arXiv subjects

Alexandru Fikl

Publications and source records attributed to Alexandru Fikl.

8 recordsLinked to original sources

arXiv2026

Macroscopic Multistability and Bifurcations in Theta-Neuron Networks with Distributed Delays

We study an all-to-all coupled network of identical theta neurons with synaptic interaction mediated by a distributed time delay. Using the Watanabe-Strogatz reduction and passing to the thermodynamic limit under the assumption of uniformly distributed constants of motion, we derive a single delay differential equation for the complex order parameter. The delay is modeled by a family of delay kernels with prescribed mean delay, allowing discrete and distributed delays to be treated in a unified framework. The equilibria of the reduced system can be classified into two geometrically distinct families: type 1 equilibria on the unit circle and type 2 equilibria on the real axis. For both families, the local stability problem reduces to scalar characteristic equations involving the Laplace-Stieltjes transform of the delay kernel. We obtain stability criteria for admissible kernels and explicit Hopf bifurcation conditions for the Dirac kernel, with additional comparison to weak and strong Gamma kernels. The results show that the delay may either preserve stability, destabilize equilibria, or produce stability switching, depending on the equilibrium branch, parameter regime, and choice of kernel. Numerical simulations for the discrete-delay case support the analytical results and illustrate the corresponding phase portraits, basins of attraction, coexistence of attractors, and delay-induced periodic dynamics.

math.DS↗

arXiv2025

Fractal geometry predicts dynamic differences in structural and functional connectomes

Understanding the intricate architecture of brain networks and its connection to brain function is essential for deciphering the underlying principles of cognition and disease. While traditional graph-theoretical measures have been widely used to characterize these networks, they often fail to fully capture the emergent properties of large-scale neural dynamics. Here, we introduce an alternative approach to quantify brain networks that is rooted in complex dynamics, fractal geometry, and asymptotic analysis. We apply these concepts to brain connectomes and demonstrate how quadratic iterations and geometric properties of Mandelbrot-like sets can provide novel insights into structural and functional network dynamics. Our findings reveal fundamental distinctions between structural (positive) and functional (signed) connectomes, such as the shift of cusp orientation and the variability in equi-M set geometry. Notably, structural connectomes exhibit more robust, predictable features, while functional connectomes show increased variability for non-trivial tasks. We further demonstrate that traditional graph-theoretical measures, when applied separately to the positive and negative sub-networks of functional connectomes, fail to fully capture their dynamic complexity. Instead, size and shape-based invariants of the equi-M set effectively differentiate between rest and emotional task states, which highlights their potential as superior markers of emergent network dynamics. These results suggest that incorporating fractal-based methods into network neuroscience provides a powerful tool for understanding how information flows in natural systems beyond static connectivity measures, while maintaining their simplicity.

nlin.CD↗

arXiv2025

A Fast Direct Solver for Boundary Integral Equations Using Quadrature By Expansion

We construct and analyze a hierarchical direct solver for linear systems arising from the discretization of boundary integral equations using the Quadrature by Expansion (QBX) method. Our scheme builds on the existing theory of Hierarchical Semi-Separable (HSS) matrix operators that contain low-rank off-diagonal submatrices. We use proxy-based approximations of the far-field interactions and the Interpolative Decomposition (ID) to construct compressed HSS operators that are used as fast direct solvers for the original system. We describe a number of modifications to the standard HSS framework that enable compatibility with the QBX family of discretization methods. We establish an error model for the direct solver that is based on a multipole expansion of the QBX-mediated proxy interactions and standard estimates for the ID. Based on these theoretical results, we develop an automatic approach for setting scheme parameters based on user-provided error tolerances. The resulting solver seamlessly generalizes across two- and tree-dimensional problems and achieves state-of-the-art asymptotic scaling. We conclude with numerical experiments that support the theoretical expectations for the error and computational cost of the direct solver.

math.NA↗

arXiv2024

Simulating neuronal dynamics in fractional adaptive exponential integrate-and-fire models

