On mathematical modelling of solitary pulses in cylindrical biomembranes

Abstract

The propagation of action potentials in nerve fibres is usually described by models based on the ionic hypotheses. However, this hypothesis does not provide explanation of other experimentally verified phenomena like the swelling of fibres and heat production during the nerve pulse propagation. Heimburg and Jackson (Proc Natl Acad Sci USA 102(28):9790-9795, 2005, Biophys Rev Lett 2:57-78, 2007) have proposed a model describing the swelling of fibres like a mechanical wave related to changes of longitudinal compressibility of the cylindrical membrane. In this paper, the possible dispersive effects in such microstructured cylinders are analysed from the viewpoint of solid mechanics, particularly using the information from the analysis of the well-known rod models. A more general governing equation is proposed which satisfies the conditions imposed by the physics of wave processes. The numerical simulations demonstrate the influence of nonlinearities, the role of various dispersion terms and the formation and propagation of solitary waves along the wall together with the corresponding transverse displacement. It is conjectured that due to the coupling effects between longitudinal and transverse displacements of a cylinder, the transverse displacement (i.e. swelling) is related to the derivative of the longitudinal displacement. In this way, the correspondence between theoretical and experimental (Tasaki in Physiol Chem Phys Med NMR 20:251-268, 1988) results can be described.

Temperature changes accompanying signal propagation in axons

“…1. It should be noted that some characteristics normally observed in experiments, like the transverse displacement (TW) of the membrane is not described by a differential equation but can be derived directly from the calculated LW as a derivative [6]. Certain oscillations of the calculated TW reflect the physical properties of the biomembrane which is described by a Boussinesq-type equation [6].…”

“…The coupled system of model equations has been previously proposed by the authors for describing the nerve pulse (action potential (AP)) propagation including the accompanying mechanical effects (pressure wave (PW) and longitudinal density change in the lipide bi-layer (LW)) [4][5][6][7]14]. In this paper, the temperature effects which accompany the propagation of an AP in the axoplasm, are also described.…”

“…In this paper, the temperature effects which accompany the propagation of an AP in the axoplasm, are also described. The dimensionless 1D model used so far [4][5][6][7]14] is the following:…”

On solitary waves in case of amplitude-dependent nonlinearity

“…is the speed of the high frequency disturbances in the system [7]. Here x is the space coordinate and t is the time.…”

“…(1) has one fundamental problem -it allows some disturbances to travel at infinite speeds, which is undesirable. For that reason the original HJ model has been modified as described in [7]. In a nutshell the idea for the improvement of the Heimburg-Jackson (HJ) model [11,12] is to add a second dispersion term for causality.…”