^{1}

^{1}

^{2}

^{*}

^{1}

^{2}

Edited by: Pedro Antonio Valdes-Sosa, Clinical Hospital of Chengdu Brain Science Institute, China

Reviewed by: Roberto C. Sotero, University of Calgary, Canada; Philippe Ciuciu, Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA), France; Gopikrishna Deshpande, Auburn University, United States

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Functional MRI (fMRI) is a popular approach to investigate brain connections and activations when human subjects perform tasks. Because fMRI measures the indirect and convoluted signals of brain activities at a lower temporal resolution, complex differential equation modeling methods (e.g., Dynamic Causal Modeling) are usually employed to infer the neuronal processes and to fit the resulting fMRI signals. However, this modeling strategy is computationally expensive and remains to be mostly a confirmatory or hypothesis-driven approach. One major statistical challenge here is to infer, in a data-driven fashion, the underlying differential equation models from fMRI data. In this paper, we propose a causal dynamic network (CDN) method to estimate brain activations and connections simultaneously. Our method links the observed fMRI data with the latent neuronal states modeled by an ordinary differential equation (ODE) model. Using the basis function expansion approach in functional data analysis, we develop an optimization-based criterion that combines data-fitting errors and ODE fitting errors. We also develop and implement a block coordinate-descent algorithm to compute the ODE parameters efficiently. We illustrate the numerical advantages of our approach using data from realistic simulations and two task-related fMRI experiments. Compared with various effective connectivity methods, our method achieves higher estimation accuracy while improving the computational speed by from tens to thousands of times. Though our method is developed for task-related fMRI, we also demonstrate the potential applicability of our method (with a simple modification) to resting-state fMRI, by analyzing both simulated and real data from medium-sized networks.

In recent years, functional magnetic resonance imaging (fMRI) has become a major tool to investigate dynamic brain networks. Earlier analysis methods focus on inferring brain regions activated by external experimental stimuli, for example using a general linear model approach (Friston et al.,

Generally speaking, there are two types of connectivity modeling: functional connectivity and effective connectivity. Functional connectivity usually models the correlations or dependencies between multiple BOLD (blood-oxygen-level dependent) time series from multiple brain regions. It can be estimated using generic statistical methods, such as correlations, partial correlations (Marrelec et al.,

Most recently, some progress has been made to refine and extend the standard DCM (Friston et al.,

In the statistical literature, inferring ODE parameters from data has been studied separately for general settings, sometimes called dynamic data analysis (DDA) or functional data analysis of ODE models, mostly for the following observation model

where the observed data _{i}) equals the noise _{i}) plus the underlying latent signal _{i}) generated from a set of ODEs. _{i}) is usually sampled at equally spaced time points _{i}, _{i}). Another complication for applying these DDA methods is that fMRI data are measured on a different time scale than the neuronal states with moderate or small signal-to-noise ratios.

One widely used yet simple model for connecting the neuronal states and BOLD responses is the linear convolution model,

where _{i}) is a noise contaminated convolution of _{i}) and a hemodynamic response function (HRF). Similar convolution formulations were employed before to model the neuronal states (Ryali et al.,

In this paper, we introduce a new statistical modeling framework for estimating effective connectivity and activations simultaneously from fMRI data. We propose a Causal Dynamic Network (CDN) method using a functional/dynamic data analysis approach, which jointly models the neuronal states modeled by a DCM-type ODE model and the observed BOLD responses. Unlike DCM or its extensions (e.g., Marreiros et al.,

The remainder of the paper is organized as follows. We start with a brief review of DCM and discuss its computational issues. We then introduce our CDN model and estimation method in section 2.2. Section 3.1 illustrates the numerical performance of our model using extensive simulations, including simulations from both CDN and DCM. A real fMRI data analysis is presented in section 3.2.

