J Comput Neurosci (2010) 28:137–154 DOI 10.1007/s10827-009-0193-z Models of cortical networks with long-range patchy projections Nicole Voges · Christian Guijarro · Ad Aertsen · Stefan Rotter Received: 13 November 2008 / Revised: 25 August 2009 / Accepted: 1 October 2009 / Published online: 29 October 2009 © Springer Science + Business Media, LLC 2009 Abstract The cortex exhibits an intricate vertical and horizontal architecture, the latter often featuring spa- tially clustered projection patterns, so-called patches. Many network studies of cortical dynamics ignore such spatial structures and assume purely random wiring. Here, we focus on non-random network structures pro- vided by long-range horizontal (patchy) connections that remain inside the gray matter. We investigate how the spatial arrangement of patchy projections influ- ences global network topology and predict its impact on the activity dynamics of the network. Since neu- roanatomical data on horizontal projections is rather sparse, we suggest and compare four candidate scenar- ios of how patchy connections may be established. To identify a set of characteristic network properties that enables us to pin down the differences between the resulting network models, we employ the framework of stochastic graph theory. We find that patchy projections Action Editor: Alessandro Treves N. Voges · A. Aertsen · S. Rotter Bernstein Center for Computational Neuroscience Freiburg, Albert-Ludwig University, Freiburg, Germany N. Voges (B ) · C. Guijarro · A. Aertsen Neurobiology & Biophysics, Faculty of Biology, Albert-Ludwig University, Freiburg, Germany e-mail: nicole.voges@incm.cnrs-mrs.fr S. Rotter Computational Neuroscience, Faculty of Biology, Albert-Ludwig University, Freiburg, Germany Present Address: N. Voges INSERM, UMR 751 Universite Aix-Marseille, 27 Bd Jean Moulin, 13385 Marseille Cedex 05, France provide an exceptionally efficient way of wiring, as the resulting networks tend to exhibit small-world prop- erties with significantly reduced wiring costs. Further- more, the eigenvalue spectra, as well as the structure of common in- and output of the networks suggest that different spatial connectivity patterns support distinct types of activity propagation. Keywords Cortical network model · Horizontal synaptic connectivity · Wiring optimization · Stochastic graph theory 1 Introduction The prevailing model for studying cortical network dynamics is based on randomly connected neurons (Brunel 2000; Kumar et al. 2008b; Kriener et al. 2008). More and more studies (Mehring et al. 2003; Kumar et al. 2008a; Roudi and Treves 2008; Kriener et al. 2009) take spatial network features into account, but are largely constrained to locally coupled neurons within the range of a cortical column (for an excep- tion see Johansson and Lansner 2007). In reality however, the cortical network exhibits a distinctive three-dimensional structure: in the vertical direction, perpendicular to the surface of cortex, it is com- posed of several layers with layer-specific connectivity (Thomson and Bannister 2003; Binzegger et al. 2004); while in the horizontal direction, parallel to the sur- face, a spatially extended system of connections exists, incorporating several interconnected columns. The dis- tance between connected neurons varies from a few micrometers to centimeters (Schüz and Braitenberg 2002; Lewis et al. 2002).