Arne Großelindemann: Self-Consistent Spike Statistics in Networks of Poisson Neurons

How microscopic neural network properties, such as properties of individual units as well as their coupling, relate to firing statistics is often not clear, especially when coupling is recurrent.
In this work we follow up on a line of research that attempts to bridge this gap by use of self-consistent mathematical network analysis. We introduce a variant of a commonly used neural network model: a network of coupled neurons where each unit is characterised through its membrane potential and emits action potentials according to an inhomogeneous Poisson process given this potential. We choose the relation of membrane potential and expected firing rate to be exponential, allowing for closed form equations where previously only integral expressions could be derived. This allows for detailed analysis, deriving explicit and closed form relations of network parameters and firing statistics such as firing rate, membrane potential auto-covariance, Fano factor or effective correlation time in the sparse mean-field limit.

We go on to specifically analyse the network under dynamic external input current and derive power spectral density of the membrane potential or coherence of membrane potential and input as a function of the stimulus power spectral density and the network parameters. Furthermore we extend the network model in various ways. For example we introduce a synaptic depression mechanism or conductance based synaptic transmission and self-consistently derive equations for the network statistics in a weak coupling regime.

Altogether this work provides various closed form equations governing the spiking statistics in terms of the model parameters under different model variations. This allows for direct understanding of relations in terms of mathematical equations and direct calculation of statistics given network parameters.