The synchronization of neural activity to external rhythms is fundamental to information processing and network coordination. However, classical integer-order models often neglect the power-law adaptation and memory effects inherent in biological tissues, such as ion channel gating and membrane impedance. This study investigates the synchronization dynamics of a periodically forced FitzHugh–Nagumo system, incorporating fractional-order calculus to explicitly model these history-dependent behaviors. We also introduce an asymmetry into the model by assigning distinct fractional orders to the fast membrane potential and the slow recovery variable to isolate their specific biophysical contributions.

The system is numerically investigated across a broad two-dimensional parameter space of forcing amplitude and frequency. We employ rotation numbers and inter-spike interval statistics to systematically map phase-locking zones—known as Arnold tongues—and to classify complex dynamical regimes versus stable periodic firing.

The investigation reveals that the two fractional orders play fundamentally different, non-interchangeable roles in shaping the entrainment landscape. Introducing memory into the voltage dynamics acts primarily as a stabilizing damping mechanism, significantly suppressing chaos and higher-order locking in favor of robust 1:1 synchronization. In contrast, increasing memory in the recovery variable functions as a timescale modulator, systematically shifting the system's resonant frequency and entrainment windows toward lower bands. Furthermore, we observe that these effects can compete; recovery memory can partially restore dynamical complexity to a system otherwise simplified by voltage memory.

We conclude that fractional-order asymmetry serves as a flexible biophysical mechanism for tuning neuronal response. By independently adjusting stability and frequency selectivity, neurons can optimize their entrainment to rhythmic stimuli, offering new theoretical insights into the modulation of neural circuits and the potential for targeted neuromodulation strategies.

This study presents a computational framework designed to analyze the intrinsic structure of fractional differential equations (FDEs) by mapping them onto optimal ordinary differential equation (ODE) architectures. While FDEs are instrumental in modeling non-local dynamics and memory effects, analyzing their fundamental structural properties—distinct from obtaining numerical solutions—remains a complex challenge. The proposed methodology transforms FDEs involving iterated Caputo derivatives into a higher-order representation comprised of two distinct elements: a parameterizable integer-order polynomial component and a residual fractional power series.

Crucially, this approach posits that the integer-order component is not a static derivation but a flexible, parameterizable architecture. By employing global optimization techniques, specifically Particle Swarm Optimization (PSO), this framework searches for a polynomial structure that minimizes a dual objective: the deviation from a high-fidelity reference solution and the magnitude of the truncated residual series. This process effectively identifies the most parsimonious ODE that captures the dominant dynamics of the fractional system.

Numerical experiments conducted on both a linear FDE and a nonlinear fractional Riccati equation demonstrate the framework's efficacy. The results reveal that the optimal integer-order representation often requires a higher polynomial degree than the forcing function of the original FDE to compensate for the transformation of the fractional operator. For instance, a quadratic ODE architecture was identified as the optimal representation for a linear FDE. This work provides a novel tool for model reduction and structural analysis, offering deeper insights into the interplay between local integer-order dynamics and non-local fractional memory effects.

The Klein–Gordon equation with cubic nonlinearity is studied in theoretical and numerical ways. The master equation is known to exhibit spontaneous symmetry breaking. Based on the high-precision numerical method [1] consisting of the Fourier spectral method for space and the implicit Runge–Kutta method for time, the detailed structure of dynamical systems around Lyapunov-stable stationary states is studied in Ref. [2]. Here, it is notable that the appearance of breather waves is suggested to be associated with the stability of Lyapunov functions for stationary solutions [3]; more precisely, the breather solution appears in the critical situation when the constant stationary state starts to lose its stability. In this paper, which is also based on the high-precision numerical method [1], finite-dimensional scatter plots are introduced to infinite-dimensional dynamical systems of coupled nonlinear Klein–Gordon equations. The proposed plot is a kind of finite-dimensional representation of originally infinite-dimensional dynamical systems [4]. Scatter plots show a wide variety of geometric shapes, which sometimes have a fractal structure. In particular, the difference between the global existence and finite-time blow-up of solutions is illustrated in a geometric way.

