remain unclear at a molecular level and a predictive theory based on mathematical physics is needed to interpret incoming data [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF].

The PNP equations serve here to compute the electric field for a single specie, where we do not assume local electro-neutrality. To derive a relation between the outward current and the voltage, we will compute the escape rate of moving particles (ions) from the steady density in a ball (which is a model for the spine head) to a small absorbing window in its boundary (sphere). We find here that the current of particles is controlled by the small absorbing window in the boundary of the ball, as predicted by the Narrow Escape Theory [START_REF] Schuss | The narrow escape problem for diffusion in cellular microdomains[END_REF][START_REF] Holcman | Time scale of diffusion in molecular and cellular biology[END_REF], while the voltage is independently regulated by the coupled PNP equations. The paper is organized as follow: in the first part we study PNP asymptotically and numerically for a ball where we found that charges accumulate at the boundary (see fig. 1.2). In the second part, we estimate the current generated in an idealized spine (head connected by a cylinder). We derive the current flowing outside the spine and demonstrate that the head geometry controls voltage, while the narrow neck radius control the current.

PNP equations in a ball

We consider the Poisson-Nernst-Planck system in a ball Ω of radius R, whose dielectric boundary ∂Ω is represented as the compatibility condition for Poisson's equation and its impermeability to the passage of ions is represented as a no-flux boundary condition for the Nernst-Planck equation. We assume that there are N positive ions Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains of valence z in Ω and that there is an initial particle density q(x) in Ω such that

Ω q(x) dx = N. (1.1)
The charge in Ω is

Q = zeN,
where e is the electronic charge. The charge density ρ(x, t) is the solution of the Nernst-Planck equation

D ∆ρ(x, t) + ze kT ∇ (ρ(x, t)∇φ(x, t)) = ∂ρ(x, t) ∂t for x ∈ Ω (1.2) D ∂ρ(x, t) ∂n + ze kT ρ(x, t) ∂φ(x, t) ∂n = 0 for x ∈ ∂Ω (1.3) ρ(x, 0) = q(x) for x ∈ Ω, (1.4)
where the electric potential in Ω is φ(x, t) is the solution of the Poisson equation

∆φ(x, t) = - zeρ(x, t) εε 0 for x ∈ Ω (1.5)
and the boundary condition

∂φ(x, t) ∂n = -σ(x, t) for x ∈ ∂Ω, (1.6)
where σ(x, t) is the surface charge density on the boundary ∂Ω. In the steady state and in spherical symmetry

σ(x, t) = Q 4εε 0 πR 2 .
(1.7)

The steady-state solution

In the steady state ∂ρ/∂t = 0 so (1.2) gives the density (1.9)

ρ(x) = N exp - zeφ(x) kT Ω exp - zeφ(x) kT dx , (1.8

PNP equations in a ball

In spherical symmetry in R d (1.9) can be written in spherical coordinates as

φ (r) + d -1 r φ (r) = - zeN exp - zeφ(r) kT S d εε 0 R 0 exp - zeφ(r kT r d-1 dr < 0, (1.10)
where S d is the surface area of the unit sphere in R d . The boundary conditions are

∂φ(0) ∂r = 0, ∂φ(R) ∂r = - Q εε 0 S d R d-1 .
(1.11)

The inequality in (1.10) means that φ(r) has a maximum at the origin and decreases toward the boundary (see Fig. 1.3A). We can normalize the radius by setting r = Rx for 0 < x < 1 and

u(x) = zeφ(r) kT , λ = (ze) 2 N εε 0 kT , (1.12)
to write (1.10) as

u (x) + d -1 x u (x) = - λ exp {-u(x)} S d R d-2 1 0 exp {-u(x))} x d-1 dx (1.13) u(0) = 0, u (0) = 0.
Note that we have dropped here the compatibility condition on the boundary at |x| = R, which is automatically satisfied by a solution. Incorporating the denominator of the RHS of (1.13) into the parameter λ by setting

λ = µS d R d-2 1 0 exp{-u(x)} x d-1 dx, (1.14)
we can write the initial value problem (1.13) as

u (x) + d -1 x u (x) = -µ exp {-u(x)} (1.15) u(0) = u (0) = 0.
First, we show that solutions exist in dimensions 1 ≤ d ≤ 3 only for µ in the range 0 ≤ µ < µ * for some positive µ * .

Solution in dimension one

We solved directly equation (1.15) in dimension 1 (see appendix 1.5.1) and we obtain (see eq. 1.54) that where I λ is solution of the implicit equation

u 1D λ (x) = log cos 2 λ 2I λ x , (1.16)
I λ = 2 λ tan 2 λ 2I λ .
(1.17)

The graph of u 1D λ (x) is shown Fig. 1.4A, while the one for λ I λ versus λ is shown in Fig. 1.4B. We have 0 < µ(λ) = λ I λ ≤ π 2 2 and lim λ→∞ µ(λ) = π 2 2 . The solution exists u 1D λ for all λ > 0 and a log-singularity develops at the boundary x = 1 when λ → ∞.

Solution in dimension two

In dimension 2, we obtain the solution in (appendix 1.5.2) .18) where

u 2D λ (x) = log (1 - λ 8I λ x 2 ) 2 . ( 1
I λ = π + 1 8 λ µ(λ) = λ I λ lim λ→∞ µ(λ) = 8.
The graph of u 2D λ (x) is shown on Fig. 1.4C, while the one for λ I λ is on Fig.

1.4D. u 2D λ (x) = log(1 -λ λ+8π x 2 ) 2
, develop a log-singularity as λ → ∞.

Analysis in dimension three

The solution of the initial value problem (1.15) in dimension d = 3 cannot be directly computed. We show now that the solution exits for all λ, while there is a critical value µ * , above which, there is no regular solution. Contrary to dimensions one and two, the value of µ * can only be estimated numerically. We first show using a phasespace analysis that the solution of equation (1.15) is unique when it exists. However

PNP equations in a ball

it is not possible to use the phase-space to study the singularity of the equation because it occurs at the initial time of the dynamics. To study the asymptotic explosion of the equation, we use an asymptotic argument. Finally, we will study the solution numerically (see appendix for the construction of a numerical scheme).

Next, we show that the problem (1.13) has a unique regular solution for all λ ≥ 0, when the solution is finite. The proof of uniqueness of the solution follows the phase-space analysis of (1.15). Indeed, using the change of variables [START_REF] Jacobsen | The Liouville Bratu Gelfand Problem for radial operators[END_REF]

s = -log r, u(r) = U (s), v(s) = dU (s) ds , w = µe -2s e -U (s) w (s) = -2w(s) -U (s)w(s) = w(s)[-2 -v(s)], (1.19)
the dynamical system becomes

v (s) = v(s) -w(s), (1.20)
w (s) = -w(s)[2 + v(s)], (1.21)
and can be written as

dw dv = -w(2 + v) v -w . (1.22)
The phase space of (1.20) contains exactly two critical points. The origin 0 is a saddle point and its stable manifold has the tangent T of equation w = 3v. The point P a = (-2, -2) is an unstable node. The initial conditions u(0) = u (0) = 0 for the solution of (1. Thus the trajectory of the solution of (1.15) in the first quadrant, which satisfies the constraints (1.23), has to be on the separatrix that converges to the saddle point. Choosing any value U (0) gives µe -U (0) the value of v(0) = U (0) has to be chosen on the separatrix. Therefore starting in the first quadrant, a trajectory of (1.20) converges to the saddle point if and only if it starts on the separatrix with the tangent T. The stable branch at the saddle point tends to infinity as s decreases toward 0. Indeed, the local expansion of (1.22) near the saddle point is

w(v) = 3v + 3 5 v 2 - 3 175 v 3 + . . . , (1.24)
which gives the phase portrait (Fig. 1.5). Finally, along the separatrix w (v) > 0, except at the origin, showing that for an initial v(0), there is a unique solution. However, it is not possible from the phase-space to study the singular solution. Indeed, as we shall see, the singularity occurs precisely at the initial value and thus the Cauchy problem cannot even start. We conclude the problem (1.13) has a finite solution and the phase diagram plotted in Fig. 1.5 ensures that for any initial Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains condition (v(0), w( 0)) (when it is finite) on the separatrix in the first quadrant, there is a unique solution to (1.20) that satisfies (1.23).

A numerical solution of (1.13) gives the graph Fig. 1.3E, that is the solution u(x) of (1.3) for µ ≤ µ * = 11.2. The graph in dashed line (µ * = 14) blows up before reaching x = 1, while the dash (small point) graph is finite throughout the interval. To estimate an upper bound for µ * , we note that whenever the solution exists for some µ near µ * , its asymptotic behavior for x close to 1 shows that u (1)

u (1) (see the blue graph in Fig. 1.3). Indeed, to show that under the assumption u (1)

u (1) the latter inequality is self-consistent, we note that near x = 1 the solution of (1.15) can be approximated by the solution of the simpler problem

ũ (x) = -µ exp {-ũ(x)} , (1.25)
given by

ũ(x) ∼ log cos 2 µ 2 x . (1.26)
Thus ũ(x) is finite in the interval as long as

µ < π 2 2 = 4.934802202 = µ * (1.27) and ũ (x) ũ (x) ≤ | √ µ - √ µ * | √ µ * 1.
(1.28)

We conclude at this stage that for fixed values of µ below and above, where above the latter they blow-up inside the interval 0 < x < 1 (frames A, C, E of Fig. 1.4). When µ varies with λ according to (1.14), the solutions exist for all values of λ (frames B, D, F of Fig. 1.4). Figure 1.4A-C-E shows that potential drop between the center and the surface of the sphere as a function of λ for 1 ≤ d ≤ 3.

In figure 1.3, Three dimensional solution obtained numerically is compared with the asymptotic expansions in two regimes. We present in appendix 1.5.3 for λ 1, the expansion u(x) = -λ x 2 8π + O(λ 2 ) (see eq. 1.73). In contrast, for λ 1, we mention above that the approximation u(x) ≈ 2 log(1 -x 2 ), which was relevant near x = 1 can be used in the entire domain [START_REF][END_REF][START_REF] Ablowitz | Method for Solving the Sine-Gordon Equation[END_REF]. The analytical approximations (red) are compared with the numerical solutions (see appendix 1.5.4).

PNP equations in a ball

The potential differences

The difference u(0) -u(1) as we shall see in the next section has a physical meaning, as it represents the difference of potential between the center and the periphery of a sphere. We have in dimension 1, 1.2. PNP equations in a ball

| u λ (1) -u λ (0) | = log cos 2 λ 2I λ , (1.29)
where λ 2I λ → π 2 4 as λ → ∞. in dimension 2, | u λ (1) -u λ (0) | = 2 log( 8π λ + 8π
), (1.30) in dimension 3, for λ 1,

| u λ (1) -u λ (0) | = 2 log(1 -f (λ)), (1.31)
where the unknown function f is increasing and f (λ) → 1 as λ → ∞. The different curves for dimension 1, 2 and 3 are shown in fig. 1.6. In all cases, the large λ asymptotic is dominated by the log-behavior.

Physical implication for the distribution of voltage and charge in a dielectric ball

The distribution of voltage and charge in a dielectric ball can be estimated from the results of the previous sections by using the dimensional relation (1.12) in a ball of radius R = 1µm. We plotted in Fig. 1.7A the voltage drop for N = 10 2 , 10 3 and 10 4 charges. Already for 1000 charges, there is a difference between the center and the surface of a ball of few milli-Volts. Moreover, the density of charge is concentrated at the periphery (Fig. 1.7B), leading also to a large field E = -∇V close to the boundary (Fig. 1.7C). Consequently most of the charges are accumulated at the boundary, as revealed by the plot of the cumulative density of charges (fig. 1.7D)

Q(r) = N r 0 exp - zeφ(r) kT r 2 dr R 0 exp - zeφ(r) kT r 2 dr . (1.32)
In summary, when the total number of charges is fixed sufficiently high, the charges accumulate at the surface. The field is only significant close to the surface and thus can trap a Brownian charged particle in such region, while outside a small boundary Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains layer of the boundary, the field is almost zero and charged particles experience no drift. This effect is discussed in the next section.

Scaling laws for the maximum number of charges

Although we found previously that for a fixed radius, the difference of potential V (0)-V (1) is bounded as a function of the total number of charge, we shall now show that the maximal number of charges increases linearly with the radius of the ball. Indeed, introducing the dimensionless radial variable ζ = r/R and u λ (r) = U λ/R (ζ), equation (1.10) becomes

U λ/R (ζ) + 2 ζ U λ/R (ζ) = - λ exp -U λ/R (ζ) 4πR 1 0 exp -U λ/R (ζ) ζ 2 dζ , (1.33)
with the initial conditions U λ/R (0) = U λ/R (0) = 0. Now, we solve the initial value problem

V µ (ζ) + 2 ζ V µ (ζ) = -µ exp {-V µ (ζ)} , V µ (0) = V µ (0) = 0 (1.34) W µ (ζ) = ζ 2 exp {-V µ (ζ)} , W µ (0) = 0 and note that u λ (r) = V µ r R , λ = 4πµRW (1). (1.35)
1.3. Ionic flux in a small absorbing window in a highly charged sphere

Thus the number of charges Q in a ball or radius R create the same distribution as a charge Q/R in a ball of radius one, which can be summarized as

Q(R) = RQ(1). (1.36)
In the next section, we will use the solution of the PNP equation for the distribution of charges and the electrical potential to study the motion of charged stochastic particles to a small hole located on the three dimensional ball.

1.3 Ionic flux in a small absorbing window in a highly charged sphere

We now discuss various consequences of distributing charges close to the boundary, in the large charge regime. The first consequence is on the Mean First Passage Time (MFPT) τ (x) from x ∈ Ω, which is the solution of the Pontryagin-Andronov-Vitt (PAV) boundary value problem [START_REF] Schuss | Theory and Applications of Stochastic Processes[END_REF],

D ∆τ (x) - ze kT ∇τ (x) • ∇φ(x) = -1 for x ∈ Ω (1.37) ∂ τ (x) ∂n + ze kT τ (x) ∂φ(x) ∂n =0 for x ∈ ∂Ω r (1.38) τ (x) =0 for x ∈ ∂Ω a , (1.39)
where ∂Ω a represents the small absorbing window and ∂Ω r = ∂Ω\∂Ω a . We consider the case of a large field |∇φ(x)| 1 near the boundary |x| = 1. The profile of φ(x) was studied in section 1.2.1 (see Figures 1.7). To study the solution of the PAV problem (1.37)-(1.39), we map the neighborhood of ∂Ω a smoothly into the upper half plane with coordinates X = (x, y, z), where z = 0 is the image of the boundary, τ (X) = τ (x), and outside a boundary layer near

∂Ω a V = ∂φ(x) ∂n |x|=1 = const, Φ(x, y) = φ(x) |x|=1 = const, so that ∇ x,y Φ(x, y) = 0. The PAV system (1.37)-(1.39) is converted to τzz (X) - ze kT Ṽ τz (X) + ∆ x,y τ (X) = - 1 D , (1.40)
A regular expansion of τ (X) for large Ṽ = ∂φ ∂z gives that to leading order τ (X) is a function of (x, y) and setting T (x, y) = τ (x, y, 0), we find that

∆ x,y T (x, y) = - 1 D . (1.41)
Thus the MFPT from x ∈ Ω to ∂Ω a is the sum of the MFPT from x to ∂Ω and the MFPT form ∂Ω to ∂Ω a on the surface ∂Ω. The MFPT to ∂Ω is negligible relative to that to ∂Ω a . This approximation means that to reach ∂Ω a in a highly charged ball a charge is first transported by the field to the reflecting part ∂Ω r of the sphere with overwhelming probability and then it finds ∂Ω a by surface diffusion.

Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains

Particle current through a small absorbing window in a highly charged ball

A second consequence of the charge distributions is the control of flux of particles through a small hole from the geometry. This has an important consequence in understanding how the electric voltage and current can be controlled in cellular microdomains such as dendritic spines (see the dictionary section 1.6). Diffusion in dendritic spines have already been investigated [START_REF] Schuss | Modeling calcium dynamics in dendritic spines[END_REF][START_REF] Biess | Diffusion in a dendritic spine: the role of geometry[END_REF], but little is known about the regulation of the electrical current since no experimental data are yet available about the voltage at a nanometer precision [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF].

We shall show that the present modeling and asymptotic analysis can lead to novel predictions about the electrical current in a dendritic spine, that can be regulated independently of the voltage. In this section we focus on the spine head. The solution T (x, y) of (1.41) is the MFPT of a Brownian motion on a sphere of radius R to an absorbing circle centered at the north-south axis near the south pole, with small radius a = R sin δ 2 . It is given by [START_REF] Cheviakov | Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps[END_REF][START_REF] Holcman | Stochatic Narrow escape. Theory and Applications[END_REF][START_REF] Singer | Narrow escape, part III: Non-Smooth Domains and Riemann Surfaces[END_REF]]

T (x, y) = 2R 2 D log sin θ 2 sin δ 2 , (1.42)
where D is the diffusion coefficient, θ is the angle between x and the north pole. Thus

τ (x) = T (x, y). (1.43)
The MFPT, averaged over the sphere with respect to a uniform distribution of x is given by

τ = 2R 2 log 1 δ + O(1) for δ 1.
(1.44)

The MFPT for

N independent charges is τN = 2R 2 N log 1 δ + O(1) for δ 1. (1.45)
It follows that the electrical current through the small window is given by

J = ze τN = QD 2R 2 log R a + O(1)
for a R.

(1.46)

The ball represents here the dendritic spine head and J is the current through the neck. Thus, once a current flows into a dielectrics ball such as a spine head, the excess of charges Q is first pushed toward the boundary, before moving by Brownian motion to the entrance of spine neck (small disk of size a). This result shows that the current J in a spine head is governed by the spine geometry (formula 1.46) and a key parameter is the radius a of the neck. When there is a conservation of charge principle (no leak), the current through the end of cylinder starting at the window (spine neck) finishing at a bigger cylinder (dendritic shaft) is the same as the one exiting the spine head. In that conditions, the spine neck length do not affect or modulate the current.

1.4. Discussion and applications of PNP to the current in a spine neck under voltage-clamp condition

Discussion and applications of PNP to the current in a spine neck under voltage-clamp condition

We have studied here the solution of the PNP equations in a ball and obtained an asymptotic expression in the large charge limit. We further computed the mean escape time of a charged stochastic particle to a small window located on the boundary. We also compared analytic solution with numerical approximations. PNP equations serve here as a model to study the current in neuronal microdomains such as dendritic spines.

We estimated the voltage in dendritic spines when the voltage in the spine head is maintained. We found that the voltage U varies nonlinearly with respect to the total number of charges Q, in contrast with the classical law of electrostatic for a conductor, where Q = CU and C is the capacitance.

We recall that a certain fraction of dendritic spines receive synaptic connections, essential for neuronal communication [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF]. Although their functions are still unclear, they are involved in regulating synaptic transmission and plasticity [START_REF] Araya | Sodium channels amplify spine potentials[END_REF][START_REF] Araya | Dendritic spines linearize the summation of excitatory potentials[END_REF][START_REF] Araya | The spine neck filters membrane potentials[END_REF][START_REF] Korkotian | Dynamic regulation of spinedendrite coupling in cultured hippocampal neurons[END_REF][START_REF] Svoboda | Direct measurement of coupling between dendritic spines and shafts[END_REF][START_REF] Bloodgood | Neuronal activity regulates diffusion across the neck of dendritic spines[END_REF]. Interestingly, most of the excitatory connections occur not on the dendrite but rather on spines and the reason is still not clear. The spine shape is quite intriguing, made of a head connected to the dendritic shaft by a cylinder. We found here that this geometry plays a key role: we predicted here that the spine head geometry determines the drop of voltage, while the current is defined by the diffusion on the surface and the mean time to find the entrance of the neck in a two dimensional Brownian motion (see Narrow escape time [START_REF] Schuss | The narrow escape problem for diffusion in cellular microdomains[END_REF][START_REF] Holcman | Time scale of diffusion in molecular and cellular biology[END_REF]). In the neck, under a voltage-clamp condition (the voltage is maintained constant), when a constant voltage difference between the head and the neck is imposed, the voltagecurrent relation follow a resistance law. Thus the spine geometry defines both the capacitance and resistance in geometrical terms, a vision that complement previous classical studies [START_REF] Svoboda | Direct measurement of coupling between dendritic spines and shafts[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF][START_REF] Segev | Computational study of an excitable dendritic spine[END_REF].

Determining the voltage drop between the membrane of the spine head and the dendrite, when a current is flowing from the head to the dendrite remains challenging because the classical cable theory cannot be applied in a system that cannot be approximated by a cable. The general scheme for modeling the electro-diffusion in the spine is the PNP model in the head and a one-dimensional conduction of ions in the neck. The neck is considered a classical ionic conductor. Thus the steady-state PNP equations have to be solved in the sphere with boundary conditions implied by the compatibility condition and the flux through the neck is determined by the mean first passage time (MFPT) of ions from the head to the neck, as discussed above.

In the case of high charge Q the potential turns out to be practically flat throughout the ball with a sharp boundary layer with negative slope at the boundary. Thus a charge diffuses and is pushed strongly toward the membrane so ionic motion is practically confined to motion on the surface. Due to spherical symmetry, the potential is constant on the boundary so ionic motion is free Brownian motion on a sphere. At high charge ions interact through the ambient potential that is determined from Poisson's equation in the ball. Therefore they can be assumed independent free Brownian particles. The MFPT τ of an ion to the small opening of the neck is determined from the two-dimensional NET theory (see previous section). Because the flux carried by a single ion is q/τ , where q is the ionic charge, the number of ions in the spine head is N = Q/q and the MFPT τN of any of the N ions is given by τN = τ N .

(1.47)

Thus the current through the neck is

I = Q τ (1.48)
and due to charge conservation, it is independent of the length of the neck. If we consider the neck to be a parallel-plate capacitor carrying a steady current I, then the voltage drop across the neck is simply V = RI, where R is the resistance of the neck, given by

R = kT L 2 q 2 nD , (1.49)
where k is Boltzmann's constant, T is absolute temperature, L is the length of the conductor, n is the number of ions in the neck, q is the charge of an ion, and D is the diffusion coefficient of the solution in the neck [START_REF] Schuss | Theory and Applications of Stochastic Processes[END_REF]. This model is valid as along as the voltage is maintained in the spine head. Finally, computing in the transient regime, the change in voltage drop between the spine head and the dendritic shaft, requires computing the time dependent PNP equations. Another open question is to study the influence of the spine head geometry on the distribution of charges. Computing the distribution of charges and the associated field in non-convex geometry is certainly the most challenging.

1.5. Appendix

Appendix

In this appendix, we first solve analytically the Liouville equation 1.13 in dimensions one and two and in the second part, we describe the numerical methods to compute the solution in dimension 3.

-u λ (r) = λ e -u λ (r) 1 0 e -u λ (r) dr (1.50)
with initial conditions

u λ (0) = 0 and u λ (0) = 0. (1.51)
This is the classical Cauchy problem. After a direct integration we get with the initial conditions

u 2 λ (r) = 2λ I λ (e -u λ (r) -1), (1.52)
where

I λ = 1 0 e -u λ (r) dr. (1.53)
A second integration gives

u λ (r) = log cos 2 λ 2I λ r. (1.54)
Now we self-consistently calculate

I λ = 1 0 e -u λ (r) dr = 1 0 dr cos 2 λ 2I λ r = 1 λ 2I λ tan λ 2I λ . (1.55)
Thus I λ > 0 is the solution of the implicit equation

I λ = 2 λ tan 2 λ 2I λ . (1.56)
The graph of λ I λ versus λ is shown in Figure 1.4. We have lim λ→∞ λ

I λ = π 2 2 , and specifically, y λ = λ 2I λ = π 2 -π 2 λ 2 + O( 1 λ 2 )
. The solution (1.54) is shown in Fig. 1.4 and is regular in the entire interval 0 < r < 1 for all values of λ. The drop between the extreme points of the interval is

u λ (1) -u λ (0) = log cos 2 λ 2I λ (1.57)
and becomes infinite as the total charge increases indefinitely.

Chapter 1. Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains

Liouville equation in dimension 2

The dimension 2 case can be transformed into the one dimensional case [START_REF] Jacobsen | The Liouville Bratu Gelfand Problem for radial operators[END_REF] using the change of variable

r = e -t ũ(t) = u λ (r) -2t. Equation (1.13) reduces to -ũ tt (t) = λ I λ e -ũ(t)+2t (1.58)
where

I λ = 2π 1 0 e -u λ (r) rdr and w(t) = ũ(t) + 2t satisfies -w tt (t) = λ e -w(t) I λ (1.59)
The initial conditions are now transform to asymptotic conditions at infinity:

lim t→∞ (w(t) -2t) = 0 (1.60) lim t→∞ ( ẇ(t) -2) e t = 0. (1.61) A first integration gives ẇ2 (t) 2 = λ e -w(t) I λ + 2. (1.62)
The solution is

w(t) = -log 8 (λe 2C+2t -1) 2 -2C -2t, (1.63)
where C is a constant. Finally, we obtain that

u λ (r) = log (1 - λ 8I λ r 2 ) 2 . (1.64)
To close the equation, we shall now compute the integral

I λ = 1 0 e -u λ (r) 2πrdr = 1 0 1 (1 -λ 8I λ r 2 ) 2 2πrdr = 8π 8 -λ/I λ (1.65)
and

I λ = π + 1 8 λ (1.66) lim λ→∞ λ I λ = 8. ( 1
u λ (r) = log 1 - λ λ + 8π r 2 2 (1.68) | u λ (1) -u λ (0) | = 2 log 1 - λ λ + 8π . (1.69)
We conclude that u λ (r) decreases smoothly and in the limit λ → ∞, the solution blow-up over the entire boundary.

1.5. Appendix

Regular expansion of solution (1.15) for small λ

We shall now study the small asymptotic expansion of solution (1.15) for small λ.

Using a regular expansion,

u(x) = u 0 (x) + u 1 (x)λ + u 2 (x)λ 2 + o(λ 2 ), (1.70)
we obtain using eq. (1.15) that u 0 (x) = 0 and u 1 is solution of

-∆u 1 (x) = 1 |Ω| on Ω (1.71) ∂u 1 ∂n = - 1 |∂Ω| on ∂Ω. (1.72) For R = 1, u 1 (r) = - r 2 8π , (1.73)
with u 1 (0) = 0. We conclude that u 1 (r) ≤ 0, Thus,

u(r) = - r 2 8π λ + O(λ 2 ). (1.74)
The second order term u 2 is solution of

-∆u 2 = - u 1 |Ω| on Ω, (1.75)
with u 2 (0) = 0 and u 2 (0) = 0. For R = 1,

u 2 (r) = - 3r 4 640π 2 .
(1.76) Thus,

u(r) = - r 2 8π λ - 3r 4 640π 2 λ 2 + O(λ 3 ).
(1.77)

1.5.4 Numerical scheme for construction the solution of eq.

(1.13)

In the different sections of this manuscript, the solution u(r) of eq. (1.13) was computed numerically using the one from the radial Neumann problem in the ball where S 3 = ∂B 3 . The relation between solutions u(r) and v(r) is expressed by the shift

B 3 v (r) - 2 r v (r) = -λ exp{-v(r)} for 0 ≤ x ≤ 1 (1.78) v (1) = - λ |S 3 | v (0) = 0,
v(x) = u(x) + β, (1.79)
where the constant β is computed from the compatibility and the boundary condition of eq. (1.78):

λ = - ∂B 3 ∂v(x) ∂n dS = -λ B 3 exp{-v(x)}dx, (1.80)
leading to

B 3 exp{-v(x)}dx = 1. (1.81)
We obtain from relations (1.79) and (1.81),

β = log(I λ ), (1.82)
where I λ is defined for the three dimensions as I λ = 1 0 e -u λ (r) 4πr 2 dr. The condition u(0) = 0 in eq. (1.13) links the value of β to the solution v(r):

β = v(0). (1.83)
In summary, the solution u(r) can be entirely computed from v(r) as

u(r) = v(r) -v(0). (1.84)
The shift in relation (1.79) permits to express the solution of a nonlinear elliptic PDE, containing the integral of the solution over the domain in terms of the solution of a classical Neumann problem eq. (1.78).

We have solved numerically eq. (1.78) using one dimensional finite element method in Matlab, and Comsol for comparison. For the application to the different physical scenario of PNP in a ball, we used an adaptative meshing to account for the stiff tangent in the region close to the boundary r = R. For example for R = 1µm the maximal element size taken was 5 • 10 -4 µm. All numerical results in the ball B 3 (Figs. 1.4,1.3,1.6,1.7) were obtained using the scheme described here.

Basic biological terminology

• Dendritic spine: neuronal microstructure located on neuronal cells. It is one of the two part of a synapse, which is a junction between two neurons. The spine geometry is approximated as a spherical head connected to a cylindrical neck see [START_REF] Schuss | Modeling calcium dynamics in dendritic spines[END_REF].

• PNP: Poisson-Nernst Planck equation.

• NET: Narrow escape time: mean time for a small particle to find a small hole.

• Voltage-clamp: condition under which the voltage is maintained constant.

D Diffusion coefficient D = 200µm 2 /sec I c Injected current I ∈ [2; 30]pA I Average Injected current I = 2.5pA Ω Spine head Ω (volume |Ω| = 1µm 3 ) a Radius of spine neck (typical) a = 0.1µm L Length of spine (typical) L = 1µm T Temperature T = 300K E Energy kT = 2.58 × 10 -2 eV e Electron charge e = 1.6 × 10 -19 C ε Dielectric constant ε = 80
Chapter 2

Geometrical effects on nonlinear electrodiffusion in cell physiology

Published as Cartailler J., Schuss Z., Holcman D. Geometrical effects on nonlinear electro-diffusion in cell physiology. Nonlinear Sciences, (2017).

We report here new electrical laws, derived from nonlinear electro-diffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nano-pipettes and help to understand the local voltage changes inside dendrites and axons with heterogenous local geometry.

Introduction

Electro-diffusion is the process by which the motion of ions in solution is driven by two physical forces: thermal motion, which is diffusion, and the electric field.

The difficulty in the mathematical description of this physical motion is due to the origin of the field, which consists of the contribution of mobile ions and of a possible external field. The dielectric membrane also affects the field by image charges. So far only few electro-diffusion systems are well understood: although the voltaic cell was invented more than 200 years ago, designing optimal configurations is still a challenge. On the other extreme, ionic flux and gating of voltage-channels [START_REF] Bezanilla | How membrane proteins sense voltage[END_REF] is now well explained by the modern Poisson-Nernst-Planck theory of electro-diffusion [START_REF] Mamonov | The role of the dielectric barrier in narrow biological channels: a novel composite approach to modeling single channel currents[END_REF], because at the nanometer scale, the cylindrical geometry approximation of protein channels reduces the computation of the electric field and of ionic diffusion to one dimension [START_REF] Eisenberg | Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube[END_REF][START_REF] Roux | Ion transport in the gramicidin channel: free energy of the solvated right-handed dimer in a model membrane[END_REF][START_REF] Horn | Permeation redux: thermodynamics and kinetics of ion movement through potassium channels[END_REF][START_REF] Eisenberg | From structure to function in open ionic channels[END_REF][START_REF] Eisenberg | Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations[END_REF][START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF][START_REF] Singer | A Poisson-Nernst-Planck model for biological ion channelsan asymptotic analysis in a three-dimensional narrow funnel[END_REF]. However, cellular domains at a micron scale involve two-and three-dimensional geometry, much more complicated than the cylindrical geometry of a channel pore, leading to a more complex electrodiffusion description [START_REF] Savtchenko | Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF]. We recall that local curvature is a key geometrical element for controlling charge distribution in various media, such as in the air (e.g., the lightning rod [START_REF] Courant | Methods of Mathematical Physics[END_REF]). The manifestation of this effect is observed in Lebesgue's thorn, which is a an inverted cusp singularity of the boundary, for which the solution of Laplace's equation blowsup inside the domain [47, p.304]. In electronics, the design of printed circuits is always pre-conditioned on corner effects [START_REF] Ruiz | A Comprehensive Study of the Corner Effects in Pi-Gate MOSFETs Including Quantum Effects[END_REF]. However, these effects are not very well known inside an electrolytic bath. Recent analysis [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF], [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF] suggests that nonelectro-neutrality in the geometry of an electrolyte confined by a dielectric membrane affects charge distribution.

