Simulating structural connectivity changes based on functional connectivity patterns in schizophrenia

Simulating structural connectivity changes based on functional

connectivity patterns in schizophrenia

Thesis MSc Computational Science

November 1, 2019

Author:

Lotte van der Wilt

Daily supervisor:

Yongbin Wei

Assessor:

Dr. Martijn P. van den Heuvel

Examinator:

Dr. Rick Quax

Abstract

Schizophrenia is a severe mental health disorder that is hypothesised to originate from

altered neural connectivity patterns. Much is still unknown about the mechanisms

un-derlying schizophrenia, and the link between structural connectivity (SC) and functional

connectivity (FC) changes.

In this study, the schizophrenic brain is simulated using

computational modelling techniques. Structural and functional networks of 74 healthy

individuals and 52 schizophrenia patients are reconstructed based on DTI and fMRI data.

The Spatial Auto-Regression (SAR) model is implemented to predict FC based on SC,

after which an Evolutionary Algorithm (EA) is implemented to evolve the SC of healthy

control individuals to a simulated SC, of which the SAR-predicted FC resembles the real

FC of schizophrenia patients. The resulting simulated structural networks of patients

showed an increase in strength, clustering coefficient and local efficiency, and a decrease

in mean shortest path length compared to that of controls. The effects were most

promi-nent in the frontal and temporal regions. Furthermore, the rich club nodes showed a

relative decrease in strength, betweenness centrality, clustering coefficient and local

effi-ciency and a decrease in mean path length in simulated patient SC, suggesting a decrease

in the central role of rich club regions in schizophrenia. Finally, the results suggested a

decrease in SC-FC coupling. EAs may provide a valuable method to gain more insight

into the structural and functional connectivity alterations underlying the disease, and in

understanding how the disease progresses over time.

Table of contents

1

Introduction

7

Box 1: Functional and Structural connectivity . . . .

9

Box 2: Graph Theory . . . .

10

2

Methods

11

2.1

Subjects . . . .

11

2.2

Imaging data . . . .

11

2.2.1

Structural connectivity

. . . .

11

2.2.2

Functional connectivity . . . .

11

2.3

Experimental setup . . . .

12

2.4

Spatial Auto-Regression (SAR) model . . . .

13

2.5

Evolutionary algorithm

. . . .

13

2.5.1

Initial population

. . . .

14

2.5.2

Parent selection . . . .

14

2.5.3

Offspring creation . . . .

15

2.5.4

Termination conditions . . . .

15

2.6

Parameter tuning . . . .

16

2.6.1

Simulated Annealing (SA)

. . . .

16

2.7

Experimental design . . . .

16

2.8

Data analysis

. . . .

17

2.8.1

Real structural connectivity . . . .

17

2.8.2

Simulated structural connectivity . . . .

17

3

Results

19

3.1

Real structural connectivity . . . .

19

3.2

Simulated structural connectivity . . . .

19

3.2.1

Fitness . . . .

19

3.2.2

Simulated SC . . . .

19

3.2.3

Rich club

. . . .

21

3.2.4

Atrophy . . . .

22

4

Discussion

23

Acknowledgements

25

References

27

Appendix A: Parameter tuning

31

Appendix B: Simulated SC thresholds

33

Appendix C: Data tables

37

1

Introduction

Schizophrenia is a chronic, severe mental health disorder, characterised by abnormal behaviour and

thought patterns, loss of affect, hallucinations and delusions. The disease affects roughly one in 200

people with significant impact on patients’ quality of life. Schizophrenia has been argued to be a

dis-ease of dysconnectivity since Carl Wernicke (1848-1905) first proposed his ‘sejunction hypothesis’ that

the loosening of association fibres in the brain, which he called sejunction, was the underlying cause

of all psychotic disorders [1][2]. Eugen Bleuler (1857-1939), who named the disease “schizophrenia”

(referring to the splitting (skhizein) of the mind (phren) [3]), also spoke of a ‘loosening of associations’

and hypothesised that the symptoms were caused by disturbances in integration within the brain [4][5].

Their ideas not only strongly influenced the way we currently view schizophrenia, but also impacted the

field of psychiatry as a whole [6]. It was not until the 1970s that modern brain imaging techniques

started to arise.

Brain imaging studies have indeed shown altered brain connectivity patterns in schizophrenia patients.

