Directed information between connected leaky integrate-and-fire neurons

Directed information is a measure that can be used to infer connectivity between neurons using their recorded time series. In this paper we develop a method of finding the directed information of a particular neural topology analytically. We assume a leaky integrate-and-fire (LIF) neuron model, and calculate the directed information between the spike train of an input neuron to the LIF model and the corresponding spike train generated by the LIF model based on this input. We show that an action potential in the LIF model causes a conditional independence of the activity before and after it, and we capture this conditional independence via a Markov model. We then use this model to find the directed information analytically. Additionally, we show how the stationary distribution and transition probabilities of the Markov model can be found using parameters of the LIF neuron. This modeling technique can thus be used to obtain the value of the directed information in a particular neuronal topology.