I'd like to thank Dr. Gewirtz for the great overview of the brain and neural anatomy and physiology. This topic, because of its complexity, often draws from and influences many disciplines of science including my own research with dynamical systems of differential equations. From these discoveries, the biological basis of epilepsy, memory, hysteresis, and other phenomena of interest to psychologists are given a mathematical descriptions.
We owe our contemporary understanding of the brain and its vast complex network of neurons to a few beautifully simple experiments and discoveries. Famed British biologist Jonathan Zachary Young, while studying the nervous system of cephalopods, discovered that some squid have rather large neurons and axons to control their water jets. You can watch a video of him dissecting a squid here.
Fellow British biologists and mathematicians Sir Alan Hodgkin and Andrew Huxley (Aldous Huxley's half brother for those fans of dystopian fiction) began working on a theory of action potential propagation via ion channels. Specifically they developed a system of 4 nonlinear differential equations, solved it numerically (in 4 dimensions by hand! A daunting task before the popularization of scientific computing), and verified it empirically using the squid's giant axon hooked up to a voltage clamp. They found that the voltage required to spike the neuron was a function of the current across the sodium and potassium channels, current leakage, the input current, and membrane capacitance. Their discovery was made possible because J.Z. Young had found an axon large enough to stand up to the rather crude methodology of the time.
In 1961, while using analogue computers to simulate and study the Hodgkin-Huxley model, Richard FitzHugh did something rather counter intuitive. He simplified the model down to its mathematical essentials which made it much easier to study the qualitative nature of action potential. Furthermore, in a scientific one-two punch, Japanese scientists Jin-ichi Nagumo et.al. created an equivalent circuit for FitzHugh's equations. This allowed them to run all sorts of tests and experiments on the nature of neuronal spiking.
The consequences and the power of simple chaotic dynamical systems to predict complex natural phenomenon in neurons reach far across the many fields of science. Nature had inspired abstract mathematical discoveries which feedback into the discoveries about the natural world. This comic becomes disingenuous:
and the world becomes more complex like this [click for link]: