Construction of functional brain connectivity networks from fMRI data with driving and modulatory inputs: an extended conditional Granger causality approach

Evangelos Almpanis; Constantinos Siettos; Evangelos Almpanis; Constantinos Siettos

doi:10.3934/Neuroscience.2020005

AIMS Neuroscience

2020, Volume 7, Issue 2: 66-88. doi: 10.3934/Neuroscience.2020005

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Research article Topical Sections

Construction of functional brain connectivity networks from fMRI data with driving and modulatory inputs: an extended conditional Granger causality approach

Evangelos Almpanis ^1,2,
Constantinos Siettos ^{3
,
,}

1.
Section of Condensed Matter Physics, National and Kapodistrian University of Athens, Greece
2.
Institute of Nanoscience and Nanotechnology, NCSR “Demokritos,” Athens, Greece
3.
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Italy

Received: 06 January 2020 Accepted: 25 March 2020 Published: 10 April 2020

We propose a numerical-based approach extending the conditional MVAR Granger causality (MVGC) analysis for the construction of directed connectivity networks in the presence of both exogenous/stimuli and modulatory inputs. The performance of the proposed scheme is validated using both synthetic stochastic data considering also the influence of haemodynamics latencies and a benchmark fMRI dataset related to the role of attention in the perception of visual motion. The particular fMRI dataset has been used in many studies to evaluate alternative model hypotheses using the Dynamic Causal Modelling (DCM) approach. Based on the use of the Bayes factor, we show that the obtained GC connectivity network compares well to a reference model that has been selected through DCM analysis among other candidate models. Thus, our findings suggest that the proposed scheme can be successfully used as a stand-alone or complementary to DCM approach to find directed causal connectivity patterns in task-related fMRI studies.

Keywords:

Citation: Evangelos Almpanis, Constantinos Siettos. Construction of functional brain connectivity networks from fMRI data with driving and modulatory inputs: an extended conditional Granger causality approach[J]. AIMS Neuroscience, 2020, 7(2): 66-88. doi: 10.3934/Neuroscience.2020005

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Abstract

1. Introduction

The stability of Hopfield neural networks is a necessary prerequisite in the practical applications of signal processing ^[1,2], pattern recognition ^[3], associative memory ^[4], nonlinear programming ^[5] and optimization ^[6]. Accordingly, the stability of various delayed Hopfield neural networks has been received enough attention ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]}. In particular, the neutral delays have been introduced into the model of neural networks and the stability of neutral-type neural networks has become a hot topic ^{[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]}. The mathematical model of neutral-type neural networks extends its application domain to a wider class of practical engineering problems, (for the detailed applications of such neural networks, the readers may refer to the references ^[34,35,36].)

The model of Hopfield neural networks of neutral type considered in this paper is expressed as

$\begin{eqnarray} \dot{x}_{i}(t) & = & - c_{i} x_{i}(t) +\sum\limits^{n}_{j = 1}a_{ij} f_j(x_{j}(t)) +\sum\limits^{n}_{j = 1}b_{ij} g_j(x_{j}(t-\tau_{ij}(t))) +\sum\limits^{n}_{j = 1}e_{ij}\dot{x}_{j}(t-\xi_{ij}(t)) +u_i , t\geq 0; \end{eqnarray}$

(1.1)

$x_i(t) = \varphi_i(t), \dot{x}_i(t) = \phi_i(t), t\in [- \max \{\tau , \xi \}, 0], i = 1, \cdots, n,$

where $\xi, \tau$ and $c_i$ are positive numbers, $\varphi_i(t)$ and $\phi_i(t)$ are continuous functions, $\xi_{ij}(t)$ and $\tau_{ij}(t)$ are delay functions, $f_j(\cdot)$ and $g_j(\cdot)$ are nonlinear continuous activation functions. These functions satisfy that for every $x, y\in R,$ $t\geq 0$ and $i, j = 1, \cdots, n,$

$\begin{eqnarray} 0\leq \xi_{ij}(t)\leq \xi, 0\leq \tau_{ij}(t)\leq \tau, \dot{\xi}_{ij}(t)\leq \overline{\xi}, \dot{ \tau}_{ij}(t)\leq \overline{\tau} ; \end{eqnarray}$

(1.2)

$\begin{eqnarray} | f_i(x)-f_i(y)| \leq l_i|x-y|, | g_i(x)-g_i(y)| \leq m_i|x-y|, \end{eqnarray}$

(1.3)

where $\overline{\xi}, \overline{\tau}, l_i$ and $m_i$ are some positive numbers. From ^[37,38], we know that $x(t) = (x_1(t), \cdots, x_n(t))^T$ of (1.1) is continuously differentiable.

The mathematical expression of system (1.1) includes some models studied in existing references. For example, when $\tau_{ij}(t) = \tau_{j}(t)$ and $\xi_{ij}(t) = \xi_{j}(t)$ , system (1.1) transforms into the following vector-matrix form studied ^[18,20]:

$\begin{eqnarray} \dot{x}(t) & = & - C x (t) +A f (x(t)) +B g(x(t-\tau(t)))+E\dot{x} (t-\xi(t)) +u , t\geq 0, \end{eqnarray}$

(1.4)

where $A = (a_{ij})_{n\times n}, B = (b_{ij})_{n\times n}, C = diag(c_1, \cdots, c_n), u = (u_1, \cdots, u_n)^T,$

$x(t) = (x_1(t), \cdots, x_n(t))^T, \dot{x}(t-\xi(t)) = (\dot{x}_1(t-\xi_1(t)), \cdots, \dot{x}_n(t-\xi_n(t)))^T,$

$f (x(t)) = (f_1(x_1(t)), \cdots, f_n(x_n(t)))^T , g (x(t-\tau(t))) = (g_1(x_1(t-\tau_1(t))), \cdots, g_n(x_n(t-\tau_n(t))))^T.$

When $\tau_{ij}(t) = \tau_{ij}, f_j = g_j$ and $\xi_{ij}(t) = \xi_{ij}$ (or $\xi_{j}$ ), system (1.1) transforms into the following system studied in ^[30]:

$\begin{eqnarray} \dot{x}_{i}(t) & = & -c_{i} x_{i}(t) +\sum\limits^{n}_{j = 1}a_{ij}f_j(x_{j}(t)) +\sum\limits^{n}_{j = 1}b_{ij} f_j(x_{j}(t-\tau_{ij} )) +\sum\limits^{n}_{j = 1}e_{ij}\dot{x}_{j}(t-\xi_{ ij} )+u_i, \end{eqnarray}$

(1.5)

or the following system studied in ^[29]:

(1.6)

Different from the systems studied in ^{[18,19,20,21,22,23,24,25,26,27,28]}, system (1.1) cannot be expressed in the vector-matrix form due to the existence of multiple delays $\tau_{ij}(t)$ and $\xi_{ij}(t)$ . Therefore, it is impossible to obtain the stability conditions of the linear matrix inequality form for the system with multiple delays ^[29,31]. In this case, it is necessary to develop new mathematical techniques and find more suitable Lyapunov-Krasovskii functional for the stability analysis of system (1.1).

