Dynamical Systems in Neuroscience:

Bifurcations 217Figure 6.48: Richard FitzHugh with analog computer, National Institute of Health,Bethesda, Maryland, ca. 1960 (photograph provided by R. FitzHugh in 2005).early as 1955, when Richard FitzHugh concluded his paper on mathematical modelingof threshold phenomena saying that many neuronal properties“... are invariant under continuous, one-to-one transformations of the coordinates ofphase space and fall within the domain of topology, a branch of mathematics whichmay be intrinsically better fitted for the preliminary description and classification ofbiological systems than analysis, which includes differential equations. This suggestionis of little practical value at present, since too little is known of the topology ofvector fields in many-dimensional spaces, at least to those interested in theoreticalbiology. Nevertheless, the most logical procedure in the description of a complexbiological system might be to characterize the topology of its phase space, then toestablish a set of physically identifiable coordinates in the space, and finally to fitdifferential equations to the trajectories, instead of trying to reach this final goal atone leap.”It is remarkable that FitzHugh was explicitly talking about topological equivalence andbifurcations, though never called them such, years before these mathematical notionswere firmly established. This book continues the line of research initiated by FitzHughand further developed by Rinzel and Ermentrout (1989).In this chapter we provided a fairly detailed exposition of bifurcation theory. Whatwe covered should be sufficient not only for understanding the rest of the book, butalso for navigating through bifurcation papers concerned with computational neuro-