Background: How specific are the synaptic connections formed as neuronal networks develop and can simple rules account for the formation of functioning circuits? These questions are assessed in the spinal circuits controlling swimming in hatchling frog tadpoles. This is possible because detailed information is now available on the identity and synaptic connections of the main types of neuron.

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This book consists of 14 chapters written by different authors, all of which are concerned with the mathematical modeling of bursting neural activity. Bursting neural activity comprises a repetitive sequence of two events: burst generation in which a neuron produces several spikes (usually from 3 to 20 spikes) with relatively short inter-spike intervals (1-5 ms) and a relatively long quiescent interval. The book is amazingly focused; in all the chapters, the authors use very similar approaches to the development and analysis of models of bursting activity. Typically, two mathematical theories are combined: the bifurcation theory and the theory of singular perturbations.The bifurcation theory is a tool which allows us to characterize the ways in which a dynamical system can undergo a qualitative change in behavior under variation of conditions, parameters, perturbations, etc. For example, the system is in a steady state and parameter variation causes a transition to another dynamical regime, e.g., the regime of regular oscillations. The bifurcation theory provides two possible scenarios for such transitions: (1) Hopf (or Andronov-Hopf) bifurcation where the steady state becomes unstable at the point of bifurcation (or critical parameter value) and the oscillations that appear have small amplitude and a pre-defined frequency of oscillation; (2) Saddle-Node on Invariant Curve (SNIC) bifurcation where the stable steady state disappears by merging with the unstable one and the resulting oscillations have a pre-defined amplitude and a small frequency of oscillation. These two mathematical mechanisms of oscillation appearance are universal and they can be applied to the modeling of a broad spectrum of dynamical systems in different scientific disciplines -a beautiful example of the generality of mathematics. Also, bifurcation theory provides critical parameter values or critical boundaries in parameter space separating different dynamical behaviors. Where no bifurcation occurs between one set of parameters and another, we can be sure that the dynamics will be qualitatively the same. 483 J. Integr. Neurosci. 2006.05:483-488. Downloaded from www.worldscientific.com by AUSTRALIAN NATIONAL UNIVERSITY on 03/14/15. For personal use only.

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