Event Abstract

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

  • 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.


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).


[1] Cohen MR, Kohn A (2011). Nat. Neurosci. (2011) 14:7 811-819.
[2] Risken H, 2nd edition, Springer Series in Synergetics, Springer.
[3] Brunel N (2000), Journal of Computational Neuroscience 8, 183–208
[4] Ginzburg I, Sompolinsky H (1994). PRE 50 (4): 3171-3191

Keywords: binary neurons, covariance matrix, LIF neurons, Ornstein-Uhlenbeck-Process, Poisson neurons, Recurrent networks

Conference: Bernstein Conference 2012, Munich, Germany, 12 Sep - 14 Sep, 2012.

Presentation Type: Poster

Topic: Neurons, networks, dynamical systems

Citation: Grytskyy D, Helias M, Tetzlaff T and Diesmann M (2012). Ornstein-Uhlenbeck-process joins and extends different theories of correlations. Front. Comput. Neurosci. Conference Abstract: Bernstein Conference 2012. doi: 10.3389/conf.fncom.2012.55.00101

Received: 12 May 2012; Published Online: 12 Sep 2012.

* Correspondence: Mr. Dmytro Grytskyy, Forschungszentrum Juelich GmbH, Institute of Neuroscience and Medicine, Juelich, 52425, Germany, d.grytskyy@fz-juelich.de

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