arXiv subjects

Eva Kaslik

Publications and source records attributed to Eva Kaslik.

At least 19 recordsLinked to original sources

arXiv2026

A distributed-delay Wilson-Cowan model of sleep-related rhythms in the corticothalamic system

The corticothalamic circuit supports rhythms with timescales that differ by orders of magnitude: sleep spindles, the sigma-band events of non-rapid-eye-movement (NREM) sleep, and infra-slow fluctuations near 0.02Hz that organize when spindles occur. Because the anatomy is the same in both cases, architecture alone cannot determine which rhythm the circuit expresses. We ask whether the temporal structure of the circuit's own feedback can. In a four-population Wilson--Cowan model comprising cortical excitatory and inhibitory populations, thalamic relay cells, and the thalamic reticular nucleus (TRN), we first establish how connectivity controls access to oscillatory behavior, and then introduce temporal coupling as either a weak Gamma distributed delay or a discrete delay. We investigate three distinct connectivity levels: recurrent cortical excitation gates whether the circuit can oscillate at all, the reciprocal relay-TRN pair determines where the oscillation lies and how it is configured, sustained, and terminated, and reticular self-inhibition limits its extent. We then examine how these connectivity-dependent regimes are affected by delayed coupling. Although delay does not change the equilibria themselves, it can substantially alter their stability and the organization of the resulting oscillatory dynamics. Under weak Gamma integration, short delays support spindle-compatible oscillations in the sigma band, while longer delays give rise to a much slower regime near 0.02Hz. The discrete-delay formulation produces a qualitatively different and more complex bifurcation structure. Together, these results show that the dynamics of the corticothalamic circuit depend not only on its connectivity, but also on the temporal organization of interactions within the circuit.

q-bio.NC↗

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

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↗

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↗

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

Stability of coupled Wilson-Cowan systems with distributed delays

Building upon our previous work on the Wilson-Cowan equations with distributed delays, we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhytms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system.

math.DS↗

arXiv2023

Dynamics of a discrete-time mixed oligopoly Cournot-type model with three time delays

The paper analyzes the interactions among one public firm and $n$ private firms on the market, in the framework of a discrete-time Cournot game with time delay. The production of the public firm is influenced by previous output levels of private firms. The productions of private companies are influenced by the past productions of the public company, as well as by the previous productions of the other private companies. The associated nonlinear system admits two equilibrium points: the positive one and the boundary equilibrium. After the stability analysis, we obtained that the boundary equilibrium point is a saddle point. If there is no delay, for the positive equilibrium point we have determined the stability region. Then, for different particular cases of delays, we found the conditions for which the positive equilibrium is asymptotically stable. The flip and Neimark-Sacker bifurcations are investigated. In addition, numerous numerical examples are performed to reveal the complex dynamic behavior of the system.

math.DS↗

arXiv2021

Stability and bifurcations in Wilson-Cowan systems with distributed delays, and an application to basal ganglia interactions

The traditional Wilson-Cowan model of excitatory and inhibitory mean field interactions in neuronal populations considers a weak Gamma distribution of time delays when processing inputs, and is obtained via a time-coarse graining technique that averages the population response. Previous analyses of the stability of the Wilson-Cowan model focused on more simplified cases, where the delays were either not present, constant or were of a specific type. Since these simplifications may significantly alter the behavior of the model, we focus on understanding the behavior of the system before time-course graining, and for a wider range of delay distributions. For these generalized delay equations, we perform stability and bifurcation analyses with respect to parameters that capture both the coupling profile, and the time delay. The investigation is done through the examination of the system's associated characteristic equation. Under mild assumptions, we give complete mathematical proofs of our theoretical results, for the model with general delay distributions and prove the transversality condition for the possible Hopf bifurcations, in a generalized context. The stability region in this parameter space is described theoretically for several types of delay kernels, and numerical simulations are presented to substantiate the theoretical results. We illustrate these theoretical principles in an application to a basal ganglia circuit, in which $β$-band oscillations have been associated with Parkinson's Disease.

math.DS↗

arXiv2021

Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterized, in terms of the main diagonal elements of the systems' matrix, as well as its determinant.

math.DS↗

arXiv2020

Stability of fractional-order systems with Prabhakar derivatives

Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings.

