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Physics shapes signals in nerves

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Abstract

In this short review, the importance of physics in understanding the signal propagation in nerve fibres is discussed. The main carrier of information is the action potential, but it is accompanied by mechanical and thermal effects. A possible model governing the ensemble of waves takes the basic laws of physics into account and describes the interactions between the dynamical effects. Such a model needs a corresponding mathematical formulation and, in this way, the interdisciplinarity combining electrophysiology, physics, and mathematics helps to understand the complex process of signal propagation in nerves measured experimentally.

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Acknowledgements

This research was supported by the Estonian Research Council under project PRG 1227. The authors thank the reviewer for suggestions on how to improve the manuscript. Jüri Engelbrecht acknowledges the support from the Estonian Academy of Sciences.

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Correspondence to Tanel Peets.

Appendix A: The mathematical model

Appendix A: The mathematical model

The general model for coupled nerve signal propagation in unmyelinated nerve axon is described in Section 4 and depicted in Fig. 2. Here, we briefly present the mathematical details of the model in dimensionless form [26].

The AP is governed by the FitzHugh-Nagumo (FHN) model [33]:

$$\begin{aligned} \begin{aligned} Z_{T}&= D Z_{XX} - J + Z \left( Z - C_1 - Z^2 + C_1 Z \right) ,\\ J_{T}&= \epsilon _1 \left( C_2 Z - J \right) . \end{aligned} \end{aligned}$$
(A1)

Here, Z is the AP, J is the ion current, D is a coefficient, \(\epsilon \) is the time-scale difference parameter and \(C_i=a_i+b_i\), where \(a_i\) is the ‘electrical’ activation coefficient and \(b_i=-\beta _i U\) is the ‘mechanical’ activation constant; \(\beta _i\) are coupling coefficients. Here and further indices T and X denote partial derivates against dimensionless time and space, respectively.

The pressure wave PW is governed by a modified wave equation (the wave equation with viscous and coupling terms):

$$\begin{aligned} P_{TT} = c_{2}^{2} P_{XX} - \mu _2 P_T + F_2(Z,J), \end{aligned}$$
(A2)

where P is the pressure, \(c^2\) is the characteristic velocity in fluid, \(\mu _2\) is the viscous coefficient and \(F_2\) models the influence form the AP and TW.

The longitudinal wave (LW) in the biomembrane is governed by the improved HJ (iHJ) model [46, 47]:

$$\begin{aligned} \begin{aligned} U_{TT}&= c_{3}^{2} U_{XX} + N U U_{XX} + M U^2 U_{XX} + N U_{X}^{2} + 2 M U U_{X}^{2}\\&\quad- H_1 U_{XXXX} + H_2 U_{XXTT} - \mu _3 U_T + F_3(Z,J,P), \end{aligned} \end{aligned}$$
(A3)

where \(U=\Delta \rho \) is the longitudinal density change, \(c_3\) is the velocity of sound in unperturbed state, NM are nonlinear and \(H_i\) are dispersion coefficients, respectively, \(\mu _3\) is a viscosity/friction coefficient, and \(F_3\) models the influence from the AP and PW.

The transverse displacement TW of the biomembrane is calculated from the LW as \(W \propto U_{X}\) (drawing inspiration from the theory of rods) [32, 48]:

$$\begin{aligned} W=KU_X, \end{aligned}$$
(A4)

where K is a coefficient.

The temperature \(\Theta \) is governed by the classical heat equation with a coupling term:

$$\begin{aligned} \Theta _{T} = \alpha \Theta _{XX} + F_4(Z,J,U,P), \end{aligned}$$
(A5)

where \(\Theta \) is the temperature, \(\alpha \) is thermal coefficient, and \(F_4\) models the influence form the AP, LW and PW.

Coupling forces following the ideas presented in [24, 49] are the following:

$$\begin{aligned} F_2 = \eta _1 Z_X + \eta _2 J_T + \eta _3 Z_T, \end{aligned}$$
(A6)
$$\begin{aligned} F_3 = \gamma _1 P_T + \gamma _2 J_T - \gamma _3 Z_T, \end{aligned}$$
(A7)
$$\begin{aligned} F_4 = \tau _{11} Z^{2} + \tau _2 \left( P_T + \varphi _2(P) \right) + \tau _3 \left( U_T + \varphi _3(U) \right) - \tau _4 \Omega , \end{aligned}$$
(A8)

where \(\eta _i\), \(\gamma _i\) and \(\tau _i\) are coefficients. Following the formalism of internal variables \(\Omega \) is determined either from

$$\begin{aligned} \Omega _T + \epsilon _4 \Omega = \zeta J, \end{aligned}$$
(A9)

or

$$\begin{aligned} \Omega _T = \varphi _4(J) - \frac{\Omega -\Omega _0}{\tau _\Omega }, \quad \Omega _0 = 0, \quad \tau _\Omega = \frac{1}{\epsilon _4}, \end{aligned}$$
(A10)

where

$$\begin{aligned} \varphi _2(P) = \lambda _2 \int P_T\, \mathrm {d} T, \quad \varphi _3(U) = \lambda _3 \int U_T\, \mathrm {d} T, \quad \varphi _4(J) = \zeta \int J \, \mathrm {d} T \end{aligned}$$
(A11)

and \(\epsilon _4\), \(\tau _\Omega \), \(\lambda _i\) and \(\zeta \) are coefficients.

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Engelbrecht, J., Tamm, K. & Peets, T. Physics shapes signals in nerves. Eur. Phys. J. Plus 137, 696 (2022). https://doi.org/10.1140/epjp/s13360-022-02883-5

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