Ornstein-Uhlenbeck-process joins and extends different theories of correlations

Abstract

Event Abstract Back to Event Ornstein-Uhlenbeck-process joins and extends different theories of correlations Dmytro Grytskyy1*, Moritz Helias1, Tom Tetzlaff1 and Markus Diesmann1, 2 1 Forschungszentrum Juelich GmbH, Institute of Neuroscience and Medicine, Germany 2 RWTH Aachen, Faculty of Medicine, Germany Different models are in use for to investigate neuronal reccurent networks and their resulting structure of covariances. The diversity of models brings up the question, which features of correlations are generic properties and which are peculiarities of the often abstracted neuronal dynamics. Currently, it is unclear how different neuron models relate to each other and whether and how results carry between models. In this work we present a unified view on pairwise correlations in recurrent random networks. We consider binary neuron models, leaky integrate-and-fire models, and linear point process models. We show the equivalence between each of these models after linear approximation to the Ornstein-Uhlenbeck (OU) process [2]. The above mentioned models split into two groups, which are distinct from each other only in a matrix prefactor scaling the noise and the choice of variables interpreted as neural activity. The known closed form solution of OU processes [2] holds for both classes. This approach enables us to map results obtained for one model to another, in particular we extend the theory of correlations of all considered models to the presence of synaptic conduction delays, and present a simpler derivations for some established results [4]. The approach is applicable to general forms of connectivities, and for the purpose of comparison to direct simulations, population averaged results are presented. The method of linearization required to map non-linear models to the OU process employs elements of a mean field approach. Furthermore, it takes into account neuron input distributions around mean field values, increasing the accuracy of the results and showing the influence of fluctuations on effective system parameters that determine e.g. the presence and parameters of oscilations. The theoretical population averages are exact for fixed out-degree network architectures and approximate for fixed in-degree. The latter, however, are beneficial for non-linear models, allowing a simpler linearization based on mean-field arguments. Finally we show that the oscillatory instability known for networks of integrate-and-fire models with delayed feedback [3] is a model-invariant feature of any of the studied dynamics: We find that an identical pole structure of the cross spectra determines the population power spectra in different models and we explain the class dependent differences between covariances in time and frequency domain. Acknowledgements Partially supported by the Helmholtz Alliance on Systems Biology, the Next-Generation Supercomputer Project of MEXT, and EU Grant 269921 (BrainScaleS). All network simulations were carried out with NEST (http://www.nest-initiative.org).