Emergence of lump-like solitonic waves in Heimburg–Jackson biomembranes and nerves fractal model

Abstract

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev–Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

1. Introduction

In the last decades, new technological imaging devices such as FMNR (functional magnetic nuclear resonance), CAT (computed axial tomography), PET (positron emission tomography), etc., have been constructed in order to better understand the local and global morphological complexity of several biological and living processes. In particular, the topological and hierarchical neuron structures in the human brain could provide neuroscientists with an experimental and a theoretical framework within which to examine the neural bases of behaviours, together with those that are frequently described in cognitive terms [1,2]. However, these experimental studies, although based on solid theoretical backgrounds, are not sufficient to comprehend the complexity of information processing in the brain and much theoretical works are still required. Recently, it was observed that two plausible mathematical tools to analyze and describe the complexity of cellular and neural brain behaviour are fractals and multifractals [3–5]. Based on experimental evidence, most biological processes may follow fractal rules and could be depicted as fractal entities. Nowadays, modern neurosciences and behavioural brain research are based on fractals analysis and have been a major archetype move in the last decades. Neurons in the brain are characterized by a self-similarity pattern, which is one of the basic properties of fractal geometry, offering a solid mathematical tool to describe neurons and the nervous system quantitatively in all their physiopathological spectrums [4]. It is noteworthy that fractals play a leading role in biology, medicine and several aspects of neurosciences [6] including morphological embryology [7], pyramidal neurons in mammalian motor cortex [8], cortical pyramidal neurons analysis [9], neuronal and glial cellular morphology [10], nonlinear behaviour of awake and sleep stages [11], analysis of states of consciousness and unconsciousness [12]–, study of cerebral cortical surface in schizophrenia and obsessive–compulsive disorder [13], characterization of atrophic changes in the cerebral cortex [14], cortical functional connectivity networks and severity of disorders of consciousness [15], study of exchange of local and global information processing in the brain [16], etc. (for a good review we suggest [17] and references therein). Fractal analysis is the most valuable mathematical tool for measuring dimensional, geometrical and functional parameters of biological cells, tissues and organs. It is noteworthy that fractals in fact cannot exist in biomembranes, since the scaling behaviour of normal objects is restricted when approaching the molecular size section. However, a recent study based on peridynamic model simulations and experiments performed on spreading double bilayers suggests that the emergence of a kind of fractal morphologies [18]. In this study, we are interested in fractals arising in the nervous system. In fact, the implications of fractal geometry in the neurosciences have been a chief paradigm move over the last decades as it allows for quantitative study and explanation of the geometric complexity of the brain, from axon to the neuronal networks [19].

In neuroscience, the classic theory to describe the propagation of nerve pulse across the axonal membrane with an amplitude of about 100 mV is the Hodgkin–Huxley model (HH) [20]. This model treats the transmitted impulses as pure electrical signals and aims to clarify the spread of signals across the nerve membrane using electrical circuits through explicit ion-conducting proteins (called ion channels) which leads to fleeting voltage changes. In other words, the HH model is used to typify the action potential of a squid axon and has been used successfully in describing and predicting a large number of neuronal properties. This is a dissipative isentropic process and based on the well-known Kirchhoff circuits with electrical currents introduced by the ion flux. The circuit involves a capacitor with constant capacity (the nerve membrane) and the channel proteins as resistors. This model plays a leading role in molecular biology, electrophysiology of biological membranes and dissipative hydrodynamic processes [20]. It is notable that under the HH paradigm the action potential is described as a purely electrical signal whereas the work done by Iwasa & Tasaki [21] is purely an experimental work. We stress that in the realm of the HH model, the action potential is described as an electrical signal whereas experimental observations describing mechanical swelling accompanying the electrical signal in a giant squid axon was not described in a HH model addressed in [21]. The main outcome of [21] based on the examination of the mechanical responses of crab and crayfish axons is that crustacean axons expand during the depolarizing phase of the action potential. Nevertheless, the HH model has been extended by several authors through various ways, e.g. incorporating complex geometries of dendrites and axons based on microscopy data [22], including ion channel populations based on experimental data [23], etc. (see [24] and reference therein). We stress that large-scale neurobiologically models have been used to perceive the presence of efficient brain networks for a broad assortment of tasks in which information processing takes place at the network level with rich temporal behaviour [25].

