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Inventory reference ISSN 1812-7231 Klin.inform.telemed. Volume 11, Issue 12, 2015, Pages 43–49

Author(s) V. P. Marceniuk, Z. V. Mayhruk

Institution(s) I. Ya. Horbachevsky Ternopil State Medical University, Ukraine

Article title Algorithms qualitative analysis Hodgkin–Huxley model axon activity

Abstract (resume)

Introduction. A model of electrical activity of the giant axon of the squid Hodgkin–Huxley.

Formulation of the problem. When the axon excitability studied by building this type of classification rules excitability calculated initial conditions. Systems of ordinary differential equations of mathematical biology are used as parameters constant speed and initial values.

The object of the study. Develop and justify bifurcation control method in electrophysiological Hodgkin–Huxley model based on the maximum principle, which is reduced to the classification rules and takes into account both speed constants and initial conditions.

Study results. The approach of qualitative analysis system Hodgkin–Huxley based multivariate method comprising sequential algorithm coating. The software environment is implemented as a package of Java-classes. It shows real example research model in the integrated software environment.

Conclusion. Qualitative analysis of the Hodgkin–Huxley model lets us design criteria for classification and prediction of the electrical excitability of nerve cells. These criteria can be expressed in terms of the structure of knowledge such as decision trees and classification rules. Multivariate method proposed in this paper is proved to program implementation. Stabilization control bifurcation in the Hodgkin–Huxley model may have important clinical application for patients suffering from Alzheimer’s disease, epilepsy, or irregular heartbeat.

Keywords Hodgkin–Huxley model; Qualitative analysis; Tree decision.


1. Hodgkin A. L., Huxley, A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiology, 1952, vol. 117, iss. 4, pp. 500–544.

2. Hassard B., Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J. Theoret. Biol., 1978, vol. 71, pp. 401–420.

3. Fukai H., Nomura T., Doi S. and Sato S. Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations, I, II, Biol. Cybern., 2000, vol. 82, pp. 215–222; 223–229.

4. Guckenheimer J. and Labouriau I. S., Bifurcation of the Hodgkin–Huxley equations: Anew twist. Bull. Math. Biol., 1993, vol. 55, pp. 937–952.

5. Hassard B. and Shiau L.-J. A special point of Z2-codimension three Hopf bifurcation in the Hodgkin-Huxley model. Appl. Math. Lett., 1996, vol. 9, pp. 31–34.

6. Labouriau S., Degenerate Hopf bifurcation and nerve impulse II, SIAM J. Math. Anal., 1989, vol. 20, pp. 1–12.

7. Mayhruk Z.V. Prohramna realizatsiya chyselnoho methodu optymalnoho keruvannya bifurkatsiyeyu v modeli Hodgkina–Haksli [Software implementation of numerical methods of optimal control bifurcation in the Hodgkin–Huxley model]. Visnyk Hmelnytskoho natsionalnoho universytetu [Bulletin of Khmelnitsky National University], 2014, iss. 1, pp. 186–194. (In Ukr.).

8. Martsenyk V. P., Mayhruk Z. V. Pobudova optymalnoho keruvannya bifurkatsiyeyu v modeli Hodzhkina–Haksli na osnovi pryntsypu maksymumu [Construction of optimal control bifurcation in the Hodgkin-Huxley model based on the maximum principle]. Matematichne ta komp’yuterne modelyuvannya Serіya: Tehnіchnі nauki [Mathematical and computer modeling. Series: Engineering], 2014, iss. 9, pp. 78–90. (In Ukr.).

9. Koch Y., Wolf T., Sorger P. K., Eils R., Brars B. Decision-Tree Based Model Analysis for Efficient Identification of Parameter Relations Leading to Different Signaling States. PLOS ONE, 2013, vol. 8, iss. 12, e82593.

10. Knayrer E., Nersett S., Vanner H. Reshenye obyknovennykh dyfferentsyal’nykh uravnenyy. Nezhestkye zadachy [Solution of ordinary differential equations. Non-rigid task], M., Myr. Publ., 1990. (In Ukr.).

11. Betts J. T. Practical Methods for Optimal Control Using Nonlinear Programming, SIAM Society for Applied and Industrial Mathematics, Philadelphia, 2001.

12. Von Stryk O., Bulirsch R. Direct and Indirect methods for trajectory optimization. Annals of Operations Research, 1992, vol. 37, pp. 357–373.

13. Bryson Jr. A. E., Ho Y.-C. Applied Optimal Control. John Wiley & Sons Publ., New York, 1975.

14. Fabien B. C. Some Tools for the Direct Solution of Optimal Control Problems. Advances in Engineering Software, 1998, vol. 29, pp. 45–61.

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