Higher-order correlations in non-stationary parallel spike trains : statistical modeling and inference

et al., 2000; Brown et al., 2004). In particular, whether or not coincident spikes of pairs of neurons participate in synchronized “cluster-events” cannot be decided on measurements of pairwise correlation alone; this can only be achieved by the systematic assessment of higher-order correlations, i.e., statistical couplings among triplets, quadruplets, and larger groups (Martignon et al., 1995; Staude et al., 2010). Importantly, the nonlinear dynamics of spike generation makes neurons extremely sensitive for synchrony in their input pools (Softky, 1995; König et al., 1996). Ignoring these higher-order correlations in the statistical description of spiking populations is therefore hardly advisable (Bohte et al., 2000; Kuhn et al., 2003). Initially, the main obstacle for assessing the higher-order structure of neuronal populations were limitations in experimental methodology, as until recently state-of-the-art electrophysiological setups allowed to record only few neurons simultaneously. The advent of multi-electrode arrays and optical imaging techniques, however, now reveals fundamental shortcomings of available analysis tools (Brown et al., 2004). Mathematical frameworks to model and estimate higher-order correlations typically assign one “interaction parameter” for every subgroup of the population, leading to a 2 − 1 dimensional model for a population comprising N neurons (Martignon et al., 1995, 2000). The associated estimation problem greatly suffers from this combinatorial explosion: the number of parameters to be estimated from the available sample size (a population of N = 100 neurons implies ∼10 parameters while 100 s of data provide only ∼10 samples) illustrates the principal infeasibility of this approach. In fact, the estimation of such higher-order IntroductIon It has long been suggested that fundamental insight into the nature of neuronal computation requires the understanding of the cooperative dynamics of populations of neurons (Hebb, 1949). A controversial issue in this debate is the role of correlations among nerve cells. On the one hand, an increasing body of both experimental (e.g., Gray and Singer, 1989; Vaadia et al., 1995; Riehle et al., 1997; Bair et al., 2001; Kohn and Smith, 2005; Shlens et al., 2006; Fujisawa et al., 2008; Pillow et al., 2008) and theoretical (Abeles, 1991; Diesmann et al., 1999; Kuhn et al., 2003) literature supports the concept of cooperative computation on various temporal and spatial scales. On the other hand, the mostly detrimental effect of correlations on rate-based information transmission and processing (Abbott and Dayan, 1999; Averbeck and Lee, 2006; Josić et al., 2009) has generated a strong opposition toward correlation-based concepts of cortical coding (Shadlen and Newsome, 1998; Averbeck et al., 2006; Schneidman et al., 2006; Ecker et al., 2010). Evidently, a thorough description of the correlation structure of neuronal populations is an indispensable prerequisite to resolve these opposing theoretical viewpoints (Brown et al., 2004). Experimental reports on coordinated activity at the level of spike trains resort almost exclusively to correlations between pairs of nerve cells (e.g., Eggermont, 1990; Vaadia et al., 1995; Kreiter and Singer, 1996; Riehle et al., 1997; Kohn and Smith, 2005; Sakurai and Takahashi, 2006; Fujisawa et al., 2008; Ecker et al., 2010). Such pairwise correlations cannot, as a matter of principle, resolve the cooperative activity of neuronal populations to the extent required for rigorous hypothesis testing (Gerstein et al., 1989; Martignon Higher-order correlations in non-stationary parallel spike trains: statistical modeling and inference