| Summary: | <p>This thesis focuses on constructing solutions to McKean--Vlasov equations with resets and common noise, which serve as a model for infinite populations of neurons based on the leaky integrate-and-fire neuron model.</p>
<p>Our study progresses through three main stages. First, we examine the limiting behaviour of a simplified McKean--Vlasov system with a smoothed feedback mechanism as the smoothing becomes singular. These equations are employed in modelling systemic risk in large interconnected financial networks and serve as a simplification of the integrate-and-fire neuron model due to the absence of resets and their simplified feedback mechanism. We prove weak convergence of the system with smoothed feedback to a system with instantaneous feedback.</p>
<p>Next, we introduce a finite particle system that generalizes the integrate-and-fire neuron model, incorporating physiologically relevant features such as common noise, refractory periods, and delayed transmission of neuronal interactions. We rigorously show the tightness of empirical measures in this interacting particle system and prove that their limit points solve the natural limiting McKean--Vlasov problem. Furthermore, we establish uniqueness within a class of processes, allowing us to conclude that our solution is measurable with respect to the common noise.</p>
<p>Finally, we synthesize these results to construct relaxed solutions to a McKean--Vlasov system with resets, common noise, and an instantaneous feedback mechanism. This is achieved by taking limits of a parameterized system as the smoothing effects become singular and refractory periods approach zero.</p>
<p>Our work provides a novel approach to constructing solutions to singular McKean--Vlasov problems under more general coefficient conditions than previously considered in the literature.</p>
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