2021/03/19 by Maria Caceres, Benoît Perthame, Caceres, Maria +7 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Biological sciences #FOS: Mathematics #Neural Networks Stability and Synchronization #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #math.AP #q-bio.NC #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2103.10737
arxiv created 2021/03/19 · openalex publication_date 2021/03/19 · arxiv updated 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The elapsed time model has been widely studied in the context of mathematical neuroscience with many open questions left. The model consists of an age-structured equation that describes the dynamics of interacting neurons structured by the elapsed time since their last discharge. Our interest lies in highly connected networks leading to strong nonlinearities where perturbation methods do not apply. To deal with this problem, we choose a particular case which can be reduced to delay equations. We prove a general convergence result to a stationary state in the inhibitory and the weakly excitatory cases. Moreover, we prove the existence of particular periodic solutions with jump discontinuities in the strongly excitatory case. Finally, we present some numerical simulations which ilustrate various behaviors, which are consistent with the theoretical results.