From a mathematical viewpoint, although a variety of properties of the dynamics of the HH vector field have been studied in the literature, nonetheless, we remain far from an inclusive theory on the dynamics displayed by this vector field [26–28]. In fact, the HH model fails to explain several features of the propagating nerve pulse, including the reversible release and reabsorption of heat and the additional mechanical, fluorescence and turbidity changes. The alternative model addressed by Heimburg and Jackson in [29,30] is in fact based on some experimental observations as a Boussinesq-type equation with an uncommon nonlinear term. However, they did not deal with the physiological heat changes but only derived a pulse-shaped localized solution for their model equation. The extension of their work to deal with the associated heat exchanges was done in [31] yet heat exchanges will not be taken into account throughout this study. One nice observation by Heimburg and Jackson is that the adiabatic pulses travelling through the membranes could be electromechanical solitons. Hence, the Heimburg and Jackson (HJ) model is able to explain the effect of anaesthetics on nerve pulse conduction and some thermodynamic processes of the neurons' membrane connected with nerve pulse such as physiological heat changes. In the HJ model, solitons propagate at a minimum velocity and besides soliton profiles are stable as a function of the soliton velocity. Solitons propagate at maximum amplitude and a minimum velocity comparable to the propagation velocity in myelinated nerves. In non-myelinated nerves, propagation velocities of solitons are considerably slower. Several models may explain the deformation of the unmyelinated axon wall. One recent model introduced in [32], is inspired by the mechanics of microstructured materials. The model also explains how dissipation may influence the process. Let us recall that there are two classical definitions for a soliton: (a) a soliton can be described as a stable particle-like state in a nonlinear system [33], (b) another way is defining through its properties as a wave in the nonlinear environment that has (i) a stable form, (ii) is localized in space, (iii) restores its speed and structure after interaction with another soliton [34]. It is also worth-mentioning that using solitons to model nerve pulses is to some extent a fundamentally flawed concept because nerve pulses annihilate each other during a head-on collision, which is backed up by a large number of experimental observations in a wide range of settings, while solitons, as a rule, do not because they need to interact elastically (restoring their shape and velocity after collision) and this is a fundamental contradiction. What Heimburg, Jackson and other authors who are investigating the ‘mechanical’ nerve pulse aspect have shown is that such a soliton-type solution can exist in a mathematical model composed for a lipid bi-layer. However, so far there has been no experimental work that would have been able to observe a ‘mechanical nerve pulse’, i.e. a mechanical signal on a test system similar enough to a real axon or neuron that behaves like a real nerve signal, meaning that it has the three key properties (1) threshold for excitation, (2) annihilation during a collision, (3) a time delay before you can propagate the next pulse. The experimental works that observe a mechanical wave in nerves [21] have been able to observe a mechanical wave accompanying the electrical signal but not a stand-alone mechanical signal that behaves like a nerve pulse.

In fact, one crucial feature of lipid bilayers is that they display phase transitions from solid-ordered to liquid-disordered states. Hence, they are considered an impulse for the proliferation of solitons is the lipid melting due to the propagation of a localized density wave in the axon membrane [35]. We stress that Heimburg et al. [30] proposed their model for a longitudinal mechanical density change as a Boussinesq-type equation (a wave equation with added nonlinear and dispersive effects) and determined some coefficients for their model by doing an experiment on a lipid mixture, which is not quite the same as a real nerve fibre. Experimental data were instead used to determine the nonlinear parameters for their model. It is noteworthy that the soliton model in nonlinear dispersive media arises due to a weak balance between the steepening effects of nonlinearity and the spreading effects of wave dispersion. Solitary waves are a subset of a family of coherent structures and could acquire a stable shape under some constraints [36]. It should be stressed again that the solution obtained by Heimburg et al. in [30] is not a ‘soliton’ since stability has not been checked if their solution is stable through the collisions with another solution. It will be better to entitle these solutions by ‘soliton-like’, or ‘solitary waves’. The question of whether the solutions described by Heimburg et al. are really ‘solitons’ has been addressed by Lautrup et al. [29] where it was concluded that ‘in particular, if multisoliton solutions were constructed it was recognized that, in spite of the strong interaction between them, solitons ‘pass through’ one another without losing their identity’. Hence, the solutions obtained in [30] are not solitons but are merely solitonic or soliton-type solutions, as they interact with each other inelastically, as was demonstrated in [29].

Despite the significant connections between fractals and physiology as discussed in [36], there is potentially very little research that treats the fractal aspects of the HH soliton model [37]. It is notable that some models based on fractional aspects have been introduced; nevertheless, these models are based on the concept of fractional analysis and not fractal calculus [38,39]. Moreover, it should be stressed that the fractal aspects of solitary waves were addressed in literature through dissimilar aspects but not within the HH model [40,41]. Yet, our approach will be based on a different aspect to those found in literature known as the ‘product-like fractal measure’.

