Neuronal Dynamics of an Intrinsically Bursting Neuron Through the Caputo–Fabrizio Fractal–Fractional Hodgkin–Huxley Model

Stress wave in the mesoscopic discontinuous medium by fractional approach

This study investigates the nonlinear and chaotic dynamics of a modified Hindmarsh–Rose (HR) neuron model through a unified framework based on piecewise fractional differential operators. The classical HR neuron model is widely used to describe the electrical activity of neuronal membranes; however, it does not fully account for memory effects and regime‐switching phenomena observed in real neurobiological processes. To address this limitation, we formulate a piecewise dynamical model in which the membrane potential evolves under different fractional operators, including the Caputo, Atangana–Baleanu, and Caputo–Fabrizio derivatives. A fractional order parameter is introduced to regulate the memory intensity of the system and to construct a generalized fractional‐order HR model. The piecewise structure enables the modeling of switching behaviors and crossover effects between distinct neuronal activity regimes. Numerical simulations are carried out to examine the influence of the fractional order and the piecewise operator structure on neuronal firing patterns, chaotic attractors, and complex oscillatory dynamics under external current stimulation. The numerical results clearly show that changes in the fractional order and the choice of switching thresholds have a pronounced influence on the stability properties, oscillation amplitudes, and chaotic behavior of the neuron model. Through a broad set of numerical experiments, the proposed framework is shown to reliably reproduce key features of neuronal dynamics, demonstrating that the adopted numerical schemes are well suited for capturing memory effects and regime switching. Overall, this study advances fractional and piecewise modeling approaches in neuroscience by providing both new theoretical perspectives and practical computational tools for the investigation of complex neuronal systems.

We investigate pattern formation driven by Turing instability in the Lengyel–Epstein model incorporating the fractional Laplacian operator. This approach simulates anomalous superdiffusion, where particle clouds spread more rapidly than predicted by the classical diffusion models. Such phase phenomena are often observed in viruses or biochemical substances that require unusually rapid diffusion. Our study allows for the estimation of superdiffusion effects in biochemistry or thermodynamics by analyzing temporal changes in fractional dimensions that exhibit abnormal patterns. Specifically, we employ the fractal dimension to quantify the complexity of these patterns using the box-counting method. Since this method is based on examining neighboring patterns, that is self-similarity, we observe that the formation of patterns varies with the bifurcations induced by Turing instability. Interestingly, despite these instabilities and variations, the fractal dimension stabilizes to a certain value once the patterns are formed. We also investigate how fractional diffusion affects the fractal dimension. Numerical experiments were performed with various initial conditions, including random values and circles centered with a fixed distance. The fractal dimension was calculated for Turing instability of the Lengyel–Epstein model, and is analyzed for both classical and fractional diffusion following a Lévy process. These tests revealed the formation of different patterns under the Turing instability by varying parameters, including a time variable [Formula: see text], and mass-conserving the chemical concentrations of the activator iodide [Formula: see text] and the inhibitor chlorite [Formula: see text]. Then, we analyzed these results in terms of the fractal dimension of the patterns. Furthermore, we analyze the consistency with the mesh refinement numerically and examine parameter sensitivity. Our numerical simulation of the fractional-Lengyel–Epstein model examined the relationship between the fractal dimension and the parameters [Formula: see text], [Formula: see text], and the fractional order [Formula: see text], while holding the scaling parameter [Formula: see text] as a constant. In general, in regions of instability, higher values of parameter [Formula: see text] are associated with lower fractal dimensions, while higher values of the parameter [Formula: see text] lead to an increase in fractal dimension. In addition, as the fractional order [Formula: see text] increases, the fractal dimension tends to decrease. Although specific parameter values around the bifurcation point influence the fractal dimension, the overall evolutionary trend follows the aforementioned patterns.

We simulated the neuronal electrical activity using the Hodgkin-Huxleymodel (HH) and a superconductor circuit, containing Josephson junctions. These HH model make possible simulate the main neuronal dynamics characteristics such as action potentials, firing thres hold and refractory period.The purpose of the manuscript is show a method to syncronize a RCL-shunted Josephson junction to a neuronal dynamics represented by the HH model. Thus the RCLSJ circuit is able to mimics the behavior of the HH neuron. We controlated the RCLSJ circuit, using and improved adaptative track scheme, that with the improved Lyapunov functions and thetwo controllable gain coefficients allowing synchronization of two neuronal models. Results will provide the path to follow forward the understanding neuronal networks synchronization about, generating the intrinsic brain behavior.

