Chap.11 Simplifying and Reducing Complex Models (in [Sefiroth/2004-02-05])

Summary

11.1 Introduction

• experimental biology 의 발전 -> challenge to create a mathematical model or simulation of a given system

• At what point does the model cease to have explanatory value?

• dozens of parameters / complexity of the model -> stochastic system 인 경우는 더욱 심함

• large simulations -> a tremendous amount of output -> useless

• simplified models
• focus on some particular level, often using heuristic approximations fo the finer details
• principle of reductionism

• microscopic <-> macroscopic level

• modelling a biological system

• the choice of parameters : an average of similar systems -> simplifed models

• Is it possible to construct simplified models that are quatitative rather than just qualitative?
• In this chapter we describe some methods that allow one to derive quantitatively correct models from more complex systems.

11.1.1 Averaging

• fast quantity <-> slow quantity

• spatial averaging : mean-field approximation

11.2 Master Equations

• many biological problems : continuous time jump processes in which a system switches from one state to the next
• e.g. random opening and closing of a channel, growth of an actin polymer, chemical reactions
• Markov process : figure11.1
• master equation

11.2.1 Application to a Model fo Fibroblast Orientation

• effect of density on the behavior of fibroblasts in culture
• fibroblasts will align with each other with a probability that depends on their relative angles of motion

11.2.2 Mean-Field Reduction of a Neural System

• The master equation essentially averages over many sample paths in a system and leads to a set of equations for the probability of any given state of the system.
• Another way to average a systme that haas intrinsic randomness is the so-called mean-field approximation.
• the effect that cortical processing has on thalamic input in the somatosensory whisker barrel area of the rat

11.3 Deterministic Systems

• There are many ways to reduce the dimension and complexity of deterministic systems.
• Here we will concentrate on techniques for reduction by exploiting time scales.

11.3.1 Reduction of Dimension by Using the Pseudo-Steady State

11.3.2 Averaged Equations

• A related way to reduce complexity is to again exploit time scales and average, keeping only the slow variables.
• Hebbian Learning
• Neural Network from Biophysics
• Weakly Coupled Oscillators

11.4 Discussion and Caveats

• Averaging is an intuitively appealing method for reducing the complexity of biological systems that operate on many different time and space scales.

• When are important details being neglected? -> This is a very difficult question to answer.

• How can you know when important details are missing? -> The only way to know is to do simulations.