Causal and Structural Connectivity of Pulse-Coupled Nonlinear Networks

Abstract

We study the reconstruction of structural connectivity for a general class of pulse-coupled nonlinear networks and show that the reconstruction can be successfully achieved through linear Granger causality (GC) analysis. Using spike-triggered correlation of whitened signals, we obtain a quadratic relationship between GC and the network couplings, thus establishing a direct link between the causal connectivity and the structural connectivity within these networks. Our work may provide insight into the applicability of GC in the study of the function of general nonlinear networks.

Analysis of sampling artifacts on the Granger causality analysis for topology extraction of neuronal dynamics

“…From this, one may conclude that GC values are linked to the adjacency matrix of the network, i.e., A yx = 1 and A xy = 0. This is also consistent with the result by performing statistical significance test (see Appendix C in Supplementary Material for details) (Zhou et al, 2013b, 2014). In the present work, we focus on the implication of sampling effects and on the assessment of the reliability of this reconstruction by the GC analysis.…”

“…Furthermore, the general strategies of overcoming these sampling issues are not limited to the bivariate time series with unidirectional connection as discussed in this work. They are also applicable to the GC analysis of bivariate time series with bidirectional connections (shown in Supplementary Figure 2) as well as multivariate time series with any general connectivity structure (Zhou et al, 2013b, 2014). …”

“…Previously, we have applied the GC analysis to reconstruct network structural connectivity and our results demonstrate that the GC analysis can be successfully used to reconstruct the coupling structures of the integrate-and-fire (I&F) networks (Zhou et al, 2013b, 2014). We note in passing that for smoothly coupled oscillator networks (with or without noisy inputs), techniques including chaotic synchronization or phase dynamics have been used to uncover the structural connectivity via random phase reset or their responses to perturbations (Yu et al, 2006; Timme, 2007; Napoletani and Sauer, 2008; Smirnov and Bezruchko, 2009; Ren et al, 2010; Levnajić and Pikovsky, 2011), whereas, for some simple non-smooth network dynamics, e.g., δ-pulse coupled, current-based integrate-and-fire systems, their topologies can be reconstructed by designing a specific set of external inputs (Bussel et al, 2011).…”

“…As can be seen below, F y → x and F x → y can be related to the directional connectivity of a network (Zhou et al, 2013b, 2014). …”

Identifying Chaotic FitzHugh–Nagumo Neurons Using Compressive Sensing

“…While many of these previous methods require complete or partial information about the dynamical equations of the isolated nodes and their coupling functions, completely data-driven and model-free methods exist. For example, the network structure can be obtained by calculating the causal influences among the time series based on, for example, the Granger causality method [15,16], the transfer-entropy method [17] or the method of inner composition alignment [18]. However, such causality-based methods are unable to reveal information about the dynamical equations of the isolated nodes.…”