Distribution of pair-wise covariances in neuronal networks

Despite the large amount of shared input between nearby neurons in cortical circuits, pairwise covariances in ensembles of spike trains are on average close to zero [1]. This has been well understood in terms of active decorrelation by inhibitory feedback in networks of infinite [2] and finite size [3]. These works derived analytical expressions that explain how structural parameters of the network as well as its operational regime determine correlations averaged over many neuron pairs. However, with the advent of massively parallel spiking recordings, experiments also show large variability of covariances across pairs of neurons. Whereas a considerable fraction of this variability may be due to the estimation of correlations from data with limited observation time [1], structural variability of direct and indirect connections between neurons contributes non-trivially. The relation between the frozen variability of the network structure and the correlated dynamics can be readily observed in networks of linear rate models, but a theoretical understanding is so far lacking. In the current study, we derive a theoretical expression for the width of the distribution of integral pairwise covariances. We make use of the fact that correlations in leaky integrate-and-fire and binary networks are well approximated by effective linear rate models [4, 5], and combine ideas from spin-glass theory [6] with a generating function representation for the joint probability distribution of the network activity [7] to obtain a functional relation of the variance of integral pairwise covariances on parameters of the network connectivity. The expression explains the divergence of the mean covariances and their width when the critical coupling w_crit is approached, the point at which the linear network looses stability. Using this relation, distributions of correlations can provide insights into the properties of the underlying network structure and its operational regime.AcknowledgementsPartly supported by Helmholtz Portfolio Supercomputing and Modeling for the Human Brain (SMHB), the Helmholtz young investigator group VH-NG-1028, EU Grant 269921 (BrainScaleS), EU Grant 604102 (HBP).References1. Ecker et al. (2010) Decorrelated neuronal firing in cortical microcircuits. Science 327: 584–587. , 10.1126/science.1179867 2. Renart et al. (2010) The asynchronous state in cortical cicuits. Science 327: 587–590. , 10.1126/science.1179850 3. Tetzlaff et al. (2012) Decorrelation of neural-network activity by inhibitory feedback. PLoS Comput Biol 8:e1002596., 10.1371/journal.pcbi.1002596 4. Grytskyy et al. (2013) A unified view on weakly correlated recurrent networks. Front Comput Neurosci 7, 10.3389/fncom.2013.00131 5. Pernice et al. (2012) Recurrent interactions in spiking networks with arbitrary topology. Phys. Rev. E 85:031916., 10.1103/PhysRevE.85.031916 6. Sompolinsky and Zippelius (1982) Relaxational Dynamics of the Edwards-Anderson Model and the Mean Field Theory of Spin Glasses. Phys. Rev. B 25, 6860. , 10.1103/PhysRevB.25.6860 7. Chow and Buice (2015) Path Integral Methods for Stochastic Differential Equations. J Math Neurosci 2015, 5:8 , 10.1186/s13408-015-0018-