Branching process

Abstract: Networks of living neurons exhibit diverse patterns of activity, including oscillations, synchrony, and waves. Recent work in physics has shown yet another mode of activity in systems composed of many nonlinear units interacting locally. For example, avalanches, earthquakes, and forest fires all propagate in systems organized into a critical state in which event sizes show no characteristic scale and are described by power laws. We hypothesized that a similar mode of activity with complex emergent properties could exist in networks of cortical neurons. We investigated this issue in mature organotypic cultures and acute slices of rat cortex by recording spontaneous local field potentials continuously using a 60 channel multielectrode array. Here, we show that propagation of spontaneous activity in cortical networks is described by equations that govern avalanches. As predicted by theory for a critical branching process, the propagation obeys a power law with an exponent of -3/2 for event sizes, with a branching parameter close to the critical value of 1. Simulations show that a branching parameter at this value optimizes information transmission in feedforward networks, while preventing runaway network excitation. Our findings suggest that “neuronal avalanches” may be a generic property of cortical networks, and represent a mode of activity that differs profoundly from oscillatory, synchronized, or wave-like network states. In the critical state, the network may satisfy the competing demands of information transmission and network stability.

Abstract: A new method is presented for inferring evolutionary trees using nucleotide sequence data. The birth-death process is used as a model of speciation and extinction to specify the prior distribution of phylogenies and branching times. Nucleotide substitution is modeled by a continuous-time Markov process. Parameters of the branching model and the substitution model are estimated by maximum likelihood. The posterior probabilities of different phylogenies are calculated and the phylogeny with the highest posterior probability is chosen as the best estimate of the evolutionary relationship among species. We refer to this as the maximum posterior probability (MAP) tree. The posterior probability provides a natural measure of the reliability of the estimated phylogeny. Two example data sets are analyzed to infer the phylogenetic relationship of human, chimpanzee, gorilla, and orangutan. The best trees estimated by the new method are the same as those from the maximum likelihood analysis of separate topologies, but the posterior probabilities are quite different from the bootstrap proportions. The results of the method are found to be insensitive to changes in the rate parameter of the branching process.

Abstract: A theorem exhibiting the duality between certain infinite systems of interacting stochastic processes and a type of branching process is proved. This duality is then used to study the ergodic properties of the infinite system. In the case of the vector model a complete understanding of the ergodic behavior is obtained.

Abstract: Review of Probability Theory and an Introduction to Stochastic Processes Introduction Brief Review of Probability Theory Generating Functions Central Limit Theorem Introduction to Stochastic Processes An Introductory Example: A Simple Birth Process Discrete-Time Markov Chains Introduction Definitions and Notation Classification of States First Passage Time Basic Theorems for Markov Chains Stationary Probability Distribution Finite Markov Chains An Example: Genetics Inbreeding Problem Monte Carlo Simulation Unrestricted Random Walk in Higher Dimensions Biological Applications of Discrete-Time Markov Chains Introduction Proliferating Epithelial Cells Restricted Random Walk Models Random Walk with Absorbing Boundaries Random Walk on a Semi-Infinite Domain General Birth and Death Process Logistic Growth Process Quasistationary Probability Distribution SIS Epidemic Model Chain Binomial Epidemic Models Discrete-Time Branching Processes Introduction Definitions and Notation Probability Generating Function of Xn Probability of Population Extinction Mean and Variance of Xn Environmental Variation Multitype Branching Processes Continuous-Time Markov Chains Introduction Definitions and Notation The Poisson Process Generator Matrix Q Embedded Markov Chain and Classification of States Kolmogorov Differential Equations Stationary Probability Distribution Finite Markov Chains Generating Function Technique Interevent Time and Stochastic Realizations Review of Method of Characteristics Continuous-Time Birth and Death Chains Introduction General Birth and Death Process Stationary Probability Distribution Simple Birth and Death Processes Queueing Process Population Extinction First Passage Time Logistic Growth Process Quasistationary Probability Distribution An Explosive Birth Process Nonhomogeneous Birth and Death Process Biological Applications of Continuous-Time Markov Chains Introduction Continuous-Time Branching Processes SI and SIS Epidemic Processes Multivariate Processes Enzyme Kinetics SIR Epidemic Process Competition Process Predator-Prey Process Diffusion Processes and Stochastic Differential Equations Introduction Definitions and Notation Random Walk and Brownian Motion Diffusion Process Kolmogorov Differential Equations Wiener Process Ito Stochastic Integral Ito Stochastic Differential Equation (SDE) First Passage Time Numerical Methods for SDEs An Example: Drug Kinetics Biological Applications of Stochastic Differential Equations Introduction Multivariate Processes Derivation of Ito SDEs Scalar Ito SDEs for Populations Enzyme Kinetics SIR Epidemic Process Competition Process Predator-Prey Process Population Genetics Process Appendix: Hints and Solutions to Selected Exercises Index Exercises and References appear at the end of each chapter.