Abstract
Outline of Lectures
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1.
Physiological Background, work of Hodgkin and Huxley, Hodgkin-Huxley equations.
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2.
Mathematical problems for the Hodgkin-Huxley equations, simplified models.
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3.
Existence theorems for travelling waves.
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4.
Stability of travelling waves, multiple pulses.
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5.
Multi-cellular phenomena, relation to patterns seen in chemistry.
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6.
Discrete models of excitable media.
These lectures will largely be devoted to the study of mathematical problems originating in the work of A. L. Hodgkin and A. F. Huxley, who, in 1952, proposed a set of partial differential equations to describe the transmission of an electrical impulse by a nerve axon [2], [3]. This set of equations is probably the most important mathematical model in physiology, and yet for nearly twenty years few mathematicians were aware of its existence. Recently, however, a number of people have studied the Hodgkin-Huxley (HH) equations, and it has become clear that they lead to several interesting and fundamental mathematical problems. In some cases resolution of these problems may increase our understanding of message transmission in the nervous system. Further, it has developed that there is an intimate connection between nerve models and equations describing other phenomena, such as certain chemical reactions which can support time periodic two dimensional spatial patterns. It is suspected that similar processes may play a role in pattern formation in many other settings, but here I will concentrate almost entirely on developments related to neurobiology.
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References
References For Lecture I
Evans, J.W. and J.A. Feroe. Local Stability Theory of the Nerve Impulse. Math, Biosciences 37 (1977), 23–50.
Hodgkin, A.L. The Conduction of the Nervous Impulse. Ryerson, Toronto; Liverpool University, and Springfield, 111; Charles C. Thomas (1964).
Hodgkin, A.L. and A. F. Huxley. A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve, J. Physiology 117 (1952), 500–544.
4. Huxley, A.F. Ion Movement during Nerve Activity, Annals of the N.Y. Academy of Sciences 8l (1959), 215–510.
Katz, B. Nerve, Muscle, and Synapse. McGraw Hill, New York, 1966.
References For Lecture II
Carpenter, G., A geometric approach to singular perturbation problems with applications to nerve impulse equations. J. Diff. Eqns. 23 (1977), 335–367.
————, Periodic solutions of nerve impulse equations. J. Math. Anal. Appl. 58. (1977), 152–173.
————, Bursting phenomena in excitable membranes. SIAM J. Appl. Math., 1979.
Casten, R., H. Cohen, and P. Lagerstrom, Perturbation analysis of an approximation to the Hodgkin-Huxley theory. Quart. Appl. Math. 32. (1975), 365–402.
Chueh, K.N., C.C. Conley, and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations. Indian Univ. Math. J. 26 (1977), 373–392.
Evans, J., Nerve axon equations, I: linear approximations. Indiana Univ. Math. J., 21 (1972), 877–885.
————, Nerve axon equations, II: stability at rest. Ibid. 22 (1972), 75–90.
————, Nerve axon equations, III: stability of the nerve impulse. Ibid. 22 (1972), 577–594.
————, Nerve axon equations, IV: The stable and the unstable impulse. Ibid. 24 (1975), 1170–1190.
Feroe, J., Local existence of the nerve impulse. Math. Biosci. 38. (1978), 259–277.
FitzHugh, R., Mathematical models of excitation and propagation in nerve, in Biological Engineering, H.P. Schwan, ed., McGraw-Hill, N.Y. (1969).
Hassard, B.D., Singular traveling wave solutions of the Hodgkin-Huxley model nerve conduction equations, to appear.
Hastings, S. P., On travelling wave solutions of the Hodgkin-Huxley equations. Arch. Rat. Mech. Anal. 60 (1976), 229–257.
McKean, H.P., Nagumo's equations, Adv. in Math. 4 (1970), 209–223.
Nagurao, J., S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962), 2061–2070.
Rauch, J. and J. Smoller, Qualitative theory of FitzHugh-Nagumo equations. Adv. Math 22 (1978), 12–44.
Schonbek, M.E., Boundary value problems for the FitzHugh-Nagumo equations. Math. Res. Ctr. Report #1739, University of Wisconsin (Madison), 1977.
References For Lecture III
Carpenter, G., (See Reference 1, Lecture II).
Casten, R., H. Cohen, and P. Lagerstrom, (Lecture II, Reference 4).
Hastings, S. P., The Existence of Periodic Solutions to Nagumo's Equation. Quart. J. Math. (Oxford) 25 (1974), 369–378.
Hastings, S.P., On the Existence of Homoclinic and Periodic Orbits for the FitzHugh-Nagumo Equations. Quart. J. Math. (Oxford) 22 (1976), 123–134.
Nemytski, V. V. and V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, N. J. (1960).
References For Lecture IV
Aronson, D. and H. Weinberger, Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Propagation. Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, 446, Springer-Verlag, 1975, 5–49.
Carpenter, G., A Geometric Approach to Singular Perturbation Problems with Applications to Nerve Impulse Equations. J, Diff. Equs. 23 (1977), 335–367.
————, Periodic Solutions of Nerve Impulse Equations, J. Math. Anal. Appl. 58. (1977), 152–173.
————, Bursting Phenomena in Excitable Membranes. SIAM. J. Appl. Math, 1978.
Cooley, J., F. Dodge, and H. Cohen, Digital Computer Solutions for Excitable Membrane Models. J. Cell. Comp. Physiol. 66 (Suppl. 2), 1965), 99–108.
