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[Chap10] |
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.