An SPDE Approach to Self-Excitatory Neuronal Models

自兴奋神经元模型的 SPDE 方法

• 批准号: 2440959
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2440959
• 关键词: SPDE; Approach; Self; Excitatory; Neuronal

This project falls within the EPSRC Statistics and Applied Probability research area. This project aims to study a system of interacting diffusions on the negative real line of the McKean-Vlasov type with a resetting component and all are influenced by an independent common source of noise. When one of the diffusions of the system hits zero, its value is reset and all other diffusions in the system will receive a continuous impulse to their value over a predetermined timescale. This system is a generalisation of noisy integrate-and-fire models in neuroscience that model the membrane potential of a neuron in biological systems. An important feature of these models is that when the membrane potential reaches a certain threshold it is instantly reset to some predetermined value and all other neurons in the system receive an impulse that affects their voltage. The primary hypothesis of noisy integrate-and-fire models is that the action potential threshold and shape are invariant from spike to spike, and thus a precise description of the spike can be omitted and simply sketched by a spiking threshold. The diffusion model captures solely spike generation and places the biological aspects of the neuron into the noise. It has been observed that neurons react in a predictable and reproducible manner to temporally structured stimuli. The main sources of stochasticity in the model arises from the innate stochasticity in the biological mechanisms which cause voltage changes in neurons and the neurons may receive inputs from other neurons in the system which are not being observed. The latter may be thought of as the common source of noise in the model.The main objective of this project is to show the existence and uniqueness of the mean-field limit to the system of diffusions described above. Moreover, as the interactions between diffusions occur over predefined time scales, this project will attempt to show that, as the time scales on which interactions occur approach zero, the models where the interaction between diffusions occur instantly are recovered. Aside from these aims, this project will also attempt to prove the convergence of an Euler-Maruyama type method for the numerical solution of the limiting system.A further novelty of this research project comes from there being a common noise source felt by all diffusions in the finite system which leads to showing well-posedness and solutions to an SPDE that describes the law of the representative particle in the limiting system with infinite diffusions. The SPDE approach which will be employed by this project leads to looking for solutions being elements of the càdlàg functions taking values in the space of tempered distributions endowed with the M1 topology. This approach differs from existing literature as generally solutions to large scale limits of systems interacting diffusions are looked for in the space of probability measures on the space of càdlàg functions endowed with the M1 topology.

该项目属于 EPSRC 统计和应用概率研究领域，旨在研究具有重置组件的 McKean-Vlasov 型负实线上的相互作用扩散系统，并且所有系统都受到独立的公共噪声源的影响。当系统的一个扩散达到零时，其值被重置，并且系统中的所有其他扩散将在预定的时间尺度内接收到其值的连续脉冲。该系统是噪声积分和激发的概括。神经科学模型对生物系统中神经元的膜电位进行建模，这些模型的一个重要特征是，当膜电位达到某个阈值时，它会立即重置为某个预定值，并且系统中的所有其他神经元都会收到一个脉冲，该脉冲会产生脉冲。噪声积分和激发模型的主要假设是动作电位阈值和形状在尖峰之间是不变的，因此可以省略尖峰的精确描述并简单地通过尖峰扩散来勾画。模型捕获据观察，神经元以可预测和可重复的方式对时间结构的刺激做出反应，模型中随机性的主要来源来自生物机制中的固有随机性。导致神经元中的电压变化，并且神经元可能会接收来自系统中未被观察到的其他神经元的输入，后者可能被认为是模型中的常见噪声源。该项目的主要目的是证明这种现象的存在。和独特性此外，由于扩散之间的相互作用发生在预定的时间尺度上，因此该项目将尝试证明，当相互作用发生的时间尺度接近于零时，模型中的除了这些目标之外，该项目还将尝试证明极限系统数值解的 Euler-Maruyama 型方法的收敛性。该研究项目的另一个新颖之处在于存在一个感受到的常见噪声源有限系统中的所有扩散，这导致显示适定性和 SPDE 的解，该 SPDE 描述了具有无限扩散的限制系统中的代表性粒子的定律。该项目将采用 SPDE 方法，导致寻找解。 càdlàg 函数的元素在赋予 M1 拓扑的缓和分布空间中取值，这种方法与现有文献不同，因为通常在空间上的概率测量空间中寻找相互作用扩散的系统的大规模限制的解决方案。的càdlàg 功能以 M1 拓扑结束。