Prototyping with tools

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# Models Reveal Main Ideas

We can develop a model at different levels of complexity. We can decide that we want to reproduce behavior at the 17th decimal point of precision, or we can decide that we are comfortable if we only get the direction of the behavior correct. The decision about the level of precision we are trying to capture with our model is a form of abstracting the problem. When we abstract a problem, we attempt to decide what is relevant and what is irrelevant.

Typically, when we create a model, we start with the simplest, first order, behavior. The goal is to try to get this right without worrying if the time and space scales are correct. This is because if we can not get the first order behavior right, then it is a waste of time to try to get the spatial and temporal scales correct. Thus, at different levels of model building, different levels of detail are relevant. Another reason for starting with a first order approximation is that sometimes that is all you need. If, when you press deeper into the problem, the first order model works, and it continues to work for each successive level of complexity, then we have stumbled on a main idea''.

Even if we are not so lucky, as we try to characterize the abstractions of multiple instances of our problem, we may begin to see common denominators. This common denominator is a main idea'', and is the scaffolding around which we can build very complex descriptions of what we observe.

For an example of this latter method of uncovering main ideas'', consider the problem from cardiology of reentrant cardiac arrhythmias. In normal circumstances, the impulse that initiates cardiac contraction forms as a continuous wave that propagates away from the sinus node. Any continuous wavefront in the heart cannot become reentrant simply because it will collide with and extinguish itself. On the other hand, if a wave breaks and becomes discontinuous1.4, then it is possible for the residual wave fragments to evolve into a spiral wave1.5. Reentrant arrhythmias, rapid uncontrollable reexcitations of the heart, are initiated from wave fragments or discontinuous waves. Therefore, forming a spiral front requires that a front arise in a region with asymmetric excitability where propagation succeeds in some directions and is blocked (or fails) in other directions - 1.6. Thus, all reentrant arrhythmias can be understood as resulting from wave formation in a region with a spatial asymmetry of cellular excitability 1.7. If you can identify the source of the asymmetry, then perhaps its possible to correct it. Since this one concept, asymmetric waves form as a result of propagation in a region with a spatial asymmetry in excitability, enables us to have a general idea about an entire class of phenomena, we will call it a main idea''.

Modeling is thus an essential step toward identifying main ideas, the recurring themes that we see as we examine different, but related, systems. We will see this when we explore excitable cells in cardiac tissue and transcription switches1.8 in the DNA of small organisms.

What follows are some of the main ideas we have developed about building and then testing models. We begin with the mathematics and physics required for model building end with statistics for model evaluation. Along the way, we'll introduce some of the software issues we have faced as we constructed tools that promoted our development of main ideas.

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Frank Starmer 2004-05-19
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