References
[1] Y. Takei, Y. Iwata, Axioms 2022, 11(1), 28
[2] Y. Takei, Y. Iwata, Springer Proceedings in Mathematics and Statistics (SPMS), accepted; arXiv:2509.12272
[3] Y. Iwata and Y. Takei, AIP conf. proc., accepted; arXiv:2309.00822
[4] Y. Iwata, Y. Takei, AIP Conf. Proc.; accepted, arXiv:2309.0082

This study presents a generalized integral transform (GIT) approach for analyzing and modeling a wide class of fractional-order mathematical systems arising in physical, engineering, and biological sciences. The investigated models include Newton’s law of cooling, the logistic population growth equation, and the blood alcohol concentration model, each formulated using distinct fractional derivatives such as the Caputo, Caputo–Fabrizio (CF), modified Atangana–Baleanu–Caputo (mABC), and constant proportional Caputo (CPC) derivatives. These fractional operators effectively describe memory-dependent and non-local characteristics inherent in many natural and engineered processes. Analytical solutions of the proposed models are derived through the generalized transform method, and graphical illustrations are provided to demonstrate the influence of various fractional orders on system dynamics. The results reveal that the GIT technique offers a unified, powerful, and efficient framework for solving a broad spectrum of fractional differential equations with diverse kernels. Furthermore, it integrates several classical and modern transforms—including the Laplace, Sumudu, Elzaki, and Formable transforms—as special cases, thereby simplifying computation and enhancing both generality and adaptability. This unified formulation provides researchers with a flexible analytical tool capable of addressing diverse problems without redefining operators for each model. The comparative analysis validates the stability, accuracy, and consistency of the proposed technique. Overall, this work highlights the efficacy, robustness, and broad applicability of the generalized integral transform, establishing a firm foundation for future explorations of hybrid fractional models and their interdisciplinary applications.

The regular logistic map was introduced in 1960s, served as an example of a complex system, and was used as an instrument to demonstrate and investigate the period-doubling cascade of bifurcations scenario of transition to chaos. The first fractional generalization was introduced in 2002. The two generalizations which are based on the continuous and discrete Caputo fractional calculus are the fractional logistic map (FLM) and the fractional difference logistic map (FDLM). They are well investigated. The finite time evolution of the FLM and FDLM is characterized by cascade of bifurcations-type trajectories (CBTTs) and strong dependance on the initial conditions and the number of iterations. The map's asymptotic behavior is described by the conditions of the asymptotic stability, the rate of convergence to the asymptotically periodic points, asymptotic bifurcation points, and the transition to chaos. The maps were used to show numerically that the fractional Feigenbaum constant δ exists and is equal to its regular value. Fractional generalizations may also be used to naturally introduce the 2D and 3D logistic maps. Applications of the FLM and the FDLM include cryptography, distribution of ageing, population biology, etc. One of the most important remaining problems is a theoretical analysis of the asymptotic universality in fractional dynamics which could be based on the derived equations defining the asymptotically periodic and bifurcation points.

This work presents a practical C++ implementation of a physics-informed neural network (PINN) for a fractional-order damped oscillator. A fully connected network outputs displacement and velocity, so the governing dynamics are enforced through a compact state-space residual involving first and second time derivatives. Integer-order derivatives are obtained via automatic differentiation, which removes finite-difference noise and preserves smooth, consistent gradients during training. The history-dependent fractional damping term is incorporated using the classical L1 discretization on a uniform time grid, which makes each residual evaluation depend on the entire predicted solution history and naturally captures memory effects. The training objective combines the squared residual norms at collocation points with a strongly weighted initial-condition penalty to control drift and stabilize early iterations. Gradients of the complete objective with respect to all network parameters are computed using reverse-mode automatic differentiation in CppAD by constructing a scalar loss function of a flat parameter vector, enabling efficient gradient-based optimization. Parameters are updated with the Adam algorithm using bias correction and double-precision moment accumulation for numerical robustness. This implementation includes deterministic parameter packing, explicit size checks, and lightweight diagnostics of boundary values during training, improving reproducibility and debuggability. Overall, the code provides an end-to-end baseline for PINN-based simulation of fractional-order oscillatory systems and can be readily extended to include external forcing, alternative loss weight schedules, and parameter identification from measurement data.

Fractional Calculus (FC) has gained attention in machine learning (ML) due to its ability to model long-term memory (LTM), complex system behavior, and non-local dynamics. Fractional machine learning models (FMLM) demonstrate improved convergence properties across diverse applications (e.g., financial forecasting, EEG, ECG, climate and environment, and robotic control theory) by combining classical learning frameworks with memory-aware dynamics, thereby improving realism, robustness, and interpretability, particularly in time-dependent settings. Thus, these advantages pose significant challenges for model explainability and interpretability. The main goal of this paper is to analyze the explainability associated with FMLM. Unlike classical models based on integer-order derivatives, Fractional Machine Learning Models (FMLMs) exhibit non-local behavior, and their predictions are influenced by the entire historical input data or by the model's optimization process. As a result, these characteristics pose important challenges for Explainable Artificial Intelligence (XAI) approaches primarily designed for local and memoryless models. Fractional neurons contribute to modeling expressiveness but also amplify the model's black-box characteristics by introducing further interpretability issues, which can be reduced by XAI that explains input contributions and highlights the roles of fractional parameters. Moreover, the limitations of existing XAI techniques for FMLMs are investigated, and significant issues related to parameter interpretability, decision traceability, and model transparency are identified. It also proposes research directions for developing explainability frameworks based on fractional learning models. By integrating fractional-order calculus in AI/ML applications, the fractional neuron retains memory effects. At the same time, XAI makes it easier to understand the impact of historical data and fractional parameters on neuronal decisions.

