# Identifying Brain Network Structure for an fMRI Effective Connectivity Study Using the Least Absolute Shrinkage and Selection Operator (LASSO) Method

^{1}

^{2}

^{*}

## Abstract

**:**

**Background:**Studying causality relationships between different brain regions using the fMRI method has attracted great attention. To investigate causality relationships between different brain regions, we need to identify both the brain network structure and the influence magnitude. Most current methods concentrate on magnitude estimation, but not on identifying the connection or structure of the network. To address this problem, we proposed a nonlinear system identification method, in which a polynomial kernel was adopted to approximate the relation between the system inputs and outputs. However, this method has an overfitting problem for modelling the input–output relation if we apply the method to model the brain network directly.

**Methods:**To overcome this limitation, this study applied the least absolute shrinkage and selection operator (LASSO) model selection method to identify both brain region networks and the connection strength (system coefficients). From these coefficients, the causality influence is derived from the identified structure. The method was verified based on the human visual cortex with phase-encoded designs. The functional data were pre-processed with motion correction. The visual cortex brain regions were defined based on a retinotopic mapping method. An eight-connection visual system network was adopted to validate the method. The proposed method was able to identify both the connected visual networks and associated coefficients from the LASSO model selection.

**Results:**The result showed that this method can be applied to identify both network structures and associated causalities between different brain regions.

**Conclusions:**System identification with LASSO model selection algorithm is a powerful approach for fMRI effective connectivity study.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Subjects, Experimental Design, MRI Collection, and fMRI Image Pre-Processing

#### 2.2. Theory

#### 2.3. Visual System

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Time series from the eight-connection visual system in response to polar angle stimulus for effective connectivity study. The fMRI response was obtained from (

**A**) (V1), (

**B**) (V2), (

**C**) (V3), (

**D**) (Vp), (

**E**) (V3a), (

**F**) (V4v), (

**G**) (V3b), and (

**H**) (V5/MT). The dotted curve shows the fMRI response, while the red curve indicates the input.

**Figure A3.**LASSO selection results and Lambda values. Lambda 1SE = 0.1045 is the regularisation corresponding to the minimum cross-validation error plus one standard deviation. The vertical dotted line on the right indicates the minimum cross-validation error value (Lambda MinDeviance = 0.0497).

**Figure A4.**The final prediction result based on the identified structure and its parameters. The dotted curve shows the fMRI response, while the red curve indicates the predicted response for the LASSO selected model. The x-axis is the magnitude of the normalized fMRI response, while the y-axis is time in terms of TR.

## References

- Ogawa, S.; Lee, T.M.; Kay, A.R.; Tank, D.W. Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proc. Natl. Acad Sci. USA
**1990**, 87, 9868–9872. [Google Scholar] [CrossRef] - Rugh, W.J. Nonlinear System Theory—The Volterra-Wiener Approach, Web version prepared in 2002; Johns Hopkins University Press: Baltimore, MD, USA, 1981; Chapter 1; pp. 1–53. ISBN O-8018-2549-0. [Google Scholar]
- Boynton, G.M.; Engel, S.A.; Glover, G.H.; Heeger, D.J. Linear Systems Analysis of Functional Magnetic Resonance Imaging in Human V1. J. Neurosci.
**1996**, 16, 4207–4221. [Google Scholar] [CrossRef] - Schetzen, M. Nonlinear system modeling based on the Wiener theory. Proc. IEEE
**1981**, 69, 1557–1573. [Google Scholar] [CrossRef] - Li, X.; Marrelec, G.; Hess, R.F.; Benali, H. A nonlinear identification method to study effective connectivity in functional MRI. Med. Image Anal.
**2010**, 14, 30–38. [Google Scholar] [CrossRef] [PubMed] - Goebel, R.; Roebroeck, A.F.; Kim, D.-S.; Formisano, E. Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and Granger causality mapping. Magn. Reason. Imaging
**2003**, 21, 1251–1261. [Google Scholar] [CrossRef] [PubMed] - Harrison, L.; Penny, W.; Friston, K.; Harrison, L.; Penny, W.; Friston, K.; Harrison, L.; Penny, W.; Friston, K. Multivariate autoregressive modeling of fMRI time series. NeuroImage
**2003**, 19, 1477–1491. [Google Scholar] [CrossRef] [PubMed] - Cao, J.; Yang, L.; Sarrigiannis, P.G.; Blackburn, D.; Zhao, Y. Dementia classification using a graph neural network on imaging of effective brain connectivity. Comput. Biol. Med.
**2024**, 168, 107701. [Google Scholar] [CrossRef] - Bressler, S.L.; Kumar, A.; Singer, I. Brain Synchronization and Multivariate Autoregressive (MVAR) Modeling in Cognitive Neurodynamics. Front. Syst. Neurosci.
**2021**, 15, 638269. [Google Scholar] [CrossRef] - Nagle, A.; Gerrelts, J.P.; Krause, B.M.; Boes, A.D.; Bruss, J.E.; Nourski, K.V.; Banks, M.I.; Van Veen, B. High-dimensional multivariate autoregressive model estimation of human electrophysiological data using fMRI priors. NeuroImage
**2023**, 277, 120211. [Google Scholar] [CrossRef] [PubMed] - Li, Q.; Calhoun, V.D.; Pham, T.D.; Iraji, A. Exploring Nonlinear Dynamics in Brain Functionality through Phase Portraits and Fuzzy Recurrence Plots. bioRxiv
**2024**. [Google Scholar] [CrossRef] - Ponce-Alvarez, A.; Deco, G. The Hopf whole-brain model and its linear approximation. Sci. Rep.
**2024**, 14, 2615. [Google Scholar] [CrossRef] [PubMed] - Abbaspourazad, H.; Erturk, E.; Pesaran, B.; Shanechi, M.M. Dynamical flexible inference of nonlinear latent factors and structures in neural population activity. Nat. Biomed. Eng.
**2024**, 8, 85–108. [Google Scholar] [CrossRef] [PubMed] - Åström, K.J.; Eykhoff, P. System identification—A survey. Automatica
**1971**, 7, 123–162. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R.; Friedman, J.H.; Friedman, J.H. The Elements of Statistical Learning: Data Mining, Inference, and Prediction; Springer: New York, NY, USA, 2009; Volume 2. [Google Scholar]
- Zou, H.; Hastie, T. Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B (Stat. Methodol.)
**2005**, 67, 301–320. [Google Scholar] [CrossRef] - Valdés-Sosa, P.A.; Sánchez-Bornot, J.M.; Lage-Castellanos, A.; Vega-Hernández, M.; Bosch-Bayard, J.; Melie-García, L.; Canales-Rodríguez, E. Estimating brain functional connectivity with sparse multivariate autoregression. Philos. Trans. R. Soc. B Biol. Sci.
**2005**, 360, 969–981. [Google Scholar] [CrossRef] [PubMed] - Liu, A.; Chen, X.; McKeown, M.J.; Wang, Z.J. A Sticky Weighted Regression Model for Time-Varying Resting-State Brain Connectivity Estimation. IEEE Trans. Biomed. Eng.
**2015**, 62, 501–510. [Google Scholar] [CrossRef] [PubMed] - Shojaie, A.; Michailidis, G. Discovering graphical Granger causality using the truncating lasso penalty. Bioinformatics
**2010**, 26, i517–i523. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Coyle, D.; Maguire, L.; McGinnity, T.M.; Benali, H. A Model Selection Method for Nonlinear System Identification Based fMRI Effective Connectivity Analysis. IEEE Trans. Med. Imaging
**2011**, 30, 1365–1380. [Google Scholar] [CrossRef] [PubMed] - Pongrattanakul, A.; Lertkultanon, P.; Songsiri, J. Sparse system identification for discovering brain connectivity from fMRI time series. In The SICE Annual Conference; IEEE: Piscataway, NJ, USA, 2013. [Google Scholar]
- Haufe, S.; Müller, K.R.; Nolte, G.; Krämer, N. Sparse causal discovery in multivariate time series. In Proceedings of the 2008th International Conference on Causality: Objectives and Assessment, Vancouver, BC, Canada, 12 December 2008; JMLR.org: Whistler, BC, Canada, 2008; Volume 6, pp. 97–106. [Google Scholar]
- Ryali, S.; Chen, T.; Supekar, K.; Menon, V. Estimation of functional connectivity in fMRI data using stability selection-based sparse partial correlation with elastic net penalty. NeuroImage
**2012**, 59, 3852–3861. [Google Scholar] [CrossRef] - Shojaie, A.; Fox, E.B. Granger Causality: A Review and Recent Advances. Annu. Rev. Stat. Its Appl.
**2022**, 9, 289–319. [Google Scholar] [CrossRef] [PubMed] - Teipel, S.J.; Grothe, M.J.; Metzger, C.D.; Grimmer, T.; Sorg, C.; Ewers, M.; Franzmeier, N.; Meisenzahl, E.; Klöppel, S.; Borchardt, V.; et al. Robust Detection of Impaired Resting State Functional Connectivity Networks in Alzheimer’s Disease Using Elastic Net Regularized Regression. Front. Aging Neurosci.
**2017**, 8, 318. [Google Scholar] [CrossRef] [PubMed] - Siggiridou, E.; Kugiumtzis, D. Granger Causality in Multivariate Time Series Using a Time-Ordered Restricted Vector Autoregressive Model. IEEE Trans. Signal Process.
**2016**, 64, 1759–1773. [Google Scholar] [CrossRef] - Tang, W.; Bressler, S.L.; Sylvester, C.M.; Shulman, G.L.; Corbetta, M. Measuring Granger Causality between Cortical Regions from Voxelwise fMRI BOLD Signals with LASSO. PLoS Comput. Biol.
**2012**, 8, e1002513. [Google Scholar] [CrossRef] [PubMed] - Das, P.; Babadi, B. Non-Asymptotic Guarantees for Reliable Identification of Granger Causality via the LASSO. IEEE Trans. Inf. Theory
**2023**, 69, 7439–7460. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Dumoulin, S.O.; Mansouri, B.; Hess, R.F. Cortical Deficits in Human Amblyopia: Their Regional Distribution and Their Relationship to the Contrast Detection Deficit. Investig. Ophthalmol. Vis. Sci.
**2007**, 48, 1575–1591. [Google Scholar] [CrossRef] [PubMed] - Nozari, E.; Bertolero, M.A.; Stiso, J.; Caciagli, L.; Cornblath, E.J.; He, X.; Mahadevan, A.S.; Pappas, G.J.; Bassett, D.S. Macroscopic resting-state brain dynamics are best described by linear models. Nat. Biomed. Eng.
**2023**, 8, 68–84. [Google Scholar] [CrossRef] - Pattanaik, R.K.; Mohanty, M.N. Nonlinear System Identification Using Robust Fusion Kernel-Based Radial basis function Neural Network. In Proceedings of the 2022 International Conference on Emerging Smart Computing and Informatics (ESCI), Pune, India, 9–11 March 2022. [Google Scholar]
- Gedon, D.; Wahlström, N.; Schön, T.B.; Ljung, L. Deep state space models for nonlinear system identification. IFAC-PapersOnLine
**2021**, 54, 481–486. [Google Scholar] [CrossRef] - Hutchison, R.M.; Womelsdorf, T.; Allen, E.A.; Bandettini, P.A.; Calhoun, V.D.; Corbetta, M.; Della Penna, S.; Duyn, J.H.; Glover, G.H.; Gonzalez-Castillo, J.; et al. Dynamic functional connectivity: Promise, issues, and interpretations. NeuroImage
**2013**, 80, 360–378. [Google Scholar] [CrossRef]

