The most conspicuous cell in the human brain is the neuron, an excitable cell specialized for long-distance signal transmission and information integration. Neurons in the brain interconnect to form networks that form a unique and massively parallel computer. Biological neurons come in great variety and even greater number. Indeed, the human neocortex is composed of roughly 28 billion neurons and the 'smaller' cerebellum has even more -- 50 billion neurons. Furthermore, these neurons have great structural and physiological complexity.
These enormous numbers, combined with the complexity of individual neurons, pose one of many challenges to building an informative simulation model of the brain. One approach to meeting this particular challenge is to use simplified models of neurons that capture the relevant `essential features' of biological neurons. This presentation focuses on the spike generation mechanism of the biological neuron. Dynamical systems analyses of the spike generation process (by mathematicians such as Fitzhugh, Rinzel, Ermentrout, Kopell, Izhikevich) reveals that spike generation can be accurately modeled by a 2-dimensional dynamical system whose behavioral regime falls into one of four classes. This makes it feasible to accurately model the spike generation mechanism of nearly any biological neuron using a system of only two coupled differential equations. The results of these mathematicians provides a clear step forward for building computationally efficient, yet biologically accurate, large scale simulations of the brain.