Dynamic causal modeling (DCM) was introduced by Friston et al. (

where

In this approximation, the entry _{mn} in _{Amn)dd} denotes the strength of intrinsic causal connection from the _{j} = (_{Bmnj)dd} denotes the influence of the _{j}(_{Cmj)dJ} denotes the effects of _{j} on the

To estimate these parameters, DCM in the first stage employs the expectation maximization (EM) procedure to fit a candidate model, usually with specific zero and nonzero patterns in (^{d}^{2}), where

We propose the following CDN model

where

Equation (5) is the same as the neuronal state model in the standard DCM (Friston et al.,

The neuronal state model (5), originally developed for task-related fMRI, is sometimes referred to as

where ω(

Equation (6) is the relationship between neuronal states and fMRI BOLD signals where ^{η}^{η−1}exp(−υ

Together, Equations (5) and (6) can be interpreted as a latent space model in statistics. Our use of a two-equation model is similar to MDS (Ryali et al.,

FMRI machines usually sample _{1}, _{2}, …, _{i}, …, _{T}. When fitting our CDN model to data, we replace Equation (6) in our CDN model with the following observation equation, for

where ⋆ denotes the convolution operation by the integral above. This observation model is different from the popular model (1) in the statistical literature. This difference warrants a new estimation approach. For example, under the classic observation model (1), a general strategy in many existing methods is to approximate _{i}) before estimating the ODE parameters. However, this is clearly not applicable to our observation model or fMRI data. As a separate note, although functional data analysis models (see review Wang et al.,

To estimate

where θ = [_{2} norm. The first part of the loss function is the data fitting error and the second part is regarded as the fidelity to our dynamic system. The ℓ_{2} norm loss was also employed before in Karahanoğlu et al. (

Following a functional data analysis technique, we represent

With the basis representation, our loss function becomes

The number of parameters in this loss function is ^{2}+^{2}+_{1}_{J}^{2}, far exceeding the fMRI data size growing linearly with _{j}(_{j}(_{j}(_{j}(

Though the loss function

Algorithm 1 Estimation for CDN

Initialize |

Update |

Solve for the minimizer |

Return |

We derive the parameter updates in our algorithm. We introduce the following notations in order to illustrate our updating procedure. Φ = (ϕ_{1}, …, ϕ_{p}) is our selected basis for estimating neuronal activity. Let

Combing these into matrix form, we denote

Setting the gradient with respect to θ to zero yields

and thus the update for θ is given by

One can derive the update for Γ by taking gradient.

To select the tuning parameter λ for our model, we use a cross-validation (CV) procedure as follows. Given two time series _{1}_{2}_{1}, using λ on a grid. Based on

Based on the fitted CDN parameters (_{kq}, _{kq}, _{kq}) from session _{kq}, _{kq}, _{kq}) and obtain the population estimate. This mixed effects approach was used before for the group level (or second stage) analysis in task activation studies (Worsley et al.,

Because our model can be computed very efficiently (usually in seconds or a few minutes), we propose to use bootstrap (across subjects) method to assess the significance of each entry in (

In this section, we evaluate the numerical performance of our CDN method using two types of simulation models: DCM and our statistical model CDN. The former provides a more realistic fMRI data generating model while it is computationally more expensive. The latter is computationally inexpensive and will serve to validate the statistical properties of our algorithm. Whenever possible, we compare with three other methods: GCA, MDS, and DCM.

We use the Python Neuroimaging toolbox (Nitime) for the GCA method, and the generative DCM model from Smith et al. (

We consider three simulation scenarios regarding the ODE parameters in our CDN model:

S1:

S2:

S3:

The first one does not consider either the stimulus activations or the stimulus modified connections, and the second one only excludes the stimulus modified connections. Because GCA models neither of these two effects, we include these two scenarios to provide biased advantages to GCA. The last scenario includes both the stimulus activations and the stimulus modified connections, and is a more realistic model for many task-related fMRI experiments.

