title: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð´ÐµÑÐ¾ÑÐ¼Ð°ÑÐ¸Ð¸ Ð¼ÐµÐ¼Ð±ÑÐ°Ð½Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð° Ð¿Ð¾Ð´ Ð´ÐµÐ¹ÑÑÐ²Ð¸ÐµÐ¼ ÑÐ»ÐµÐºÑÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¸Ð¼Ð¿ÑÐ»ÑÑÐ°
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Mathematical model of neuronal membrane mechanical deformation during electrical stimulation
Creators: name: ÐÐ¾ÑÐ¸ÐºÐ¾Ð²Ð°, ÐÐ°ÑÐ¸Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2021
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐÐµÐ¹ÑÐ¾Ð½Ñ
ÐÐµÑÐ¾ÑÐ¼Ð°ÑÐ¸Ñ
Ð­Ð»ÐµÐºÑÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¸Ð¼Ð¿ÑÐ»ÑÑÑ
Ð²Ð¾Ð»Ð½Ð° Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ñ
Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð» Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ
pressure wave
action potential

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸Ð·ÑÑÐµÐ½Ð¸Ñ ÑÐ²ÑÐ·Ð¸ Ð²Ð»Ð¸ÑÐ½Ð¸Ñ ÑÐ»ÐµÐºÑÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÑÐ¸Ð¼ÑÐ»ÑÑÐ¸ÐµÐ¹ Ð½Ð° Ð¼ÐµÑÐ°Ð½Ð¸ÑÐµÑÐºÐ¸Ðµ Ð´ÐµÑÐ¾ÑÐ¼Ð°ÑÐ¸Ð¸ Ð¸ Ð²Ð¾Ð»Ð½Ñ Ð´Ð°Ð²Ð»ÐµÐ½Ð¸Ñ Ð²Ð½ÑÑÑÐ¸ÐºÐ»ÐµÑÐ¾ÑÐ½Ð¾Ð¹ Ð¶Ð¸Ð´ÐºÐ¾ÑÑÐ¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð°. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐµÐ´Ð»Ð¾Ð¶ÐµÐ½Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ, Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÑÑÐ°Ñ Ð¸Ð·Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÐ¾ÑÐ¼Ñ Ð°ÐºÑÐ¾Ð½Ð° Ð¿ÑÐ¸ Ð¸Ð·Ð¼ÐµÐ½ÐµÐ½Ð¸Ð¸ Ð¼ÐµÐ¼Ð±ÑÐ°Ð½Ð½Ð¾Ð³Ð¾ Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð»Ð° Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ ÑÐ¿Ð»Ð¾ÑÐ½Ð¾Ð¹ ÑÑÐµÐ´Ñ, ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¸Ð¿Ð¿Ð¼Ð°Ð½Ð°, Ð° ÑÐ°ÐºÐ¶Ðµ Ð·Ð°ÐºÐ¾Ð½Ð¾Ð² ÐÐ°Ð¿Ð»Ð°ÑÐ° Ð¸ ÐÑÐºÐ°. Ð ÑÐ¾Ð´Ðµ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð¾ ÑÐµÑÐµÐ½Ð¾ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¾ Ð·Ð°Ð´Ð°Ñ: 1. ÐÐ»Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð°-ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐ° Ð²ÑÐ²ÐµÐ´ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ 1D Ð´ÐµÑÐ¾ÑÐ¼Ð°ÑÐ¸Ð¸ Ð² ÑÑÐ»Ð¾Ð²Ð¸ÑÑ Ð½ÐµÑÑÐ°ÑÐ¸Ð¾Ð½Ð°ÑÐ½Ð¾Ð¹ ÑÐ¿ÑÑÐ³Ð¾ÑÑÐ¸ Ð¼ÐµÐ¼Ð±ÑÐ°Ð½Ñ; 2. ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð° Ð°Ð½Ð°Ð»Ð¾Ð³Ð¸Ñ Ñ Ð·Ð°Ð´Ð°ÑÐµÐ¹ Ð¾ Ð³Ð¸Ð´ÑÐ°Ð²Ð»Ð¸ÑÐµÑÐºÐ¾Ð¼ ÑÐ´Ð°ÑÐµ. ÐÐ¾Ð»ÑÑÐµÐ½Ð° Ð¾ÑÐµÐ½ÐºÐ° ÑÐºÐ¾ÑÐ¾ÑÑÐ¸ ÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð¸Ñ Ð²Ð¾Ð·Ð¼ÑÑÐµÐ½Ð¸Ð¹ Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¾ ÐµÐµ ÑÐ¾Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ Ñ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ°Ð»ÑÐ½ÑÐ¼Ð¸ Ð´Ð°Ð½Ð½ÑÐ¼Ð¸; 3. Ð¡ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð° Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð° ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð°Ñ ÑÑÐµÐ¼Ð° ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹. Ð¡Ð¾Ð·Ð´Ð°Ð½Ð° ÑÐ´Ð¾Ð±Ð½Ð°Ñ Ð´Ð»Ñ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¸Ñ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð² Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð° Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ; 4. ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð¼ ÑÐ°Ð±Ð¾ÑÑ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ñ Ð¸Ð·Ð²ÐµÑÑÐ½ÑÐ¼Ð¸ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ°Ð»ÑÐ½ÑÐ¼Ð¸ Ð´Ð°Ð½Ð½ÑÐ¼Ð¸. Ð ÑÐ°Ð±Ð¾ÑÐµ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð¾ ÑÑÐ¸ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð·Ð°Ð´Ð°ÑÐ¸: Ð¾ ÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð¸Ð¸ Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð»Ð° Ð´ÐµÐ¹ÑÑÐ²Ð¸Ñ Ð²Ð´Ð¾Ð»Ñ Ð°ÐºÑÐ¾Ð½Ð°, Ð¾ ÑÑÑÐ¿ÐµÐ½ÑÐ°ÑÐ¾Ð¼ Ð¸Ð·Ð¼ÐµÐ½ÐµÐ½Ð¸Ð¸ Ð¼ÐµÐ¼Ð±ÑÐ°Ð½Ð½Ð¾Ð³Ð¾ Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð»Ð°, Ð¸ Ð¾ Ð¼ÐµÑÐ°Ð½Ð¸ÑÐµÑÐºÐ¾Ð¼ Ð²Ð¾Ð·Ð¼ÑÑÐµÐ½Ð¸Ð¸ Ð°ÐºÑÐ¾Ð½Ð°. ÐÐ»Ñ Ð²ÑÐµÑ ÑÑÐµÑ Ð·Ð°Ð´Ð°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ ÐºÐ°ÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾ Ð¸ ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾ ÑÐ¾Ð²Ð¿Ð°Ð´Ð°ÐµÑ Ñ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ°Ð»ÑÐ½ÑÐ¼Ð¸ Ð´Ð°Ð½Ð½ÑÐ¼Ð¸.
description: This work is devoted to the study of the effect of electrical stimulation on mechanical deformations and pressure waves in neuronal intracellular fluid. Mathematical model describing axonal shape deformations accompanying the membrane potential changes is based on the equations of dynamics of a continuous medium, the Lippmann equation, as well as Laplaceâs and Hookeâs laws. In the work, several tasks have been solved: 1. For a neuron-cylinder, the equation of 1-D deformation under the conditions of nonstationary elasticity of the membrane has been derived; 2. The 1-D deformation problem has been found to be analogous to that about hydraulic pressure jump. The speed of propagation of the disturbances has been estimated and compared with experimental data; 3. A numerical scheme for solving equations has been formulated and implemented. A simulation program convenient for numerical experiments has been created; 4. A comparison has been made between the model and known experimental data. We considered three different tasks: the propagation of an action potential along an axon, the stepwise change of the membrane potential and the mechanical disturbance of the axon. For all three tasks, the model qualitatively and quantitatively matches the experimental data.