The human brain is an endlessly fascinating organ from either the perspective of its structural complexity, originating in the astronomical number of neurons and connections, or the complexity of the tasks which it executes seemingly effortlessly, challenging science and engineering. Understanding the dynamics of individual neurons and neuronal networks is a key ingredient for understanding how the brain works. Neurons can operate in a wide variety of regimes of electrical activity, which roughly could be described as rest states, sub-threshold oscillations, spiking, and bursting. The latter four types of regimes represent oscillatory activities. They could be periodic, quasi-periodic, or chaotic. The bursting regime is an oscillatory activity consisting of intervals of repetitive spiking separated by intervals of rest. It embodies a prominent manifestation of the complexity of neural dynamics based on ionic currents operating on different time scales. This special issue of the Journal of Biological Physics highlights approaches using biophysically accurate modeling, thorough analysis of the dynamics, and classification of the parameter regimes. A number of articles presented here apply bifurcation analysis to gain new insights into how neural systems operate. This assembly of articles substantiates the assertion that methods developed in the theory of dynamical systems are essential components of neuroscience research. In this context, I would also like to underscore prime advantages that are presented by the invertebrate nervous system, where single neurons can be identified by location, morphology, and activity from preparation to preparation and their dynamics can be analyzed individually. Three articles in this issue investigate the dynamics of specific neurons and neuronal networks in the medicinal leech. This special issue spotlights the bursting regime and its role in the nervous system’s functions and pathologies. Brain functions such as information processing, memory formation, and motor control frequently engage oscillatory dynamics of neurons. The functional and pathological roles of bursting regimes have been intensively investigated. Most clearly, it is the key regime for the control of rhythmic movements. Bursting activity is ubiquitously recorded in central pattern generators, oscillatory neuronal networks controlling motor behaviors such as breathing, locomotion, or heartbeat in invertebrates. Also, bursting has been widely observed in sleep and pathological brain states, like those associated with epilepsy syndromes. The main theme of this special issue is the elucidation of the roles of bursting regimes in the operation of the central nervous system under normal and pathological conditions. In conclusion, it clearly required expertise developed in different fields of science and to that end this issue deliberately gathered articles from neuroscientists with different backgrounds, ranging from physics, chemistry, and mathematics to ethology, who are interested in the dynamics of bursting regimes. In acknowledgment, I would like to express my greatest appreciation for the tireless hard work and constant support and encouragements of Sonya Bahar, Editor-in-Chief, Maria Bellantone, Springer senior publishing editor, and Mieke van der Fluit, Springer senior publishing assistant.

The components of complex bioelectrical activity ? action potential (AP), interspike interval (ISI) and the quiet interburst interval (IBI), along with the effects of 2.7 mT and 10 mT static magnetic fields, were identified and examined in the spontaneously active Br neuron of the subesophageal ganglion complex of the garden snail Helix pomatia by fractal analysis using Higuchi?s fractal dimension (FD). The normalized mean of the empirical FD distributions of bursting activity of the Br neuron were formed under different experimental conditions: before (Control), during (MF), and after exposure to the static magnetic field (AMF). Using the fractal analysis method for the first time, a separation of the AP, ISI and IBI components was successfully achieved. Our results show that fractal analysis with deconvolution of the normalized mean of the empirical FD distributions into Gaussian functions is a useful tool for revealing the effects of magnetic fields on the complexity of neuronal bioelectric activity and its AP, ISI and IBI components.

In this brief, indirect design estimation of fractional order systems is proposed. In indirect fractional order approach, fractional order plant is shifted in the frequency domain and the equivalent plant is modeled by employing binomial approximation. The equivalent fractional order plant obtained is used for the design of the controller. While designing fractional order controllers, robustness and handling sensitivity to parametric variations are of key importance. Therefore, indirect fractional order approach is used, which gives the flexibility to adjust the maximum sensitivity according to the system requirements and eliminates the need for external IMC filter. IMC filter added externally results in an additional phase lag of the system. The proposed design method excludes external IMC filter and the binomial expansion used helps in achieving the internal filter which results in better transient response. This is termed as Indirect Fractional order IMC based fractional order PID controller.

Mathematical models such as Fitzhugh-Nagoma and Hodgkin-Huxley models have been used to understand complex nervous systems. Still, due to their complexity, these models have made it challenging to analyze neural function. The discrete Rulkov model allows the analysis of neural function to facilitate the investigation of neuronal dynamics or others. This paper introduces a fractional memristor Rulkov neuron model and analyzes its dynamic effects, investigating how to improve neuron models by combining discrete memristors and fractional derivatives. These improvements include the more accurate generation of heritable properties compared to full-order models, the treatment of dynamic firing activity at multiple time scales for a single neuron, and the better performance of firing frequency responses in fractional designs compared to integer models. Initially, we combined a Rulkov neuron model with a memristor and evaluated all system parameters using bifurcation diagrams and the 0-1 chaos test. Subsequently, we applied a discrete fractional-order approach to the Rulkov memristor map. We investigated the impact of all parameters and the fractional order on the model and observed that the system exhibited various behaviors, including tonic firing, periodic firing, and chaotic firing. We also found that the more I tend towards the correct order, the more chaotic modes in the range of parameters. Following this, we coupled the proposed model with a similar one and assessed how the fractional order influences synchronization. Our results demonstrated that the fractional order significantly improves synchronization. The results of this research emphasize that the combination of memristor and discrete neurons provides an effective tool for modeling and estimating biophysical effects in neurons and artificial neural networks.