For these scenarios, we consider a medium-sized network with 10 nodes with boxcar stimuli and different signal-to-noise (SNR) levels (3, 1, 0.5). We use the piece-wise linear basis. The simulations are repeated 50 times for each setting to represent fMRI data from 50 subjects, assuming for simplicity that they have the same underlying structure of brain connectivity, i.e., θ. We set that repetition time (TR) to be 0.72 s to demonstrate an ideal setting for GCA, because longer TR usually leads to low identification accuracy for directionality (Smith et al.,

AUC (area under the ROC curve) is used to evaluate the performance of recovering nonzero entries parameters. The AUC is calculated based on comparing the estimates against the true zero/non-zero statuses for all the entries of

Comparison of AUCs by GCA and CDN for recovering nonzero intrinsic network connections under three simulation scenarios (S1–S3 described in section 3.1.1) and varying signal-to-noise ratios (SNRs). For better visualization, we subtract random uniform jitters (between 0 and 0.05) for those cases when all points equal to 1.

To assess the statistical estimation accuracy of CDN, we use AUC and a scaled Frobenius norm for the connectivity-related parameters

where

For the sake of space, we only report the average performance metrics for box-car stimuli with varying SNR (3,1) in

Estimation accuracy of CDN under the simulation scenarios described in section 3.1.1.

S1 | 10 | 0.5 | 1.00 | – | – | 0.52 | – | – |

S1 | 10 | 1 | 1.00 | – | – | 0.11 | – | – |

S2 | 10 | 0.5 | 1.00 | – | 0.98 | 0.38 | – | 0.56 |

S2 | 10 | 1 | 1.00 | – | 0.98 | 0.26 | – | 0.44 |

S3 | 1 | 0.5 | 0.95 | 0.71 | – | 0.55 | 0.92 | 0.56 |

S3 | 1 | 1 | 1.00 | 0.74 | – | 0.36 | 0.88 | 0.54 |

S3 | 1 | 1 (50% duration) | 1.00 | 0.70 | – | 0.25 | 1.07 | 0.55 |

To test the robustness of our proposed model, we also compared the estimation performance using simulated data from DCM. We used the code and a simulation setting from Smith et al. (

The network structure used to simulate DCM data in section 3.1.2. All nodes are influenced by the same stimulus.

A representative example of the neuronal state time series of five nodes

Applying DCM usually requires specifying a candidate model of stimulus activation and latent/induced connections. Based on the model, only selected entries in (

Comparison of the AUCs, Frobenius losses, computation times of CDN, DCM and MDS. All computations are conducted in a computer with 8 Intel CPU cores (at 2.6 GHz) and sufficient memory for both algorithms.

DCM | 0.56 | 1.42 | 24 h and 15 min |

MDS | 1.57 | 580 s |

As a simple extension, we use the modification described in section 2.2.1 to fit our model to a simulated resting-state fMRI dataset (simulation scenario 4) from Smith et al. (

Simulated (

In this section, we apply our method to analyze a block-design fMRI dataset from the Human Connectome Project (HCP). The task is a language processing task developed by Binder et al. (

We are interested in modeling a language network of regions that were implicated in a previous study (Turken and Dronkers,

Estimated intrinsic connections in the math-story task experiment. Blue solid lines denote the positive causal connections, and red dashed lines denote negative connections. All connections and positive stimulus projections drawn are statistically significant at level 0.01, FDR corrected.

Connections induced by the

The directionality recovered by CDN help better elucidate the roles of these ROIs. One key region highlighted by our results is MTG, because it is the converging point of various directional pathways in this network. MTG is regarded as a high-level region contributing to language comprehension. The key role of MTG is supported by various types of prior evidence. For example, MTG was a shared node of six different networks analyzed by a conjunction analysis (Koyama et al.,

Comparison of the task-dependent connections (

We apply our CDN approach to a publicly available event-related fMRI dataset, downloaded from OpenfMRI.org under the access number ds000030. In the experiment, healthy subjects perform the stop-go response inhibition task inside the fMRI scanner. This task consists of two types of trials: go and stop. Specifically, on a go trial, subjects were instructed to press a button quickly when a go stimulus was presented on a computer screen; on a stop trial, subjects were to withhold from pressing when a go stimulus is followed shortly by a stop signal. We preprocess data using a suggested preprocessing pipeline based on FSL, a standard fMRI analysis software. See Poldrack et al. (

