FUNCTIONAL CONNECTOME OF THE HUMAN BRAIN WITH TOTAL CORRELATION Qiang Li

2025-05-08 0 0 5.36MB 21 页 10玖币

侵权投诉

FUNCTIONAL CONNECTOME OF THE HUMAN BRAIN WITH

TOTAL CORRELATION

Qiang Li

Image Processing Laboratory

University of Valencia

Valencia, 46980

qiang.li@uv.es

Greg Ver Steeg

Information Sciences Institute

University of Southern California

Marina del Rey, CA 90292

gregv@isi.edu

Shujian Yu

Machine Learning Group

UiT - The Arctic University of Norway

9037 Tromsø, Norway

shujian.yu@uit.no

Jesus Malo

Image Processing Laboratory

University of Valencia

Valencia, 46980

ABSTRACT

Recent studies proposed the use of Total Correlation to describe functional connectivity among brain

regions as a multivariate alternative to conventional pair-wise measures such as correlation or mutual

information. In this work we build on this idea to infer a large scale (whole brain) connectivity network

based on Total Correlation and show the possibility of using this kind of networks as biomarkers

of brain alterations. In particular, this work uses Correlation Explanation (CorEx) to estimate Total

Correlation. First, we prove that CorEx estimates of total correlation and clustering results are

trustable compared to ground truth values. Second, the inferred large scale connectivity network

extracted from the more extensive open fMRI datasets is consistent with existing neuroscience studies

but, interestingly, can estimate additional relations beyond pair-wise regions. And ﬁnally, we show

how the connectivity graphs based on Total Correlation can also be an effective tool to aid in the

discovery of brain diseases.

Keywords Total Correlation ·CorEx ·fMRI ·Functional Connectivity ·Large Scale Connectome ·Biomarkers

1 Introduction

The human brain is a complex system comprised of interconnected functional units. Millions of neurons in the brain

interact with each other at both a structural and functional level to drive efﬁcient inference and processing in the brain.

Furthermore, the functional connectivity among these regions also reveals how they interact with each other in speciﬁc

cognitive tasks. Functional connectivity refers to the statistical dependency of activation patterns between various brain

regions that emerges as a result of direct and indirect interactions [1,2]. It is usually measured by how similar neural

time series are to each other, and it shows how the time series statistically interact with each other.

A variety of ways to analyze functional connectivity exist. A seed-wise analysis can be performed by selecting a

seed-driven hypothesis and analyzing its statistical dependencies with all other voxels outside its limits. It’s a common

tool for studying how different parts of the brain are connected to one another. Connectivity is determined by calculating

the correlation between the time series of each voxel in the brain and the time series of single seed voxel. Another

option is to perform a wide analysis of the voxel or region of interest (ROI), where statistical dependencies on all voxels

or ROIs are studied [3]. Structural connectivity refers to the anatomical organization of the brain by means of ﬁber

tracts [4]. The sharing of communication between neurons in multiple regions is coordinated dynamically via changes

in neural oscillation synchronizations [5]. When it comes to the brain connectome, functional connectivity refers to how

different areas of the brain communicate with one another during task-related or resting-state activities [6]. The use of

arXiv:2210.03231v4 [q-bio.NC] 14 Nov 2022

November 15, 2022

information-theoretic metrics can efﬁciently detect their interaction in dynamical brain networks, and it’s widely used

in the ﬁeld of neuroscience [7]. For instance, quantify information encoding and decoding in the neural system [8

–

11],

measure visual information ﬂow in the biological neural networks [12,13], and color information processing in the

neural cortex [14], and so on. However, although functional connectivity has already become a hot research topic in

neuroscience [15,16], systematic studies on the information ﬂow or the redundancy and synergy amongst brain regions

remain limited. One extreme type of redundancy is full synchronization, where the state of one neural signal may be

used to predict the status of any other neural signal, and this concept of redundancy is thus viewed as an extension of

the standard notion of correlation to more than two variables [17]. Synergy, on the other hand, is analogous to those

statistical correlations that govern the whole but not its constituent components [18]. High-order brain functions are as-

sumed to require synergies, which give simultaneous local independence and global cohesion, but are less suitable for

them under high synchronization situations, such as epileptic seizures [19]. Most functional connectivity approaches

until now have mainly concentrated on pairwise relationships between two regions. The conventional approach used to

estimate indirect functional connectivity among brain regions is Pearson correlation (CC) [20] and Mutual Information

(I) [8,21

–

23]. However, real brain network relationships are often complex, involving more than two regions, and the

pairwise dependencies measured by correlation or mutual information could not reﬂect these multivariate dependencies.