**Figure 1.**(

**A**) An example of a three-connection network structure. (

**B**) An example of an eight-connection network structure. Line arrow indicates the direction of information flow (causality).

**Figure 2.**The fMRI time series from the eight-connection visual system in response to an eccentricity stimulus for the effective connectivity study. (

**A**) (V1), (

**B**) (V2), (

**C**) (V3), (

**D**) (Vp), (

**E**) (V3a), (

**F**) (V4v), (

**G**) (V3b), and (

**H**) (V5/MT) were included in the brain network. The dotted curve shows the fMRI response, while the red curve indicates the input from the experimental design. The x-axis is the normalised fMRI signal, while the y-axis is time (in terms of TR = 3 s).

**Figure 4.**LASSO selection results. The blue circle and (

**left**) dotted line locate the point with the minimum cross-validation error plus one standard deviation (Lambda 1SE = 0.1643). The green circle and (

**right**) dotted line locate the Lambda with minimum cross-validation error (LambdaMinDeviance = 0.0648).

**Figure 5.**A visual cortex (V1 region) response to the stimuli and final prediction results from the identified model structure and its parameters. The red curve shows the prediction result from the V1 region.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, X.; Zhang, Y.
Identifying Brain Network Structure for an fMRI Effective Connectivity Study Using the Least Absolute Shrinkage and Selection Operator (LASSO) Method. *Tomography* **2024**, *10*, 1564-1576.
https://doi.org/10.3390/tomography10100115

**AMA Style**

Li X, Zhang Y.
Identifying Brain Network Structure for an fMRI Effective Connectivity Study Using the Least Absolute Shrinkage and Selection Operator (LASSO) Method. *Tomography*. 2024; 10(10):1564-1576.
https://doi.org/10.3390/tomography10100115

**Chicago/Turabian Style**

Li, Xingfeng, and Yuan Zhang.
2024. "Identifying Brain Network Structure for an fMRI Effective Connectivity Study Using the Least Absolute Shrinkage and Selection Operator (LASSO) Method" *Tomography* 10, no. 10: 1564-1576.
https://doi.org/10.3390/tomography10100115