We are interested in studying the regions and their interconnections under either the go or stop stimuli. We select six brain regions implicated in prior publications, which are also validated by the meta-analysis tool from neurosynth.org. These six regions include M1 (primary motor cortex, MNI coordinate: −41, −20, 62), pos-preSMA (posterior presupplementary motor area, MNI: −4, −8, 60), ant-preSMA (anterior presupplementary motor area, MNI: −4, 36, 56), SMA (supplementary motor area, MNI: −3, 6, 50), Thalamus (MNI: −12, −13, 7), and STN (subthalamic nucleus, MNI: 6, −18, −2). The exact brain MNI coordinates for these regions are taken from Aron and Poldrack (

Estimated intrinsic connections in the stop-go task experiment. Blue solid lines denote the positive causal connections, and red dashed lines denote negative connections. All connections and positive stimulus projections drawn are statistically significant at level 0.01, FDR corrected.

Connections induced by the

To check the model fit of CDN, we plot the representative BOLD signal and neuronal state time series from all the six regions in

Real BOLD signals (

In this section, we test the applicability of our method for recovering a medium-sized network using resting-state fMRI. The dataset is collected by the Human Connectome Project (Smith et al.,

Brain region names and MNI coordinates of the 36 ROIs used in section 3.2.3.

1 | Posterior cingulate/Precuneus | 0 −52 7 |

2 | Medial Prefrontal | −1 54 27 |

3 | Left lateral parietal | −46 −66 30 |

4 | Right lateral parietal | 49 −63 33 |

5 | Left inferior temporal | −61 −24 −9 |

6 | Right inferior temporal | 58 −24 −9 |

7 | Medial dorsal thalamus | 0 −12 9 |

8 | Left posterior cerebellum | −25 −81 −33 |

9 | Right posterior cerebellum | 25 −81 −33 |

10 | Left frontal eye field | −29 −9 54 |

11 | Right frontal eye field | 29 −9 54 |

12 | Left posterior IPS | −26 −66 48 |

13 | Right posterior IPS | 26 −66 48 |

14 | Left anterior IPS | −44 −39 45 |

15 | Right anterior IPS | 41 −39 45 |

16 | Left MT | −50 −66 −6 |

17 | Right MT | 53 −63 −6 |

18 | Dorsal medial PFC | 0 24 46 |

19 | Left anterior PFC | −44 45 0 |

20 | Right anterior PFC | 44 45 0 |

21 | Left superior parietal | −50 −51 45 |

22 | Right superior parietal | 50 −51 45 |

23 | Dorsal anterior cingulate | 0 21 36 |

24 | Left anterior PFC | −35 45 30 |

25 | Right anterior PFC | 32 45 30 |

26 | Left insula | −41 3 6 |

27 | Right insula | 41 3 6 |

28 | Left lateral parietal | −62 −45 30 |

29 | Right lateral parietal | 62 −45 30 |

30 | Left motor cortex | −39 −26 51 |

31 | Right motor cortex | 38 −26 48 |

32 | Supplementary motor area | 0 −21 48 |

33 | Left V1 | −7 −83 2 |

34 | Right V1 | 7 −83 2 |

35 | Left A1 | −62 −30 12 |

36 | Right A1 | 59 −27 15 |

To assess the validity of our method,

Averaged effective (

Because the values of correlation, CDN, lrDCM estimates are not directly comparable, as they are fitted from different models, we use the following binarization step to convert all the numerical estimates to network connections before the comparison. The rationale is that larger estimated values (in magnitude) in all the models are typically interpreted as stronger connections, and neuroscientists usually resort to these connectivity methods to investigate if connections between certain brain nodes exist or not. Our binarization step converts the largest 100 ×

Comparison of identified connections recovered by different methods:

We develop a novel causal dynamic method to study brain activations and causal connections simultaneously from fMRI. Driven by the data nature of fMRI, CDN uses a functional data analysis approach to fit ODEs of neuronal states from BOLD time series. Unlike DCM, our model is estimated by an optimization algorithm, and this allows data-driven estimation of the ODE parameters, without DCM's requirement for hypothesizing the connections and stimulus inputs. The high computational efficiency of our algorithm also reduces the computation time.

