Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918398770806784 |
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| author | Dou, Xu'an Tao, Louis Xue, Zhe Zhou, Zhennan |
| author_facet | Dou, Xu'an Tao, Louis Xue, Zhe Zhou, Zhennan |
| contents | In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset. In this work, we propose a robust {multiscale numerical framework based on time dilation}. By transforming the governing equation into a dilated timescale proportional to the firing activity, we desingularize the blow-up, effectively stretching the instantaneous synchronization event into a resolved mesoscopic process. This approach is shown to be physically consistent with the {microscopic cascade mechanism} underlying MFEs and the system's inherent fragility. To implement this numerically, we develop a hybrid scheme that utilizes a {mesh-independent flux criterion} to switch between timescales and a semi-analytical ``moving Gaussian'' method to accurately evolve the post-blowup Dirac mass. Numerical benchmarks demonstrate that our solver not only captures steady states with high accuracy but also efficiently reproduces periodic MFEs, matching Monte Carlo simulations without the severe time-step restrictions associated with particle cascades. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks Dou, Xu'an Tao, Louis Xue, Zhe Zhou, Zhennan Numerical Analysis Analysis of PDEs Neurons and Cognition In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset. In this work, we propose a robust {multiscale numerical framework based on time dilation}. By transforming the governing equation into a dilated timescale proportional to the firing activity, we desingularize the blow-up, effectively stretching the instantaneous synchronization event into a resolved mesoscopic process. This approach is shown to be physically consistent with the {microscopic cascade mechanism} underlying MFEs and the system's inherent fragility. To implement this numerically, we develop a hybrid scheme that utilizes a {mesh-independent flux criterion} to switch between timescales and a semi-analytical ``moving Gaussian'' method to accurately evolve the post-blowup Dirac mass. Numerical benchmarks demonstrate that our solver not only captures steady states with high accuracy but also efficiently reproduces periodic MFEs, matching Monte Carlo simulations without the severe time-step restrictions associated with particle cascades. |
| title | Resolving the Blow-Up: A Time-Dilated Numerical Framework for Multiple Firing Events in Mean-Field Neuronal Networks |
| topic | Numerical Analysis Analysis of PDEs Neurons and Cognition |
| url | https://arxiv.org/abs/2603.18475 |