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Stochastic Simulation of Neurons, Axons, and Action Potentials

A. Aldo Faisal

in Stochastic Methods in Neuroscience

Variability is inherent in neurons. To account for variability we have to make use of stochastic models. We will take a look at this biologically more rigorous approach by studying the fundamental ... More

Variability is inherent in neurons. To account for variability we have to make use of stochastic models. We will take a look at this biologically more rigorous approach by studying the fundamental signal of our brain’s neurons: the action potential and the voltage-gated ion channels mediating it. We will discuss how to model and simulate the action potential stochastically. We review the methods and show that classic stochastic approximation methods fail at capturing important properties of the highly nonlinear action potential mechanism, making the use of accurate models and simulation methods essential for understanding the neural code. We will review what stochastic modelling has taught us about the function, structure, and limits of action potential signalling in neurons, the most surprising insight being that stochastic effects of individual signalling molecules become relevant for whole-cell behaviour. We suggest that most of the experimentally observed neuronal variability can be explained from the bottom-up as generated by molecular sources of thermodynamic noise.Less

Stochastic-Simulation Tests of Nonlinear Econometric Models

Roberto S. Mariano and Bryan W. Brown

in Comparative Performance of U.S. Econometric Models

Stochastic simulations of nonlinear dynamic econometric models have been used in various ways in the past. This chapter discusses how stochastic simulations can be exploited to develop appropriate ... More

Stochastic simulations of nonlinear dynamic econometric models have been used in various ways in the past. This chapter discusses how stochastic simulations can be exploited to develop appropriate system-specification tests for nonlinear systems. The approach is through auxiliary regressions of stochastic simulation errors to develop asymptotically valid significance tests of the predictive performance of the model. The first section discusses Adrian Pagan's critique of the use of simulations in testing nonlinear models for misspecification. The related issue of the informational content of multi-period-ahead predictions is also analyzed in this section. The stochastic simulations that it uses to form the prediction-based tests and their basic asymptotic properties are reviewed in the second section. The last section then develops the auxiliary regressions leading to our prediction-based tests.Less

Master equations and simulation algorithms for the discrete‐stochastic approach

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

The discrete-stochastic model of diffusion introduced in the preceding chapter implies that the system state evolves in time as what is known as a jump Markov process. The time evolution of such a ... More

The discrete-stochastic model of diffusion introduced in the preceding chapter implies that the system state evolves in time as what is known as a jump Markov process. The time evolution of such a process is generally governed by what is called a master equation. In this case, the assumption that the solute molecules move about independently of each other gives rise to two master equations, one for a single solute molecule and one for the entire collection of solute molecules. In this chapter the chapter derives both of those master equations. The chapter obtains the exact time-dependent solution of the single-molecule master equation, and the exact time-independent (equilibrium) solution of the many-molecule master equation. The chapter also derives companion stochastic simulation algorithms for the two master equations, and the chapter applies those algorithms to the microfluidic diffusion experiment of Chapter 5. The chapter uses simulation to test the validity of the discrete-stochastic version of Fick's Law.Less

Numerical Simulation for Biochemical Kinetics

Daniel T. Gillespie and Linda R. Petzold

in System Modeling in Cellular Biology: From Concepts to Nuts and Bolts

This chapter discusses concepts and techniques for mathematically describing and numerically simulating chemical systems that into account discreteness and stochasticity. The chapter is organized as ... More

This chapter discusses concepts and techniques for mathematically describing and numerically simulating chemical systems that into account discreteness and stochasticity. The chapter is organized as follows. Section 16.2 outlines the foundations of “stochastic chemical kinetics” and derives the chemical master equation (CME)—the time-evolution equation for the probability function of the system’s state. The CME, however, cannot be solved, for any but the simplest of systems. But numerical realizations (sample trajectories in state space) of the stochastic process defined by the CME can be generated using a Monte Carlo strategy called the stochastic simulation algorithm (SSA), which is derived and discussed in Section 16.3. Section 16.4 describes an approximate accelerated algorithm known as tau-leaping. Section 16.5 shows how, under certain conditions, tau-leaping further approximates to a stochastic differential equation called the chemical Langevin equation (CLE), and then how the CLE can in turn sometimes be approximated by an ordinary differential equation called the reaction rate equation (RRE). Section 16.6 describes the problem of stiffness in a deterministic (RRE) context, along with its standard numerical resolution: implicit method. Section 16.7 presents an implicit tau-leaping algorithm for stochastically simulating stiff chemical systems. Section 16.8 concludes by describing and illustrating yet another promising algorithm for dealing with stiff stochastic chemical systems, which is called the slow-scale SSA.Less

