MATCHING DENDRITIC NEURON MODELS TO EXPERIMENTAL-DATA

We are very pleased to contribute to this special issue commemorating the pioneering research contributions of Hodgkin, Huxley, and Katz. All of us have been greatly influenced and encouraged by their example of using theory to design experiments that yield values for key model parameters. W. Rall was privileged to get an early glimpse of this research on the special occasion when Alan Hodgkin visited K. S. Cole at the University of Chicago, in the spring of 1948, and presented a biophysics seminar about research done at Plymouth, in September 1947, with Bernard Katz (51, 52; see Ref. 16, p. 267, and Ref. 49, p. 17). Later, W. Rall and J. J. B. Jack were in Dunedin, New Zealand, on the special occasion when J. C. Eccles reported the exciting new results and interpretations of Hodgkin, Huxley, and Katz that he had learned from attending a Cold Spring Harbor Symposium on Quantitative Biology (June, 1952). Thus the classic series of papers, published in the Journal of Physiology (London) in 1952, did not come as a surprise to us in Dunedin; it came rather as an impressive fulfillment of expectations already raised. The concept of separating the ionic fluxes and treating them as electric currents governed by a time-varying conductance and a time-varying difference of potential (membrane potential minus ionic reversal potential) was of critical importance. It facilitated both qualitative thinking about membrane phenomena and quantitative treatment of experimental data. This concept also was central to thinking about neuromuscular junctions (33) and synapses (19) and to the modeling of synaptic conductance transients in dendrites (97, 99). The subsequent realization of ionic conductance in terms of specific ion channels is a remarkable development that is covered in other reviews in this commemorative issue. The explicit mathematical model presented by Hodgkin and Huxley (50) provided a giant step forward in capturing the nonlinear properties of nerve membrane. The effort to increase functional insight into this class of nonlinear dynamical systems was significantly advanced by FitzHugh (35) with his introduction of a nonlinear two-variable system (cubic BVP), which captured the essential qualitative nonlinear dynamics of nerve membrane. By studying this system in a two-dimensional phase space, FitzHugh was able to distinguish different physiological states: the resting point, the active region, the refractory regions (both absolute and relative), and regions of depressed and enhanced excitability, as well as "no man's land" (35). Many valuable insights related to the physiological states of classic physiology were communicated in this paper and in an outstanding review chapter (36). Fruitful study of this reduced system (now known as the FitzHugh-Nagumo equations) has continued. For example, by using a piecewise-linear approximation to the cubic nonlinearity, Rinzel and Keller (116) succeeded in solving this system of equations explicitly, providing pulse and periodic solutions together with a rigorous analysis of the stability of these solutions. Insights gained by using the phase-plane approach of the qualitative theory of differential equations can be found in two reviews (115, 116). In addition, the impact of the Hodgkin-Huxley equations on mathematical biology and applied mathematics has been reviewed recently (114).