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ISSN 2227-6017 (ONLINE), ISSN 2303-9868 (PRINT), DOI: 10.18454/IRJ.2227-6017
ПИ № ФС 77 - 51217, 16+

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Понетаева Е. Г. ФУНКЦИЯ ИМПУЛЬСНОГО ОТКЛИКА ПОТЕНЦИАЛА ДЕЙСТВИЯ В НЕРВНОМ ВОЛОКНЕ / Е. Г. Понетаева, Н. М. Богатов, Л. Р. Григорьян и др. // Международный научно-исследовательский журнал. — 2019. — №2 (21) Часть 1. — С. 5—8. — URL: https://research-journal.org/physics-mathematics/impulse-response-function-of-the-action-potential-in-nerve-fiber/ (дата обращения: 13.12.2019. ).
Понетаева Е. Г. ФУНКЦИЯ ИМПУЛЬСНОГО ОТКЛИКА ПОТЕНЦИАЛА ДЕЙСТВИЯ В НЕРВНОМ ВОЛОКНЕ / Е. Г. Понетаева, Н. М. Богатов, Л. Р. Григорьян и др. // Международный научно-исследовательский журнал. — 2019. — №2 (21) Часть 1. — С. 5—8.

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ФУНКЦИЯ ИМПУЛЬСНОГО ОТКЛИКА ПОТЕНЦИАЛА ДЕЙСТВИЯ В НЕРВНОМ ВОЛОКНЕ

Понетаева Е.Г.1, Богатов Н.М.2, Григорьян Л.Р.3, Богатов Н.Е.4, Богатов Е.М.5, Щербаков А.С.6

1Аспирант, Кубанский государственный университет; 2Доктор физико-математических наук, профессор, Кубанский государственный университет; 3Кандидат физико-математических наук, доцент, Кубанский государственный университет; 4Магистрант, Кубанский государственный университет; 5 Магистрант, Кубанский государственный университет; 6 Магистрант, Кубанский государственный университет

ФУНКЦИЯ ИМПУЛЬСНОГО ОТКЛИКА ПОТЕНЦИАЛА ДЕЙСТВИЯ В НЕРВНОМ ВОЛОКНЕ

Аннотация

Проанализировано поведение импульсного сигнала при его распространении в нервном волокне. Сигнал бесконечно малой длительности на входе аксона имеет вид асимметричного импульса в пространственном и временном сечениях аксона. Максимум импульса уменьшается, а ширина увеличивается с увеличением пространственной координаты и времени.

Ключевые слова: нервное волокно, импульсный сигнал, функция импульсного отклика.

Ponetaeva E.G.1, Bogatov N.M.2, Grigoryan L.R.3, Bogatov N.E.4, Bogatov E.M.5, Csherbakov A.S.6

1Postgraduate student, Kuban State University;  2Doctor in Physics and mathematics, Professor, Kuban State University; 3PhD in Physics and mathematics, associate professor, Kuban State University; 4Master student, Kuban State University; 5Master student, Kuban State University; 6Master student, Kuban State University

IMPULSE RESPONSE FUNCTION OF THE ACTION POTENTIAL IN NERVE FIBER

Abstract

Behavior of the impulse signal was analyzed during the propagation process in the nerve fiber. An infinitesimal duration signal which is used as an axon input value has a view of asymmetrical impulse in space and time cross section. The maximum of pulse decreases and width of signal increases when coordinate and time increase.

Keywords: nerve fiber, impulse signal, impulse response function.

Regular functioning of the living organism is impossible without the exchange of information between its subsystems. One of the ways of information transfer is propagation of electrical impulses in the nerve fiber. Electric character of nerve pulses is proved in researches done by Hodgkin A.L., Huxley A.F. and others [1-3].

Modeling of bioelectrical effects is widely used in modern electrophysiology to study the processes occurring in living electroexcitable structures [4-7]. Soliton model of nerve fiber transmembrane potential change occurring in process of excitation propagation was developed in works [8-11]. Accurate analytical solution of nerve impulse propagation within Hodgkin-Huxley’s model, based on the integral Laplace transformation and Efros theorem when the input excitation pulse deviates from the Heaviside step function was obtained in [12].

Simulated results are generally consistent with the experimental data. Applied problems, e.g. bioprosthetics, require a solution of a general problem: an analysis of changes of arbitrary signals in nerve fiber. The problem of action potential propagation in nerve fiber for arbitrary input pulses was successfully solved in [13].

The aim of this research is to calculate the impulse response function of the action potential in nerve fiber.

Action potential is the change of the membrane potential between intracellular medium and extracellular substance, which moves with nerve signal propagation, when nerve cells are excited. Equation (1) described an action potential, V(x,t),  propagation in nerve fiber [1].

Error! Objects cannot be created from editing field codes.,                                                     (1)

where r is a radius of the axon, ra is a resistivity of the axoplasm, Сm is a capacitance per unit area of membrane, rm is a resistivity of membrane material, l is a membrane thickness, the action potential is measured from the resting potential.