Mathematical and computational modelling tools

Baltazar D. Aguda and Avner Friedman

in Models of Cellular Regulation

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary ... More

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary differential equations; but coupling of chemical reactions to physical processes such as diffusion requires the formulation of partial differential equations to describe the spatiotemporal evolution of the system. Mathematical analysis of the dynamical models involves basic concepts from ordinary and partial differential equations. Computational methods, including stochastic simulations and sources of computer software programs available free on the internet are also summarized.Less

Notional Defined Contribution Pension Systems in a Stochastic Context: Design and Stability

Ronald Lee

Alan J. Auerbach (ed.)

in Social Security Policy in a Changing Environment

This chapter explores a new approach to Social Security reform that is known as “Notional Defined Contribution” or “Nonfinancial Defined Contribution” (NDC). This chapter uses a stochastic ... More

This chapter explores a new approach to Social Security reform that is known as “Notional Defined Contribution” or “Nonfinancial Defined Contribution” (NDC). This chapter uses a stochastic macroeconomic model for forecasting and simulating the long-term finances of NDC-type public pension programs in the context of demographic and economic trends in the United States. While NDC plans are seen as having various potential advantages over traditional pay-as-you-go (PAYGO) systems, the focus of this chapter is on their financial stability over the long term. Around the world, PAYGO public pension programs are facing serious long-term fiscal problems due primarily to actual and projected population aging and most appear unsustainable as currently structured. The stochastic population model is based on a Lee-Carter mortality model and a somewhat similar fertility model. The feasible internal rate of return for a PAYGO system with stable population structure equals the rate of growth of the population plus the rate of growth of output per worker. Evidently, stochastic simulation of the system's finances can reveal aspects of its performance that are not otherwise obvious and can assist in improving system design. This promises to be a valuable use for stochastic simulation models of pension systems.Less

Computing Chemical Reactions

Wolfgang Banzhaf and Lidia Yamamoto

in Artificial Chemistries

This chapter examines various reactor algorithms that are frequently used in artificial chemistries. We contrast macroscopic, deterministic reaction algorithms and microscopic, stochastic reaction ... More

This chapter examines various reactor algorithms that are frequently used in artificial chemistries. We contrast macroscopic, deterministic reaction algorithms and microscopic, stochastic reaction algorithms. The Chemical Master Equation is introduced, followed by well-known simulation algorithms such as Gillespie’s direct method and tau-leaping. An overview of algorithms that simulate spatial and multicompartmental chemistries then follows, from numerical integration to stochastic reaction-diffusion algorithms based on subvolumes.Less

Implications and limitations of the Einstein theory of diffusion

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

Chapter 3 showed how Einstein's analysis of Brownian motion shifted the focus of the classical diffusion equation from the average behavior of many solute molecules to the probabilistic behavior of ... More

Chapter 3 showed how Einstein's analysis of Brownian motion shifted the focus of the classical diffusion equation from the average behavior of many solute molecules to the probabilistic behavior of an individual solute molecule. This chapter deduces some further implications of the Einstein theory of diffusion, using not only analytical reasoning but also numerical simulation. This chapter shows how simulation can be used to construct plots of a single diffusing molecule's position probability density function for some simple boundary conditions. The chapter also shows how simulation can be used to construct “snapshots” of an unrestricted solute molecule's position at successive instants of time. But the chapter will discovers that such snapshots expose a serious physical limitation of the Einstein theory. The chapter rationalizes a quick fix that allows us to complete the derivation which has begun in Chapter 3 of a formula for the stochastic rate of a diffusion-controlled bimolecular chemical reaction. But from a broader view, it will become apparent that, while the Einstein theory of diffusion has a wide range of practical utility, a physically more accurate theory is needed.Less