In fact, this new concept was introduced recently by Li and Ostoja-Starzewski in order to describe dynamics in anisotropic and continuum media [42–44] and was motivated by Tarasov fractal calculus arguments [45,46]. It is considered a successful approach, which has proved to have several successful implications in sciences and engineering at different scales [47–58]. In the Li and Ostoja-Starzewski approach (LOSA henceforth), the dynamic equations of motion hold mathematical forms involving integer-order integrals, whereas their local forms are expressed through partial differential equations with integer-order derivatives except that they contain coefficients involving fractal dimensions. Hence, this model is suitable to describe dynamics in fractal dimensions and at all scales. We set-up in brief the basic mathematical aspects of LOSA: we assume we have a parallelepiped of different lengths x₁, x₂, x₃, mass $m=m(x_1,\hspace{0.17em}{x}_2,\hspace{0.17em}{x}_3)\approx x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}{x}_3^{{\alpha }_3}$ and density ρ = ρ(x₁, x₂, x₃) in a medium of dimension D = α₁ + α₂ + α₃ where 0 < α_k ≤ 1 is the fractal dimension in the direction x_k. Mathematically, one can define the mass of the parallelepiped by the following triple integral $m={\int }_V{\prod }_{k=1}^3{c}_1^k({\alpha }_k,\hspace{0.17em}{x}_k)\hspace{0.17em}\mathrm{d}{x}_k\equiv {\int }_V\rho {\prod }_{k=1}^3\hspace{0.17em}\mathrm{d}{\mu }_k(x_k)$ over the volume V. Here $\mathrm{d}{\mu }_k(x_k):={\prod }_{k=1}^3{c}_1^k({\alpha }_k,\hspace{0.17em}{x}_k)\mathrm{d}{x}_k$ is the length measurement and $c_1^{(k)}({\alpha }_k,\hspace{0.17em}{x}_k)={\alpha }_k{((l_k-x_k)/l_{k0})}^{{\alpha }_k-1}$ is a coefficient function of the length l_k along axis 0 < x_k < l_k obtained from the Jumarie's fractional integral and l_k0 is a typical length where its meaning depends on the physical problem under study. The parallelepiped is, therefore, characterized by the volume coefficient $c_3={\prod }_{i=1}^3{c}_1^{(i)}$. Li and Ostoja-Starzewski has proved that for $c_2^{(k)}=c_1^{(i)}{c}_1^{(j)}=c_3/c_1^{(k)},i\ne j,i,j\ne k$, the vector calculus identities and the following fundamental theorems of vector calculus hold accordingly:

• — the Gauss theorem ${\int }_{\mathrm{\partial }W}\hspace{0.17em}{f}_k{n}_k\hspace{0.17em}\mathrm{d}{S}_d={\int }_W\hspace{0.17em}{f}_{k,k}{c}_2^{(k)}{c}_3^{-1}{V}_d=$ ${\int }_W\hspace{0.17em}{f}_{k,k}{c}_1^{-1(k)}\mathrm{d}{x}_1\mathrm{d}{x}_2\mathrm{d}{x}_3$,
• — the Stokes theorem ${\int }_A\mathbf{n}\cdot \mathrm{\nabla }\times f\mathrm{d}{S}_d={\int }_l\mathbf{f}\cdot \mathrm{d}{l}_{{\alpha }_1}$,
• — the divergence theorem ${\int }_V{\mathrm{\nabla }}^D\cdot \mathbf{f}\mathrm{d}{V}_D={\int }_V{\mathrm{\nabla }}^D\cdot \mathbf{f}{c}_1^{(1)}{c}_2^{(2)}{c}_3^{(3)}$ $\mathrm{d}x\mathrm{d}y\mathrm{d}z={\int }_S\mathbf{f}\cdot \mathbf{n}\mathrm{d}{S}_d$.

Here $\mathrm{d}{S}_d^{(k)}=c_2^{(k)}\mathrm{d}{S}_2=c_1^{(i)}{c}_1^{(j)}\mathrm{d}{S}_2=(c_3/c_1^{(k)})\hspace{0.17em}\mathrm{d}{S}_2,\hspace{0.17em}i\ne j,i,j\ne k$ is the planar surface element, dS_d is the infinitesimal fractal surface element and $\mathrm{d}{V}_D=c_1^{(1)}{c}_2^{(2)}{c}_3^{(3)}\mathrm{d}x\mathrm{d}y\mathrm{d}z$ is the infinitesimal fractal volume element. As a result, the gradient and the Laplacian operators hold respectively the unique forms ${\mathrm{\nabla }}_k^D={(c_1^{(k)})}^{-1}{\mathrm{\nabla }}_k$, ${\mathrm{\Delta }}_k^D={\mathrm{\nabla }}_k^D\cdot {\mathrm{\nabla }}_k^D$ and ${\mathrm{\nabla }}^D={\mathbf{e}}_k{\mathrm{\nabla }}_k^D$ with e_k are base vectors. LOSA mathematical structures are in fact comparable to more than a few approaches discussed in literature yet based on fractional calculus [59,60]. It is noteworthy that the extended fractional HH model has been addressed in the literature [39]. Besides, it was revealed in [61] that a fractional capacitance model may overcome the following drawbacks of the HH model: ignorance of dielectric losses in the membrane and the assumption of an ideal membrane capacitance. Let us recall also that there exists a physically based connection between fractional calculus and fractal geometry [62]. Moreover, there are many indications that ion channel proteins exhibit memory and are based on a power-law self-similar function [63,64]. Hence fractal kinetics may help to understand how ion channel conductance profiles are affected by fractal dimension [65,66]. These power-law and correlation characteristics may be practical in the coding and decoding process of HH neuron [67]. These outcomes motivate the studies of HH and HJ models in fractal dimensions. It is notable that neurons possess a wide variety of shapes, sizes and electromechanical properties [68]. There exist also several observations based on experimental studies which indicate that an Action Potential (AP) is accompanied with a number of physiological changes in the nerve membrane such as, production and absorption of heat, variation of axon diameter, density, pressure and length [69]. Hence, the neurons are subject to an increase and decrease in axon diameter and density. Hence, the LOSA and its associated parameters, such as characteristic lengths and density, may be a successful attempt to describe the HH and HJ models in fractal dimensions.