We search for the solution of the equation (1) when 19-08-2019 13-38-04 meeting next conditions:

19-08-2019 13-40-24

We impose following conditions on the function V0(t)

19-08-2019 13-40-33

 

and denote

19-08-2019 13-40-54

where λ is a constant of the neuron fiber length, τ is a constant of signal attenuation.

Then equation (1) takes a form

19-08-2019 13-43-30

We introduce dimensionless variables:

19-08-2019 13-43-39

where v0 is a resting potential.

In terms of dimensionless variables (9) the equation (8) takes a form

19-08-2019 13-43-51

Then we represent the solution of the equation (10) as a Fourier integral

 

19-08-2019 13-44-05

In equation (11) 19-08-2019 13-47-27  is a Fourier transform of the function 19-08-2019 13-47-34 which satisfies the condition (5),

19-08-2019 13-52-03   (12)

The characteristic equation follows from the equations (10) and (11)

19-08-2019 13-52-32

We can find imaginary part of 19-08-2019 13-54-12 from the equation (14) considering condition (3). The result is:

19-08-2019 13-54-27

Similarly, we can find the real part of 19-08-2019 13-54-12

19-08-2019 13-55-02

The equation (15) shows that 19-08-2019 14-01-50  is an odd function of 19-08-2019 14-01-58 and the equation (16) shows that 19-08-2019 14-02-04 is an even function. We will have a solution of the equation (10) with 19-08-2019 14-02-12 for input arbitrary excitation pulse 19-08-2019 14-02-23 if we insert (14 – 16) to the (11).

19-08-2019 13-59-55

Let 19-08-2019 14-14-04, where 19-08-2019 14-14-10 is delta function. We denote the corresponding solution of (17) as 19-08-2019 14-14-18 and name it impulse response function. From (12) we have 19-08-2019 14-14-28 then:

19-08-2019 14-00-24

According to the Plancherel theorem, for arbitrary input excitation impulse 19-08-2019 14-16-34 we have:

19-08-2019 14-16-43   (19)

Fig. 1 – 4 show the function  changes with different argument values. The function for x’ = const is shown in Fig. 1 and Fig. 2. The function for t’ = const is shown in Fig. 3 and Fig. 4.

Behavior of the impulse signal was analyzed during the propagation process in the nerve fiber. An infinitesimal duration signal 19-08-2019 14-19-42 which is used as an axon input value has a view of asymmetrical impulse in space and time cross section. The maximum of pulse decreases and width of signal increases when x’  and t’ increase.

19-08-2019 14-21-09

19-08-2019 14-21-20

19-08-2019 14-21-33

19-08-2019 14-21-46

References

  1. Hodgkin A. L., Rushton W. A. H. The electrical constants of a crustacean nerve fiber // Proc. Roy. Soc. London. 1946. Ser B. V. 133. P. 444-479.
  2. Hodgkin A. L., Huxley A. F. A quantative description of membrane current and its application to conduction and excitation in nerve // J. Physiol. (London). 1952. V. 117. N4. P. 500-544.
  3. Hodgkin A.L. Nerve impulse. М.: Mir, 1965. 128 с. (Rus)
  4. Frankenhaeuser В., Huxley A. The action potential in the myelinated nerve fibre of Xenopus Laevis as computed on the basis of voltage clamp data // J.Physiol. 1964. N P. 302-315.
  5. Fomin S.V., Berkinblit M.B. Mathematical problems in biology. М.: Nauka, 1973. 200 с. (Rus)
  6. Abbott L. F., Kepler Т. B. Model neurons: from Hodgkin-Huxley to Hopfield // Statistical Mechanics of Neural Networks, L. Garrido, ed., no. 368 in Lecture notes in Physics, Springer-Verlag, 1990. P. 5-18.
  7. Dominique Debanne. Information processing in the axon // Nature Reviews Neuroscience. V.5. N4. P. 304-316.
  8. Maksimenko E.V. The analytical model of the nerve impulse // Obozrenie prikladnoy i promishlennoy matematiki. Т. 10. Vol.3. pp. 696-697. (Rus)
  9. Maksimenko E.V. The Korteweg-de Vries equation using for transmembrane potential modeling in nerve fiber // Vestnik Severo-Kavkazskogo gosudarstvennogo tehnicheskogo universiteta, seriya «Estestvennonauchnaya». №1(7). pp. 234-235. (Rus)
  10. Maksimenko E.V. The nerve impulse propagation modeling with computer using // Obozrenie prikladnoy i promishlennoy matematiki. Т. 11. Vol.2. pp.368-369. (Rus)
  11. Maksimenko E.V. About mathematical methods using in biological researches // Obozrenie prikladnoy i promishlennoy matematiki. Т. 12. Vol. 2. pp. 431-432. (Rus)
  12. Selezov I.Т., Morozova L.V. Generalization of the action potential excitation and propagation in nerve fiber problem // Prikladna gidrimehanika. Т. 12. N. 3. pp. 75-83. (Ua)
  13. Bogatov N.М., Grigoryan L.R., Ponetaeva E.G. The modeling of electrical impulses propagation in nerve fiber // Collective monograph. Sovremennie problemy fiziki, biofiziki i infokommunikacionnyh tehnologiy. Krasnodar: Krasnodarskiy CNTI, 2012. pp. 33 – 44. (Rus)

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