Mathematical Modelling in Neuroscience

我的研究工作包括开发数学模型，以帮助更好地理解神经元的正常和病理行为。我与几位实验研究人员合作，使我的模型在神经科学方面有切实的应用。我的研究计划分为以下几个轴。1)建立常微分方程（ode）模型来描述神经细胞中离子浓度波动和体积变化。与我的团队和合作者一起，我们开发了描述离子浓度波动的数学模型，特别是氯化物，在神经性疼痛研究的背景下。我们目前正在努力改进我们的模型，以便它们也能描述细胞膜上的水通量和体积变化。体积波动在细胞信号传导中起着至关重要但往往被忽视的作用。它们很少被研究，部分原因是难以进行准确的实验测量。我们将使用合作者的数字全息显微测量来验证和校准我们的数学模型。2)四合神经的建模。为了充分理解突触信号传导，有必要了解突触的四个部分：1)突触前按钮，2)突触后受体，3)细胞外空间和4)胶质细胞。与博士候选人李晓婷和我的合作者Antoine Godin一起，我们开发了一个数学模型来研究纳米柱的影响，纳米柱是一种纳米结构，可以促进突触后受体与突触前囊泡的开放位点对齐。从数学的角度来看，该模型使用蒙特卡罗方法，离散马尔可夫链和使用有限体积方法描述突触电位。在这个项目的继续，我们将开发模型，考虑到突触周围的细胞外空间，固定电荷在这个空间的分布和神经胶质细胞对神经递质扩散的影响。3)应用有限元方法研究Ranvier节点。我们开发了使用泊松-能-普朗克偏微分方程组的模型来计算神经结构中离子种类的电场分布和电扩散。这些模型允许，除其他外，研究髓磷脂损失对以动作电位形状为特征的电信号的影响。我们将利用与Antoine Godin的合作，对离子浓度在动作电位期间的波动进行实验测量。在这三个项目中，非线性动力学模型参数的确定是一个重要问题。博士生Pierre-Louis Gagno正在开发新的数学方法来更好地解决这个问题。

My research work consists in developing mathematical models that help better understand the normal and pathological behiavor of neurons. I  work in collaboration with several experimental researchers so that my models have tangible applications in neuroscience. My research program is divided into the following axes. 1) Develop models of ordinary differential equations (ODEs) to describe ionic concentration fluctuations and volume changes in neural cells. With my team and collaborators, we have developed mathematical models that describe the fluctuations of ionic concentrations, especially chloride,  in the context of the study of neuropathic pain. We are currently working to improve our models so that they also describe water fluxes across cell membranes and volume changes. Volume fluctuations play a crucial but often overlooked role in cellular signaling.  They are little investigated partly due to difficulties in performing accurate experimental measurements.  We will use digital holography microscopy measurements from our collaborators to validate and calibrate our mathematical models.   2) Modeling the tetrapartite synapse. To fully understand synaptic signaling, it is essential to understand the four parts of the synapse that is 1) the presynaptic button, 2) the postsynaptic receptors, 3) the extracellular space and 4) the glial cells.  With the PhD candidate Xiaoting Li and my collaborator Antoine Godin, we have developed a mathematical model to study the impact of nanocolumns which are nanoscopic structures that promote the alignment of postsynaptic receptors with the opening sites of presynaptic vesicles. From a mathematical point of view, this model uses a Monte Carlo approach, discrete Markov chains and a description of the synaptic electrical potential using a finite volume approach. In the continuation of this project, we will develop models that take into account the extracellular space surrounding the synapse, the distribution of fixed electrical charges in this space and the impact of glial cells on neurotransmitters diffusion. 3) Application of the finite element method to study the node of Ranvier. We have developed models that use the Poisson Nernst-Planck system of partial differential equations to calculate the electric field distribution and the electrodiffusion of ionic species in neural structure. These models allow, among other things, to study the impact of a myelin loss on the electrical signaling as characterized by the shape of an action potential. We will take advantage of a collaboration with Antoine Godin to obtain experimental measurements on the fluctuations of ionic concentrations during action potentials. In these three projects, the determination of parameters in non-linear dynamical models is an important issue. The PhD student Pierre-Louis Gagno is developing new mathematical methods to better solve this problem.