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].

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where r is a radius of the axon, r_a is a resistivity of the axoplasm, С_m is a capacitance per unit area of membrane, r_m 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  meeting next conditions:

We impose following conditions on the function V₀(t)   and denote where λ is a constant of the neuron fiber length, τ is a constant of signal attenuation. Then equation (1) takes a form We introduce dimensionless variables: where v₀ is a resting potential. In terms of dimensionless variables (9) the equation (8) takes a form Then we represent the solution of the equation (10) as a Fourier integral   In equation (11)  is a Fourier transform of the function  which satisfies the condition (5),    (12) The characteristic equation follows from the equations (10) and (11) We can find imaginary part of from the equation (14) considering condition (3). The result is: Similarly, we can find the real part of 

The equation (15) shows that  is an odd function of  and the equation (16) shows that  is an even function. We will have a solution of the equation (10) with  for input arbitrary excitation pulse  if we insert (14 – 16) to the (11).

Let , where  is delta function. We denote the corresponding solution of (17) as  and name it impulse response function. From (12) we have  then:

According to the Plancherel theorem, for arbitrary input excitation impulse we have:

   (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  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.

References

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