On the other hand, the results of ^{[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]} have only provided the sufficient conditions of global asymptotic stability and have not considered the exponential stability. Actually, in the stability analysis of neural networks, exponential stability is a better property than asymptotic stability since it can converge to the equilibrium point faster and provides information about the decay rate of the networks. Moreover, the exponential stability property can ensure that whatever transformation occurs, the network stability of fast storage activity mode remains unchanged by self-organization ^[39]. Therefore, the exponential stability analysis of neural networks with multiple time-varying delays is worth investigating.

In addition, based on our careful review of recently almost all the existing stability results for system (1.1), we have realized that the research on the exponential stability of system (1.1) has not received enough attention. These facts have been the main motivations of the current paper to focus on the exponential stability of system (1.1).

The primary contributions and innovations of this work are summarized as follows: (1) novel sufficient conditions of exponential stability are established for Hopfield neural networks of neutral type with multiple time-varying delays; (2) a modified and suitable Lyapunov-Krasovskii functional is provided to study the exponential stability; (3) novel sufficient conditions of global asymptotical stability are provided for the systems studied in the existing references; (4) compared with the existing results, the established conditions are less conservative.

2. Exponential stability

From ^[18,20], we know that system (1.1) has at least one equilibrium point $x^* = (x^*_1, \cdots, x^*_n)^T$ under the conditions (1.2) and (1.3). By employing the formula $y_i(t) = x_i(t)-x^*_i (i = 1, \cdots, n)$ , system (1.1) can be transformed into the following equivalent system

$\begin{eqnarray} \dot{y}_{i}(t) & = & -c_{i} y_{i}(t) +\sum\limits^{n}_{j = 1}a_{ij} \widetilde{ f}_j(y_{j}(t)) +\sum\limits^{n}_{j = 1}b_{ij} \widetilde{ g}_j(y_{j}(t-\tau_{ij}(t))) +\sum\limits^{n}_{j = 1}e_{ij}\dot{y}_{j}(t-\xi_{ij}(t)), \end{eqnarray}$

(2.1)

where

$\widetilde{ f}_{j}( y_{j}(t)) = f_{j}( y_{j}(t)+x^*_j)-f_{j}( x^*_j), \widetilde{g}_j(y_{j}(t-\tau_{ij}(t))) = g_j(y_{j}(t-\tau_{ij}(t))+x^*_j)-g_j( x^*_j ).$

Similarly, system (1.5) can be transformed into the following equivalent system

$\begin{eqnarray} \dot{y}_{i}(t) & = & -c_{i} y_{i}(t) +\sum\limits^{n}_{j = 1}a_{ij} \widetilde{ f}_j(y_{j}(t)) +\sum\limits^{n}_{j = 1}b_{ij} \widetilde{f}_j(y_{j}(t-\tau_{ij} )) +\sum\limits^{n}_{j = 1}e_{ij}\dot{y}_{j}(t-\xi_{ij} ). \end{eqnarray}$

(2.2)

It is obvious that if the origin of system (2.1) is exponentially stable, then the equilibrium point of system (1.1) is also exponentially stable. Meanwhile, the origin of system (2.1) and the equilibrium point of system (1.1) are also globally asymptotically stable. Now, we state the main stability result for system (2.1).

Theorem 1. Suppose that there exist some positive numbers $\gamma, p_1, p_2, \cdots, p_n$ such that $\gamma < 1, \max\{ \bar{\xi}, \; \bar{\tau}\} < 1-\gamma,$

$\gamma p_i {c}_{i} - \sum\limits^{n}_{j = 1} p_j (l_i |a_{ji}|\gamma+ m_i|b_{ji}| ) > 0, p_i\gamma- \sum\limits^{n}_{j = 1} p_j|e_{ji}| > 0, i = 1, 2, \cdots, n .$

Then the origin of system (2.1) is exponentially stable.

Proof. Let

$h_1(z) = 1 - e^{ \xi z}\overline{\xi}- e^{ \xi z}\gamma, h_2(z) = 1- e^{ \tau z}\overline{\tau}-\gamma,$

$h_3(z) = \gamma p_i {c}_{i}- z p_i \gamma -z \sum\limits^{n}_{j = 1}p_j |e_{ji}|- \sum\limits^{n}_{j = 1} p_j (l_i|a_{ji}|\gamma+ m_i|b_{ji}| e^{ \tau z} ) .$

Then, $\dot{h}_i(z) < 0, h_i(+\infty) < 0, i = 1, 2, 3,$ and

$h_1(0) = 1 - \overline{\xi}- \gamma > 0, h_2(0) = 1- \overline{\tau}-\gamma > 0,$

$h_3(0) = \gamma p_i {c}_{i}- \sum\limits^{n}_{j = 1} p_j (l_i|a_{ji}|\gamma+ m_i|b_{ji}| ) > 0.$

Therefore, there exist some positive numbers $\lambda_1, \lambda_2$ and $\lambda_3$ such that $h_1(\lambda_1) = h_2(\lambda_2) = h_3(\lambda_3) = 0$ , which implies that there must exist a positive number $\lambda\in (0, \min\{\lambda_1, \lambda_2, \lambda_3 \})$ such that

$\begin{eqnarray} 1 - e^{\lambda \xi}\overline{\xi}- e^{\lambda \xi}\gamma > 0, 1- e^{\lambda \tau }\overline{\tau}-\gamma > 0, \end{eqnarray}$

(2.3)

$\begin{eqnarray} \gamma p_i {c}_{i}- \lambda p_i \gamma -\lambda \sum\limits^{n}_{j = 1}p_j |e_{ji}|- \sum\limits^{n}_{j = 1} p_j (l_i|a_{ji}|\gamma+ m_i|b_{ji}| e^{\lambda \tau } ) > 0. \end{eqnarray}$

(2.4)

We construct the following Lyapunov-Krasovskii functional ^[40]

$\begin{eqnarray} V (t)& = & e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} [p_i \gamma- \sum\limits^{n}_{j = 1}p_j |e_{ji}| sgn(y_i(t)) sgn(\dot{y}_i(t))]| y_i(t)| +\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}p_i|e_{ij}| \int^t_{t-\xi_{ij}(t)}e^{\lambda (s+\xi)} |\dot{y}_{j}(s)|ds\\&& +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j\int^t_{t-\tau_{ij}(t)} e^{\lambda (s+\tau)}|y_{j}(s)| ds, \end{eqnarray}$

(2.5)

and derive

$\begin{eqnarray} \min\limits_{1\leq i\leq n}\{ p_i \gamma-\sum\limits^{n}_{j = 1}p_j|e_{ji}|\} e^{\lambda (t+\xi)} \| y(t) \|_1 &\leq& e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} [p_i \gamma-\sum\limits^{n}_{j = 1}p_j|e_{ji}| ]| y_i(t) | \\ &\leq& e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} [p_i \gamma-\sum\limits^{n}_{j = 1}p_j |e_{ji}| sgn(y_i(t)) sgn(\dot{y}_i(t))]| y_i(t)| \leq V (t) , \end{eqnarray}$

(2.6)

