Estimation of brain connectivity through Artificial Neural Networks

Abstract

Among different methods available for estimating brain connectivity from electroencephalographic signals (EEG), those based on MVAR models have proved to be flexible and accurate. They rely on the solution of linear equations that can be pursued through artificial neural networks (ANNs) used as MVAR model. However, when few data samples are available, there is a lack of accuracy in estimating MVAR parameters due to the collinearity between regressors. Moreover, the assessment procedure is also affected by the lack of data points. The mathematical solution to these problems is represented by penalized regression methods based on l1 norm, that can reduce collinearity by means of variable selection process. However, the direct application of l1 norm during the training of an ANN does not result in an efficient learning. With the introduction of the stochastic gradient descent-L1 (SGD-L1) it is possible to apply l1 norm directly on the estimated weights in an efficient way. Even if ANNs has been used as MVAR model for brain connectivity estimation, the use of SGD-L1 algorithm has never been tested to this purpose when few data samples are available. In this work, we tested an approach based on ANNs and SGD-L1 on both surrogate and real EEG data. Our results show that ANNs can provide accurate brain connectivity estimation if trained with SGD-L1 algorithm even when few data samples are available.

Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks

“…In this context, our results document that the LASSO regression performs better in challenging conditions when the number of model parameters approaches the sample size ( ). In these conditions it has been pointed out how OLS is not suitable for the solution of a regression problem and that its solution could even not exist [ 40 , 65 ]. On the other hand, LASSO shows high robustness to the lack of data points, which results in limited values of bias and variance [ 66 ].…”

“…Consequently, in this case the solution to the DARE equation necessary to convert the SS model into the ISS form did not converge, thus impeding OLS-based estimation of the cTE. In such conditions it is necessary to move to the use of penalized regression techniques [ 34 , 38 , 40 ]. Here we document that the LASSO regression leads to trends of the cTE bias which are consistently very low for any value of K in the estimation of the null links ( Figure 5 a), and rise with K but without exhibiting abrupt increases even for in the estimation of the non-null links ( Figure 5 b).…”

“…This can be seen as a binary classification task where the existence (class 1) or absence (class 0) of each estimated connection is assessed (using surrogate data for OLS and looking for zero/non-zero estimated coefficients for LASSO) and compared with the underlying ground-truth structure. Performances were assessed through the computation of the false positive rate (FPR, measuring the fraction of null links for which a statistically significant cTE was detected), false negative rate (FNR, measuring the fraction of non-null links for which the cTE was detected as non-significant) and accuracy (ACC, measuring the fraction of false detections) parameters [ 40 , 60 ]. Each of these performance measures was obtained across the network links for each individual network, and its distribution across the 100 simulated network structures was then presented separately for OLS and LASSO.…”

“…Among them, the Least Absolute Shrinkage and Selection Operator (LASSO) uses a constraint based on the norm that if applied directly on the regression problem, yields to a sparse coefficients matrix which leads to a reduction of the mean square error in conditions of data paucity [ 38 ]. Penalized regression techniques implemented for GC analysis have been successfully applied in many different contexts, ranging from simulation studies [ 39 ] to the analysis of electroencephalographic signals [ 34 , 40 , 41 ], neuroimaging data [ 42 ] and Macroeconomic data [ 43 ]. In the present work, the LASSO regression is embedded in the VAR-SS framework for the computation of information dynamics, and is compared with the traditional OLS regression as regards its capability to estimate conditional information transfer and PID measures both in benchmark networks of simulated multivariate processes and in real networks of multiple physiological time series.…”

Estimation of Granger causality through Artificial Neural Networks: applications to physiological systems and chaotic electronic oscillators

“…However, regardless of the methodology used to approach the regression problem, the estimation may be problematic in the setting of many observed processes and short time series available ( Antonacci et al, 2019a ). The literature reports that the stability and the existence of the solution for a linear regression problem are ensured when the number of data points is an order of magnitude greater than the number of VAR coefficients to be estimated ( Schlögl & Supp, 2006 ; Lütkepohl, 2013 ).…”