Modeling and analysis of the polluted lakes system with various fractional approaches

An indicator for the electrode aging of lithium-ion batteries using a fractional variable order model

<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.</p> </abstract>

In this paper, we design and simulate a Single-Ended Colpitts Oscillator (CO) in integer and fractional order domains. The oscillators are realized in 32 nm node conventional MOSFET and carbon nanotube field effect transistor (CNTFET) technologies. Therefore, four COs have been designed, simulated and rigorously compared. These include integer order conventional MOSFET-based CO, fractional order MOSFET-based CO, CNTFET-based integer order CO and CNTFET-based fractional order CO, all based on 32-nm technology nodes. The fractional order approach has been used as it results in better control over the phase and frequency of the oscillator. Herein, fractional order capacitors of various orders, used in realizing the fractional order COs, are realized and their frequency responses are studied. This is being done to ensure whether the designed pseudo-capacitances have fractional behavior or not in the desired frequency and phase spectrum. It has been observed that the variation of fractional order [Formula: see text] from 0.4 to 0.81 has resulted in a slight reduction of oscillation frequency from 1.68 GHz ([Formula: see text]) to 1.351 GHz ([Formula: see text]) keeping the pseudo-capacitance same at 0.3 nF in MOS-based topology. Since the fractional order realization increases the circuit complexity and power consumption, therefore, CNTFET-based integer order as well as fractional order COs have been designed. The CNTFET-based fractional order CO retains the advantages of the fractional order domain as well as the power efficiency of CNTFETs. Further, it has been observed that integrating fractional-order capacitor (FOC) with the CNTFET CO results in a much larger constant phase zone (CPZ), an important performance measuring parameter. A rigorous comparative analysis of the four COs designed in this work has been performed.

A physical interpretation of fractional viscoelasticity based on the fractal structure of media: Theory and experimental validation

Efficient simulation of large-scale mammalian brain models provides a crucial computational means for understanding complex brain functions and neuronal dynamics. However, such tasks are hindered by significant computational complexities. In this work, we attempt to address the significant computational challenge in simulating large-scale neural networks based on biophysically plausible Hodgkin-Huxley (HH) neuron models. Unlike simpler phenomenological spiking models, the use of HH models allows one to directly associate the observed network dynamics with the underlying biological and physiological causes, but at a significantly higher computational cost. We exploit recent commodity massively parallel graphics processors (GPUs) to alleviate the significant computational cost in HH model based neural network simulation. We develop look-up table based HH model evaluation and efficient parallel implementation strategies geared towards higher arithmetic intensity and minimum thread divergence. Furthermore, we adopt and develop advanced multi-level numerical integration techniques well suited for intricate dynamical and stability characteristics of HH models. On a commodity GPU card with 240 streaming processors, for a neural network with one million neurons and 200 million synaptic connections, the presented GPU neural network simulator is about 600X faster than a basic serial CPU based simulator, 28X faster than the CPU implementation of the proposed techniques, and only two to three times slower than the GPU based simulation using simpler phenomenological spiking models.

In this paper, we apply the fractal-fractional derivative in the Atangana-Baleanu sense to a model of the human immunodeficiency virus infection of CD$ 4^{+} $ T-cells in the presence of a reverse transcriptase inhibitor, which occurs before the infected cell begins producing the virus. The existence and uniqueness results obtained by applying Banach-type and Leray-Schauder-type fixed-point theorems for the solution of the suggested model are established. Stability analysis in the context of Ulam's stability and its various types are investigated in order to ensure that a close exact solution exists. Additionally, the equilibrium points and their stability are analyzed by using the basic reproduction number. Three numerical algorithms are provided to illustrate the approximate solutions by using the Newton polynomial approach, the Adam-Bashforth method and the predictor-corrector technique, and a comparison between them is presented. Furthermore, we present the results of numerical simulations in the form of graphical figures corresponding to different fractal dimensions and fractional orders between zero and one. We analyze the behavior of the considered model for the provided values of input factors. As a result, the behavior of the system was predicted for various fractal dimensions and fractional orders, which revealed that slight changes in the fractal dimensions and fractional orders had no impact on the function's behavior in general but only occur in the numerical simulations.