Moreover, in this study we will extend the LOSA by using the Hausdorff fractal time derivative operator (HFTDO), which is derived from fractal calculus (FC). We will prove that the combination of the LOSA with HFTDO will give rise to a nonlinear partial differential equation that holds a number of particular features and will be used to describe soliton wave propagation characteristics. In fact, FC is a relatively new field introduced to deal with kinetic fractal theory where the continuous time is replaced by the fractal time [70,71]. Such a derivative substitution is effective at very low scales [72]. Based on FC, the HFTDO takes the general form $D^{\beta }:={(T_0/\overline{\tau })}^{\beta -1}D$ where $D^{\beta }=\mathrm{\partial }/\mathrm{\partial }{\overline{\tau }}^{\beta }$ and $D=\mathrm{\partial }/\mathrm{\partial }\overline{\tau }$, $\overline{\tau }=t-\tau$, 0 < τ < t, T₀ is a characteristic time and β is a real parameter close to unity and is interpreted in FC as the fractal dimension of time [73–77]. We can associate τ to the locally recorded time and t to the final measured value of the moving body time. When t varies, the complete earlier time interval varies as well. This operator is also obtained from the fractional velocity arguments which has the mathematical form ${\nu }_{\pm }^{\beta }f\hspace{0.17em}(x)\triangleq \underset{\epsilon \to 0}{lim}{\nu }_{\beta }^{\epsilon \pm }[f](x)$ and is defined in terms of the fractal variations operators ${\nu }_{\beta }^{\epsilon +}[f](x)=(f\hspace{0.17em}(x+\epsilon )-f\hspace{0.17em}(x))/{\epsilon }^{\beta }$ and ${\nu }_{\beta }^{\epsilon -}[f](x)=(f\hspace{0.17em}(x)-f\hspace{0.17em}(x-\epsilon ))/{\epsilon }^{\beta }$. Two cases were discussed in [78–81]: 0 < β ≤ 1 and 1 ≤ β < 2, which proves dissimilar dynamics. Another comparable definition has been also introduced in [82–84] by using indefinite limit and L'Hospital rule such that f^(β)(x)≜lim _ɛ→0Δ^βf(x)/L^β where Δ^βf(x) = f^β(x + ɛ) − f^β(x) and L^β = (x + ɛ)^β − x^β. These approaches led to the definition of the HFTDO. Note that:

$$D^{\beta }{D}^{\beta }={\left(\frac{T_0}{\overline{\tau }}\right)}^{\beta -1}\left(D^2+\frac{1-\beta }{\overline{\tau }}D\right):={\left(\frac{T_0}{\overline{\tau }}\right)}^{\beta -1}{(B_{1-\beta })}_{\overline{\tau }},$$

where ${(B_{1-\beta })}_{\overline{\tau }}:=D^2+\frac{1-\beta }{\overline{\tau }}D$ is the Bessel operator [85]. Several applications of HFTDO were discussed in [86–93]. However, for very large time, we can use the following asymptotic rule ${(B_{1-\beta })}_{\overline{\tau }}\approx D^2$. It is notable that fractal time has been used in studying nonlinear dynamics of microtubes in cells which are considered as a network for solitary waves [94]. In addition, multifractal time series has been used to study the topological properties of both experimental and simulated microtubes [95]. In fact, the implications of fractal time have been discussed in several complex systems including living cells [96–98]. There exist also several arguments which indicate that biological time is fractal [99] and in addition, fractal time has been used to study the movement activity of Drosophila [100]. The importance of spatial and temporal scaling to the study of biological systems and physiological processes [101] motivate us to consider fractal time in this study.

This study is organized as follows in §2, we introduce the basic set-ups of the theory; in §3, we discuss some of its main computational properties and features; finally conclusions and perspectives are given in §4.

2. Basic set-ups of the fractal Heimburg-Jackson model

In order to implement LOSA + HFTDO in HJ theory, we let Δρ^A be the area density change in the plane (τ, z) and we introduce the following nonlinear partial differential equation in fractal dimensions (α, β) which is derived from the Euler equations of compressible media (comparable to a variable-coefficient wave equation) [102–107]:

$$\begin{array}{rl}& S(z){\left({\displaystyle \frac{T_0}{\overline{\tau }}}\right)}^{\beta -1}{(B_{1-\beta })}_{\overline{\tau }}\mathrm{\Delta }{\rho }^A\equiv S(z){\left({\displaystyle \frac{T_0}{\overline{\tau }}}\right)}^{\beta -1}\left({\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }\overline{\tau }{}^2}+{\displaystyle \frac{1-\beta }{\overline{\tau }}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\overline{\tau }}}}}\right)(\mathrm{\Delta }{\rho }^A),\\ & ={\mathrm{\nabla }}_z^{\alpha }({\mathit{c}}^2{\mathrm{\nabla }}_z^{\alpha }(\mathrm{\Delta }{\rho }^A))\equiv {\left({\displaystyle \frac{l_0}{l-z}}\right)}^{\alpha -1}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(c^2{\left({\displaystyle \frac{l_0}{l-z}}\right)}^{\alpha -1}(\mathrm{\Delta }{\rho }^A)\right),}\end{array}$$ 2.1