Evaits, J. Nerve Axon Equations: I. Linear Approximations. Indiana Univ. Math. J. 21 (1972), 677–885.
Evans, J., Nerve Axon Equations III. Stability of the Nerve Impulse. Ibid. 22 (1972), 577–593. See also IV, The Stable and Unstable Impulse, Ibid. 24 (1975), 1171–1190.
Evans, J. and N. Fenichel, to appear.
Evans, J. and J. Feroe, Local Stability Theory of the Nerve Impulse, Math. Biosci. 37 (1977), 23–50.
Feroe, J., Existence and Stability of Multiple Impulse Solutions of a Nerve Equation, to appear.
Fife, P. and J. B. McLeod, The Approach of Solutions of a Nonlinear Diffusion Equation to Travelling Wave Solutions. Bull. Am. Math. Soc. 81 (1975), 1076–1078.
Guttman, R., S. Lewis, and J. Rinzel, Control of Repetitive Firing in Squid Axon Membrane as a Model for a Neuron Oscillator, to appear.
Hassard, B. D., Bifurcation of periodic solutions of the Hodgkin Huxley model for the squid giant axon, J. Theo. Biol. 71 (1978), 401–419.
Rinzel, J. On Repetitive Activity in Nerve. Federation Proceedings, 1979.
Rinzel, J. and J. Keller, Travelling Wave Solutions of a Nerve Conduction Equation. Biophys. J.,13 (1973), 1313–1337.
Sabah, N. H. and R. Spangler, Repetitive Response of the Hodgkin-Huxley Model for the Squid Giant Axon. J. Theor. Bio. 29 (1970), 155–171.
Troy, W. C, Bifurcation Phenomena in Fitzhugh's Nerve Conduction Equations, J. Math. Anal, and Appl., 54 (1976), 678–690.
————, Oscillation Phenomena in the Hodgkin-Huxley Equations. Proc. Roy. Soc. Edinburgh 74 A (1975), 299–310.
References For Lecture V
Cowan, J. D., Mathematical Models of Large-Scale Nervous Activity, in Some Mathematical Questions in Biology V, Amer. Math. Soc., Providence, 1974. (This describes work of Wilson and Cowan.)
Field, R. and R. Noyes. Oscillations in Chemical Systems IV, J. Chem, Phys. 160, 1877–1884.
Field, R. and W. Troy, The Amplification before Decay of Perturbations around Stable Steady States in a Model of the Zhabotinskii Reaction, SIAM J. Appl. Math. 32. (1977), 306–322.
————, The Existence of Solitary Travelling Wave Solutions of a Model of the Belousov-Zhabotinskii Reaction. To appear in SIAM J. Appl. Math.
Hartline, H. K. and F. Ratliff, Inhibitary Interaction of Receptor Units in the Eye of Limulus, J. Gen. Physiol. 10 (1957), 357–376.
Knight, B. W., J. Toyoda and F. A. Dodge. A Quantitative Description of the Dynamics of Excitation and Inhibition in the Eye of Limulus, J. Gen. Physiol. 56 (1970), 421–437.
Leão, A.A.P., Spreading Depression of Activity in the Cerebral Cortex. J. Neurophysiol. 7 (1944), Pg, 359.
Shibata, M. and J. Bures. Reverberation of Cortical Spreading Depression along Closed-loop Pathways in Rat Cerebral Cortex. J. Neurophysiol. 315 (1972), pg. 381.
Troy, W. C. Threshold Phenomena in the Field-Noyes Model of the Belousov-Zhabotinskii Reaction. J. Math. Anal, and Appl. 58 (1977), 233–248.
Tuckwell, M. and R. Muira. A Mathematical Model for Spreading Cortical Depression, Biophys. J. 23 (1978), 257–276.
Zaikin, A. N. and A. M. Zhabotinskii, Nature 255 (1970), pg. 535.
Zhabotinskii, A. M. Dokl. Akad. Nauk SSR 157 (1964), pg. 392.
References For Lecture VI
Cohen, D.S., J. Neu and R. Rosales. Rotating Spiral Wave Solutions of Reaction - Diffusion Equations. To appear.
Greenberg, J. M. and S. P. Hastings, Spatial Patterns for Discrete Models of Diffusion in Excitable Media. SIAM J. Appl. Math. 34 (1978).
Greenberg, J. M., B. D. Hassard and S. P. Hastings, Pattern Formation and Periodic Structures in Systems Modeled by Reaction-Diffusion Equations. Bulletin Amer. Math. Soc. 84 (1978), 1296–1327.
Greenberg, J. M., C. Greene and S. P. Hastings. A Combinatorial Problem Arising in the Study of Reaction-Diffusion Equations. SIAM J. Appl. Math.. To appear.
Hastings, S. P., A Winding Number Principle for Some Semi-discrete Models of Excitable Media. To appear.
Wiener, N. and A. Rosenblueth. The Mathematical Formulation of the Problem of Conduction of Impulses in a Network of Connected Excitable Elements, Specifically in Cardiac Muscle. Arch. Inst. Cardiol. Mexico 16 (1946), 205–265.
Winfree, A. T., Rotating Solutions to Reaction/Diffusion Equations in Simply Connected Media. SIAM-AMS Proceedings 8, Am. Math. Soc. Providence, 1974.
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Hastings, S. (2010). Some Mathematical Problems Arising in Neurobiology. In: Iannelli, M. (eds) Mathematics of Biology. C.I.M.E. Summer Schools, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11069-6_4
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