We use the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. The entire boundary is impermeable to particles (ions) and the electric field satisfies the compatibility condition of Poisson's equation. Phenomenological descriptions of electro-diffusion, such as the cable equation or the reduced electrical engineering approximation by resistance, capacitance, and even electronic devices, are not sufficient to describe noncylindrical geometry [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF], because they assume a simple reduced one-dimensional or reduced geometry. We present here results about charge and field distributions in electro-diffusion in various geometrical microdomains, when the condition of electro-neutrality is not satisfied. We recall that under the non-electro-neutrality assumption, and with charge distributed in bounded domains confined by a dielectric membrane, Debye's concept of charge screening decaying exponentially away from a charge [START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF], do not apply and long-range correlation leads to a gradient of charges in a ball with no inward current. A new capacitance law was derived for an electrolyte ball [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF], where the difference of potential between the center C and the surface S, that is, V (C) -V (S), increases, first linearly and then logarithmically, when the total number of charges in the ball increases.

Our aim here is to understand the effect of boundary curvature on an electrical cell, such as neuron. In particular, we explore the effect of boundary curvature on the charge and field distribution at steady state. The curvature of membranes 2.2. The PNP equations of dendrites and axons of neurons have many local maxima that can modulate the channel's local electric potential [START_REF] Yuste | Dendritic Spines[END_REF]. In this article, we study the effects of local curvature on the distribution of charge in bounded domains with no electroneutrality. The effect of non-electro-neutrality was recently studied in [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF] and a long-range electrostatic length, much longer than the Debye length was found. This effect is due to the combined effects of non-electro-neutrality and boundary, which lead to charge accumulation near the boundary.

The cusp-shaped funnel geometry was studied in [START_REF] Holcman | Brownian motion in dire straits[END_REF], however this paper presents several crucial mathematical differences with [START_REF] Holcman | Brownian motion in dire straits[END_REF], in particular, we are solving a nonlinear equation, while it was linear in [START_REF] Holcman | Brownian motion in dire straits[END_REF]. Furthermore, the boundary condition at the end of the cusp-shaped funnel: while it is the Dirichlet condition in [START_REF] Holcman | Brownian motion in dire straits[END_REF], it is the Neumann condition here. This means that in [START_REF] Holcman | Brownian motion in dire straits[END_REF] the absorption flux at the end of the funnel is computed, whereas here the stationary voltage and charge distribution are computed in the absence of flux. We develop here new boundary layer analysis, different than the classical matched asymptotics method [START_REF] Lindsay | First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps[END_REF][START_REF] Pillay | An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains[END_REF][START_REF] Delgado | Conditional Mean First Passage Times to Small Traps in a 3-D Domain with a Sticky Boundary[END_REF]. The manuscript is organized as follow: first, we consider a bounded domain with an uncharged narrow cusp-shaped funnel on the boundary, which is a singular geometrical effect. Second, we further study the case of charge distribution in a charged narrow cusp.

The PNP equations

The Poisson-Nernst-Planck system of equations in a domain Ω, whose dielectric boundary ∂Ω is represented as the compatibility condition for Poisson's equation, and its impermeability to the passage of ions is represented as a no-flux boundary condition for the Nernst-Planck equation. We assume that the charge in Ω consists of N identical positive ions (see Appendix) with initial particle density q(x) in Ω, their valence is z, and the total number of particles is fixed, equal to Ω q(x) dx = N.

(2.1)

Thus the charge in Ω is

Q = zeN,
where e is the electronic charge. The charge density ρ(x, t) is the solution of the initial and boundary value problem for the Nernst-Planck equation

D ∆ρ(x, t) + ze kT ∇ (ρ(x, t)∇φ(x, t)) = ∂ρ(x, t) ∂t for x ∈ Ω (2.2) D ∂ρ(x, t) ∂n + ze kT ρ(x, t) ∂φ(x, t) ∂n = 0 for x ∈ ∂Ω (2.3) ρ(x, 0) = q(x) for x ∈ Ω. (2.4)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

Here φ(x, t) is the electric potential in Ω and is the solution of the Neumann problem for the Poisson equation

∆φ(x, t) = - zeρ(x, t) ε r ε 0 for x ∈ Ω (2.5) ∂φ(x, t) ∂n = -σ(x, t) for x ∈ ∂Ω, (2.6)
where ε r ε 0 is the permittivity of the medium and σ(x, t) is the surface charge density on the boundary ∂Ω. In the steady state,

σ(x, t) = Q ε r ε 0 |∂Ω| . (2.7)

Steady solution in a ball with a cusp-shaped funnel

Local boundary curvature is a key geometrical feature that controls charge distribution in the domain. Specifically, we study the effect of a narrow funnel attached to a sphere. In various media, such as air (e.g., the lightning rod, [START_REF] Courant | Methods of Mathematical Physics[END_REF]), the manifestation of this effect is observed in Lebesgue's thorn, which is a an inverted cusp singularity of the boundary, for which the solution of Laplace's equation blows-up inside the domain [47, p.304]. In the steady state (2.2) gives the particle density

ρ(x) = N exp - zeφ(x) kT Ω exp - zeφ(x) kT dx , (2.8) hence (2.5) gives Poisson equation ∆φ(x) = - zeN exp - zeφ(x) kT ε r ε 0 Ω exp - zeφ(x) kT dx .
(2.9) and (2.6) gives the boundary condition

∂φ(x) ∂n = - Q ε r ε 0 |∂Ω| , (2.10)
for |x| = R, which is the compatibility condition, obtained by integrating Poisson's equation (2.5) over Ω. Changing variables to

u(x) = zeφ(x) kT , λ = (ze) 2 N ε r ε 0 kT , ( 2
(x) = - λ exp {-u(x)} Ω exp {-u(x)} dx (2.12)
and the boundary condition (2.10) becomes

∂u(x) ∂n = - λ |∂Ω| for x ∈ ∂Ω. (2.13)
The translation ũ = u + ln λ/ Ω exp{v(x)} dx , converts (2.12) into

-∆ũ(x) = exp{-ũ(x)} for x ∈ Ω (2.14) ∂ ũ(x) ∂n = - λ |∂Ω| for x ∈ ∂Ω.
We consider a dimensionless planar domain Ω with a cusp-shaped funnel formed by two bounding circles A and B of dimensionless radii 1 (see Fig. 2.1(left)). The opening of the funnel is ε 1. We construct an asymptotic solution in this limit to the nonlinear boundary value problem (BVP) (2.14) by first mapping the domain Ω conformally with the Möbius transformation of the two osculating circles A and B into concentric circles (see Fig. 2.1(right)). To this end, we move the origin of the complex plane to the center of the osculating circle B and set

w = w(z) = z -α 1 -αz , (2.15)
where

α = -1 - √ ε + O(ε). (2.16)
The Möbius transformation (2.15) maps the circle B (dashed blue) into itself and Ω is mapped onto the domain Ω w = w(Ω) in Figure 2.1(right). The straits in Figure 2.1(left) are mapped onto the ring enclosed between the like-style arcs and the large disk is mapped onto the small red disk in Figure 2.1(right). The radius of the small disk and the elevation of its center above the real axis are O( √ ε). The short black segment AB of length ε in Figure 2.1(left) is mapped onto the segment AB of length 2 √ ε+O(ε) in Figure 2.1(right). This mapping (see [START_REF] Holcman | Narrow escape through a funnel and effective diffusion on a crowded membrane[END_REF]), transforms the PNP equations as well and thus leads to a new non-linear effect. Setting u(z) = v(w) converts (2.12) to

∆ w v(w) = - exp {-v(w)} |w (z)| 2 = - (4ε + O(ε 3/2 )) |w(1 - √ ε) -1 + O(ε)| 4 exp {-v(w)} for w ∈ Ω w .
(2.17)

Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

∂u(z) ∂n z = - ∂v(w) ∂θ w=-1 ∂θ ∂Y , (2.18)
where

ie iθ ∂θ ∂Y = w (z) = 1 -α 2 (1 -αz) 2 .
(2.19)

For θ = π (for z = -1), we obtain ∂θ/∂Y = -2/ √ ε and the boundary condition at ∂Ω w,a is

∂v(w) ∂n = - λ √ ε 2|∂Ω| for w ∈ ∂Ω w,a . (2.20)
We show in Fig. 2.2A-B numerical solution of the fields lines obtained with (2.14) (left) and (2.17) (right).

Reduced PNP equations in an uncharged cusp-shaped funnel

Approximating the banana-shaped domain Ω w by a one-dimensional circular arc, we use a one-dimensional approximation of the solution in Ω w [START_REF] Holcman | Brownian motion in dire straits[END_REF][START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF]. This approximation assumes that there are no non-neutralized charges on the surface of the cusp (Fig. 2.3A). The boundary condition for the approximate one-dimensional solution of (2.17) is zero at angle θ Lim = c √ ε, where c is a constant (see details in [START_REF] Holcman | Brownian motion in dire straits[END_REF][START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF]) and represents the solution inside the disk in Figure 2.1(left), away from the cusp. Thus, (2.17) in the conformal image Ω w becomes the boundary value problem

v + 4ε |e iθ -1 -e iθ √ ε| 4 exp -v(e iθ ) = 0 (2.21) v (c √ ε) = 0 (2.22) v (π) = - λ √ ε 2|∂Ω| .
Our goal is now to estimate the difference of potentials between the north pole N and the end of the funnel C,

∆u = u(N ) -u(C) = v(c √ ε) -v(π). (2.23)
To construct an asymptotic approximation to the solution of (2.22) in the limits ε → 0 and λ → ∞, we first construct the outer-solution in the form of a series in powers of ε, which is an approximation valid away from the boundary. In the limit of small ε, the first term in the series vanishes, exponential terms drop out, and the second order term is

y outer (θ) = M θ + M , (2.24)
where M and M are yet undetermined constants. The outer solution cannot satisfy all boundary conditions, so a boundary layer correction is needed at the reflecting boundary at θ = c √ ε. Thus, we set θ = √ εξ and expand

ε 2 |e iθ -1 -e iθ √ ε| 4 = 1 (1 + ξ 2 ) 2 + O( √ ε).
Writing the boundary layer solution as y bl (θ) = Y (ξ), we obtain to leading order the boundary layer equation

Y (ξ) + 4 (1 + ξ 2 ) 2 exp {-Y (ξ)} = 0, (2.25)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology with Y (c) = 0. The solution is decaying for large ξ and develops a singularity at finite ξ. However, a Taylor expansion near ξ = 0,

Y (ξ) = A + B 2 ξ 2 + B 4 ξ 4 + . . . , (2.26)
gives in (2.25)

B 2 = -2e -A . (2.27)
In general, the coefficients satisfy B k = O(e -A ), for A 1. For small ξ, we obtain the approximate solution of (2.25) by considering the leading term in a regular expansion of the solution in powers of ξ. The equation for the leading term is

Y (ξ) + 4e -A (1 + ξ 2 ) 2 = 0 (2.28)
and the solution is defined up to an additive constant. Setting Y appr (0) = 0, which does not affect the potential difference, we find that

Y appr (ξ) = -2ξe -A arctan ξ. (2.29)
It follows that the boundary layer solution at c √ ε is

y bl (θ) = A - 2θ √ ε e -A arctan θ √ ε . (2.30)
The boundary layer near π is needed, because A → ∞ as ε → 0 (see (2.45) below).

An approximation of the solution can be obtained by freezing the power-law term in (2.22), for which the equation is for a generic parameter b > 0,

d 2 dθ 2 v(θ) + be -v(θ) = 0, dv(0) dθ = v(0) = 0.
The solution is

v b (θ) = ln cos 2 b 2 θ. (2.31)
Putting the outer and boundary layer solutions together gives the uniform asymptotic approximation

y unif (θ) = A - 2θ √ ε e -A arctan θ √ ε + ln cos 2 b 2 θ, (2.32)
where the parameters A and b are yet undetermined constants. The condition at

c √ ε = o(1) for ε 1 is satisfied, because y unif (0) = 0.
The condition at θ = π gives that

y unif (π) = - πe -A √ ε -b tan b 2 π = - λ √ ε 2|∂Ω| .

Steady solution in a ball with a cusp-shaped funnel

The compatibility condition for (2.14),

λ = Ω exp{-ũ(x)}dS x , (2.33)
gives in Ω w that

λ = Ωw exp{-ṽ(w)} dw |φ (φ -1 (w))| = 8 √ ε π c √ ε exp {-v(θ)} |e iθ (1 - √ ε) -1| 4 dθ. (2.34)
Using the uniform approximation (2.32) in the compatibility condition (2.34), we obtain the second condition

λ = 8 √ εe -A π c √ ε 1 cos 2 b 2 θ exp e -A 2θ √ ε arctan θ √ ε |e iθ (1 - √ ε) -1| 4 dθ ≈ 8e -A ε π/ √ ε 0 1 cos 2 b 2 √ εξ exp 2e -A ξ arctan ξ |1 + ξ 2 | 2 dξ, (2.35)
where we used the change of variable θ = √ εξ. Integrating by parts, we get for ε 1

λ ∼ 8e -A ε        2 b √ ε tan b 2 π exp 2e -A π √ ε π 2 1 + π √ ε 2 2 - π/ √ ε 0 2 b √ ε tan b 2 θ Ψ(θ) dθ        , (2.36)
where

Ψ(ξ) = d dξ exp 2e -A ξ arctan ξ |1 + ξ 2 | 2 .
(2.37) Thus, it remains to solve the asymptotic equation

λ ∼ 8e -A ε 1/2 2 bπ 4 tan πb 2 exp π 2 e -A √ ε + O ln cos πb 2 . (2.38)
for A and b in the limit ε → 0. We consider the limiting case where

e -A √ ε = O(1) = C for λ → ∞, (2.39)
for which condition (3.87) can be simplified and gives to leading order

b tan πb 2 = λ √ ε 2|∂Ω| , (2.40)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

that is, for λ √ ε 1 (2.40) gives b ≈ 1 - 4 π |∂Ω| λ √ ε , tan b 2 π ∼ λ √ ε 2|∂Ω| .
With condition (2.38), we get

λ ≈ 8e -A ε 1/2 2 π 4 λ √ ε 2|∂Ω| exp π 2 e -A √ ε + O ln cos πb 2 , (2.41)
To leading order in large C, we obtain

π 4 |∂Ω| 8ε 3/2 = C exp Cπ 2 . (2.42)
The solution is expressed in terms of the Lambert-W function,

Cπ 2 = W π 6 |∂Ω| 2 3 ε 3/2 , (2.43)
and for small ε, using the asymptotics of the Lambert function,

Cπ 2 = ln π 6 |∂Ω| 2 3 ε 3/2 -ln ln π 6 |∂Ω| 2 3 ε 3/2 + o(1). (2.44)
Finally,

e -A √ ε = C ∼ 1 π 2 ln π 6 |∂Ω| 2 3 ε 3/2 , A = ln 1 √ ε -ln 1 π 2 ln π 6 |∂Ω| 2 3 ε 3/2 → ∞ as ε → 0. (2.45)
It follows that a uniform asymptotic approximation (2.32) in the limits λ → ∞ ε → 0 is given by

y unif (θ) = ln 1 √ ε -ln 1 π 2 ln π 6 |∂Ω| 2 3 ε 3/2 (2.46) -2θ 1 π 2 ln π 6 |∂Ω| 2 3 ε 3/2 arctan θ √ ε + ln cos 2 1 -4 π |∂Ω| λ √ ε 2 θ .
The uniform approximation (2.46) is plotted for different values of ε and λ in Figure 2.3 against the numerical solution of (2.21), with the boundary conditions v (c √ ε) = v (0) = 0. The numerical solutions are computed with the software COMSOL, based on an adaptive mesh refinement and a relative tolerance of 10 -3 , that we validated on known analytical results of steady state PNP equations in a disk [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF]. We find that the asymptotic expansion is particularly good in the limit ε → 0 and λ → ∞ (Fig. 2.3A-D). However, for λ = O(1) the log-term approximation in (2.46) is nonmonotonic in θ. We estimated numerically the difference between the asymptotic solution y unif (2.32) and the numerical estimation V num (2.21), averaged over the The asymptotic solution y unif (θ) of (2.32) (blue dashed lines) is compared to the numerical solution of (2.21) (red line). The four panels A-B-C-D are obtained for different pairs of parameters (λ, ε). E. shows a 3D plots the difference between the asymptotic solution y unif (eq. 2.32) and numerical results V num (eq. 2.21), averaged over the domain Ω w .

domain Ω w , for 10 3 ≤ λ ≤ 5 • 10

The voltage drop between the end of the funnel and the center of the ball

We can now use (2.32) to compute the potential drop in (2.23). It is given by

∆SC u = u(S) -u(C) = -v(c √ ε) + v(π) = -ln π 6 |∂Ω| 2 3 ε 3/2 + 2 ln 2|∂Ω| λε 1/2 = ln 2 5 |∂Ω| √ ε π 6 λ 2 .
(2.47)

Next, we compare the potential drop (2.23) with the one between the center and the north pole. Numerical solution of the PNP equations shows that the voltage and charge distribution in a disk with a funnel do not differ from the ones in a disk in the upper sphere (Fig. 2.2). This result is compared next to the difference between the north pole and the center evaluated from the exact analytical expression derived for a disk. The expression for the voltage in the two-dimensional disk of radius R is given

Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology by (see [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF])

u 2D λ (x) = ln 1 - λ D 8π + λ D r R 2 2
, where λ D is a parameter. We calibrate λ D so that the solutions of the PNP equations in a disk with a funnel have the same total charge as a disk. The Neumann boundary conditions for the disk and the funnel are, respectively,

∂u(x) ∂n = - λ D 2πR , ∂u(x) ∂n = - λ |∂Ω| .
The calibration is

λ D = λ 2πR |∂Ω| . (2.48)
We compare in Figure 2.4D the two-dimensional numerical solution of the PNP equation (2.14) in the domain Ω (blue line), with the analytical solution (2.48) in a disk with no cusp (dashed red). The numerical solution of the PNP equation (2.14) is plotted along the main axis 0y in the interval [0, y 0 ] (where the point y 0 is defined by the condition ∇u(y 0 ) = 0). In the range [y 0 , y cusp ], where y cusp is the coordinate of the cusp, we compare the solution of (2.14) with the uniform solution y unif of (2.32) in the funnel (dashed green). We conclude that in the cusp, the two-dimensional approximation in a disk is in good agreement with the numerical solution of equation (2.14), confirming that the solution in the bulky head does not influence the one in the cusp (as already shown in Fig. 2.2). This result also confirms the validity of the analytical formula to predict the large λ asymptotics. For a disk of radius R, the potential drop is given by

∆NC u = u(N ) -u(C) = ln 8π 8π + λ D 2 = -2 ln λ -2 ln R 4|∂ Ω| + O( 1 λ ) (2.49) (see section 2.3.2)
. The two differences of potential ∆SC u (2.49) and ∆NC u (2.47) have the same logarithmic behavior ln 1/λ 2 for λ 1 and u(N ) -u(S) = O(1) as shown in Fig. 2.4E. A numerical solution in two-dimensions shows that u(N ) -u(S) may converge to zero as λ increases (Fig. 2.4F), thus having a local maximum for small values of λ. This maximum cannot be analyzed by the uniform expression (2.32), because it appears outside the domain of validity of (2.32). This result is in agreement with the two-dimensional numerical solution of (2.14) for the difference between u(N ) (potential at the north pole) and u(S) (potential at the end of the funnel) (Fig. 2.3F). The potential drop calculated above is non-dimensionalized by the radius of curvatures R f at the right and left of the funnel,

ε = ε R f ,
where ε is the length of the absorbing arc AB. The non-dimensionalized volume and boundary measure are, respectively, Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

|Ω| = | Ω| R 2 , |∂Ω| = |∂ Ω| R .
In dimensional units (2.47) gives the potential drop in the dimensional disk with a funnel as

∆SC u = u(S) -u(C) = ln 2 5 |∂ Ω| √ ε π 6 R 3/2 f λ 2 .
(2.50)

We conclude in the limit of λ 1, ε → 0 that the difference of potential between the end of cusp S and the north pole N in the domain is obtained by adding (2.49) and (2.50) and we get

∆SN u = u(S) -u(C) + u(C) -u(N ) = ln 2 5 |∂ Ω| √ ε π 6 R 3/2 f + 2 ln R 4|∂ Ω| + O( 1 λ ). (2.51)
We recall that R is the radius of the entire ball, while R f is the radius of curvature of the funnel.

The PNP equations in a charged domain with a cusp-shaped funnel

Due to the Neumann boundary conditions (2.6) on the lateral part of the funnel,(2.17) in the transformed domain cannot be reduced to one dimension. Thus we derive a different one-dimensional approximation for the mapped PNP equations in the banana-shaped domain Ω w by averaging over the radius r. Rewriting (2.17) in polar coordinates w = re iθ , we obtain

1 r ∂ ∂r r ∂v(w) ∂r + 1 r 2 ∂ 2 v(w) ∂θ 2 = - (4ε + O(ε 3/2 )) exp {-v(w)} |re iθ (1 - √ ε) -1 + O(ε)| 4 for w ∈ Ω w .(2.52)
In the section Ω w ∩ {1 -√ 2ε < r < 1]}, the boundary conditions are

∂v(r, θ) ∂r r=1 = -λ √ ε |∂Ω|(cos θ -1) , for θ ∈ [c √ ε, π] (2.53) ∂v(r, θ) ∂r r=1- √ 2ε = 0, for θ ∈ [c √ ε, π] ∂v(r, θ) ∂θ θ=π = -λ √ ε 2|∂Ω| , ∂v(r, θ) ∂θ θ=c √ ε = 0.
Taylor's expansion of v in the section gives

v(r, θ) = v 0 (θ) + (r -1)v 1 (θ) + O((r -1) 2 ), (2.54)
and because |r -

1| = O( √ ε), we obtain the approximation, exp {-v(w)} = exp {-v 0 (θ)} 1 - √ εv 1 (θ) + O(ε) .
2.4. The PNP equations in a charged domain with a cusp-shaped funnel Multiplying (2.52) by r 2 and integrating over the radius, we get r ∂v(r, θ) ∂r

r=1 1- √ ε + ∂ 2 ∂θ 2 1 1- √ ε v(r, θ) dr = - 1 1- √ ε (4r 2 ε + O(ε 3/2 )) e -v(r, θ) |re iθ (1 - √ ε) -1 + O(ε)| 4 dr. (2.55)
The boundary conditions (2.53) give, to leading order in √ ε, that

- λ √ ε |∂Ω|(cos θ -1) + √ ε ∂ 2 v 0 (θ) ∂θ 2 = (2.56) - 1 1- √ ε (4r 2 ε + O(ε 3/2 )) |re iθ (1 - √ ε) -1 + O(ε)| 4 e -v 0 (θ) 1 - √ εv 1 (θ) + O(ε) dr.
that is, the BVP (2.52) in the section becomes the ODE (with respect to θ),

v 0 (θ) = - (4ε + O(ε 3/2 )) |e iθ (1 - √ ε) -1 + O(ε)| 4 exp {-v 0 (θ)} - λ |∂Ω|(1 -cos θ) , (2.57)
v 0 (θ)| θ=π = -λ √ ε 2|∂Ω| , v 0 (θ)| θ=c √ ε = 0.
Equation (2.57) is obtained by averaging over the radial direction and its solution seems to be a good approximation to (2.52) only for small λ. A different approach for large λ is discussed in the next section.

A regular expansion for λ 1,

v 0 (θ) = w 0 (θ) + λw 1 (θ) + o(λ), (2.58)
gives in (2.57) that w 0 = O(ε) and w 1 is the solution of the BVP

w 1 (θ) = - (4ε + O(ε 3/2 )) |e iθ (1 - √ ε) -1 + O(ε)| 4 - 1 |∂Ω|(1 -cos θ) , (2.59
)

w 1 (θ)| θ=π = - √ ε 2|∂Ω| , (2.60
)

w 1 (θ)| θ=c √ ε = 0.
Direct integration with respect to θ gives

w 1 (θ) = - 2θ ε √ ε arctan θ √ ε + 1 |∂Ω| ln sin 2 θ 2 + Aθ + B. (2.61)
Equation (2.60) gives A as

A = π ε 3/2 - 4 3π 3 - √ ε 2|∂Ω| . (2.62)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

The zero Neumann boundary condition cannot be satisfied and a boundary layer appears, leading to the local expansion

v 0 (θ) = λw 1 (θ) + o(λ). (2.63)
It follows that for λ 1, the solution increases with λ. It is shown below that it decreases for λ 1, demonstrating that there is at least one maximum in the variable λ.

PNP asymptotics in a charged disk with a charged funnel

In the limit of λ 1, ε → 0, the asymptotic expansion of the potential found above for a charged disk with a funnel is no longer valid. Some insight can be gained by observing the field lines in the domain Ω w , described in Figure 2.2A-B. These lines are parallel to the radius vector, except in a small region near the funnel. Two sections can be distinguished,

A ={(r, θ) ∈ Ω w : |θ - √ ε| > π, |r -1| ≤ √ ε} (2.64) B ={w = (1 - √ ε)e iθ : |θ -π| ≤ √ ε}.
The two sections A and B are illustrated in Figure 2.5A. Note that the boundary of section B contains a circular arc (marked magenta). Next, the approximate solutions u A (r, θ) and u B (θ) of (2.52) in the two sections are constructed and used to construct a uniform approximation u unif in Ω w (Fig. 2.5B).

Asymptotics of u A (r, θ) in section A

The boundary conditions (2.53) for the potential equation (2.52) (see Fig.

= const = θ 0 for r ∈ [1 - √ ε, 1]. Thus, to leading order in λ √ ε, u A (r, θ 0 ) + 1 r u A (r, θ 0 ) = -4ε exp(-u A ) |re iθ 0 (1 - √ ε) -1| 4 for r ∈ [1 - √ ε, 1] (2.65) u A (r, θ 0 )| r=1-√ ε = 0 u A (r, θ 0 )| r=1 = -λ √ ε |∂Ω|(1 -cos θ 0 )
.

For ε 1, we get |re iθ 0 (1 - √ ε) -1| 4 = |e iθ 0 -1| 4 + O( √ ε). Setting h(θ 0 ) = 4ε |e iθ 0 -1| 4 , (2.66)
and

v A,θ 0 (r) = -u A (r, θ 0 ) + ln h(θ 0 ), (2.67)
we get

v A,θ 0 (r) + 1 r v A,θ 0 (r) = exp(v A,θ 0 ) (2.68) v A,θ 0 (r) r=1-√ ε =0 v A,θ 0 (r) r=1 = λ √ ε |∂Ω|(1 -cos θ 0 )
.

The general solution of (2.68) is given by [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF] v A,θ 0 (r) = ln

C 2 2 2r 2 -ln cos 2 C 2 2 (ln r -C 1 ), (2.69)
where the constants C 1 and C 2 are determined from the boundary conditions (2.68). Using

v A,θ 0 (r) = C 2 r tan C 2 2 (ln r -C 1 ) - 2 r , (2.70)
we find the constant C 1 from (2.70) and from the boundary condition (2.68) at the point r = 1 -√ ε, getting

C 1 = - 2 C 2 arctan 2 C 2 + √ ε + O(ε). (2.71)
This gives in (2.70) at r = 1 the transcendental equation for

C 2 , C 2 tan -C 2 C 1 2 = λ √ ε |∂Ω|(1 -cos θ 0 ) + 2, (2.72)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

hence lim λ→∞ - C 2 C 1 2 = π 2 . (2.73)
Now, it follows from (2.71) that

- C 2 C 1 2 = arctan 2 C 2 + C 2 2 √ ε. (2.74)
Note that lim λ→∞ C 2 = 0, because otherwise we would get the asymptotic expansion

- C 2 C 1 2 = π 2 + C 2 2 ( √ ε -1) + O(C 3 2 ), (2.75)
which leads to

C 2 tan -C 2 C 1 2 = 2 1 - √ ε + O(C 2 2 ) (2.76)
and contradicts the condition (2.72) in the limit λ → ∞. Then (2.73) and (2.74) would imply that

C 2 √ ε 2 = O(1) (2.77)
and (2.77) would give C 2 1, so that the arctan term in (2.74) drops out, and we would be left with

- C 2 C 1 2 ∼ C 2 2 √ ε, (2.78)
hence

C 1 ∼ - √ ε. (2.79)
Expanding the left hand side of (2.72), using (2.73) and (2.78), we obtain that tan

C 2 √ ε 2 = - 2 C 2 √ ε -π + O C 2 √ ε 2 -π/2 . (2.80)
Together with (2.80), the solution of (2.72) is

C 2 ∼ λπ √ ε 2|∂Ω|(1 -cos θ 0 ) + λε . (2.81)
With the values of C 1 and C 2 computed in (2.71) and (2.81), the solution v A,θ 0 of (2.69) is given by

v A,θ 0 (r) = ln ε 2r 2 λπ 2|∂Ω|(1 -cos θ 0 ) + λε 2 (2.82) -ln cos 2 λπ 2 √ ε [ln r + √ ε] 2|∂Ω|(1 -cos θ 0 ) + λε .
2.4. The PNP equations in a charged domain with a cusp-shaped funnel Finally, using (2.67) and (2.82), we obtain for (r, θ) ∈ A,

u A (r, θ) = -ln |e iθ -1| 4 8r 2 λπ 2|∂Ω|(1 -cos θ) + λε 2 (2.83) + ln cos 2 λπ 2 √ ε [ln r + √ ε] 2|∂Ω|(1 -cos θ) + λε .
The asymptotic solution u A is plotted in Figure 2.5B (blue dashed line). Comparison with numerical solutions for various values of λ and ε is shown in Figure 2.6 below.

The asymptotics of u B in section B

The asymptotic solution u A (r, θ) in section A cannot satisfy the boundary conditions (2.53) at θ = π. Indeed, (2.83) gives ∂u A (r, θ)/∂θ| θ=π = 0, while the boundary condition (2.57) is ∂v/∂θ| θ=π = -λ √ ε/2|∂Ω|, so a boundary layer correction is needed. The boundary layer u B (θ) is an asymptotic solution of (2.52) in section B, where the θ derivatives dominate the radial ones. The right-hand-side of (2.52) can be simplified for ε

1. For r = 1 - √ ε the approximation -4ε |re iθ (1 - √ ε) -1| 4 ∼ -ε 4 (2.84)
holds, which does not depend on r and θ. With this simplification in (2.52), we rewrite u B (θ) as

u B (θ) = ũB (η) + C 0 , (2.85)
where C 0 is an additive constant and ũB is a function of η = θ -(π -√ ε) and solves the BVP

∂ 2 ũB (η) ∂η 2 = -exp {-ũ B (η)} (2.86) ũ B (η)| η= √ ε = - λ √ ε 2|∂Ω| ũ B (η)| η=0 =0.
The solution of (2.86) (see [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF]) is

ũB (η) = ln cos 2 λ 2I λ η, (2.87)
where I λ is the solution of the transcendental equation

I λ = 2|∂Ω| 2 λε tan 2 λε 2I λ . (2.88)
Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

We obtain to leading order for λ 1 that

I λ = 2λε π 2 1 + 8|∂Ω| λε + O 1 λε . (2.89)
It follows from (2.89), (2.87), and (2.85) that for θ ∈ B, the asymptotic solution is

u B (θ) = ln cos 2 π 2 (θ -(π - √ ε)) 2 ε 1 - 2|∂Ω| λε + C 0 (2.90) (see (2.85))
. It is shown in Figure 2.5B (red dots).

A uniform approximation of u(r, θ) in Ω w

A uniform asymptotic approximation u unif (r, θ) of the voltage u(r, θ) in the entire mapped domain Ω w can be now constructed by matching the the leading term u A (r, θ), given in (2.83) in section A, with that of u B (θ), given in (2.90) in section B.

These approximations agree at θ = π -√ ε, so we obtain that

C 0 = u A (1 - √ ε, π - √ ε). (2.91)
Thus

u unif (r, θ) = u A (r, θ) for θ ∈ [0, π - √ ε] u B (θ) for θ ∈ [π - √ ε, π].
(2.92)

The numerical solution of (2.14) in Ω w and the approximation u unif (r, θ) of (2.92) are shown Fig. 2.6B-D.

