Nuclei discovered new practical insights via optimized soliton-like pulse analysis in a space fractional-time beta-derivatives equations

This paragraph specifically aims to justify and elucidate the dynamic theory analysis of Eq. (5), which characterizes the behavior of interconnected models of nerve fibers through methodologies including Modulational Instability (MI), fixed points (FPs), and the Jacobian matrix (JM), among others. The analysis of bundles of nerve fibers highlights the critical inquiry into the fusion of the fibers. The interactions of muscles examined in this theoretical context focus on understanding the interconnections of nerve fiber clusters, a characteristic of the nervous systems of higher mammals, while also aiming to generate concepts for empirical research. Additionally, they provide both qualitative and quantitative approaches for the design of electrode and stimulus prototypes utilized in functional electrical stimulation, a process that involves transmitting signals, information, and pulses through two interconnected nerve fibers.

Unstable modulation

An electrical cable model can serve as a precise representation of the nerve fibers within the nervous system that facilitate the transmission of excitation pulses. The waveguide model, recognized in existing literature, effectively describes the propagation of nonlinear excitation during its transit in these systems. The generation and transmission of a nerve impulse depend on variations in the electrical conductivity of the local fiber tissue. Consequently, specific pathological processes, such as self-modulation instability (SMI), arise when typical nonlinearities, presumed to be weak, validate the application of averaged equations in the context of nonlinearly amplified propagation. The amplification process leads to the generation of additional, side-by-side waves, ultimately resulting in the disruption of the original waveform into a series of high waves. An electrical cable model can serve as a precise representation of the nerve fibers within the nervous system that facilitate the transmission of excitation pulses. The waveguide model, extensively documented in existing literature, effectively elucidates the behavior of nonlinear excitation during propagation in these systems. The generation and conduction of a nerve impulse relies on the localized changes in the electric conductivity of the fiber-tissue. Consequently, specific pathological processes, including self-modulation instability (SMI), occur when a sinusoidal wave exhibits nonlinearity and can be amplified via periodic feedback mechanisms. The amplification process leads to the emergence of extra spectral sidebands, ultimately causing the waveform to break down into a sequence of pulses rather than maintaining a continuous signal wave:

In the context of anti-symmetry, the value is represented as\(\:\:\sigma\:=-1\). In the context of symmetry, the value is represented by the symbol\(\:\:\sigma\:=1\). The term\(\:\sqrt{{P}_{0}}\) stands for incident power, while \(\:h\) is the smallest perturbation parameter. Moreover, the symbols\(\:\:\psi\:\left(x,t\right)\), \(\:\:\phi\:\left(x,t\right)\) represent the perturbation terms. The substitution of Eq. (6) into Eq. (5) is followed by the execution of analytical calculus operations. Using the linearizing principle, we obtain the following equations for the perturbed solutions \(\:\psi\:\left(x,t\right)\), \(\:\phi\:\left(x,t\right)\).

Let us acknowledge that these sought perturbed solutions, denoted as \(\:\psi\:\left(x,t\right)\), \(\:\phi\:\left(x,t\right)\), have the following form^61,62:

The symbols \(\:q\) and \(\:\varOmega\:\) represent the normalized wave number and the frequency of the perturbation, respectively. By performing substitution of Eq. (8) into Eq. (7) and isolating the coefficients in terms of\(\:{e}^{i\left(qx-{\Omega\:}t\right)}\) and \(\:{e}^{-i\left(qx-{\varOmega\:}^{*}t\right)}\), it leads to the formation of the following system of homogeneous equations:

To obtain the nontrivial solution of the system of equations developed within the context of Eq. (9), it is essential to compute the determinant formed by the matrix of coefficients derived from (9). The desired solutions are obtained by solving the relevant polynomial equation \(\:\varOmega\:=\varOmega\:\left(q\right)\) represented by:

with\(\:\gamma\:={R}_{f}C\), \(\:b=-M{\delta\:}^{2}{q}^{2}\), \(\:A=b+M{\delta\:}^{2}{q}^{2}-2\beta\:\sqrt{{P}_{0}}\left(1+a\right)+a\beta\:+3\beta\:{P}_{0}\), \(\:{m}_{4}={\gamma\:}^{4}\), \(\:{m}_{3}=-2i{\gamma\:}^{3}\left(A+B\right)\), \(\:B=b+M{\delta\:}^{2}{q}^{2}-2\beta\:\sigma\:\sqrt{{P}_{0}}\left(1+a\right)+a\beta\:+3\beta\:{\sigma\:}^{2}{P}_{0}\), \(\:{m}_{2}=-\left({\left(A+B\right)}^{2}+2AB-b+{b}^{2}\right)\), \(\:{m}_{1}=-i\gamma\:\left(\left(A+B\right)\left(2AB-b-{b}^{2}\right)\right)\), \(\:{m}_{0}={A}^{2}{B}^{2}-AB\left(b+{b}^{2}\right)+{b}^{4}\). Analytically solving Eq. (10) can be quite labor-intensive and require significant time investment. The subsequent steps involve numerical computations utilizing MATLAB software, which serves as an efficient platform for these mathematical applications. The linear equation presented above (10) illustrates the linear stability analysis of the steady-state solution, which is consistent with physical principles. The modulational instability domain encompasses a specific range of instability that exists under certain conditions. A steady-state is considered stable when \(\:\varOmega\:\) is a real number. Conversely, the presence of an imaginary \(\:\:\varOmega\:\) indicates a perturbation in the steady state solution, characterized by the exponential growth of disturbances. The gain spectrum associated with modulational instability is analyzed:

In this scenario, it is evident that the solution \(\:\varOmega\:\) is defined by Eq. (10). This section outlines several significant inter-information dependencies identified in specific experiments and subjected to thorough analysis.

Symmetric case: \(\:\sigma\:=1\)

The defined parameters for obtaining the MI graphs are reasonable and invite debates about certain physiological mechanisms. In fact, we carry out a deep examination to the mutual information graphs shown in the panels of Figs. 2 and 3. In fact, with certain constraints on the parameters of the nervous system, a zone of MI is obtained with a positive gain. In the given scenario where\(\:\:\sigma\:=1\) (denotes symmetry), \(\:a=0.25\) (\(\:a\) less than 1) and \(\:M=0.5\) (\(\:M\) less than 1), we notice the presence of a zone of MI as Binczak et al. have suggested³. From³, we have also noticed a jump like threshold that could be interpreted as a blockage of the impulse current (which actually can be a bad feature of real nerves), to this effect see Fig. 2a. In addition, sometimes at the same premises which were indicated above, when the value of \(\:a\) was equal to 0.75, one can see a stability region for \(\:g\left(q\right)=0\) as \(\:q\) approaches 0.75 or – 0.75 in Fig. 2b. Instead the gain \(\:g\left(q\right)\) is also reduced. As a result, when \(\:a=0.99\) (or \(\:a=1.01\)) \(\:M\) is lower than 1 \(\:\left(M=0.5\right)\) and MI lobes are clearly visible. And in Fig. 2c this occurrence happens when \(\:\eta\:<0\) considering a narrower band of disturbance of wave number \(\:q\) which is \(\:-0.25<q<0.25\). Likewise, on Fig. 2c, in the case of \(\:-0.75<q<-0.25\) \(\:\left(0.25<q<0.75\right)\), the \(\:\eta\:\) value is negative giving stable nervous impulse as. Also, for any value of the increased \(\:\eta\:\) or any value of \(\:q\) in the range \(\:q\in\:\left[-0.75,\:0.75\right]\), it is apparent from the physiological point of view that the operating parameters of the currently available range of the nervous system is the safety net for the system as seen in Fig. 2c. For complex phenomena, when the value of \(\:M\) is negative (regressive propagating impulse) two symmetric patterns of MI will occur for a value of \(\:a=0.25\) at which \(\:\eta\:\) is greater than 0 and \(\:q\) in the range − 2 to 2. When the value of \(\:\eta\:\) becomes negative, the gain \(\:g\left(q\right)\) takes on the value of zero. This means, the stability condition is still maintained even in the absence of MI area (as we can see in panels Fig. 2d–f). Additionally, when the value of the parameter \(\:a\) sticks at 0.75, the lobe of MI and the distance between these two lobes is said to go up. Usage of parameter \(\:a\to\:1\) also suggests that there exists a Brillouin’s zone, which corresponds to the stability with \(\:g\left(q\right)=0\), lying in between the two MI zones. There emerges a threshold value of the parameter above which the Brillouin’s zone corresponds to one of the MI zones. See Fig. 2 (e), (f) for this. We consider MI for such abnormal states of nerve fibers \(\:\left(M<0,\:\eta\:>0.5\right)\) and show that there is no risk of any injury of nerve impulses even under such conditions.