Physics-Informed Neural Networks (PINNs) provide a mesh-free, optimization-based framework for the numerical solution of partial differential equations by enforcing the governing equations and boundary conditions through a unified loss functional. In parallel, classical numerical methods such as the Galerkin Boundary Element Method (BEM) offer a well-established discretization strategy based on boundary integral formulations and boundary-only representations.

In this work, we present a systematic comparison between PINNs and a Galerkin BEM for the numerical solution of the Laplace equation posed on Koch prefractal domains. These geometries form a sequence of polygonal approximations converging to a fractal limit, with increasing geometric complexity and decreasing boundary regularity as the prefractal level increases. Both approaches are implemented independently and tested on identical geometries and discretizations in order to ensure a fair and consistent comparison.

The numerical experiments investigate the influence of boundary refinement and prefractal level on the performance of each method. The comparison focuses on solution accuracy, computational cost, sensitivity to the prescribed boundary data, and robustness with respect to increasing geometric irregularity.

The goal of this study is to provide a balanced assessment of the capabilities and limitations of PINNs and classical boundary integral techniques when applied to non-smooth and highly structured computational domains, and to highlight their respective advantages in the context of fractal-like geometries.

Introduction: Data-driven modelling in the fields of science, engineering, and health has been revolutionized by the rapid development of machine learning (ML). However, the majority of current machine learning frameworks rely on traditional integer-order calculus, which frequently falls short of capturing memory effects, long-term relationships, and inherited characteristics present in many real-world systems.

Method: The goal of this study is to develop and analyze fractional-order machine learning architectures, including fractional dynamic systems for time-series prediction, fractional neural networks, and fractional gradient-based optimization methods. Applications in biomedical signal processing, epidemiological modelling, and socio-behavioral systems, where memory effects and delayed reactions are critical, will receive special attention. The project aims to improve model accuracy, stability, and interpretability in complicated situations by integrating Caputo and Riemann–Liouville fractional derivatives into learning dynamics.

Results: Despite its potential, several obstacles hinder the application of fractional calculus in machine learning, including the lack of standardized training frameworks, increased computational costs, numerical instability, and parameter identifiability issues. In order to facilitate practical implementation, this study will methodically investigate these difficulties and suggest effective numerical methods, regularization techniques, and scalable algorithms. A unified theoretical framework for fractional-order learning systems, validated application case studies, and open-source computational tools to facilitate additional study in this developing subject are among the anticipated results.

Conclusion: This project intends to contribute to the next generation of intelligent systems that can more accurately model memory-dependent phenomena by bridging cutting-edge mathematical theory with contemporary artificial intelligence techniques. This will advance machine learning theory and practice in complex, real-world settings.

The circuit realization of fractional neural models is a well-known and essential approach in neuromorphic computing, advanced encryption methods, intelligent robotics, self-tuning control, and massively parallel computation, among other areas. Indeed, a crucial step toward moving such applications forward in real-world scenarios is the electronic implementation of neural networks. However, fractional-order Hopfield neural networks (FO-HNNs), with and without memristors, pose two main issues for circuit implementation. The first is that FO-HNNs rely on hyperbolic tangent activation functions, which lead to bulky and cumbersome hardware implementations. As the second issue, FO-HNNs require, on the one hand, frequency-approximation methods to emulate a fractional capacitor in analog circuit designs. But those methods suffer from equilibrium points and memory inconsistencies. On the other hand, FPGA and ARM hardware require appropriate numerical methods to address the inherent memory requirements of fractional derivatives.
Both issues limit the applications of FO-HNNs. Therefore, in this talk, we present two novel approaches for achieving feasible, robust, and efficient implementation of FO-HNNs. To address the first problem, a novel approach employing piecewise-linear activation functions, rather than complex hyperbolic functions, is discussed. To address the second drawback, a new decomposition method is applied to obtain a semi-analytical solution for FO-HNNs. Similar to the Adomian approach, the Caputo-based fractional-order differential equation describing the underlying system is decomposed into n-pwl subsystems without explicit nonlinearities. As a result, we obtain an algorithm with low computational complexity, optimized simulation time, and no massive amounts of generated data, in contrast to standard numerical approaches such as Adams-Bashforth-Moulton and short-memory Grunwald-Letnikov. Based on the results and comparisons presented, both the piecewise-linear activation function and decomposition are excellent choices for the experimental realization of memristive and nonmeristive FO-HNNs, offering low hardware complexity, reduced form factor, and the fewest electronic components.