In recent years, techniques in systems neuroscience and electrophysiology to record from many neurons in parallel became widely available (see, e.g., Georgopoulos et al., 1986
; Nicolelis et al., 1997
; Santhanam et al., 2006
; Jarosiewicz et al., 2008
). While in the beginning, this was seen merely as a way to perform many single-neuron experiments simultaneously, the full potential of such parallel recordings was soon realized and led to a paradigm shift in the analysis from single-spike statistics (Perkel et al., 1967a
; Brillinger, 1988
; Bialek et al., 1991
; Rieke et al., 1999
; Gerstner and Kistler, 2002
) toward the analysis of neural population activity and interactions between neurons. Early attempts focused on pair-wise cross-correlation analysis (Perkel et al., 1967b
; Aertsen et al., 1989
), but recent statistical models have tried to capture the global network activity (Iyengar, 2001
; Brown et al., 2004
; Truccolo et al., 2005
; Schneidman et al., 2006
; Pillow et al., 2008
; Paninski et al., 2010
) in an attempt to understand how neurons collectively encode and process information. The access to this high-dimensional type of data has also triggered a challenge for statistical data analysis and modeling (Kass et al., 2005
): What is an adequate low-dimensional description of the structure in the spiking activity of neural populations?
One possible solution is to extract the dynamic coupling (or effective connectivity) structure based on the recorded spike trains. The effective coupling between two neurons quantifies how the spiking activity of one neuron enhances or reduces the spiking probability of the second one and should in general be directed and causal. After thresholding, such a coupling structure reduces to a binary, directed connectivity matrix in which a non-zero entry indicates a directed coupling between two neurons. The connectivity matrix defines a graph and its structural properties can be further quantified. Using this analysis scheme, the overall network activity can be summarized in a few quantitative parameters of the inferred network topology (Figure ).
Figure 1 Schematic overview of our approach. (A) The activity of a number of neurons has been recorded in the form of a set of spike trains, e.g., using extracellular electrodes. (B) A statistical framework is applied to extract effective coupling filters between (more ...)
The construction of the network graph proceeds therefore in three steps. First, we need to extract couplings from time-series observations. Several statistical methods have been proposed that extend beyond simple cross-correlation analysis. A particularly interesting statistical model is the generalized linear model (GLM; McCullagh and Nelder, 1989
). GLMs allow the extraction of directed, i.e., causal coupling filters that are conditioned on the whole population response (Figure B). This is in contrast to many other proposed schemes that are based only on pair-wise measures or very crude approximations to the overall population activity.
As a second step, each coupling can be further reduced to a binary entry in a connectivity matrix by thresholding (Figure C). Since GLMs are set up on the basis of maximum-likelihood estimation, the threshold can be chosen by cross-validation and is therefore not a free parameter of the estimation procedure.
Third, once a graph of the effective connectivity is estimated (Figure D), its structural properties can be studied (Bullmore and Sporns, 2009
; Rubinov and Sporns, 2010
). Two notions have been influential in the analysis of biological networks: small-world (Watts and Strogatz, 1998
) and scale-free networks (Barabási and Albert, 1999
). The latter are characterized by the fact that their degree distribution, i.e., the distribution of the number of incoming or outgoing links at each node, can be described by a power-law.
Small-world networks show a highly clustered structure while retaining an overall low average path length between any two nodes (see Section 2.4
for a formal definition). In simulation studies, it has been shown that small-world topology can evolve from certain optimality considerations (Sporns et al., 2000
) and synaptic plasticity rules (Cho and Choi, 2010
). A functional role of small-world structures has been hypothesized for improved memory recall in associative networks (Morelli et al., 2004
) and faster and more reliable synchronizability (Lago Fernández et al., 2000
Both concepts of scale-free and small-world networks have been empirically studied in the context of neuroscience, especially in systems neuroscience where connectivity between different brain areas was studied using coarse-grained signals such as fMRI, EEG, or MEG (see, e.g., Reijneveld et al., 2007
; Bullmore and Sporns, 2009
; Chavez et al., 2010
; Sepulcre et al., 2010
). Only very few studies have attempted to quantify small-world properties of neural networks on the level of individually recorded neurons (Bettencourt et al., 2007
; Yu et al., 2008
). These studies suffer from a number of flaws: First, they consider only networks that were obtained from in vitro
preparations or anesthetized animals. Furthermore, their estimates of the connectivity matrix were either based on pair-wise measures (Bettencourt et al., 2007
) or used very crude approximations to the population activity (Yu et al., 2008
). Both studies used undirected (i.e., correlative) measures that did not offer any notion of causality.
In this paper, we attempt to overcome most of these limitations by using multi-electrode recordings from the awake monkey on a larger data set than in previous approaches. The monkey was exposed to naturalistic video sequences while activity was recorded from a set of neurons in the visual cortex, using multiple electrodes. We investigate scale-free and small-world properties of neuronal networks using directed couplings estimated by GLMs. For this, the quantity proposed by Humphries and Gurney (2008
) for quantifying small-world-ness has to be generalized to directed networks. We find that the networks under consideration lack scale-free behavior, but show a small, but significant small-world structure.
We show that the design of typical electrophysiological experiments imposes a particular sampling scheme that can have a considerate impact on how the small-world structure of the network is evaluated (Bialonski et al., 2010
). Random graphs that take the geometry of the experiment into account can serve as a more refined null model than the homogeneous random graphs that are usually proposed as reference models to evaluate small-world properties. Using such refined reference models, we find that most of the small-world structure can be attributed to a simple distance-dependent connection probability between neurons. Finally, typical experimental methods are based on sub-sampling from a larger network. We investigate how this sub-sampling is likely to affect the estimation of small-world properties by simulating multi-electrode experiments in a virtual network of neurons that are coupled in a physiologically plausible way.