The effect of intra- and inter-subject variability of hemodynamic responses on group level Granger causality analyses

Abstract

Granger causality analyses aim to reveal the direction of influence between brain areas by analyzing temporal precedence: if a signal change in area A consistently precedes a signal change in area B, then A Granger-causes B. fMRI-based Granger causality inferences are mediated by the hemodynamic response function which can vary across brain regions. This variability might induce a bias in Granger causality analyses. Here we use simulations to investigate the effect of hemodynamic response variability on Granger causality analyses at the level of a group of twenty participants. We used a set of hemodynamic responses measured by Handwerker et al. (Neuroimage, 2004) and simulated 200 experiments in which time series with known directions of influence are convolved with these hemodynamic responses and submitted to Granger causality analysis. Results show that the average chance to find a significant Granger causality effect when no actual influence is present in the data stays well below the p-level imposed on the second level statistics. Most importantly, when the analyses reveal a significant directed influence, this direction was accurate in the vast majority of the cases. The sensitivity of the analyses however depended on the neuronal delay between the source and target regions and their relative hemodynamic delay. Influences flowing from regions to one with the same or a slower hemodynamic response function were detected in over 80% of the cases when the neuronal delay was at least 100 ms. Influences flowing to a region with a faster hemodynamic delay were detected in over 80% of the cases when delays are above 1 s.

Introduction

Granger causality is a measure of directed influence between two time series. Originally conceptualized by Wiener and formalized by Granger (Wiener, 1956, Granger, 1969), it was introduced as a connectivity analysis for fMRI data in 2003 by Goebel et al. (2003). Granger formalized causality between two time series using the concept of temporal precedence: if a signal change in A is consistently followed by a signal change in B, A Granger-causes B.

When applied to fMRI, Granger causality indicates the direction of influence between BOLD time series of different brain areas. Results of Granger causality analyses are interpreted as indicating connectivity on a neuronal level. However, fMRI measures BOLD responses rather than neuronal activity directly, therefore this inference is mediated by the hemodynamic response in the brain. The hemodynamic response is not equal across brain regions (Rajapakse et al., 1998, Aguirre et al., 1998, Kruggel and von Cramon, 1999, Handwerker et al., 2004) and this regional variability could cause problems for Granger causality analyses (David et al., 2008, de Marco et al., 2009, Roebroeck et al., 2005, Friston, 2009, Chang et al., 2008). On the one hand, it is feared that spurious Granger causality findings could be reported as a difference in hemodynamic response might introduce temporal relations where there are none. On the other hand a difference in hemodynamic response might invert the reported direction of Granger causality. The intuitive idea behind this last problem is as follows: If region A causes activity changes in region B and region B has a faster hemodynamic response than region A, a Granger causality analysis might indicate a net influence going from B to A rather than the true underlying causality from A to B (see Fig. 1). Roebroeck et al. (2005) warned against this possible confound and suggested using the modulation of connectivity between different conditions, rather than within one condition.

Deshpande et al. (2010) investigated the effect of differences in hemodynamic response function on the sensitivity of Granger causality analyses in single subjects. They found that even when intra-subject differences in hemodynamic response function are present, Granger causality is still sensitive to influences in the order of a hundred milliseconds. This result seems counterintuitive with differences in hemodynamic delay as big as 2.5 s (Handwerker et al., 2004). However, as Deshpande et al. (2010) note, differences between hemodynamic responses are not just due to a temporal shift of the whole response. Rather, parameters that varied most between regions were onset time and time-to-peak (Handwerker et al., 2004). This means that convolution of time series of neuronal activity with a hemodynamic response function is not mathematically equivalent to a shift in time. The difference in hemodynamic response does not remove the characteristic temporal relation between the two time series, which could be an explanation why the effect of regional variability of the hemodynamic response was not as disastrous as initially expected.

The study of Deshpande et al. (2010) shows how sensitivity of Granger causality is affected by variability in hemodynamic response at the level of the single subject. However, Granger causality at group level is of more interest than at single subject level for most studies. Furthermore, as within subject variability varies across subjects (Aguirre et al., 1998) results from Deshpande et al. (2010) cannot be extrapolated trivially to group results. In this article, we investigate whether differences in hemodynamic response have an effect on group level Granger causality results. We use simulations to answer the following three questions. (1) When no actual directed influence is present between two time series, what is the chance to find a significant Granger causality result (i.e. a false positive)? (2) When an actual directed influence is present between two time series, how often will differences in hemodynamic response lead to a significant inverted Granger causality direction and (3) how often will the true direction be detected (sensitivity)?

Hemodynamic response shapes as measured by Handwerker et al. (2004) are used in the current simulations. These hemodynamic response shapes were measured subject by subject for 20 individuals, permitting us to simulate data for an appropriate sized ‘group study’ that will take realistic inter-subject variability into account. Unfortunately, due to the fact that participants were performing a task while their hemodynamic responses were measured, the BOLD latencies conflate neuronal latencies and hemodynamic latencies. This inflates the actual variability of the hemodynamic responses. Ideally, one would use data where local field potentials were measured simultaneously to disentangle these sources of variance. Awaiting such a dataset, the current work will most likely represent a scenario worse than reality.