生成函数法在生化反应系统、随机HH神经元中的应用

发布时间：2018-05-11 22:33

本文选题：生化反应系统 + 随机性　；参考：《清华大学》2016年博士论文

【摘要】：精密而复杂的生化反应网络调控着各种各样的生命活动,研究模拟生命系统的反应动力学行为无疑是了解生命活动的重要途径。生命活动的基本单元是细胞,传统的确定性反应速率方程不再能够描述细胞这种介观体系。如何准确描述随机生化反应系统的动力学过程是当前研究的热点之一。信息以动作电位的形式编码并在神经网络中传递,动作电位的产生和传播都是电流通过细胞膜上的离子通道实现的。HH模型给出动作电位的演化方程。而由于通道噪声、突触噪声的存在,这种确定性的描述需要做进一步修正。随机生化反应系统以及HH神经元通道的随机动力学过程均是马尔可夫过程,这种过程可以由描述系统状态概率分布演化的主方程准确描述。Gillespie算法是一种常用的能精确求解它的蒙特卡洛模拟方法,但对于很多复杂系统而言,Gillespie算法需要的计算量过大。本文介绍生成函数法,将由极多的常微分方程(ODE)组成的主方程改写为由一个决定概率分布演化的波动方程的量子场论(QFT)形式,并等价转换为生成函数的描述方式。我们将生成函数法应用到非线性的双元反应系统中,并给出不同的尝试解,利用变分法将生成函数满足的偏微分方程(PDE)近似为有限个ODEs。在不同的反应参数条件下,无论分子数多少,都能和Gillespie算法符合得非常好。而且,由于这种方法仅仅计算有限个ODEs,且计算时间和分子数多少无关,算法高效。我们利用生成函数法对神经元动作电位进行随机模拟。通道打开关闭的过程是一个典型的马尔可夫过程,可以用主方程描述。由于电压的含时演化只和通道打开数有关,我们同时提出了只需对通道打开数进行取样的两种加速算法。计算结果表明,除了计算效率高之外,生成函数法均能提供与准确的Gillespie算法相近的结果,然而大多数Langevin方法差距较大。生成函数法为动作电位在轴突乃至整个神经网络中的传播提供一种新的途径。
[Abstract]:Precise and complex biochemical reaction networks regulate all kinds of life activities. It is undoubtedly an important way to understand life activities to study the dynamic behavior of simulated life systems. The basic unit of life activity is cell, and the traditional deterministic reaction rate equation can no longer describe the mesoscopic system of cell. How to accurately describe the dynamic process of stochastic biochemical reaction system is one of the hot research topics at present. The information is encoded in the form of action potential and transmitted in the neural network. The generation and propagation of action potential are all realized by the current through the ion channel on the cell membrane. The evolution equation of action potential is given by using the. HH model. Due to the presence of channel noise and synaptic noise, this deterministic description needs to be further modified. The stochastic biochemical reaction system and the stochastic dynamics of HH neuronal channel are all Markov processes. This process can be accurately described by the master equation describing the evolution of the system state probability distribution. The Gillespie algorithm is a common Monte Carlo simulation method that can accurately solve it, but for many complex systems, the Gillespie algorithm requires too much computation. In this paper, the method of generating function is introduced. The main equation, which consists of a lot of ordinary differential equations (ODEs), is rewritten into the quantum field theory (QFTT) form of a wave equation which determines the evolution of the probability distribution, and is equivalent to the description of the generating function. In this paper, we apply the generating function method to the nonlinear binary reaction system, and give different solutions. By using the variational method, we approximate the PDE of the generating function to a finite number of ODEs. Under different reaction parameters, no matter the number of molecules, it can agree well with the Gillespie algorithm. Moreover, the algorithm is efficient because only a finite number of ODEs is calculated, and the calculation time is independent of the number of molecules. We use the generating function method to simulate the action potential of neurons at random. The process of opening and closing the channel is a typical Markov process, which can be described by the master equation. Since the time-dependent evolution of voltage is only related to the number of open channels, we also propose two acceleration algorithms that only need to sample the number of open channels. The results show that except for the high computational efficiency, the generative function method can provide the same results as the accurate Gillespie algorithm. However, most of the Langevin methods are far behind each other. The method of generating function provides a new way for the propagation of action potential in axon and even the whole neural network.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：Q61;O241.8

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