arXiv subjects

Tanel Peets

Publications and source records attributed to Tanel Peets.

17 recordsLinked to original sources

arXiv2026

Action potentials and solitons

During the last decade, the notion of solitons has been mentioned in neuroscience related to the propagation of action potentials (AP). In this paper, based on many studies in mathematical physics and neuroscience, the clear differences between the APs and solitons are summarised. It is stressed that the physics of the emergence of APs and solitons is fundamentally different. The numerical examples collected from earlier studies demonstrate the differences in the process of generation and interaction of corresponding waves explicitly. It is also noted that although the longitudinal mechanical waves in biomembranes can be described by the Boussinesq-type equation, the time-scale of emerging solitons from an arbitrary initial excitation exceeds the time-scale of the usual AP existence considerably.

physics.bio-ph↗

arXiv2026

On phenomenology of physical effects in axons

This paper deals with the mathematical modelling of signal propagation in nerve fibres. Due to the complexity of the processes where electrical, mechanical, and thermal effects are coupled, a phenomenological approach helps to build mathematical models. The ideas of phenomenology are briefly presented, and their application is described. These applications cover the modelling of ion currents (the Hodgkin-Huxley model), temperature effects, and inductance. This means that the ion currents through the biomembrane, the influence of endo- and exothermic reactions on temperature, and the influence of energy in a non-electrical form are taken into account using phenomenological variables, i.e., observables. Such an approach brings the mathematical models closer to reality. Using the concept of phenomenological inductance helps us better understand the propagation of an action potential in myelinated axons. In principle, contemporary mathematical models describing the process in axons are hybrid in nature, combining physical laws with phenomenology, i.e., with observables.

physics.bio-ph↗

arXiv2024

From modelling to understanding: the signals in nerves

This paper attempts to review our studies on the propagation of signals in nerves over the past decade. The need for interdisciplinary studies is stressed that helps to understand the physical mechanisms of coupling the electrical, mechanical, and thermal effects in nerves. Based on the analysis of structural properties of axons and possible mechanisms of interaction between different physical phenomena, a set of assumptions and hypotheses is formulated. As a proof of concept, a rather general mathematical model is presented for describing a wave ensemble in unmyelinated axons. This model is composed of several governing equations ("building blocks") which are coupled by forces describing the interaction between the effects. The numerical simulation using the dimensionless variables demonstrated a rather good qualitative match with experiments. The further generalisation of this model in physical units for the processes in myelinated axons permits a closer match to measurements. Based on modelling and in silico experiments, the guidelines for modelling such a complex electrophysiological process are formulated. These guidelines reflect the importance of following the physical principles in modelling together with interdisciplinary knowledge from continuum mechanics and mathematics.

physics.bio-ph↗

arXiv2024

The modelling of the action potentials in myelinated nerve fibres

The initial version of the planned paper has gone through a major revision in 2025. First, the paper ended up growing a bit too long, and as a result of that, we decided to split it into two parts. The first part focuses on the model for the unmyelinated case and its behaviour, and the second part focuses on including the influence of myelination into the model. Second, when the initial version of the manuscript was going through the review process, it became evident that the way the content was presented was somewhat confusing for readers with a background in the experimental side of research into nerve processes. As a result, we went through a major revision, redoing all the numerical simulations with parameters that are closer to what Hodgkin and Huxley used in their classical paper from 1952, where the Hodgkin-Huxley model was initially introduced. The second major change was to change the logic how the specific inductance value is chosen for the numerical example - in the previous version it was chosen by aiming for a specific propagation velocity when the axon radius was chosen as 1 micrometre, in the updated version the value is chosen to get the AP propagation velocity which was experimentally observed in the HH 1952 paper at the same parameters as were used in that paper. Part 1 - On hyperbolicity for nerve pulse propagation in axons. Part 2 - The modelling of the action potentials in myelinated nerve fibres.

