Theory of individual pairwise correlations in stochastic binary networks

Pairwise correlations between the activities of neurons exhibittime-dependent modulations with respect to behavior [1], influence theinformation-content of population signals [2,3], determine theinteraction of neuronal activity and spike-timing dependent synapticplasticity [4].Correlations averaged over neuron pairs are closely linked tofluctuations of the population activity and are hence readilyaccessible by population-averaged mean-field theory in the large Nlimit [5], but also in finite-sized networks [6].Here we leave the population description and extend the theory ofsecond order correlations in stochastic binary networks [7] toindividual neuron pairs. We show how a systematic truncation of themoment hierarchy that consistently neglects third order cumulantsyields a non-linear system of equations for individual mean activitiesand pairwise covariances, which after linearization, leads to amodified Lyapunov equation. We show that the method of solution,sufficient for the population-averaged case, yields a systematicover-estimation of the spread of individual pairwise covariances,while an adapted method of solution provides quantitative predictions ina balanced random network model down to hundreds of neurons.As a corollary we show how the covariance matrix together with itsslope at zero time lag determine the effective synaptic interactionstrength between neurons only modulo the total synaptic noise level ofthe receiving neuron. The presented theory renders the investigationof distributed pairwise correlations analytically accessible and mayprove useful for network reconstruction as well as to foster ourunderstanding of correlation-sensitive synaptic plasticity inrecurrent networks.Partially supported by the Helmholtz Association: Young investigator'sgroup VH-NG-1028, portfolio theme SMHB, the Jülich Aachen ResearchAlliance (JARA), and 604102 (HBP).[1] Kilavik BE, Roux S, Ponce-Alvarez A, Confais J, Gruen S, et al. (2009) J Neurosci 29: 12653--12663.[2] Zohary E, Shadlen MN, Newsome WT (1994). Nature 370: 140--143.[3] Shamir M, Sompolinsky H (2001). In: Advances in Neural Information Processing Systems. MIT Press, pp. 277--284.[4] Morrison A, Diesmann M and Gerstner W (2008). Biol. Cybern. 98 459--78[5] Renart A, De La Rocha J, Bartho P, Hollender L, Parga N, et al. (2010). Science 327: 587–590.[6] Tetzlaff T, Helias M, Einevoll GT, Diesmann M (2012). PLoS Comput Biol 8(8): e1002596.[7] Ginzburg I, Sompolinsky H (1994). Phys Rev E 50: 3171--3191