Brain connectivity can be defined as functional connectivity (FC), showing the synchronisation in brain

activity, or structural connectivity (SC), which represents the anatomical connections in the brain. For

a detailed explanation of FC and SC, see Box 1. Both functional and structural connectivity have been

found to be affected in schizophrenia patients [4]. The region that shows the most consistent decrease in

functional connectivity is the frontal cortex, specifically, the prefrontal cortex [7][8]. The frontal-temporal

connections were most affected, followed by the frontoparietal links [8]. Nonetheless, a few studies have

shown increases in functional connectivity [9][10][11]. The exact cause of such contradictory results is

unknown, but methodological differences and symptomatic variations may play a role [4]. Similar to

the functional alterations, the frontal and temporal lobe show the most stable decrease in structural

connectivity when the fractional anisotropy is measured [12][13]. A study using white matter

tractogra-phy shows the most significant alterations in the white matter tracts attached to frontal cortex as well,

including areas with both increased and decreased SC [14]. Further evidence of the dysconnectivity in

schizophrenia were given by studies in the field of connectomics, where the brain is studied as a network

[15], and network organisation and integration are focused on rather than the function of specific brain

areas or single interareal connectivity.

A brain network consists of nodes and edges. Depending on the scale of the network, network nodes

can represent single neurons or entire brain areas, whereas the edges can stand for connections, for

example, the synapses connecting two neurons or white matter tracts connecting different brain areas.

More information on how these networks are analysed using graph-theoretical analysis can be found

in Box 2. Connectome studies have presented evidence of decreased network efficiency [16] and less

centralised frontal and parietal hubs in schizophrenia patients [17]. The impaired rich club organization,

which describes a group of densely interconnected hub regions with many long-range connections that

play an important role in whole-brain communication and integration, was also observed in

schizophre-nia [18][7][19]. A recent research involving medication-na¨ıve first-episode schizophreschizophre-nia patients already

showed altered rich-club connectivity, suggesting this pattern is present at an early stage and is not

mediated by medication [20]. Furthermore, the medication-naive patients showed a decrease in SC-FC

coupling, whereas both increased [7] and decreased [21] SC-FC coupling have been found in chronic

patients.

Much is still unclear about the mechanisms underlying the whole course of schizophrenia, and the

coherence between structural and functional connectivity changes. One of the methods that allows us

to gain new insights into disease-related connectivity changes, as well as the connectivity pattern of the

healthy brain, is computational modelling. Various computational models have been applied in attempts

to predict FC based on the SC. A model that stands out in its simplicity, as well as its accuracy, is

the Spatial Auto-Regression (SAR) model. Although a multitude of more complex models has been

developed, the SAR model tends to (nearly) parallel them in accuracy while the computational costs

are significantly lower [22][23]. It is important to note that no computational model has been able to

explain all the variance contained in real FC matrices, but the models can explain more of the variation

than SC alone. A different computational model that has not been used for the prediction of FC, but

has shown promising results in a wide range of scientific fields, is the Evolutionary Algorithm (EA) [24].

EAs are optimisation algorithms using techniques inspired by genetics and evolution.

In this study, we apply the SAR model in an attempt to simulate the schizophrenic brain. By

do-ing so, we aim to gain new insights into the development of brain network changes in schizophrenia

patients and the coupling between structural and functional alterations. First, structural and functional

networks of 74 healthy individuals and 52 schizophrenia patients are reconstructed based on DTI and

fMRI data. Then, the SAR model is implemented to predict the functional connectivity matrix based

on structure. An Evolutionary Algorithm (EA) is applied to evolve the structural connectivity - starting

with real structural network of healthy participants - towards a simulated SC that results in a predicted

functional connectivity matrix matching the average FC of schizophrenia patients. Finally, the changes

in structural connectivity over time as well as in the final population will be analysed using statistical

methods and graph-theoretical measures. Our study may provide novel insights into neural mechanisms

underlying schizophrenia and prevention and intervention methods.

1: Box 1: Functional and Structural connectivity

Brain connectivity can be estimated via multi-modal neuroimaging techniques. Although multiple measurements

are available, here we focus on the MRI methods that are most often used: Diffusion Tensor Imaging (DTI) to

define the structural connectivity, and functional MRI (fMRI) to define the functional connectivity.

Diffusion Tensor Imaging. DTI is a form of MRI that measures the diffusion of water molecules. In an open

space, the diffusion of water molecules is random, and therefore uniform in all directions.

Some structures

prevent this random movement of water molecules, allowing us to visualise their shape. Axons, the neuronal

structures forming the white matter tracts that connect the different brain areas, are covered in myelin sheaths.

The myelin provides a sort of insulation, playing an essential part in the transmission of signals to other cells

[25]. Myelin sheaths prevent water molecules from freely diffusing through the brain. The stronger the myelin

around the axons, the more restricted particles are in diffusion.

