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:
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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.
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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.
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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.