$\begin{eqnarray} V(0) & = & \nonumber e^{\lambda \xi } \sum\limits^{n}_{i = 1} [p_i \gamma-\sum\limits^{n}_{j = 1}p_j |e_{ji}| sgn(y_i(0)) sgn(\dot{y}_i(0))]| y_i(0)|\\&&\nonumber + e^{\lambda \xi }\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}p_i|e_{ij}| \int^0_{-\xi_{ij}(0)}e^{\lambda s} |\dot{y}_{j}(s)|ds + e^{\lambda (\xi+\tau)} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j\int^0_{-\tau_{ij}(0)} e^{\lambda s}|y_{j}(s)| ds \\ \nonumber &\leq& \nonumber e^{\lambda \xi }\max\limits_{1\leq i\leq n}\{p_i \gamma+\sum\limits^{n}_{j = 1}p_j |e_{ji}| \} \| y(0)\|_1 +e^{\lambda \xi }\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}p_i|e_{ij}| \int^0_{-\xi} |\dot{y}_{j}(s)|ds \\&& +e^{\lambda (\xi+\tau)} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j\int^0_{-\tau} |y_{j}(s)| ds. \end{eqnarray}$

(2.7)

Taking the right upper Dini derivative of the first term and the time derivatives of the all other terms in the Lyapunov-Krasovskii functional $V(t)$ along the trajectories of system (2.1), we derive

$\begin{eqnarray} \dot{V}(t)& = &\nonumber \lambda e^{\lambda (t+\xi)}\nonumber \sum\limits^{n}_{i = 1} [p_i \gamma- \sum\limits^{n}_{j = 1}p_j |e_{ji}| sgn(y_i(t)) sgn(\dot{y}_i(t))] | y_i(t)|\\&&\nonumber +e^{\lambda t}\sum\limits^{n}_{i = 1} e^{\lambda \xi} [ p_i \gamma- \sum\limits^{n}_{j = 1}p_j |e_{ji}| sgn(y_i(t)) sgn(\dot{y}_i(t))]sgn(y_i(t)) \dot{ y}_i(t)\\&&\nonumber +e^{\lambda (t+\xi)}\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}p_i|e_{ij}| |\dot{y}_{j}(t)| -\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}(1-\dot{\xi}_{ij} (t)) e^{\lambda (t-\xi_{ij} (t)+\xi)}p_i|e_{ij}| |\dot{y}_{j}(t-\xi_{ij}(t))| \\ && \nonumber +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j \bigg( e^{\lambda (t+\tau)}|y_{j}(t)| -(1-\dot{\tau}_{ij}(t) )e^{\lambda (t-\tau_{ij}(t)+\tau)}|y_{j}(t-\tau_{ij}(t))|\bigg) \\&\leq &\nonumber \lambda e^{\lambda (t+\xi)}\nonumber \sum\limits^{n}_{i = 1} (p_i \gamma+ \sum\limits^{n}_{j = 1}p_j |e_{ji}| ) | y_i(t)| +e^{\lambda t}\sum\limits^{n}_{i = 1} \bigg\{ e^{\lambda \xi} p_i \gamma sgn(y_i(t)) \dot{ y}_i(t) \\&&\nonumber- e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j |e_{ji}| (sgn(y_i(t)) )^2 | \dot{ y}_i(t)| +e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}| |\dot{y}_{i}(t)|\bigg\}\\&& \nonumber + \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}( \dot{\xi}_{ij} (t)e^{\lambda (t-\xi_{ij} (t)+\xi)} -e^{\lambda (t-\xi_{ij} (t)+\xi)}) p_i|e_{ij}| |\dot{y}_{j}(t-\xi_{ij}(t))| \\ && \nonumber +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j \bigg( e^{\lambda (t+\tau)}|y_{j}(t)| +[ \dot{\tau}_{ij}(t) e^{\lambda (t-\tau_{ij}(t)+\tau)}- e^{\lambda (t-\tau_{ij}(t)+\tau)} ]|y_{j}(t-\tau_{ij}(t))|\bigg) \\&\leq &\nonumber \lambda e^{\lambda (t+\xi)}\nonumber \sum\limits^{n}_{i = 1} (p_i \gamma+ \sum\limits^{n}_{j = 1}p_j |e_{ji}| ) | y_i(t)| +e^{\lambda t}\sum\limits^{n}_{i = 1} \bigg\{ e^{\lambda \xi} p_i \gamma sgn(y_i(t)) \dot{ y}_i(t) \\&&\nonumber - e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j |e_{ji}| (sgn(y_i(t)) )^2 | \dot{ y}_i(t)| +e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}| |\dot{y}_{i}(t)|\bigg\}\\&& \nonumber + \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}( \bar{\xi} e^{\lambda (t+\xi)} -e^{\lambda t}) p_i|e_{ij}| |\dot{y}_{j}(t-\xi_{ij}(t))| \\ && +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j \bigg( e^{\lambda (t+\tau)}|y_{j}(t)| +( \bar{\tau} e^{\lambda (t +\tau)}- e^{\lambda t})|y_{j}(t-\tau_{ij}(t))|\bigg) \nonumber \\ & = &\nonumber\lambda e^{\lambda (t+\xi)}\nonumber \sum\limits^{n}_{i = 1} (p_i \gamma+ \sum\limits^{n}_{j = 1}p_j |e_{ji}| ) | y_i(t)| +e^{\lambda t}\sum\limits^{n}_{i = 1} \bigg\{ e^{\lambda \xi} p_i \gamma sgn(y_i(t)) \dot{ y}_i(t) \\&&\nonumber- e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j |e_{ji}| (sgn(y_i(t)) )^2 | \dot{ y}_i(t)| +e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}| |\dot{y}_{i}(t)|\bigg\} \\ &&\nonumber + e^{\lambda t}( \overline{\xi}e^{\lambda \xi}-1)\sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}p_i|e_{ij}| |\dot{y}_{j}(t-\xi_{ij}(t))| \\&& +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j \bigg ( e^{\lambda (t+\tau)}|y_{j}(t)| +( \overline{\tau}e^{\lambda \tau} -1)e^{\lambda t}|y_{j}(t-\tau_{ij}(t))|\bigg). %\\&& +\eta \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1}(e^{\lambda (t+\tau)}|y_{j}(t)|-(1-\overline{\tau})e^{\lambda t}| y_{j}( t-\tau_{ij}(t))| ). \end{eqnarray}$

(2.8)