Potential drop in Ω w

The potential drop ∆funnel u between the center of mass C and the tip of the funnel S, is

∆ f unnel u = u(C) -u(S). (2.93)
Due to the axial symmetry of the domain Ω, the center of mass

C is at r = 1 - √ ε, hence (2.92) gives u(S) = u(1 - √ ε, π) and u(C) = u(1 - √ ε, c √ ε). (2.94)
Recall that the constant c depends on the domain geometry only, and is defined by the conformal mapping w (see relation (2.15)). The potential drop ∆Cusp u in the funnel can be decomposed as the sum of difference of potential between the two sections, A and B. First, the approximations are

∆u A = u A (1 - √ ε, π) -u A (1 - √ ε, c √ ε). ( 2
u A (1 - √ ε, θ 0 ) = -ln |e iθ 0 -1| 4 8(1 - √ ε) 2 λπ 2|∂Ω|(1 -cos θ 0 ) + λε 2 -ln cos 2 λπ 2 √ ε(ln(1 - √ ε) + √ ε) 2|∂Ω|(1 -cos θ 0 ) + λε . (2.98)
For ε 1, we get from (2.98) that

-ln cos 2 λπ 2 √ ε(ln(1 - √ ε) + √ ε) 2|∂Ω|(1 -cos θ 0 ) + λε = O(ε).
(2.99)

Hence, using (2.99) in (2.98), we get

u A (1 - √ ε, θ 0 ) = -ln |e iθ 0 -1| 4 8(1 - √ ε) 2 λπ 2|∂Ω|(1 -cos(θ 0 )) + λε 2 + O(ε). (2.100)
The approximate solution u A (S) at the tip of the funnel S (south pole at θ 0 = π) is (2.100)

u A (S) = -ln 2λ 2 π 2 (4|∂Ω| + λε) 2 + 2 ln(1 - √ ε) + O(ε). (2.101)
At the center C, where θ 0 = c √ ε, equation (2.98) gives for ε 1 the θ 0 -dependent terms in (2.100) as

|e iθ 0 -1| 4 = c 4 ε 2 + O(ε 3 ), (2.102)
and

2|∂Ω|(1 -cos c √ ε) + λε = ε(|∂Ω|c 2 + λ) + O(ε 2 ). ( 2
u A (C) = -ln c 4 8 λπ |∂Ω|c 2 + λ 2 + 2 ln(1 - √ ε) + O (ε) . (2.104)
For λ 1, (2.104) becomes 2.6E (red) and compared to ln λ 2 + const (green) and to a two-dimensional numerical solution. The good agreement confirms the validity of the asymptotic expansion and thus confirming the new asymptotic formulas derived here. We conclude with the general formula for a dimensional cusp-shaped funnel where |∂Ω| = |∂ Ω| Rc and R c is the radius of curvature at the cusp ∆u ∼ -ln λ 2 + 2 ln Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiology

u A (C) = -ln π 2 c 4 8 + 2 ln(1 - √ ε) + O ε, 1 λ . ( 2
πc 2 |∂ Ω| 4R c + O 1 λ . ( 2

Expansion of the potential drop between N and S

To expand the potential difference u(N ) -u(S) between the funnel tip S and the north pole N of Ω, we first use the results (2.114) computed above, to expand the difference u(C) -u(S), and then subtract (2.114) and (2.49). The the terms 2 ln(λ) drop out and we have

u(N ) -u(S) = 2 ln 4|∂Ω| R -2 ln πc 2 |∂Ω| 4R c + O 1 λ , (2.115)
where R is the distance between the north pole N and the center of mass C and R c is the radius of curvature at the cusp. We obtain to leading order 2.5. Discussion and conclusion

u(N ) -u(S) ∼ -2 ln πc 2 R 16R c , ( 2

Discussion and conclusion

In this thesis work we have derived new electrostatic laws in non-neutral confined electrolytes from nonlinear electro-diffusion theory (PNP equations). The effect of local geometrical structure, such as the local curvature of the boundary, emerges from the asymptotic solution of the model. The PNP equations describe the charge concentration and electric potential. The new electrical laws are derived in the context of non-electro-neutrality and we use a single ionic species. The approximation of the steady-state solution in a ball with an attached cusp-shaped funnel on its surface is new and the construction of the asymptotic expansion uses a new boundary layer analysis.

Using asymptotic and numerical solution of the PNP equation we found that, for a sufficiently high number of charges, the charge concentration peaks at the end of the funnel in a charged funnel boundary domain; but this is not the case for an uncharged funnel domain (Fig. 2.7A-C). This effect is clearly the result of the cusp-shaped geometry. The present analysis reveals that the curvature affects the membrane potential. We also found that the voltage increases logarithmically in the total number of excess charges N , which is valid for uncharged (2.47) and charged (2.52) cusp-shaped funnel on the boundary. We studied the voltage changes and electro-diffusion under an excess of positive ions. The voltage difference in the limit λ → ∞ is probably attenuated in a mixed ionic solution, but the electro-neutrality remains broken. Cytoplasmic ions are characterized by the following concentrations Na + = 148ml, K + = 10ml and Cl -= 4ml. There is a clear unbalance toward positive charges, however there are probably molecules of various sizes with negative charges to re-balance the charges. We can note that the motility of these proteins should be driven by a diffusion coefficient smaller than the one of the ions. This difference of motility is certainly a key feature in maintaining non-electro-neutrality and then tuning the value of λ.

We conclude that local geometrical properties, such as curvature, can modulate the local voltage in biological cellular electrolytes when electro-neutrality is violated. This result generalizes the case of a ball, where the distribution of charges accumulates on the surface as the total charge increases [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF]. Following a non uniform boundary curvature, we expect that charges will be non-uniformly distributed, leading to a difference of potential across the membrane with charges on its surface. Since this difference of potential plays a key role in information processing at synapses, we conclude that the spine geometry, in particular its curvature, may impact the coding or decoding of voltage through current [START_REF] Yuste | Dendritic Spines[END_REF]. This effect may as well influence the propagation and genesis of local depolarization [START_REF] Rall | Methods in Neuronal Modeling: from Synapses to Networks[END_REF][START_REF] Qian | An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF]. More realistic funnels with two different curvature radii can be incorporated to the formalism presented by modifying the parameter α (2.15) as shown in [START_REF] Holcman | Brownian motion in dire straits[END_REF]. The formalism presented in this paper can be applied beyond physiology, in particular in the design of nanopipettes with an optimal shape [START_REF] Perry | Characterization of Nanopipettes[END_REF][START_REF] Holcman | Brownian motion in dire straits[END_REF] by modulating α (2.15) or with a patterned surface [START_REF] Sparreboom | Eijkel Principles and applications of nanofluidic transport[END_REF] by changing the surface charge density via λ in region A (2.65).

Appendix

Regular expansion of PNP with two types of charge

In this appendix we show that, for biological concentrations of cations and anions found in literature [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF], the first order behavior of the electrical potential is obtained considering positive charges only. We assume that the charge of a (1, 1) electrolyte confined in Ω consists of identical N p positive and N m negative ions with initial particle density q p (x) and q m (x) such as

N i = Ω q i (x) dx, for i ∈ {p , m}.
(2.117)

Where p and m shall refer to positive and negative species, respectively. The total charge in Ω is then given by

Q = e(N p -N m ). (2.118)
The charge densities ρ p (x, t) and ρ m (x, t), for positive and negative charges respectively, are solutions of the initial and boundary value problem for the Nernst-Planck equation

D i ∆ρ i (x, t) + z i e kT ∇ (ρ i (x, t)∇φ(x, t)) = ∂ρ i (x, t) ∂t for x ∈ Ω (2.119) D i ∂ρ i (x, t) ∂n + z i e kT ρ i (x, t) ∂φ(x, t) ∂n = 0 for x ∈ ∂ Ω (2.120) ρ i (x, 0) = q i (x) for x ∈ Ω, (2.121)
where z i is the valence and D i is the diffusion coefficient for the ion specie i. Here φ(x, t) is the electric potential in Ω and is the solution of the Neumann problem for the Poisson equation

∆φ(x, t) = - e ε r ε 0 (ρ p (x) -ρ m (x)) for x ∈ Ω (2.122) ∂φ(x, t) ∂n = -σ(x, t) for x ∈ ∂ Ω,
where σ(x, t) is the surface charge density on the boundary ∂ Ω. In steady state, (2.119) gives

ρ i (x) = ρ i,0 exp - z i eφ(x) k B T for i ∈ {p , m}, (2.123)
where ρ i,0 is obtained from no-flux boundary condition (2.120), it yields

ρ i (x) = N i exp - z i eφ(x) k B T Ω exp - z i eφ(s) k B T ds for i ∈ {p , m}.
(2.124)

Appendix

We introduce the adimensionalized potential ũ(x) = e φ(x) k B T , then equation (2.123) becomes

ρ i (x) = N i e -z i ũ(x)
Ω e -z i ũ(s) ds for i ∈ {p , m}.

(2.125)

Then, using (2.122) and (2.125) we obtain

-∆ũ(x) = l B N p e -ũ(x) Ω e -ũ(s) ds - l B N m e ũ(x) Ω e ũ(s) ds in Ω (2.126) ∂u(x) ∂n = - (N p -N m ) |∂ Ω| l B on ∂ Ω,
where l B is the Bjerrum length. We adimentionalize (2.126) setting x = x R c , and

ũ(x) = u(x)
where R c is the cusp curvature radius, it yields

-∆u(x) = l B N p e -u(x) R c Ω e -u(s) ds - l B N m e u(x) R c Ω e u(s) ds in Ω (2.127) ∂u(x) ∂n = - l B (N p -N m ) R c |∂Ω| on ∂Ω.
Remark that Ω and ∂Ω have no dimensions. We translate u(x) = v(x) + ln R c l B N p in (2.127), and we introduce ζ = N m N p , then we obtain

-∆v(x) = l 2 B N 2 p e -v(x) R 2
c Ω e -u(s) ds

- ζe v(x)
Ω e u(s) ds

in Ω (2.128)

∂v(x) ∂n = - l B N p R c |∂Ω| + ζ l B N p R c |∂Ω| on ∂Ω.
Since the concentration of chloride is about 4 mM while potassium and sodium account together roughly for 167 mM [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF], we assume

ζ = N p N m 1. A regular expansion of v(x) in this limit is v(x) = v 0 (x) + ζv 1 (x) + • • • (2.129)
Using (2.129) in (2.128) gives to leading order

-∆v 0 (x) = l 2 B N 2 p e -v 0 (x) R 2
c Ω e -u(s) ds

in Ω (2.130)

∂v 0 (x) ∂n = - l B N p R c |∂Ω| on ∂Ω.
We can notice that the equation (2.130) does not depend on the negative charges N m . We study the electro-diffusion properties of a domain containing a cusp-shaped structure in three dimensions when one ionic specie is dominant. The mathematical problem consists in solving the steady-state Poisson-Nernst-Planck (PNP) equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary conditions are imposed on the boundary. We construct an asymptotic approximation for certain singular limits that agree with numerical simulations. Finally, we analyse the consequences of non-homogeneous surface charge density. We conclude that the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three dimensional electrostatic laws for non-electroneutral electrolytes. We discuss applications to dendritic spines in neuroscience.

Keywords. Electro-diffusion, Cusp-shaped Funnel, Poisson-Nernst-Planck, Electroneutrality, Möbius conformal map; Asymptotics; Nonlinear, Partial differential equation, Biophysics.

AMS subject classification. 35J66, 35B44, 35B25, 92C05, 92C37, 92C05

Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

Introduction

We study here the Poisson-Nernst-Planck (PNP) equations in three dimensional domains containing a cups-shaped funnel. These equations are used to model electrodiffusion in neurobiology [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF][START_REF] Bezanilla | How membrane proteins sense voltage[END_REF], where the charge concentration is coupled to the electric potential. We consider here a convex domain formed of a ball with an attached cusp-shaped funnel on its surface. Such geometry is common in cellular neurobiology, for instance dentritic spines [START_REF] Bourne | Balancing structure and function at hippocampal dendritic spines[END_REF], structure that cannot be reduced to 1D geometry [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF]. Phenomenological descriptions of electro-diffusion, such as the linear cable theory or RC circuit representation, and even electronic devices, are not sufficient to describe non-cylindrical geometry [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF], since they assume a simple reduced one-dimensional or overly simplified geometry.

We present here results about voltage distribution of based on electro-diffusion in various geometrical microdomains, when the condition of electro-neutrality is not satisfied and one ionic specie dominates. The boundary is impermeable to particles (ions) and the electric field satisfies the compatibility condition resulting from Poisson's equation. We recall that under the non-electro-neutrality assumption, and with charge distributed in bounded domains confined by a dielectric membrane, Debye's concept of charge screening decaying exponentially away from a charge [START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF] does not apply and long-range correlation leads to a gradient of charges in a ball with no inward current. A new capacitance law was derived for an electrolyte ball [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF] and for a two-dimensional cusp [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF], where the difference of potential V (C) -V (S) between the center C and the surface S increases, first linearly and then logarithmically when the total number of charges in the ball increases.

Our aim here is to estimate the effect of boundary curvature on three-dimensional electrical domains such as dendritic spines. In particular, we explore the effect of boundary curvature on the charge and field distribution at steady state. The curvature of neuronal dendrites and axons membranes possesses many local maxima that can modulate the channel's local electric potential [START_REF] Bezanilla | How membrane proteins sense voltage[END_REF][START_REF] Cartailler | Nanophysiology of dendritic spines: electro-diffusion for voltage modulation and conduction[END_REF][START_REF] Yuste | Dendritic Spines[END_REF]. In this article, we study the effects of local curvature on the distribution of charge in bounded domains with no electro-neutrality. The effect of non-electro-neutrality was recently studied in [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF][START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] and a long-range electrostatic length, much longer than the Debye length was found. This effect is due to the combined effects of non-electro-neutrality and di-electric boundary, which lead to charge accumulation. The cusp-shaped funnel geometry was studied in [START_REF] Holcman | Brownian motion in dire straits[END_REF], however this paper presents several crucial mathematical differences with [START_REF] Holcman | Brownian motion in dire straits[END_REF], in particular, we are solving in the present paper a nonlinear problem with non-homogeneous Neumann boundary conditions. Besides, we used an analysis based on conformal mapping and a new boundary layer analysis, different from the classical matched asymptotics method [START_REF] Pillay | An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains[END_REF][START_REF] Delgado | Conditional Mean First Passage Times to Small Traps in a 3-D Domain with a Sticky Boundary[END_REF][START_REF] Lindsay | First Passage Statistics for the Capture of a Brownian Particle by a Structured Spherical Target with Multiple Surface Traps[END_REF].

The manuscript is composed of three parts: in sections 1 and 2, we extend the results we have obtained in [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF], that describe the voltage in a planar cusp with homogeneous surface charge density. We then focus on an uncharged cusp for a 3D cusp-shaped funnel. In the third section, we extend the results derived in section 1 to a non-homogeneous surface charge density. We summarized now the new electrostatic laws we derived here for the difference of potential V (C) -V (S) where C is the center of mass of the domain and S is located at the bottom of funnel For a constant surface charge density (section 3.2, eq. (3.76)), the voltage difference is given by

V (C) -V (S) = kT e    ln sin 2 π|∂ Ω| (e 2 /kT )N ε -ln 2 (e 2 /kT ) 2 π 2 N 2 R 2 c 4|∂ Ω| + (e 2 /kT )N ε 2 + O(1)    ,
that depends the number N of ions enclosed in the domain Ω, the thermal energy kT and the elementary charge e of the electron (1.602 • 10 -19 C), the cusp-shaped funnel width at the base ε, and its curvature radius R c (see. Fig. 3.1A).

When the surface of the cusp does not carry any charges, the voltage difference (section 3.3, eq. (3.97)) is

V (C) -V (S) = kT e -ln 8R c ε π 4 |∂ Ωε | (1 + N bulk /N ε ) -ln sin 2 2|∂ Ωε | (e 2 /kT )N ε R c ε + O(1) ,
which depends on the surface |∂Ω ε | at the end of the funnel, the number of charges N bulk and N ε in bulk and at the end of the funnel respectively. When the surface charge density is non-homogeneously distributed, the potential differences (section 3.4, formula (3.103))

V (C) -V (S) = kT e    ln sin 2 π|∂ Ωε | (e 2 /kT )N ε ε -ln 2 (e 2 /kT ) 2 π 2 N 2 cusp R 2 c 4|∂ Ωcusp | + (e 2 /kT )N cusp ε 2 + O(1)    ,
which depends on the total surface charge density N cusp on the cusp.

These new electrostatics expressions are asymptotic formula derived in the limit ε 1 and for a large number of charge. There are the main results of the present study. Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

The Poisson-Nernst-Planck equations

The Poisson-Nernst-Planck system of equations is a model of electro-diffusion. In a domain Ω, the total charge in Ω results from the sum of the positive N p and negative N m charges. The concentration of mobile ions [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF] shows an imbalance of positive negative ions N p N m , such that the charge in Ω can be approximated [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] by N identical positive ions with an initial density q(x) in Ω. The valence is z and the total number of particles is fixed equal to Ω q(s) ds = N.

(3.1)
It follows that the charge in Ω is

Q = zeN,
where e is the electronic charge. The charge density ρ(x, t) is the solution of the initial and boundary value problem for the Nernst-Planck equation

D ∆ρ(x, t) + ze kT ∇ (ρ(x, t)∇φ(x, t)) = ∂ρ(x, t) ∂t for x ∈ Ω (3.2) D ∂ρ(x, t) ∂n + ze kT ρ(x, t) ∂φ(x, t) ∂n = 0 for x ∈ ∂ Ω (3.3) ρ(x, 0) = q(x) for x ∈ Ω, (3.4)
where kT represents the thermal energy. Here φ(x, t) is the electric potential in Ω and is the solution of the Neumann problem for the Poisson equation

∆φ(x, t) = - zeρ(x, t) ε r ε 0 for x ∈ Ω (3.5) ∂φ(x, t) ∂n = -σ(x, t) for x ∈ ∂ Ω, (3.6)
where ε r ε 0 is the permitivity of the medium and σ(x, t) is the surface charge density on the boundary ∂ Ω.

Steady solution in a ball with a cusp-shaped funnel

To study the effect of a narrow funnel attached to a sphere filled with an electrolyte as illustrated Fig. 3.1A, we study the solution of the steady state equation (3.2) We consider now the PNP problem (3.11)-(3.12) in the solid of revolution (Fig. 3.1A), obtained by rotating the symmetric planar domain Fig. 3.1B about its axis of symmetry (z-axis). Consequently, Ω represents now a ball with a cusp-shaped funnel, with a radius curvature R c at the entrance of the funnel (blue dashed circles in Fig. 3.1A-B).

ρ(x) = N exp - zeφ(x) kT Ω exp - zeφ(s) kT ds , (3.7
Using x = x R c , ∂Ω = ∂ Ω R c 2 and Ω = Ω R c 3 and u(x) = ū(x)+ln λR 2 c / Ω exp{-u(s)} ds converts (3.11) into -∆u(x) = exp{-u(x)} for x ∈ Ω (3.13) ∂u(x) ∂n = - λ |∂Ω|R c for x ∈ ∂Ω.
The non-dimensional surface charge density is

σ = λ |∂Ω|R c . (3.14)
We first consider a uniform surface charge density in (3.13) and then study the consequence of a non-homogeneously distributions.

Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

Poisson-Nernst-Planck solutions in a 3D cusp-shaped funnel

The cylindrical symmetry of the Neumann boundary value problem (BVP) (3.13) in the (r, z, φ) coordinates (Fig. 3.1A) centered on the axis of symmetry, implies that ũ(x) is independent of the angle φ in the domain Ω. It follows that (3.13) in the domain Ω can be written as

∂ 2 u(r, z) ∂r 2 + 1 r ∂u(r, z) ∂r + ∂ 2 u(r, z) ∂z 2 = -exp(-u(r, z)) (3.15) ∂u(r, z) ∂n = -σ,
where n = [n r , n z ] T is the outward normal unit vector to the surface ∂Ω and r is the distance to the symmetry axis of Ω. The opening at the cusp funnel is small AB = ε 1 (green line Fig. 3.1B), so the funnel is a narrow passage. To remove the cusp singularity, we use first the transformation to the rotated and translated coordinates given by r = r -1 -ε/2 and z = -z + 1. Setting u(r, z) = ũ(r, z), eq. (3.15) becomes,

∂ 2 ũ(r, z) ∂r 2 + ∂ 2 ũ(r, z) ∂ z2 + 1 (r + 1 + ε/2) ∂ ũ(r, z) ∂r = -exp(-ũ(r, z)) (3.16) ∂ ũ(r, z) ∂ ñ = -σ.
We shall construct an asymptotic expansion of the solution ũ(r, z) for small ε by first mapping the cross section in the (r, z)-plane conformally into its image under the Möbius transformation [START_REF] Holcman | Brownian motion in dire straits[END_REF] w

(ξ) = ρe iθ = ξ -α 1 -αξ , (3.17)
where

α = -1 - √ ε + O(ε), (3.18)
and ξ = r +iz. In the dimensionless domain Ω, the parameter ε is also dimensionless and R c ε = ε. Möbius transformation maps the two osculating circles A and B (dashed blue) into concentric circles (see Fig. 3.1B-C). The Möbius transformation (3.17) maps the right circle B (dashed blue) into itself and Ω is mapped onto the banana-shaped domain Ω w = w(Ω) as shown in Figure 3.1C. The second order derivative for ũ(ξ) = v(w) is computed using (3.17) in (3.16) [80]

∂ 2 ũ ∂r 2 + ∂ 2 ũ ∂ z2 = |w (ξ)| 2 ∆ w v(w). (3.19)
In the small ε limit, we have

|w (ξ)| 2 = |(1 - √ ε)e iθ -1 + O(ε)| 4 4ε + O(ε 3/2 ) . (3.20)

The Poisson-Nernst-Planck equations

The 3D BVP (3.16) differs from the 2D problem [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] by the extra first order radial derivative. For small ε limit, we have

r + 1 + ε/2 = ε 1 -cos(θ) + O(ε 3/2 ). (3.21)
In complex coordinates we have ∂u(r, z) ∂r = e (∇u(ξ)) ,

where e(•) is the real part. Under the conformal mapping (3.17), the gradient from (3.22) transforms as follows [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF] ∇u(ξ) = ∇ w v(w) w (ξ).

(3.23)

Using polar coordinates (ρ, θ) in the mapped domain Ω w , we write

w (ξ) = w 1 (ρ, θ) + i w 2 (ρ, θ), (3.24)
where i 2 = -1. Using (3.17), we obtain To leading order, using (3.21) and (3.26), we get (Appendix)

w1 (ρ, θ) = 1 -α 2 ρ 2 + 2αρ cos(θ)(1 + αρ cos(θ)) 1 -α 2 (3.25) w2 (ρ, θ) = -2αρ sin(θ) 1 + αρ cos(θ) 1 -
1 r ∂ ũ(r, z) ∂r = - ρ(1 -cos(θ)) 2 ε 3/2 ∂ṽ(ρ, θz) ∂ρ - sin(θ)(1 -cos(θ)) ε ∂ṽ(ρ, θz) ∂θ .(3.27)
In summary, using (3.19) in polar (ρ, θ)-coordinates, eq. (3.27) and (3.16) in Ω w , are changed to

|(1 - √ ε)e iθ -1| 4 4ε ∂ 2 ṽ(ρ, θ) ∂ρ 2 + 1 ρ ∂ṽ(ρ, θ) ∂ρ + 1 ρ 2 ∂ 2 ṽ(ρ, θ) ∂θ 2 - ρ(1 -cos(θ)) 2 ε 3/2 ∂ṽ(ρ, θz) ∂ρ - sin(θ)(1 -cos(θ)) ε ∂ṽ(ρ, θz) ∂θ = -exp {-ṽ(ρ, θ)} ∂ṽ(ρ, θ) ∂n = - σ √ ε 1 -cos(θ) . ( 3

Asymptotic analysis of the PNP equations in a cuspshaped funnel

To analyse eq. (3.28) in the limit of σ 1, ε → 0 [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF], we approximate the domain Ω w by two subregions

A = {(ρ, θ) ∈ Ω w : |θ - √ ε| > π, |ρ -1| ≤ √ ε} (3.29) B = {w = (1 - √ ε)e iθ : |θ -π| ≤ √ ε},
as illustrated in Fig. 3.2A. The regions B consists of a circular arc (dashed red).

We construct now the solution u A (r, θ) and u B (θ) of (3.13) in each subregion.

= θ 0 = const, for ρ ∈ [1 - √ ε, 1]. Setting u A (ρ, θ 0 ) = v(ρ, θ 0 ), to leading order in σ √ ε, equation (3.28) reduces to -e -u A (ρ, θ 0 ) = |(1 - √ ε)e iθ 0 -1| 4 4ε ∂ 2 u A (ρ, θ 0 ) ∂ρ 2 + 1 ρ ∂u A (ρ, θ 0 ) ∂ρ (3.30) - ρ(1 -cos(θ 0 )) 2 ε 3/2 ∂ ũA (ρ, θ 0 ) ∂ρ du A (ρ, θ 0 ) dρ ρ=1 = - √ ε 1 -cos(θ 0 ) du A (ρ, θ 0 ) dρ ρ=1- √ ε = 0.

The Poisson-Nernst-Planck equations

In the limit ε 1, we note that

|ρe iθ 0 (1 - √ ε) -1| 4 = |e iθ 0 -1| 4 + O( √ ε)
and using the change of variable ρ = ρ√ ε and setting u A (ρ, θ 0 ) = v A (ρ, θ 0 ), to leading order in ε 1, eq. (3.30) becomes

- 4ε 2 e -v A (ρ, θ 0 ) |e iθ 0 -1| 4 = ∂ 2 v A (ρ, θ 0 ) ∂ ρ2 - √ ε ∂v A (ρ, θ 0 ) ∂ ρ 1 - 4(1 -cos(θ 0 )) 2 |e iθ 0 -1| 4 . (3.31)
Using the function,

h(θ 0 ) = 4ε 2 |e iθ 0 -1| 4 (3.32)
and ṽA (ρ, θ 0 ) = v A (ρ, θ 0 ) -ln(h(θ 0 )), eq. (3.31) is transformed into Using a regular expansion in the small ε limit (in the regime

∂ 2 ṽA (ρ, θ 0 ) ∂ ρ2 = -e -v A (ρ, θ 0 ) + √ ε ∂v A (ρ, θ 0 ) ∂ ρ 1 - (1 -cos(θ 0 )) 2 |e iθ 0 -1| 4 . ( 3
σε 3/2 = O(1)) ṽA (ρ, θ 0 ) = ṽA,0 (ρ, θ 0 ) + √ εṽ A,1 (ρ, θ 0 ) + O(ε) (3.34) in (3.33), we get ∂ 2 ṽA,0 (ρ, θ 0 ) ∂ ρ2 = -e -ṽ A,0 (ρ, θ 0 ) (3.35) ∂ṽ A,0 (ρ, θ 0 ) ∂ ρ ρ=0 = σε 1 -cos(θ 0 ) ∂ṽ A,0 (ρ, θ 0 ) ∂ ρ ρ=1 = 0.
Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels A direct integration of (3.35) is [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] ṽA,0 (ρ,

θ 0 ) = ln 2C 1 (θ 0 ) 2 cos 2 ρ + C 2 (θ 0 ) 2C 1 (θ 0 ) , (3.36)
where C 1 (θ 0 ) and C 2 (θ 0 ) are two constants that depend on θ 0 . To compute these constants, we differentiate (3.36)

ṽ A,0 (ρ, θ 0 ) = -1 C 1 (θ 0 ) tan ρ + C 2 (θ 0 ) 2C 1 (θ 0 ) . ( 3
σεC 1 (θ 0 ) (1 -cos(θ 0 )) = tan 1 2C 1 (θ 0 ) . (3.39)
In the regime σ = O(ε -3/2 ), we have

C 1 (θ 0 ) = 2(1 -cos(θ 0 )) + σε πσε + O 1 σε . ( 3
v A (ρ, θ 0 ) = ln 2 2(1 -cos(θ 0 )) + σε πσε 2 + ln 4ε 2 |e iθ 0 -1| 4 (3.42) + ln cos 2 πσε(ρ -1) 2(2|(1 -cos(θ 0 )) + σε) +O( √ ε).
In particular the solution at ρ

= 1 - √ ε is u A (1 - √ ε, θ 0 ) = ln 8 2(1 -cos(θ 0 )) + σε πσ|e iθ 0 -1| 2 2 + O( √ ε). (3.43)
We note that the three dimensional solution (3.43) is identical to the one obtained inside a planar cusped-shaped domain [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF].

The Poisson-Nernst-Planck equations

Asymptotics of u B (θ) in region B

The asymptotic solution u A (ρ, θ) in A does not satisfy the boundary condition (3.28) at θ = π. Indeed, ∂u A (ρ, θ)/∂θ| θ=π = 0, while the boundary condition (3.28) is ∂v/∂θ| θ=π = -σ √ ε/2 1, thus a boundary layer should develop. The boundary layer solution u B (θ) is derived by taking into account the θ derivatives in eq. (3.28):

|(1 - √ ε)e iθ -1| 4 4ρ 2 ε ∂ 2 u B (θ) ∂θ 2 + sin(θ)(1 -cos(θ)) ε ∂ ũB (θ) ∂θ = -e -u B (θ) . (3.44)
In small ε limit, for ρ = 1 -√ ε, we have

4ε |ρe iθ (1 - √ ε) -1| 4 = ε 4 , (3.45)
which is constant. Using (3.45) in (3.44) and η = π -θ, we define u B (θ) = ũB (η), leading to

∂ 2 ũB (η) ∂η 2 - 1 4 sin(η)(1 + cos(η)) ∂ ũB (θ) ∂η = - ε 4 e -ũ B (η) . (3.46)
Since 0 ≤ η ≤ √ ε, we shall approximate the first order term and thus eq. (3.46) reduces to

∂ 2 ũB (η) ∂η 2 - η 2 ∂ ũB (θ) ∂η = - ε 4 e -ũ B (η) . (3.47)
Using v(η) = u B (η) -ln (4/ε), eq. (3.47) is transformed to

- ∂ 2 ṽ(η) ∂η 2 + η 2 ∂ṽ(η) ∂η = e -ṽ(η) . (3.48)
Using the boundary condition (3.28), we further reduce the solution v(η) to the equation

- ∂ 2 ṽ(η) ∂η 2 = e -ṽ(η) + O(λε 2 ) (3.49) ∂v(η) ∂η η=0 = σ √ ε 2 ∂v(η) ∂η η= √ ε = 0.
The solution is ṽ (3.52)

(η) = ln 2 C2 1 cos 2 η + C2 2 C1 , ( 3
In the limit σ 1, we have

C1 = 2 π √ ε 2 + 2 σ √ ε + O 1 (σ √ ε) 3 . (3.53)
We note that η 2 ∂ṽ(η) ∂η is small, justifying our simplifications. We conclude from

u B (θ) = ln cos 2 π 2 (θ -(π - √ ε)) 2 ε 1 - 4 σε + C 0 , (3.54)
where C 0 is a constant that we find in the next paragraph by matching the solution in two regions A and B.

A uniform approximation of u(ρ, θ) in Ω w

We now construct a uniform asymptotic approximation u unif (ρ, θ) in the region A ∪ B (Fig. 3.3A) using u A (ρ, θ) with u B (ρ, θ) that match for θ = π -√ ε, leading to

C 0 = u A 1 - √ ε, π - √ ε . (3.55)
Using the analytical expression (3.43) of u A , we get

C 0 = ln (4 + σε) 2 2(πσ) 2 . (3.56) Thus, u B (θ) = ln cos 2 π 2 (θ -(π - √ ε)) 2 ε 1 - 4 σε + ln (4 + σε) 2 2(πσ) 2 . (3.57)
Consequently, using (3.43) and (3.57) the solution in the funnel is

u unif (ρ, θ) =              ln 8 2(1 -cos(θ)) + σε πσ|e iθ -1| 2 2 , for θ ∈ [0, π - √ ε] ln cos 2 π 2 (θ -(π - √ ε)) 2 ε 1 - 4 σε + ln (4 + σε) 2 2(πσ) 2 , for θ ∈ [π - √ ε, π].
(3.58)

The numerical solution of eq. (3.13) in Ω w and the approximation u unif (ρ, θ) of (3.58) are shown in Fig. 3.3B.