Anti-symmetric case: \(\:\sigma\:=-1\)

For example, in the case of anti-symmetry, when denoted by the would-be \(\:\sigma\:=-1\), and when \(\:M\) is greater than 0, the panels in Fig. 3a and b show that MI domains exist inside the Brillouin’s zone displays by\(\:-0.75<q<0.75\). It can be noticed that the observed domains are positively proportional to the amplitude of gain that the parameter \(\:a\) will take. But, while the parameter \(\:a\) comes close to 1, there is active interest in the range where \(\:q\) is limited from − 0.75 to 0.75, regardless of the \(\:\eta\:\) parameter value (which can be positive or negative), Fig. 3c will be referred to for further details. In addition, when the value of \(\:\eta\:\) is greater than \(\:0.5\) it is found out that the interconnected nerve fibers transmit nervous impulses through the assisting nerve fibers, without the help of amplification devices regardless of the value of \(\:q\) mentioned in the range of – 2 to 2. While \(\:M\) is less than 0, the MI zones drop down together with the amplitude of the gain which is lower in value of 0.25. As the value of \(\:a\) turns out to be bigger (for instance \(\:a=0.75\) so does the amplitude of the gain. This feature could be treated as a somatic mechanism for transmitting more complex signals (for example, signal integration) among axons and between hemispheres of the brain. Also, for\(\:\:0<\eta\:<2\) and\(\:-2\le\:q\le\:2\) signal transmitted through the two, inter-connecting nerve fibers does not diverge from steady state as \(\:g\left(q\right)=0\) validate. For\(\:\:\eta\:>2\), the condition that the Modulational Instability plot in Fig. 3f indicates two regions of instability which are surrounded by one stable area for all \(\:q\) values lying within \(\:\left[-0.75,\:0.75\right]\). With increase in the parameter \(\:a\) the area of instability decreases hence the maximum gain attained as displayed in Fig. 3c when compared with Fig. 3d and e. The limit of amplification region constricts however extend stability region tends that increasing the values of \(\:q\) that cluster on size-extreme interactively moderate area dilation or depletes the region, while values of \(\:q\) that clustering pose on the stability region sails, however. The MI analysis has as its principal purpose the act of verifying the model under study with respect to the existence of soliton-like solutions. In this, we have demonstrated that, under realistic circumstances, the MI zones are quite large, providing enough scope for the physiological system to generate solitonic impulses over a wide range of \(\:M\), \(\:\eta\:\), and \(\:q\) values. They have hitherto been seen in both symmetric and asymmetric cases, \(\:\sigma\:=1\) and \(\:\sigma\:=-1\) respectively, showing the presence of MI zones. They are caused by non-very normal values of \(\:M\) and \(\:\eta\:\), and possibly, in relation to anatomical, medical, and neurological diseases. In addition to these, we also offer some form of experimental evidence supporting the notion of MI lobes as being discernible, within particular limited parameter regimes of \(\:M\), \(\:\eta\:\), and \(\:q\). It confirms that, solitonic impulses can indeed be triggered towards the very specific parameters at which MI zones come into being. Hence, the pathological physiology of the nerve fibers gives rise to a different form of solitonic impulse.

Jacobian matrix and fixed points theory

Equation (5), as defined by the applicable mathematical model, allows for the determination of equilibrium points and the requisite Jacobian matrix, which aids in the analysis of the linear stability of the two interacting components governed by the system. From a mathematical perspective, conducting numerical experiments related to Eq. (5) or its transformed variants aids in establishing and understanding the existence of various useful solutions. In this context, we will examine the wave transformation characterized by the variables\(\:\:\zeta\:=k\left(x-\lambda\:t\right)\),\(\:\:u\left(x,t\right)=U\left(\zeta\:\right)\), \(\:v\left(x,t\right)=V\left(\zeta\:\right)\). The provided expressions generate a comparable autonomous dynamical system in the specified format:

By setting\(\:\frac{dU}{d\zeta\:}=0\),\(\:\:\frac{dY}{d\zeta\:}=0\),\(\:\:\frac{dV}{d\zeta\:}=0\) and \(\:\frac{dZ}{d\zeta\:}=0\) within the framework of Eq. (12), The solutions of (12) identify the equilibrium points, which are also referred to as fixed points. Upon successfully solving Eq. (12), the resulting fixed points \(\:{X}_{i}\left(i=\text{1,2},3,.,6\right)\) can be articulated as follows:

All fixed points collected are real and do not bear any peculiar conditions. Nevertheless, it is possible to use the determinant of the Jacobian matrix to determine the stability of the obtained fixed points. The study described in this paper is a linear stability analysis, and the mode of solution employs the usage of the Jacobian matrix \(\:\left({M}_{J}\right)\). This suggests that positive values for the determinant of the Jacobian matrix pertaining to any fixed point \(\:{X}_{i}\) would continue to positively support that fixed point irrespective of the movement of the system being considered. In case the determinant of the Jacobian matrix \(\:\left({\left|{M}_{J}\right|}_{{X}_{i}}\right)\) pertaining to this particular point is negative then, that particular point or the fixed point \(\:{X}_{i}\) is said to be unstable. The parameters of the equilibrium state represented in Eq. (12) can be restated in the following form:

