Search Results (10)

Transcranial electrical stimulation (tES) is increasingly recognized for its potential to modulate cerebral blood flow (CBF) and evoke cerebrovascular reactivity (CVR), which are crucial in conditions like mild cognitive impairment (MCI) and dementia. This study explores the impact of tES on the neurovascular unit (NVU), employing a physiological modeling approach to simulate the vascular response to electric fields generated by tES. Utilizing the FitzHugh–Nagumo model for neuroelectrical activity, we demonstrate how tES can initiate vascular responses such as vasoconstriction followed by delayed vasodilation in cerebral arterioles, potentially modulated by a combination of local metabolic demands and autonomic regulation (pivotal locus coeruleus). Here, four distinct pathways within the NVU were modeled to reflect the complex interplay between synaptic activity, astrocytic influences, perivascular potassium dynamics, and smooth muscle cell responses. Modal analysis revealed characteristic dynamics of these pathways, suggesting that oscillatory tES may finely tune the vascular tone by modulating the stiffness and elasticity of blood vessel walls, possibly by also impacting endothelial glycocalyx function. The findings underscore the therapeutic potential vis-à-vis blood-brain barrier safety of tES in modulating neurovascular coupling and cognitive function needing the precise modulation of NVU dynamics. This technology review supports the human-in-the-loop integration of tES leveraging digital health technologies for the personalized management of cerebral blood flow, offering new avenues for treating vascular cognitive disorders. Future studies should aim to optimize tES parameters using computational modeling and validate these models in clinical settings, enhancing the understanding of tES in neurovascular health. Full article

In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order delay-coupled FitzHugh–Nagumo neural model is acquired. By designing a proper $P{D}^p$ controller, we can efficaciously control the stability domain and the time of emergence of the bifurcation phenomenon of the considered fractional delay-coupled FitzHugh–Nagumo neural model. By exploiting a reasonable hybrid controller, we can successfully adjust the stability domain and the bifurcation onset time of the involved fractional delay-coupled FitzHugh–Nagumo neural model. This study shows that when the delay crosses a critical value, a Hopf bifurcation will arise. When we adjust the control parameter, we can find other critical values to enlarge or narrow the stability domain of the fractional-order delay-coupled FitzHugh–Nagumo neural model. In order to check the correctness of the acquired outcomes of this article, we present some simulation outcomes via Matlab 7.0 software. The obtained theoretical fruits in this article have momentous theoretical significance in running and constructing networks. Full article

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters. Full article

This paper presents sufficient conditions for the existence of a bifurcation point for nonlinear periodic third-order fully differential equations. In short, the main discussion on the parameter s about the existence, non-existence, or the multiplicity of solutions, states that there are some critical numbers ${\sigma }_0$ and ${\sigma }_1$ such that the problem has no solution, at least one or at least two solutions if $s<{\sigma }_0$, $s={\sigma }_0$ or ${\sigma }_0>s>{\sigma }_1,$ respectively, or with reversed inequalities. The main tool is the different speed of variation between the variables, together with a new type of (strict) lower and upper solutions, not necessarily ordered. The arguments are based in the Leray–Schauder’s topological degree theory. An example suggests a technique to estimate for the critical values ${\sigma }_0$ and ${\sigma }_1$ of the parameter. Full article

In this paper, the transport phenomena of synaptic electric impulses are considered. The FitzHugh–Nagumo and FitzHugh–Rinzel models appear mathematically appropriate for evaluating these scientific issues. Moreover, applications of such models arise in several biophysical phenomena in different fields such as, for instance, biology, medicine and electronics, where, by means of nanoscale memristor networks, scientists seek to reproduce the behavior of biological synapses. The present article deals with the properties of the solutions of the FitzHugh–Rinzel system in an attempt to achieve, by means of a suitable “energy function”, conditions ensuring the boundedness and existence of absorbing sets in the phase space. The results obtained depend on several parameters characterizing the system, and, as an example, a concrete case is considered. Full article

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters. Full article

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences. Full article

In this paper, we consider the second order discontinuous differential equation in the real line, ${\left(a\left(t,u\right)\varphi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u,u^{\prime }\right),a.e.t\in \mathbb{R},u(-\infty )={\nu }^{-},u(+\infty )={\nu }^{+},$ with $\varphi$ an increasing homeomorphism such that $\varphi \left(0\right)=0$ and $\varphi \left(\mathbb{R}\right)=\mathbb{R}$ , $a\in C({\mathbb{R}}^2,\mathbb{R})$ with $a(t,x)>0$ for $(t,x)\in {\mathbb{R}}^2$ , $f:{\mathbb{R}}^3\to \mathbb{R}$ a $L^1$ -Carathéodory function and ${\nu }^{-},{\nu }^{+}\in \mathbb{R}$ such that ${\nu }^{-}<{\nu }^{+}$ . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities $\varphi$ and $f$ . To the best of our knowledge, this result is even new when $\varphi \left(y\right)=y$ , that is for equation ${\left(a\left(t,u\left(t\right)\right)u^{\prime }\left(t\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),a.e.t\in \mathbb{R}$ . Moreover, these results can be applied to classical and singular $\varphi$ -Laplacian equations and to the mean curvature operator. Full article

The aim of this paper is to show that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. By using the concept of Kolmogorov–Nagumo quasi-linear means, we prove this with the help of Darótzy’s mapping theorem. Our point is further illustrated with a number of explicit examples. Other salient issues, such as connections of conditional entropies with the de Finetti–Kolmogorov theorem for escort distributions and with Landsberg’s classification of non-extensive thermodynamic systems are also briefly discussed. Full article

In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D_4^-symmetries. Full article