The Poisson-Nernst-Planck equations

Estimating the potential drop in Ω w

The difference of potential between the center of mass C and the tip of the funnel S (see Fig. 3.3A) is defined as

∆funnel u = u(C) -u(S), (3.59)
where

u(S) = u(1 - √ ε, π) and u(C) = u(1 - √ ε, c √ ε), (3.60)
u is solution of eq. 3.13 and the constant c depends on the domain geometry and is defined by the conformal mapping w (relation (3.17)). To compute ∆funnel u, we use the two differences

∆u A = u A (1 - √ ε, π) -u A (1 - √ ε, c √ ε), (3.61)
and

∆u B = u B (π) -u B (π - √ ε). (3.62) It follows that ∆funnel = ∆u A + ∆u B . (3.63)
To compute ∆u A , we use the analytical expression (3.43) for ρ = 1 -√ ε and any θ 0 ,

u A (1 - √ ε, θ 0 ) = -ln |e iθ 0 -1| 4 8(1 - √ ε) 2 σπ 2(1 -cos(θ 0 )) + σε 2 + O(ε). (3.64) At the point S (θ 0 = π), u A (S) = -ln 2σ 2 π 2 (4 + σε) 2 + 2 ln(1 - √ ε) + O(ε). (3.65)
To estimate u A (C) for which θ 0 = c √ ε, we observe that for ε 1 in relation (3.64),

|e iθ 0 -1| 4 = c 4 ε 2 + O(ε 3 ), (3.66)
and

2(1 -cos(c √ ε)) + σε = ε(c 2 + σ) + O(ε 2 ). (3.67)
We use (3.66) and (3.67), so eq. (3.64) reduces to

u A (C) = -ln c 4 8 σπ c 2 + σ 2 + 2 ln(1 - √ ε) + O (ε) . (3.68)
In the large σ limit, Equation (3.74) shows that for σ 1, the potential drop in the cusp-shaped funnel is dominant in region B. We compare (Fig. 3.3C) expression (3.76) with the numerical solution of 3.13. We note that the distribution of the potential inside a 3D solid funnel is to leading order identical to the one we obtained inside a planar cusp [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF].

u A (C) = -ln π 2 c 4 8 + 2 ln(1 - √ ε) + O ε, 1 σ . ( 3

The PNP equations in a cusp-shaped domain with non-homogeneous surface charge density

When the surface charge density is not homogeneously distributed over the surface ∂Ω, we expect a re-organization of the potential u of (3.13). we subdivide the surface ∂Ω into three regions (Fig. 3.4), we obtain that

∂Ω = ∂Ω ε ∪ ∂Ω cusp ∪ ∂Ω bulk , (3.78
λ = λ ε + λ cusp + λ bulk . (3.81)
We will use the notation

σ j = λ j |∂Ω j | , (3.82)
where j ∈ {ε , cusp , bulk}.

PNP solutions for

σ cusp = 0, σ bulk = σ ε = σ in 3D
To compute the solution of (3.13) for an uncharged funnel (σ cusp = 0), we will use the same conformal mapping (3.17) as describe above with now reflecting boundary condition on ∂Ω cusp , which are invariant under the conformal mapping. As a result Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels the boundary conditions on the two like-style arcs of the domain Ω w are also reflective. Consequently, instead of searching a solution in the banana-shaped domain Ω w , we will construct it in the circular arc as a one-dimensional solution.

The boundary value problem (3.28) in the conformal image Ω w becomes ṽ -

4 sin(θ)(1 -cos(θ)) |e iθ -1 -e iθ √ ε| 4 ṽ = - 4ε |e iθ -1 -e iθ √ ε| 4 exp -ṽ(e iθ ) (3.83) ṽ (c √ ε) = 0 ṽ (π) = - σ √ ε 2 .
To construct an asymptotic approximation to the solution of (3.83) in the limits ε → 0 and σ → ∞, we first construct the outer-solution in the form of a series in powers of ε, which is an approximation valid away from the boundary θ = c √ ε. After dropping the terms in ε in 3.83, we obtain the outer solution by a direct integration v 1 (θ) = -A(θ -sin(θ)) + ṽ(0), (3.84) where ṽ(0) and A are constants. The outer solution (3.84) cannot satisfy all boundary conditions, consequently a boundary layer correction is needed at θ = π. An approximation of the solution can be obtained by freezing the power-law term and neglecting the first order derivatives in (3.83), for which the equation is for a generic parameter b > 0,

d 2 dθ 2 v b (θ) + be -v b (θ) = 0, dv b (0) dθ = v b (0) = 0.
The solution is [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF] v b (θ) = ln cos 2 b 2 θ .

(3.85)

Putting the outer and boundary layer solutions together gives the uniform asymptotic approximation

y unif (θ) = -A(θ -sin(θ)) + ṽ(0) + ln cos 2 b 2 θ . (3.86)
The condition at θ = π gives that

y unif (π) = -2A -b tan b 2 π = - σ ε √ ε 2 .
The compatibility condition for (3.13),

λ ε + λ bulk = Ω exp{-u(x)}dS x , (3.87)
3.3. The PNP equations in a cusp-shaped domain with non-homogeneous surface charge density

gives in Ω w that

λ ε + λ bulk = Ωw exp{-ṽ(w)} dw |φ (φ -1 (w))| . (3.88)
Using the uniform approximation (3.86) in the compatibility condition (3.88), we obtain the second condition

λ ε + λ bulk = 8 √ ε e -ṽ(0) π c √ ε 1 cos 2 b 2 θ exp {A(θ -sin(θ))} |e iθ (1 - √ ε) -1| 4 dθ ≈ 8 e -ṽ(0) ε π/ √ ε 0 1 cos 2 b 2 √ εξ exp{A( √ εξ -sin( √ εξ))} |1 + ξ 2 | 2 dξ, (3.89)
where we used the change of variable θ = √ εξ. Integrating by parts, we get for ε 1

λ ε + λ bulk ∼ 8 e -ṽ(0) ε        2 b √ ε tan b 2 π e Aπ 1 + π √ ε 2 2 - π/ √ ε 0 2 b √ ε tan b 2 θ Ψ(θ) dθ        , (3.90)
where

Ψ(ξ) = d dξ exp{A( √ εξ -sin( √ εξ))} |1 + ξ 2 | 2 . (3.91)
Thus, it remains to solve the asymptotic equation

λ ε + λ bulk ∼ 8 e -ṽ(0) ε 1/2 2 bπ 4 tan πb 2 exp{Aπ} + O ln cos πb 2 . (3.92)
for A and b in the limit ε → 0. We consider the limiting case where

A σ ε √ ε 1 for σ ε → ∞, (3.93)
for which condition (3.87) can be simplified and gives to leading order

b tan πb 2 = σ ε √ ε 2 , (3.94) that is, for σ ε √ ε 1 (3.94) gives b ≈ 1 - 4 π 1 σ ε √ ε , tan b 2 π ∼ σ ε √ ε 2 .
Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels It follows from (3.92) using (3.82) that

A = - 1 π     ln     8ε π 4 |∂Ω ε | 1 + λ bulk λ ε     -ṽ(0)     .
(3.95)

We conclude from expression 3.86 that

y unif (θ) = 1 π     ln     8ε π 4 |∂Ω ε | 1 + λ bulk λ ε     -ṽ(0)     (θ -sin(θ)) (3.96) + ln cos 2     1 - 4 π 1 σ ε √ ε 2 θ     + ṽ(0).
We compare in Fig. 3.5A-D, the uniform approximation (3.96) with numerical simulations of the reduced eq. (3.83) and the three-dimensional numerical solution (eq3. [START_REF] Bandle | Asymptotic behaviour of large solutions of quasilinear elliptic problems[END_REF]). The difference of potential ∆y unif = y unif (0)-y unif (π), can now be estimated using (3.96) and we obtain

∆y unif = -     ln     8ε π 4 |∂Ω ε | 1 + λ bulk λ ε     -ṽ(0)     -ln sin 2 2 σ ε √ ε .(3.97)
In the small ε limit, the constant ṽ(0) = O(1) can be neglected. We compare the analytical expression for difference of potential (3.97) with the result of the reduced equation (3.83) computed numerically in Fig. 3.5E. We note that the solution in 3D differs from 2D, as shown in Fig. 3.5F.

3.4 PNP solution for σ ε = σ cusp and σ bulk = O(1)

Analytical representation of the PNP solution

We study here the effect of the charge density σ cusp located on the cusp-shaped funnel on the solution u(x) of in the small ε and large σ cusp limits, such as 1) and σ bulk = O(1).

-∆u(x) = exp{-u(x)} for x ∈ Ω (3.98) ∂u(x) ∂n = -σ ε on ∂Ω ε ∂u(x) ∂n = -σ cusp on ∂Ω cusp ∂u(x) ∂n = -σ bulk on ∂Ω bulk , 3
σ cusp √ ε 1, σ ε /σ cusp = O(
In the large σ cusp √ ε limit, we have shown (section 3.2) that for θ in the range [c √ ε, π -√ ε] (region A, Fig. 3.2), the angular derivatives of u unif can be neglected. We thus use the result of eq. (3.43) by changing σ by σ cusp to obtain

u cusp (ρ, θ) = ln 2 2(1 -cos(θ)) + σ cusp ε πσ cusp ε 2 + ln 4ε 2 |e iθ -1| 4 (3.99) + ln cos 2 πσ cusp (ρ -ε) 2(2|(1 -cos(θ)) + σ cusp ε) +O( √ ε). For ρ = 1 - √ ε and θ ∈ [c √ ε, π - √ ε], we get u cusp (1 - √ ε, θ) = ln 8 2(1 -cos(θ)) + σ cusp ε πσ cusp |e iθ -1| 2 2 + O( √ ε). (3.100)
To construct a uniform solution u unif , we match to a solution u B in region

B = {(ρ, θ), θ ∈ [π - √ ε, π] and ρ = 1 - √ ε}.
We obtain the general expression Thus the difference of potential u(C) -u(S) between the center of mass C and the funnel base S is then

u unif (ρ, θ) =            ln 8 2(1 -cos(θ)) + σ cusp ε πσ cusp |e iθ -1| 2 2 for θ ∈ [0, π - √ ε] u B (θ) for θ ∈ [π - √ ε, π].
V (C) -V (S) = -ln 2σ cusp π 2 (4 + σ cusp ε) 2 + ln π 2 c 4 8 + ∆u B + O ε, 1 σ cusp , (3.102)
where c √ ε is the angular coordinate of the mapped center of mass C in Ω w . We compare in Fig. 3.6A-B the analytical expression (dashed) of (3.101) with the threedimensional numerical simulations (solid) of u (eq. (3.98)). When u B is given by expression (3.57) with condition σ ε √ ε 1, then the difference of potential is given by

u(C) -u(S) = kT e ln sin 2 π σ ε ε -ln 2σ cusp π 2 (4 + σ cusp ε) 2 + O(1).
(3.103)

The two conditions σ ε /σ cusp = O(1) and σ bulk = O(1) imply that the uniform solution is not affected by the bulk or the tip of the cusp. This is in contrast with the results computed for σ cusp = 0 (section 3.3.1) for which the solution in the cusp is entirely defined by the surface charge densities σ cusp and σ bulk (see eq. (3.96)). However, when the previous conditions are not satisfied (σ ε /σ cusp = O(1) is not verified), the numerical solution (red) and the analytical expression (3.101) (dashed blue) do not agree (Fig. 3.6A-B).

PNP solution with reflecting boundary at the end of the funnel

When we impose a reflecting boundary condition at the end of the cusp ∂Ω ε (σ ε = 0), we construct an approximation of equation (3.98) in the regimes ε 1 and σ cusp 1 in the following regime of parameters σ cusp √ ε 1 and σ bulk = O(1). To construct the approximation u unif in Ω w , we use expression in the cusp (3.101), where the solution u B is constructed by extending u cusp (ρ, θ) to region B. We have

∂u unif (ρ, θ) ∂θ θ=π = 0. (3.104)
To show that u cusp satisfies the same boundary condition, we differentiate u cusp (ρ, θ), (eq. (3.99)), in θ at θ = π:

∂u cusp (ρ, θ) ∂θ θ=π = 0. (3.105)
We conclude that u cusp matches at θ = π the boundary condition satisfied by the solution u(x) for σ ε = 0. Consequently,

u unif (ρ, θ) = ln 8 2(1 -cos(θ)) + σ cusp ε πσ cusp |e iθ -1| 2 2 . (3.106)
Thus the difference of potential between the funnel base S and the center of mass C is

u(C) -u(S) = -ln 2σ cusp a 2 π 2 (4 + σ cusp ε) 2 + ln π 2 c 4 8 .
(3.107)

Discussion and conclusion

We obtain a good agreement between the analytical expression (eq. (3.106)) and the three dimensional numerical solution of (3.98) (Fig. 3.6C).

The result obtained from (3.107) can be used to model the voltage in a domain with a cusp-shaped funnel connecting a reservoir with a fixed electrical potential and zero electric field at the of funnel-reservoir junction. This no field condition is satisfied when σ ε = 0. This result can be applied to the electrical properties of dendritic spines with a short neck (see [START_REF] Yuste | Dendritic Spines[END_REF], p.28, Fig. 3.9, spine 7), approximated by a cusp and the parent dendrite as a reservoir.

Discussion and conclusion

We have studied here, based on the steady-state solution of the Poisson-Nernst-Planck equations, the electrostatic properties of non electro-neutral electrolytes confined in a cusp-shaped funnel geometry. We showed that the local curvature and the distribution of surface charge density shape the electrical landscape within small domains. The new electrical properties have been obtained for a dominant ionic specie, in an electrolyte having an excess of charges [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF]. The new mathematical methods consist here in the construction of an asymptotic expansion of the nonlinear PNP equations inside 3D domains, with non-homogeneous Neumann boundary conditions.

Using asymptotics methods validated by numerical solutions of the PNP equations, we found several explicit voltage drops: first, for a surface charge density homogeneously distributed, the electrical potential distribution in 3D and 2D domains is quite similar to leading order potential inside a planar cusp (Fig. 3.3). However, the voltage inside an uncharged funnel (Fig. 3.5), associated to the condition σ cusp = 0 varies significantly between a 2D and 3D. We summarize in table 3.1 the results we have obtained in the three sections above, where we use the dimensional physical units and used σ i = σi zeR c /kT (section 3.2). The presence of negative ions in Ω may slightly reduces the voltage. However, as shown in [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF], accounting for negative charges carried by chloride anions present in the cytosol at physiological concentration [START_REF] Hille | Ionic Channels of Excitable Membranes[END_REF], does not alter the voltage to leading order. Consequently the voltages summarized in table 3.1 provide insights for understanding the electro-diffusion properties. The present results could be used in the design of quartz nanopipettes with an optimal shape [START_REF] Holcman | Brownian motion in dire straits[END_REF][START_REF] Perry | Characterization of Nanopipettes[END_REF][START_REF] Jayant | Targeted intracellular voltage recordings from dendritic spines using quantum-dot-coated nanopipettes[END_REF]. It would be interesting to vary the surface charge densities [START_REF] Sparreboom | Eijkel Principles and applications of nanofluidic transport[END_REF] in some sub-regions σ bulk , σ cusp or σ ε (3.79).

These analytical results can be used to predict the voltage drop in neuronal microdomains such as dendritic spines [START_REF] Yuste | Dendritic Spines[END_REF]. The local curvature is certainly a key factor in modulating the voltage and thus we are beginning to understand how nanoand micrometer geometry can encode synaptic modulation, that underlyes learning and memory in the Brain. Indeed, in compartment such as dendritic spines, the high curvature variation play a major role in converting injected current into voltage. This effect may as well influences the propagation and genesis of local depolarization in excitable cells [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Rall | Methods in Neuronal Modeling: from Synapses to Networks[END_REF][START_REF] Qian | An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons[END_REF]. We introduce X and Y such as w = X + iY and

Conditions V (C) -V (S) σ 1 kT e ln sin 2 kT π eεσ -2 ln √ 2 eπR c σ 4kT + eεσ + O(1) σcusp = 0 σε 1 σbulk = C ste kT e -ln 8R c ε π 4 |∂ Ωε | (1 + σbulk /σ ε ) -ln sin 2 2kT eσ ε √ R c ε + O(1) σcusp = σε σcusp √ ε 1 σbulk = C ste kT e ln sin 2 kT π eεσ ε -2 ln √ 2 eπR c σcusp 4kT + eεσ cusp + O(1)
∇ w v(w) = ∂v(X, Y ) ∂X + i ∂v(X, Y ) ∂Y . (3.111)
In the same way we define the real functions w 1 (X, Y ) and w 2 (X, Y ) that satisfy

w (ξ) = w (w -1 (X, Y )) = w 1 (X, Y ) + i w 2 (X, Y ). (3.112)
Using (3.17) (Möbius transformation), it yields

w (w -1 (X, Y )) = (1 + αw) 2 1 -α . ( 3
w 1 (X, Y ) = (1 + αX) 2 -α 2 Y 2 1 -α 2 (3.114) w 2 (X, Y ) = - 2αY (1 + αX) 1 -α 2 .
It where we set wi (ρ, θ) = w i (X, Y ) for i ∈ {1 , 2}, such as

w1 (ρ, θ) = 1 -α 2 ρ 2 + 2αρ cos(θ)(1 + αρ cos(θ)) 1 -α 2 (3.119) w2 (ρ, θ) = -2αρ sin(θ) 1 + αρ cos(θ) 1 -α 2 .
Using (3.119) and (3.118), to leading order we obtain

1 r ∂ ũ(r, z) ∂r = - ρ(1 -cos(θ)) 2 ε 3/2 ∂ṽ(ρ, θz) ∂ρ - sin(θ)(1 -cos(θ)) ε ∂ṽ(ρ, θz) ∂θ .
(3.120)

The numerical procedure

Numerical solutions were constructed by the COMSOL Multiphysics 5.0 (BVP problems), Maple 2015 (Shooting problems) and Matlab R2015 (Conformal mapping).

The boundary value problems in 1D, 2D, and 3D were solved by the finite elements method in the COMSOL 'Mathematics' package. We used an adaptive mesh refinement to ensure numerical convergence for large value of the parameters σ, σ ε , σ bulk and σ cusp . We solved the PDEs by the shooting procedure for boundary value problems using Runge-Kutta fourth-order method.

Chapter 4

Poisson-Nernst-Planck equation in narrow ellipses and ellipsoids.

Introduction

Biological and physical sciences show a rising interest toward electro-diffusion in small compartments [START_REF] Cartailler | Electrostatics of non-neutral biological microdomains[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Savtchenko | Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules[END_REF][START_REF] Cervera | Ionic conduction, rectification, and selectivity in single conical nanopores[END_REF], for example to develop technological applications such as the design of desalinization devices [START_REF] Mani | Deionization shocks in microstructures[END_REF], batteries [START_REF] Smith | Multiphase Porous Electrode Theory[END_REF] and also to improve information storage [START_REF] Han | Resistive Switching in Aqueous Nanopores by Shock Electrodeposition[END_REF]. At the same time in biology many compartments have a characteristic length of a hundred of nanometers, sometimes even less, such as dendritic spine, nanoscopic processes, or filopodia. Most of these structures have an elongated elliptical shape. For example, astrocytes -star-shaped cells that mostly support neurons -have nanoscopic protrusions that can be as thin as 30-50nm and that help regulate the synaptic transmission [START_REF] Heller | Probing nanoorganization of astroglia with multi-color super-resolution microscopy[END_REF][START_REF] Rusakov | Disentangling calcium-driven astrocyte physiology[END_REF]. An other example is the neural growth cone, a protrusion neurons use to operate in their environment. It is mostly composed of thin 100-300nm finger-like structures called filopodia and whose length is about 10µm [START_REF] Mattila | Filopodia: molecular architecture and cellular functions[END_REF] (even to 40µm in one peculiar sea-urchin [START_REF] Welch | Cellular control of actin nucleation[END_REF]). We use the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. The entire boundary is impermeable to particles (ions) and the electric field satisfies the compatibility condition of Poisson's equation. Our goal is to understand the effect of boundary curvature on the electrical properties of a biological elliptically shaped domain. In particular, we explore the effect of boundary curvature on the charge and field distribution at steady state. The effects of non-electro-neutrality and curvature were studied in [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF][START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF][START_REF] Cartailler | Electrostatics of non-neutral biological microdomains[END_REF] and a long-range electrostatic length, much longer than the Debye length, was found. This effect is due to the combined effects of non-electro-neutrality and boundary, which lead to charge accumulation near the boundary. The present paper is organized in three parts. First, we introduce the physical and the mathematical framework based on the PNP equations we shall use. Then, in the second and third parts, we study two domains: a the planar ellipse and the solid ellipsoid. For both scenarii, the ratio of the minor on the major axis is small. We construct in this limit an asymptotic solution for the voltage computed from the stationary PNP.

Poisson-Nernst-Planck equation

The Poisson-Nernst-Planck system of equations in a domain Ω, whose dielectric boundary ∂ Ω is represented as the compatibility condition for Poisson's equation, has an impermeability to the passage of ions represented as a no-flux boundary conditions for the Nernst-Planck equation. We assume that the charge of the (z p , z m ) electrolyte confined in Ω consists of identical N p positive and N m negative ions with initial particle density q p (x) and q m (x) such as

N i = Ω q i (x) dx, for i ∈ {p , m} (4.1)
The total charge in Ω is then given by

Q = e(z p N p + z m N m ), ( 4
ρ i (x, 0) = q i (x) for x ∈ Ω, (4.5)
where k B T represents the thermal energy, and D i is the diffusion coefficient for the ion specie i. Here φ(x, t) is the electric potential in Ω and is the solution of the Neumann problem for the Poisson equation

∆φ(x, t) = - e ε r ε 0 (z p ρ p (x) -z m ρ m (x)) for x ∈ Ω (4.6) ∂φ(x, t) ∂n = -σ(x, t) for x ∈ ∂ Ω, (4.7)
where ε r ε 0 is the permitivity of the medium and σ(x, t) is the surface charge density on the boundary ∂ Ω. In steady state, (4.3) gives

ρ i (x) = ρ i,0 exp - z i eφ(x) k B T for i ∈ {p , m}, (4.8)
where ρ i,0 is obtained from no-flux boundary condition (4.4), it yields

ρ i (x) = N i exp - z i eφ(x) k B T Ω exp - z i eφ(s) k B T ds for i ∈ {p , m}, (4.9)
We introduce the adimensionalized potential ũ(x) = e φ(x) k B T , then equation (2.123) becomes

ρ i (x) = N i e -z i ũ(x) Ω e -z i ũ(s) ds for i ∈ {p , m}, (4
∂ ũ(x) ∂n = - l B N p |∂ Ω| on ∂ Ω.

Poisson-Nernst-Planck solutions for a 2D elliptic domain

We consider a dimensionless planar elliptical domain Ω ⊂ R 2 defined by the general equation,

x b 2 + y a 2 ≤ 1 for (x, y) ∈ Ω, (4.14)
where a and b are the lengths of the major and the minor axis respectively. One can note that in dimension two, the units of N p is charge per unit of length, such that l B N p in dimension two is dimensionless. Indeed, a planar ellipse can be seen as the cross section of an infinite cylinder with an elliptical cross-section. We nondimensionalize the BVP (4.

∂ 2 u(x, y) ∂x 2 + ∂ 2 u(x, y) ∂y 2 = -exp(-u(x))
in Ω (4.17) ∂u(x) ∂n = -σ on ∂Ω.

Poisson-Nernst-Planck solutions for a 2D elliptic domain

We remark that the case N p = 0 in (4.15) yields to the trivial solution u(x) = C ste . The particular case a = b corresponds to a circular geometry for the domain Ω for which the BVP (4.17) has been solved analytically in [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF]. However, the method used to solve (4.15) in a disk consists in reducing the problem to a one dimensional equation by the means of the radial symmetry [START_REF] Jacobsen | The Liouville Bratu Gelfand Problem for radial operators[END_REF]. Unfortunately, in the ellipse we have in general a = b so the previous approach fails.

In this paper, we construct an asymptotic solution of (4.17) in the limit b a 1.

We apply the method we have developed in [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] for the domains with cusp-shaped funnels. The method consists into dividing the domain Ω into subregions, based on the direction of the flow lines ∂u(x, y) ∂x , ∂u(x, y) ∂y . We then use a conformal mapping in order to simplify the flow but not necessarily the geometry. We construct an asymptotic solution to eq. (4.17 are discriminated. We introduce the outward normal unit vector n(y) to the ellipse surface ∂Ω at y:

n(y) = 1 b 2 y 2 + 1 -y 2   1 -y 2 by   . (4.19)
In the b small limit, eq. ( 4. [START_REF] Bazant | Conformal mapping of some non-harmonic functions, in transport theory[END_REF] becomes In other words, the equation (4.25) shows that the region A is characterized by an outward normal unit vector n(y), to leading order, perpendicular to the y-axis (4.22). It is the argument we shall use to approximate the solution u(x, y) in the region A . We now have distinguished the regions A and B. We will estimate asymptotic solutions u A (x, y) and u B (x) from (4.17) computed in each region A and B respectively. We will then use the two solutions to construct a uniform solution u unif (x, y) in the region A ∪ B. We will use u unif (x, y) to estimate the potential drop along the major and the minor axis. Finally we will compute the potential difference between the ellipse tips (a, 0) and (0, b).

n(y) =     1 by 1 -y 2     + O b 2 1 -y 2 . ( 4

4.2.

Poisson-Nernst-Planck solutions for a 2D elliptic domain

Asymptotic solution u A in region A

We compute u A (x, y) in the region A observing first that the boundary conditions (4.25) indicates that the derivatives ∂u ∂y are negligible relative to the derivatives ∂u ∂x .

Then, a regular expansion of the solution gives that the equation is solved along the lines y = const for x ∈ [0, b 1 -y 2 ], where b 1 -y 2 represents half of the ellipse width at the point y. Thus, in the small σ limit, the leading order solution in region A is obtained solving the following BVP:

∂ 2 ∂x 2 u A (x, y) = -exp(-u A (x, y)) for x ∈ [0 , b 1 -y 2 ] (4.26) ∂u A (x, y) ∂x x=0 = 0 ∂u A (x, y) ∂x x=b √ 1-y 2 = -σ.
Equation (4.26) represents the stationary PNP evaluated on a segment of length b 1 -y 2 [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF]. Integrating twice (4.26) and using the homogeneous boundary conditions from (4.26), it yields We assume that in large σ limit, σC 1 (y) → ∞, such that arctan (σC

u A (x, y) = -ln     tan 2 1 2 x C 1 (y) + 1 2C 2 1 (y)     .
1 (y)) = π 2 - 1 σC 1 (y) + O 1 σC 1 (y) 3 .
(4.29)

Using (4.29) in (4.28), to the leading order we obtain

C 1 (y) = 2 + bσ 1 -y 2 πσ . (4.30)
Using in (4.27) we conclude that

u A (x, y) = ln   2 2 + bσ 1 -y 2 πσ 2 cos 2   1 2 πσx 2 + bσ 1 -y 2     . (4.31)
From the solution u A (x, y) we can estimate the potential along the Ω minor axis,

but not yet along the major axis. Indeed, to obtain this second potential, we shall now derive an asymptotic solution in region B. The ellipse tip (left) is mapped onto the drop shaped domain (right).

Solution in region B

We construct the asymptotic expansion u B (y) in the region B in the small b limit. We first move the origin to the major axis lower end (x, y) = (-1, 0) (Fig. 4.2B). Then, we map the domain Ω conformally by the Schwarz-Christoffel (SC) transformation [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF][START_REF] Nehari | Conformal mapping[END_REF] w(z) = arcsin(αz), (4.32) where α is a parameter and z = x + iy is a complex coordinate. The SC transformation (4.32) maps Ω into the bell bottom-shaped domain Ω w = w(Ω) and the region B into

B w = w(B) = [0 , w(ỹ 0 )], (4.33)
where ỹ0 = 1 -y 0 . The regions are shown in Fig. 4.2B-C. Setting u(z) = v(w), under the conformal map (4.32), the PDE (4.17) transforms in

∆ w v(w) = - | cos(w)| 2 α 2 exp(-v(w)) for w ∈ Ω w . (4.34)
We remind that the gradient, under the map w(z), reads as the dot product [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF]

∇u(z) = w (z) • ∇ w v(w), (4.35)
where w (z) is the w (z) conjugate. Setting w = X + iY , using (4.32), we have 1

w (z) = 1 α (cos(X) cosh(Y ) + i sin(X) sinh(Y )) . ( 4
∆ w v(w) = - | cos(w)| 2 α 2 exp(-v(w)) for w ∈ Ω w . (4.39) ∂v(w) ∂Y X=0,Y =0 = - σ α ∂v(w) ∂Y X=0,Y =w(ỹ 0 ) = -ỹ 0 ∂u A (0, y) ∂y y=y 0 ∂v(w) ∂n w = φ bdv (X, Y, σ) on ∂Ω w \ {(0, 0) ∪ (0, w(ỹ 0 ))} ,
where φ bdv encompasses the transformed boundary conditions whose relative boundaries do not intersect the region B w .

Solution in the region B in the large σ limit

We now use the free parameter α in the SC transformation (4.32) to find the optimal shape of the domain Ω w that shall linearize v(X, Y ) on B w . Thus, we search α such that we have the equality

∂v ∂Y Y =0 = ∂v ∂Y Y =w(ỹ 0 ) . ( 4

.40)

Using the boundary conditions in (4.39), we obtain

α = σ ν ỹ0 , (4.41)
where ν = ∂u A (0, y) ∂y

y=y 0 .
We construct an asymptotic expansion of (4.39) in the limits σ 1 and α 1 such that σ α = O(1). We expand v(w) in those limits, We calculate the constant Ã0 by matching the solutions u A (0, y) in region A and u B (y) in region B at y = ỹ0 , we obtain

v(w) = v 0 (w) + α -1 v 1 (w) + α -2 v 2 (w) + • • • (4.
Ã0 = u A (ỹ 0 ) + σ α arcsinh(α ỹ0 ). (4.48)
We conclude, using (4.32), (4.46) and (4.48), that the solution in region B is to the leading order

u B (y) = u A (1 -y 0 ) + σ α (arcsinh(α (1 -y 0 )) -arcsinh(α(1 -y))) . (4.49)

Uniform approximation of PNP in the elliptic domain Ω

We construct an uniform solution u unif (x, y) in the region A ∪ B by matching the solution u A (x, y) (4.31) in region A with u B (y) (4.49) in region B. The two functions u B and u A , as well as their derivatives with respect to y, match at y = y 0 (see eq.4.39 and 4.49). Thus, the uniform approximation of u unif (x, y) in A ∪ B is

u unif (x, y) =                ln   2 2 + bσ 1 -y 2 πσ 2 cos 2   1 2 πσx 2 + bσ 1 -y 2     for 0 ≤ y ≤ y 0 u A (1 -y 0 ) + νy 0 arcsinh σ(1 -y 0 ) ν -arcsinh σ(1 -y) ν for y 0 ≤ y ≤ 1. (4.50)
We compare numerics of (4.17

    (bσ + 2) cos πbσ 4 + 2bσ bσ 1 -(1 -b) 2 + 2     + arcsinh(bσ(1 + b/2)) 1 + b/2 . (4.59)
As a summary, we find that the potential in the ellipse Ω decays logarithmically from the center to the border of the domain. Yet, the potential does not decay with the same rate. Indeed, the point with the higher curvature (0, 1) has a lower potential than (b, 0) whose curvature is smaller.