It is noted that for $y_i(t)\neq 0,$

and for $y_i(t) = 0,$

$\begin{eqnarray*} && \nonumber e^{\lambda \xi} p_i \gamma sgn(y_i(t)) \dot{ y}_i(t)-e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j |e_{ji}| (sgn(y_i(t)) )^2 | \dot{ y}_i(t)| +e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}| |\dot{y}_{i}(t)| \\& = & e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}|\nonumber | \dot{ y}_i(t)| \\& = &\nonumber e^{\lambda \xi}\bigg\{ - \sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) {c}_{i} y_{i}(t) + \sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) \sum\limits^{n}_{j = 1}a_{ij} \widetilde{f}_j(y_{j}(t)) \\&& \nonumber + \sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) \sum\limits^{n}_{j = 1}b_{ij} \widetilde{g}_j(y_{j}(t-\tau_{ij}(t))) + \sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t))\sum\limits^{n}_{j = 1}e_{ij}\dot{y}_{j}(t-\xi_{ij}(t)) \bigg\} \\&\leq &\nonumber e^{\lambda \xi}\bigg\{ - \gamma p_i {c}_{i} |y_i(t)| + |\sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) \sum\limits^{n}_{j = 1}a_{ij} \widetilde{f}_j(y_{j}(t))| \\&& \nonumber + | \sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) \sum\limits^{n}_{j = 1}b_{ij} \widetilde{g}_j(y_{j}(t-\tau_{ij}(t)))| + |\sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t))\sum\limits^{n}_{j = 1}e_{ij}\dot{y}_{j}(t-\xi_{ij}(t)) |\bigg\} \\&\leq&\nonumber e^{\lambda \xi}\bigg\{ - \gamma p_i {c}_{i} |y_i(t)|+ \sum\limits^{n}_{j = 1}p_j|e_{ji}| \sum\limits^{n}_{j = 1}|a_{ij}| l_j|y_{j}(t) | + \sum\limits^{n}_{j = 1}p_j|e_{ji}| \sum\limits^{n}_{j = 1}|b_{ij}|m_j| y_{j}(t-\tau_{ij}(t)) |\bigg\} \\&& \nonumber + e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j|e_{ji}|\sum\limits^{n}_{j = 1}|e_{ij}||\dot{y}_{j}(t-\xi_{ij}(t))| \\ &\leq&\nonumber e^{\lambda \xi}\bigg\{ - \gamma p_i {c}_{i} |y_i(t)|+ \gamma p_i \sum\limits^{n}_{j = 1}|a_{ij}| l_j|y_{j}(t) | + \gamma p_i \sum\limits^{n}_{j = 1}|b_{ij}|m_j| y_{j}(t-\tau_{ij}(t)) |\bigg\}\\&& \nonumber + e^{\lambda \xi}\gamma p_i\sum\limits^{n}_{j = 1}|e_{ij}||\dot{y}_{j}(t-\xi_{ij}(t))|, \end{eqnarray*}$

where we use $\sum\limits^{n}_{j = 1} p_j|e_{ji}| < p_i\gamma,$ and $\sum\limits^{n}_{j = 1}p_j|e_{ji}| sgn(\dot{y}_i(t)) {c}_{i} y_{i}(t) = \gamma p_i {c}_{i} |y_i(t)| = 0$ when $y_i(t) = 0.$

Therefore, for every $y_i(t) \in R$ , we have

$\begin{eqnarray} && e^{\lambda \xi} p_i \gamma sgn(y_i(t)) \dot{ y}_i(t)-e^{\lambda \xi} \sum\limits^{n}_{j = 1}p_j |e_{ji}| (sgn(y_i(t)) )^2 | \dot{ y}_i(t)| +e^{\lambda \xi } \sum\limits^{n}_{j = 1}p_j|e_{ji}| |\dot{y}_{i}(t)| \\ &\leq& e^{\lambda \xi}\bigg\{ - \gamma p_i {c}_{i} |y_i(t)|+ \gamma p_i \sum\limits^{n}_{j = 1}|a_{ij}| l_j|y_{j}(t) | + \gamma p_i \sum\limits^{n}_{j = 1}|b_{ij}|m_j| y_{j}(t-\tau_{ij}(t)) |\bigg\}\\&& + e^{\lambda \xi}\gamma p_i\sum\limits^{n}_{j = 1}|e_{ij}||\dot{y}_{j}(t-\xi_{ij}(t))|. \end{eqnarray}$

(2.9)

Then, from (2.3), (2.4), (2.8) and (2.9), we have

$\begin{eqnarray} \dot{V}(t)&\leq & \lambda e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} (p_i \gamma+ \sum\limits^{n}_{j = 1}p_j |e_{ji}| ) | y_i(t)| +e^{\lambda t}\sum\limits^{n}_{i = 1} \bigg\{ e^{\lambda \xi}\bigg( - \gamma p_i {c}_{i} |y_i(t)| + \gamma p_i \sum\limits^{n}_{j = 1}|a_{ij}| l_j|y_{j}(t) | \\&& + \gamma p_i \sum\limits^{n}_{j = 1}|b_{ij}|m_j| y_{j}(t-\tau_{ij}(t)) |\bigg) +( e^{\lambda \xi}\gamma + e^{\lambda \xi}\overline{\xi} -1) p_i\sum\limits^{n}_{j = 1}|e_{ij}||\dot{y}_{j}(t-\xi_{ij}(t))|\bigg\} \\&& +e^{\lambda \xi} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}|m_j \bigg ( e^{\lambda (t+\tau)}|y_{j}(t)| +( \overline{\tau}e^{\lambda \tau}-1)e^{\lambda t}|y_{j}(t-\tau_{ij}(t))|\bigg) \\& = & -e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} \bigg( \gamma p_i {c}_{i}- \lambda p_i \gamma -\lambda \sum\limits^{n}_{j = 1}p_j |e_{ji}|- \sum\limits^{n}_{j = 1} p_j (l_i |a_{ji}|\gamma+m_i|b_{ji}| e^{\lambda \tau }) \bigg) | y_i(t)| \\&& -e^{\lambda t}(1 - e^{\lambda \xi}\overline{\xi}- e^{\lambda \xi}\gamma )\sum\limits^{n}_{i = 1}\sum\limits^{n}_{j = 1}p_i|e_{ij}||\dot{y}_{j}(t-\xi_{ij}(t))| \\&& - (1- e^{\lambda \tau }\overline{\tau}-\gamma) e^{\lambda (t+\xi)} \sum\limits^{n}_{i = 1} \sum\limits^{n}_{j = 1} p_i |b_{ij}| m_j |y_{j}(t-\tau_{ij}(t))| \leq 0. \end{eqnarray}$

(2.10)

Finally, (2.6), (2.7) and (2.10) imply that there must exist a number $\delta > 1$ such that

$\| y(t) \|_1 \leq \delta e^{- \lambda t} (\|\varphi\|_1+\|\phi\|_1), t\geq 0,$

where

$\|\varphi\|_1 = \sup\limits_{t\in [-\max \{\tau , \; \xi \}, \;0]}\|\varphi(t)\|_1, \|\phi\|_1 = \sup\limits_{t\in [- \max \{\tau , \; \xi \}, \; 0]}\|\phi(t)\|_1.$

Generally speaking, it is not easy to find the values of the positive constants $p_1, \cdots, p_n$ . Therefore, it is necessary to give a result without involving the constants $p_1, \cdots, p_n.$ The following result is a special case of Theorem 1 for $p_1 = \cdots = p_n$ , which is easier to validate.

Theorem 2. Suppose that there exists a positive number $\gamma$ such that

$\max\{ \bar{\xi} , \bar{\tau}\} < 1-\gamma, \sum\limits^{n}_{j = 1} |e_{ji}| < \gamma < 1, \gamma {c}_{i} - \sum\limits^{n}_{j = 1} (l_i |a_{ji}|\gamma+m_i |b_{ji}| ) > 0, i = 1, 2, \cdots, n.$

Then the equilibrium point of system (1.1) is globally asymptotically stable, even exponentially stable.

