A multiple time renewal equation for neural assemblies with elapsed time model
arXiv:2108.10577 · doi:10.1088/1361-6544/ac8714
Abstract
We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since the last discharge, i.e., the refractory period. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin's theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both classical and extended models.
References in corpus (13)
- Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size
- Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
- Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions
- Asymptotic behaviour of neuron population models structured by elapsed-time
- Measure solutions to the conservative renewal equation
- Spectral gap for the growth-fragmentation equation via Harris's Theorem
- Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons
- The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states
- Weak and strong connectivity regimes for a general time elapsed neuron network model
- Distributed synaptic weights in a LIF neural network and learning rules
- Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model
- Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective
- On a finite-size neuronal population equation