Poisson-Nernst-Planck solutions in a 3D ellipsoid

We shall now extend the method we applied above to an ellipsoid obtained by rotating the symmetric planar ellipse Fig. 4.1A about its axis of symmetry (major axis) as shown in Fig. 4.4A. In the (r, θ, z) cylindrical coordinates, the ellipsoid Ω3 is characterized by

r2 cos 2 (θ) b2 + r2 sin 2 (θ) b2 + z2 a 2 ≤ 1, (4.60)
where a and b are the length of the major and minor axis respectively. We adimen- The potential u(r, z, θ) is solution of the stationary PNP problem (4.17). In the cylindrical coordinates, where the longitudinal Ω 3 axis is aligned with the z-axis, the problem becomes independent of the angle θ. The BVP that corresponds to (4.17) in Ω 3 reads as

∂ 2 u 3 (r, z) ∂r 2 + 1 r ∂u 3 (r, z) ∂r + ∂ 2 u 3 (r, z) ∂z 2 = -exp(-u 3 (r, z)) (4.62) ∂u 3 (r, z) ∂n 3 = -σ.
We use the same method as in section 4.2, which consists in dividing Ω 3 into the two regions Ā = {(r, z) ∈ Ω 3 such that r ≥ 0 and 0 ≤ z < z 0 } (4.63) B = {(r, z) ∈ Ω 3 such that r = 0 and z ≥ z 0 } where z 0 is a parameter. The normal outward unit vector n 3 (z, θ) to the surface ∂Ω 3 evaluated at coordinates (z, θ) is given in Cartesian coordinates by

n 3 (z, θ) = 1 √ 1 -z 2 + b 2 z 2       √ 1 -z 2 cos(θ) √ 1 -z 2 sin(θ) bz       . (4.64)
In small b limit, (4.64) reduces to

n 3 (z, θ) =         cos(θ) sin(θ) bz √ 1 -z 2         + O b 2 1 -z 2 .
(4.65)

Once back in cylindrical coordinates, eq. ( 4.65) becomes

n 3 (z) =     1 bz √ 1 -z 2     + O b 2 1 -z 2 .
(4.66)

We define z 0 such that for 0 ≤ z ≤ z 0 , we have

O b √ 1 -z 2 = O √ b . ( 4
∂u 3 (r, z) ∂r (1 + O(b)) + ∂u 3 (r, z) ∂z O √ b = -σ. (4.68)
Using (4.68) in the large σ limit, we find that to the leading order, for 0 ≤ z ≤ z 0 < 1,

∂u 3 ∂n = ∂u 3 (r, z) ∂r r∈∂Ω 3 . (4.69)
In the following sections we will estimate asymptotic solutions u 3,A (r, z) and u 3,B (z) for the BVP (4.62) in each region Ā and B respectively. We will then use the two solutions to construct a uniform solution ūunif (r, z) in Ā ∪ B.

Asymptotic solution u 3,A (r, z) in region Ā

The discrepancy between z and r-derivatives in the region Ā relies on the same arguments as in section 4.2.2. It then follows that the BVP problem (4.62) in the 4.3. Poisson-Nernst-Planck solutions in a 3D ellipsoid

Ω 3 circular cross sections is solved for z = const and r ∈ [0, b √ 1 -z 2 ]
. The leading order u 3,A (r, z), in the large σ limit, is solution of

∂ 2 ∂r 2 u 3,A (r, z) + 1 r ∂ ∂r u 3,A (r, z) = -exp(-u 3,A (r, z)) for r ∈ [0 , b √ 1 -z 2 ] (4.70) ∂u 3,A (r, z) ∂r r=0 = 0 ∂u 3,A (r, z) ∂r r=b √ 1-z 2 = -σ,
One can remark that (4.70) differs from (4.26) (see section 4.2) only by the term 1 r ∂u 3,A (r, z) ∂r . Furthermore, the BVP (4.70) corresponds the stationary PNP problem is a disk, which can be solved analytically [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF], such that

u 3,A (r, z) = ln 8C 1 (z) λ -2 ln(r 2 -C 1 (z)). (4.71)
We compute C 1 (z) from the boundary conditions at r = b √ 1 -z 2 in (4.70), it yields

C 1 (z) = b √ 1 -z 2 (4 + b √ 1 -z 2 σ) σ . (4.72)
Finally, using the solution above in (4.71), we conclude

u 3,A (r, z) = ln 8|∂Ω|b √ 1 -z 2 (4 + bσ √ 1 -z 2 ) (4.73) -2 ln(λr 2 -|∂Ω|b √ 1 -z 2 (4 + bσ √ 1 -z 2 )).

Solution in region B

We use the SC transformation (4.32) to map conformally the domain Ω 3 cross-section into a bell bottom-shaped domain Ω 3,w = w(Ω 3 ) and we have

Bw = w( B) = [0 , w(z 0 )], (4.74)
where z0 = 1 -z 0 . We then compute the Laplace operator in (4.62) under the SC map (4.32), in two step. We set u(z) = v(w), such that the second order derivatives in (4.62) transform as in (4.34), it yields,

∂ 2 u 3 (r, z) ∂r 2 + ∂ 2 ū3 (r, z) ∂z 2 = α 2 | cos(w)| 2 ∆ w v(w). (4.75)
The extra term ∂u 3 (r, z) ∂r that result from the expression of the Laplace operator in the cylindrical coordinates transforms as follows [START_REF] Henrici | Applied and Computational Complex Analysis, Volume 1: Power Series Integration Conformal Mapping Location of Zero[END_REF]:

∂u 3 ∂r = ∇ w v(w) w (ξ) , (4.76)
where ξ = r + iz. In the Ω 3,w Cartesian coordinates (X, Y ), such that we write w = X + iY , it follows

w (ξ) = w (w -1(X + iY )) = α | cos(w)| 2 (w 1 (X, Y ) + iw 2 (X, Y )), (4.77) with    w 1 (X, Y ) = cos(X) cosh(Y ) w 2 (X, Y ) = sin(X) sinh(Y ).
Using (4.76) and (4.77), it yields

∂u 3 ∂r = α | cos(w)| 2 w 1 (X, Y ) ∂v ∂X -w 2 (X, Y ) ∂v ∂Y (4.78)
On the other hand, using (4.32), we have ξ = 1 α sin(w), such that the radial coordinate reads as:

r = sin(X) cosh(Y ) α . (4.79) Using (4.78), (4.78) and (4.79), we obtain 1 r ∂u(r, z) ∂r
= α 2 | cos(w)| 2 1 tan(X) ∂v(X, Y ) ∂X -tanh(Y ) ∂v(X, Y ) ∂Y (4.80)
The two expressions (4.75) and (4.80) combined correspond to the Laplace operator in Ω 3,w . We conclude that

∆ w v(w) + 1 tan(X) ∂v(X, Y ) ∂X -tanh(Y ) ∂v(X, Y ) ∂Y = - | cos(w)| 2 α 2 e -v(w) (4.81)

Construction of an asymptotics of u 3,B

To construct an asymptotics for (4.62) in the region B, we used the same method as in region B. However, the equation (4.81) reveals a dependency in tanh(Y ) which diverges from the case in 2D. Nevertheless, we will show this extra term does not affect the first order solution of (4.62). We start by searching α such as the two boundary conditions at the ends of Bw match

∂v ∂Y Y =0 = ∂v ∂Y Y =w( z0 ) . (4.82)
We choose z0 = b which satisfies the condition (4.67), and we introduce ν such as

ν = ∂u A (r, z) ∂z r=0,z=1-b . ( 4
∂u A (r, z) ∂z r=0,z=1-b = - z 1 -z 2 2(bλ √ 1 -z 2 + 2|∂Ω 3 |) bλ √ 1 -z 2 + 4|∂Ω 3 | (4.84)
Using (4.84) in (4.83), it follows

ν = 2 b 3/2 λ √ 2 + 4 |∂Ω 3 | (b -1) b 3/2 λ √ 2 + 4 |∂Ω| b (b -2) . (4.85)
From boundary conditions (4.40), we have

∂v ∂Y Y =0 = - σ α , (4.86)
and

∂v ∂Y Y =w( z0 ) = - ν α cosh(w(i z0 )) = -ν 1 + (α z0 ) 2 α z0 . (4.87)
We conclude from (4.86) and (4.87),

α = √ σ 2 -ν 2 νb (4.88) = 1 b λ(b 3/2 λ √ 2 + 2 |∂Ω|) |∂Ω|(b 3/2 λ √ 2 + 4 |∂Ω|) 2 -1 (1 + O(b)) . (4.89)
We remark that in large λ limit we have α 1. We now introduce η = x + iy, and scale w such as

w = η α . (4.90)
Setting v(w) = ṽ(η), using the scaling (4.90) in (4.81), it yields

∆ η ṽ(η) + 1 α tan x α ∂ṽ(η) ∂x - 1 α tanh y α ∂ṽ(η) ∂y = - | cos(η/α)| 2 α 4 exp(-ṽ(η)). (4.91)
We expand ṽ(η

) in α -1 , ṽ(η) = ṽ0 (η) + α -1 ṽ1 (η) + O(α -2 ). (4.92)
We inject (4.92) in (4.91), and since we have tanh y α < 1, the first order solution is obtained solving

∆ η ṽ0 (η) + 1 α tan x α ∂ṽ 0 (η) ∂x = 0. (4.93)
Chapter 4. Poisson-Nernst-Planck equation in narrow ellipses and ellipsoids.

We assume ṽ0 (x, y) = h(x) φ(y), where h and φ are two smooth functions, consequently (4.93) becomes φ(y) h (x) + h (x)

α tan x α = -φ (y) h(x). (4.94)
Since the y-axis is a symmetry axis, for x = 0, it imposes that h (0) = h (0) = 0, it follows φ (y) = 0. (

Integrating twice (4.95), we immediately obtain

φ(y) = A 0 + B 0 y (4.96)
we conclude that ṽ0 (0, y) = Ã0 + B0 y (4.97)

where, from the boundary condition (4.86) we obtain

B0 = - σ α 2 (4.98)
and à will be used to connect the region A 3D and the region B 3D . We conclude that to leading order, the solution in region B is given by,

u 3,B (z) = Ã - σ α arcsinh(αz) (4.99)

Uniform approximation of PNP in the elliptic domain Ω

We construct an uniform solution u 3,unif (x, y) in the region Ā ∪ B following exactly the same pattern as for the planar ellipse. We match the solutions u 3,A (x, y) (4.73) in region Ā with u 3,B (y) (4.99) in region B such that the both function sas well as their derivatives -with respect to z -match at z = z 0 . It follows

u 3,unif (x, y) =              ln 8b √ 1 -z 2 (4 + bσ √ 1 -z 2 ) |∂Ω|(σr 2 -b √ 1 -z 2 (4 + bσ √ 1 -z 2 )) 2 for 0 ≤ z ≤ z 0 u 3,A (1 -z 0 ) + νy 0 arcsinh σ(1 -z 0 ) ν -arcsinh σ(1 -z) ν for z 0 ≤ z ≤ 1. (4.100)
We compare numerics of (4.100) (solid lines) with the asymptotic solution u unif (x, y) from (4.50) in Fig. 4.4B-C, where we chose z 0 = 1 -b. V (b) -V (a), each versus σ, obtained numerically (blue) from (4.17) and analytically from (4.107) (dashed green).

Discussion and conclusion

Discussion and conclusion

To conclude, we have constructed an asymptotic approximation for a Liouville problem obtained from the stationary PNP to model the electro-diffusion in a non-neutral confined electrolytes. We imposed non-homogeneous Neumann boundary conditions derived from Poisson's equation compatibility condition. Neumann conditions contrast with other works that use with Dirichlet conditions [START_REF] Chipot | On the solutions of Liouville systems[END_REF][START_REF] Gogny | Sur les états déquilibre pour les densités électroniques dans les plasmas[END_REF] or large solutions [START_REF] Bandle | Asymptotic behaviour of large solutions of quasilinear elliptic problems[END_REF]. We considered two geometries: a planar ellipse and a 3D ellipsoid. For both domains the width -represented by the minor axis -is a small parameter. We aimed to construct an asymptotic approximation in this limit.

We used a method we developed in [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF]: we split the domain into two subregions (see Fig. 4.1) such that in one region the problem reduces to the resolution of a 1D or 2D Liouville equation, and in the second region we use the SC conformal transformation (see Fig. 4.2) to find an optimal shape such that, to the leading order, the problem reduces to a Laplace equation. The solutions in each subregions are then matched to construct an uniform solution. We found that this asymptotic approximations agrees with numerical simulations (Fig. 4.3 and Fig. 4.4). Finally, we used the uniform solution to estimate the voltage drop along the two main axises. Then we computed the potential difference between the major and minor axises tips. Surprisingly we found that such voltage difference had a maximum for a certain value of σ (see Fig. 4.3F and Fig. 4.4F).

Those results can be used to better understand electrical properties of nanoscale compartments in biology. Indeed, so far, measuring key electrophysiological quantities such as voltage or ionic concentrations in such small compartments is impossible due to experimental limitations. Although, new methods have been developed to explore such quantities, for instance based on genetically encoded voltage sensor [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF] or nanodiamonds [109], but these methods are indirect. However, a direct method exists [START_REF] Jayant | Targeted intracellular voltage recordings from dendritic spines using quantum-dot-coated nanopipettes[END_REF]: it relies on a nanopipette that turns out to be also a nanocompartment. As the characteristic length becomes small, non linear effects that arise from the electro-diffusion of charge carrier are no longer negligible [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF][START_REF] Sparreboom | Eijkel Principles and applications of nanofluidic transport[END_REF][START_REF] White | Ion Current Rectification at Nanopores in Glass Membranes[END_REF]. The method used in this paper provides new tool to address such questions.

Finally, the method we used in this paper deals with the construction of solution for the PNP non-linear partial differential equations, and it could actually be extended to different elliptical equations. An interesting example is the sinh-Poisson equation ∆u = µ sinh(u), µ ∈ R [START_REF] Ablowitz | Method for Solving the Sine-Gordon Equation[END_REF][START_REF] Chow | Doubly periodic and multiple pole solutions of the sinh-Poisson equation: Application of reciprocal transformations in subsonic gas dynamics[END_REF][START_REF] Mcdonald | E Numerical calculation of nonunique solutions of a twodimensional sinh-Poisson equation[END_REF][START_REF] Ting | Exact solutions of a nonlinear boundary value problem: the vortices of the two-diemsional sinh-Poisson equation[END_REF]. Indeed, it describes the stationary distribution of an electrolyte composed of two species having opposite charge sign [START_REF] Andelman | Electrostatic Properties of Membranes: The Poisson-Boltzmann Theory[END_REF][START_REF] Rubinstein | Electro-Diffusion of ions[END_REF]. Such theory aims to a better comprehension of the electrical phenomena in biology inasmuch they are described by the theory of electro-diffusion.

Chapter 5

Electrostatics of non-neutral biological microdomains

Introduction

Electro-diffusion in cellular microdmains remains difficult to study due to the absence of specific sensors and the theoretical hurdle of understanding the dynamics of charged particle in shaped domains. The diffusion of charged particles is largely influenced by the interaction of diffusing ions with the electrical field generated by all charges in the solution and possibly with external field. The dielectric membrane of a charged biological cell also affects the electric field, because it creates image charges (see [START_REF] Abrashkin | Dipolar Poisson-Boltzmann equation: ions and dipoles close to charge interfaces[END_REF] for an infinite plan). So far, only a few electro-diffusion systems are well understood. For example, although the electrical battery was invented more than 200 years ago, designing optimal configurations is still a challenge. On the other extreme, ionic flux and gating of voltage-channels [START_REF] Bezanilla | How membrane proteins sense voltage[END_REF] is now well explained by the modern Poisson-Nernst-Planck (PNP) theory of Chapter 5. Electrostatics of non-neutral biological microdomains electro-diffusion, because at the nanometer scale, cylindrical symmetry of a channel model reduces computations to a one-dimensional model for the electric field and charge densities in the channel pore [START_REF] Eisenberg | Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube[END_REF][START_REF] Eisenberg | From structure to function in open ionic channels[END_REF][START_REF] Eisenberg | Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations[END_REF]. However, cellular microdomains involve two-and three-dimensional neuronal geometry [START_REF] Lee | Computer simulation of voltage sensitive calcium ion channels in a dendritic spine[END_REF][START_REF] Lopreore | Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier[END_REF][START_REF] Savtchenko | Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules[END_REF][START_REF] Sylantyev | Electric fields due to synaptic currents sharpen excitatory transmission[END_REF][START_REF] Sylantyev | Spike-driven glutamate electrodiffusion triggers synaptic potentiation viahomer-dependent mGluR-NMDAR link[END_REF], which makes the analysis of the PNP equations much more complicated than in the cylindrical geometry of a channel pore [START_REF] Qian | An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons[END_REF].

We report here recent results about the distribution of charges and field, obtained from the analysis of the nonlinear PNP model of electro-diffusion in various geometries of microdomains in the absence of electro-neutrality. In our model the entire boundary is impermeable to particles (ions) and the electric field satisfies the compatibility condition of Poisson's equation. Phenomenological descriptions of the electro-diffusion, such as cable equations or the reduced electrical-engineering approximation by resistance, capacitance, and even electronic devices, are insufficient for the description of non-cylindrical geometry [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF], because they assume simple one-dimensional or reduced geometry.

Results

Electrostatic theory with no electro-neutrality

In the absence of electro-neutrality and with N charges distributed in a bounded domain Ω surrounded by a dielectric membrane, the PNP model for total charge Q = zeN , where e is the electron charge (1.6 • 10 -19 C), z -the valence, and the charge density ρ(x, t) is given by [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF] for x ∈ Ω,

D ∆ρ(x, t) + ze kT ∇ (ρ(x, t)∇V (x, t)) = ∂ρ(x, t) ∂t (5.1) D ∂ρ(x, t) ∂n + ze kT ρ(x, t) ∂V (x, t) ∂n = 0 , x ∈ ∂Ω (5.2) ρ(x, 0) = q(x) , x ∈ Ω, (5.3)
where D is the diffusion coefficient, kT represents the thermal energy and V (x, t) is the electric potential in Ω. It is the solution of the Poisson equation

∆V (x, t) = - zeρ(x, t) εε 0 for x ∈ Ω (5.4)
with the boundary condition

∂V (x, t) ∂n = -σ(x, t) for x ∈ ∂Ω, (5.5)
where the surface charge density σ(x, t) is defined on the boundary ∂Ω using the the electrical permitivity εε 0 of the electrolyte solution. In the steady state,

σ(x) = Q εε 0 |∂Ω| . ( 5
ρ(x) = N exp - zeV (x) kT Ω exp - zeV (x) kT dx , (5.7) hence (5.4) becomes ∆V
(x) = - zeN exp - zeV (x) kT εε 0 Ω exp - zeV (x) kT dx , (5.8)
and (5.5) gives the boundary condition

∂V (x) ∂n = - zeN εε 0 |∂Ω| for x ∈ ∂Ω, (5.9)
which is the compatibility condition, obtained by integrating Poisson's equation (5.4) over Ω. Debye's [START_REF] Debye | Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen[END_REF] concept of charge screening, which makes the induced field decay exponentially fast away from a charge, does not apply when electroneutrality is broken and long-range correlations lead to a gradient of charges, as is the case, for example, in a ball without inward directed current. By solving (5.8) numerically and asymptotically, a new capacitance law was derived for an electrolytic solution in a ball of radius R [START_REF] Cartailler | Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains[END_REF], where the difference of potentials between the center C and any point of the spherical surface S, V (C) -V (S), increases with the total number of charges, first linearly and then logarithmically

V (C) -V (S) ≈ -2 kT ze log 2πR(ze) 2 N εε 0 kT |∂Ω| ,
The effect of the geometry on the voltage and the charge distribution for other cell shapes is described below.

Local boundary curvature affects field and charge distribution

Axons and dendrites are not perfect cylinders and the curvature of their surfaces has many local maxima [START_REF] Bourne | Balancing structure and function at hippocampal dendritic spines[END_REF]. It turns out that this local curvature can influence the local voltage significantly, as shown in numerical solutions (using Comsol classical packages) of the PNP equations (Fig. 5.1), which reveal that regions of high curvature correspond to local charge accumulation. This effect should be sufficient to influence the voltage by creating measurable local voltage increase of the order of a few millivolts. The voltage inside the cylinder and along curved surfaces can vary (Fig. 5.1A-D) and can also depend on changes in curvature (Fig. 5.1F-I) and in the total number of charges N (Fig. 5.1E-J). Curvature creates a narrow boundary layer in the voltage (Fig. 5.1H-I). Note that the previous computations assume one specie only and in practice, due to the presence of N other ions (i.e., λ = (ze) 2 N/εε 0 kT ), should only represent

Results

the excess of positive charges. Thus, we explore a large range for the parameter λ (a generalization of Bjerrum length l B = (ze) 2 /εε 0 kT , but has no physical unit in dim 2 and is a length in dim 3). Note that λ is not simply given by the difference of positive and negative charges, due to the non-linear PNP equations. From the exploration of the graphs in Fig. 5.1-5.2, we expect λ ≈ 1 -500, which corresponds to a range N ≈ 10 2 -5•10 4 , leading to voltage fluctuations of few tens of mV. A more physiological range for the parameter λ is shown in fig. 5.1E (inset). Consequently, the resting cross-membrane potential and the resting voltage across voltage-gated channels along a dendrite may vary, depending on curvature. This may affect the propagation and genesis of local depolarization or back-propagation action potential in dendrites of neuronal cells [START_REF] Markram | Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs[END_REF].

A cusp-shaped funnel can influence charge distribution

The passage of ions and other particles between different cellular compartments is done through narrow passages that form cusp-shaped funnels, which have negative curvature (see the example described in Fig. 2A) We consider here the case of dendritic spines, whose geometry is approximated as a convex domain (head) connected by a cylindrical neck to the dendritic shaft. This geometrical model approximation was already validated for diffusion [START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF]. We observe that the cusp-shaped funnel prevents the entrance of charges, at least when their number does not exceed a given threshold (Fig. 5.2C-F). Therefore, there is a difference of steady state potential drop between the end of the funnel and the bulk of the domain.

Assume that for a dimensionless domain Ω ⊂ R 2 with a cusp-shaped funnel F formed by two bounding circles of equal radii R c (see Fig. 5.2A), the opening of the funnel is ε 1 (in unit of micrometers). This radius is used in changing to dimensionless coordinates, so we may assume R c = 1. The surface area of the domain boundary is |∂Ω|. For an uncharged funnel domain, the condition (5.5) becomes ∂V (x)/∂n = 0 for x ∈ F . The potential drop between the center of Ω and the end of the cusp is found by solving (5.8) in a domain obtained by mapping Ω conformally by the Möbius transformation [START_REF] Holcman | Stochastic Narrow Escape in Molecular an Cellular Biology, Analysis and Applications[END_REF]

w = w(z) = z -α 1 -αz , (5.10)
where α = -1 -ε/R c + O(ε) and R c is the (dimensional) radius of curvature at the cusp domain (Fig. 2A). In the variable w = Re iθ = X + iY , the dimensional solution in the image of the cusp is given by

V (θ) = ln R c ε -ln 1 π 2 ln π 6 |∂Ω|R 1/2 c 2 3 ε3/2 - 2θ π 2 ln π 6 |∂Ω|R 1/2 c 2 3 ε3/2 (5.11) × arctan R c ε θ + ln cos 2 1 -4|∂Ω| λπ √ εRc 2 θ .
Chapter 5. Electrostatics of non-neutral biological microdomains Note that in the planar case λ has no dimensional units. It leads for λ 1, to the dimensional voltage drop

V (S) -V (C) = kT ze log 2 5 |∂Ω| √ ε π 6 λ 2 R 3/2 c + O(1). (5.12)
This difference should be compared with that between the north pole N and the center C, given in [START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF], is given by

V (N ) -V (C) = kT ze log 8π 8π + λ D 2 .
(

where λ D = 2πλR l /|∂Ω| (R l is the (dimensional) radius of the external domain Ω). For a three-dimensional symmetric domain with a cusp-funnel, λ 3d = λR C and V 3d (θ) = V (θ) + O( ε/R c ). We conclude in the limit of λ 1 and ε → 0 that the difference of potential between the end of cusp S and the north pole N in the domain is obtained by adding (5.13) and (5.12) and we get

V (S) -V (N ) = kT ze ln 2 5 |∂Ω| √ ε π 6 R 3/2 c (5.14) + 2 kT ze ln R l 4|∂Ω| + O 1 λ .
When Ω is charged, the boundary condition is still (5.9) and the potential drop is

V (S) -V (N ) = 2 kT ze ln πc 2 R l 16R c , (5.15)
The constant c depends only on the geometric center of mass C such as w(C) = c ε Rc , where w is the Möbius transformation (5.10), R c is the curvature at the cusp and ε the width at the base of the cusp. The analytical solution in the charged cusp-funnel domain is (in the same conformal coordinates as in eq. 5.11)

V ch (θ) =              -kT ze ln |e iθ -1| 4 8 λπR c 2|∂Ω|(1 -cos θ) + λε 2 , for θ ∈ [0, π - ε Rc ] kT ze ln cos 2 π √ Rc 2 θ-π+ ε Rc √ ε 1 -2|∂Ω| λε -2 ln √ 2Rcλπ 4|∂Ω|+λε , for θ ∈ [π - ε Rc , π] (5.16)
which is in very good agreement with the numerical solution (fig. 5.2C dotted orange). The present results are quite different from the Gouy-Chapman theory which concerns the exponential decay of voltage away from the double layer on a flat surface and derived for electro-neutral bulk, which is not the case here (see wiki "double layer"). It follows that the funnel curvature is an interesting free parameter for the design of optimized nano-pipettes to regulate the molecular and ionic fluxes. These pipettes were recently designed to record voltage in dendritic spines [START_REF] Jayant | Targeted intracellular voltage recordings from dendritic spines using quantum-dot-coated nanopipettes[END_REF]. Thus considering the shape of the pipette as a new parameter to control the ion

Results

fluxes injected in small biological domain, could impact the design and the precision of future biological experiments.

Voltage distribution in an elongated ellipse

Numerical solutions of the PNP equations 5.8 (Fig. (5.3)A) show the effect of local charge accumulation in an elongated ellipse, in particular the potential difference V (A) -V (B) is maximal on the ellipse for A and B the ends of the small and long axes (fig. 5.3B-C). The voltage along each axis is shown in fig. 5.3D-G. The elliptic domain can represent the cross section of an axon, and confirms the effect of curvature discussed in the previous sections. Specifically, that charges accumulate near the boundary of highest curvature. The equipotential contours are shown in Fig. 5.3G-L.

Discussion and conclusions.

As shown in the present study, inside a non-flat electrolyte domain characterized by significant changes in membrane curvature, the difference of voltage varies with the log of the curvature. Specifically, we derived various formula 5.14, 5.15 and 5.16 at a cusp-shaped funnel (curvature) R c and we showed that the changes are significant in the range R c ∈ [0.1 -2] µm (Fig. 5.2E). These formula should be considered as predictions to study the effect of curvature inside neuronal microdomains, such as dendritic spines or part of dendrites. Local changes in geometry, non-electro-neutrality, and a dielectric boundary affect charge distribution in electro-diffusion of electrolytes, as shown here and in [START_REF] Mamonov | The role of the dielectric barrier in narrow biological channels: a novel composite approach to modeling single channel currents[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF][START_REF] Cartailler | Geometrical effects on nonlinear electrodiffusion in cell physiology[END_REF]. This paper considers a single ionic species, whereas in reality the solution contains multiple positive and negative ions. The present electro-diffusion model captures the effect of the excess of positive ions. Consequently, the large value of voltage differences that are found in the large charge limit λ → ∞ are probably attenuated in a mixed ionic solution, when electro-neutrality remains broken. For large λ, the largest voltage differences occur near the boundary as shown in Fig. 5.1B-D-I-H, Fig. 5.2B, and Fig. 5.3K. The cytoplasmic ions are characterized by the following concentrations Na + = 12mM , K + = 155mM and Cl -= 4.2mM . Thus depending on ionic species, there is a clear unbalance toward positive charges. Are negative charges missing? Probably there are molecules of various sizes with negative charges, but their diffusion coefficients are certainly smaller than these of the ions. This difference of mobility is certainly a key feature in maintaining nonelectro-neutrality. The present results should be of significant consequences for neurons, sensory and glial cells, and many more. Indeed, local curvature is associated with gradient of charge density that can affect the electrical properties of micro-compartments [START_REF] Yuste | Dendritic Spines[END_REF][START_REF] Araya | The spine neck filters membrane potentials[END_REF][START_REF] Araya | Activity-dependent dendritic spine neck changes are correlated with synaptic strength[END_REF][START_REF] Lan | Voltage-Rectified Current and Fluid Flow in Conical Nanopores Acc[END_REF]. In all cases, charge accumulates near boundary points of locally maximal curvature. These results can further be used to design nano-devices such as pipettes and to better understand voltage changes inside dendrites and axons. Future analysis should reveal charge distribution during transient current. Inward current flowing inside the neuronal post-synaptic terminal modulates the membrane voltage potential. Most of synaptic excitatory connections are made on dendritic spines, characterized by a large diversity in their geometry. How the voltage in a spine is modulated by its geometry remains unclear due in part to the absence of direct measurements. To investigate the voltage-current relation and determine the electrical properties, we combine live cell fluorescent imaging data with electro-diffusion model. We first deconvolve the genetically encoded voltage sensor expressed in hippocampal neurons, then use the electro-diffusion theory to compute the electric field and the current-voltage conversion. We further extract the effective spine neck resistance and find R = 99.2 ± 34.5M Ω. We investigate numerically using the electro-diffusion model in non-cylindrical geometry the effect of changing the geometry. We found that contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius, when the injected current is large. We conclude that the postsynaptic current cannot only be modulated by changing the number of receptors, but also by changing the spine geometry, that could be a key step in synaptic transduction and plasticity. Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines

Part II

Electro-diffusion modeling of synaptic input in dendritic spines

Introduction

Neurons communicate via synaptic microdomains, where an input current generates a voltage change in the post-synaptic neuron. This voltage change, induced by excitatory current, reflects the strength of the synaptic connection between two interacting neurons [START_REF] Jack | Electric current flow in excitable cells[END_REF][START_REF] Holmes | Interpretation of time constant and electrotonic length estimates in multicylinder or branched neuronal structures[END_REF][START_REF] Rall | Methods in Neuronal Modeling: from Synapses to Networks[END_REF][START_REF] Qian | An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons[END_REF][START_REF] Yuste | Dendritic Spines[END_REF] and depends on two components: the first is the number of glutamatergic receptors for excitatory neurons and the second is the geometry of the post-synaptic terminal. However the relative contribution between these two factors is still unclear. For example, the post-synaptic structure is often a dendritic spine, the geometry of which is involved in modulating the time scale of diffusion [START_REF] Svoboda | In vivo dendritic calcium dynamics in neocortical pyramidal neurons[END_REF][START_REF] Korkotian | Dynamic regulation of spinedendrite coupling in cultured hippocampal neurons[END_REF][START_REF] Bloodgood | Neuronal activity regulates diffusion across the neck of dendritic spines[END_REF][START_REF] Tonnesen | Spine neck plasticity regulates compartmentalization of synapses[END_REF][START_REF] Holcman | Control of flux by narrow passages and hidden targets in cellular biology[END_REF]. In parallel, increasing the number of receptors on that terminal leads to a larger synaptic current [START_REF] Huganir | AMPARs and synaptic plasticity: the last 25 years[END_REF][START_REF] Elias | Synaptic trafficking of glutamate receptors by MAGUK scaffolding proteins[END_REF][START_REF] Kessels | Synaptic AMPA receptor plasticity and behavior[END_REF]. But diffusion alone is not sufficient to interpret the synaptic response because it is driven by electro-diffusion involving an electric field and diffusing ions. Electro-diffusion was applied successfully for studying ionic fluxes and gating of voltage-channels [START_REF] Mc Laughlin | The role of electroosmosis in the electric field-induced movement of charged macromolecules on surface of cell[END_REF][START_REF] Bezanilla | How membrane proteins sense voltage[END_REF][START_REF] Eisenberg | Look at biological systems through an engineer's eyes[END_REF]. Those work were all based on the fact that at the nanometer scale, cylindrical symmetry of a channel model reduces to a one-dimensional segment, allowing to study the electric field and charge densities in the channel pore [START_REF] Eisenberg | From structure to function in open ionic channels[END_REF][START_REF] Eisenberg | Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations[END_REF]. Moreover, recent studies have clearly shown that the current in the synaptic cleft reflects the coupling between moving ions and the local electrical field [START_REF] Sylantyev | Spike-driven glutamate electrodiffusion triggers synaptic potentiation viahomer-dependent mGluR-NMDAR link[END_REF][START_REF] Savtchenko | Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules[END_REF]. In a dendritic spine, voltage changes during the synaptic response are generated by the interactions between the ionic flow and the spine geometry. Dendritic spines are heterogenous microdomains at the limit of optical resolution and for that reason, voltage changes were estimated for many years using modeling and numerical simulations of the cable equations, the basis for the Hodgkin-Huxley model [START_REF] Rall | Methods in Neuronal Modeling: from Synapses to Networks[END_REF]. This approach is however not appropriate for spines because their micro-geometry composed of a bulby head connected to a thin neck, is significantly different from that of a cable. In addition cable theories break down when applied to small neuronal compartments, such as dendritic spines, because they assume spatial and ionic homogeneity. Linear approximations of electro-diffusion that couples the electric field with the ionic flow have been used to improve the estimation of the voltage changes in spines, approximated as cylinders of various sizes, but assuming local electroneutrality [START_REF] Qian | An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons[END_REF]. This method was later on generalized to study the electrical effect of Ranvier node [START_REF] Lopreore | Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier[END_REF]. Recent advent in monitoring the voltage changes at a submicrometer resolution [START_REF] Araya | The spine neck filters membrane potentials[END_REF][START_REF] Araya | Activity-dependent dendritic spine neck changes are correlated with synaptic strength[END_REF][START_REF] Hochbaum | Alloptical electrophysiology in mammalian neurons using engineered microbial rhodopsins[END_REF] revealed the electrical properties of dendritic spines [START_REF] Harnett | Synaptic amplification by dendritic spines enhances input cooperativity[END_REF][START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF], but the heterogeneity of the results [START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF][START_REF] Acker | EPSPs Measured in Proximal Dendritic Spines of Cortical Pyramidal Neurons[END_REF] and the absence of a robust computational framework and theory to interpret data challenges our understanding of electrical properties of these structures and cellular microdomains in general.