The DTI scan measures the diffusion of water molecules in three dimensions, resulting in a diffusion ellipsoid,

or tensor, for each voxel. Then, the fibres are traced to see which regions are connected, how many streamlines

connect the areas (number of streamlines: NOS), as well as how elongated the ellipsoids are on average (fractional

anisotropy: FA). This elongation of the diffusion ellipsoid, known as fractional anisotropy, shows how strongly

myelinated the fibres are, and is often used as a measure of white matter integrity. The measures mentioned above

can be used as the connection weight in the structural connectivity graph. One disadvantage of this technique is

that we cannot trace the direction of the axon, and therefore, the data is undirectional.

Figure 1: Displays the principles of Diffusion Tensor Imaging (DTI) [26]. A. When diffusion is entirely unrestrained,

diffusion in all directions will be equal: isotropic diffusion. B. In white matter tracts, diffusion is restricted, leading

to anisotropic diffusion tensors. C. By tracing the diffusion tensors in three-dimensional space, the white matter

structure can be derived.

Functional Magnetic Resonance Imaging. A popular imaging method to study the functional connectivity in

the brain is fMRI. In fMRI studies, blood-oxygen-level-dependent (BOLD) imaging is used. The BOLD signal

represents the relative percentages of oxyhemoglobin and deoxyhemoglobin in the blood, showing the ratio of

oxygenated blood compared to deoxygenated blood. As such, BOLD imaging is used as a measure of brain

activity. When a specific area is active, more oxygen is extracted from the blood, resulting in changes in the

local oxygenation level over time.

When the term functional connectivity is used, this usually refers to the

synchronisation in the BOLD signal between different areas in resting-state. The correlation coefficient between

BOLD signals is used as the connectivity weight.

SC-FC coupling. Nearly all studies that investigated the relationship between SC and FC show a significant

correlation between these two ends of connectivity [27]. However, functional connectivity differs over time, as

well as between tasks [28]. Structural connectivity, on the other hand, is much more stationary. Also, functional

connectivity frequently presents itself in the absence of a direct anatomical connection [29][30], which can be at

least partially the result of indirect structural connectivity [28]. In addition, current DTI resolutions might lead to

an underestimation of SC in areas where many fibres cross [22]. A study that investigated the relationship between

structural connectivity and variability in eight functional connectivity measurements taken over the course of one

day, found that the SC matrix limits the same-day variability as well as the strength in FC [31]. Furthermore,

characteristics of individual FC patterns have been shown to be unique and stable over multiple years [32]. Hence,

although the FC is variable and there is no one-to-one SC-FC coupling, both SC and FC are thought to contain

highly valuable information.

2: Box 2: Graph Theory

Graph theory is the mathematical study of network graphs, often used to gain insight into large, complex systems

and their organisational properties. Graph theory is often involved in studying social networks, transportation and

road networks, and even molecular systems. Networks can be classified into different types, namely undirected

and directed, and binary and weighted networks. In an undirected network, edges are bidirectional. When a

network includes one-way connections, the network is directional. Binary networks only differentiate between

connected and unconnected nodes, whereas weighted networks allow for the discrimination between strong and

weak connections.

Figure 2: Displays some of the different network types (A-C), and illustrates various graph-theoretical metrics

(D-F, in blue). A. Binary undirected matrix, B. Binary directed matrix, C. Weighted undirected matrix, D. Shows

a hub with high degree, E. Maximum shortest path between two nodes; eccentricity, F. High clustering.

Graph theory provides us with some general measures to describe networks. The network density represents the

number of connections that are present, relative to the number of connections the graph would have if it were

fully connected. The clustering coefficient is the likelihood that any nodes’ neighbours are interconnected. We

can also calculate the shortest path between any two nodes in the network. This metric is often used as a measure

of network efficiency. The maximum shortest path length for a specific node is called its eccentricity. Highly

related to the path length, are the local- and global efficiency measures, representing the inverse mean shortest

path length of a node’s nearest neighbours and of the whole network, respectively. The betweenness centrality of

a node signifies how often a node is present on the shortest path between any two other nodes. Nodes that have a

high degree - meaning they have many connections – and also have a central position in the network are known as

hubs. When these hubs are more densely interconnected than would be expected based on chance, we speak of a

rich club organisation [33]. In a weighted network, the sum of all the weights of a node is called the node strength.

