MÉTHODES

Increasing robustness of pairwise methods for
effective connectivity in magnetic resonance
imaging by using fractional moment series
of BOLD signal distributions

Natalia Z. Bielczyk

1,3, Jan K. Buitelaar
Jeffrey C. Glennon1,2, and Christian F. Beckmann 1,2,3,4

1,2, Alberto Llera

1,2,

un accès ouvert

journal

1Donders Institute for Brain, Cognition and Behavior, Nijmegen, the Netherlands
2Department of Cognitive Neuroscience, Radboud University Nijmegen Medical Centre, Nijmegen, the Netherlands
3Radboud University Nijmegen, Nijmegen, the Netherlands
4Oxford Centre for Functional MRI of the Brain, University of Oxford, Oxford, ROYAUME-UNI

Mots clés: Causal
Pairwise causal inference

inference, Effective connectivity, Functional magnetic resonance imaging,

ABSTRAIT

Estimating causal interactions in the brain from functional magnetic resonance imaging
(IRMf) data remains a challenging task. Multiple studies have demonstrated that all current
approaches to determine direction of connectivity perform poorly when applied to synthetic
fMRI datasets. Recent advances in this field include methods for pairwise inference, lequel
involve creating a sparse connectome in the first step, and then using a classifier in order to
determine the directionality of connection between every pair of nodes in the second step. Dans
this work, we introduce an advance to the second step of this procedure, by building a
classifier based on fractional moments of the BOLD distribution combined into cumulants.
The classifier is trained on datasets generated under the dynamic causal modeling (DCM)
generative model. The directionality is inferred based on statistical dependencies between
the two-node time series, Par exemple, by assigning a causal link from time series of low
variance to time series of high variance. Our approach outperforms or performs as well as
other methods for effective connectivity when applied to the benchmark datasets. Surtout, it
is also more resilient to confounding effects such as differential noise level across different
areas of the connectome.

RÉSUMÉ DE L'AUTEUR

Estimating causal interactions from functional magnetic resonance imaging (IRMf) data is a
formidable task. Recent advances in this field include methods for pairwise inference. In the
first step of this procedure, connections are revealed by means of functional connectivity. Dans
the second step, every detected connection is analyzed separately to reveal the direction of
the causal links. We introduce an advance to the second step of this procedure by building a
classifier based on the novel concept of fractional moments of the BOLD distribution
combined into cumulants. The classifier is trained on datasets generated under the dynamic
causal modeling (DCM) generative model. Using fractional cumulants gives a measure
resilient to confounding effects such as differential noise levels across different areas of the
connectome.

Citation: Bielczyk, N. Z., Llera, UN.,
Buitelaar, J.. K., Glennon, J.. C., &
Beckmann, C. F. (2019). Increasing
robustness of pairwise methods for
effective connectivity in magnetic
resonance imaging by using fractional
moment series of BOLD signal
distributions. Neurosciences en réseau,
3(4), 1009–1037. https://est ce que je.org/
10.1162/netn_a_00099

EST CE QUE JE:
https://doi.org/10.1162/netn_a_00099

Informations complémentaires:
https://doi.org/10.1162/netn_a_00099

Reçu: 11 May 2018
Accepté: 3 Juin 2019

Intérêts concurrents: Les auteurs ont
a déclaré qu'aucun intérêt concurrent
exister.

Auteur correspondant:
Natalia Bielczyk
natalia.bielczyk@gmail.com

Éditeur de manipulation:
Shella Keilholz

droits d'auteur: © 2019
Massachusetts Institute of Technology
Publié sous Creative Commons
Attribution 4.0 International
(CC PAR 4.0) Licence

La presse du MIT

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Improving pairwise inference in fMRI using fractional moments of BOLD

Effective connectivity:
Causal relations between nodes of
the investigated network. In fMRI,
effective connectivity (as opposed to
directed functional connectivity) est
typically derived from a model that
additionally considers the underlying
neuronal processes (voir: dynamic
causal modeling).

Dynamic Causal Modeling:
The most popular generative model
in the fMRI connectivity research. Il
assumes that the observable BOLD
fMRI signal is a product of an
underlying neuronal communication
smoothed with a slow hemodynamic
response.

Functional connectivity:
Statistical associations between
activity in nodes of the investigated
réseau. In fMRI, functional
connectivity is most often evaluated
by means of Pearson/partial
correlation or mutual information.

Pairwise inference:
A two-step causal inference
procedure that reduces causal
inference in a large graph to studying
two-node interactions.

INTRODUCTION

In the context of functional magnetic resonance research, effective connectivity refers to the
process of estimating causal interactions between distinct regions within the brain. Several
characteristics of fMRI data impose severe restrictions on the possibility of estimating such
effective connectivity (Valdes-Sosa, Roebroeck, Daunizeau, & Friston, 2011; Friston, 2011;
Bielczyk et al., 2019). D'abord, the temporal resolution of the image acquisition is low (sampling
rate typically <1 Hz). Furthermore, blood oxygen level–dependent (BOLD) activity is delayed with respect to neuronal firing, with a delay of 3–6 s in the adult human brain (Arichi et al., 2012). The delayed hemodynamic response can also induce spurious cross-correlations between two BOLD time series (Ramsey al., 2010; Bielczyk, Llera, Buitelaar, Glennon, & Beckmann, 2017). Both subject-to-subject and region-to-region variability shape hemodynamic response (Devonshire 2012) provide general limitation methods for effective connectivity research fMRI: when one region faster than in another, temporal precedence peak easily be mistaken causation. Secondly, fMRI data characterized by relatively low signal-to- noise ratio. Within gray matter at field strengths 3 T, task-induced signal changes range within 2–3% mean depending on task (Kruger, 2018). Furthermore, the stochastic noise has been shown have scale-free spectral characteristic (He, 2014; Bédard, Kröger, Destexhe, 2006; Dehghani, Cash, Halgren, Destexhe, 2010), which additionally hinders identifiability causal structures derived from data (Bielczyk, Beckmann, 2017). Moreover, typical protocols in- volve short (a few hundred samples), estimation conditional probabilities between variables, higher order statistic becomes difficult. Multiple were proposed estimate effective connectivities (Friston, 2011). In computational study Smith al. (2011), range con- nectivity tested synthetic datasets dynamic modeling forward model (DCM; Friston, Harrison, Penny, 2003). this study, most estimating causal interactions remained chance level. One method highlighted as successful at identifying links based Patel’s tau measure (PT; Patel, Bowman, Rilling, 2006; Smith 2011). PT entails two-step approach first step involves identifying the (undirected) connections means functional connectivity. This achieved basis of correlations different regions, referred Patel’s kappa (Patel 2011). One note make that, other pairwise inference procedures assume that causation implies correlation. assumption necessary perform infer- ence procedure, is, select reliable further classification into upstream and downstream regions. However, although often true, not always the case under certain circumstances (e.g., control systems) causation might be as- sociated correlation (Kennaway, 2015). recent Di Biswal (2019), the authors investigated task-modulated whole-brain connectivity six block-design and event-related cognitive task. By using psychophysiological pairs of regions interest (ROIs) whole brain, authors identified statistically significant task modulations connectivity, reported, “task modulated was found only activated during but were not or deactivated.” suggests considering pairs which activity correlated neglect some underlying connections. There are multiple strategies implementing thresholding connectivity estimates (Varoquaux Craddock, 2013). Since Pearson typically returns dense Network Neuroscience 1010 l D o w n o a d e d f r o m h t t p : > 0).