Remark 1. Theorem 1 and Theorem 2 give novel sufficient conditions of global asymptotical stability for the systems studied in ^{[18,20,29,30]} since these systems are some special cases of system (1.1).

For system (1.5) or (1.6), Theorem 1 and Theorem 2 give the following results.

Corollary 1. Suppose that there exist some positive numbers $\gamma, p_1, p_2, \cdots, p_n$ such that $\gamma < 1,$

$\gamma p_i {c}_{i} - \sum\limits^{n}_{j = 1} p_j l_i ( |a_{ji}|\gamma+ |b_{ji}| ) > 0, p_i\gamma- \sum\limits^{n}_{j = 1} p_j|e_{ji}| > 0, i = 1, 2, \cdots, n .$

Then the equilibrium point of system (1.5) (or (1.6)) is globally asymptotically stable, even exponentially stable.

Corollary 2. Suppose that there exists a positive number $\gamma$ such that $\gamma < 1,$

$\gamma {c}_{i} - \sum\limits^{n}_{j = 1} l_i ( |a_{ji}|\gamma+ |b_{ji}| ) > 0, \gamma- \sum\limits^{n}_{j = 1} |e_{ji}| > 0, i = 1, 2, \cdots, n .$

Then the equilibrium point of system (1.5) (or (1.6)) is globally asymptotically stable, even exponentially stable.

Remark 2. It is noted that $\gamma p_i {c}_{i} - \sum\limits^{n}_{j = 1} p_j l_i (|a_{ji}|\gamma+ |b_{ji}|) > 0$ and $0 < \gamma < 1$ can deduce that $p_i {c}_{i} - \sum\limits^{n}_{j = 1} p_j l_i (|a_{ji}| + |b_{ji}|) > 0.$ Therefore, the conditions of Corollary 2 are less conservative than those of the result in ^[29].

Remark 3. For system (1.5), the conditions of Theorem 1 in ^[30] are as follows:

$\begin{eqnarray*} \alpha_i& = & \nonumber c^2_i-l^2_i\sum\limits^{n}_{j = 1}|\sum\limits^{n}_{k = 1}a_{ki}a_{kj}|-l^2_i\sum\limits^{n}_{j = 1} \sum\limits^{n}_{k = 1}(|a_{ji}||b_{jk}|+|b_{ji}||a_{jk}| +|b_{ji}||b_{jk}|)\\&&\nonumber -l^2_i\sum\limits^{n}_{j = 1} \sum\limits^{n}_{k = 1}c_j |e_{jk}|(|a_{ji}|+|b_{ji}|)-\sum\limits^{n}_{j = 1} \sum\limits^{n}_{k = 1}c_j |e_{ji}|(|a_{jk}|+|b_{jk}|) \\&&-\sum\limits^{n}_{j = 1}c^2_i |e_{ij}|-\sum\limits^{n}_{j = 1}c^2_j |e_{ji}| > 0, \end{eqnarray*}$

and $1-\sum^{n}_{j = 1} |e_{ji}| > 0, i = 1, \cdots, n.$ Example 2 demonstrates the above conditions are not satisfied while the conditions of Corollary 1 and Corollary 2 can be satisfied.

Remark 4. Global asymptotical stability of a more general class of multiple delayed neutral type neural network model has been studied in ^[31,32]. The results of ^[31,32] can be particularized for system (2.2) as follows:

i) The sufficient conditions of global asymptotical stability are given in ^[31]:

$\begin{eqnarray} \sigma_i& = & 2c_i- \sum\limits^{n}_{j = 1}(l_j| a_{ij}|+l_i|a_{ji}|)- \sum\limits^{n}_{j = 1}(l_j| b_{ij}|+l_i|b_{ji}|)- \sum\limits^{n}_{j = 1}(| e_{ij}| + |e_{ji}| ) \\&& -\sum\limits^{n}_{j = 1}\sum\limits^{n}_{k = 1}(l_i|a_{ki}| |e_{kj}| +l_i|b_{ki}||e_{kj}|)-\sum\limits^{n}_{j = 1}\sum\limits^{n}_{k = 1}(l_k|a_{jk}| |e_{ji}| +l_k|b_{jk}||e_{ji}|) > 0, \end{eqnarray}$

and $1-\sum^{n}_{j = 1} |e_{ji}| > 0, i = 1, \cdots, n.$

ii) The sufficient conditions of global asymptotical stability are given in ^[32]:

$\begin{eqnarray*} \epsilon_i& = & \nonumber c^2_i- \sum\limits^{n}_{j = 1}|\sum\limits^{n}_{k = 1}a_{ki}a_{kj}|- \sum\limits^{n}_{j = 1} \sum\limits^{n}_{k = 1}(|a_{ji}||b_{jk}|\\&&\nonumber+|a_{ji}||e_{jk}| +|b_{ji}||a_{jk}| +|b_{ji}||e_{jk}|+|b_{ji}||b_{jk}|) > 0, \end{eqnarray*}$

$\begin{eqnarray*} \varepsilon_{ij} = \frac{1}{n}- \sum\limits^{n}_{k = 1}(|e_{ji}||e_{jk}| +|a_{jk}||e_{ji}|+|b_{jk}| |e_{ji}|) > 0, i, j = 1, \cdots, n. \end{eqnarray*}$

Example 3 demonstrates the above conditions are not satisfied while the conditions of Corollary 1 and Corollary 2 can be satisfied.

Example 1. Consider system (1.1) with the following conditions:

$A = \left(\begin{array}{cccc} 1 & -1 & 1 & -1\\ -1 & 1& 1 & -1\\ 1 & 1& 1& 1 \\ 1 & -1 &-1 & -1 \end{array}\right), B = \left(\begin{array}{cccc} 1 & 1 &-1 & 1\\ -1 & -1 &1 &-1\\ 1 & 1& 1 & 1\\ -1 & -1 &-1 & 1 \end{array}\right), E = \left(\begin{array}{cccc} 0.1 & -0.1 &-0.1 & -0.1\\ -0.1 & 0.1 &0.1 &0.1\\0.1 & -0.1& 0.1 & -0.1\\ -0.1 & -0.1 &-0.1 & 0.1 \end{array}\right),$

$c_1 = c_2 = c_3 = 9, c_4 = 8, f_i(x) = tanh(x), g_i(x) = 0.5tanh(x), \xi_{ii}(t) = 0.1sin t+0.1, \tau_{ii}(t) = 0.1cos t+0.1; \xi_{ij}(t) = 0.1cos t+0.1, \tau_{ij}(t) = 0.1sin t+0.1, i\neq j; i, j = 1, 2, 3, 4.$

We calculate $\bar{\xi} = \bar{\tau} = 0.1, l_i = 1, m_i = 0.5, \sum\limits^{4}_{j = 1} |e_{ji}| = 0.4$ and

$\begin{eqnarray*} {c}_{i}\gamma - \sum\limits^{4}_{j = 1} (l_i |a_{ji}|\gamma +m_i |b_{ji}| ) = \begin{cases} 5\gamma-2 , & i = 1, 2, 3; \\ 4\gamma-2 , & i = 4. \end{cases} \end{eqnarray*}$

It is clear that when $\gamma \in (0.5, 0.9),$ Theorem 2 holds.

