概率性脉冲神经网络：为计算机时代撰写的神经数学

English | 2024 | ASIN: B0DK6F98MY | | Epub PDF (True) | 19 MB

This book provides a self-contained introduction to a new class of stochastic models for systems of spiking neurons. These systems have a large number of interacting components, each one evolving as a stochastic process with a memory of variable length. Several mathematical tools are put to use, such as Markov chains, stochastic chains having memory of variable length, point processes having stochastic intensity, Hawkes processes, random graphs, mean field limits, perfect sampling algorithms, the Context algorithm, and statistical model selection. The book’s focus on mathematically tractable objects distinguishes it from other texts on theoretical neuroscience. The biological complexity of neurons is not ignored, but reduced to some of its main features, such as the intrinsic randomness of neuronal dynamics. This reduction in complexity aims at explaining and reproducing statistical regularities and collective phenomena that are observed in experimental data, an approach that leads to mathematically rigorous results. With an emphasis on a constructive and algorithmic point of view, this book is directed towards mathematicians interested in learning about stochastic network models and their neurobiological underpinning, and neuroscientists interested in learning how to build and prove results with mathematical models that relate to actual experimental settings.

这本书提供了一个新的类神经元爆发系统随机模型的自包含介绍。这些系统具有大量相互作用的组件，每个组件均以可变长度的记忆演变为一个随机场过程。数学工具被用来处理这些问题，包括马尔科夫链、具有可变记忆长度的随机链、具有随机强度的过程、霍克赫兹过程、随机图、平均场极限、完美采样算法、Context算法和统计模型选择。 这本书将数学上可解的对象作为焦点，使它与其他神经科学理论类文本有所不同。尽管没有忽视神经元的生物复杂性，但将其简化为其主要特征之一，即神经动力学中的固有随机性。这种简化的目标在于解释并重现实际实验数据中观察到的统计规律性和集体现象，这种方法引向了数学上严格的成果。 强调建设性和算法观点，这本书旨在对那些希望了解随机网路模型及其神经生物学基础的数学家和对如何使用与实际实验设置相关的数学模型进行学习和证明结果感兴趣的神经科学家。

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