Well-Posedness and Numerical Analysis of an Elapsed Time Model with Strongly Coupled Neural Networks

Core Concepts

The elapsed time equation is an age-structured model that describes the dynamics of interconnected spiking neurons through the elapsed time since their last discharge. This work establishes well-posedness results and a numerical analysis for this model, considering both instantaneous and distributed delay transmission cases.

Abstract

The content discusses the well-posedness and numerical analysis of an elapsed time model, which is an age-structured model used to describe the dynamics of interconnected spiking neurons.
Key highlights:

The elapsed time model considers the time elapsed since a neuron's last discharge, which determines the evolution of the neural network.
Two variants of the model are studied: the instantaneous transmission model (ITM) and the distributed delay model (DDM).
For the ITM equation, well-posedness results are proved for the inhibitory case and the weak interconnections regime, improving upon previous works.
A numerical scheme based on an explicit upwind finite-volume discretization is introduced for both the ITM and DDM equations.
Convergence of the numerical scheme is established through BV-estimates on the discrete solutions.
Numerical simulations are presented to compare the behavior of the ITM and DDM models under different parameter regimes and delay kernels.
The article provides a comprehensive analysis of the well-posedness and numerical approximation of the elapsed time model, which is an important tool in mathematical neuroscience for understanding the dynamics of interconnected spiking neurons.

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Well-posedness and numerical analysis of an elapsed time model with strongly coupled neural networks

by Mauricio Sep... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2310.02068.pdf

Deeper Inquiries

How can the well-posedness and numerical analysis results be extended to more general forms of the hazard rate function p(s,N) beyond the inhibitory and weakly excitatory regimes

The well-posedness and numerical analysis results for the elapsed time model can be extended to more general forms of the hazard rate function p(s, N) by considering a broader class of functions that satisfy certain regularity and growth conditions. Instead of restricting the analysis to just the inhibitory and weakly excitatory regimes, the key lies in ensuring that the hazard rate function p(s, N) meets specific criteria related to its behavior with respect to the age s and the activity N. By relaxing some of the assumptions on the non-linearity and incorporating a wider range of hazard rate functions, the well-posedness and numerical analysis results can be generalized to encompass a more diverse set of scenarios in neural network modeling.

What are the implications of the multiple solution branches that can arise in the excitatory case, and how can the numerical scheme be adapted to capture these different solution profiles

In the excitatory case where multiple solution branches can arise, the numerical scheme can be adapted to capture these different solution profiles by incorporating a method to handle the non-uniqueness of solutions. By considering the existence of multiple steady-states and periodic solutions in the excitatory regime, the numerical scheme can be modified to explore and track these different branches of solutions over time. This adaptation may involve implementing a strategy to identify and follow specific solution paths based on the initial conditions and the behavior of the system. By incorporating this flexibility into the numerical scheme, the model can effectively capture the diverse dynamics that arise in the excitatory case.

Can the elapsed time model be coupled with other neuronal models or network structures to provide a more comprehensive description of brain dynamics

The elapsed time model can be coupled with other neuronal models or network structures to provide a more comprehensive description of brain dynamics by integrating additional elements that influence neural activity. By incorporating spatial dependencies, connectivity patterns, synaptic plasticity mechanisms, or feedback loops into the elapsed time model, a more detailed and realistic representation of brain dynamics can be achieved. This coupling allows for a more nuanced understanding of how different factors interact to shape neural activity and network behavior. By combining the elapsed time model with other models, researchers can gain insights into complex brain processes and phenomena that emerge from the interactions within neural networks.