# 1. Modeling Introduction

## 1.1. Objective

• To introduce a simplified framework for thinking about how computation fits in the scientific process

## 1.2. Motivation

 Over the next few weeks, we will follow the described scientific process from end to end. To predict phenomenon like bacterial growth [1] [2] [3], we need to use equations.

## 1.3. Introduction

In this course, I will refer to three types of models

Models

1. Science (or observational) models
2. Mathematical models
3. Computational models

## 1.4. Process

A common way that science gets done:

Modeling

1. Domain specialists provide a statement about how a system works
2. Statement is translated into equations
3. Equations are implemented in a computer program

Experiment

Once the above is done: Perform computational experiments

## 1.5. Details

1. "Domain Specialists" (e.g., biologists) come up with a statement about how a system works (what are the major processes, what is the physics, etc.).
• The domain specialists collaborate with mathematicians to translate the statement into an equation that will give numerical values.
• That is, they develop a conceptual representation of the system. Example: The change in population from this year to the next is proportional to the population this year.
2. The domain specialists and mathematicians work with computational scientists to implement the equations and explore the results using a computer.
• That is, they develop a mathematical representation that corresponds to the science model.
• Example: P(next year) - P(this year) = aP(this year). From this equation, we can predict an exact numerical value for population.
3. The domain specialists and mathematicians work with computational scientists to implement the equations and explore the results using a computer.
• In the same way that the mathematical representation of the science model is an approximation, so is the computational representation of the mathematical model.
• These mathematical equations are written in a computer language (e.g., MATLAB/Octave/Java/C/Fortran). Writing the program is only part of the process (but is the only part of the process that we will cover in this class).

Experiment

• Once the computer representation is created, we can experiment: What if the proportionality constant a changes? What if we change it to use "next day" and "this day" instead of "next year" and "this year"?

## 1.6. Details cont.

In reality, the above step-by-step process is not always followed.

• Similar to the scientific method, which has feedback steps.
• In addition, one can start at any point: A mathematician will sometimes come up with formulas which have no know science application until many years later. This happened many times in the development of Quantum Mechanics. (Part of the science model for Quantum Mechanics includes the statements that energy is quantized and that light has both particle and wave properties.)

## 1.7. Example

The McCulloch-Pitts Neuron model is an example of the above process of doing science.

In 1943 Warren S. McCulloch, a neuroscientist, and Walter Pitts, a logician, published "A logical calculus of the ideas immanent in nervous activity" in the Bulletin of Mathematical Biophysics 5:115-133. In this paper McCulloch and Pitts tried to understand how the brain could produce highly complex patterns by using many basic cells that are connected together. These basic brain cells are called neurons, and McCulloch and Pitts gave a highly simplified model of a neuron in their paper. [4]

# 2. Problems

## 2.1. Examples of Process

Find an example from any field in science, similar to that for the McCulloch Pitts example given above, that highlights several part of the process described above. Identify the science model, the mathematical model, and the computational experiments that were performed. (Don't worry about finding out about the program that was used or the details about how the mathematical model was represented in a computer language.)