[Paper Review] Towards a realistic NNLIF model: Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods

María J. Cáceres, Ricarda Schneider|arXiv (Cornell University)|May 5, 2017

Neural dynamics and brain function3 citations

TL;DR

This paper presents a comprehensive analytical and numerical study of a realistic two-population Network of Noisy Leaky Integrate-and-Fire (NNLIF) model with transmission delays and refractory periods. It proves the existence and uniqueness of steady states, demonstrates that blow-up occurs without excitatory-to-excitatory transmission delay, and introduces a high-order numerical solver using WENO schemes and TVD Runge-Kutta methods to simulate blow-up, synchrony, and stability across diverse network dynamics.

ABSTRACT

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 Cáceres-Carrillo-Perthame(2011), Cáceres-Perthame(2014), Cáceres-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 Poincaré'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.

Motivation & Objective

• To develop a mathematically rigorous model of neural networks that incorporates key biological features: two populations (excitatory/inhibitory), transmission delays, and refractory periods.
• To analyze the number and stability of steady states in the full NNLIF system, particularly under varying connectivity and delay parameters.
• To investigate blow-up phenomena and identify the critical role of excitatory-to-excitatory transmission delay in preventing finite-time blow-up.
• To design and implement a high-order numerical solver capable of simulating complex network behaviors such as blow-up, synchrony, and stability transitions.
• To provide a deterministic numerical framework for exploring open problems in NNLIF dynamics, including global existence and stability of solutions.

Proposed method

• Formulates a coupled system of two Fokker-Planck-type PDEs and two ODEs for excitatory and inhibitory populations, incorporating membrane potential densities and refractory states.
• Applies the entropy method and Poincaré’s inequality to prove exponential convergence to steady states under small connectivity parameters and absence of transmission delay.
• Uses high-order flux-splitting WENO schemes for spatial discretization of the PDEs to ensure accuracy and stability in advection-dominated regimes.
• Employs an explicit third-order TVD Runge-Kutta method for temporal integration, preserving monotonicity and avoiding oscillations.
• Implements an efficient data-saving and recovery strategy to handle time delays in synaptic transmission, particularly for delayed terms in the connectivity kernels.
• Validates the solver through numerical experiments showing blow-up with zero excitatory delay, convergence to steady states, and emergence of periodic solutions under specific parameter regimes.

Experimental results

Research questions

• RQ1 Under what conditions does the NNLIF model with two populations, refractoriness, and transmission delays admit steady states?
• RQ2 How do transmission delays—particularly between excitatory neurons—affect the stability and long-time behavior of the network?
• RQ3 What role does the entropy method play in proving exponential convergence to steady states in the absence of delays and for small connectivity?
• RQ4 Can the model exhibit blow-up in finite time, and if so, under what parameter conditions?
• RQ5 What determines whether solutions converge to a steady state or evolve into synchronous, periodic, or unstable dynamics?

Key findings

• The NNLIF model with refractory states always admits at least one steady state, in contrast to models without refractoriness, which may lack steady states for certain parameter values.
• Uniqueness of the steady state is proven under specific parameter constraints, and exponential convergence to it is established for small connectivity parameters when no transmission delays are present.
• Blow-up in finite time occurs if the transmission delay between excitatory neurons is zero, even when other delays (inhibitory-to-excitatory or inhibitory-to-inhibitory) are nonzero.
• Numerical simulations confirm that nonzero excitatory-to-excitatory delay prevents blow-up and allows solutions to converge to steady states or periodic, synchronous solutions.
• The numerical solver successfully captures diverse dynamics: blow-up, asynchronous and synchronous solutions, and instability/stability transitions, with accurate tracking of firing rates, refractory states, and probability density functions.
• Stability analysis of multiple steady states reveals that only the lowest firing rate state is stable, while higher ones are unstable, indicating no bistability in the observed parameter regime.