Hodgkin and Huxley's legacy: the science of neural supercomputing

Abstract

Reviews the books, Mathematics for Neuroscientists by Fabrizio Gabbiani and Stephen Cox (2010) and Mathematical Foundations of Neuroscience by G. Bard Ermentrout and David H. Terman (2010). Mathematics for neuroscientists shows that the sophisticated mathematical methods involved in understanding and modelling the brain are not necessarily out of reach for the ‘ordinary’ neurobiologist. Based on courses taught by the authors, the book is a superb introduction to quantitative methods that take the reader on a journey from ion channel to network. In doing so, it follows the history of many pivotal neuroscience discoveries along the way, asking the same questions and using the same analysis as was used by the pioneering neuroscientists that discovered them. Mathematical foundations of neuroscience set out in a similar, albeit more concise, fashion covering the fundamentals of the Hodgkin–Huxley equations and Cable Theory. However, they soon lay the groundwork for a specific approach where they have been at the forefront—applying dynamical systems theory at multiple levels of analysis. They start by introducing phase planes—a geometric representation of the solution to the membrane ion channel equations, which depict the dynamic evolution of the voltage trace following an injection of current. They then describe how fixed points and limit cycles emerge, reflecting periodic solutions that result in oscillatory firing of the model neuron. Within this framework, transitions between different dynamic states can then be studied with powerful analytical tools such as bifurcation theory. (PsycINFO Database Record (c) 2016 APA, all rights reserved)