We introduce an efficient discretization of a novel fractional-order adaptive exponential (FrAdEx) integrate-and-fire model, which is used to study the fractional-order dynamics of neuronal activities. The discretization is based on extension of L1-type methods that can accurately handle the exponential growth and the spiking mechanism of the model. This new method is implicit and uses adaptive time stepping to robustly handle the stiff system that arises due to the exponential term. The implicit nonlinear system can be solved exactly, without the need for iterative methods, making the scheme efficient while maintaining accuracy. We present a complete error model for the numerical scheme that can be extended to other integrate-and-fire models with minor changes. To show the feasibility of our approach, the numerical method has been rigorously validated and used to investigate several different spiking oscillations of the model. We observed that the fractional-order model is capable of predicting biophysical activities, which are interpreted through phase diagrams describing the transition from one firing type to another. This simple model shows significant promise, as it has sufficient expressive dynamics to reproduce several features qualitatively from a biophysical dynamical perspective.

physics.bio-ph↗

arXiv2023

Complex dynamics in two-dimensional coupling of quadratic maps

This paper examines the structure and limitations of equi-M sets in two-dimensional Complex Quadratic Networks (CQNs). In particular, we aim to describe the relationship between the equi-M set and the parameter domains where the critical orbit converges to periodic attractors (pseudo-bulbs). The two-node case serves as a foundational testbed: its analytical tractability enables the identification of critical phenomena and their dependence on coupling, while offering insight into more general principles. The two-node case is also simple enough to allow for explicit coupling conditions that govern phase transitions between synchronized and desynchronized behavior. Using a combination of analytical and numerical methods, the study reveals that while the period-1 pseudo-bulb closely tracks the boundary of the equi-M set near its main cusp, this correspondence breaks down for higher periods and in regions supporting coexisting attractors. These discrepancies highlight key differences between single-map and coupled dynamics, where equi-M sets no longer provide a full encoding of system combinatorics. These findings clarify the topological and dynamical behavior of low-dimensional CQNs and point toward a sharp increase in complexity as the number of nodes grows, laying the groundwork for future studies of high-dimensional dynamics.

nlin.CD↗

arXiv2022

Adjoint-based Control of Three Dimensional Stokes Droplets

We develop a continuous adjoint formulation and implementation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow through farfield velocity boundary conditions. The focus is on dynamics where surface tension plays an important role through the Young-Laplace law. To perform the optimization, we require access to first-order gradient information, which we obtain from the linearized sensitivity equations and their corresponding adjoint by applying shape calculus to the space-time tube formed by the interface evolution. We show that the adjoint evolution equation can be efficiently expressed through a scalar adjoint transverse field. The optimal control problem is discretized by high-order boundary integral methods using Quadrature by Expansion coupled with a spherical harmonic representation of the droplet surface geometry. We show the accuracy and stability of the scheme on several tracking-type control problems.

physics.comp-ph↗

arXiv2022

A Comprehensive Study of Adjoint-Based Optimization of Non-Linear Systems with Application to Burgers' Equation

In the context of adjoint-based optimization, nonlinear conservation laws pose significant problems regarding the existence and uniqueness of both direct and adjoint solutions, as well as the well-posedness of the problem for sensitivity analysis and gradient-based optimization algorithms. In this paper we will analyze the convergence of the adjoint equations to known exact solutions of the inviscid Burgers' equation for a variety of numerical schemes. The effect of the non-differentiability of the underlying approximate Riemann solver, complete vs. incomplete differentiation of the discrete schemes and inconsistencies in time advancement will be discussed.

physics.comp-ph↗

arXiv2017

Experimenting with the p4est library for AMR simulations of two-phase flows

Many physical problems involve spatial and temporal inhomogeneities that require a very fine discretization in order to be accurately simulated. Using an adaptive mesh, a high level of resolution is used in the appropriate areas while keeping a coarse mesh elsewhere. This idea allows to save time and computations, but represents a challenge for distributed-memory environments. The MARS project (for Multiphase Adaptative Refinement Solver) intends to assess the parallel library p4est for adaptive mesh, in a case of a finite volume scheme applied to two-phase flows. Besides testing the library's performances, particularly for load balancing, its user-friendliness in use and implementation are also exhibited here. First promising 3D simulations are even presented.

math.NA↗