physics.bio-ph↗

arXiv2023

On concepts of mathematical physics for modelling signals in axons

In this short paper, the results of the paper by Drab et al. (Eur. Phys. J. E (2022) 45:79) are described in the framework of wave mechanics and mathematical physics based on common understandings. The attention is focused on properties of Boussinesq-type equations, solitons, and peakons. These concepts are supported by several experimental observations.

physics.bio-ph↗

arXiv2021

Mechanical waves in myelinated axon wall

The propagation of an action potential is accompanied by mechanical and thermal effects. Several mathematical models explain the deformation of the unmyelinated axon wall. In this paper, the deformation of the myelinated axon wall is studied. The mathematical model is inspired by the mechanics of microstructured materials. The model involves the improved Heimburg-Jackson equation together with another equation of wave motion that describes the process in the myelin sheath. The dispersion analysis of such a model explains the behaviour of group and phase velocities. In addition, it is shown how dissipative effects may influence the process. Numerical calculations demonstrate the changes in velocities and wave profiles in the myelinated axon wall.

physics.bio-ph↗

arXiv2020

On physical background of nerve pulse propagation: heat and energy

Recent studies have revealed the complex structure of nerve signals in axons. Besides the electrical signal, mechanical and thermal effects are also detected in many experimental studies. In this paper, the mathematical models of heat generation are analysed within the framework of a general model derived earlier by the authors. The main mechanisms of the heat generation are seemingly the Joule heating and endo- and exothermic reactions. The concept of internal variables permits to model the heat relaxation typical to these reactions. The general energy balance of the whole signal is analysed based on physical mechanisms responsible for emerging the components of a signal. Some open questions are listed for further studies.

physics.bio-ph↗

arXiv2019

On mechanisms of electromechanophysiological interactions between the components of nerve signals in axons

Recent studies have revealed the complex structure of nerve signals in axons. There is experimental evidence that the propagation of an electrical signal (action potential) is accompanied by mechanical and thermal effects. In this paper, first an overview is presented on experimental results and possible mechanisms of electromechanophysiological couplings which govern the signal formation in axons. This forms a basis for building up a mathematical model describing an ensemble of waves. Three physical mechanisms responsible for coupling are (i) electric-lipid bi-layer interaction resulting in the mechanical wave in biomembrane; (ii) electric-fluid interaction resulting in the mechanical wave in the axoplasm; (iii) electric-fluid interaction resulting in the temperature change in axoplasm. The influence of possible changes in variables which could have a role for interactions are analysed and the concept of internal variables introduced for describing the endothermic processes. The previously proposed mathematical model is modified reflecting the possible physical explanation of these interactions.

physics.bio-ph↗

arXiv2019

On numerical modeling of dispersive mechanical waves in lipid bi-layers

We investigate different mechanical effects which accompany the nerve pulse propagation by using mathematical modeling. The propagation process is composed by three connected phenomena: (i) the action potential (electrical signal) which is usually considered when nerve pulses are discussed, (ii) the mechanical wave propagating in the biomembrane and (iii) the pressure wave in the axoplasm inside the axon. The main goal of the present study is to investigate numerically how the mechanical wave is generated by the action potential and how the characteristics of the system are reflected in the emerging wave ensemble. %the existence and characteristics of the mechanical wave influence the parameters of an action potential part of the wave ensemble. The key characteristics for the coupled model system are: (i) the velocity of the peak of the mechanical pulse is associated with the velocity of the action potential regardless of the sound velocity value in the lipid bi-layer, (ii) the velocity of the front and the shape of the mechanical wave depends on sound velocity in the lipid bi-layer, (iii) the shape of the mechanical wave can have an effect on the velocity and shape of the action potential.

physics.bio-ph↗

arXiv2019

Modelling of processes in nerve fibres at the interface of physiology and mathematics

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 the Appendix. This model generates an ensemble of waves including the electrical signal and mechanical and thermal effects.