Fait intéressant, the range of high discriminability is different for real and imaginary compo-
nents. Maximal discriminability in both real and imaginary cumulants exists in the range of k
+ l > 5.0. En plus, in real cumulants there is a region of high discriminability in the range
k + je < 2.0, and in imaginary cumulants there is such region in the range 2.0 < k + l < 5.0 (Figure 3, both boundary lines marked with a white line). This different characteristic along the imaginary axis illustrates that moving from integer to fractional moments of the distribution provides additional predictive power. Note also that as we are deriving the discriminability maps from the DCM generative model. These maps are fingerprints of the particular problem of effective connectivity in fMRI; when derived from another generative model simulating another dataset, the maps would be different. Network Neuroscience 1016 Improving pairwise inference in fMRI using fractional moments of BOLD Figure 3. Discriminative power for all cumulants in range k, l in [0.0, 5.0], in the ideal case of a very long BOLD time series and no background neuronal noise. Maximal discriminability in both real and imaginary cumulants exists in the range of k + l > 5.0. En plus, in real cumulants
there is a region of high discriminability in the range k + je < 2.0, and in imaginary cumulants there is such region in the range 2.0 < k + l < 5.0 (both boundary lines marked with a white line). In order to investigate how the performance of classification based on single cumulants changes when a single connection is embed in a bigger network, we evaluated their success rate in estimating connectivity for benchmark synthetic datasets (Smith et al., 2011). Figure 4 presents the grand mean success rate achieved by every cumulant separately, across all 28 benchmark synthetic datasets (Smith et al., 2011). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d . t / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Success rate for all the individual cumulants, averaged over 28 simulations from the Figure 4. synthetic benchmark datasets (Smith et al., 2011). White-edged square denotes a single cumulant used by Hyvärinen and Smith (2013). The performance of this cumulant is shown in the color bar, white band. Black-edged squares denote cumulants that give the highest performance on this dataset. Their performance is presented in the color bar, black band. For the cumulants of high indexes k + l > 4.0 (white line), the success rate is not as high as the discriminability presented
in Figure 3 would suggest. The high success rate is not preserved for high-indexed cumulants that
achieved high discriminability on two-node simulations. The maximal grand mean performance
equals 0.847 for the real components and 0.814 for imaginary components.

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Improving pairwise inference in fMRI using fractional moments of BOLD

The success rate for each of the 28 separate synthetic datasets (Smith et al., 2011) is pre-
sented in Supporting Information 4. The maps of simulation-dependent success rate relate to
the maps of discriminative power (Chiffre 3), but they are not identical and differ between sim-
ulations. One difference is that for the cumulants of high indexes k + l > 4.0, the success
rate is not as high as the discriminability presented in Figure 3 would suggest. This is because
Chiffre 3 represents the limit of a system of two isolated nodes with infinite SNR, and a very
long BOLD time series, whereas benchmark synthetic datasets refer to a more realistic case
when for each pair of nodes, the time series is short, there are confounding signals from
other nodes in the network, and there is a certain degree of noise in the communication (voir
Informations complémentaires 2). Altogether, these factors make the high moments hard to estimate
in practice.

Combining Fractional Cumulants into a Classifier

We propose to combine information contained in multiple cumulants by building the classifier
based on a “voting” scheme between the cumulants. This classifier determines whether the
map of cumulants obtained for a pair of time series X(t), Oui(t) is closer to the benchmark maps
presented in Figure 2A (which is an evidence for a connection X → Y), or their inverse (lequel
is an evidence for a flipped connection Y → X). Each of the cumulants Ckl votes due to sign
Srk,je, Sik,je (Figure 2C). If the sign of the cumulant is the same as in Figure 2C, it adds to the
evidence for a connection X → Y, and against this connection otherwise.

Since in realistic conditions (short datasets, large TRs), high index cumulants, k + l > 3.0,
yield the aforementioned estimation problem, we discount their contribution in the voting by
using a nonlinearity of a form (further referred to as weighting throughout the manuscript):

F (X) = log(cosh(maximum(X, 0)))

(3)

A similar function was proposed to discount the outliers present in the BOLD time series in
the work by Hyvärinen and Smith (2013). The final classifier yields:

{ X → Y i f ∑k,je[Srk,l f (Crk,je) + Sik,l f (Cik,je)] ≥ 0

Y → X

otherwise

in the weighted case, et

{ X → Y i f ∑k,je[Srk,lCrk,je + Sik,lCik,je] ≥ 0

Y → X

otherwise

(4)

(5)

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in the unweighted case.

Supervised Learning Using Synthetic Benchmark Datasets

In this work, we derive the classifier by using sign maps Srk,je, Sik,je (Figure 2C) developed using
multiple realizations of a two-node simulation under the DCM generative model, which is a
form of supervised learning. In a different application and under a different generative model,
these maps would look differently.

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Improving pairwise inference in fMRI using fractional moments of BOLD

