Modelling of processes in nerve fibres at the interface of physiology and mathematics
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ORIGINAL PAPER
Modelling of processes in nerve fibres at the interface of physiology and mathematics Jüri Engelbrecht1 · Kert Tamm1 · Tanel Peets1 Received: 27 February 2020 / Accepted: 23 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The in silico simulations are widely used in contemporary systems biology including the analysis of nerve pulse propagation. As known from numerous experiments, the propagation of an action potential is accompanied by mechanical and thermal effects. This calls for an analysis at the interface of physics, physiology and mathematics. In this paper, the background of the model equations governing the effects in nerve fibres is analysed from a physical viewpoint and then discussed how to unite them into a system by using the coupling forces. The leading hypothesis associates the coupling to the changes of variables, not to their values or amplitudes. This hypothesis models actually the physiological mechanisms of energy transductions in a fibre. The general assumptions in modelling the processes and the properties of the coupled system of equations are described. The dimensionless mathematical model which couples the action potential with mechanical waves together with temperature effects is presented in “Appendix”. This model generates an ensemble of waves including the electrical signal and mechanical and thermal effects. Keywords Action potential · Mathematical modelling · Interdisciplinarity
1 Introduction Signal propagation in nervous systems is extremely important for all multicellular animals including human beings. Nerve signals control motion, behaviour and consciousness in many respects. The rich history of studies into the function of nerves is described in many overviews like Bishop (1956), Nelson et al. (2004), and a concise history of neuroscience spiced with mathematics is given by Scott (2002). Contemporary studies are characterised by attempts to describe interactions between the biological and physiological functions of a nervous system across multiple scales (Noble 2002b; Gavaghan et al. 2006). The better knowledge about functional behaviour of normal signalling in nerves helps to understand also the axon dysfunction, neuronal * Jüri Engelbrecht [email protected] Kert Tamm [email protected] Tanel Peets [email protected] 1
Department of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia
communication, synaptic efficacy, etc. needed for understanding neuronal activity (Debanne et al. 2011). For this purpose, in silico studies of biological processes are gaining more and more attention (Noble 2002a). It means developing interdisciplinary approaches combining ideas of physiology, physics and mathematics in studies on nervous function. In this paper, a possible approach is envisaged for mathematical modelling of the signal propagation in axons. In Sect. 2, the general background of basic model equations is analysed followed in Sect. 3 by physiological description of signals in nerve fibres. The assumptions made i
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