Graph-theoretical approaches are often used in connectomics research to analyse brain networks. Network measures

suggest that brain networks are shaped by the trade-off between wiring costs and network efficiency [34]. Brain

wiring is expensive. In an area with significant spatial constraints, connections require physical space as well

as energy. The longer the fibres, and the higher the density, the higher the wiring costs are [35]. Therefore,

efficient use of connections is critical to the functioning of the network. Some costly, inter-modular, long-range

connections are required to pass down information efficiently, to increase conduction speed and enable efficient

integration. This efficiency is achieved by a rich club organisation. The rich club provides a backbone for efficient

functional integration [36].

2

Methods

2.1

Subjects

In this study, imaging data from 52 schizophrenia patients (11 females, 41 males, age µ ± SD =

38.17 ± 14.13) and 74 healthy control individuals (20 females, 54 males, age µ ± SD = 37.88 ± 11.91)

between the ages of 18 and 65 were included. The study was approved by the Institutional Review Board

of the University of New Mexico, and all participants provided written informed consent. Individuals

with a history of neurological disorder or severe head trauma or with a history of substance abuse within

the past 12 months were excluded.

2.2

Imaging data

The imaging data that was used in this study originates from the Center for Biomedical Research

Excel-lence (COBRE) [37], and was downloaded from the SchizConnect database (http//schizconnect.org).

The investigators within SchizConnect contributed to the design and implementation of SchizConnect

and/or provided data but did not participate in analysis or writing of this report. All data was recorded

using 3 Tesla Siemens Magnetom Trio scanner (Siemens, Erlangen, Germany).

2.2.1

Structural connectivity

Data aqcuisition

A Diffusion Weighted Imaging protocol was used to record structural connectivity data. A five-echo

MPRAGE T1-weighted image was used, with the following parameters: TR = 2539, TE = [1.64, 3.5,

5.36, 7.22, 9.08ms], TI = 900ms, flip angle = 7

◦

, 1mm isotropic voxel size, FOV = 256x256mm, slab

thickness = 176mm. Additionally, a DWI set was measured including 30 diffusion weighted volumes

(b-value = 800 1000 s/mm2) and 5 diffusion-unweighted volumes equally inter-spread between the 30

diffusion weighted volumes (parameters: TE = 84 ms, 2 mm isotropic voxel size). FLIRT (FMRIB’s

Linear Image Registration Tool) was used for all registration steps [38].

Data preprocessing

The T1-weighted images were processed using FreeSurfer (http://surfer.nmr.mgh.harvard.edu/) [39].

Images were corrected for eddy-current, realigned to the b=0 image, and automatically segmented into

68 cortical regions based on a subdivision of the Desikan-Killiany atlas (DK-68) [40]. Affine

transfor-mation mapping was used to map the individual T1-weighted image to the DWI image, in order to

co-register the parcellation maps. Then, a robust tensor fitting algorithm [41] was applied to fit tensors

to the diffusion signals measured in each voxel, after which FA was calculated [42]. Fiber Assignment

by Continuous Tracking (FACT) [43] was used to reconstruct DTI white matter tracts. Streamline

reconstruction started from eight seeds in every white matter voxel. Streamlines continued unless they

showed high curvature (> 45

◦

), entered a voxel with an FA below 0.1, or exited the brain mask.

Network reconstruction

The streamline reconstructions and parcellation of the cortical structures were integrated for each subject,

resulting in an individual weighted network. If one or more streamlines were reconstructed touching both

regions, the regions were considered to be connected. The number of streamlines between the regions

was used as the edge weight. This process resulted in a 68x68 connectivity matrix for each subject,

showing the connection weight between each combination of nodes.

2.2.2

Functional connectivity

Data aqcuisition

Resting state fMRI data was collected using the following parameters: 33 axial slices, TR = 2000 ms,

TE = 29 ms, flip angle = 75

◦

_{, slice thickness = 3.5 mm, slice gap = 1.05 mm, acquisition matrix = 64}

150 volumes of functional images were obtained. Subjects were asked to keep their eyes open and stare

at a fixation cross.

Data preprocessing

Like the aforementioned preprocessing of the structural imaging data, T1-weighted images were

pro-cessed using FreeSurfer (http://surfer.nmr.mgh.harvard.edu/) [39], and automatically segmented into

68 cortical regions based on a subdivision of the Desikan-Killiany atlas (DK-68) [40]. The resting-state

fMRI time-series were then realigned with the T1 image, and parcellation maps were co-registered.

Lin-ear trends and first order drifts were removed from the time-series, and time-series were corrected for

global effects (regressing out the white matter, ventricle, and global mean signals, as well as 6 motion

parameters) and band-pass filtered (0.01 - 0.1 Hz).