Based on the task fMRI simulation studies, we show that our CDN approach is robust and accurate for recovering the ODE parameters for modeling dynamic brain networks. Compared with other effective connectivity methods, including GCA, DCM, and MDS, CDN performs the best in terms of parameter estimation and network identification across a wide range of scenarios. In the low SNR settings, CDN has the least variability followed by MDS, suggesting the robustness of our method. In the high SNR settings, both CDN and MDS achieve close to perfect network identification, while CDN yields a smaller parameter estimation loss in a realistic DCM simulation. GCA is relatively sensitive to SNRs, and its performance usually has substantial variability. Regarding the computational speed, CDN requires only a small fraction of the computation times of DCM and MDS. This makes it a very competitive approach for inferring effective connectivity. Patel's tau was a top method for recovering directional connectivity in a previous large-scale simulation study (Smith et al.,

For the analysis of the story/math task fMRI data, our activation and connectivity results are consistent with prior evidence on the language comprehension process and network. Our effective connectivity finding complements the structural and functional connectivity findings (Turken and Dronkers,

CDN in this paper is presented as a purely data-driven method. Other data-driven methods for effective connectivity were proposed in the literature, including latent-space GCA (David et al.,

The number of parameters in CDN grows with the numbers of stimuli and nodes. This can be a potential issue to scale our model to large-scale networks with multiple stimuli, especially when the number of parameters exceeds the sample size. This relatively small sample size setting yields a so-called under-determined system, because there might exist multiple sets of parameters that yield the same fit of data. This is further complicated by the colinearity issue introduced by the fMRI task designs. One possible direction is to introduce informative Bayesian priors to help invert such large systems with colinearity. We leave to future research on extending our model for large-scale networks.

The bilinear approximation used in this initial paper is a simplified model for the biophysical mechanisms. It neglects several micro and macro neurophysicological processes (see a review Daunizeau et al.,

To extend our method to biophysically more realistic models, there remain several challenges for future research. More model parameters associated with these more sophisticated models will increase the computation burden. One should also be aware of the overfitting issue, especially for task-related fMRI where the recording sessions tend to be relatively short with limited repetitions of certain stimuli of interest. There is clearly a trade-off between model complexity and data evidence, in addition to the concern of computation time. Bayesian priors, for example based on anatomical evidence, could be helpful for constraining the model complexity and computation time. However, these priors should be chosen carefully to ensure reliability and robustness (Frässle et al.,

Another limitation of this first paper is its canonical model for HRF. Though our model with the canonical HRF yields relatively robust results for one dataset simulated with HRF variations (Smith et al.,

Our method here is mainly developed for task-related fMRI. It shows some promising results on analyzing resting-state fMRI with medium sized networks. However, our method, like others based on the deterministic DCM principal, can be computationally challenging for inverting large-scale networks, partly because our method needs to estimate the hidden neuronal states. For resting-state fMRI, the neuronal states recovered by CDN may serve as input data to other connectivity methods, in order to minimize the confounding effect of HRF variability (Handwerker et al.,

The stop/go task-related fMRI dataset analyzed for this study can be found on openfmri.org under accession number ds000030 (

XC and XL designed and carried out the research, drafted the manuscript, analyzed the data, and interpreted the results. XC implemented the new statistical procedure. XL and BS obtained funding, supervised the research, and provided critical revision of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The stop/go task dataset was provided by the Consortium for Neuropsychiatric Phenomics LA Study from University of California, Los Angeles. The language/math task and resting-state fMRI datasets were provided by the Human Connectome Project (HCP; Principal Investigators: Bruce Rosen, M.D., Ph.D., Arthur W. Toga, Ph.D., Van J. Weeden, MD). The HCP was funded by the National Institute of Dental and Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH), and the National Institute of Neurological Disorders and Stroke (NINDS). HCP is the result of efforts of researchers from the University of Southern California, Martinos Center for Biomedical Imaging at Massachusetts General Hospital (MGH), Washington University, and the University of Minnesota. The HCP data are disseminated by the Laboratory of Neuro Imaging at the University of Southern California. These studies that provided the data have been approved by their governing institutional review boards.