Therefore, recent studies in neuroscience focus on the development of information-theoretic measures that can handle

more than two regions simultaneously such as the Total Correlation [24,25].

Total Correlation (TC) [26] (also known as multi-information [27

–

29]) mainly describes the amount of dependence

observed in the data and, by deﬁnition can be applied to multiple multivariate variables. Its use to describe functional

connectivity in the brain was ﬁrst proposed as a empirical measure in [24], but in [25] the superiority of TC over

mutual information was proved analytically. The consideration of low-level vision models allows to derive analytical

expressions for the TC as a function of the connectivity. These analytical results show that pairwise I cannot capture the

effect of different intra-cortical inhibitory connections while the TC can. Similarly, in analytical models with feedback,

synergy can be shown using TC, while it is not so obvious using mutual information [25]. Moreover, these analytical

results allow to calibrate computational estimators of TC.

In this work we build on these empirical and theoretical results [24, 25] to infer a larger scale (whole brain) network

based on TC for the ﬁrst time. As opposed to [24,25] where the number of considered nodes was limited to the range of

tens and focused on specialized subsystems, here we consider wider recordings [30,31] so we use signals coming from

hundreds of nodes across the whole brain. Additionally, applying our analysis to data of the same scale for regular

and altered brains

. We also show the possibility of using this kind of wide-range networks as biomarkers. From the

technical point of view, here we use Correlation Explanation (CorEx) [32,33] to estimate TC in these high-dimensional

scenarios. Furthermore, graph theory and clustering [15,16] are used here to represent the relationships between the

considered regions.

The rest of this paper is organized as follows: Section 2 introduces the necessary information-theoretic concepts and

explains CorEx. Sections 3 and 4 show two synthetic experiments that prove that CorEx results are trustable. Section 5

estimates the large-scale connectomes with fMRI datasets that involve more than 100 regions across the whole brain.

Moreover, we show how the analysis of these large scale networks based on TC may indicate brain alterations. Sections

6 and 7 give a general discussion and the conclusion of the paper, respectively.

2 Total Correlation as neural connectivity descriptor

2.1 Deﬁnitions and Preliminaries

Mutual Information:

Given two multivariate random variables

and

, the mutual information between them,

I(X1;X2), can be calculated as the difference between the sum of individual entropies, H(Xi)and the entropy of the

variables considered jointly as a single system, H(X1, X2)[34]:

I(X1;X2) = H(X1) + H(X2)−H(X1, X2)(1)

where for each (multivariate) random variable

, the entropy is

H(v) = h− log2p(v)i

and the brackets represent

expectation values spanning random variables. The mutual information also can be seen as the information shared by

the two variables or the reduction of uncertainty in one variable given the information about the other [35].

Mutual information is better than linear correlation:

For Gaussian sources mutual information reduces to linear

correlation because the entropy factors in Eq. 1 just depend on

|hX1·X>

2i|

. However, for more general (non-Gaussian)

sources mutual information cannot be reduced to covariance and cross-covariance matrices. In these (more realistic)

1http://fcon_1000.projects.nitrc.org/indi/ACPI/html/

November 15, 2022

Figure 1:

Conceptual scheme of information theoretic measures of neural information ﬂow.

The left circle areas

represent amounts of information, and intersections represent shared information among the corresponding variables,

X0, X1, X2

. Examples of entropy,

H(X0), H(X1), H(X2)

, total correlation (red color), and

T C[X0, X1, X2]

are

given. The middle ﬁgures show some neural time series are extracted from brain regions, which correspond to the nodes

in the right ﬁgure. The right ﬁgures illustrate large-scale time series in the brain and how the coupled information is

transmitted among the brain regions. The blue and green lines show linear correlation (CC) and mutual information (I),

respectively, between different parts of the brain. The modules represent the lobes of the human brain. Each module has

speciﬁc brain regions, and each module works with the others.

situations I is better than the linear correlation because I captures nonlinear relations that are ruled out by

|hX1·X>

2i|

For an illustration of the qualitative differences between I and linear correlation see the examples in Section 2.2 of [24].

As a result, mutual information has been proposed as a good alternative to linear correlation for estimating functional

connectivity [8,21]. However, mutual information cannot not capture dependencies beyond pairs of nodes. And this

may be a limitation in complex networks [36].

Total Correlation:

This magnitude describes the dependence among

variables and it is a generalization of the mutual

information concept from two parties to

parties. The Venn Diagram in Fig. 1 qualitatively illustrates this for three

variables. The deﬁnition of total correlation from Watanabe [26] can be denoted as,

TC (X1,...,Xn)≡

i=1

H (Xi)−H (X1,...,Xn)=DKL p (X1,...,Xn)k

i=1

p (Xi)!(2)

where

X≡(X1,...,Xn)

, and TC can also be expressed as the Kullback Leibler divergence,

DKL

between the joint

probability density and the product of the marginal densities. From these deﬁnitions, if all variables are independent

then TC will be zero.

