Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical Justification, Dynamical Analysis and Neurocomputational Implications

In this chapter, the dynamical behavior of the incommensurate fractional-order FitzHugh-Nagumo model of neuron is explored in details from local stability analysis. First of all, considering that the FitzHugh-Nagumo model is a mathematical simplification of the Hodgkin-Huxley model, the considered model is derived from the fractional-order Hodgkin-Huxley model obtained taking advantage of the powerfulness of fractional derivatives in modeling certain biophysical phenomena as the dielectrics losses in cell membranes, and the anomalous diffusion of particles in ion channels. Then, it is shown that the fractional-order FitzHugh-Nagumo model can be simulated by a simple electrical circuit where the capacitor and the inductor are replaced by corresponding fractional-order electrical elements. Then, the local stability of the model is studied using the Theorem on the stability of incommensurate fractional-order systems combined with the Cauchy’s argument Principle. At last, the dynamical behavior of the model are investigated, which confirms the results of local stability analysis. It is found that the simple model can exhibit, among others, complex mixed mode oscillations, phasic spiking, first spike latency, and spike timing adaptation. As the dynamical richness of a neuron expands its computational capacity, it is thus obvious that the fractional-order FitzHugh-Nagumo model is more computationally efficient than its integer-order counterpart

Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations

[EN] In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Mixed mode oscillations in a conceptual climate model

Much work has been done on relaxation oscillations and other simple oscillators in conceptual climate models. However, the oscillatory patterns in climate data are often more complicated than what can be described by such mechanisms. This paper examines complex oscillatory behavior in climate data through the lens of mixed-mode oscillations. As a case study, a conceptual climate model with governing equations for global mean temperature, atmospheric carbon, and oceanic carbon is analyzed. The nondimensionalized model is a fast/slow system with one fast variable (corresponding to ice volume) and two slow variables (corresponding to the two carbon stores). Geometric singular perturbation theory is used to demonstrate the existence of a folded node singularity. A parameter regime is found in which (singular) trajectories that pass through the folded node are returned to the singular funnel in the limiting case where $\epsilon = 0$. In this parameter regime, the model has a stable periodic orbit of type $1^s$ for some $s>0$. To our knowledge, it is the first conceptual climate model demonstrated to have the capability to produce an MMO pattern.Comment: 28 pages, 11 figure

Emergence of mixed mode oscillations in random networks of diverse excitable neurons: the role of neighbors and electrical coupling

In this paper, we focus on the emergence of diverse neuronal oscillations arising in a mixed population of neurons with different excitability properties. These properties produce mixed mode oscillations (MMOs) characterized by the combination of large amplitudes and alternate subthreshold or small amplitude oscillations. Considering the biophysically plausible, Izhikevich neuron model, we demonstrate that various MMOs, including MMBOs (mixed mode bursting oscillations) and synchronized tonic spiking appear in a randomly connected network of neurons, where a fraction of them is in a quiescent (silent) state and the rest in self-oscillatory (firing) states. We show that MMOs and other patterns of neural activity depend on the number of oscillatory neighbors of quiescent nodes and on electrical coupling strengths. Our results are verified by constructing a reduced-order network model and supported by systematic bifurcation diagrams as well as for a small-world network. Our results suggest that, for weak couplings, MMOs appear due to the de-synchronization of a large number of quiescent neurons in the networks. The quiescent neurons together with the firing neurons produce high frequency oscillations and bursting activity. The overarching goal is to uncover a favorable network architecture and suitable parameter spaces where Izhikevich model neurons generate diverse responses ranging from MMOs to tonic spiking

Mechanisms leading to bursting oscillations in the system of predator–prey communities coupled by migrations

The purpose is to study the periodic regimes of the dynamics for two non-identical predator–prey communities coupled by migrations, associated with the partial synchronization of fluctuations in the abundance of communities. The combination of fluctuations in neighboring sites leads to the regimes that include both fast bursts (bursting oscillations) and slow oscillations (tonic spiking). These types of activity are characterized by a different ratio of synchronous and non-synchronous dynamics of communities in certain periods of time. In this paper, we describe scenarios of the transition between different types of burst activity. These types of dynamics differ from each other not so much in size, shape, and number of spikes in a burst, but in the order of these bursts relative to the slow-fast cycle. Methods. To study the proposed model, we use the bifurcation analysis methods of dynamic systems, as well as geometric methods based on the division of the full system into fast and slow equations (subsystems). Results. We showed that the dynamics of the first subsystem with a slow-fast limit cycle directly determines the dynamics of the second one with burst activity through a smooth dependence of regime on the number of predators and a non-smooth dependence on the number of prey. We constructed the invariant manifolds on which there are parts of dynamics with tonic (slow manifold) and burst (fast manifold) activity of the full system. Conclusion. We described the scenario for bursting with different waveforms, which are determined by the appearance of the fast invariant manifold and the location of its parts relative to the slow-fast cycle. The transitions between different types of burst are accompanied by a change in the oscillation period, the degree of synchronization, and, as a result, the dynamics becomes quasi-periodic when both communities are not synchronous with each other

Abrupt Changes and Intervening Oscillations in Conceptual Climate Models

This thesis examines the role of fast/slow dynamics in understanding the mechanisms behind oscillatory patterns found in paleoclimate data. Fast/slow systems often exhibit rapid transitions between metastable states, and understanding these transitions is important to understanding climate phenomena. However, these rapid changes in the state of the system implicitly require examining trajectories that enter a region of phase space where the basic theory used to analyze fast/slow systems no longer applies. The content of this thesis examines the non-standard behavior arising from the break down of the theory, typically appearing in the form of small amplitude oscillations due to canard trajectories. First, canard theory is extended to piecewise-smooth systems. Conditions are found in which canard behavior is similar to that of smooth systems. Additionally, the dynamics are classified when these conditions are not met. Second, the new theory is used to analyze a variation on Stommel's model of large-scale ocean circulation, showing that the model is capable of exhibiting both canards and relaxation oscillations. Another variation of Stommel's model with an extra phase-space dimension is also demonstrated to exhibit relaxation oscillations. Finally, a model for glacial-interglacial cycles is analyzed through the lens of mixed-mode oscillations. The model is demonstrated to exhibit complicated oscillations due to a generalized canard phenomenon.Doctor of Philosoph

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Six Decades of Flight Research: An Annotated Bibliography of Technical Publications of NASA Dryden Flight Research Center, 1946-2006

Titles, authors, report numbers, and abstracts are given for nearly 2900 unclassified and unrestricted technical reports and papers published from September 1946 to December 2006 by the NASA Dryden Flight Research Center and its predecessor organizations. These technical reports and papers describe and give the results of 60 years of flight research performed by the NACA and NASA, from the X-1 and other early X-airplanes, to the X-15, Space Shuttle, X-29 Forward Swept Wing, X-31, and X-43 aircraft. Some of the other research airplanes tested were the D-558, phase 1 and 2; M-2, HL-10 and X-24 lifting bodies; Digital Fly-By-Wire and Supercritical Wing F-8; XB-70; YF-12; AFTI F-111 TACT and MAW; F-15 HiDEC; F-18 High Alpha Research Vehicle, F-18 Systems Research Aircraft and the NASA Landing Systems Research aircraft. The citations of reports and papers are listed in chronological order, with author and aircraft indices. In addition, in the appendices, citations of 270 contractor reports, more than 200 UCLA Flight System Research Center reports, nearly 200 Tech Briefs, 30 Dryden Historical Publications, and over 30 videotapes are included