Biological systems are characterized by many interwoven processes over multiple scales where interactions between the various phenomena play a decisive role. These phenomena can be chemical, electrical, and/or mechanical, all embedded into a whole. This paper discusses the propagation of signals in nerve fibres, which exhibits clear signs of complexity: electrical signals (action potentials) are coupled to mechanical waves in the internal axoplasmic fluid and in the surrounding biomembrane. Here the underlying microstructure affects strongly the processes: the existence of ion currents changes the balance of ions within fibres, the opening of ion channels in the surrounding biomembrane is a crucial process, and the biomembrane itself has a microstructure composed of lipids. The whole process is governed by interactions, and the analysis of single processes has demonstrated the importance of nonlinearities. The main challenge is to build up a general model where the coupling of all related phenomena is taken into account. It is proposed that three processes – the propagation of an action potential and mechanical waves in the biomembrane and in the axoplasmatic fluid – be united into a general model with additional interaction forces for reflecting coupling. Such a model results in the emerging of a mutually interacting ensemble of waves. The preliminary numerical simulations cast light onto the possible validation of this general model reflecting the complexity of signal propagation in nerve fibres. The mathematically consistent modelling will allow not only the prediction of process characteristics but gives a possibility of understanding the role of governing factors in the whole complex process.

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.