Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model. Fractional differentiation by neocortical pyramidal neurons. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Impulses and physiological states in theoretical models of nerve membrane. In Abstract and Applied Analysis Hindawi: London, UK, 2020 Volume 2020. On One Method of Studying Spectral Properties of Non-selfadjoint Operators. Asymptotics of eigenvalues for differential operators of fractional order. Stability analysis on a class of nonlinear fractional-order systems. Stability properties of two-term fractional differential equations. Fractional dynamical system and its linearization theorem. On stability of commensurate fractional order systems. In Proceedings of the 2018 41st International Conference on Telecommunications and Signal Processing (TSP), Athens, Greece, 4–6 July 2018 pp. Stability of Linear Systems with Caputo Fractional-, Variable-Order Difference Operator of Convolution Type. ![]() Fractional Linear Equations with Discrete Operators of Positive Order. Stability by linear approximation and the relation between the stability of difference and differential fractional systems. Explicit criteria for stability of fractional h-difference two-dimensional systems. Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model. Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model. The role of fractional calculus in modeling biological phenomena: A review. Fractional Relaxation-Oscillation and Fractional Phenomena. ![]() Fractal rheological models and fractional differential equations for viscoelastic behavior. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. On the role of fractional calculus in electromagnetic theory. A novel exact representation of stationary colored Gaussian processes (fractional differential approach). The authors declare no conflict of interest. Still, when compared to fractional-order differential equations, their discrete-time counterparts, namely fractional-order difference equations, have not received the same amount of attention, especially regarding their qualitative theory and, in particular, their stability and instability properties. Stability results for linear systems with Caputo fractional-order difference equations with variable order of convolution type have been studied in, along with the recurrence formulas for the solutions of linear initial value problems for the considered fractional operators. ![]() The solutions of initial value problems that involve fractional-order Caputo-type and Riemann-Liouville-type difference equations with positive orders were given in, but a qualitative study regarding the stability properties of these types of equations or systems of several equations have not been explored. Stability and linearization results have been recently explored for continuous-time fractional-order systems in, as well as for discrete-time fractional systems. 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. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. 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. 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.
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