where the function S(z) = S₀((l − z)/l₀)^m is, in general, associated with density [108], m is a real parameter, S₀ > 0, c is the celerity of sound, which usually is a function of $\mathrm{\Delta }{\rho }^A={\rho }^A-{\rho }_0^A$ in particular close to melting transitions in membranes with ρ^A being the lateral density of the membrane and ${\rho }_0^A$ the equilibrium lateral density in the fluid phase of the membrane [102,109]. We recall that we are consider a one-spatial dimensional model where τ is the time and z is the position along the nerve axon. In the HJ model, the soliton-like model considers the wave equation for area density changes Δρ^A that originates from the Euler equations of compressible media. In general, the celerity of sound takes the form ${\mathit{c}}^2\approx {\mathit{c}}_0^2+p\mathrm{\Delta }{\rho }^A+q{(\mathrm{\Delta }{\rho }^A)}^2+\cdots$ where p and q describe the reliance of the sound velocity on density in the vicinity of the melting transition and are subject to experimental statistics and c₀ ≈ 176.6 m × s⁻¹ is the low frequency sound velocity [24]. It is noteworthy that equation (2.1) is a fractal generalization of the Euler–Poisson–Darboux equation, which is used in many problems of modern mathematics and physics [110]. Besides, the variable-coefficients (linear or nonlinear) partial differential equations supply us, in several dynamical systems or physical theories, with more plausible information on the inhomogeneities of media than analogous constant-coefficient counterparts. It is notable that neurons exhibit a kind of spatially inhomogeneity in biological bodies. The brain, for example, exhibits an inhomogeneous allocation of neurons and astrocytes throughout the diverse brain areas [111,112]. Inhomogeneities have also been observed in biomembranes and bionanocomposites [113–115] and have important implications in bioengineering.

In order to take into account the dispersion effect, a four-order fractal derivative term − hΔ_zΔ_z(Δρ^A) ≡ −hΔ_zz(Δρ^A) is added to equation (2.1) in the HJ model. Δ_z = ∂²/∂z², Δ_zz = ∂⁴/∂z⁴, h in the HJ model is assumed to be a small constant describing the frequency dependence of the speed of sound. In this study, we will extend the HJ model by adding the term − Δ_zΔ_z(h(z)(Δρ^A)) since in general, the medium is subject to various external constraints which may be give rise to higher-order solitonic waves with variable coefficients [116–122]. Some authors added a fourth-order mixed partial derivate term to model the inertial properties of the membrane as a modification to the original model and have interpreted the dispersive term included in the original HJ model as representing the elastic properties of the membrane [123,124]. Hence, the additional element may represent physically or physiologically in the present model describing a density pulse in a biomembrane.

For very large evolution time, the HJ equation in fractal dimension 0 < α ≤ 1 takes the form (fractal Boussinesq-type equation):

$$\begin{array}{rl}{S}_0& {\left({\displaystyle \frac{Z}{l_0}}\right)}^m{\left({\displaystyle \frac{T_0}{\overline{\tau }}}\right)}^{\beta -1}{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }\overline{\tau }{}^2}\mathrm{\Delta }{\rho }^A={\mathrm{\nabla }}_Z^{\alpha }({\mathit{c}}^2{\mathrm{\nabla }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A))-{\mathrm{\Delta }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }(h(Z)\mathrm{\Delta }{\rho }^A)}\\ & \equiv {\mathrm{\nabla }}_Z^{\alpha }{\mathit{c}}^2{\mathrm{\nabla }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A)+{\mathit{c}}^2{\mathrm{\Delta }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A)-{\mathrm{\Delta }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }(h(Z)\mathrm{\Delta }{\rho }^A),\end{array}$$ 2.2

It is noteworthy that one is able to solve equation (2.2) numerically or using the Sonine–Poisson–Delsarte transmutation method, which are the transmutations of the hyper-Bessel operators and functions [125] keeping the Bessel operator ${(B_{1-\beta })}_{\overline{\tau }}:=D^2+\frac{1-\beta }{\overline{\tau }}D$, however, our aim is to give an asymptotic solution of the HJ theory of biomembranes and nerves system. Since we have:

$$\begin{array}{rl}{\mathrm{\Delta }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }(h(Z)(\mathrm{\Delta }{\rho }^A))& =h(Z){\mathrm{\Delta }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A)+4{\mathrm{\nabla }}_Z^{\alpha }h(Z){\mathrm{\nabla }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A)\\ & +6{\mathrm{\Delta }}_Z^{\alpha }h(Z){\mathrm{\Delta }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A)\\ & +4{\mathrm{\nabla }}_Z^{\alpha }(\mathrm{\Delta }{\rho }^A){\mathrm{\nabla }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }h(Z)\\ & +(\mathrm{\Delta }{\rho }^A){\mathrm{\Delta }}_Z^{\alpha }{\mathrm{\Delta }}_Z^{\alpha }h(Z),\end{array}$$ 2.3