The conditional total correlation, which is similar to the deﬁnition of total correlation but with a condition appended to

each term, The Kullback-Leibler divergence of the two conditional probability distributions can also be used to deﬁne

the conditional total correlation. The estimation method used in this work (CorEx presented in the next subsection) uses

the TC after conditioning on some other variable Y, which can be deﬁned as [34],

T C(X|Y) = X

H(Xi|Y)−H(X|Y) = DKL(p(x|y)k

i=1

p(xi|y)) (3)

Total correlation is better than Mutual information:

This superiority is not only due to the obvious

-wise versus

pair-wise deﬁnitions in Eqs. 1 and 2. It also has to do with the different properties of these magnitudes. To illustrate this

point let us recall one of the analytical examples in [25]. Consider the following feedforward network:

X1//X2//e

//X3(4)

where the nodes

, and

can have any number of neurons, the ﬁrst two transforms,

X1//X2//e

are linear and affected by additive noise, and the last transform,

f(·)

, is nonlinear but deterministic. Imagine that in this

network one is interested in the connectivity between the neurons in the hidden layer,

, but the nonlinear function

f(·)

is unknown and one only has experimental access to the signal in the regions

and

. In this situation one

November 15, 2022

could think on measuring

I(X1, X3) = I(X1, f(e))

I(X2, X3) = I(X1, f(e))

. However, the invariance of

under arbitrary nonlinear re-parametrization of the variables [35] implies that these measures are insensitive to

and

the connectivity there in. On the contrary, as pointed out in [25], using the expression for the variation of TC under

nonlinear transforms [13, 37], the variation of

under nonlinear transforms [34], and the deﬁnition in Eq. 2, one

obtains

T C(X1, X2, X3) = [T C(X1, X2,e)−T C(e)] + T C(X3)

, where the term in the bracket does not depend on

f(·), but the last term deﬁnitely does, which proves the superiority of T C over Iin describing connectivity.

In [25] the network in Eq. 4 speciﬁcally refers to the ﬂow from the retina,

, to the LGN,

, and ﬁnally to the visual

cortex,

and

. However, the result of the superiority of

T C

over

to describe the connectivity in the hidden layer is

totally general for every network with the generic properties listed after Eq. 4.

2.2 Total Correlation estimated from CorEx

Straightforward application of the direct deﬁnition of TC is not feasible in high dimensional scenarios, and alternatives

are required [28,29]. A practical approach to estimate total correlation is via latent factor modelling. A latent factor

model is a statistical model that relates a set of observable variables to a set of latent variables. The idea is to explicitly

construct latent factors,

, that somehow capture the dependencies in the data. If we measure dependencies via total

correlation,

T C(X)

, then we say that the latent factors explain the dependencies if

T C(X|Y) = 0

. We can measure

the extent to which Yexplains the correlations in Xby looking at how much the total correlation is reduced

T C(X)−T C(X|Y) =

i=1

I(Xi;Y)−I(X;Y)(5)

The total correlation is always non-negative, and the decomposition on the right in terms of mutual information can be

veriﬁed directly from the deﬁnitions. Any latent factor model can be used to lower bound total correlation, and the

terms on the right-hand side of Eq. 5 can be further lower-bounded with tractable estimators using variational methods,

and Variational Autoencoders (VAEs) are a popular example [38].

Although latent factor models do not give a direct total correlation estimation as the Rotation-based Iterative Gaussian-

ization (RBIG) [28,29] and the Matrix-based Rényi’s entropy [39] did, the approach can be complementary because the

construction of latent factors can help in dealing with the curse of dimensionality and for interpreting the dependencies

in the data. Compared to CorEx, the main goal of RBIG

is to convert any non-Gaussian distribution data into a

Gaussian distribution through marginal Gaussization and rotation to get TC. The Matrix-based Rényi’s entropy

mainly used for estimating multivariate information based on Shannon’s entropy, which is Rényi’s

α−

order entropy [40].

With these goals in mind, we now describe a particular latent factor approach known as Total Cor-relation Ex-planation

(CorEx4) [32].

CorEx constructs a factor model by reconstructing latent factors using a factorized probabilistic function of the input

data,

p(y|x) = Qm

j=1 p(yj|x)

, with

discrete latent factors,

. This function is optimized to give the tightest lower

bound possible for Eq. 5.