We know that cumulants differ with respect to discriminability (Chiffre 3), and that the suc-
cess rate of cumulants differs depending on the range k + je <= Indmax (Figure 4). Therefore, we optimize the performance of the classifier with respect to these two dimensions by finding a combination that gives the best grand mean performance across the 28 simulations from the synthetic benchmark datasets, as they represent the variety of experimental conditions en- countered in real-life fMRI setups. First, we fix Indmax = max(k, l) = 3.1, and consider cumulants on a triangle k; l >= 0.1, k +
je < Indmax. We then choose only cumulants of discriminability exceeding a particular value to be fed into the classifier. For instance, a cutoff value of 0.1 means that we include the vote from all cumulants for which the discriminative value is not less than 0.1. We can then evaluate the grand mean success rate (as the mean success rate over all 28 benchmark synthetic datasets) in the function of the thresholding discriminability value. Figure 5A, demonstrates that including l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t / . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 5. Dependence of the grand mean performance on synthetic datasets on the choice of cumulants. (A) Grand mean performance for unweighted cumulants in range k + l <= 3.1, in the function of the cutoff discriminative value. The higher cutoff, the less cumulants we take into account while voting for the directionality of the link. The results clearly show that in order to maximize the success rate in estimating effective connectivity, all the cumulants should be taken into account, except for the diagonal of k = l. (B) The grand mean performance based on cumulants of indexes k, l between 0.1 and k + l <= Indmax, in the function of that maximal index. The optimal performance in unweighted case equals 0.835 for [IndRmax; IndImax] = (2.4, 1.7), and 0.886 for [IndRmax; IndImax] = (2.1, 3.7) in weighted case. Network Neuroscience 1019 Improving pairwise inference in fMRI using fractional moments of BOLD all the cumulants with a positive discriminative value (all cumulants except for k = l, for which discriminability is always zero) gives the best classification performance. Second, we optimize the window Indmax for indexes k, l and compare between classifier with and without weighting with the discount function introduced in Equation 3. Since dis- criminability is generally higher for low indexes k, l (Figure 3), we will evaluate the grand mean performance based on cumulants of indexes between 0 and a maximum Indmax in the function of that maximum. We consider the maximal indexes along real and imaginary di- mension separately. The results are presented in Figure 5B. The optimal performance in an unweighted case equals 0.835 for [IndRmax, IndImax] = (2.4, 1.7), and 0.886 for [IndRmax, IndImax] = (2.1, 3.7) in a weighted case, which exceeds both the grand mean performance of the “PW-LR skew r” method by Hyvärinen (0.845) and the maximal grand mean performance of any single cumulant in our study (Figure 4, the maximum of 0.847). Selection of Other Approaches for Effective Connectivity Research in fMRI In order to benchmark our classifier, we compare the performance against other methods. As mentioned in the Introduction, the field of effective connectivity in fMRI is very wide (Smith et al., 2011); therefore, we restricted ourselves to the most popular approaches (other than DCM itself; Friston et al., 2003): 1. State-space implementation of Granger causality (GC; Granger, 1969; Seth, Barrett, & Barnett, 2015) is a multivariate method inferring effective connectivity between a pair of time series under the assumption that both of them can be expressed as autoregressive processes. We used a simple version of GC featuring ordinary least square regression with lag of 1, implemented in Multivariate Granger Causality Toolbox (Barnett & Seth, 2014), obtained from http://www.sussex.ac.uk/sackler/mvgc. For GC based on VAR pro- cess, as in our study, the state-space implementation is more robust than spectral GC (Geweke, 1982, 1984), because the frequency-domain version has a bias-variance trade- off (a function of the VAR model order that can induce spurious conditional GG causality estimates, such as erroneous peaks in the frequency domain, as indicated in the recent work by Stokes and Purdon, 2017). Furthermore, the state-space formulation of GC is the most robust, mitigating effects of bias and variance due to the fact that the reduced model is VAR rather than VARMA (Barnett & Seth, 2015). In order to compare performance with the methods for pairwise inference, we used GC in a bivariate rather than multivariate fashion: by applying GC to each of the previously found connections separately 2. Partial Directed Coherence (PDC; Baccalá & Sameshima, 2001) is known as a method conceptually close to Directed Transfer Function (Kami ´nski & Blinowska, 1991; Baccalá & Sameshima, 2001). Both these methods are derivatives from Geweke spectral measures of GC (Geweke, 1982, 1984), and all three methods have similar limitations (Chicharro, 2011). However, PDC is used substantially more often in fMRI studies than the other two methods, especially when compared with spectral GC. For this reason, we chose PDC as a method representing this class of approaches. We used PDC implementation from the Extended Multivariate Autoregressive Modelling Toolbox (Faes, Erla, Porta, & Nollo, 2013; http://www.lucafaes.net/emvar.html). As in case of GC, we applied PDC in a bivariate fashion 3. Patel’s tau (PT; Patel et al., 2006), as described in the Introduction, is implemented sim- ilarly as in Smith et al. (2011) by recalculating each time series into the range [0, 1], setting samples under the 10th percentile to 0, over the 90th percentile to 1, and linearly Network Neuroscience 1020 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d / . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD mapping the remaining samples to the range [0, 1]. Then, we infer the directionality of connection from the difference between P (X|Y) and P (Y|X). In addition to the previous implementation, however, we also integrate the results over all the possible thresholds in order to eliminate the thresholding problem while calculating the conditional proba- bilities P (X|Y), P (Y|X). 4. Pairwise likelihood ratios methods (PW-LR; Hyvärinen & Smith, 2013) are represented by “PW-LR r skew,” as it gives the highest performance among all the PW-LR methods when applied to synthetic fMRI data. We obtained the codes for the PW-LR methods from https://www.cs.helsinki.fi/u/ahyvarin/code/pwcausal/ (Hyvärinen used the cumulant k; l = (2; 1) weighted with covariance for synthetic benchmark datasets). Therefore, for this comparison, we use the classifier based on fractional cumulants weighted with covari- ance. For our study, we chose a PW-LR r skew version of the method, which involves an inference based on a third cumulant with a discount for outliers (Hyvärinen & Smith, 2013) As in Hyvärinen & Smith (2013), we performed the first step of the inference as inverse co- variance thresholded with permutation testing. All the methods, including multivariate meth- ods such as GC and PDC, were then applied in a pairwise fashion (i.e., separately for each two-node network representing a single connection found in the previous step). Furthermore, we did not include the DCM procedure in this comparison, for the same reasons as Smith et al. (2011): DCM is not an exploratory method and using it in this context, namely for exploratory causal research on the set of benchmark synthetic datasets (where the smallest network consists of five nodes) is not computationally feasible. Furthermore, the characteristics of this synthetic benchmark dataset is that input signals (Figure 1A) represent random events and can therefore emulate all types of fMRI experiments: classic task-fMRI studies, event-related responses or resting-state BOLD time series. In DCM, however, the inputs must be strictly specified in the block design, otherwise DCM inference cannot be initiated. Therefore, assumptions of DCM do not fit a research problem formulated in this particular way. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d . t / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Testing Robustness of the Methods Against Confounds In addition to evaluating our approach against the existing simulations from Smith et al. (2011), we further evaluate the performance under additional yet typical modes of variation in the data. Specifically we are interested in characterizing the discriminative performance relative to (i) more complex forms of stochastic noise in the data and (ii) unequal levels of SNR per node. The benchmark synthetic datasets involve temporally uncorrelated, white background noise of a low magnitude on the neuronal level (Smith et al., 2011). This type of noise is not physio- logically plausible, as it is known from physiological studies that in the neuronal net- works, the background noise has a scale-free power spectrum (He, 2014; Bédard et al., 2006; Dehghani et al., 2010; Bielczyk et al., 2017). Therefore, we simulated a two-node system and introduced scale-free (pink) noise to the system. Then, we varied the variance of the noise in the range of [0.2, 5.0] while keeping the amplitude of the inputs si(t) fixed to 1.0. We per- formed 500 realizations of 10 min simulation at high temporal resolution of Fs = 200 Hz, for each configuration of the noise variances. Furthermore, in the original version of the DCM procedure (Friston et al., 2003), as well as in most computational studies (Smith et al., 2011), equal stimulus strengths to both nodes si(t) are assumed. This assumption might not hold true in the real fMRI datasets. Therefore, we performed another, noiseless simulation, in which we varied signal strengths between the Network Neuroscience 1021 Improving pairwise inference in fMRI using fractional moments of BOLD upstream and downstream region. We performed 500 realizations of 10-min simulation at high temporal resolution of Fs = 200 Hz for each configuration of input strengths in the range of [0.2, 5.0]. RESULTS Supervised Learning Using Synthetic Benchmark Datasets The best version of the classifier was obtained for voting between cumulants in the range [IndRmax; IndImax] = (2.1; 3.7), and with the discount for high moment indexes (Equation 3). The comparison of this classifier against four other methods, GC, PDC, PT, and PW-LR r skew, on the benchmark simulation no. 2 is presented in Figure 6. The violin plots denote the distri- bution of the z-scores for connections as compared to the null distribution. Blue dots denote the percentage of correct assignments for the true connections, as in Smith et al. (2011). In most of the other 27 benchmark datasets, our classifier outperforms all the other methods (Support- ing Information 5. As in the original study by Smith et al. (2011), lagged methods, GC, and PDC perform worse than the structural methods. PW-LR r skew and fractional cumulants both outperform PT, most probably because PT is based on the thresholded signal and therefore contains a free parameter. In general, in the benchmark synthetic datasets, fractional cumulants outperform all other techniques in almost all cases, although the difference between performance of the fractional cumulants and PW-LR methods is small. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d . t / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 6. Comparison between the classifier based on the fractional cumulants and several other methods on benchmark simulation no. 2. The violin plots denote the distribution of the z-scores for connections as compared with the null distribution. Blue dots denote the percentage of correct assignments for the true connections (Smith et al., 2011). The difference in performance between the classifier based on fractional cumulants and PW-LR r skew (Hyvärinen & Smith, 2013) is small. Network Neuroscience 1022 Improving pairwise inference in fMRI using fractional moments of BOLD Robustness of the Methods with Respect to Confounds Figure 7 presents the comparison between the fractional cumulant classifier and various other methods on a two-node simulation under the addition of varying levels of physiologically plausible, scale-free neuronal noise across various levels of SNR for the upstream and down- stream regions. The results suggest that all previously tested methods show low levels of ro- bustness toward such additional sources of variability in the data. GC as well as PDC are at the lowest performance, and give results on a chance level across the whole parameter space. In PT, the success rate in proper classification between upstream and downstream node rapidly drops toward 50% along with a decrease in SNR in the upstream node. However, in case SNR in the upstream node is higher than 1 (left half of the Patel’s tau panel, Figure 7), PT is resilient to the variance of SNR in the downstream node. The performance of PW-LR r skew drops down to the chance level with respect to both the noise level in the upstream and downstream region, whereas GC and PDC are performing almost equally poorly under any combination of noise variances (probably because the variance of the noise from the fitted autoregressive model is used to establish the directionality of the causal influence in GC). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t / . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 7. Robustness of the methods against the background scale-free neuronal noise. The variance of the background noise s(t) differs between upstream and downstream region in the range of [0:2; 5:0]. As signal magnitude is constant and equal to 1 in these simulations, the signal-to-noise ratio (SNR) was calculated as 1/var(σ). Network Neuroscience 1023 Improving pairwise inference in fMRI using fractional moments of BOLD l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t . / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Figure 8. Robustness of different methods to the change in signal strengths. The variance of the signal differs between upstream and down- stream region, both in the range of [0:2, 5:0]. GC and PDC give performance around the chance level across the whole parameter space, whereas PW-LR r skew and PT exhibit certain resilience toward this variability in the inputs. However, the classifier based on fractional cumulants is the only method whose performance does not fall toward the chance level within the parameter space. Figure 8 presents the comparison between the classifier based on fractional cumulants and other methods, given noiseless simulation and varying signal magnitudes. The classifier based on fractional cumulants is the only method where the performance does not decrease down to a chance level within the chosen parameter space. GC and PDC give performance around the chance level across the whole parameter space, whereas PW-LR r skew and PT exhibit certain resilience toward this variability in the inputs. However, the performance breaks down to the chance level in cases when signal magnitudes between the upstream and downstream node becomes highly disproportionate (higher than 3.0). DISCUSSION Summary This work provides an advance to the effective connectivity research in fMRI, by utilizing the additional information contained in the BOLD time series with use of fractional moments of the BOLD distribution combined into cumulants. Usage of this additional information (embedded within a classifier) significantly increases the robustness toward plausible sources of variability Network Neuroscience 1024 Improving pairwise inference in fMRI using fractional moments of BOLD in fMRI, namely presence of physiologically realistic (scale-free) background noise as well as disproportion in the inputs strength, either due to differences in the amount of neuronal activity locally induced and/or due to effective differences induced by, for example, regional variations in the coil sensitivity profiles. This is where the value of added information coming from fractional cumulants becomes apparent: among the methods tested in this work, only the classifier based on the fractional cumulants gives a performance better than chance across the whole parameter space in these experimentally realistic conditions. Effective connectivity is a research problem directly related to the notion of causality. Causal- ity is, in general, difficult both to define (Pearl, 2000) and to measure. In the most basic for- mulation, in X causes Y, it means that without X, Y would not occur. The picture is far less clear in complex dynamic systems such as brain networks: for any event, a high number of potential causes can be defined, and these causes most often interfere with each other. This research problem was recently discussed by Albantakis, Marshall, Hoel, and Tononi (2017) who decomposed causality into independent dimensions: realization, composition, informa- tion, integration, and exclusion. Also, interpretation of a causal component in a given process depends on the context. For example, respiratory movement is typically considered a con- found in fMRI experiments, unless we are interested in the influence of respiration speed on the activity of neuronal populations. Furthermore, in brain networks, temporal ordering of the cause of cause and effect is hard to maintain, as information is circulating in recurrent rather than feed forward networks (Schurger & Uithol, 2015). Furthermore, If we have an interventional study at disposal, establishing causality becomes straightforward, but this is rarely the case in human brain research. In human fMRI, all the studies are observational rather than interventional. In such cases, causation can never be observed directly, just correlation (Hume, 1772). When a correlation is highly stable, we are inclined to infer a causal link. Additional information is then needed to assess the direction of the assumed causal link, as correlation indicates for association and not for causation (Altman & Krzywinski, 2015). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d . t / In the simulations, we used multiple realizations of the dynamics for every network pattern. We needed to run multiple instantiations of a noisy system in order to evaluate the mean success rate of the methods under noisy circumstances. We also simulated long runs of the dynamics aiming to find an upper bound on the methods’ performance in a function of the SNR disproportion and background noise levels. We did not investigate the effects of the sample length on the results. This is because in our study, we focused on the confounding factors which—unlike the duration of the study—cannot be influenced by the researcher. f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Although fractional moments of a distribution, as a mathematical concept, were studied before (Dremlin, 1994; Matsui& Pawlas, 2013), this concept was not applied to biomedical sciences to date. One reason for this lack of applications might be that the fractional moments become complex numbers for the normalized time series, and that subsequently, the features characterized by these moments cannot be conceptualized as easily as the features character- ized by the integer moments (e.g., skewness can be interpreted as a measure of “asymmetry” of the distribution, and kurtosis can be interpreted as its “flatness”). However, although the fractional moments of a distribution are a mathematical concept with limited practical in- terpretation, nevertheless they could still contain valuable, in our case causal, information. In this work, we demonstrate that fractional moments provide important additional informa- tion about the distribution of BOLD values. We first perform supervised learning of the clas- sifier on the set of benchmark synthetic datasets, and then validate the classifier on two-node Network Neuroscience 1025 Improving pairwise inference in fMRI using fractional moments of BOLD simulations with biologically realistic confounds. We believe that confounding factors such as a physiological background noise of a magnitude varying between the nodes is important to overcome for any method for causal inference in fMRI. This is because every network in the brain is embedded in larger networks, and therefore the background activity of other inter- connected networks can be interpreted as “noise” (Deco, Jirsa, & McIntosh, 2011). We demon- strate that our approach can increase the robustness of the methods for pairwise inference in fMRI to the main sources of variability in BOLD fMRI. therefore it Unlike the previous methods for pairwise inference in fMRI (Hyvärinen & Smith, 2013), the classifier defined in this study is informed by the dynamic causal modeling generative incorporates the priors derived from the neurophysiological studies model, (Buxton, Wong, & Frank, 1998). Deriving benchmark signum maps for the classifier from the multiple instantiations of the DCM generative model allows for marginalizing out all the param- eters unimportant for the effective connectivity research: the classification procedure focuses only on classifying a pair of regions into upstream and downstream instead of fitting all the hyperparameters as is done in the classic DCM inference procedure. Therefore, this approach is a reduction of the problem of effective connectivity in a large network to a two-node classi- fication problem on one hand, and an extension of the feature space from integer to fractional moments on the other hand. The signum maps derived in the training process are dependent on the generative model. In this study, we chose the canonical, original formulation of the DCM (Friston et al., 2003). There are also newer formulations of the DCM, for example, the canonical microcircuit ap- proach (Pinotsis et al., 2017; Friston et al., 2017), in which layer-specific neuronal populations in the cortex are modeled with use of a neural mass model. In this case, we assume that the inference is performed on mesoscale level, in which ROIs represent brain regions (cortex re- gions or subcortical nuclei) rather than cortex components. Furthermore, although versions of DCM containing higher order, nonlinear effects (Stephan et al., 2008) are also developed, we believe that a (bi)-linear model is a good simplification to describe the underlying neuronal dynamics, as it refers to the linear part of the sigmoidal transfer functions between neuronal populations in the brain (Silver, 2010; Bielczyk, Buitelaar, Glennon, & Tiesinga, 2015). Mod- eling communication between nodes in the network with use of linear transfer functions is a common practice in modeling effective connectivity in fMRI (see, e.g., structural equation models (Mclntosh & Gonzalez-Lima, 1994) or LiNGAM-ICA (Shimizu et al., 2006; Smith et al., 2011). The bilinear version of DCM, often referred to as the general linear model approach, is still a very popular tool for finding effective connectivity patterns from fMRI in clinical practice (see, e.g., recent work by Zhang et al., 2018; Nackaerts et al., 2018; Arioli et al., 2018; Pool et al., 2018). We reproduced the DCM generative model after Smith et al. (2011). This implementation is acknowledged in the field as a benchmark setting for testing new methods for functional and effective connectivity in fMRI (see, e.g., Hyvärinen & Smith, 2013; Hinne et al., 2015; Bielczyk et al., 2019). In this implementation, bilinear effects (namely, modulation of connections by experimental inputs) are not modeled. In pairwise inference this omission is justified, as mod- ulation of connection only affects the strength of connection weight A12, which will influence cumulant values quantitatively and not qualitatively, which does not influence the outcome signum maps. Evaluating methods with the use of synthetic datasets as the ground truth is typically the first step in the validation of any new data analytic framework. Validating new methods with use of Network Neuroscience 1026 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / / t e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d . / t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD synthetic datasets is a state of the art technique across the whole field of neuroimaging, from single cell imaging to whole-brain imaging with fMRI or EEG/MEG. This tradition has a long history, starting from the Nobel-winning Hodgkin and Huxley model for initiation and propa- gation of action potential (Hodgkin & Huxley, 1952). Today, methods for effective connectivity between neuronal assemblies measured with multielectrode arrays are still validated on syn- thetic datasets generated from this classical model, including recent approaches: nonlinear data assimilation (Hamilton, Berry, Peixoto, & Sauer, 2013) and differential covariance (Lin, Das, Krishnan, Bazhenov, & Sejnowski, 2017). In cognitive neuroimaging, testing methods on synthetic datasets from generative models is also a standard. In EEG/MEG research, there are multiple classes of generative models generating different type of dynamics, depending on the purpose of the modeling study, for example, the nonlinear lumped-parameter model for generating alpha rhythms and its neural mass extension by (David & Friston, 2003), the Wong-Wang model for winner-take-all dynamics (Wong & Wang, 2006), the Hindmarsh-Rose model for epileptor dynamics (Hindmarsh & Rose, 1984), and DCM for EEG/MEG (Kiebel, Garrido, Moran, Chen, & Friston, 2009; Steen, Almgren, Razi, Friston, & Marinazzo, 2018; Moran, Pinotsis, & Friston, 2013). Furthermore, the Human Neocortical Neurosolver simula- tor developed at Brown University (HNN, https://hnn.brown.edu) is a complex tool simulating local field potentials measured with EEG/MEG by bottom-up modeling of clusters of neurons. All these tools can serve to validate new methods for functional and effective connectivity in EEG/MEG (Valdes-Sosa et al., 2011; Wang et al., 2014). In fMRI, the selection of generative models is narrower than in EEG/MEG: DCM (Friston, Moran, & Seth, 2013; Smith et al., 2011) achieved a status of the standard generative model. With use of this synthetic data generated from this model, new methods for effective connectiv- ity in fMRI are validated, for example, the third- and fourth-cumulant method by Hyvärinen and Smith (2013) and artificial immune algorithm combined with the Bayes net method (AIAEC; Ji, Liu, Liang, & Zhang, 2016). To date, DCM is the most biologically relevant generative model proposed in the field of functional magnetic resonance imaging. The implementation of benchmark synthetic datasets based on DCM by Smith et al. (2011) has gained a lot of attention and following in the field, but it has also gained its critics. For instance, according to Smith’s results, PT (Patel, Bowman, & Rilling, 2006) is one of the methods giving best performance in retrieving directed con- nectivity patterns from synthetic benchmark datasets. In a recent work, Wang, David, Hu, and Deshpande (2017) performed a modeling study on datasets derived from an experiment by David et al. (2008) in which fMRI activity was measured in genetically modified rats suffering from epilepsy. Activity from the same set of regions was recorded in an associated intracere- bral EEG study in order to provide ground truth information flow. The authors chose primary somatosensory cortex barrel field (S1BF), thalamus, and striatum (caudate-putamen; CPu) as ROIs and demonstrated that Patel’s tau is no better than chance in recovering directional con- nectivity patterns from this data in both raw and deconvolved fMRI datasets. On the contrary, DCM and Granger causality proved to correctly estimate the directionality of the information flow on the group level and on the deconvolved data. There are more caveats to be noted with respect to the benchmark DCM simulations by Smith et al. (2011). First, the synthetic datasets derived by the authors of the study involve a low noise condition, in which the background noise in the networks in as low as 5% of the signal magnitude. Given that the background activity in the brain networks includes not only noise but also echo of cognitive processes unrelated to the experiment, 5% of back- ground activity seems to be on the lower end of the spectrum of possibilities. Second, networks Network Neuroscience 1027 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d / . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD investigated by Smith et al. are sparse (a number of connections in a network of size N is of order of N) and almost acyclic, which also seems to be a very optimistic scenario. Third, Smith et al. used a TR of 3 s and time series of 200 data points. This TR is too long, and the time series length too short for generalizing the empirical datasets used today. Today, shorter TRs (1 s or less, e.g., 0.72 s as in Human Connectome Project datasets; Van Essen et al., 2013), and longer time series have become the norm (e.g., 4,800 samples in resting-state datasets from Human Connectome Project datasets; Van Essen et al., 2013). Last, Smith et al. used a fixed delays of 50 ms in the first layer of the DCM model, representing the underlying neuronal commu- nication. This delay represents synaptic transmission delays and axonal transmission delays between nodes of the network. The constant value of delay is a crude estimation, especially given that pairs of brain regions positioned at different distances from each other should have different axonal transmission delays. Also, given polysynaptic connections, effective delays between neuronal populations might be much higher than the aforementioned 50 ms. For ex- ample, P300 potential appears after 300 ms (Polich, 2007), and some other cortical potentials have even slower latencies. This lack of attention toward modeling neuronal delays might fa- vor nonlagged methods (i.e., PT or LiNGAM) and might be preferred in this analysis over the lagged methods. Altogether, there are reasons to believe that benchmark datasets derived by Smith et al. are, to some extent, not representative of the real fMRI datasets. For these reasons, results of validation on the benchmark datasets should be interpreted with care. There are also other, competitive generative models in the field, for example, the model proposed by Seth et al. (2013). In this model, the authors used a simple VAR generative model in order to simulate neuronal dynamics in the testing network of five regions, based on work by Baccalá and Sameshima (2001). Subsequently, the VAR model output is convolved with five different HRF kernels generated with use of the difference-of-gamma approach as imple- mented in SPM8 (http://www.fil.ion.ucl.ac.uk/spm/software/spm8/). Another possibility, would be to use local field potentials (LFPs) instead of neuronal dynamics simulated as a system of differential equations with delay, and convolve LFPs with HRF (Deshpande, Sathian, & Hu, 2010). However, to date, Smith’s synthetic datasets derived from the DCM generative model remains the benchmark datasets. Furthermore, in this work, we performed the inference on the full BOLD response, without deconvolving the BOLD time series into the neuronal time series. It has been shown in synthetic and empirical data that incorporating a physiologically based model of spatiotemporal hemo- dynamic response function into the preprocessing pipeline leads to an improvement in the esti- mated neuronal activation (Aquino, Robinson, Schira, & Breakspear, 2014). It was also shown that it is generally difficult to accurately recover true task-evoked changes in BOLD fMRI time series irrespectively of the method chosen for modeling HRF response (Lindquist et al., 2009). Hence, there is a long-lasting debate in the field of connectomics on whether or not a (blind) hemodynamic deconvolution is necessary to perform the (effective) connectivity research in fMRI (Wu et al., 2013). For instance, structural equation models (Mclntosh & Gonzalez-Lima, 1994) are often applied without deconvolution to fMRI datasets (Schlösser et al., 2003; Zhuang, LaConte, Peltier, Zhang, & Hu, 2005). Our previous theoretical research in synthetic datasets generated from the DCM model suggests that deconvolution is not necessary in effective connectivity research in fMRI if the method used in the study is not lag dependent (Bielczyk et al., 2017). This is because under the assumption that the underlying signal on the neuronal level is in the low-frequency range, the hemodynamic response—as a low-pass filter—does not affect the signatures of different connectivity patterns present in the outcome BOLD re- sponse. Therefore, in this work, we did not perform the deconvolution step before assessing effective connectivity with any of the tested methods, including Granger causality. In fMRI Network Neuroscience 1028 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d / t . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD literature, GC is applied both with (David et al., 2008; Ryali, Supekar, Chen, & Menon, 2011; Ryali et al., 2016; Hutcheson et al., 2015; Wheelock et al., 2014; Sathian, Deshpande, & Stilla, 2013; Goodyear et al., 2016) and without (Zhao et al., 2016; Regner et al., 2016; Chen et al., 2017) use of deconvolution. Recent research suggests that all connectivity methods (includ- ing functional connectivity) will improve their estimation accuracy post-HRF deconvolution (Rangaprakash, Wu, Marinazzo, Hu, & Deshpande, 2018). However, in this work, we chose for implementation without deconvolution, to be consistent with (Smith et al., 2011). Here, we would like to mention that in recent years a lot of progress has been made in the area of modeling local hemodynamics from fMRI datasets. For example, Havlicek et al. (2011) proposed a new approach to modeling hemodynamic response functions based on cubature Kalman filtering. Furthermore, Bush et al. (2015) proposed and validated a meta-algorithm for performing semiblind deconvolution of the BOLD fMRI by using bootstrapping. This method allows for estimating the timing of the underlying neural events stimulating BOLD responses, together with confidence levels. Sreenivasan et al. (2015) proposed a nonparametric blind BOLD deconvolution method based on homomorphic filtering. Last, in our simulations we have set the connection strength to a fixed value of a = 0.9. On this stage, the output of the classifier (Equation 4 and Equation 5) is a binary response, that is, an indication for a connection, either X → Y or Y → X. This indication is based on a linear combination the binary signum maps Sr, Si (Figure 2C) with the values of the cumulants Cr, Ci computed for the given dataset X(t), Y(t), in either weighted or unweighted form. If the connection strength varies, the strength of the coupling between fractional moments in X(t) and Y(t) varies accordingly. Therefore, the absolute values of the associated fractional cumulants will also adjust. If cumulant values scale, then the RHS sum in Equations 4 and 5 will scale accordingly, but the signum of this sum should stay the same. Therefore, in a noiseless case, this classifier should give the same response regardless of the connection strength a. Limitations of the Method It is also important to remember that there are always two independent aspects to a method for causal inference. First, the method should have assumptions grounded in a biologically plau- sible framework relevant for the given research problem. For instance, a method for causal inference in fMRI should respect (1) the confounding, region-, and subject-specific BOLD dy- namics (Handwerker, Ollinger, & D’Esposito, 2004) and (2) co-occurrence of cause and effect (since the time resolution of the data is low compared with the underlying neuronal dynamics; the causes and their effects most likely happen within the same frame in the fMRI data). The new methods for pairwise inference such as classifying on the basis of fractional cumulants ad- dress this issue by (1) breaking the time order, and performing causal inference on the basis of statistical properties of the distribution of the BOLD samples, and not from the timing of events; and (2) using correlation in order to detect connections. A good counterexample here is GC, which has been proven useful in multiple disciplines. However, there is an ongoing discus- sion on whether or not GC is suited for causal interpretations of fMRI data. On the one hand, theoretical work by Seth, Chorley, and Barnett (2013) and Roebroeck, Formisano, & Goebel (2005) suggest that despite the slow hemodynamics, GC can still be informative about the directionality of causal links in the brain. Second, an estimation procedure needs to be com- putationally stable. Even if the generative model faithfully describes the data, it still depends on the estimation algorithm as to whether the method will return correct results. However, the face validity of the algorithms can only be tested in particular paradigms, in which the ground truth is known. If in the given paradigm the ground truth is unknown, which is most often the case in fMRI experiments, only reliability can be tested. Network Neuroscience 1029 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t . / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD Our approach requires certain assumptions. For instance, we assume that on the neuronal level, effects of directed connectivity are linear. This is also an assumption underlying the orig- inal DCM model used in this study. However, it is known that this is not always the case in the neuronal dynamics. For instance, shunting inhibition (Alger & Nicoll, 1979) is a phenomenon in which excitatory potential is reduced by division rather than by subtraction. However, effects such as shunting inhibition typically happen in microscale and should not affect large-scale neuronal dynamics as measured by BOLD fMRI. Therefore, we do not consider effects such as shunting inhibition as plausible confounds to our approach. One crucial limitation of our approach (as well as previous methods such as pairwise like- lihood ratios) is that these techniques only retrieve the net connectivity. Namely, what these methods effectively pick up on is the difference between connectivity strengths, and in a sce- nario where the connectivity strengths X → Y and Y → X are equal, the outcome cumulant map for the system will have lower amplitudes than in case of a unidirectional connection X → Y with the same connection strength. The significance of the cumulant values (whether or not the values are significantly different from zero) can be established with use of permuta- tion testing. For more details, see Supporting Information 6, Figure 12B. However, since in the first step of the inference we select only strong functional links for further classification, we can interpret ambiguous output from the classifier as a bidirectional connection. Since the quality of the classifier depends on the ability to determine the directionality of a unidirectional con- nection, we used only unidirectional connections in the validation stage. One point to make here is that bidirectional connections are hard to disambiguate for many methods for effec- tive connectivity, as they represent cyclic nets. This is an issue, for example, in applications of Bayesian nets (Pearl, 2000), in which joint probability for a certain graph is not tractable when propagation of information in the network is cyclic. Also, the third- and fourth-cumulant method by Hyvärinen and Smith (2013) falls into this category, as the asymmetry between third cumulants in the case of bidirectional connections will be equal to zero. One interesting direction for the method development would be also classification between excitatory and inhibitory connectivity. This aspect is missing in our study, as we focus on the connections of a positive sign only deriving a classifier able to distinguish between excitatory and inhibitory connections would require deriving an additional set of benchmark Sr and Si maps built on the basis of repetitive simulations of an inhibitory connection, and creating a new set of benchmark synthetic datasets, as the original datasets by Smith et al. (2011) involve excitatory links only. The reason why inhibition is not implemented yet, is because the cu- mulant patterns for inhibitory connection are different from patterns given in Figure 2 (we did not include the pictures in this manuscript though). The classifier in its current form gives a binary decision on whether the connection is going in the direction of X → Y or Y → X. The decision is based on whether the cumulant pattern obtained from the given dataset X(t), Y(t) is more similar to the signum maps derived from DCM for connection X → Y (Figure 2C) or to the inverse of these signum maps. In order to add inhibition to the picture, one would need to extend the number of possible classes by adding signum maps derived from DCM for inhibitory connection X → Y and introducing some metrics of distance to excitatory/inhibitory signum maps. This is the next step to take. One thing to note in addition is whether or not inhibitory effective connectivity is expected in large-scale networks investigated with fMRI; this is a mat- ter for debate. On one hand, it is known that long-distance projections in the brain are mostly excitatory as inhibition is typically local, presented by groups of tightly connected interneu- rons within single brain regions (Markram et al., 2004), which is also modeled in the DCM generative model with use of the self-inhibition term (Friston et al., 2003). On the other hand, several anatomical and physiological studies indicate that there are also long-range inhibitory Network Neuroscience 1030 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d / . t f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD Confounder: A factor latent in the experiment that can influence the causal inference. It can be a node that projects information to two other nodes in the network, causing a spurious causal association between them, or a background neuronal noise of magnitude varying across the