Now we choose $\gamma = 0.424$ and require that

$\gamma p_i {c}_{i} - \sum\limits^{4}_{j = 1} p_j (l_i |a_{ji}|\gamma+ m_i|b_{ji}| ) = 0.01 , i = 1, 2, 3, 4.$

Then, we calculate $p_1 = p_2 = p_3 = 2.2222, p_4 = 2.5000,$ and

$\begin{eqnarray*} p_i\gamma- \sum\limits^{4}_{j = 1} p_j|e_{ji}| = 0.424 p_i - 0.1\sum\limits^{4}_{j = 1} p_j = \begin{cases} 0.025552, & i = 1, 2, 3; \\ 0.14334, & i = 4. \end{cases} \end{eqnarray*}$

Thus, all conditions of Theorem 1 are satisfied.

If we choose $\gamma = 0.423$ and still require that

$\gamma p_i {c}_{i} - \sum\limits^{4}_{j = 1} p_j (l_i |a_{ji}|\gamma+ m_i|b_{ji}| ) = 0.01 , i = 1, 2, 3, 4.$

Then $p_i < 0, i = 1, 2, 3, 4.$ Therefore, when $\gamma\in [0.424, 0.9),$ the system (1.1) in this example is globally asymptotically stable, even exponentially stable.

Example 2. Consider system (1.5) with the conditions: $c_1 = c_2 = c_3 = 9, c_4 = 8.5, f_i(x) = tanh(x), \xi_{ii} = 0.2, \tau_{ii} = 0.1; \xi_{ij} = 0.3, \tau_{ij} = 0.4, i\neq j; i, j = 1, 2, 3, 4,$ the matrices $A, B, E$ are the same as in Example 1.

Then, we calculate $l_i = 1, \sum\limits^{4}_{j = 1} |e_{ji}| = 0.4$ and

$\begin{eqnarray*} {c}_{i}\gamma - \sum\limits^{4}_{j = 1} l_i ( |a_{ji}|\gamma + |b_{ji}| ) = \begin{cases} 5\gamma-4 , & i = 1, 2, 3; \\ 4.5\gamma-4 , & i = 4. \end{cases} \end{eqnarray*}$

It is clear that when $\gamma \in (8/9, 1),$ Corollary 2 holds.

Now we choose $\gamma = 0.85$ and require that

$\gamma p_i {c}_{i} - \sum\limits^{4}_{j = 1} p_j l_i (|a_{ji}|\gamma+|b_{ji}| ) = 0.01 , i = 1, 2, 3, 4.$

Then, we calculate $p_1 = p_2 = p_3 = 0.0708, p_4 = 0.0750,$ and

$\begin{eqnarray*} p_i\gamma- \sum\limits^{4}_{j = 1} p_j|e_{ji}| = 0.85 p_i - 0.1\sum\limits^{4}_{j = 1} p_j = \begin{cases} 0.03144, & i = 1, 2, 3; \\ 0.03501, & i = 4. \end{cases} \end{eqnarray*}$

Thus, all conditions of Corollary 1 are satisfied.

On the other hand, we calculate

$\begin{eqnarray*} \alpha_1& = & \nonumber 81- \sum\limits^{4}_{j = 1}|\sum\limits^{4}_{k = 1}a_{k1}a_{kj}|- 3\times 4\times 4 - 35.5\times 4 \times 0.2 -35.5\times 4 \times 0.2 \\&&-81 \times 0.4-315.25 \times 0.1 < 0, \end{eqnarray*}$

which shows the conditions of Theorem 1 in ^[30] are not satisfied. Therefore, Theorem 1 in ^[30] is invalid for the system (1.5) in this example.

Example 3. Consider system (2.2) with all parameters and functions are the same as in Example 2. Then, we calculate $l_i = 1, \sum^{4}_{j = 1} |e_{ji}| = 0.4,$

$\varepsilon_{ij} = \frac{1}{4}-0.21\times 4 = -0.59 < 0,$

$\sigma_{1} = 2\times 9-2\times 4-2\times 4-0.2\times 4-0.2\times 4^2-0.2\times 4^2 = -5.2 < 0.$

Therefore, the sufficient conditions of the results in ^[31,32] are not satisfied. Meanwhile, all conditions of Corollary 1 and Corollary 2 are staisfied for system (2.2) since system (1.5) is equivalent to system (2.2).

3. Conclusions

This paper has investigated the exponential stability of neutral-type Hopfield neural networks involving multiple time-varying delays. Different from some existing results, linear matrix inequality approach cannot be used to determine the stability conditions of such networks since the networks studied here can not be expressed in vector-matrix form. By using a modified and suitable Lyapunov-Krasovskii functional and inequality techniques, novel algebraic conditions are established to ensure the exponential stability and the global asymptotic stability of neutral-type Hopfield neural networks involving multiple time-varying delays. Compared with some references, the networks studied here is more general and the established conditions are less conservative. Three examples are given to demonstrate the effectiveness of the theoretical results and compare the established stability conditions to the previous results.

Acknowledgments

The authors would like to thank the editor and the reviewers for their detailed comments and valuable suggestions. This work was supported by the National Natural Science Foundation of China (No: 11971367, 11826209, 11501499, 61573011 and 11271295), the Natural Science Foundation of Guangdong Province (2018A030313536).

Conflict of interest

All authors declare no conflicts of interest in this paper.