We develop here an electro-diffusion framework to compute the voltage-current relation and the local voltage variations generated by synaptic inputs. We present a deconvolution procedure to recover the time scale of voltage responses from voltagesensitive indicators in hippocampal neurons. The deconvolution procedure transforms the voltage dye Arclight response into voltage dynamics. To interpret the drop of voltage between the spine head and dendrite we use the Poisson-Nernst-Planck theory for electrodiffusion. We further compute how a current flowing inside the spine head is converted into voltage and determine the electrical resistance. Nu-

Results

merical simulations of the voltage drop in the entire spine reveal how a change in the neck length and radius alters voltage in the entire spine. The manuscript is divided in two main parts: first we present the general method of time series deconvolution applied here to voltage dyes and we construct an electro-diffusion model to extract parameters from data. In the second part, we use our electro-diffusion model to generate three dimensional simulations to investigate how the spine geometry modulate its electrical properties. We conclude here that while the numbers and the types of synaptic receptors determine the injected current, the geometry of a dendritic spine controls the conversion of current into voltage.

Results

Converting the Arclight fluorescent signal into a voltage response

Arclight dye indicators respond to voltage with an intrinsic delay [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF], such that a fast voltage response leads to a convolve fluorescence response. In that context, a synaptic input entering a dendritic spine generates a fluorescent response that needs to be deconvolved in order to recover the electrical genuine time course. As the voltage intensity has already been deconvolved in [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF], we focus here on the temporal response. The basis of the method is to find the causal kernel K(t), which is computed empirically by comparing the electrophysiology and the fluorescence responses in the soma (see Methods). Once the kernel is found, we will use it to recover the noisy voltage dynamics in much smaller structures such as dendritic spines and portions of dendrites. We apply the deconvolution procedure (Method and Fig. S1 in SI) on the soma region (Fig. 6.1A), which transforms the fluorescence dye signal (dashed green) into the voltage response (black) (Fig. 6.1B). The deconvolved signal is the shown by a line (green) which superimposes with the electrophysiological recordings (continuous black line). This result confirms the validity of the method. Indeed, by using the direct convolution of the electrophysiological recording by the kernel we obtain the response (black curve) that exactly super-imposes with the fluorescence soma signal (Fig. S2 in SI). We note that the second bump in fluorescence (dashed green) in Fig. 6.1C is removed by the fitting procedure before the deconvolution kernel K(t), as it should because the electrophysiological recordings does not exhibit this second bump.

Based on the deconvolution kernel K(t), we shall now deconvolve the fluorescent responses in the regions of interest (ROIs) R1 and R2 shown in Fig. 6.1C. Indeed, after we found the deconvolution kernel K(t) from the soma data, we apply it to deconvolve the signal in spines because we assume that properties and distribution of dyes are the same in the entire neuron. The results are shown in Fig. 6.1D-E for the voltage in the spine head and dendrite. We note that the original fluorescence (thin line) contains fluctuations (Fig. 6.1D-E) and an additional step is needed to remove the fluctuations (see Method section and in SI, Fig. S2. We found that the voltage time scale in the dendritic spines, recovered from the present deconvolution procedure, is slower than the expected direct synaptic response. Probably, this slow Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines time scale compared to direct synaptic electrical stimulation is due to glutamate uncaging, that can last for hundred of milliseconds. To conclude, we can now recover the voltage in region R1 that represents the head and R2, which refers to the adjacent portion of the dendritic shaft at the base of the spine. This deconvolution step is crucial because it allows us a direct comparison with the electro-diffusion model, that we shall describe in the next section.

Electro-diffusion theory for ionic flows in dendritic spines

To interpret the voltage dynamics in a dendritic spine, we use the electro-diffusion model that couples positive c p (x, t) and negative c m (x, t) charge concentrations with the electrical potential V (x, t). The model is based on the phenomenological Poisson-Nernst-Planck (PNP) equations, where the ionic flow is driven by diffusion and the electric field. The voltage is described by the classical Poisson equation [START_REF] Jackson | Classical Electrodynamics[END_REF] (see Methods). We use the PNP equations to model the flow of ions when a current I stim (t), composed exclusively of positive charges, is injected at the top of a dendritic spine. We recall that the electrical potential generated by a flow of ions is defined up to an additive constant.

We start with the description of the voltage response produced by an input current I(t) inside the spine neck. The current I(t) that arrives at the neck entrance results from the stimulation current I stim (t) received by the head. We applied the electro-diffusion approach at the nano-micrometer scale, by reducing the neck geometry to a one dimensional wire of length L (Fig. 6.2A-C). Due to its large size, compared to a dendritic spine, the dendrite constitutes a ionic reservoir (fixed concentrations).

Furthermore, using for the internal resistivity the value 109 Ω.cm [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF], we find that the fluctuations of the voltage in the portion of the dendrite R2 (Fig. 6.1C) are neglected, and in that case, the region R2 is expected to be isopotential. Consequently, we fix in the dendrite the value for the voltage V (0, t) = 0 mV. We thus interpret the potential V (t) = V (L, t) computed at the end of the neck (Fig. 6.2A-E) as the difference of potential between the entrance and the end of the spine neck. Consequently, we describe the electro-diffusion inside the neck by the ensemble of equations 6.9-6.13-6.14, where I(t) has yet to be estimated.

From the classical theory of electricity [START_REF] Jackson | Classical Electrodynamics[END_REF], it is not possible to extract the current passing through a passive device from the difference of potential when the resistance is unknown. However, using an electrical model for the axial current, escaping from the spine head to the neck, we shall reconstruct the voltage in the neck and recover the current in the entire spine. Because there is no direct measurement of the current I(t), we develop here a procedure (see SI) to estimate this current from the measured membrane potential φ(t) in the spine head. We assumed that the current can be written as the sum of a resistive and a capacitive term:

I(t) = Gφ(t) + C dφ(t) dt . (6.1)
The conductance G and the capacitance C reflect how the current I stim (t), injected in the spine, is processed by the head, depending on the intrinsic properties of the

Results

spine head (in the next section, we will simulate the entire voltage using PNP starting from the current entering through the channels). Consequently, we will refers to G and C as intrinsic conductance and intrinsic capacitance, respectively. We now estimate the two constants C and G and the voltage drop across a spine neck by solving numerically eqs. 6.9-6.13-6.14 and compare the results to the simulations with the deconvolved Arclight fluorescent voltage response. The response is generated following glutamate uncaging stimulations at the top of the spine head (Fig. 6.1C).

We solve numerically the PNP equations for the distribution of positive c p (x, t) and negative c m (x, t) charges, as well as the potential difference V (x, t) (Fig. 6.2D-E).

To estimate the voltage difference ∆ Ṽ (t) across the neck, we grounded the potential to 0 mV at the dendritic shaft (before stimulations, the voltage is described by eq. 6.12).

To assess whether the potential difference ∆ Ṽ (t) = V head (t) -V dend (t) can be predicted from the electro-diffusion model, we fix the input voltage φ(t) = V head (t). We then compare the voltage obtained by solving eqs. 6.9-6.13-6.14 (Fig. 6.2D-E) to the measured voltage V dend (t) in region R2 (blue) at the dendritic shaft (Fig. 6.2C). Although region R1 includes the head and the neck, we neglected the fluorescence in the neck due to its small thinness ≤ 100nm. We found a good agreement between the experimental data and numerical simulations (Fig. 6.2D) showing that the difference of voltages between the head and dendrite can be predicted from the input voltage V head (t). In addition, we estimated the injected current (Fig. 6.2E, see Methods and eq. 6.1) directly without any direct electrophysiological recordings. We can remark on this same figure that the deconvolved voltage in the head (blue) and the computed current (red) are almost perfectly proportional. This is because the capacitance C reported in table 6.1 and used to compute I(t) in eq. 6.1 is negligible such that the current is simply proportional to the current. We conclude at this stage that the electro-diffusion theory allows estimating the electrical properties of the spine red neck and the injected current (of the order of tens of pA) in the spine neck, triggered by a synaptic current I stim (t).

We apply systematically the electro-diffusion approach, based on (6.1), to extract the capacitance C and the conductance G of several spines (SI Fig. S3-5). Using an optimization procedure, we explore the parameter space for computing C and G (SI Fig. S3-4). We minimize the error between the solution of the electrodiffusion equation and the voltage output of the dendrite during a small time interval at the beginning of the response (SI Fig. S4). The resistance is computed by averaging the voltage changes in time. We use for the estimator the expression R neck = V (0,t)-V (L,t)

I(t)
for t ∈ [0 , 0.4]s. In all cases, we find a good agreement (SI Fig. S5) between the measured and computed voltage drop across the spine neck, where we estimated the current injected in the neck from the head of several spines.

In summary, we found that the average spine neck resistance is R = 99.2 ± 34.5M Ω (table 6.1). We also reported here a large variability in the spine resistance, while the intrinsic capacitance is negligible. To conclude, the electro-diffusion model allows computing the current injected in the spine neck after a synaptic stimulation, and thus recover the neck resistance. Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines

Voltage transduction in a spine and predictions of electro-diffusion

We have shown in the previous section that electro-diffusion PNP based model can be used to analyze Arclight fluorescence recording and this model allows estimating the current flowing in the spine neck. We shall now extend the PNP model to geometries that characterize the peculiar shape of the dendritic spines. To analyze how a dendritic spine influences the voltage response to a synaptic input, we simulate the three-dimensional PNP equations (see Methods) for two geometries: a ball of radius r head and a spine-like geometry (Fig. 6.3). We computed the distribution of the electrical potential for short spine necks (Fig. 6.3A), where the head contains two narrow openings: one of radius r o = 100nm representing the junction with the neck, and the other of radius r i = 10nm that receives the steady current I stim of positive charges (Fig. 6.3B). Note that we have dropped the time t in the expression of the current I stim , since we consider now steady-state current. Besides, the membrane capacitance is neglected because the current I stim represents the net contribution to the ionic flow entering the head. We computed the distribution of positive c p (x) and negative c m (x) charge concentrations as well as the voltage V (x) when it is grounded to 0 volt at the end of the spine neck, which represents the connection to the dendrite, considered as a large ionic reservoir. Consequently, V (x) represents the voltage difference induced by the injected current I stim . We find the distribution of the voltage along the x-axis (blue) Fig. 6.3B when I stim = 150pA is injected in the spherical geometry as shown in Fig. 6.3A-B. We found that there are two narrow layers due to the small entry and exit, where the injected current I stim induces a 15 mV drop that can propagate over small distance (≈ 100nm) inside the spine head. Those small regions of large convection are at the limit of actual resolution (≈ 0.116µm/pixel). Outside these layers, the voltage is quite uniform, leading to a reduced field convection e k B T ∇φ (eq. 6.9), demonstrating that diffusion is dominant inside the spine head. At this stage, we demonstrated numerically using PNP equations that the voltage drop in the spine head is negligible (less than a quarter of mV), in contrast with the classical cable theory (SI Fig. S7), which suggests that the motion of ions is driven by the voltage gradient.

To conclude, inside a spherical domain, diffusion is the dominant driving force and the potential drop is reduced significantly, which is equivalent of having a small spine head resistance. Indeed, applying the electro-diffusion model (eqs. 6.9-6.10-6.11) to a spine-like geometry (Fig. 6.3C), we observe that most of the voltage drop is carried by the spine neck (Fig. 6.3D). Interestingly, it is not equivalent to decrease the neck length to compensate for a decrease in the injected current (see result with an injected current of 150 versus 300 pA), suggesting that changing the synaptic weight by adding or removing receptors or modifying spine neck length have different consequences on the spine voltage. At this stage, electro-diffusion theory predicts that the voltage in the spine head V head (t) is spatially homogeneous, confirming the approximation of eq. 6.1, except near the post-synaptic density or at the entrance of the spine neck. The spine head resistance is thus negligible since the potential drop occurs just at the end of the neck and thus the entire spine resistance is carried by the neck: R spine ≈ R neck .

Results

Spine geometry determines the I-V conversion

To study the influence of the geometrical parameters on the electrical property of a spine, we first estimated the effect of the spine head radius r head for five spines (SI Fig. S5). By measuring their projected area S head from the two-photon images, we use relation

r head = S head π (6.2)
(Fig. 6.4A) to extract the equivalent radius (blue stars). We then use the PNP model associated to the short spine with no neck (Fig. 6.3A) to estimate the average voltage difference V ball (for a current of 100 pA) between the north and the south pole of a spine head. We find that the mean voltage varies in a range of 1.5-1.6mV , when the radius of the head varies in the range 0.3 -1.5µm. This result shows that the head radius had little influence on the mean voltage.

We then estimated how the spine neck resistance R neck depends on the neck length and width, usually unaccessible using classical microscopy approaches: we find both theoretically and experimentally that the resistance increases (blue stars) with the neck length L (Fig. 6.4B and Fig. 6.2D-E. Note that the size of the head is not correlated with the resistance (table 1). Furthermore, electro-diffusion simulations in a segment predicts that for the mean current input extracted from data, there is a significant change in the local concentration of positive charges along the segment of length L = 0.7µm at the time-to-peak (t 0 = 55ms). Indeed, the difference in concentration is 33mM (the imposed concentration on the other end is 163 mM), leading to a concentration gradient between the dendrite and the spine head (Fig. 6.4C-D). Although the injected current I stim (t) is composed of positive charges, the concentration gradient in the neck is driven by positive and negative charges. Such phenomenon is called concentration polarization [START_REF] Strathmann | Ion-Exchange Membrane Separation Processes, Membrane science and technology series[END_REF].

Finally, using the electrodiffusion theory and the spine parameter r head = 0.5µm, L = 1µm, we estimated the I-V relation for various neck radius, showing a saturation for large current (Fig. 6.4E). These curves show that the neck radius is one of the most critical parameter in defining the conversion of current into voltage. Then to clarify the dependency of the spine neck resistance R neck on its radius, we simulated the PNP equations by solving equations (6.15), where a steady-state current I stim = 20; 50; 100; 150pA is injected inside the spine head (Fig. 6.3C). We obtained the average voltage inside the head V head . We plotted R neck = V head /I stim in Fig. 6.4F. In particular, we predict that for an injected current of I = 50pA and a spine radius of r = 50nm (resp. 100nm), the overall spine resistance, which is mostly carried by the neck would be R spine ≈ R neck = 120 (resp. 350)M Ω). Surprisingly, we could fit the curve in Fig. 6.4F with power laws 1 r 3/2 0 (dotted blue) and 1 r 0 (dotted red) for an injected current I stim = 20pA and 150pA respectively. This results show a clear deviation from the classical Ohm's law for a resistance that depends on 1/r 2 0 and also a limitation of the diffusion approach [START_REF] Svoboda | In vivo dendritic calcium dynamics in neocortical pyramidal neurons[END_REF] In summary, we used the electro-diffusion theory and the Arclight fluorescent data to characterize the electrical properties of a dendritic spine. With respect to a synaptic input, a spine can be electrically characterized as a diode device (Fig.

Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines 6.5A-B) with a finite resistance (for a small current), saturating for large currents (Fig. 6.5C). The voltage difference varies from few to tens of mV. However, from the perspective of a Back Propagation Action potential, the equivalent circuit of a spine is a diode with zero resistance (no leak current [START_REF] Horowitz | The Art of Electronics[END_REF]) Fig. 6.5D.

Discussion

We developed here a computational approach based on the electro-diffusion theory to estimate the electrical properties of dendritic spines. We first deconvolved the Arclight fluorescent signal and then applied the electro-diffusion theory to estimate the resistance and the capacitance from hippocampal neuron data. Our approach contrasts with classical estimation of the spine neck resistance and dendritic spine electric properties, extracted in the context of the electrical circuit approximation, cable theory and even diffusion approximation [START_REF] Svoboda | In vivo dendritic calcium dynamics in neocortical pyramidal neurons[END_REF][START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF][START_REF] Acker | EPSPs Measured in Proximal Dendritic Spines of Cortical Pyramidal Neurons[END_REF][START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. We found here that the electro-diffusion coupling is the main driving force for the ionic current in the spine neck (Fig. 6.3), while diffusion is sufficient to describe the motion of ions inside the head. The electric field is negligible in the head, except very close to the entrance of the synaptic input and at the exit with the neck. Electro-diffusion theory reveals that the spine head geometry imposes that the voltage is almost constant in the head, while the neck is responsible for most of the voltage drop. This is in contrast with the predictions of the cable theory or previous approximations of electro-diffusion [START_REF] Rall | Methods in Neuronal Modeling: from Synapses to Networks[END_REF], based on electroneutrality and no gradient of charges. We also demonstrated here that the ion conduction is mostly driven by diffusion in the spine head, suggesting that the head resistance is negligible compared to the neck.

It remains difficult to study the exact local balance of positive by negative charges, because in transient regimes or at equilibrium, positive charges are all the time in excess. Possibly the sum of negatively ionic charges plus the negative charges located on immobile proteins can balance positive charges at a tens to hundreds of nanometers. Long-range electro-diffusion effects have already been described for directing the current flow in the synaptic cleft into the post-synaptic terminal [START_REF] Sylantyev | Spike-driven glutamate electrodiffusion triggers synaptic potentiation viahomer-dependent mGluR-NMDAR link[END_REF][START_REF] Sylantyev | Electric fields due to synaptic currents sharpen excitatory transmission[END_REF], showing in a different context that electro-diffusion drives ionic flows and the voltage in neuronal microdomains.

Time deconvolution of the Arclight fluorescent signal

Neuronal voltage is reported by recording electrodes and we showed here how the Arclight signal can be deconvolved in small and large microdomains, so that we can now access the voltage dynamics and electrical properties from microdomains. Genetically encoded activity sensors combined with microscopies are now classically used [START_REF] Emiliani | All-Optical Interrogation of Neural Circuits[END_REF] to record and manipulate the activity of neural circuits. We show here how fluctuations contained in the fluorescent signal can be filtered and the voltage time course is recovered from the empirical kernel K(t) (Methods). This approach can be apply to any encoded activity sensors expressed in neurons and only requires to compare the electrophysiological recording with the florescence in the soma. The

Discussion

present deconvolution could also be used to recover the electrical activity from slow calcium indicators [START_REF] Emiliani | All-Optical Interrogation of Neural Circuits[END_REF].

Influence of the neck radius on the spine resistance

The spine neck radius cannot be spatially resolved, so any geometrical fluctuation is likely to result in a drastic change in the resistance. For diffusion alone, the rate of extrusion [START_REF] Holcman | Control of flux by narrow passages and hidden targets in cellular biology[END_REF] (for Brownian particle) from a dendritic spine is given in first approximation as

|Ω spine |L Dπr 2 o + C 2 |Ω head |R 3/2 c Dr 3/2 o
, where D is the diffusion coefficient, |Ω spine | is the volume of the spine, L the length of the neck, R c the radius of curvature at the base of the neck-head junction, |Ω head | the volume of the head and C 2 a constants and r o is the radius of the neck. This expression shows that a small change in the radius r o (dividing by two for example) leads to a significant change of at least 4 for the diffusion time scale. We addressed the radius neck uncertainty here in the context of electro-diffusion by computing the neck resistance for different radii (Fig. 6.4F and SI, S6). Spine intrinsic electrical characteristics are revealed by the impedance which is the ratio of the voltage to the injected current. For example, for a steady state current of I = 50pA, the Ohmic resistance of a spine of radius 100nm (resp. 50nm) is R neck = 120M Ω, (resp. R neck = 350M Ω). Interestingly, we also found here that the resistance of a dendritic spine is inversely proportional to the radius of the neck r 0 , and not by the square r 2 0 , as classically described for electrical devices (SI Fig. S6) [START_REF] Svoboda | In vivo dendritic calcium dynamics in neocortical pyramidal neurons[END_REF]. This result shows that the neck size has a key effect in modulating the spine electrical resistance. Another prediction of the present theory is that a synaptic current injected in a spine head should be of the order of 100 pA (as suggested in Fig. 6.2E). The shortest diameter of a spine neck along its length is certainly a key factor that could drastically affect its resistance. Indeed, the critical geometrical parameter is the minimal shortest constriction along the neck [START_REF] Holcman | Control of flux by narrow passages and hidden targets in cellular biology[END_REF], that could further be influenced by the crowding due to the internal endoplasmic reticulum [START_REF] Holcman | The new nanophysiology: regulation of ionic flow in neuronal subcompartments[END_REF].

The geometry of dendritic spines modulate the voltage changes independently of the input current

Dendritic spines are involved in modulating two-and three dimensional receptor trafficking [START_REF] Huganir | AMPARs and synaptic plasticity: the last 25 years[END_REF][START_REF] Elias | Synaptic trafficking of glutamate receptors by MAGUK scaffolding proteins[END_REF][START_REF] Kessels | Synaptic AMPA receptor plasticity and behavior[END_REF], molecular post-synaptic density composition, calcium diffusion [START_REF] Yuste | Dendritic Spines[END_REF][START_REF] Korkotian | Dynamic regulation of spinedendrite coupling in cultured hippocampal neurons[END_REF], synaptic transmission and plasticity. We have shown here using the electro-diffusion framework that the voltage in dendritic spines can also be controlled by changing the neck length geometry. This modulation obtained by changing the geometry is complementary to the possible changes in the number of receptors resulting in a long-term modification of the synaptic current, reflecting synaptic plasticity.

Changing the spine neck length can thus regulate the local dendritic voltage, that contributes to the genesis of an action potential. We further confirm previous experimental findings [START_REF] Araya | The spine neck filters membrane potentials[END_REF][START_REF] Araya | Activity-dependent dendritic spine neck changes are correlated with synaptic strength[END_REF], showing that the synaptic voltage amplitude is inversely correlated with the neck length, but we found here a much stronger effect compared Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines to previously evaluated [START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF]. However, in agreement with [START_REF] Popovic | Electrical behaviour of dendritic spines as revealed by voltage imaging[END_REF], we do not need to use any additional active channels in the electro-diffusion model to account for the voltage in the spine, suggesting that they might not play a predominant role.

To conclude, voltage changes in dendrites can now be detected at the nanometer scale and the electro-diffusion theory allows interpreting these data and predicts a nonlinear current-voltage relation imposed by the specific geometry of dendritic spines. While the spine geometry controls voltage, the synaptic current is set by the number of receptors [START_REF] Huganir | AMPARs and synaptic plasticity: the last 25 years[END_REF][START_REF] Elias | Synaptic trafficking of glutamate receptors by MAGUK scaffolding proteins[END_REF][START_REF] Kessels | Synaptic AMPA receptor plasticity and behavior[END_REF]. These two mechanisms are supposed to be independent and they are both involved in controlling the synaptic response. It would certainly be interesting to study how changes in one affects the other. Moreover as thought in [START_REF] Crick | Do dendritic spines twitch?[END_REF] then shown in [START_REF] Korkotian | Spike-associated fast contraction of dendritic spines in cultured hippocampal neurons[END_REF][START_REF] Korkotian | Regulation of dendritic spine motility in cultured hippocampal neurons[END_REF], spine twitching may impact the electrical property of spines. Indeed, the consequence of the spine head constriction might be negligible for the voltage inside the head, but could influence the current inside the neck and thus reduce the effective resistance. This could be investigated by adding a fluidic component, a heavy analysis that would require to couple the Poisson-Nernst-Planck equation to the Navier-Stokes equations. Yet, such computation could emphasize even more the key role that plays geometry in defining electrical properties of dendritic spines.

6.4. Methods

Methods

Arclight signal

We briefly described here the experimental data we have used for our electrodiffusion theory and time deconvolution. There are fully described in [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF]. The protein-based voltage indicator ArcLight is injected in primary cultured hippocampal neurons. ArcLight expressing dissociated hippocampal culture neurons in DIV 12-16 were recorded in artificial cerebrospinal fluid (ACSF) containing ions of various concentration. Two-photon glutamate uncaging was done with a custom-made two-photon laser scanning microscope. In glutamate uncaging, the location of stimulation was selected with 1-2 µm distance from dendritic spines, not closer than 1 µm. The whole-cell patch clamp and the glutamate uncaging were performing while doing the wide-field one photon imaging of ArcLight fluorescence. Finally, we used the voltage deconvolved from the fluorescence signal, based on a two-state model of voltage dependent ArcLight fluorescence described in [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF].

Deconvolution Kernel

To recover the intrinsic voltage dynamics h(t) from the slow Arclight signal A rc (t), we compare the electrophysiological patch-clamp recording in the soma with the Ar-cLight fluorescence extracted from the somatic region delimited in the image (Figure 6.1A). This comparison is at the basis of the deconvolution method of the causal fluorescent signal. Indeed, the slow Arclight reporter convolves the fast electrical voltage signal, modeled by a kernel function K(t) with the intrinsic dye dynamics, leading to a slow fluorescent response. The kernel K(t) describes the time delay of the fluorescence activation compared to the voltage dynamics. We model the kernel by the function

K(t) = A τ 2 te -t τ , (6.3)
where the value of the parameters A and τ are obtained by comparing the Arclight response in the soma with the convolution of the electrophysiological recordings (Fig. 6.2). Indeed, for a voltage signal h, the Arclight signal A rc (t) is expressed by the convolution product

A rc (t) = t 0 K(t -s)h(s)ds (6.4)
To recovered h from the Arclight signal A rc , we first calibrated the kernel so that the Arclight signal peaks exactly at the one monitored by the electrophysiological signal (Figure 6.1B) and we obtain τ = .05s. The other parameter A is a scaling that will be adjusted for each experimental data. We denote the normalized kernel by K n (t) = te -t τ (plotted in SI Fig. 1).

Noise filtering and approximation

In small dendrite and dendritic spine regions, the Arclight data contains a large noise that should be removed. For that purpose, we use a Savitzky-Golay filter [START_REF] Savitzky | Smoothing and Differentiation of Data by Simplified Least Squares Procedures[END_REF], to Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines increase the signal-to-noise ratio. The detail of that procedure is explained in the SI. Once the noise is removed, we define a new step which consists in approximating the signal using a family of analytical functions

f β 1 ,β 2 ,γ 1 ,γ 2 ,α (t) = t α (β 1 e -γ 1 t + β 2 e -γ 2 t ), (6.5)
where the parameters β 1 , β 2 , γ 1 , γ 2 , α are obtained by a best approximation (see SI Fig. S2).

Microdomain Arclight deconvolution

In the final step, we shall use the deconvolution procedure to compute the voltage. Indeed, once the kernel K(t) is determined (see above) from the somatic signal and after the step of noise filtering, we shall retrieve the voltage dynamics from dendrites and dendritic spines, where direct electrophysiological recordings are not possible. Using the analytical approximation A rc (t) = f β 1 ,β 2 ,γ 1 ,γ 2 ,α (t) of the Arclight fluorescent response (relation 6.5) described in the previous subsection, we shall now compute the voltage h using K(t) (eq. 6.3) by inverting equation 6.4 using the Laplace's transform: ĥ

(ω) = fβ 1 ,β 2 ,γ 1 ,γ 2 ,α (ω) K(ω) , (6.6)
where the Laplace's transform of the kernel is K

(s) = ∞ 0 K(t)e -t s dt = A (sτ +1) 2 and fβ 1 ,β 2 ,γ 1 ,γ 2 ,α (s) = Γ(α + 1) β 1 (s + γ 1 ) α+1 + β 2 (s + γ 2 ) α+1 , (6.7)
where Γ(•) is the Gamma function [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]. The final expression for the voltage is derived in the SI and is given by

h(t) = β 1 e -γ 1 t A t α-1 (t(1 -2τ γ 1 ) + 2τ α) + τ 2 t α-2 (tγ 1 ) 2 + α 2 -α -2γ 1 αt + β 2 e -γ 2 t A t α-1 (t(1 -2τ γ 2 ) + 2τ α) + τ 2 t α-2 (tγ 2 ) 2 + α 2 -α -2γ 2 αt . (6.8)
Note that in practice the value of the parameter A is calibrated so that the maximum amplitude of the voltage before and after the deconvolution are identical. We applied this procedure to recover the voltage h(t) in dendritic spines. This procedure is the time deconvolution of the voltage dynamics from the Arclight fluorescent signal.

Electro-diffusion model in the spine neck

The Poisson-Nernst-Planck (PNP) equations express the coupling between the ionic flow and the voltage (Poisson equation). We present here the one-dimensional version of these equations. They reduce for the voltage V and the concentration of Where D p , D m are diffusion coefficients, e the electronic charge, the valencies for each specie is z = ±1 and k B T is the thermal energy. Equations 6.9-6.10-6.11 are used to compute the voltage drop when a current I(t) is injected at the tip of the spine neck. During the simulations, the ionic concentrations in the dendrite (ionic reservoir) are the boundary conditions fixed at the values C p and C m (see table 2). We recall that the electrical potential is defined to an additive constant. The initial and boundary conditions are

V (x, 0) = 0 (6.12) c p (x, 0) = C p and c m (x, 0) = C m . ∂V ∂x (x, t) x=0 = 0 (6.13) ∂c m ∂x (x, t) x=0 = 0 ∂c p ∂x (x, t) + e k B T c p (x, t) ∂V ∂x (x, t) x=0 = I(t) D p F πr 2 i , (6.14)
V (L, t) = 0, c p (L, t) = C p and c m (L, t) = C m ,
where r i is the radius of the circular opening where the current is injected in our simulations (see Fig. 6.3A). In summary, eqs. 6.9-6.13-6.14 describe the ionic response of an input current I(t) inside a thin cylinder reduced to a one dimensional segment. We simulate these equations using Comsol to determine the voltage drop (Fig. 6.2).

3 dimensional PNP-equations in a ball and a dendritic spine shape

We present now the steady-state PNP equations, that describe the concentration of positive c p (x) and negative c m (x) charge concentrations and the voltage φ(x) inside a three dimensional bounded domain that we use in Figure 6.

∂c p ∂n (x) = ∂c p ∂n (x) = 0 on ∂Ω r ∂c p ∂n (x) + e k B T c p (x) ∂φ ∂n (x) = I stim πr 2 i F D p on ∂Ω i ∂c m ∂n (x) - e k B T c m (x) ∂φ ∂n (x) = 0 on ∂Ω i φ(x) = 0 on ∂Ω o c p (x) = C p on ∂Ω o c m (x) = C m on ∂Ω o .
In that model, only positive charges can enter the spine domain. We use the Comsol platform to solve numerically equations 6.15 presented in Fig. 6.3.

Arclight genetically encoded voltage indicator

We used here the genetically encoded voltage indicators (GEVIs) [START_REF] Brinks | Two-Photon Lifetime Imaging of Voltage Indicating Proteins as a Probe of Absolute Membrane Voltage[END_REF][START_REF] Gong | High-speed recording of neural spikes in awake mice and flies with a fluorescent voltage sensor[END_REF][START_REF] Han | Fluorescent protein voltage probes derived from ArcLight that respond to membrane voltage changes with fast kinetics[END_REF][START_REF] Jin | Single action potentials and subthreshold electrical events imaged in neurons with a fluorescent protein voltage probe[END_REF][START_REF] St-Pierre | High-fidelity optical reporting of neuronal electrical activity with an ultrafast fluorescent voltage sensor[END_REF], which can be expressed in neuronal membrane through plasmid transfection. This dye is used for an optical measurements of the membrane potential. ArcLight was expressed in cultured mouse hippocampal neurons and the fluorescence was imaged with an upright fluorescence microscope and a fast sCMOS camera. To monitor somatic electrophysiology, imaged neurons also were patched in whole-cell, current clamp mode. In neurons expressing ArcLight, fluorescent signals were clearly visualized in dendritic spines as well as soma.