we can write equation (2.1) as:

$$\begin{array}{rl}& S_0{\left(\frac{T_0}{\overline{\tau }}\right)}^{\beta -1}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\overline{\tau }}^2}\mathrm{\Delta }{\rho }^A={\left(\frac{Z}{l_0}\right)}^{2(1-\alpha )-m}\\ & \left({\mathrm{\nabla }}_Z{c}^2{\mathrm{\nabla }}_Z(\mathrm{\Delta }{\rho }^A)+c^2\left({\mathrm{\Delta }}_Z+\frac{\alpha -1}{Z}{\mathrm{\nabla }}_Z\right)(\mathrm{\Delta }{\rho }^A)\right)\\ & -{\left(\frac{Z}{l_0}\right)}^{4(1-\alpha )}h(Z)({\mathrm{\Delta }}_{ZZ}(\mathrm{\Delta }{\rho }^A)+\frac{{(\alpha -1)}^2}{Z^2}{\mathrm{\Delta }}_Z(\mathrm{\Delta }{\rho }^A)\\ & -\frac{(1-\alpha )(2{\alpha }^2-\alpha +1)}{Z^3}{\mathrm{\nabla }}_Z(\mathrm{\Delta }{\rho }^A))-{\left(\frac{Z}{l_0}\right)}^{4(1-\alpha )}(\mathrm{\Delta }{\rho }^A)\\ & \left({\mathrm{\Delta }}_{ZZ}h(Z)+\frac{{(\alpha -1)}^2}{Z^2}{\mathrm{\Delta }}_{Z}h(Z)-\frac{(1-\alpha )(2{\alpha }^2-\alpha +1)}{Z^3}{\mathrm{\nabla }}_{Z}h(Z)\right)\\ & -4{\left(\frac{Z}{l_0}\right)}^{4(1-\alpha )}{\mathrm{\nabla }}_{Z}h(Z)({\mathrm{\nabla }}_{ZZZ}(\mathrm{\Delta }{\rho }^A)+\frac{1-\alpha }{Z}{\mathrm{\Delta }}_Z(\mathrm{\Delta }{\rho }^A)\\ & +\frac{(1-2\alpha )(\alpha -1)}{Z^2}{\mathrm{\nabla }}_Z(\mathrm{\Delta }{\rho }^A))-4{\left(\frac{Z}{l_0}\right)}^{4(1-\alpha )}{\mathrm{\nabla }}_Z(\mathrm{\Delta }{\rho }^A)\\ & \left({\mathrm{\nabla }}_{ZZZ}h(Z)+\frac{1-\alpha }{Z}{\mathrm{\Delta }}_{Z}h(Z)+\frac{(1-2\alpha )(\alpha -1)}{Z^2}{\mathrm{\nabla }}_{Z}h(Z)\right)\\ & -6{\left(\frac{Z}{l_0}\right)}^{4(1-\alpha )}\left({\mathrm{\Delta }}_{Z}h(Z)+\frac{\alpha -1}{Z}{\mathrm{\nabla }}_{Z}h(Z)\right)\\ & \left({\mathrm{\Delta }}_Z(\mathrm{\Delta }{\rho }^A)+\frac{\alpha -1}{Z}{\mathrm{\nabla }}_Z(\mathrm{\Delta }{\rho }^A)\right).\end{array}$$ 2.4

The scales ${\mathit{c}}^2:={\mathit{c}}^2(\mathrm{\Delta }{\rho }^A)={\overline{\mathit{c}}}^2(\mathrm{\Delta }{\rho }^A){(Z/l_0)}^{1-\alpha }$ and h(Z) = h₀(Z/l₀)^α−1 (h₀ being a tiny constant) reduce equation (2.4) for very large propagation distance to:

$$\begin{array}{rl}& S_0{\left(\frac{T_0}{\overline{\tau }}\right)}^{\beta -1}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\overline{\tau }}^2}\mathrm{\Delta }{\rho }^A\approx {\left(\frac{Z}{l_0}\right)}^{3(1-\alpha )-m}.\\ & {\mathrm{\nabla }}_Z({\overline{c}}^2{\mathrm{\nabla }}_Z)\mathrm{\Delta }{\rho }^A(-h_0({\mathrm{\nabla }}_{ZZZ}(\mathrm{\Delta }{\rho }^A)))+\mathrm{O}\left(\frac{1}{Z^2}\right)\end{array}$$ 2.5

