2025 Activity Report

Alexandru Fikl — Fri, 12 Dec 2025 00:00:00 +0000

Introduction🔗

Our second year (January-December 2025) built on the first year's foundations. Results now include five published papers, a growing suite of open-source tools, and initial bridging from theory to brain-imaging data.

What we've done so far🔗

From two nodes to whole-brain connectomes🔗

Last year we mentioned studying simplified quadratic networks in the complex plane. This year we extended this analysis: for two-node networks, we characterized how the adjacency matrix and coupling weights determine whether the system converges to a stable attractor or exhibits chaotic dynamics. This research was consolidated in the paper "Asymptotic dynamics in systems of two coupled quadratic maps" and published in Chaos, Solitons & Fractals.

Furthermore, we scaled this up by introducing the equi-M set, a generalization of the classic Mandelbrot set for networks of any size. A key finding is that fractal and geometric invariants of these sets correlate robustly with the network's wiring. Applied to Human Connectome Project data, these invariants differentiate structural connectomes (anatomical wiring with only positive weights) from functional connectomes (activity-based correlations with both positive and negative weights), and distinguish between resting-state and emotion-processing task states. Notably, this method succeeds where classical graph-theoretic measures applied separately to the positive and negative subnetworks cannot. This work is part of the "Fractal geometry predicts dynamic differences in structural and functional connectomes" paper published in Chaos.

Wilson-Cowan with delays: Parkinson's disease and beyond🔗

We extended our stability and bifurcation analysis of Wilson-Cowan networks, first from 3-node networks to an arbitrary number of nodes. As a first step, we assumed identical delay distributions on all connections, which enabled a systematic analysis of how delays and connectivity jointly affect stability and bifurcations. We then generalized to the more realistic case where intra-node and inter-node connections have different delay distributions (two distinct delay kernels). We also systematically compared how the shape of the delay distribution (exponential, Gamma, Dirac, etc.) influences the linear spectrum and the onset of Hopf bifurcations. Even small perturbations to the delay profile can produce significant transitions between stationary behavior, periodic oscillations, and complex dynamics - an insight with direct relevance to the abnormal brain rhythms seen in Parkinson's disease.

This work was presented by E. Kaslik at NODYCON (2025), with two full papers in preparation for next year.

Generalized fractional operators and periodicity🔗

In related work, we explored generalized fractional operators based on Sonine kernels as an alternative to distributed-delay modeling. These operators have a Volterra-type representation, where the Sonine kernel acts mathematically as a delay distribution over the entire past interval. By choosing different kernel families (power-type, Prabhakar, distributed-order, etc.), one can model various long-term memory profiles.

In the paper "Generalized fractional operators do not preserve periodicity ", published in Fractional Calculus and Applied Analysis, we proved that these operators never preserve periodicity: for any non-constant periodic function, application of the operator yields an aperiodic function. The proof relies on a decomposition of the fractional derivative into a local periodic term and a nonlocal completely monotonic term responsible for the loss of periodicity. At the level of differential equations, this means fractional-order dynamical systems cannot admit exact periodic solutions - only asymptotically periodic ones. This is relevant for neural modeling because brain oscillations observed experimentally are generally only approximately periodic, influenced by slow adaptation and long-term memory processes. In future work, we plan to use generalized fractional operators with different Sonine kernels to construct fractional neural network models and compare the results with the distributed-delay approach.

Emotion, dopamine, and schizophrenia🔗

After an extensive revision based on reviewer feedback, our 4-dimensional delay-differential model of the prefrontal-limbic circuit in schizophrenia has been accepted for publication in Chaos, Solitons & Fractals (originally submitted in the first project stage). The paper shows how dopamine-modulated interactions in this emotion-regulation circuit can give rise to pathological dynamics - a result that connects theoretical analysis to clinical questions.

These findings were also presented by M. Neamțu at NODYCON (2025).

Software tools🔗

The software toolkit expanded this year:

netbrot now includes Fourier-mode parametrization, area and perimeter computation, fractal dimension estimation, and more. It powered the simulations in both the coupled quadratic maps and the fractal geometry papers. The full reproducible code for the latter is on Zenodo.
pycaputo gained four new Integrate-and-Fire neuron models (Perfect, Leaky, Exponential, Adaptive-Exponential). The exponential variants required new implicit time-stepping methods with adaptive step-size control to accurately capture action potential generation. Due to the structure of the exponential models, the implicit solvers were solved exactly via the Lambert W function. These methods and the supporting analysis are described in the paper "Simulating neuronal dynamics in fractional adaptive exponential integrate-and-fire models", published in Fractional Calculus and Applied Analysis.
orbitkit is a new library developed along two directions: analysis of large dynamical systems (especially those defined on networks or graphs) and code generation for efficient simulation. It supports backends for jitcdde, JAX, and a NumPy-based testing backend. It was used for the numerical results presented at NODYCON 2025 and will power the large-scale Wilson-Cowan simulations planned for the next phase.
polsys-rs continues to be developed for solving the high-degree polynomial systems that arise when computing fixed points in high-dimensional networks.

These tools are open source and available on GitHub. For an up-to-date listing, see our Software page.

