Dynamical Systems in Neuroscience:

Electrophysiology of Neurons 51Bibliographical NotesOur summary of the membrane electrophysiology is limited: We present only those conceptsthat are necessary to understand the Hodgkin-Huxley description of generation ofaction potentials. We have omitted such important topics as Goldman-Hodgkin-Katzequation, cable theory, dendritic and synaptic function, etc., although some of thosewill be introduced later in the book.The standard textbook on membrane electrophysiology is the second edition ofIon Channels of Excitable Membranes by B. Hille (2001). An excellent introductorytextbook with an emphasis on the quantitative approach is Foundations of CellularNeurophysiology by D. Johnston and S. Wu (1995). A detailed introduction into mathematicalaspects of cellular biophysics can be found in Mathematical Physiology byJ. Keener and J. Sneyd (1998). The latter two books complement rather than repeateach other. Biophysics of Computation by Koch (1999) and Chapters 5 and 6 of TheoreticalNeuroscience by Dayan and Abbott (2001) provide a good introduction intobiophysics of excitable membranes.The first book devoted exclusively to dendrites is Dendrites by Stuart et al. (1999).It emphasizes the active nature of dendritic dynamics. Arshavsky et al. (1971; Russianlanguage edition - 1969) make the first and probably still the best theoretical attempt tounderstand the neuro-computational properties of branching dendritic trees endowedwith voltage-gated channels and capable of generating action potentials. Had theypublished their results in the 90s, they would have been considered classics in the field;Unfortunately, the computational neuroscience community of the 70s was not ready toaccept the “heretic” idea that dendrites can fire spikes, that spikes can propagate backand forward along the dendritic tree, that EPSPs can be scaled-up with distance, thatindividual dendritic branches can perform coincidence detection and branching pointscan perform non-linear summation, and that different and independent computationscan be carried out at different parts of the neuronal dendritic tree. We touch some ofthese issues in Chap. 8.Exercises1. Determine Nernst equilibrium potentials for the membrane of the squid giantaxon using the following dataand T = 20 ◦ C.Inside (mM) Outside (mM)K + 430 20Na + 50 440Cl − 65 5602. Show that a non-selective cation currentI = ḡ Na p (V − E Na ) + ḡ K p (V − E K )