# Toy models

The world is a complex place and creating a mathematical model to describe a complex mechanism can be a challenging task.

In any real world event, while there are often many factors, events, and interactions that are important, it is often the case that a small number of factors are quantitatively more significant than the others. When this is the case, looking at a highly simplified model that *only* includes those factors can provide an excellent starting point for building a more complete model. We refer to this as a *toy model*. For this reason, we choose a stick figure as our icon for toy models. Even in drawing a realistic image of a person, it's often an excellent starting point to start with a stick figure to get the proportions and alignment of the limbs right before filling in the details.

While the name sounds like it is trivializing the model, don't be misled! The term "toy model" is used with pride in theoretical physics and many advanced research papers include the term. It does not appear to be explicitly used in biology, though such models as single allele inheritance, phylogentic trees, direct assembly of proteins by RNA are what I would call toy models.

A better name might be "core model" since the idea is to find a simple core hiding in a complex process. Doing this is often among the most creative and insightful advances is science. One powerful example is the model of chemistry based on atomic orbitals. Simple orbital filling is only a starting point for more realistic models but it provides an excellent starting point. A second powerful example is the Bardeen-Cooper-Schrieffer paired electron model of superconductivity. This is a highly simplified model, but it won a Nobel prize!

In this class, our toy models will be simpler — projectiles treated as if they were moving in a vacuum in flat earth gravity, or electric circuits with perfect batteries and resistanceless wires.

Even when a toy model doesn't dominate a phenomenon it can still be quite useful. Giving a toy model is not saying that this is exactly what is going on. What it says is that we have identified some factors that play an important role in the mechanism we are exploring and we want to see what its influence is by letting it act all by itself. And learning just how the math of that toy model behaves gives us a lot of insight as to how that particular set of factors behave, and what kind of corrections we might want to make to the model. For example, our discussions of electric circuits have a series of models of how electric currents on membranes behave that begin a batteries and resistors, then move to a model with multiple batteries, and then add capacitors to the model. More realistic models (such as the Hodgkin-Huxley model) add variable resistance and non-linear effects.

Working through the details of the math of one or more toy models can also serve as organizing tools for your thinking! When we've built a mathematical toy model of a part of a system, there can be an organizing core equation that you can use as a starting point for thinking about a system and solving problems. (See the follow-on page on Anchor Equations.)

Joe Redish 5/12/19

#### Follow-on

Last Modified: June 14, 2021