# Towards a realistic NNLIF model: Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods

arXiv: https://arxiv.org/abs/1705.02205 · HTML: https://mlsanity.com/paper/1705.02205/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2017.

## Built on by (most cited first, 1 of 1)

- [Emergent Bio-Functional Similarities in a Cortical-Spike-Train-Decoding Spiking Neural Network Facilitate Predictions of Neural Computation](https://mlsanity.com/paper/2303.07830/) (2023)

## Abstract (arXiv)

The Network of Noisy Leaky Integrate and Fire (NNLIF) model describes the behavior of a neural network at mesoscopic level. It is one of the simplest self-contained mean-field models considered for that purpose. Even so, to study the mathematical properties of the model some simplifications were necessary C'aceres-Carrillo-Perthame(2011), C'aceres-Perthame(2014), C'aceres-Schneider(2017), which disregard crucial phenomena. In this work we deal with the general NNLIF model without simplifications. It involves a network with two populations (excitatory and inhibitory), with transmission delays between the neurons and where the neurons remain in a refractory state for a certain time. We have studied the number of steady states in terms of the model parameters, the long time behaviour via the entropy method and Poincar'e's inequality, blow-up phenomena, and the importance of transmission delays between excitatory neurons to prevent blow-up and to give rise to synchronous solutions. Besides analytical results, we have presented a numerical resolutor for this model, based on high order flux-splitting WENO schemes and an explicit third order TVD Runge-Kutta method, in order to describe the wide range of phenomena exhibited by the network: blow-up, asynchronous/synchronous solutions and instability/stability of the steady states; the solver also allows us to observe the time evolution of the firing rates, refractory states and the probability distributions of the excitatory and inhibitory populations.