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 The human brain contains approximately 1011 neurons, each of which makes on the average 1000 connections to other neurons. It is natural, and comforting to think of the brain as a computational device, much like the computers that we build and understand. In this electrophysiological view of the brain, the neurons are the elementary devices and the way they are connected up determines the functioning of the brain. But the brain is a much more flexible, adaptive, and complicated organ than this point of view suggests. The brain is made up of cells and there are ten times as many glial cells as there are neurons. The glial cells help the neurons make glutathione, and astrocytes store glucose as glycogen. Neurons synthesize and release more than 50 different kinds of neurotransmitters and myriad different receptor types allow neurons to influence each other’s electrophysiology by volume transmission in which cells in one nucleus change the local biochemistry in a distant nucleus. This is the pharmacological view of the brain. We discuss volume transmission on a separate page.

This is just the beginning of the full complexity of the problem. The functioning of neurons and glial cells is affected by an individual’s genotype and dynamic changes of gene expression levels, both on the short term and the long term. These dynamic changes are influenced by the endocrine system, because the brain is an endocrine organ and is influenced by other endocrine organs like the gonads and the adrenal glands. And, although we think of the brain as producing behavior, in fact our behavior influences the electrophysiology, the pharmacology, and endocrine status of the brain, and therefore the gene expression levels. This is true both on the short term and the long term. Individuals who exercise in their 30s and 40s are 30% less likely to get Parkinson’s disease and the progression of Parkinson’s symptoms is slower in those who exercise. Thus the functioning of an individual brain depends on the history of environmental inputs and behavior throughout the individual’s lifetime. And, we haven’t even mentioned the complicated and changing anatomy of the brain.

Mathematical models are an important tool for understanding complicated biological systems. A model gives voice to our assumptions about how something works. Every biological experiment or psychology experiment is designed within the context of a conceptual model and its results cause us to confirm, reject, or alter that model. Conceptual models are always incomplete because biological systems are very complex and incompletely understood. Moreover, and as a purely practical matter, experiments tend to be guided by small conceptual models of only a very small part of a system, with the assumption (or hope) that the remaining details and context do not matter or can be adequately controlled.

Mathematical models are formal statements of conceptual models. Like conceptual models, they are typically incomplete and tend to simplify some details of the system. But what they do have, which experimental systems do not, is that they are completely explicit about what is in the model, and what is not. Having a completely defined system has the virtue of allowing one to test whether the assumptions and structure of the model are sufficient to explain the observed, or desired, results.

Since 2008, we have been working with Janet Best, a mathematician at Ohio State, making mathematical models that shed light on various aspects of brain function in health and disease. We are particularly interested in how the electrophysiology affects the pharmacology and how the pharmacology affects the electrophysiology of the brain. Thus, many of the questions we address are on the interface between the electrophysiological and pharmacological views of the brain.

In the past two years, Janet, Fred, and Mike have been collaborating with Parry Hashemi, an electrochemist at the University of South Carolina, who can measure the time course of neurotransmitters in vivo in the extracellular space.