We let $Z=l_0^{1-n}{X}^n$ and $\overline{\tau }=T_0^{1-\chi }{\hat{\tau }}^{\chi }$(n and χ are real parameters) which convert equation (2.5) for h₀/X G ≪ 1 and for very large time to:

$$\begin{array}{rl}& \frac{S_0}{{\chi }^2}{\left(\frac{T_0}{\hat{\tau }}\right)}^{\chi \beta +\chi -2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\hat{\tau }}^2}(\mathrm{\Delta }{\rho }^A)\approx {\left(\frac{X}{l_0}\right)}^{2-n(m+3\alpha )}\\ & {\mathrm{\nabla }}_X({\overline{c}}^2{\mathrm{\nabla }}_X-h_0{\mathrm{\nabla }}_{XXX})(\mathrm{\Delta }{\rho }^A).\end{array}$$ 2.6

Obviously, for n = 2/(m + 3α − 1) and χ(1 + β) = 2 equation (2.6) is considerably simplified to:

$${\displaystyle \frac{(1+{\beta }^2)S_0}{4}{\displaystyle \frac{{\mathrm{\partial }}^2}{\mathrm{\partial }\tilde{\tau }{}^2}\mathrm{\Delta }{\rho }^A\approx {\mathrm{\nabla }}_X({\overline{\mathit{c}}}^2{\mathrm{\nabla }}_X-h_0{\mathrm{\nabla }}_{XXX})(\mathrm{\Delta }{\rho }^A)+\mathrm{O}\left({\displaystyle \frac{1}{X}}\right),}}$$ 2.7

where $\tilde{\tau }=4\hat{\tau }/\sqrt{S_0(1+{\beta }^2)}$. Again, the change of variables $u=\mathrm{\Delta }{\rho }^A/{\rho }_0^A$, $Y=({\mathit{c}}_0/\sqrt{h_0})X$, $t=({\mathit{c}}_0^2/\sqrt{h_0})\tilde{\tau }$, $B_1=({\rho }_0/{\overline{\mathit{c}}}_0^2)p$ and $B_2=({\rho }_0^2/{\overline{\mathit{c}}}_0^2)q$ transform equation (2.7) to roughly the conventional form:

$${\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }t{}^2}\approx {\mathrm{\nabla }}_Y(B(u){\mathrm{\nabla }}_{Y}u)-{\mathrm{\Delta }}_{YY}u+\mathrm{O}\left({\displaystyle \frac{1}{Y}}\right)}$$ 2.8

where B(u) = 1 + B₁u + B₂u² with B₁ ≈ −16.6 and B₂ ≈ 79.5 as determined experimentally for a synthetic lipid membrane [24]. The coordinate transformation ξ = Y − ηt (η being a dimensionless propagation velocity of the density pulse) converts equation (2.8) to:

$${\eta }^2{\mathrm{\Delta }}_{\xi }u\approx {\mathrm{\nabla }}_{\xi }(B(u){\mathrm{\nabla }}_{\xi }u)-{\mathrm{\Delta }}_{\xi }{\mathrm{\Delta }}_{\xi }u+\mathrm{O}\left({\displaystyle \frac{1}{\xi }}\right).$$ 2.9

The solution of this equation is given by [23]:

$$u(\xi )={\displaystyle \frac{2\mathit{P}}{\mathit{S}+\mathit{N}\cosh \left(\xi \sqrt{1-{\eta }^2}\right)}.}$$ 2.10

Here P = a₊ a₋, S = a₊ + a₋, N = a₊ − a₋, $a_{\pm }=-(B_1/B_2)$ $(1\pm \sqrt{({\eta }^2-{\eta }_0^2)/(1-{\eta }_0^2)}$ with ${\eta }_0=\sqrt{1-B_1^2/6B_2}\approx$ $0.649851$, 1 > |η| > η₀ and $h_0\approx 2\hspace{0.17em}{\mathrm{m}}^4\times {\mathrm{s}}^{-2}$, i.e. a₊ ≈ 0.39, a₋ ≈ 0.08, P ≈ 0.007371, S ≈ 0.4089 and N ≈ 0.3711 [24]. In term of the original coordinates, the solution is given by:

$$\begin{array}{r}u(z,\tau )={\displaystyle \frac{2\mathit{P}}{\mathit{S}+\mathit{N}\cosh \left({\overline{\mathit{c}}}_0/\sqrt{h_0}\sqrt{1-{\eta }^2}\left(l_0{(l-z/l_0)}^{m+3\alpha -1/2}-(4\eta {\overline{\mathit{c}}}_0/\sqrt{S_0{(1+\beta )}^2})T_0{(t-\tau /T_0)}^{1+\beta /2}\right)\right)}}\end{array}.$$ 2.11

In the next section, we will analyse some of the computational properties of equation (2.11).

3. Computational aspects of the fractal Heimburg–Jackson model

For m = 0 and α = β = 1, equation (2.11) is reduced to its conventional form. However, a plausible solution exists for α > 1/2. Besides, when t ≈ τ, the solution u(Z) is roundly spatial and is plotted in figure 1 (u₁(Z)) and figure 2 (u₂(Z)) for η = 0.95, l₀ = 1, $h_0\approx 2{\mathrm{m}}^4\times {\mathrm{s}}^{-2}$ and different values of m and α. In fact, u₁(Z) and u₂(Z) are solutions of u(Z) for different values of the parameters,

Figure 1.