Acknowledgments

E.A. acknowledges fruitful discussions with Dr. Panagiotis Athanasopoulos.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	Friston KJ (1994) Functional and effective connectivity in neuroimaging: A synthesis. Hum Brain Mapp 2: 56-78. doi: 10.1002/hbm.460020107
[2]	Friston KJ (2011) Functional and effective connectivity: A review. Brain Connect 1: 13-36. doi: 10.1089/brain.2011.0008
[3]	Friston K, Moran R, Seth AK (2013) Analysing connectivity with granger causality and dynamic causal modelling. Curr Opin Neurobiol 23: 172-178. doi: 10.1016/j.conb.2012.11.010
[4]	Siettos C, Starke J (2016) Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools. Wiley Interdiscip Rev Syst Biol Med 8: 438-458. doi: 10.1002/wsbm.1348
[5]	Smith SM, Miller KL, Salimi-Khorshidi G, et al. (2011) Network modelling methods for fmri. Neuroimage 54: 875-891. doi: 10.1016/j.neuroimage.2010.08.063
[6]	Brazier MA (1972) Spread of seizure discharges in epilepsy: Anatomical and electrophysiological considerations. Exp Neurol 36: 263-272. doi: 10.1016/0014-4886(72)90022-2
[7]	Gerstein GL, Perkel DH, Subramanian K (1978) Identification of functionally related neural assemblies. Brain Res 140: 43-62. doi: 10.1016/0006-8993(78)90237-8
[8]	Zalesky A, Fornito A, Bullmore E (2012) On the use of correlation as a measure of network connectivity. NeuroImage 60: 2096-2106. doi: 10.1016/j.neuroimage.2012.02.001
[9]	Siggiridou E, Kugiumtzis D, Kimiskidis V (2014) Correlation networks for identifying changes in brain connectivity during epileptiform discharges and transcranial magnetic stimulation. Sensors 14: 12585-12597. doi: 10.3390/s140712585
[10]	Hockett CF (1953) The mathematical theory of communication.
[11]	Vlachos I, Kugiumtzis D (2010) Nonuniform state-space reconstruction and coupling detection. Phys Rev E 82: 016207. doi: 10.1103/PhysRevE.82.016207
[12]	Vicente R, Wibral M, Lindner M, et al. (2011) Transfer entropy—a model-free measure of effective connectivity for the neurosciences. J Comput Neurosci 30: 45-67. doi: 10.1007/s10827-010-0262-3
[13]	Savva AD, Mitsis GD, Matsopoulos GK (2019) Assessment of dynamic functional connectivity in resting-state fMRI using the sliding window technique. Brain Behav 9: e01255. doi: 10.1002/brb3.1255
[14]	Lachaux JP, Rodriguez E, Martinerie J, et al. (1999) Measuring phase synchrony in brain signals. Hum Brain Mapp 8: 194-208. doi: 10.1002/(SICI)1097-0193(1999)8:4<194::AID-HBM4>3.0.CO;2-C
[15]	Stam CJ, Nolte G, Daffertshofer A (2007) Phase lag index: Assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources. Hum Brain Mapp 28: 1178-1193. doi: 10.1002/hbm.20346
[16]	Glerean E, Salmi J, Lahnakoski JM, et al. (2012) Functional magnetic resonance imaging phase synchronization as a measure of dynamic functional connectivity. Brain Connect 2: 91-101. doi: 10.1089/brain.2011.0068
[17]	Mylonas DS, Siettos CI, Evdokimidis I, et al. (2015) Modular patterns of phase desynchronization networks during a simple visuomotor task. Brain Topogr 29: 118-129. doi: 10.1007/s10548-015-0451-5
[18]	Calhoun VD, Adali T, Pearlson GD, et al. (2001) A method for making group inferences from functional mri data using independent component analysis. Human Brain Mapp 14: 140-151. doi: 10.1002/hbm.1048
[19]	Calhoun VD, Liu J, Adali T (2009) A review of group ica for fmri data and ica for joint inference of imaging, genetic, and erp data. Neuroimage 45: S163-S172. doi: 10.1016/j.neuroimage.2008.10.057
[20]	Reidl J, Starke J, Omer DB, et al. (2007) Independent component analysis of high-resolution imaging data identifies distinct functional domains. Neuroimage 34: 94-108. doi: 10.1016/j.neuroimage.2006.08.031
[21]	Anderson A, Cohen MS (2013) Decreased small-world functional network connectivity and clustering across resting state networks in schizophrenia: an fmri classification tutorial. Front Hum Neurosci 7: 520.
[22]	Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290: 2319-2323. doi: 10.1126/science.290.5500.2319
[23]	Duncan D, Talmon R, Zaveri HP, et al. (2013) Identifying preseizure state in intracranial EEG data using diffusion kernels. Math Biosci Eng 10: 579-590. doi: 10.3934/mbe.2013.10.579
[24]	Granger CW (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 424-438. doi: 10.2307/1912791
[25]	Granger C, Newbold P (1974) Spurious regressions in econometrics. J Econom 2: 111-120. doi: 10.1016/0304-4076(74)90034-7
[26]	Geweke J (1982) Measurement of linear dependence and feedback between multiple time series. J Am Stat Assoc 77: 304-313. doi: 10.1080/01621459.1982.10477803
[27]	Seth AK (2010) A matlab toolbox for granger causal connectivity analysis. J Neurosci Methods 186: 262-273. doi: 10.1016/j.jneumeth.2009.11.020
[28]	Barnett L, Seth AK (2014) The mvgc multivariate granger causality toolbox: A new approach to granger-causal inference. J Neurosci Methods 223: 50-68. doi: 10.1016/j.jneumeth.2013.10.018
[29]	Seth AK, Barrett AB, Barnett L (2015) Granger causality analysis in neuroscience and neuroimaging. J Neurosci 35: 3293-3297. doi: 10.1523/JNEUROSCI.4399-14.2015
[30]	Friston KJ, Harrison L, Penny W (2003) Dynamic causal modelling. Neuroimage 19: 1273-1302. doi: 10.1016/S1053-8119(03)00202-7
[31]	Keller CJ, Bickel S, Honey CJ, et al. (2013) Neurophysiological investigation of spontaneous correlated and anticorrelated fluctuations of the BOLD signal. J Neurosci 33: 6333-6342. doi: 10.1523/JNEUROSCI.4837-12.2013
[32]	Megumi F, Yamashita A, Kawato M, et al. (2015) Functional MRI neurofeedback training on connectivity between two regions induces long-lasting changes in intrinsic functional network. Front Hum Neurosci 9: 160. doi: 10.3389/fnhum.2015.00160
[33]	Wiener N (1956) The theory of prediction. Modern Math Eng 1: 125-139.
[34]	Barrett AB, Murphy M, Bruno MA, et al. (2012) Granger causality analysis of steady-state electroencephalographic signals during propofol-induced anaesthesia. PloS One 7: e29072. doi: 10.1371/journal.pone.0029072
[35]	Barnett L, Seth AK (2011) Behaviour of granger causality under filtering: theoretical invariance and practical application. J Neurosci Methods 201: 404-419. doi: 10.1016/j.jneumeth.2011.08.010
[36]	Gow DW, Segawa JA, Ahlfors SP, et al. (2008) Lexical influences on speech perception: a granger causality analysis of meg and eeg source estimates. Neuroimage 43: 614-623. doi: 10.1016/j.neuroimage.2008.07.027
[37]	Keil A, Sabatinelli D, Ding M, et al. (2009) Re-entrant projections modulate visual cortex in affective perception: Evidence from granger causality analysis. Hum Brain Mapp 30: 532-540. doi: 10.1002/hbm.20521
[38]	Nicolaou N, Hourris S, Alexandrou P, et al. (2012) Eeg-based automatic classification of ‘awake’ versus ‘anesthetized’ state in general anesthesia using granger causality. PloS One 7: e33869. doi: 10.1371/journal.pone.0033869
[39]	De Tommaso M, Stramaglia S, Marinazzo D, et al. (2013) Functional and effective connectivity in eeg alpha and beta bands during intermittent flash stimulation in migraine with and without aura. Cephalalgia 33: 938-947. doi: 10.1177/0333102413477741