To test how the ArcLight fluorescence responsed to bAPs generated by somatic current injection, the eletrophysiological signals were recorded from areas of interest (ROI) in somata, proximal dendrites and spines. To quantify optical signals, we measured the relative change in fluorescence intensity -∆F/F , a quantity directly proportional to membrane potential [START_REF] Peterka | Imaging voltage in neurons[END_REF]. Intracellular ArcLight proteins located in ER and Golgi may contaminate baseline fluorescence (F ) because they are fluorescent, but insensitive to membrane potential, as they are too far from the Debye length of the membrane's electric field. Although ArcLight is in a dark state at the lower pH of intracellular organelles [START_REF] Han | Mechanistic studies of the genetically encoded fluorescent protein voltage probe ArcLight[END_REF],

6.4. Methods its contribution to background could be critical to determine -∆F/F , if averaged together with a membrane responding ArcLight. To evaluate this possibility, it was calculated [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF] the activity probability of each pixel in response to voltage by using a constrained non-negative matrix factorization (CNMF) algorithm [START_REF] Pnevmatikakis | Simultaneous Denoising, Deconvolution, and Demixing of Calcium Imaging Data[END_REF] and compared the resulted weight matrix image to the base fluorescence image. Through this comparison, we identified regions of non-responding intracellular ArcLight in the soma, which has strong base fluorescence yet low activity weight, while there were no such regions in dendrites and spines. This result implies that the trafficking mechanism of ArcLight proteins is by ER translation at inactive sites of soma, and then targeting to adjacent somatic cell membrane and by diffusion to the rest of the cell along the cell membrane. In summary, the Arclight dye is now inserted in the membrane where the voltage sensor is located in the cytoplasm and thus it reports internal changes of the membrane, that we analyzed here. Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines Column 3 presents the radius r head of the spine head computed using formula 2. Column 4: intrinsic capacitance C computed from the method described in section 2 (Fig. S3) of the SI. Column 5: Intrinsic resistance of the spine 1/G defined in Section Electro-diffusion theory for ionic flows in dendritic spines and computed according to the method described in section 2 of the SI (Fig. S3). Column 6: Spine neck resistance computed from formula R neck = V (0,t)-V (L,t)

I(t)
(see section S3 for the definition and computation). We use here the notation V (0, t) and V (L, t) to represent the voltage at 0 (entrance of the neck) and L (based of the spine) respectively and • represent the average in time between 0 and 0.5s. We computed V (0, t) and V (L, t) from formula (6.9) and the current I(t) from equation (6.1). The averaged voltage in the head V head is computed from system of PNP equations (6.15). The computed resistance R neck is fitted to power laws a 1 /r 2 o (dashed green) and a 2 /r We present in this section a deconvolution procedure for the time series of the Arclight fluorescent signal. The procedure has been summarized in the Method section and we provide now additional details. The first part of the deconvolution uses the input signal recorded in the soma. This part allows computing the deconvolution kernel K(t) as described in the Method section. In Fig. 7.2A, we confirm the result of the deconvolution by comparing the direct convolution of the voltage (continuous black) with the kernel K. We recover the fluorescent signal (dashed green). We show in Fig. 7.2B the kernel K(t).

Parameter

We shall describe now how we have removed the high frequency fluctuations present in the fluorescence signal (Fig. 7.1A). We use the Savitsky-Golay (SG) filter, which is based on local least-squares polynomial approximation. The filter reduces the fluctuations while maintaining the shape and height of the initial signal [START_REF] Savitzky | Smoothing and Differentiation of Data by Simplified Least Squares Procedures[END_REF].

Savitsky-Golay filter

The Savitsky-Golay filter is based on decomposing the input signal into polynomials. We start with the sampling,

F (t) = N k=1 F (t k )δ(t -t k ), (7.1)
where N is the number of points. We define the discrete time subinterval I k = {t k -n∆t • • • , t k • • • t k + n∆t} of size 2n + 1 time points, separated by a time step ∆t. The filter is constructed by finding an ensemble of polynomials iteratively on each time window I k . We define the sequences F q of functions q = 0..N , where F 0 = F . To compute the polynomial of degree

(7.9)

The system of equations (7.9) is a linear matrix equation,

  

F k-1 (t k -n∆t) . . . where V is the (2n + 1, p + 1) matrix We now segment the fluorescence signal. The first time interval starts t unc = 0 which is the initiation of the voltage response until the maximal response at time t peak , Figure 7.1. Because the physical modeling of the dye kinetics [START_REF] Kwon | Attenuation of Synaptic Potentials in Dendritic Spines[END_REF] predict a single exponential decay, we shall treat, after the time t peak , any fluctuations as noise and we will apply the iteratively the SG filter (7.13). In summary, we divide the fluorescent response F (t) into three subregions Fig. 7.1A:

V =      1 t k -n∆t (t k -n∆t) 2 • • • (t k -n∆t) p 1 t k -(n -1)∆t (t k -(n -1)∆t)
1. before the time of glutamate uncaging t unc , 2. from the time t unc to the time-to-peak t peak , 3. after the time-to-peak t peak .

In region 1 (before t unc ), we use the following parameters: polynomial degree p = 2, the size of the window |I k | = 40 points and the number of iteration of the filter N SG = 1 to 3 (depending on the signal). This filtering reduces the noise amplitude in this first region by 80%. The filtered signal is f 1 . In regions 2 and 3, to preserve the signal peak amplitude, we use a hight degree polynomial and the parameters p ≥ 25, |I k | = 100, and N SG = 6 ± 2 to remove small fluctuations (Fig. 7.1A). However, we can note that the amplitude of the noise leads in the fitting procedure to a second bump of smaller amplitude around t = 0.65s, yet this does not impact our analysis. In region 3, we remove large fluctuations using parameters p ≥ 5, |I k | = 80, and N SG = 5. The filtered signals in regions 2 and 3 are f 2 and f 3 respectively.

To glue continuously the results on each region, we use the sigmoid function, The results are shown (Fig. 7.1A).

σ

Details of the fluorescence time deconvolution

We now described in detail the fit approximation by analytical function of the filtered signal. The goal of this part is to prevent any amplification of fluctuation in the deconvolution procedure. After the voltage is filtered, we use a family of function f γ 1 ,γ 2 ,α,β 1 ,β 2 (t) = t α (β 1 exp(-γ 1 t) + β 2 exp(-γ 2 t)) (7.16) to fit the voltage. This step eliminates any small fluctuations that could have been amplified in the next deconvolution step (Fig. 7.1B). Another advantage of having an analytical representation is to obtain an explicit Laplace's transformation. Indeed, fγ 1 ,γ 2 ,α,β 1 ,β 2 (s) = Γ(α + 1) β 1 (s + γ 1 ) α+1 + β 2 (s + γ 2 ) α+1 , (7.17 16. The results are given for a dendrite (dashed green) and a spine head (dashed magenta). C. Deconvolution of the fitted signal (dashed lines) using the deconvolution kernel eq. 7.19. This procedure recovers the voltage time course in the dendrite (red) and the spine head (blue).

Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series where Γ is the Gamma function [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]. We now use the exact form of the kernel to obtain an explicit formula for the deconvolution. We recall that the convolution kernel (see Material and Method) is given by To conclude, we obtain here the expression for the deconvolution of eq. 7.25 from the analytical approximation of the Arclight fluorescent signal, described in Figure 7.1C. Note that the value of A is calibrated for each set of data such that the maximum amplitude between the Arclight response and the deconvolved voltage remains identical.

K(t) = A τ 2 t
Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series C. The signal from panel B (blue) is fitted by a sum of terms (t -t j ) α j e -γ j (t -t j ))H(t -t j ). D. Deconvolved voltage obtained from the fitted trace (dashed green) in panel C. E. The deconvolved voltage (red) is compared to the original voltage (dashed black).

7.2. Optimization procedure to extract the electrical capacitance and conductance C and G of a dendritic spine from the voltage drop between the head and the base of spine where t j are the time sequences where the signal has a local minimum (see Fig. 7.3A, red dashed lines), β j , α j > 2 and γ j are positive constants and H(t -t j ) is the Heaviside function defined such as H(t -t j ) = 1, if t ≥ t j 0, if t < t j (7.28)

We can remark that the t j are provided by the electro-physiology and not the fluorescence signal. Following the procedure described in the previous section, to recover the voltage V (t) from the fluorescence A rc (t) , we need to invert the convolution equation

A rc (t) = (V * K)(t) (7.29)
Applying Laplace transform leads to the analytical representation of the solution

V (s) = A rc (s) K(s) = n j=0
β j Γ(α j + 1) A

(1 + τ s) 2 (s + γ j ) α j +1 e -s t j (7.30)

The voltage V (t) is recovered computing the inverse Laplace transform of the expression above, we get

V (t) = n 0
β j A e -γ j (t -t j ) H(t -t j ) P τ,α j ,γ j (t -t j ). (7.31) where P τ,α,γ (t) = t α (τ γ -1) 2 -2ατ (γτ -1)t α-1 + ατ 2 (α -1)t α-2 (7.32)

The deconvolved voltage is shown Fig. 7.3D. Note that the fluctuations disappear completely because the fitting procedure (7.27) remove them completely. The result of the deconvolution is compared to the initial voltage response (dashed black) in Fig. 7.3E. We conclude that the present method can be used to deconvolve several inputs.

7.2 Optimization procedure to extract the electrical capacitance and conductance C and G of a dendritic spine from the voltage drop between the head and the base of spine

We present now an optimization procedure for extracting the capacitance C and conductance G from the measured voltage time series. The procedure consists in fitting a very small portion of the voltage time response curve and to predict from this short time interval, the entire time response. This procedure allows also to compute the current I(t) injected in the spine neck, based on equation 1 (main text) that we recall here The electrical potential is defined to an additive constant in the dendrite, which is modeled as a ionic reservoir where we fix the voltage V = 0 mV. The initial conditions are where r i is the radius of the narrow opening ∂Ω i . To determine the couple of unknown parameters (C, G), we use an iterative algorithm, where we solve numerically the boundary value problem eqs. 7.34-7.35, from which we obtain a ionic current I C,G (t) (eq. 7.33) injected at the tip of the neck and the voltage V C,G (x, t) in the neck at position x and at time t. Because V C,G (L, t) = 0, the voltage V C,G (0, t) represents the difference of potentials between the head and the dendrite. We start the iteration algorithm with a value for the parameters C 0 , G 0 : C 0 = 0.1pF and G 0 = 1nS (Fig. We first look for the minimum of expression 7.39 for all entries of the matrix. For a given precision (ε G , ε C ), we evaluate whether or not the conditions

I(t
V
C * -C 0 < ε C and G * -G 0 < ε G , (7.41)
7.4. Limitation of the cable theory

Limitation of the cable theory

To confirm the limitation of the cable theory in describing the potential change across the spine neck, we use the cable equation 7.48 below for the potential V (x, t) at position x and time t (see Methods) along a one dimensional segment [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. We impose a first condition for the voltage at the entrance of the spine neck from the head V (0, t) = V head (t), while the boundary condition at x = L assumes a zero electric field at the connection with the dendrite. We use for V head (t) the deconvolved voltage measured in the head.

The linear cable equation

The where the two independent parameters λ = rm ra and τ = r m c m [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF] are related to the membrane capacitance c m , the resistance r m and the intracellular resistance r a . By definition, r m = Rm πd and c m = C m • πd, where R m is the specific membrane resistance and C m the specific membrane capacitance and d is the diameter of the cable [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF].

Conclusion of the linear cable equation

Voltage changes computed from the cable model (green dashed) when the input is the measured voltage in the head (blue) does not match the measured response in the dendrite (red). To obtain a response comparable to the output, the intracellular resistivity should be increased by a factor 4.10 5 , showing the limitation of the cable equation to account for the voltage propagation in a dendritic spine (Fig. 7.7A). This can be explained by the fact that one of the assumptions the cable theory is based on consists into neglecting longitudinal fluctuation of the concentration c i : ∂c i /∂x = 0, an assumption that was expected to be a drawback in small compartments [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. On the contrary, in the PNP model, the voltage drop in the neck results from the longitudinal gradient of concentration (concentration polarization) in the neck, as show in the main text Fig. 4D, where ∂c i /∂x ≈ 50mM/µm = 0. We conclude that the classical cable equation cannot describe accurately in the normal range of parameters the voltage change in a dendritic spine neck, contrary to the PNP model, as described in the main text. The simulated voltage response (green dashed) using the cable equation eq. 7.48 and the spine head input. To obtain the response that should be comparable to the voltage in the dendrite, the intracellular resistivity should be increased by a factor 4.10 5 from the basal value R 0 i = 105Ωcm [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. The other cable parameters are given by C m = µF/cm 2 and R m = 2 • 10 4 Ωcm 2 [START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF].

Conclusion

To conclude my PhD I would like to discuss three possible future directions.

The first possible direction is the development of new asymptotic methods for nonlinear partial differential equations such as transient Poisson-Nernst-Planck equations. We have been concerned here with the solution of the stationary PNP equations inside domains containing singularities such as cusp-shaped funnels. A possible direction is the extension of our approach to general domains to evaluate the effect of the local mean curvature of the boundary on the maximal drop of the voltage between two points. Another direction is to estimate the voltage drop in the boundary layer of the surface where a steady-state current is injected. Finally, it would be interesting to study fast transient regimes, and in particular, to obtain possible asymptotic solutions in a short time asymptotic.

The second direction concerns the signal processing and the deconvolution of the fluorescence time series. I developed a deconvolution method to recover the voltage from the fluorescence of an organic dye (Arclight). It would be interesting to generalize this method to deconvolve multiple fast spikes. In parallel, a new technology has emerge based on nanodiamonds that can report voltage changes at a resolution of sub-milliseconds timescale and hundred of nanometers [109,[START_REF] Hall | High spatial and temporal resolution wide-field imaging of neuron activity using quantum NV-diamond[END_REF]. It would be interesting to generalized our method to extract the voltage below the technical resolution of such method through averaging of the signal.

Finally, the last direction that could be interesting is the exploration of the molecular dynamic simulations of ions in micro-domains. So far, molecular simulations have been limited to few hundreds of ions, to study channels [START_REF] Schuss | Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model[END_REF][START_REF] Kalugin | Microscopic Structure and Dynamics of Molecular Liquids and Electrolyte Solutions Confined by Carbon NanoTubes: Molecular Dynamics Simulations, Carbon Nanotubes -Synthesis[END_REF], or they are limited in time (microsecond). The challenge here is to simulate thousand of ions in a time scale of milliseconds. New molecular coarse-graining approach using homogenization techniques starting with a Langevin equation could lead to stochastic coarse-grained equations. This approach would be important to clarify the extension of non-electro-neutrality, which was the subject of this thesis.

•

  : Average f * g: Convolution product between f and g ∇: Differential operator (nabla) δ(x): Dirac's delta function ∂u(x) ∂n : Dot (scalar) product ∇u(x) • n Γ(•): Euler gamma function H(•): Heaviside function Im(•): Imaginary part W (•): Lambert's function ∆: Laplace operator (Delta) f : Laplace transform of f R, R d : Real line and d-dimensional Euclidean space Re(•): Real part x, f (x): Vectors -bold letter

Figure 1 :

 1 Figure 1: Anatomy of a neuron and dendritic spine A. Microscopy imaging of a neuron, adapted from [148]. B. Schematic representation of a chemical synapse. C-D 3D reconstruction from electron microscopy adapted from [210]. The red structure is the Post-Synaptique Density (PSD) [210].

0. 1 .

 1 Part I: Electrostatics of non-neutral biological micro-domains

Figure 2 :

 2 Figure 2: The narrow escape problem. A Brownian trajectory (red) is reflected on the boundary of a domain Ω, and is absorbed at a small circular target site ∂Ω a (green).

  Fig. 3C. One can remark that the banana-shaped domain shown in Fig 3C is 2 √ ε width. If the Neumann boundary condition is zero at the cusp boundary -mapped into the two circular arc in Fig 3C -then one can simply reduce the babana-shaped domain Ω w to a 1D circular arc [86, 92]. The boundary conditions for the approximated one-dimensional solution in the banana-shaped domain are zero at angle θ Lim = c √ ε, where c is a constant (see

Figure 3 :

 3 Figure 3: Ball with a cusp-shaped funnel and image Ω w of the domain Ω under the Möbius transformation A. Schematic representation of the domain Ω, with the funnel curvature radius R C , the north pole N , the funnel tip S, and the center of mass C. B-C. The neck in the domain Ω (B) is mapped onto the semi-annulus in the image domain Ω w (C). The large disk in Ω is mapped onto the small red filled disk in Ω w . The short green segment AB (of length ε) in Ω is mapped onto the thick green segment AB in the image domain.

Figure 4 :

 4 Figure4: Influence of the cusp on the field lines (orthogonal to the level lines). The field lines inside the original domain Ω (A) and its image domain Ω w (B), are computed numerically from equation[START_REF] Bazant | Conformal mapping of some non-harmonic functions, in transport theory[END_REF]. The blue lines originate from the bulk, while the orange ones start in the cusp. The domain Ω w is subdivided into three regions: the bulk Ω 1 w , the region Ω 2 w connecting the bulk to the region Ω 3 w inside the funnel. (C) Representation of the two subregions A (blue) and B (green) of Ω w .

Figure 5 :

 5 Figure 5: Thin ellipse Ω and its image Ω w under the Schwartz-Christofell map A. Representation of the two subregions A (blue) and B (dotted green) of Ω. B. Magnification of the ellipse tip from A. C. Image Ω w = w(Ω) of the domain Ω in B under the Schwarz-Christoffel conformal mapping. The region B is mapped into the region B w .

Introduction [ 74 ,Figure 6 :

 746 Figure 6: Voltage sensitive dye fluorescence A. Schematic representation of the GEVI Arclight (red frame) inside the membrane. The green and the blue cylinders represent the green fluorescent protein (GFP) and the transmembrane segments respectively [139]. B. A region of interest is selected around the soma to estimate the fluorescence during a synaptic stimulation. C. Regions of interest (ROIs) R1 and R2 representing the spine and the dendrite respectively. The red dot shows the location where glutamate is uncaged.converts the change in the membrane potential into a fluorescent emission. Arclight critical asset is to produce a more intense fluorescent signal compared to other dyes. Yet, it is also a slow indicator, so the recorded fluorescent signal will represent a distorted version of genuine fast voltage dynamic. Consequently, one mandatory task is to construct a deconvolution method to recover the voltage dynamic from slow Arclight fluorescence.

) 0 . 2 .Figure 7 :

 027 Figure 7: Modeling electro-diffusion in the spine neck A-B Schematic representation of a dendritic spine, divided into three regions: the head R1, the neck of length L and the close dendrite R2. The current I(t) represents the axial current escaping from the head to the dendrite.

0. 2 .

 2 Part II: Electro-diffusion modeling of synaptic input in dendritic spines

Figure 1 . 1 :

 11 Figure 1.1: Representation of a dendritic spine (Electron-Microscopy image) (courtesy of J. Spacek). Abbreviations G: glial cells, S:spine, D:dendrite, A: axon.

Chapter 1 .Figure 1 . 2 :

 112 Figure 1.2: Schematic representation of the distribution of a single unscreened ionic specie in a dielectric ball.

  15) impose lim s→∞ U (s) = u(0) = 0 and lim s→∞ U (s) = -lim r→0 ru (r) = v(0) = 0, hence the constraints lim s→∞ v(s) = 0, lim s→∞ w(s) = lim s→∞ µe -2s e -U (s) = 0.(1.23)

Figure 1 . 3 :

 13 Figure 1.3: Asymptotics behavior of the solution u(x) in dimension 3. (A) change in the profile u(x) for 3 values of the parameter λ = 10 2 , 10 3 , 10 4 . (B,C) We present two regimes: for λ = 0.1 1, we have u(x) = -λ x 2 8π + O(λ 2 ) (see eq. 1.73) and λ 1 where u(x) ≈ 2 log(1 -x 2). The analytical approximations (red) are compared with the numerical solutions (see appendix).

Figure 1 . 5 :

 15 Figure 1.5: Phase-space solution of (1.20). The separatrix is shown in red, while the other trajectories are in blue.

Chapter 1 .Figure 1 . 4 :

 114 Figure 1.4: Numerical solutions u(x) of the initial value problem (1.15). (A), (C), and (E) correspond to different profile values of λ in dimensions 1, 2, and 3, respectively. The dotted curves are solutions that blow-up for x < 1. (B), (D), and (F) are plots of the ratio λ I λ in dimensions 1,2 and 3, respectively.

Figure 1 . 6 :

 16 Figure 1.6: Asymptotics of u λ (1) -u λ (0) for dimensions 1,2 and 3.

Figure 1 . 7 :

 17 Figure 1.7: Distribution of (A) the potential, (B) charge and the field (C) and cumulative density of charges (D) inside a dielectric ball. Parameters of simulations are given in table 1.

Chapter 1 .

 1 Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains

1. 5 . 1

 51 Solution of the minus sign Liouville-Bratu-Gelfand equation in a unitary segment Liouville equation in the interval [0 1] is

. 67 )

 67 The curve λ I λ is shown on Fig. 1.4 and | u λ (1) -u λ (0) | in Fig. 1.6. Finally,

Chapter 1 .

 1 Poisson-Nernst-Planck equations analysis in a ball for modeling the Voltage-Current relation in neurobiological microdomains

Figure 2 . 1 :

 21 Figure 2.1: Image Ω w = w(Ω) of the domain Ω (A.) under the conformal mapping (2.15). The neck (left) is mapped onto the semi-annulus enclosed between the like-style arcs and the large disk in Ω is mapped onto the small red disk. The short green segment AB (left) (of length ε) is mapped onto the thick green segment AB (of length 2 √ ε+O(ε)). The letters S and N designate the south and the north pole respectively.

2. 3 .Figure 2 . 2 :

 322 Figure 2.2: Influence of the cusp on the field lines (orthogonal to the level lines). The field line inside the original domain Ω (A) and its image domain Ω w (B), computed numerically from equation (2.14). The blue lines originate from the bulk, while the orange starts in the cusp. The domain Ω w is subdivided into three regions: the region Ω 1w inside the funnel, the region Ω 2 w connecting the end of the funnel to the bulk Ω 3 w .

2. 3 .

 3 Figure 2.3:

  4 and 5 • 10 -3 ≤ ε ≤ 10 -1 . The difference is almost constant in the range [0.01, 0.025] as shown Fig.2.3E. Finally, to further validate the uniform asymptotic expansion, we compared the numerical solutions of the full equation (2.23) in the initial domain Ω (see Fig. 2.4A) with the reduced PNP equation (2.14) with zero Neumann boundary conditions, except at the end of the funnel for the mapped domain Ω w . The result is shown in Figure 2.4B-C, showing good agreement between the one-dimensional PNP approximation in Ω w and the numerical solution of the full equation.

2. 3 .Figure 2 . 4 :

 324 Figure 2.4: Comparison of the numerical solutions of the full and reduced PNP equations (2.14) with zero Neumann boundary conditions, except at the end of the funnel. A. Schematic representation of the domain Ω with an uncharged cusp (blue). The letters N , S, and C refer to the north pole, the funnel tip, and the center of mass respectively. B-C Numerical solutions of (2.14) (solid) and the solution of (2.57) in the funnel (dashed) in the mapped domain Ω w . The solution have been obtained for ε = 0.01. D. Comparison of (2.14) (blue) with the numerical solution (2.21) inside the funnel (dashed green) and (2.48) in the bulk (dashed red). E. Solution u(S)-u(C) (dashed blue) obtained numerically from (2.47) and compared to the logarithmic function -2 ln(λ) (greed dotted). F. Two-dimensional numerical solutions of the difference |u(N ) -u(C)| vs λ. The inset in panel F. is a magnification showing a maximum for small λ.

Figure 2 . 5 :

 25 Figure 2.5: Decomposition of the banana-shaped domain Ω w into two subregions regions A and B. A. Representation of the two subregions A (blue) and B (magenta) of Ω w . B. Solutions of (2.83) (dashed blue), (2.90) (red dots), and the uniform approximation u unif of (2.32) (green) for r = 1 -√ ε.

  2.6A) indicate that the radial derivative are O(λ √ ε) → ∞. Thus the angular derivatives 2.4. The PNP equations in a charged domain with a cusp-shaped funnel are negligible relative to the radial ones. It follows in a regular expansion of the solution that the θ derivatives can be neglected relative to the r derivative and the equation is then solved along the rays θ

.114) 2 . 4 .Figure 2 . 6 :

 2426 Figure 2.6: Comparison of numerical solution of (2.14) in the plane with the approximations u unif (x) in (2.92). A. Schematic representation of the domain Ω with a charged funnel (red). The letters N , S, and C refer to the north pole, the funnel tip, and the center of mass, respectively. B-C Numerical solutions of (2.14) (solid) and the solution of (2.92) in the funnel (dashed) in the mapped domain Ω w for several values of λ and for ε = 0.01. D. Comparison of (2.14) (blue) with analytical solutions (2.32) inside the funnel (dashed green) and (2.48) in the bulk (dashed red). E. Solution u(S) -u(C) obtained numerically (dashed blue) from (2.47) and analytically from (2.32) (red), compared to the logarithmic function -2 ln λ+const (green dots). F. Two-dimensional numerical solutions of the difference |u(N ) -u(C)| vs λ compared to the analytical solutions (2.112) (red). The inset in panel F. is a magnification showing a maximum for small λ.

Figure 2 . 7 :

 27 Figure 2.7: Normalized charge distribution ρ(y) in charged and uncharged funnel domains. A. ρ(y) is computed numerically from (2.8) with ∂u/∂n = 0 at the funnel boundaries (λ = 1 (blue), λ = 10 (red), λ = 1500 (green), and λ = 1000 (dashed magenta)). B. Representation of Ω and the funnel boundary conditions. Left: uncharged funnel domain ∂u/∂n = 0 (blue), and Right: charged funnel domain ∂u/∂n = -λ/|Ω| (red). C. ρ(y) in a charged funnel domain. The same color code is used as in panel A.

  -Planck in three-dimensional microdomains with cusp-shaped funnels Submitted in Multiscale Modeling and Simulation as Cartailler, J., Holcman D., Voltage laws for three-dimensional microdomains with cusp-shaped funnels derived from Poisson-Nernst-Planck equations, (2017).

Figure 3 . 1 :

 31 Figure 3.1: Ball with a cusp-shaped funnel and image Ω w of the domain Ω cross-section under the conformal mapping (3.17). A. schematic representation of the domain Ω, with the funnel curvature radius R C , the north pole N , the funnel tip S, and the center of mass C. B-C The neck (B) is mapped onto the semi-annulus enclosed between the like-style arcs and the large disk in Ω is mapped onto the small red disk. The short green segment AB (left) (of length ε) is mapped onto the thick green segment AB (of length 2 √ ε + O(ε)).

) 3 . 2 .r ε 0

 320 The Poisson-Nernst-Planck equations hence (3.5) gives the Poisson equation ∆φ(x) = -|∂ Ω| for x ∈ ∂ Ω. (3.9) The equation (3.9) represents the compatibility condition obtained by integrating Poisson's equation (3.5) over Ω, assuming the surface charge density is constant. Using non dimensional variables, we define ū(x) = zeφ(x) kT , λ = (ze) 2 N ε r ε 0 kT , (3.10) where λ generalizes the Bjerrum length l B = e 2 /kT . The Poisson's equation (3.8) reduces to ∆ū(x) = -λ exp {-ū(x)} Ω exp {-ū(s)} ds

α 2 .

 2 Using(3.22) and(3.25), in polar coordinates (see Appendix), it follows that ∂ ũ(r, z) ∂r = ∂ṽ(ρ, θ) ∂ρ (cos(θ) w1 (ρ, θ) -sin(θ) w2 (ρ, θ)) θ) w1 (ρ, θ) + cos(θ) w2 (ρ, θ)) .

. 28 ) 3 .

 283 Chapter Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

Figure 3 . 2 :

 32 Figure 3.2: Decomposition of the domain Ω w into two subregions regions A and B A. Representation of the two subregions A (blue) and B (dotted red) of Ω w . B. Solutions of (3.43) (dashed blue), (3.54) (red dots), and the uniform approximation u unif (3.58) (green) for r = 1 -√ ε.

Figure 3 . 3 :

 33 Figure 3.3: PNP solution (3.13) in a 3D domain with a cusp-shaped funnel A.Representation of the domain Ω with a surface charge density σ, the north pole N , the funnel tip S, and the center of mass C, respectively. B. Numerical (3.13) (solid) and analytical (3.58) (dashed) solutions in the domain Ω w for several values of σ = 10, 100, 1000 and 4000 for ε = 0.01. C. Difference u(C) -u(S) computed numerically (solid blue) from (3.13) and analytically (dashed green) from(3.76).

. 37 )

 37 Using the Neumann boundary condition at ρ = 1 in (3.35), we get C 2 (θ 0 ) = -1. (3.38) Using (3.38) and (3.37) and the boundary condition at ρ = 0 in (3.35), we find that C 1 is solution of the transcendental equation,

. 40 ) 2 ( 3 . 41 )

 402341 Using(3.36),(3.38) and(3.40) in(3.36), we obtain to leading order ṽA,0 (ρ, θ 0 ) = ln 2 2(1 -cos(θ 0 )) + σε πσε+ ln cos 2 πσε(ρ -1) 2(2(1 -cos(θ 0 )) + σε) .Using (3.41), (3.34) and (3.32), we conclude

Chapter 3 .

 3 Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels and C1 is solution of the transcendental

( 3 .

 3 53)-(3.51)-(3.36) that for θ ∈ B, the asymptotic solution is

Figure 3 . 4 :

 34 Figure 3.4: Schematic representation of the ∂Ω boundary subregions. Subregions of the boundary ∂Ω: the cusp ∂Ω cusp (red), the bulk ∂Ω bulk (blue) and (as shown in the inset panel) the funnel bottom ∂Ω ε (orange). Their respective surface charge densities are σ bulk , σ cusp and σ ε .

. 4 .Figure 3 . 5 :

 435 Figure 3.5: Numerical (3.13)-(3.83) versus analytical (3.96) solutions with zero Neumann boundary conditions, except at the end of the funnel. A-D Analytical (dashed green) obtained from (3.96) and numerical solutions (3.13) (blue) computed in 3D and the 1D reduced equation (3.83) (dashed red). E. Potential difference v(0) -v(π) computed numerically from (3.83) (blue) and the asymptotics (3.97). F. Comparison of eq. (3.13) numerical solutions in 2D (red) and 3D (blue).

(3. 101 ) 3 .

 1013 Chapter Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

Figure 3 . 6 :

 36 Figure 3.6: Comparison of numerical and analytical solutions for nonhomogeneous surface charge density A. Numerical (eq. (3.98)) in 3D (solid) and analytical (eq. (3.101)) (dashed) solutions for σ cusp = 1000 and σ cusp = 1, σ bulk = σ ε = 2500. B. Magnification of panel A in the region of θ = π. C. 3D Numerical (solid) from eq. (3.98) and analytical (eq.(3.106)) (dashed) solutions computed for σ cusp = 10, 100, 1000 and 4000, where σ ε = 0 and σ cusp = σ bulk . Here ε = 0.01.

Chapter 3 .

 3 Voltage laws for Poisson-Nernst-Planck in three-dimensional microdomains with cusp-shaped funnels

. 113 )

 113 Equations (3.112) and (3.113) lead to

. 10 )

 10 where ũ(x) = e φ(x) k B T is the adimensionalized potential. Then, using (4.10) and (4.6)we obtain,-∆ũ(x) = l B N p e -z p ũ(x) Ω e -z p ũ(s) ds -l B N m e -z m ũ(x) Ω e -z m ũ(s) ds in Ω (4.11) ∂u(x) ∂n = -(N p -N m ) |∂ Ω| l B on ∂ Ω. (4.12) where l B is the Bjerrum length. In this paper we shall consider the degenerate case N m = 0, consequently (4.11) models the electrical potential created by positive charges exclusively. The formulation of the Poisson equation becomes the boundary value problem (BVP), -∆ũ(x) = l B N p e -ũ(x) Ω e -ũ(s) ds (4.13)

2 .

