Abstract
Our collaborative proposal is centered around developing new conceptual frameworks and methods for analyzing networks of Wilson–Cowan oscillators with distributed delays. The motivation is three-fold.
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Firstly, our investigation is driven directly by the vital importance of both network coupling and temporal aspects in neural functioning, and the necessity of studying them together to observe their interplay.
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Secondly, it is motivated by the absence of general analytical results or even frameworks for the simultaneous study of these two aspects.
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Thirdly, the work is timely in terms of human imaging data currently available for analysis, allowing the investigators to explore modeling the translation between neural meanfield dynamics captured by Wilson-Cowan equations and empirically-driven applications to clinical neuroscience and mental health.
Goals
The PIs launch in this endeavor on the bases of a long-term successful collaboration in modeling time-delay equations and dynamic networks, collaboration that has materialized over the years in four joint publications, three co-mentored Ph.D. students and many conference presentations. The proposed work will be performed along the following steps:
- AIM 1: ANALYTICAL EXTENSIONS. In a preliminary joint study on low-dimensional delay networks, the PIs found specific conditions on the network geometry which simplified formal dynamic analyses. These analyses can be more broadly extended to generic networks and delay distributions, and then subsequently illustrated in wide classes of networks with specific desired architectures and delay structure, in particular in brain circuit models.
- AIM 2: NUMERICAL SIMULATIONS. Numerical methods will be carefully selected to complement the formal approaches and potentially inform further analyses. We will use methods that we have developed specifically to capture dynamic behavior in arbitrary and stochastic networks (e.g., swarm algorithms and “probabilistic” bifurcation diagrams). This aim will engage the efforts of a postdoctoral fellow at WUT, and one undergraduate student at New Paltz.
- AIM 3: APPLICATIONS TO EMPIRICAL BRAIN NETWORKS. We will explore applications of our modeling approach to questions in clinical neuroscience, driven jointly by network and delay aspects. This step will engage the work of a graduate student at WUT, and of an undergraduate at New Paltz.