[40]	Protopapa F, Siettos CI, Evdokimidis I, et al. (2014) Granger causality analysis reveals distinct spatio-temporal connectivity patterns in motor and perceptual visuo-spatial working memory. Front Comput Neurosci 8: 146. doi: 10.3389/fncom.2014.00146
[41]	Protopapa F, Siettos CI, Myatchin I, et al. (2016) Children with well controlled epilepsy possess different spatio-temporal patterns of causal network connectivity during a visual working memory task. Cogn Neurodyn 10: 99-111. doi: 10.1007/s11571-015-9373-x
[42]	Al-Aidroos N, Said CP, Turk-Browne NB (2012) Top-down attention switches coupling between low-level and high-level areas of human visual cortex. Proc Natl Acad Sci 109: 14675-14680. doi: 10.1073/pnas.1202095109
[43]	Florin E, Gross J, Pfeifer J, et al. (2010) The effect of filtering on granger causality based multivariate causality measures. Neuroimage 50: 577-588. doi: 10.1016/j.neuroimage.2009.12.050
[44]	Zhou Z, Wang X, Klahr NJ, et al. (2011) A conditional granger causality model approach for group analysis in functional magnetic resonance imaging. Magn Reson Imaging 29: 418-433. doi: 10.1016/j.mri.2010.10.008
[45]	Wu G, Liao W, Stramaglia S, et al. (2013) Recovering directed networks in neuroimaging datasets using partially conditioned granger causality. BMC Neurosci 14: P260. doi: 10.1186/1471-2202-14-S1-P260
[46]	Tang W, Bressler SL, Sylvester CM, et al. (2012) Measuring granger causality between cortical regions from voxelwise fmri bold signals with lasso. PLoS Comput Biol 8: e1002513. doi: 10.1371/journal.pcbi.1002513
[47]	Roebroeck A, Formisano E, Goebel R (2005) Mapping directed influence over the brain using granger causality and fmri. Neuroimage 25: 230-242. doi: 10.1016/j.neuroimage.2004.11.017
[48]	Sabatinelli D, McTeague LM, Dhamala M, et al. (2015) Reduced medial prefrontal–subcortical connectivity in dysphoria: Granger causality analyses of rapid functional magnetic resonance imaging. Brain Connect 5: 1-9. doi: 10.1089/brain.2013.0186
[49]	Miao X, Wu X, Li R, et al. (2011) Altered connectivity pattern of hubs in default-mode network with alzheimer's disease: an granger causality modeling approach. PloS One 6: e25546. doi: 10.1371/journal.pone.0025546
[50]	Liao W, Ding J, Marinazzo D, et al. (2011) Small-world directed networks in the human brain: Multivariate granger causality analysis of resting-state fMRI. Neuroimage 54: 2683-2694. doi: 10.1016/j.neuroimage.2010.11.007
[51]	Gao Q, Duan X, Chen H (2011) Evaluation of effective connectivity of motor areas during motor imagery and execution using conditional granger causality. Neuroimage 54: 1280-1288. doi: 10.1016/j.neuroimage.2010.08.071
[52]	Lin FH, Ahveninen J, Raij T, et al. (2014) Increasing fMRI sampling rate improves granger causality estimates. PLoS One 9: e100319. doi: 10.1371/journal.pone.0100319
[53]	Duggento A, Passamonti L, Valenza G, et al. (2018) Multivariate granger causality unveils directed parietal to prefrontal cortex connectivity during task-free MRI. Sci Rep 8: 1-11. doi: 10.1038/s41598-018-23996-x
[54]	Penny WD, Stephan KE, Mechelli A, et al. (2004) Comparing dynamic causal models. Neuroimage 22: 1157-1172. doi: 10.1016/j.neuroimage.2004.03.026
[55]	Friston KJ, Bastos A, Litvak V, et al. (2012) Dcm for complex-valued data: cross-spectra, coherence and phase-delays. Neuroimage 59: 439-455. doi: 10.1016/j.neuroimage.2011.07.048
[56]	Guo S, Seth AK, Kendrick KM, et al. (2008) Partial granger causality—eliminating exogenous inputs and latent variables. J Neurosci Methods 172: 79-93. doi: 10.1016/j.jneumeth.2008.04.011
[57]	Roelstraete B, Rosseel Y (2012) Does partial granger causality really eliminate the influence of exogenous inputs and latent variables? J Neurosci Methods 206: 73-77. doi: 10.1016/j.jneumeth.2012.01.010
[58]	Ge T, Kendrick KM, Feng J (2009) A novel extended granger causal model approach demonstrates brain hemispheric differences during face recognition learning. PLoS Comput Biol 5: e1000570. doi: 10.1371/journal.pcbi.1000570
[59]	Bajaj S, Adhikari BM, Friston KJ, et al. (2016) Bridging the gap: Dynamic causal modeling and granger causality analysis of resting state functional magnetic resonance imaging. Brain Connect 6: 652-661. doi: 10.1089/brain.2016.0422
[60]	Dhamala M, Rangarajan G, Ding M (2008) Analyzing information flow in brain networks with nonparametric granger causality. Neuroimage 41: 354-362. doi: 10.1016/j.neuroimage.2008.02.020
[61]	Fox MD, Snyder AZ, Vincent JL, et al. (2005) The human brain is intrinsically organized into dynamic, anticorrelated functional networks. Proc Natl Acad Sci 102: 9673-9678. doi: 10.1073/pnas.0504136102
[62]	Biswal BB (2012) Resting state fMRI: A personal history. Neuroimage 62: 938-944. doi: 10.1016/j.neuroimage.2012.01.090
[63]	Seth AK, Chorley P, Barnett LC (2013) Granger causality analysis of fMRI BOLD signals is invariant to hemodynamic convolution but not downsampling. Neuroimage 65: 540-555. doi: 10.1016/j.neuroimage.2012.09.049
[64]	Ding M, Chen Y, Bressler SL (2006) Granger causality: basic theory and application to neuroscience. Handbook of time series analysis: recent theoretical developments and applications 437: 437-460. doi: 10.1002/9783527609970.ch17
[65]	Baccalá LA, Sameshima K (2001) Partial directed coherence: A new concept in neural structure determination. Biol Cybern 84: 463-474. doi: 10.1007/PL00007990
[66]	Dumoulin S, Wandell B (2008) Population receptive field estimates in human visual cortex. Neuroimage 39: 647. doi: 10.1016/j.neuroimage.2007.09.034
[67]	Büchel C, Friston K (1997) Modulation of connectivity in visual pathways by attention: cortical interactions evaluated with structural equation modelling and fmri. Cereb Cortex 7: 768-778. doi: 10.1093/cercor/7.8.768
[68]	Büchel C, Josephs O, Rees G, et al. (1998) The functional anatomy of attention to visual motion. a functional mri study. Brain 121: 1281-1294. doi: 10.1093/brain/121.7.1281
[69]	Penny WD, Stephan KE, Mechelli A, et al. (2004) Modelling functional integration: a comparison of structural equation and dynamic causal models. Neuroimage 23: S264-S274. doi: 10.1016/j.neuroimage.2004.07.041
[70]	Penny WD, Friston KJ, Ashburner JT, et al. (2011) Statistical parametric mapping: the analysis of functional brain images Academic press.
[71]	Friston KJ, Holmes AP, Worsley KJ, et al. (1994) Statistical parametric maps in functional imaging: a general linear approach. Human Brain Mapping 2: 189-210. doi: 10.1002/hbm.460020402
[72]	Worsley KJ, Friston KJ (1995) Analysis of fmri time-series revisited—again. Neuroimage 2: 173-181. doi: 10.1006/nimg.1995.1023
[73]	Zou C, Feng J (2009) Granger causality vs. dynamic bayesian network inference: A comparative study. BMC Bioinformatics 10: 122. doi: 10.1186/1471-2105-10-122

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