Bridging Between Experiments and Equations: A Tutorial on Modeling Excitability

The goal of this chapter is to empower collaboration across the disciplines. It is aimed at mathematical scientists whowant to better understand neural excitability and experimentalists who want to better understand mathematical modeling and analysis. None of us need to be expert in both disciplines, but each side needs to learn the other’s language before our conversations can spark the exciting new collaborations that enrich both disciplines. Learning is an active process:

[1]  B. Katz,et al.  The electrical properties of crustacean muscle fibres , 1953, The Journal of physiology.

[2]  J. Nerbonne,et al.  IA Channels , 2014, The Neuroscientist : a review journal bringing neurobiology, neurology and psychiatry.

[3]  Daniela Fischer.  Differential Equations Dynamical Systems And An Introduction To Chaos , 2016 .

[4]  R. Latorre,et al.  K(+) channels: function-structural overview. , 2012, Comprehensive Physiology.

[5]  D. P. McCobb,et al.  Pituitary Control of BK Potassium Channel Function and Intrinsic Firing Properties of Adrenal Chromaffin Cells , 2001, The Journal of Neuroscience.

[6]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990 .

[7]  Richard Bertram,et al.  Ion channels and signaling in the pituitary gland. , 2010, Endocrine reviews.

[8]  B. Bean.  The action potential in mammalian central neurons , 2007, Nature Reviews Neuroscience.

[9]  J. Trimmer,et al.  Diverse roles for auxiliary subunits in phosphorylation-dependent regulation of mammalian brain voltage-gated potassium channels , 2011, Pflügers Archiv - European Journal of Physiology.

[10]  D. P. McCobb,et al.  Acute modulation of adrenal chromaffin cell BK channel gating and cell excitability by glucocorticoids. , 2004, Journal of neurophysiology.

[11]  R. Bertram,et al.  Topological and phenomenological classification of bursting oscillations. , 1995, Bulletin of mathematical biology.

[12]  R. Bertram,et al.  Topological and phenomenological classification of bursting oscillations , 1995 .

[13]  J. NAGUMOt,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 2006 .

[14]  A. Dolphin.  Calcium channel diversity: multiple roles of calcium channel subunits , 2009, Current Opinion in Neurobiology.

[15]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[16]  D. P. McCobb,et al.  Bovine versus rat adrenal chromaffin cells: big differences in BK potassium channel properties. , 2000, Journal of neurophysiology.

[17]  M. Zeeman,et al.  β2 and β4 Subunits of BK Channels Confer Differential Sensitivity to Acute Modulation by Steroid Hormones , 2006 .

[18]  T. Jegla,et al.  Evolution of the human ion channel set. , 2009, Combinatorial chemistry & high throughput screening.

[19]  R. Keynes,et al.  Calcium and potassium systems of a giant barnacle muscle fibre under membrane potential control , 1973, The Journal of physiology.

[20]  O. Pongs,et al.  Ancillary subunits associated with voltage-dependent K+ channels. , 2010, Physiological reviews.

[21]  Vivien A. Casagrande,et al.  Biophysics of Computation: Information Processing in Single Neurons , 1999 .

[22]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[23]  B. Rudy,et al.  Molecular Diversity of K+ Channels , 1999, Annals of the New York Academy of Sciences.

[24]  Arthur Sherman,et al.  Dynamical systems theory in physiology , 2011, The Journal of general physiology.

[25]  A. Hodgkin.  The local electric changes associated with repetitive action in a non‐medullated axon , 1948, The Journal of physiology.

[26]  S. Terakawa,et al.  Fenestration nodes and the wide submyelinic space form the basis for the unusually fast impulse conduction of shrimp myelinated axons. , 1999, The Journal of experimental biology.

[27]  Martin Golubitsky,et al.  An unfolding theory approach to bursting in fast–slow systems , 2001 .

[28]  M. Shipston,et al.  Characterization of hyperpolarization-activated cation currents in mouse anterior pituitary, AtT20 D16:16 corticotropes. , 2000, Endocrinology.

[29]  H. Zakon.  Adaptive evolution of voltage-gated sodium channels: The first 800 million years , 2012, Proceedings of the National Academy of Sciences.

[30]  T. W. Barrett,et al.  Catastrophe Theory, Selected Papers 1972-1977 , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[31]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .

[32]  D. P. McCobb,et al.  Control of hypothalamic–pituitary–adrenal stress axis activity by the intermediate conductance calcium-activated potassium channel, SK4 , 2011, The Journal of physiology.

[33]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[34]  David Terman,et al.  Mathematical foundations of neuroscience , 2010 .

[35]  James P. Keener,et al.  Comprar Mathematical Physiology · I: Cellular Physiology | Keener, James | 9780387758466 | Springer , 2009 .

[36]  Leah Edelstein-Keshet,et al.  Mathematical models in biology , 2005, Classics in applied mathematics.

[37]  J. Ruppersberg.  Ion Channels in Excitable Membranes , 1996 .

[38]  Michel Crucifix,et al.  Oscillators and relaxation phenomena in Pleistocene climate theory , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[39]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[40]  Y. Jan,et al.  Voltage‐gated potassium channels and the diversity of electrical signalling , 2012, The Journal of physiology.

[41]  D. Lipscombe,et al.  Alternative splicing: functional diversity among voltage-gated calcium channels and behavioral consequences. , 2013, Biochimica et biophysica acta.