T C(X)≥max

p(Yj|x)

i=1

I(Xi;Y)−I(X;Y) =

j=1 n

i=1

αi,j I(Xi;Yj)−I(Yj;X)!(6)

The factorization of the latent leads to the terms

I(X;Y) = PjI(Yj;X)

which can be directly calculated. The term

I(Xi;Y)

is still intractable and is decomposed using the chain rule into

I(Xi;Y)≈Pαi,j I(Xi;Yj)

. Each

I(Xi;Yj)

can then be tractably estimated [32,33]. There are free parameters

αi,j

that must be updated while searching for latent

factors and achieving objective functions. When t= 0, the αi,j initializes and then updates according to:

αt+1

i,j = (1 −λ)αt

i,j +λα∗∗

i,j (7)

The second term

α∗∗

i,j = exp (γ(I(Xi:Yj)−maxjI(Xi:Yj)))

and

are constant parameters. This decompo-

sition allows us to quantify the contribution to the total correlation bound from each latent factor, which we can aid

interpretability.

2https://isp.uv.es/RBIG4IT.htm

3http://www.cnel.uﬂ.edu/people/people.php?name=shujian

4https://github.com/gregversteeg/CorEx

November 15, 2022

CorEx can be further extended into a hierarchy of latent factors [33], helping to reveal hierarchical structure that we

expect to play an important role in the brain. The latent factors at layer

explain dependence of the variables in the

layer below.

T C(X)≥

k=1 



j=1 



i=1

αk

i,j I(Yk−1

i;Yk

j)−

j=1

I(Yk

j;Yk−1)



(8)

Here

gives the layer and

Y0≡X

denotes the observed variables. Ultimately, we have a bound on TC that gets tighter

as we add more latent factors and layers and for which we can quantify the contribution for each factor to the bound.

We exploit this decomposition for interpretability [41] as illustrated in Fig. 2. CorEx prefers to ﬁnd modular or tree-like

latent factor models which are beneﬁcial for dealing with the curse of dimensionality [42]. For neuroimaging, we

expect this modular decomposition to be effective because functional specialization in the brain are often associated

with spatially localized regions. We explore this hypothesis in the experiments.

Figure 2:

CorEx learns a hierarchical latent factor as illustrated above.

Edge thickness indicates strength of

relationship between factors, and node thickness indicates how much total correlation is explained by each latent factor.

3 Experiment 1: Total Correlation for independent mixtures

In this experiment, we estimate the total correlation of three independent variables

and

, and each follows a

Gaussian distribution. For this setup, the ground truth of TC should satisfy

T C(X, Y, Z)=0

, and generated various

samples with different lengths. Then estimated total correlation values are shown in the Fig 3. Here, we compared

CorEx with other different total correlation estimators, such as, RBIG [28,29], Matrix-based Rényi’s entropy [39],

Shannon discrete entropy

, and ground truth. The left ﬁgure (2 dimensional) is mutual information, and the middle

(3 dimensional) and right ﬁgure (4 dimensional) are total correlation. As we mentioned above, the simulation data is

totally Gaussian distributed. Therefore, their dependency should be zero. We ﬁnd that CorEx and RBIG both perform

very well and are very stable, and matrix-based Renyi entropy performance becomes more and more nice with increased

dimensions, while Shannon discrete entropy becomes more and more accurate with an increase of samples. All these

make sense, and it also explains the accuracy of total correlation estimation with CorEx. Here, compared to other

estimators, the main functionality goal of CorEx is to cluster statistical dependency variables based on total correlation.

However, other estimators mainly focus on directly getting the total correlation value and do not supply very nice

visualization results. The CorEx gives us a nice connection with graph theory to visualize and show their functional

relationship.

4 Experiment 2: Clustering by Total Correlation for dependent and independent mixtures

To evaluate the performance of CorEx in clustering tasks. The elements in group

include

, and

, which

satisfy Gaussian distributions and are completely independent from each other and from group

, and variables in

5https://github.com/nmtimme/Neuroscience-Information-Theory-Toolbox

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FUNCTIONALCONNECTOMEOFTHEHUMANBRAINWITHTOTALCORRELATIONQiangLiImageProcessingLaboratoryUniversityofValenciaValencia,46980qiang.li@uv.esGregVerSteegInformationSciencesInstituteUniversityofSouthernCaliforniaMarinadelRey,CA90292gregv@isi.eduShujianYuMachineLearningGroupUiT-TheArcticUniversityofNorway90...

展开>> 收起<<

FUNCTIONAL CONNECTOME OF THE HUMAN BRAIN WITH TOTAL CORRELATION Qiang Li.pdf

共21页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

FUNCTIONAL CONNECTOME OF THE HUMAN BRAIN WITH TOTAL CORRELATION Qiang Li

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: