Benjamin Staude

How Structure Determines Correlations in Neuronal Networks

Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks.

CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains

Recent developments in electrophysiological and optical recording techniques enable the simultaneous observation of large numbers of neurons. A meaningful interpretation of the resulting multivariate data, however, presents a serious challenge. In particular, the estimation of higher-order correlations that characterize the cooperative dynamics of groups of neurons is impeded by the combinatorial explosion of the parameter space. The resulting requirements with respect to sample size and recording time has rendered the detection of coordinated neuronal groups exceedingly difficult. Here we describe a novel approach to infer higher-order correlations in massively parallel spike trains that is less susceptible to these problems. Based on the superimposed activity of all recorded neurons, the cumulant-based inference of higher-order correlations (CuBIC) presented here exploits the fact that the absence of higher-order correlations imposes also strong constraints on correlations of lower order. Thus, estimates of only few lower-order cumulants suffice to infer higher-order correlations in the population. As a consequence, CuBIC is much better compatible with the constraints of in vivo recordings than previous approaches, which is shown by a systematic analysis of its parameter dependence.

Recurrent interactions in spiking networks with arbitrary topology.

The population activity of random networks of excitatory and inhibitory leaky integrate-and-fire neurons has been studied extensively. In particular, a state of asynchronous activity with low firing rates and low pairwise correlations emerges in sparsely connected networks. We apply linear response theory to evaluate the influence of detailed network structure on neuron dynamics. It turns out that pairwise correlations induced by direct and indirect network connections can be related to the matrix of direct linear interactions. Furthermore, we study the influence of the characteristics of the neuron model. Interpreting the reset as self-inhibition, we examine its influence, via the spectrum of single-neuron activity, on network autocorrelation functions and the overall correlation level. The neuron model also affects the form of interaction kernels and consequently the time-dependent correlation functions. We find that a linear instability of networks with Erdös-Rényi topology coincides with a global transition to a highly correlated network state. Our work shows that recurrent interactions have a profound impact on spike train statistics and provides tools to study the effects of specific network topologies.

Higher-Order Correlations in Non-Stationary Parallel Spike Trains: Statistical Modeling and Inference

The extent to which groups of neurons exhibit higher-order correlations in their spiking activity is a controversial issue in current brain research. A major difficulty is that currently available tools for the analysis of massively parallel spike trains (N >10) for higher-order correlations typically require vast sample sizes. While multiple single-cell recordings become increasingly available, experimental approaches to investigate the role of higher-order correlations suffer from the limitations of available analysis techniques. We have recently presented a novel method for cumulant-based inference of higher-order correlations (CuBIC) that detects correlations of higher order even from relatively short data stretches of length T = 10–100 s. CuBIC employs the compound Poisson process (CPP) as a statistical model for the population spike counts, and assumes spike trains to be stationary in the analyzed data stretch. In the present study, we describe a non-stationary version of the CPP by decoupling the correlation structure from the spiking intensity of the population. This allows us to adapt CuBIC to time-varying firing rates. Numerical simulations reveal that the adaptation corrects for false positive inference of correlations in data with pure rate co-variation, while allowing for temporal variations of the firing rates has a surprisingly small effect on CuBICs sensitivity for correlations.

Can spike coordination be differentiated from rate covariation

There has been a long and lively debate on whether rate covariance and temporal coordination of spikes, regarded as potential origins for correlations in cortical spike signals, fulfill different roles in the cortical code. In this context, studies that report spike coordination have often been criticized for ignoring fast nonstationarities, which would result in wrongly assigned spike coordination. The underlying hypothesis of this critique is that spike coordination is essentially identical to rate covariation, only on a shorter timescale. This study investigates the validity of this critique. We provide a decomposition for the cross-correlation function of doubly stochastic point processes, where each of the components corresponds precisely to the concepts of dependence under investigation. This allows us to correct the correlation function for rate effects, which implies that spike coordination and rate covariation are statistically separable concepts of dependence. Furthermore, we present direct and intuitive model implementations of the discussed concepts and illustrate that their difference is not a matter of timescale. Analysis of data generated by our models and analytical description of the relevant estimators reveals, however, that spike coordination dramatically influences the accuracy of rate covariance estimation. As a consequence, extreme parameter combinations can lead to situations where the concept of dependence cannot be identified empirically. However, for a wide range of parameters, the concept of dependence underlying a given data set can be identified regardless of its timescale.

Higher-Order Correlations and Cumulants

Recent advances in electrophysiological and imaging techniques have highlighted the need for correlation measures that go beyond simple pairwise analyses. In this chapter, we discuss cumulant correlations as natural and intuitive higher-order generalizations of the covariance. In particular, we show how cumulant correlations fit to a frequently used additive model of correlation, an idea that mimics correlations among spike trains that originate from overlapping input pools. Finally, we compare the cumulant correlations to the interaction parameters of the well-known exponential family by computing the respective parameters for two different models. We find that the different frameworks measure entirely different aspects, so that populations can have higher-order correlations in one framework but none in the other.

Modeling and analyzing higher-order correlations in non-Poissonian spike trains.

Measuring pairwise and higher-order spike correlations is crucial for studying their potential impact on neuronal information processing. In order to avoid misinterpretation of results, the tools used for data analysis need to be carefully calibrated with respect to their sensitivity and robustness. This, in turn, requires surrogate data with statistical properties common to experimental spike trains. Here, we present a novel method to generate correlated non-Poissonian spike trains and study the impact of single-neuron spike statistics on the inference of higher-order correlations. Our method to mimic cooperative neuronal spike activity allows the realization of a large variety of renewal processes with controlled higher-order correlation structure. Based on surrogate data obtained by this procedure we investigate the robustness of the recently proposed method empirical de-Poissonization (Ehm et al., 2007). It assumes Poissonian spiking, which is common also for many other estimation techniques. We observe that some degree of deviation from this assumption can generally be tolerated, that the results are more reliable for small analysis bins, and that the degree of misestimation depends on the detailed spike statistics. As a consequence of these findings we finally propose a strategy to assess the reliability of results for experimental data.

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

A new method to infer higher-order spike correlations from membrane potentials

What is the role of higher-order spike correlations for neuronal information processing? Common data analysis methods to address this question are devised for the application to spike recordings from multiple single neurons. Here, we present a new method which evaluates the subthreshold membrane potential fluctuations of one neuron, and infers higher-order correlations among the neurons that constitute its presynaptic population. This has two important advantages: Very large populations of up to several thousands of neurons can be studied, and the spike sorting is obsolete. Moreover, this new approach truly emphasizes the functional aspects of higher-order statistics, since we infer exactly those correlations which are seen by a neuron. Our approach is to represent the subthreshold membrane potential fluctuations as presynaptic activity filtered with a fixed kernel, as it would be the case for a leaky integrator neuron model. This allows us to adapt the recently proposed method CuBIC (cumulant based inference of higher-order correlations from the population spike count; Staude et al., J Comput Neurosci 29(1–2):327–350, 2010c) with which the maximal order of correlation can be inferred. By numerical simulation we show that our new method is reasonably sensitive to weak higher-order correlations, and that only short stretches of membrane potential are required for their reliable inference. Finally, we demonstrate its remarkable robustness against violations of the simplifying assumptions made for its construction, and discuss how it can be employed to analyze in vivo intracellular recordings of membrane potentials.

Effect of network structure on spike train correlations in networks of integrate-and-fire neurons.

Balanced networks of excitatory and inhibitory neurons are a popular paradigm to describe the ground state of cortical activity. Although such networks can assume a state of asynchronous and irregular activity with low firing rates and low pairwise correlations, recurrent connectivity inevitably induces correlations between spike trains [1]. To elucidate the influence of network topology on correlations, we have recently employed the framework of linearly interacting point processes [2] as an analytically tractable model for network dynamics [3]. A power series of the connectivity matrix can be used to disentangle the different contributions to pairwise correlations from direct and indirect interactions between neurons. In the present study we show that this framework can be applied to approximate dynamics of networks of integrate-and-fire neurons, if the reset after each spike is formally described as self-inhibition. The reset then effectively decreases overall correlations. We study ring networks, where we are able to derive analytical expressions for the distance dependence of correlations and fluctuations in population activity. Rates and correlations in simulated networks are predicted accurately, provided that spike train correlations are reasonably small and the linear impulse response of single neurons is known.