brain. connections, for example, between hippocampus and entorhinal cortex in rats (Melzer et al., 2012). Furthermore, we consider local variability in brain physiology by varying hemodynamic responses between realizations of the simulation and by studying the effects of the scale-free background noise on the resulting effective connectivity measures. The DCM generative model summarizes the current state of knowledge about the mechanisms underlying generation of the BOLD response (Friston et al., 2017; Havlicek et al., 2015; Havlicek, Ivanov, Roebroeck, & Uluda˘g, 2017). Therefore, we do not have efficient ways of incorporating human brain phys- iology into our consideration in any more depth than this model allows for. However, at the same time, we do not consider the influence of artifacts from the data ac- quisition, such as the effects of movement in the scanner. We believe that the influence of such confounders should be limited by a proper data preprocessing. For instance, motion artifacts can be reduced with use of new, data-driven protocols for motion artifacts removal such as ICA-AROMA (Pruim et al., 2015) or a censoring-based artifact removal strategy based on vol- ume censoring (Power et al., 2014). Therefore, developing efficient strategies for controlling such confounders is beyond the scope of this paper. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . / t / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t / . f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Future Research In the context of fMRI research, increasing the granularity of moments in order to better char- acterize the distribution is an especially useful application because the experimental fMRI datasets are short (a few thousand samples at most); therefore, the estimation error for the high-order integer moments of the distribution becomes high. However, the subsequent cu- mulants contain information redundant to a certain extent, as they are correlated for any given time series x(t). We chose the granularity that gives smooth patterns of discriminability (Figure 3), which is Δ k = 0.1. Choosing the optimal moment resolution is a subject to future research; although, we believe that increasing index resolution to substantially less than 0.1 would not be beneficial, yet it would substantially increase the computational cost for the method. In this work, we validated our approach by using synthetic benchmark datasets derived from the DCM generative model. Using generative models is valuable in neuroimaging, in terms of validating new methods as mentioned in point 2, but also in applied research. Especially in the fields of applied computational psychiatry (Frässle et al., 2018) and network neuroscience in general (Betzel & Bassett, 2017), using generative models is valuable because these models enable inference on model parameters, which afford for some degree of mechanistic inter- pretability on the putative processes underlying the studied phenomena. Generative models, especially DCM, are acknowledged in multiple contexts in the field of cognitive neuroimaging, from method validation to application in clinical datasets. However, a valuable method should also give predictions testable in clinics (e.g., lead to more reliable estimation of directed causal influences during cognitive tasks, lead to better stratification of clinical populations in resting state, etc.), which we will also further investigate. In this work, we faithfully reproduced the pipeline after Smith et al. (2011). However, since the original version of DCM (Friston, Harrison, & Penny, 2003) based on the original Balloon- Windkessel model (Buxton, Wong, & Frank, 1998) was published, substantial advancements to the hemodynamic model have been proposed. First, Obata et al. (2004) reported an error in the expression for the outcome BOLD response (Equation 8, Supporting Information 1: DCM forward model). Stephan, Weiskopf, Drysdale, Robinson, & Friston (2007) proposed a more Network Neuroscience 1031 Improving pairwise inference in fMRI using fractional moments of BOLD accurate expression for one of the terms in this formula. Second, Heinzle, Koopmans, den Ouden, Raman, & Stephan (2016) proposed an updated formula for field strengths higher than 1.5 T. In the following work, we will take these improvements into account. Furthermore, the approach needs to further be tested and compared against other methods with use of experimental fMRI datasets, such as, for example, the Human Connectome Project data (Van Essen et al., 2013; Barch et al., 2013). Since little is known about causal connections in large-scale brain networks, especially in the resting state, such a validation might be quan- titative (i.e., by means of reliability) rather than qualitative. Alternatively, one could perform such a validation in particular pathways in which one can make assumptions about the direc- tionality of information flow, such as the dorsal and the ventral stream in the visual cortex. One possibility is testing using datasets coming from an interventional study in which neural activity was evoked and, therefore, the ground truth is known: a fused fMRI-optogenetic ex- periment (Ryali et al., 2016), in which intervention with use of optogenetics guarantees causal link between investigated neuronal populations. CONCLUSIONS The field of effective connectivity research in fMRI is still growing. For instance, in recent years, multiple algorithms for the graph network search have been developed and applied to fMRI datasets, including independent multiple-sample greedy equivalence search (IMaGES; Ramsey et al., 2010), group iterative multiple model estimation (GIMME; Gates & Molenaar, 2012), or fast greedy equivalence search (FGES; Ramsey et al., 2016). It is material for debate whether or not bivariate or multivariate inference serves a better purpose for effective connectivity research in fMRI. In our work, we focused on the pairwise inference, and we achieved a significant im- provement over previous approaches for pairwise estimation of functional causal interactions. Most importantly, the robustness against known sources of variability (possible differences be- tween up- and downstream noise magnitude and possible presence of non-Gaussian scale-free noise) significantly increases due to the simultaneous incorporation of multiple aspects of the associated BOLD distributions. We believe that, as this approach based on fractional moments of a distribution increases resilience of the methods for pairwise connectivity to potential con- founds in the experimental data, it can become a generic method to increase the power of causal discovery studies, both in cognitive neuroimaging and beyond. SUPPORTING INFORMATION Supporting information for this article is available at https://doi.org/10.1162/netn_a_00099. ACKNOWLEDGMENTS We thank the whole SIN group at the Donders Institute for Brain, Cognition and Behavior for advice and engagement in the project. AUTHOR CONTRIBUTIONS Natalia Bielczyk: Conceptualization; Formal analysis; Methodology; Validation; Visualization; Writing – Original Draft; Writing – Review & Editing. Alberto Llera: Conceptualization; Formal analysis; Supervision; Writing – Review & Editing. Jeffrey Glennon: Funding acquisition. Jan Buitelaar: Supervision; Writing – Review & Editing. Christian Beckmann: Conceptualization; Formal analysis; Methodology; Supervision; Writing – Review & Editing; Funding acquisition. Network Neuroscience 1032 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . t / / e d u n e n a r t i c e - p d l f / / / / 3 4 1 0 0 9 1 8 6 6 8 0 3 n e n _ a _ 0 0 0 9 9 p d t . / f b y g u e s t t o n 0 7 S e p e m b e r 2 0 2 3 Improving pairwise inference in fMRI using fractional moments of BOLD FUNDING INFORMATION Jeffrey Glennon, FP7 Ideas: European Research Council, Award ID: 305697 (OPTIMISTIC). Jeffrey Glennon, European Community’s Seventh Framework Programme (FP7/2007-2013), Award ID: 278948 (TACTICS). Jeffrey Glennon, European Union’s Seventh Framework Pro- gramme, Award ID: 603016 (MATRICS). Christian Beckmann, Wellcome Trust UK Strategic Award, Award ID: 098369/Z/12/Z. Christian Beckmann, Netherlands Organization for Interna- tional Cooperation in Higher Education (NL) Netherlands Organisation for Scientific Research (NWO-Vidi, Award ID: 864-12-003. REFERENCES Albantakis, L., Marshall, W., Hoel, E., & Tononi, G. (2017). What caused what? 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