Beyond Blow-Up for Nonlinear Noisy Leaky Integrate and Fire neuronal models: numerical approach to the "plateau" state

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• Alejandro Ramos-lora (University of Granada)
• Thursday 27 January 2022, 13:30-14:30
• Venue to be confirmed.

If you have a question about this talk, please contact Daniel Boutros.

The Nonlinear Noisy Leaky Integrate and Fire neuronal models are mathematical models that describe the activity of neural networks. These models have been studied at a microscopic level, using Stochastic Differential Equations, and at a mesoscopic/macroscopic level, through the mean field limits using Fokker-Planck type equa- tions. To advance in the understanding of the NNLIF models, we have analyzed in depth the behaviour of the classical and physical solutions of the Stochastic Differential Equations and we compare it with what is already known about the Fokker-Planck equation, using a numerical study of their particle systems. This allows us to under- stand what happens in the neural network when an explosion occurs in finite time, which is one of the most important open problems about this kind of models. This allows us to go beyond the mesoscopic/macroscopic description. We answer one of the most important open questions about these models: what happens after all the neurons in the network fire at the same time? We find that the neural network converges towards its unique steady state, if the system is weakly connected. Otherwise, its behaviour is more complex, tending towards a stationary state or a “plateau” distribution (membrane potentials are uniformly distributed between reset and threshold values). To our knowledge, these distributions have not been described before for these nonlinear models.

This talk is part of the Junior Analysis and PDE seminar series.

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