Internal variables used for describing the signal propagation in axons
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O R I G I NA L A RT I C L E
Jüri Engelbrecht · Kert Tamm
· Tanel Peets
Internal variables used for describing the signal propagation in axons
Received: 19 October 2019 / Accepted: 21 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, the concept of internal variables developed in continuum mechanics is briefly described and then applied to the modelling of temperature change accompanying the propagation of signals in nerve fibres. The internal variable describes the exothermic and endothermic processes and involves the relaxation time measured in experiments. The model is based on using the FitzHugh and Nagumo idea for calculating the action potential with one generalised ion current. In the case of more detailed modelling like the Hodgkin– Huxley approach, sodium and potassium ion currents could be taken into account separately which dictates two internal variables. Such a phenomenological modelling of functional dependencies should be checked by experimental studies. Keywords Nerve signals · Temperature · Internal variables · Evolution equations
1 Introduction Signals in nerve fibres mean an ensemble of waves propagating with synchronised velocities. It has been demonstrated experimentally that an electrical signal is accompanied by mechanical and thermal effects. In addition to the action potential (AP), the mechanical effects are generated in the axoplasm and in the biomembrane which separates the axoplasm within the fibre from the extracellular environment. These electromechanical effects are also accompanied by the change in the temperature. Such experiments have been described in several studies [1,12,21,23], and the main mechanisms responsible for the coupling of essential physical effects are recently summed up [7]. A mathematical model which couples the effects into one system is presented by Engelbrecht et al. [5,6]. The recent analysis [7] has improved the system concerning the modelling of temperature changes, but the measured influence of possible exo- and endothermic processes [1] was not taken into account. These temperature changes are small (in order of μK) but measurable. The AP is not affected significantly by the accompanying temperature change [21] at normal conditions, but such an effect might occur in pathological situations caused by changes in ion-exchange processes [24] or related to CO2 output and oxygen consumption Communicated by Andreas Öchsner. J. Engelbrecht · K. Tamm (B) · T. Peets Laboratory of Solid Mechanics, Department of Cybernetics, School of Science, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia E-mail: [email protected] J. Engelbrecht E-mail: [email protected] T. Peets E-mail: [email protected]
J. Engelbrecht et al.
[3]. An inclusion of possible temperature changes completes the analysis of the signal propagation in axons from the thermodynamic viewpoint. The challenge is that there are several mechanisms proposed to model the temperature changes [1], and there is presently no consensus about the main mechan
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