Empirical data analysis🔗

We also began exploring empirical data:

Alzheimer's MRI: We compared nonlinear dimensionality reduction techniques (t-SNE, UMAP) on structural MRI data, evaluating clustering quality via standard indices and the impact on Random Forest classifier accuracy. The study shows that t-SNE can reduce to just 2-3 dimensions while maintaining or improving classification performance, and provides recommendations for choosing dimensionality reduction methods in future structural connectomics analyses. This work was accepted at NODYCON 2025 (Springer).
EEG for epilepsy and multiple sclerosis: A systematic comparison of ML architectures (kNN, Random Forest, ARIMA + logistic regression, CNN, GRU, LSTM) on two EEG datasets showed that the best model depends heavily on dataset size. On the large epilepsy dataset, Random Forest and deep recurrent models (GRU, LSTM) reach accuracies above 90%; on the smaller MS dataset, simpler nonparametric classifiers such as kNN prove more robust. The study underscores the need to align model complexity with data size and structure, and opens directions for integrating EEG and fMRI within a unified modeling framework.

Both of these were presented at NODYCON 2025 by D. Jianu and S.-A. Folta, respectively.

Team activities🔗

Collaboration with our US partners at SUNY New Paltz (Anca Rădulescu) and University at Buffalo (Thomas J. Covey) was extensive. The Romanian team visited the US in June 2025; Anca Rădulescu visited Timișoara in July 2025. Weekly online meetings - in both extended teams and in focused subgroups for specific subprojects - maintained progress between visits.

Collectively the team gave presentations at three international conferences: NODYCON 2025 in Hoboken, ENUMATH 2025 in Heidelberg, and SYNASC 2025 in Timișoara.

By the numbers🔗

What we said we'd do	What we did
3+ papers submitted	5 published, 1 conference paper accepted
1 intermediate report	Done

The five journal papers appeared in Chaos, Solitons & Fractals (2), Fractional Calculus and Applied Analysis (2), and Chaos (1) - all Q1-ranked venues. Three of them are open access.

What's coming up🔗

With the analytical foundations solid and the software stack maturing, year three will focus on heterogeneous delays across network architectures, scaling the numerical simulations to larger networks, deepening the connection to empirical brain data, comparing fractional and distributed-delay approaches to neural modeling, and working toward integrating EEG and fMRI within a unified framework.

2024 Activity Report

Alexandru Fikl — Fri, 13 Dec 2024 00:00:00 +0000

Introduction🔗

The project was just accepted by the NSF (see here for more details) and UEFISCDI in October 2024, so this first activity report is sparse. However, we outline here a few ongoing directions.

What we've done so far🔗

1. Stability and bifurcations in delayed neural networks🔗

We started by studying simplified networks of coupled nodes whose dynamics follow quadratic iterations in the complex plane. Even small changes in how the nodes are connected can lead to qualitatively different overall behavior -- a finding with implications for complex systems more broadly.

On the biological modeling side, we've been analyzing Wilson--Cowan type networks with distributed delays. These are well-known models of neural population activity, and we are investigating how the combination of network connectivity and signal delays affects their dynamics. A concrete application is the abnormal brain oscillations seen in Parkinson's disease.

Three papers are in preparation from this line of research.

2. A new model of emotional circuits in schizophrenia🔗

We revisited a mathematical model of the prefrontal-limbic network -- a brain circuit deeply involved in emotion regulation. By incorporating recent findings about midbrain dopamine systems, we arrived at a 4-dimensional delay-differential model of emotional excitation in schizophrenia. We carried out stability analysis, studied parameter sensitivity, and identified conditions that give rise to oscillations (Hopf bifurcations). The numerical simulations support the theory and offer biologically meaningful interpretations.

This work has since been published in the paper "A time-delay model of dopamine-modulated prefrontal-limbic interactions in schizophrenia" in Chaos, Solitons & Fractals.

3. Building the tools🔗

We have also developed supporting software tools:

netbrot -- a high-performance fractal renderer for coupled quadratic networks. It is parallelized using SIMD instructions and can render an 8000×8000 image in seconds. We are using it to investigate how network architecture shapes dynamic behavior.
polsys-rs -- a work-in-progress solver for polynomial systems, aimed at finding fixed points in high-dimensional networks with better performance than general-purpose methods.
We've also started working with the jitcdde library for simulating large Wilson--Cowan networks, including minor contributions back to the project.

These tools are open source and available on GitHub. For an up-to-date listing of our code output, see our Software page.

The team🔗

The project is coordinated by Eva Kaslik (West University of Timișoara) and Anca Rădulescu (SUNY New Paltz), building on a long-standing collaboration that has already produced four joint papers and three co-advised PhD students. The research team also includes Mihaela Neamțu, Alexandru Fikl, and Diana Jianu.

What's next?🔗

All planned activities for this initial stage have been completed. With the foundations in place, the next phase will focus on extending the analytical results to more general network architectures and heterogeneous delays, scaling up the numerical simulations, and moving closer to empirical brain network data.

Project Accepted

Alexandru Fikl — Tue, 01 Oct 2024 00:00:00 +0000

Early Days🔗

The project has been accepted by the NSF (see here for more details). We're currently in the stage of starting things up and signing contracts.

If you're interested in the project, the code or the ideas, don't hesitate to reach out!

caDDEnce - News

2025 Activity Report

Introduction🔗

What we've done so far🔗

From two nodes to whole-brain connectomes🔗

Wilson-Cowan with delays: Parkinson's disease and beyond🔗

Generalized fractional operators and periodicity🔗

Emotion, dopamine, and schizophrenia🔗

Software tools🔗

Empirical data analysis🔗

Team activities🔗

By the numbers🔗

What's coming up🔗

2024 Activity Report

Introduction🔗

What we've done so far🔗

1. Stability and bifurcations in delayed neural networks🔗

2. A new model of emotional circuits in schizophrenia🔗

3. Building the tools🔗

The team🔗

What's next?🔗

Project Accepted

Early Days🔗