Variations of u₁(Z): the blue plot corresponds to m = 4 and α = 1/3, the yellow plot corresponds to m = 2 and α = 1/3, the red plot corresponds to m = 2 and α = 0.83.

Figure 2.

Variations of u₂(Z): the blue plot corresponds to m = 1 and α = 1/2, the yellow plot corresponds to m = 1 and α = 1/3, the red plot corresponds to m = 4 and α = 0.66.

Figure 1 appears to show the HJ equation solution, which is similar to the one published and analysed by Lautrup et al. in [29], whereas figure 2 describes the evolution for different parameters in the positive domain. These wave profiles are different from wave profiles analysed in [24]. We observe deformations or distortions in the shape of these profiles compared to the results obtained in [24] due to the fractal dimensions of the medium although the solution displays exponentially localized solitary solutions for a finite range of subsonic velocities. Comparing the positive domains of figures 1 and 2, we observe the presence of long tails for lower fractal dimensions and for large scales, which represent one of the main features of fractal and fractional wave equations [126].

In figures 3–6, we plot the three-dimensional numerical variations of u(Z) after fixing the numerical values of (l₀, T₀) and for different values of m, α and β to observe the physical behaviour of the system (Z = l − z, T = t − τ and the base of figures 3–6 correspond to (Z, T)):

Figure 4.

Variations of u(Z) for $(m,\hspace{0.17em}\alpha ,\hspace{0.17em}\beta )=(7,\hspace{0.17em}\frac{1}{2},\hspace{0.17em}0.9)$.

Figure 5.

Variations of u(Z) for (m, α, β) = (2, 0.83, 0.9).

Figure 3.

Variations of u(Z) for $(m,\hspace{0.17em}\alpha ,\hspace{0.17em}\beta )=(2,\hspace{0.17em}0.83,\hspace{0.17em}\frac{1}{2})$.

Figure 6.

Variations of u(Z) for $(m,\hspace{0.17em}\alpha ,\hspace{0.17em}\beta )=(7,\hspace{0.17em}\frac{2}{3},\hspace{0.17em}0.9)$.

These graphs describe dissimilar cone wave solitonic solutions which reveal a symmetric surface and peaks in the centre depending on the fractal dimensions of the medium. We observe how our obtained solutions change their wave structure via appropriate choices of the numerical values of (m, α, β). These solutions illustrated graphically clarify the new features of the model in question. The novelty of our solutions is shown by comparing these numerical solutions with figure 7 obtained in the conventional HJ approach [24]. In particular, we have the emergence of a lump-like soliton wave whereas in the approach of [24], such a solution is not present (as shown in figure 7).

Figure 7.

Variations of u(Z) for the HJ approach.

In fact, the lump solution is a kind of special rational function solution localized along all directions in the space and by the lump-type solitonic solution we refer to as standard hyperbolic secant shaped solitary waves.

4. Conclusion and perspectives

The result, equation (2.11), which is due to the combination of LOSA + HFTDO, describes a lump-type solitonic wave which decays algebraically to the background wave in space direction. Our solutions represent solitonic or soliton-like solutions (i.e. localized in space and reasonably stable) [31,34,127]. Hence, it is viewed as a particular class of rational localized waves, which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev–Petviashvili I equation, which describes small-amplitude shallow-water waves in particular when the Bond number is greater than 1/3 [128]. More explicitly, lump solutions with kink backgrounds are used to describe nonlinear patterns on a shallow water surface with dominating surface tension [129]. The lump soliton solutions have been regularly used in different fields of sciences including biology and chemistry [130–132]. It is interesting to obtain lump-type soliton solutions in the HJ model for biomembranes and nerves. In the present model, the lump-type solitonic wave is due to the fractal dimensions of the medium and the influence of the dispersion and nonlinear effects. Varying the numerical values of (l₀, T₀) can lead to a variation of the range of the lump wave but has no effect on the amplitude. Hence, the amplitude of the lump is not affected by the numerical values of (m, α, β) and (l₀, T₀). This is comparable to the lump wave solutions for the (3 + 1) Kadomtsev–Petviashvili I equation in fluid [133]. It is notable that, in general, the origin of the lump waves is still a matter of debate even through research argues that solitons can release or swallow lump waves [134].

It will be of interest to extend this work to analyse collision phenomena among lump waves with periodic and single-, double-kink soliton solutions of the present model and to compare the outcomes with findings obtained in literature within the (2 + 1)-dimensional Kadomtsev–Petviashvili soliton model [132] and peakon model [135,136]. The solutions obtained in this study show that the proposed model is consistent and simple and so, we believe that the recommended idea could be extended for supplementary nonlinear models in neurosciences. It will also of interest to estimate fractal dimensions directly from the observed behaviour of the neuro-system.

Data accessibility

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Authors' contributions

R.A.E.: Conceptualization, methodology, resources.

Competing interests

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

The author would like to thank Chiang Mai University for funding this research.