CNS*2015 Tutorial - theory of correlations in recurrent networks

In the first part of this tutorial, we introduce the mathematical tools to determine firing statistics of neurons receiving fluctuating input form a network. We show how one can apply an efficient Fokker-Planck method to derive the neurons’ output statistics whenever the input can be assumed to be Gaussian white (iid) noise. We further study more realistic cases, where the input fluctuations depart from the iid assumptions. Using the integrate-and-fire neuron model, we will demonstrate how to compute the firing rate, auto-correlation and cross-correlation functions of the output spike trains. The transfer function of the output correlations given the time scale of the input correlations will be discussed [Moreno-Bote and Parga, 2006, Brunel et al 2001]. In particular, we will show that the output correlations are generally weaker than the input correlations and how the working regime of the neuron shapes the cross-correlation functions [Ostojic et al., 2009; Helias et al., 2013]. We conclude the first part by investigating the relation between neurons’ pairwise correlation due commen fluctuations and their firing rates [de la Rocha et al., 2007].In the second part, we will consider correlations in recurrent random networks. Using a binary neuron model [Ginzburg & Sompolinsky, 1994], we explain how mean-field theory determines the stationary state and how the network-generated noise linearizes the single neuron response. The resulting linear equation for the fluctuations in recurrent networks is then solved to obtain the correlation structure in balanced random networks. We discuss two different points of view of the recently reported active suppression of correlations in balanced networks by fast tracking [Renart et al., 2010] and by negative feedback [Tetzlaff et al., 2012]. Finally, we consider extensions of the theory of correlations of linear Poisson spiking models [Hawkes, 1971, Pernice et al. 2011] to the leaky integrate-and-fire model [Trousdale et al. 2012, Pernice et al. 2012] and present a unifying view of linear response theory of weak correlations [Grytskyy et al, 2013]