  There is a wide range of mathematical models that describe populations of
large numbers of neurons. In this article, we focus on nonlinear noisy leaky
integrate and fire (NNLIF) models that describe neuronal activity at the level
of the membrane potential of neurons. We introduce a set of novel states, which
we call "pseudo-equilibria", and give evidence of their defining role in the
behaviour of the NNLIF system when a significant synaptic delay is considered.
The advantage is that these states are determined solely by the system's
parameters and are derived from a sequence of firing rates that result from
solving a recurrence equation. We propose a new strategy to show convergence to
an equilibrium for a weakly connected system with large transmission delay,
based on following the sequence of pseudo-equilibria. Unlike with the direct
entropy dissipation method, this technique allows us to see how a large delay
favours convergence. We also present a detailed numerical study to support our
results. This study explores the overall behaviour of the NNLIF system and
helps us understand, among other phenomena, periodic solutions in strongly
inhibitory networks.