math.NA↗

arXiv2020

On initial conditions for fractional delay differential equations

Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining integer-order derivatives. Although the two approaches lead (under standard smoothness assumptions) to equivalent operators, the first one does not involve the function at the left of the initial point where, instead, the latter forces the function to assume selected values. With fractional delay differential equations new problems arise: the presence of the delay imposes to assign the solution not just at the initial point but on an entire interval. Due to the freedom in the choice of the initial function, some inconsistencies with the values forced by the fractional derivative are possible and the operators may no longer be equivalent. In this paper we discuss the initialization of fractional delay differential equations, we investigate the effects of the initial condition not only on the solution but also on the fractional operator as well and we study the difference between solutions obtained by incorporating or not the initial function in the memory of the fractional derivative. The exact solution of a family of linear equations is obtained by the Laplace transform whilst numerical methods are used to solve nonlinear problems; the different results are therefore shown and commented.

math.CA↗

arXiv2020

Stability analysis of multi-term fractional-differential equations with three fractional derivatives

Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases, fractional-order-dependent as well as fractional-order-independent characterisations of stability and instability properties are obtained, in terms of the coefficients of the multi-term fractional differential equation. The theoretical results are exemplified for the particular cases of the Basset and Bagley-Torvik equations, as well as for a multi-term fractional differential equation of an inextensible pendulum with fractional damping terms, and for a fractional harmonic oscillator.

math.AP↗

arXiv2017

Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model

Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional order derivative and a classical first order differential equation. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional order of the Caputo derivative. In this setting, we obtain a generalization of the well known Routh-Hurwitz conditions. These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris-Lecar neuronal model, focusing on stability and instability properties. This fractional order model is built up taking into account the dimensional consistency of the resulting system of differential equations. The occurrence of Hopf bifurcations is also discussed. Numerical simulations exemplify the theoretical results, revealing rich spiking behavior. The obtained results are also compared to similar ones obtained for the classical integer-order Morris-Lecar neuronal model.

math.DS↗

arXiv2017

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

A parallel numerical simulation algorithm is presented for fractional-order systems involving Caputo-type derivatives, based on the Adams-Bashforth-Moulton (ABM) predictor-corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractional-order version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior.

cs.MS↗

arXiv2017

Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model

For two-dimensional autonomous linear incommensurate fractional-order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh-Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two-dimensional fractional-order FitzHugh-Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer-order FitzHugh-Nagumo model.

math.DS↗

arXiv2017

Stability and Hopf bifurcation analysis of a four-dimensional hypothalamic-pituitary-adrenal axis model with distributed delays

A four-dimensional mathematical model of the hypothalamus-pituitary-adrenal (HPA) axis is investigated, incorporating the influence of the GR concentration and general feedback functions. The inclusion of distributed time delays provides a more realistic modeling approach, since the whole past history of the variables is taken into account. The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that the considered four-dimensional system has at least one equilibrium state and a detailed local stability and Hopf bifurcation analysis is given. Numerical results reveal the fact that an appropriate choice of the system's parameters leads to the coexistence of two asymptotically stable equilibria in the non-delayed case. When the total average time delay of the system is large enough, the coexistence of two stable limit cycles is revealed, which successfully model the ultradian rhythm of the HPA axis both in a normal disease-free situation and in a diseased hypocortisolim state, respectively. Numerical simulations reflect the importance of the theoretical results.

math.DS↗

arXiv2017

Analysis of two- and three-dimensional fractional-order Hindmarsh-Rose type neuronal models

A theoretical analysis of two- and three-dimensional fractional-order Hindmarsh-Rose neuronal models is presented, focusing on stability properties and occurrence of Hopf bifurcations, with respect to the fractional order of the system chosen as bifurcation parameter. With the aim of exemplifying and validating the theoretical results, numerical simulations are also undertaken, which reveal rich bursting behavior in the three-dimensional fractional-order slow-fast system.

math.DS↗

arXiv2016

Stability analysis of a hypothalamic-pituitary-adrenal axis model with inclusion of glucocorticoid receptor and memory

This paper analyzes a four-dimensional model of the hypothalamic-pituitary-adrenal (HPA) axis that includes the influence of the glucocorticoid receptor in the pituitary. Due to the spatial separation between the hypothalamus, pituitary and adrenal glands, distributed time delays are introduced in the mathematical model. The existence of the positive equilibrium point is proved and a local stability and bifurcation analysis is provided, considering several types of delay kernels. The fractional-order model with discrete time delays is also taken into account. Numerical simulations are provided to illustrate the effectiveness of the theoretical findings.

math.OC↗