## Stochastic Neuron ModelsThis book describes a large number of open problems in the theory of stochastic neural systems, with the aim of enticing probabilists to work on them. This includes problems arising from stochastic models of individual neurons as well as those arising from stochastic models of the activities of small and large networks of interconnected neurons. The necessary neuroscience background to these problems is outlined within the text, so readers can grasp the context in which they arise. This book will be useful for graduate students and instructors providing material and references for applying probability to stochastic neuron modeling. Methods and results are presented, but the emphasis is on questions where additional stochastic analysis may contribute neuroscience insight. An extensive bibliography is included. Dr. Priscilla E. Greenwood is a Professor Emerita in the Department of Mathematics at the University of British Columbia. Dr. Lawrence M. Ward is a Professor in the Department of Psychology and the Brain Research Centre at the University of British Columbia. |

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### Contents

1 | |

2 Single Neuron Models | 9 |

3 Population and Subpopulation Models | 33 |

4 Spatially Structured Neural Systems | 49 |

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### Common terms and phrases

2-dimensional action potential activity behavior binary neuron Biosciences Institute Lecture bistability brain Brownian motion Comput described deterministic deterministic model diffusion eigenvalues excitatory and inhibitory Figure frequency function Gillespie algorithm HH model Hodgkin-Huxley individual neurons inhibitory neurons Institute Lecture Series interactions ion channel ISI distribution ISI histogram Izhikevich neurons Kolmogorov equation Kuramoto Kuramoto model LIF model LIF neuron linear noise approximation local field potential Longtin Markov process Mathematical Biosciences Institute ML model model neuron neural networks neural systems neuron firing neuron model neuronal noise Neurosci Ornstein-Uhlenbeck process output peak Phys Poisson population model populations of neurons power spectrum probabilists problem quasicycles quiescent reaction–diffusion system Reprinted with permission reset Section 3.2 simulation single neuron models spatial structure spectra spike stable limit cycle stochastic analysis stochastic differential equation stochastic dynamics stochastic facilitation stochastic ML stochastic model Stochastic Neuron Models stochastic patterns subthreshold oscillations synchronization variable voltage Wilson-Cowan