physics.bio-ph↗

arXiv2019

Mathematics of nerve signals

Mathematical models describing the signals propagating in nerve fibres are described. The emphasis is on the mathematical structures of governing equations while the extremely rich physiological aspects are here not analysed. Based on models of single waves, a joint coupled model is presented which is able to describe the action potential and the accompanying mechanical effects togehter with temperature changes within one system of partial differential equations. The whole signal is an ensemble which includes primary and secondary components. The primary components of a signal are the action potential itself and longitudinal mechanical waves in axoplasm and surrounding biomembrane. These components are characterized by corresponding velocities. The secondary components of a signal are derived from primary components and include transverse displacement of a biomembrane and the temperature change. These secondary components have no independent velocities in the presented model.

physics.bio-ph↗

arXiv2018

Primary and secondary components of nerve signals

The action potential propagating in a nerve fibre generates accompanying mechanical and thermal effects. The whole signal is therefore an ensemble which includes primary and secondary components. The primary components of a signal are the action potential itself and longitudinal mechanical waves in axoplasm and surrounding biomembrane. These components are characterized by corresponding velocities. The secondary components of a signal are derived from primary components and include transverse displacement of a biomembrane and the temperature -- these have no independent velocities but have been measured in several experiments. A robust mathematical model is presented based on differential equations describing the signal primary components which are coupled into a system by coupling forces. The model includes also mathematical formulation for establishing the secondary components following the ideas from experimental studies.

physics.bio-ph↗

arXiv2018

Temperature changes accompanying signal propagation in axons

In this paper mathematical models are formulated in order to simulate heat production and corresponding temperature changes which accompany the propagation of an axon potential. Based on earlier experimental results, several models are proposed. Together with the earlier system of coupled differential equations derived by the authors for describing the electrical and mechanical components of signalling in nerve fibres, the novel results permit to cast the whole process of signalling into one system. The emphasis is on the mathematical description of coupling forces. The numerical results are qualitatively similar to experiments.

physics.bio-ph↗

arXiv2018

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: a soliton doublet

In this paper the permanent profile waves governed by a Boussinesq-type wave equation are analysed. The model involves displacement-type nonlinearities and dispersion terms. Physically such a model equation describes longitudinal waves (density change) in biomembranes which have an internal structure composed by lipid molecules. The possible solutions are constructed and analysed. The phase plane analysis and numerical simulation reveal a novel phenomenon: the possible existence of a soliton doublet.

nlin.PS↗

arXiv2018

Electromechanical coupling of waves in nerve fibres

The propagation of an action potential (AP) in a nerve fibre is accompanied by mechanical and thermal effects. In this paper an attempt is made to build up a mathematical model which couples the AP with a possible pressure wave (PW) in the axoplasm and waves in the nerve fibre wall (longitudinal - LW and transverse - TW) made of a lipid bilayer (biomembrane). A system of differential equations includes the governing equations of single waves with coupling forces between them. The single equations are kept as simple as possible in order to carry out the proof of concept. An assumption based on earlier studies is made that the coupling forces depend on changes (the gradient, time derivative) of the voltage. In addition it is assumed that the transverse displacement of the biomembrane can be calculated from the gradient of the LW in the biomembrane. The computational simulation is focused to determining the influence of possible coupling forces on the emergence of mechanical waves from the AP. As a result, an ensemble of waves (AP, PW, LW, TW) emerges. The further experiments should verify assumptions about coupling forces. In the Appendix, the numerical scheme used for simulations, is presented.

physics.bio-ph↗

arXiv2016

On solutions of a Boussinesq-type equation with amplitude-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with amplitude-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson (2005) for describing longitudinal waves in biomembranes and later improved by Engelbrecht et al. (2015) 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 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.

nlin.PS↗

arXiv2016

On modelling of physical effects accompanying the propagation of action potentials in nerve fibres

The recent theoretical and experimental studies have revealed many details of signal propagation in nervous systems. In this paper an attempt is made to unify various mathematical models which describe the signal propagation in nerve fibres. The analysis of existing single models permits to select the leading physiological effects. As a result, a more general mathematical model is described based on the coupling of action potentials with mechanical waves in a nerve fibre. The crucial issue is how to model coupling effects which are strongly linked to the ion currents through biomembranes.

physics.bio-ph↗