 2 13) setting x = x a and ũ(x) = ū(x), it becomes -∆ū(x) = l B N p e -ū(x) Ω e -ū(s) ds for x We introduce the dimensionless surface charge σ = l B N p |∂Ω| . (4.16) Then, we assume N p > 0, we get rid of the integral in (4.15) with the translation u(x) = ū(x) + β, where β = ln l B N p Ω e -u(s) ds is a constant. We conclude that the problem for u(x, y) is

4. 2 . 1 Figure 4 . 1 :

 2141 Figure 4.1: The regions A and B. The schematic representation shows the two regions A (blue) and B (dashed green) defined in (4.18). The green circles represent the tips of the ellipse Ω major and minor axises. The region limited by the red lines represents the part of Ω where the approximation (4.25) (∂u/∂n ≈ ∂u/∂x) is not valid.

) in the limit b 1 .

 1 We consider only the upper right quadrant of the ellipse because of the domain symmetries. We divide the domain Ω in two subregions, A = {(x, y) ∈ Ω such that x ≥ 0 and 0 ≤ y < y 0 } (4.18) B = {(x, y) ∈ Ω such that x = 0 and y ≥ y 0 }, where y 0 is a parameter. The two regions are shown in Fig. 4.2A. We can remark that the region B is a segment. We shall now explain how two regions A and B Chapter 4. Poisson-Nernst-Planck equation in narrow ellipses and ellipsoids.

. 20 )

 20 Using (4.20), we characterize y 0 such as for 0 ≤ y ≤ y 0 < 1, we have in the b small limit 20) and (4.21), for 0 ≤ y ≤ y 0 , we have for y ∈ [0, y 0 ].(4.24) In particular, from (4.24), we have to leading order that for 0 ≤ y ≤ y 0

(4. 27 )

 27 We estimate the constant C 1 (y) from the non-homogeneous boundary conditions in (4.26) which leads to an implicit equation:2C 1 (y) arctan (σC 1 (y)) = b 1 -y 2 .(4.28)

Chapter 4 .Figure 4 . 2 :

 442 Figure 4.2: Image Ω w = w(Ω) of the domain Ω (left) under the conformal mapping (4.32).

42 )Using ( 4 .

 424 [START_REF] Chow | Doubly periodic and multiple pole solutions of the sinh-Poisson equation: Application of reciprocal transformations in subsonic gas dynamics[END_REF] in(4.39), we obtain that the leading order term v 0 (X, Y ) is solution of the Laplace equation ∆v 0 (X, Y ) = 0.(4.43)Using the variable separation method, it yieldsv 0 (X, Y ) = h(X)g(Y ),(4.44)where h(X) and g(Y ) are two smooth functions. Integrating twice (4.43) we obtaing(Y ) = A 0 + B 0 Y,(4.45)where A 0 and B 0 are constants. We can remark that the condition (4.40) is satisfied by the function g(Y ). Using (4.44)-(4.45), for X = 0, we obtainv 0 (0, Y ) = Ã0 + B0 Y,(4.46)where Ã0 = A 0 h(0) and B0 = B 0 h(0). We introduce ṽ0 (Y ) = v 0 (0, Y ), which is defined on B w . We compute B0 from boundary conditions in(4.40)

  ) (solid lines) with the asymptotic solution u unif (x, y) from (4.50) in Fig.4.3B-C, where we chose y 0 = 1 -b.

4. 2 .Figure 4 . 3 :

 243 Figure 4.3: Comparison of PNP numerics (4.17) and analytics (4.50) in Ω A. Schematic representation of the domain Ω. The letters a and b represent the tip as well as the length of the major and minor axis respectively. B-C. Numerical solutions of (4.17) (solid) and solution (4.50) (dashed) along the major axis (B) and the minor axis (C). The inset in panels B-C represents a magnification close to the boundary ∂Ω. bf D. Solution V (0) -V (a), E. and solution V (0) -V (b) and F. V (b) -V (a), each versus σ, obtained numerically (blue) from (4.17) and analytically from (4.50) (dashed green).

3 .

 3 sionalize the domain Ω3 setting (r, θ, z) The domain Ω 3 becomes an ellipsoid with a major axis equal to 1 and a non-

Chapter 4 .

 4 Poisson-Nernst-Planck equation in narrow ellipses and ellipsoids.

Figure 4 . 4 :

 44 Figure 4.4: Comparison of numerics and V unif(4.50) in Ω 3D A. Schematic representation of the domain Ω 3D . The letters a and b represent the tip as well as the length of the major and minor axis respectively. B-C Numerical solutions of (4.17) (solid) and solution (4.100) (dashed) along the major axis (B.) and the minor axis (C.) The inset in panels B-C represents a magnification close to the boundary ∂Ω. D. Solution V (0) -V (a), E. and Solution V (0) -V (b) and F. V (b) -V (a), each versus σ, obtained numerically (blue) from (4.17) and analytically from (4.107) (dashed green).

Figure 5 . 1 :

 51 Figure 5.1:

Figure 5 . 2 :

 52 Figure 5.2: Distribution of charge and voltage in a domain with a narrow funnel. A-B. Voltage distribution obtained for λ = 1 (A) and λ = 2000 (B), both for ε = 0.01µm. C. Voltage distribution evaluated along the z-axis, for λ = 1 (red), λ = 10 (green), λ = 50 (dashed blue) and obtained analytically (dotted orange) with formula (5.16). D. Difference of voltage between V (C) -V (S) versus λ (blue) compared to the logarithmic function kT e ln λ 2 + C ste , where S, N and C are the south, north pole and the center of mass respectively. E. Difference of voltage between V (C) -V (S) versus the curvature radius R c of the cusp-shape funnel, computed from eq. (5.16) for ε = 0.01 and λ = 25. F. Voltage difference V (N ) -V (S) between the two poles versus λ. The inset in panel E is a magnification in the small λ region.

Chapter 6 Deconvolution

 6 of voltage sensor time series and electro-diffusion modeling of synaptic input in dendritic spines Submitted in Neuron Cartailler J., Kwon T., Yuste R., Holcman D., Nanophysiology of dendritic spines: electro-diffusion for voltage modulation and conduction . (2017).

Figure 6 . 1 :

 61 Figures

Figure 6 . 3 :Chapter 6 .Figure 6 . 4 :

 63664 Figure 6.3: Simulation of PNP equations for electro-diffusion in a dendritic spine. A. Representation of a spine head where a current is injected in a 3D spherical cavity of radius r head = 0.5µm. B. Voltage profile (blue) along the x-axis obtained from 3dimensional simulations of the PNP equations, that we compare to the potential averaged over the entire head (dashed green line) when the injected current is I stim = 150pA. The south pole is grounded at V = 0V . C. Representation of a 3D-spine geometry composed by a spherical head of radius r head = 0.5µm and a neck of length L = 1µm. The head has two narrow openings, one of radius 10nm where the steady current I stim is injected (north) and a second one (south) of 100nm at the junction with the neck. D. Potential drop along the x-axis computed from the top of the head to the bottom of the spine. We compare the voltage drop between a spine where L = 1µm and I stim = 300pA (red) with L = 0.5µm and I stim = 150pA (blue).

3 / 2 oFigure 6 . 5 :Chapter 6 .Figure 6 . 6 :

 3265666 Figure 6.5: Summary of dendritic spine electrical responses. A. Schematic rep-resentation of a dendritic spine. Left: a current is injected in the head. Right: modeling the effect of the BPAP on the spine voltage. The inset in panel A is an electrical circuit representation of the spine electrical properties, composed by a diode (green) with a resistance R neck in the direction spine to dendrite and by ideal diode in the opposite direction (orange). B. Electrical response of a spine (Length and radius of the head are L = R = 1µm) and the radius of the neck a = 0.1µm, following a synaptic input (I = 100pA) and a BPAP, where the value depend on the voltage in the dendrite, but it is constant in the spine. C. Modulation of the voltage between the spine head and the dendrite: the voltage attenuation can be modeled as a diode to account for the saturation behavior (Fig.6.3F). D. Response to a BPAP showing no voltage change between the head and the dendrite, as predicted by the electro-diffusion model.

F k- 1

 1 (t k + n∆t)

  )

7. 1 .Figure 7 . 1 :

 171 Figure 7.1: Filtering and deconvolution of the Arclight fluorencent signal at a single spine level. A. The fluorescence signal (thick lines) in the dendrite (red) and the spine head (blue) is passed on the Savitsky-Golay filter. The outputs (thick lines) are super-imposed over the fluorescence. B.The smooth signal output from (A) is approximated by a family of functions eq. 7.16. The results are given for a dendrite (dashed green) and a spine head (dashed magenta). C. Deconvolution of the fitted signal (dashed lines) using the deconvolution kernel eq. 7.19. This procedure recovers the voltage time course in the dendrite (red) and the spine head (blue).

Figure 7 . 3 :

 73 Figure 7.3: Deconvolution of several peaks A. Voltage trace (red) and convolved voltage (blue). B. Normalized fluorescence -∆F/F signal. C. The signal from panel B (blue)is fitted by a sum of terms (t -t j ) α j e -γ j (t -t j ))H(t -t j ). D. Deconvolved voltage obtained from the fitted trace (dashed green) in panel C. E. The deconvolved voltage (red) is compared to the original voltage (dashed black).

V

  (x, 0) = 0 c p (x, 0) = C p and c m (x, 0) = C m .The boundary conditions are given by (L, t) = 0, c p (L, t) = C p and c m (L, t) = C m ,

Figure 7 . 5 :

 75 Figure 7.5: Iteration step of the algorithm for extracting the value of C and G from voltage time seriesA.-B.-C. Simulated voltage differences (yellow, orange, red) based on the current reconstruction (eq. 7.33) following several iterations of the optimizing procedure (Fig.7.4). We use a linear approximation ξ(t) (magenta) (see eq. 7.39) of the measured potential (blue) on a short time window t f -t i = 20ms.

Chapter 7 .Figure 7 . 7 :

 777 Figure 7.7: Voltage change computed from the linear cable model Measured voltage in a dendritic spine head (blue) and the response in the dendrite (red). The simulated voltage response (green dashed) using the cable equation eq. 7.48 and the spine head input. To obtain the response that should be comparable to the voltage in the dendrite, the intracellular resistivity should be increased by a factor 4.10 5 from the basal value R 0 i = 105Ωcm[START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF]. The other cable parameters are given by C m = µF/cm 2 and R m = 2 • 10 4 Ωcm 2[START_REF] Koch | Biophysics of Computation: Information Processing in Single Neurons[END_REF].

Table 1 :

 1 Differences of potential inside a 2D ellipse

0.2 Part II: Electro-diffusion modeling of synaptic input in dendritic spines

  Chapter 2. Geometrical effects on nonlinear electrodiffusion in cell physiologyFinally, the approximate potential difference ∆u A in (2.95), is the difference between (2.105) and (2.101), potential drop in the funnel domain occurs mostly in the region B. The expression (2.112) is plotted in Figure

		∆u A = -ln	2λ 2 π 2 (4|∂Ω| + λε) 2 + ln	π 2 c 4 8	+ O ε,	1 λ	.	(2.106)
	For λ	1 (2.106) becomes to leading order
			∆u A ∼ -ln	2 4 c 4 ε 2 ,	(2.107)
	which is independent of λ. (2.90) shows that the approximate potential in section
	B is
			u B (π -	√	ε) = C 0	(2.108)
	and
		u B (π) = ln sin 2 π|∂Ω| λε	+ C 0 .	(2.109)
	Using (2.108) and (2.109) in (2.96), we obtain
			∆u B = ln sin 2 π|∂Ω| λε	.	(2.110)
	For λ	1, (2.110) shows that ∆u B is
		∆u B = -2 ln λ + 2 ln	|∂Ω|π ε	+ O	1 λ 2 .	(2.111)
	Finally, using (2.106), (2.110) and (2.97), we find that the potential drop is
		∆u = ln sin 2 π|∂Ω| λε	-ln	2λ 2 π 2 (4|∂Ω| + λε) 2 + ln	π 2 c 4 8
		+ O ε,	1 λ	.			(2.112)
	Again, using (2.107), (2.111) and (2.97) for λ	1 limit, we get the approximate
	potential drop as
		∆u ∼ -ln λ 2 + 2 ln	πc 2 |∂Ω| 4	+ O	1 λ	.	(2.113)
	Equation (2.110) shows that for λ		1, the

.

[START_REF] Kaiser | Mechanisms of Connectome Development[END_REF]

  .69)

	Chapter 3. Voltage laws for Poisson-Nernst-Planck in three-dimensional
		microdomains with cusp-shaped funnels
	Using u A (C) and u A (S), we conclude that
		∆u A = -ln	2σ 2 π 2 (4 + σε) 2 + ln	π 2 c 4 8	+ O ε,	1 σ	.	(3.70)
	For σ	1, to leading order, the solution of eq. (3.70) does not depend on σ
				∆u A ∼ -ln	2 4 c 4 ε 2 .	(3.71)
	We now estimate the difference ∆u B . We have from (3.54) that
				u B (π -	√	ε) = C 0	(3.72)
	and
		u B (π) = ln sin 2 π σε	+ C 0 .	(3.73)
	Using (3.72) and (3.73) in (3.62), we obtain
			∆u B = ln sin 2 π σε	.	(3.74)
	For σ	1, eq. (3.74) reduces to
		∆u B = -2 ln σ + 2 ln	π ε	+ O	1 σ 2 .	(3.75)
	Finally, using (3.70), (3.74) and (3.63), we find that the difference in the funnel is
		∆u = ln sin 2 π σε	-ln	2σ 2 π 2 (4 + σε) 2 + ln	π 2 c 4 8	+ O ε,	1 σ	.	(3.76)
	The results in large σ limit found in (3.71), (3.75) and leads to
		∆u = -ln σ 2 + 2 ln	πc 2 4	+ O	1 σ	.	(3.77)

  ) 3.3. The PNP equations in a cusp-shaped domain with non-homogeneous surface charge densitywhere ∂Ω ε is the bottom of the funnel, ∂Ω cusp the funnel area and ∂Ω bulk the bulk surface. The Neuman boundary conditions on each sub-regions are defined by

	∂u(x) ∂n	=	-λ ε |∂Ω ε |	on ∂Ω ε	(3.79)
	∂u(x) ∂n	= -	λ cusp |∂Ω cusp |	on ∂Ω cusp
	∂u(x) ∂n	=	-λ bulk |∂Ω bulk |	on ∂Ω bulk .
	Using the compatibility condition obtained by integrating the Poisson equation
	(3.13)
	∂Ω	∂u(x) ∂n	dS = -λ.	(3.80)

Table 3 .

 3 1: Electrodiffusion laws for voltage drop for various surface charge densities

	3.6 Appendix
	3.6.1 Radial derivative under the Mobius map (3.17)
	We shall describe in this appendix the computation step to reduce the first order
	radial derivative from (3.16) leading to the result (3.28) in section 3.2.2. First, we
	note that in complex coordinates, we have
	∂u(r, z) ∂r	=	e (∇u(ξ)) ,	(3.108)
	where we define
	∇u(ξ) =	∂u(r, z) ∂r	+ i	∂u(r, z) ∂ z .	(3.109)

  .2) 4.1. Poisson-Nernst-Planck equation where e is the electronic charge. The charge densities ρ p (x, t) and ρ m (x, t) are solutions of the initial and boundary value problem for the Nernst-Planck equation

	D i ∆ρ i (x, t) +	z i e k B T	∇ (ρ i (x, t)∇φ(x, t)) =	∂ρ i (x, t) ∂t	for x ∈ Ω	(4.3)
	D i	∂ρ i (x, t) ∂n	+	z i e k B T	ρ i (x, t)	∂φ(x, t) ∂n	= 0 for x ∈ ∂ Ω	(4.4)

  The second boundary conditions at Y = w(ỹ 0 ) are obtained by matching the y-dependent partial derivatives of the two solutions u A (x, y) and u B (y) at y = y 0 ,

	4.2. Poisson-Nernst-Planck solutions for a 2D elliptic domain
	we get
	∂v ∂Y Y =w(ỹ 0 )	= -	∂u A (0, y) ∂y	y=y 0	cosh(w(iỹ 0 )) α	= -ỹ 0	∂u A (0, y) ∂y	y=y 0	. (4.38)
	We conclude from (4.34), (4.37) and (4.38) that the BVP in Ω w is
										.36)
	Using (4.17), (4.35) and (4.36), the transformed boundary conditions is
		∂v(w) ∂Y Y =0	=	1 (w (0))	∂u(z) ∂y y=0	= -	σ α	.	(4.37)

  3. The equations areThe boundary is decomposed into three subdomains: the current is injected into ∂Ω i . Charges can exit in ∂Ω o and the impermeable membrane is represented by ∂Ω r . The boundary conditions are

	Chapter 6. Deconvolution of voltage sensor time series and electro-diffusion
	modeling of synaptic input in dendritic spines
	given by
	∆φ(x) =	-F εε 0	(c p (x) -c m (x))	(6.15)
	0 = D p ∇ ∇c p (x) +	e k B T	c p ∇φ(x)
	0 = D m ∇ ∇c m (x) -	e k B T	c m ∇φ(x) .
			∂φ ∂n	(x) = 0 on ∂Ω r ∪ ∂Ω i	(6.16)

Table 6 . 1 :

 61 Parameters extracted from modeling. Columns 1 is the index of the spine. Column 2 is the measured neck length L.

	6.5 Tables
	Spine Neck Length L Head radius	Intrinsic	Intrinsic	Effective neck
		(µm)	r head (µm) Capa. C(pF) Res. 1/G (M Ω) Res. R neck (M Ω)
	1	0.6	1.2	<0.5	128.5	73.3
	2	0.7	0.8	<0.5	163.4	57.1
	3	0.8	0.7	18	212	99.15
	4	1.0	0.8	10	252	132.6
	5	1.2	0.6	<0.5	261	134.1

Table 6 . 2 :

 62 Biophysical an geometrical parameters.

		Description	Value
	z	Valence of ions	1
	D	Diffusion coefficient	200µm 2 /s [39]
	D p	Diff. coeff. for + charges D
	D m	Diff. coeff. for -charges D
	C p	+ charge concentration	167mol/m 3 [81]
	C m	-charge concentration	167mol/m 3 [81]
	Ω	Spine head	Ω (volume |Ω| ≈ 1fL) [210]
	a	Spine neck radius	(typical) 0.1µm[194]
	L	Spine neck length	(typical) 1µm
	T	Temperature	293.15K
	E	Energy	kT = 2.58 × 10 -2 eV
	e	Electron charge	1.6 × 10 -19 C
	ε	Dielectric constant	ε = 80
	ε 0	Abs. Dielectric constant 8.8 • 10 -12 F/m
	k	Boltzmann constant	1.38 • 10 -23 J/K
	F	Faraday constant	96485 As/mol

  (7.2) that approximate the function F , we start with k = 1: the coefficients a 1,i are computed by minimizing on the interval I 1 (t 1 + s∆t) -P 1 (t 1 + s∆t))2 . Signal processing and filtering method for the Arclight fluorescent signal In general, the k th -iteration is obtained by minimizing on I k ,(F k-1 (t k + s∆t) -P k (t k + s∆t)) 2 .(7.7)The filtered function F k at the k th -iteration is given byF k (t) = {P 1 (t 1 ), • • • , P k (t k ), F (t k+1 ), • • • , F (t N )}. (7.8)The coefficients {a k,i }, for i = 0..p are obtained by differentiating R 2 k (eq. 7.6) with respect to the coefficients a k,i ,

	where where	min t∈I 1 t∈I k n s=-n  t k +s∆t∈I k 1 (t) = R 2 n k (t) = R 2 min       ∂R 2 k ∂a k,0 R 2 1 (t), R 2 k (t), = 0 . . . (F 0 min    ∂R 2 k    ∂a k,p	(7.3) (7.6)
		s=0

p, P k (t) = p i=0 a k,i t i , t ∈ I k (7.4)

The function F 1 is constructed from F , by replacing F (t 1 ) by P 1 (t 1 ),

F 1 (t) = {P 1 (t 1 ), F (t 2 ), • • • , F (t N )}.

(7.5)

7.1.

  2

  • • • (t k -(n -1)∆t) p Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series 7.1.2 Preliminary treatment before using the Savitsky-Golay filter

							
	. . . 1	. . . t k + n∆t	. . . (t k + n∆t) 2	. . . • • •	. . . (t k + n∆t) p	    (7.11) .
	This matrix is inverted numerically in matlab. To conclude the filtering procedure
	is summarized by
			N
		S G (F (t)) =	P k (t k )δ(t -t k ).		(7.12)
			k=1
	We can iterate N SG times the SG filter,

f (t) = (S G ) N SG (F (t)),

(7.13)

where

(S G ) N SG = S G • • • • • S G , N SG times.

  ,γ 2 ,α,β 1 ,β 2 (t) = (K * h)(t)(7.20) can be written in the Laplace's coordinatef γ 1 ,γ 2 ,α,β 1 ,β 2 (s) = K(s) h(s).(7.21)Thus using expression 7.19 and 7.17, we obtainh(s) = f γ 1 ,γ 2 ,α,β 1 ,β 2 (s) (s + γ 2 ) α+1 . (7.22)We can now compute h using the analytical result that the inverse Laplace for α ≥ 2 and 0 ≤ γ of the function-2τ γ) + 2τ α) + Cτ 2 e -γt t α-2 Γ(α + 1) (tγ) 2 + α 2 -α -2γαt . (7.24)Finally, we obtain the expression for the voltage signalh(t) = β 1 e -γ 1 t A t α-1 (t(1 -2τ γ 1 ) + 2τ α) + τ 2 t α-2 (tγ 1 ) 2 + α 2 -α -2γ 1 αt + β 2 e -γ 2 t A t α-1 (t(1 -2τ γ 2 ) + 2τ α) + τ 2 t α-2 (tγ 2 ) 2 + α 2 -α -2γ 2 αt . (7.25)

				exp(-	t τ	)	(7.18)
	and its Laplace's transform is
			K(s) =	A (sτ + 1) 2 .	(7.19)
	The convolution product
		f γ 1 K(s) = Γ(α + 1)(sτ + 1) 2 A	β 1 (s + γ 1 ) α+1 +	β 2
			M (s) =	C(sτ + 1) 2 (s + γ) α+1	(7.23)
	is given by
	m(t) =	Ce -γt t α-1 Γ(α + 1)	(t(1

  Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series where G is the conductance of the spine and C a capacitance term. We recall (see Material and methods) that the one dimensional electro-diffusion equations to model the current and voltage in the spine neck is∂ 2 V ∂x 2 (x, t) = -Where D p , D m are diffusion coefficients, e the electronic charge, F the Faraday constant, the valency for each specie is z = ±1 and k B T thermal energy. The steady state ionic concentrations are fixed at C p and C m (see table 2 main text).

		F εε 0	(c p (x, t) -c m (x, t))
	∂c p ∂t	(x, t) = D p	∂ ∂x	∂c p ∂x	(x, t) +	e k B T	c p (x, t)	∂V ∂x	(x, t)	(7.34)
	∂c m ∂t	(x, t) = D m	∂ ∂x	∂c m ∂x	(x, t) -	e k B T	c m (x, t)	∂V ∂x	(x, t) .
										dφ(t) dt	,	(7.33)

) = Gφ(t) + C

  7.4). Following each iteration, we compare the computed voltage V C,G (0, t) (from eqs. 7.34 ) with the measured potential difference V head (t) -V dend (t) extracted on a short time interval [t i , t f ] (usually [0.2 -0.4]ms). We chose this small time window because it allows us differentiating clearly the response from the background noise during glutamate uncaging (Fig.7.5).The value for the parameters C, G are the minimizers of the variational problem min 7.2. Optimization procedure to extract the electrical capacitance and conductance C and G of a dendritic spine from the voltage drop between the head and the base of spine Figure 7.4: Minimization procedure for computing the capacitance C and conductance G and current from the deconvolved voltage time series. The initial values (red) for (C 0 , G 0 ) = (0.1pF, 1nS. We use V head (t) and V dend (t) on a small subinterval |[t i , t f ]| = 0.2s, where t i varies from 0 to half of the time to peak (no more than 50ms). Initial values are injected in the iterative loop (blue). the current I(t) is first computed from eq.7.33 and it is then used to estimate numerically V (x, t) from eq.7.34. The voltage is compared to the initial approximation ξ(t) of V head (t) -V dend (t) on the interval [t i , t f ]. For the next iteration, the output (green) is the difference of the potential V (x, t) estimated on the entire time interval t max -t min ∈ (20 -50ms). Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series where S is given below. We approximated V head (t) -V dend (t) by a linear function ξ(t) = a(t -t i ) + b, (7.38) a and b are constants fitted to data in the interval [t i , t f ]. This approximation reduces possible fluctuations (see fig. 7.5C-F). Finally, we score the different couple

	6 A		Entries ,kC iterate solution V head -V dend Iteration k	6 B	measured 6 ,k+1C iterate solution Iteration k+1 C	Numerical Solution Final result
	Voltage ,mVC	0 2 4	0	{C,G∈S} Computed voltage for Output 0.1 0.2 0.3 t f t i Time ,msC t Capacitance Conductance Linear approx. ,G k ,C k C t f	0.4	0 4 2	Measured potential difference: to experimental Computed Computed on subinterval Comparing: 0.4 0 0.1 0.2 0.3 0 t f t i Match Time ,msC Estimation of the current Optimal and Algorithm 0 0.1 0.2 0.3 t f t i Time ,msC Coupled PNP equations 4 2 ,G k+1 ,C k+1 C ,G,CC	0.4

i |V C,G (0, t) -(V head (t) -V dend (t))| 2 dt,

(7.37)

  diffusion equation for voltage is

	τ	∂V (x, t) ∂t	= λ 2 ∂ 2 V (x, t) ∂x 2	-V (x, t)	(7.48)
		V (0, t) = V 1 (t)		(7.49)
	∂V (x, t) ∂x	x=L	= 0.		(7.50)

Remerciements

Acknowledgement: DH thanks the hospitality of Department of Applied Mathematics and Theoretical Physics (DAMTP), Churchill College CB3 0DS, this work is supported by a Marie-Curie and a Simons fellowship Award. This research is supported by a DEQ20160334882 Equipes FRM 2016 grant and a PSL excellence laboratory grant. RY and TK are supported by the NIMH (R01MH101218, R01-MH100561). This material is based upon work supported by, or in part by, the U. S. Army Research Laboratory and the U. S. Army Research Office under contract number W911NF-12-1-0594 (MURI).

The numerical procedure Numerical solutions were constructed by the Comsol Multiphysics 5.0 (BVP problems), Maple 2015 (Shooting problems) and Matlab R2015 (Conformal mapping). The boundary value problems in 1D, 2D, and 3D were solved by the finite elements method in the Comsol 'Mathematics' package. We used an adaptive mesh refinement to ensure numerical convergence for large value of the parameter λ. We solved the PDEs by the shooting procedure for boundary value problems using Runge-Kutta fourth-order method, as well as solvers from Maple packages.

Estimating the potential drop in Ω

We now study how the ellipse shape of the domains affects the voltage. We estimate from the uniform approximation u unif (4.50) three characteristic potential differences inside the ellipse:    ∆Maj U = u unif (0, 0) -u unif (0, 1) ∆Min U = u unif (0, 0) -u unif (b, 0) ∆a,b U = u unif (b, 0) -u unif (0, 1). (4.51) where we chose y 0 = 1 -b that satisfies the condition (4.21). We first estimate the potential drop ∆Maj U which is the sum of the longitudinal potential drop in regions A and B. In the region A, using (4.31), we have

Similarly, in region B, we first remark that using (4.41) we have

In small b limit, since 1 -y 0 = b, it follows

In the limit b 2 σ 1, using (4.54) in (4.49), we obtain

We conclude using (4.52) and (4.55) that, to the leading order, it yields

We compare in Fig. 4.3D the potential drop ∆Maj U computed numerically (blue line) from of eq. (4.17) with the analytical expression (4.56) (dashed green). We found that the solutions almost fully overlap. We now compute the voltage drop ∆Min U straightforwardly from the expression found for u A (x, y) ( We compare, in Fig. 4.3E, ∆Min U computed numerically from of (4.17) (blue) with the analytical expression (4.57) (dashed green). We find a very good matching. We underline the effect of the domain curvature on the local voltage by computing, from the uniform approximation u 3,unif (4.50), the three characteristic potential differences:

where we chose z 0 = 1 -b that satisfies the condition (4.21). We follow the exact same steps as in section 4.2.6. The difference ∆Maj U 3 is the sum of the longitudinal potential drop in regions Ā and B. In the region A, using (4.31), we have

In region B, The solution in Ω 3 and Ω are, to the leading order, identical. Hence using (4.55), we obtain

We conclude using (4.102) and (4.103) that to the leading order, it yields

We compare in Fig. 4.4D the potential drop ∆Maj U 3 computed numerically (blue line) from of eq. (4.17) with the analytical expression (4.104) (dashed green). We find good agreement between the two solutions. We now compute the voltage drop ∆Min V straightforwardly from the expression found for u A (x, y) (4.31) by setting y = 0 and x = b, it follows ∆Min

We compare, in Fig. 4.4E, ∆Min U computed numerically from of (4.17) (blue) with the analytical expression (4.57) (dashed green). We find a very good matching. Chapter 7

Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series

This SI is divided into five sections: first, we present the filtering and the deconvolution procedures to recover the voltage dynamics from the Arclight fluorecent signal. In the second part, we describe an optimization procedure for extracting the resistance and capacitance from voltage time series. We compute the current and voltage for five spines. In part three, we briefly discuss the relation between the intrinsic and effective neck resistances. In part four, we describe the role of the spine neck radius in the electrical resistance. In section five, we show the limitation of the cable theory for extracting the resistance parameter of a dendritic spine.

Deconvolution method applied on several peaks

We present shortly how the deconvolution procedure described above can easily be extended to signals with several peaks and/or faster dynamic. To illustrate the result of the deconvolution method, we use as an input signal a two successive spikes from a post-synaptic voltage response (see for example: [START_REF] Tsodyks | The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability[END_REF]). The two spikes are shown in Fig. 7.3A (dashed black). We first simulate the fluorescent response by convolution of this data with the kernel (7.18) K(t) = A τ 2 te -t/τ where we choose τ = 18ms. To account for possible fluorescence fluctuations, we added to the convolved voltage a colored noise ζ(t) that aims to represent fluctuations of the fluorescence signal. This colored noise is solution of the Ornstein-Uhlenbeck process [START_REF] Gardiner | W Stochastic Methods A Handbook for the Natural and Social Sciences[END_REF] of the form

where W (t) is a Wiener process and µ = 0.045 and σ = 0.025 are parameters we use to tune the resulting color of the noise. The resulting fluorescence signal is shown Fig. 7.3A (blue). We will then consider this signal as the input of the deconvolution method.

The deconvolution starts with the fluorescence A rc (t) shown Fig. 7.3B (same as in 7.3A). Following the method of described above for a single peak, we then approximate the input A rc (t) (Fig. 7.3C (dashed green), that represents the fluorescence response) using a sum of exponential terms,

7.2. Optimization procedure to extract the electrical capacitance and conductance C and G of a dendritic spine from the voltage drop between the head and the base of spine are satisfied and in that case, we have V C * ,G * (x, t) = V (x, t), otherwise we continue to iterate the algorithm by replacing the initial condition by the new values: .43) This iteration shows how the new 'S' matrix (eq. 7.40) is refinement near (C * , G * ).

In practice, we always find a unique solution for the final parameter C * and G * . We show in Fig. 7.6 several examples of dendritic spines, where we applied the present algorithm to extract the current, G and C from the voltage deconvolved time series. Fig. 7.6A shows the different region of interest ROIs. In Fig. 7.6B, we compare the deconvolved voltage signal with the one computed from PNP. In Fig. 7.6C, we plot the computed injected current (from voltage) inside the spine neck. Chapter 7. Supplementary Informations: Electro-diffusion simulations of synaptic input in dendritic spines and deconvolution of voltage sensor time series 7.3 Relation between the intrinsic 1/G and the effective neck R neck resistance

We discuss here the relation between the spine effective and intrinsic resistance. The effective neck resistance is computed from the difference of voltage V (t) between the entrance and the exit of the neck, when the injected current is I(t) . By definition, averaging over time, we have

When the injected current follows the model equation

where V 1 (t) is the voltage in the entrance of the neck, the intrinsic conductance and capacitance are G and C respectively. We thus obtain the following relation with

where V 2 is the voltage at the end of the neck. In particular, when C = 0, .47)