The Jacobian matrix serves as a mathematical instrument commonly utilized to analyze the dynamics of autonomous systems, exemplified by the system represented in Eq. (14):

According to the results of the current study, it has been noted:

With \(\:\frac{\partial\:{F}_{1}}{\partial\:U}=0\), \(\:\frac{\partial\:{F}_{1}}{\partial\:Y}=1\), \(\:\frac{\partial\:{F}_{1}}{\partial\:V}=0\), \(\:\frac{\partial\:{F}_{1}}{\partial\:Z}=0\), \(\:\frac{\partial\:{F}_{2}}{\partial\:U}=\frac{\left(1-\eta\:\right)\beta\:\left(\left(U-a\right)\left(U-1\right)+U\left(U-1\right)+U\left(U-a\right)\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\frac{\partial\:{F}_{2}}{\partial\:Y}=\frac{{-R}_{f}Ck\lambda\:\left(1-\eta\:\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\),\(\:\:\frac{\partial\:{F}_{2}}{\partial\:V}=\frac{-\eta\:\beta\:\left(\left(V-a\right)\left(V-1\right)+V\left(V-1\right)+V\left(V-a\right)\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\frac{\partial\:{F}_{2}}{\partial\:Z}=\frac{{R}_{f}Ck\lambda\:\eta\:}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\frac{\partial\:{F}_{3}}{\partial\:U}=0\), \(\:\frac{\partial\:{F}_{3}}{\partial\:Y}=0\), \(\:\frac{\partial\:{F}_{3}}{\partial\:V}=0\), \(\:\frac{\partial\:{F}_{3}}{\partial\:Z}=1\),\(\:\:\frac{\partial\:{F}_{4}}{\partial\:U}=\frac{\eta\:\beta\:\left(\left(U-a\right)\left(U-1\right)+U\left(U-1\right)+U\left(U-a\right)\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\:\frac{\partial\:{F}_{4}}{\partial\:Y}=\frac{{-R}_{f}Ck\lambda\:\eta\:}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\frac{\partial\:{F}_{4}}{\partial\:V}=\frac{-\left(1-\eta\:\right)\beta\:\left(\left(V-a\right)\left(V-1\right)+V\left(V-1\right)+V\left(V-a\right)\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\), \(\:\frac{\partial\:{F}_{4}}{\partial\:Z}=\frac{{R}_{f}Ck\lambda\:\left(1-\eta\:\right)}{\left(1-2\eta\:\right)M{k}^{2}{\delta\:}^{2}}\). By evaluating the determinant of the Jacobian matrix \(\:{M}_{J}\), it yields:

In the context of analyzing the behavior of interconnected neural fibers and their fixed-point structures, the establishment of equilibrium points can be achieved through the application of directional matrix methods, which involve determining the sign of the Jacobian matrix’s determinant at each invariant point. In this framework, calculating the determinant of a Jacobian matrix is crucial as it elucidates the stability of each of the analytical conditions established. The determinant of the Jacobian matrix in each iteration addresses the constraints imposed by the injection of \(\:2\eta\:-1\) and a\(\:-1\), thereby establishing the values of specific parameters within the nervous system, namely \(\:\eta\:\) and \(\:a\). As an example,

Case 1

η < 0.5 (2η – 1 < 0). According to the value of, a (a < 1, a > 1), |M_J|_Xi(i=1,2,3)<0, then, the related fixed points are overall unstable, whereas, |M_J|_X6 > 0 makes up X₆ a stable fixed point. In addition, for a < 1, |M_J|_X4 < 0, so X₄ is an unsteadyequilibrium point, while, |M_J|_X5 > 0 and consequently, X₅ is a stable fixed point.Unlike, if a > 1, |M_J|_X4 > 0, thus X₄ is a steady fixed point, however, |M_J|_X5 < 0therefore, X₅ is an unstable fixed point.

Case 2

η > 0.5 (2η – 1 > 0). Whatever the value of a (a < 1, a > 1), |M_J|_Xi(i=1,2,3) > 0,thus,, the linked fixed points are overall stable, whereas, |M_J|_X6 < 0, X₆ stands anunsteady equilibrium point. Furthermore, if a < 1, |M_J|_X4 > 0, so X₄ is a steadyequilibrium point, whereas, |M_J|_X5 < 0 and accordingly, X₅ is an unstable fixed point.However, for a > 1, |M_J|_X4 < 0, then, X₄ is an unsteady fixed point, whereas, X₅ is astable fixed point because |M_J|_X5 > 0.