Network reconstruction

For each node, the fMRI voxels overlapping with the region were selected and the time-series of the

selected voxels were averaged. Then, for each combination of nodes, the average regional time-series

were correlated. The Pearson’s correlation coefficient was used as the network weight. Again, this

resulted in a 68x68 connectivity matrix for each subject, showing the connection weight between each

combination of nodes.

2.3

Experimental setup

The experimental design incorporated two main parts, as shown in Figure 3. First, we implemented

the SAR model to predict functional connectivity based on structural connectivity. The model allows

us to compare simulated functional connectivity (FCsim) to the real, measured FC. Then, we applied

an evolutionary algorithm to evolve the input of the SAR model - the real, individual control SC –

towards an SC matrix that creates a simulated FC matching the real FC. Both steps involved parameters

that required tuning (Figure 3). Once the optimal parameter values were found, the model was run

repeatedly, and the resulting simulated structural connectivity matrices were analysed.

2.4

Spatial Auto-Regression (SAR) model

In the SAR model, the activity of a node can be calculated from a linear combination of its SC matrix,

the neighbours’ activity, an auto-regression parameter (k) that determines the synchronisation strength,

and random noise (Equation 1). Although in some situations it can be valuable to simulate activity over

time, in this study we are merely interested in the FC matrix, which can be calculated directly using

Equation 2 and 3. In the equations below, k represents an auto-regression parameter, determining the

synchronisation strength, σ is the noise level, v

i

represents uncorrelated white Gaussian noise with mean

0 and unit variance. D is the structural connectivity matrix, and I is the identity matrix. Finally,

t

_is

the matrix transposition. The σ parameter was set to 1, based on earlier research [22][23]. The input

of the SAR model needs to be normalised. In this study row normalisation was applied, meaning that

each value in the matrix was divided by the sum of its row.

y

i

= k

X

j6=i

D

i,j

y

j

+ σv

i

(1)

cov = σ

2

(I − kD)

−1

(I − kD)

−t

(2)

cor

ij

=

cov

ij

√

cov

ii

cov

jj

(3)

2.5

Evolutionary algorithm

The goal of the Evolutionary Algorithm in this study is to evolve the structural connectivity matrix,

the input of the SAR model, towards a simulated structural connectivity matrix that leads to simulated

functional connectivity resembling the real FC of the schizophrenia group.

The idea behind EAs originates from the biological and genetical basis of evolution. The algorithm aims

to find the optimal solution to a problem. Usually, this concerns either the input or the parameters of a

model, while the desired output is known. In this case, the input of the SAR model will be evolved. A

possible set of inputs for the model is known as a genotype, and the resulting model output is called its

phenotype. Each separate input is known as an allele. The model iterates through generations (Figure

4). In each generation, the fitness of all individuals is tested, and parents are selected based on their

fitness values. Offspring are created through crossover, or recombination, and mutation of genotypes.

The new individuals replace the previous population, and this process repeats itself until the termination

conditions are met.

Figure 4: Main components of an evolutionary algorithm.

In this report, the genotypes were the individual SC matrices, and the resulting simulated FC matrices

were their genotypes. Since the connectivity matrices are symmetrical, only the upper triangular matrix

was used, which was represented as an array of floating-point numbers. As a measure of fitness, the

Pearson correlation coefficient between the individual simulated FC (F C

i

) and the population mean

(F C

real

) was calculated. The functional connectivity matrices were normalised before calculating the

mean F C

real

, using the z-score. Since it is customary to minimise the fitness in an EA, where the optimal

value is close to zero, this correlation coefficient was subtracted from 1 to represent the individual fitness

score (Equation 4).

F itness

i

= 1 −

Cov(F C

i

, F C

real

)

σ

F C

σ

F C

(4)

2.5.1

Initial population

The real individual structural connectivity matrices were used to create the initial population. This

population was rounded out to the intended number by adding recombinations of random individuals.

2.5.2

Parent selection

The parents were selected using rank-based stochastic uniform sampling (SUS). First, the individuals

were ranked based on fitness, and the rank was used to calculate a scaled fitness value. The scaled fitness

then depended on rank rather than the exact score (Equation 6). This method ensures that selection

is stronger among high-fitness individuals. The difference in scaled fitness is much more significant

between the top individuals than for the lower-ranked individuals. From this scaled fitness, a selection

probability can be calculated using Equation 6.

F itness

scaled

=

1

√

rank

i

(5)

P (i) =

F itness

scaled

P

N

j=1

F itness

scaled

(6)

In SUS, the parents are chosen based on their respective probabilities. All parents are chosen at once,

using only one random number. This method ensures that, although there is a random effect, the

parents composing the next generation are a good representation of the scaled fitness distribution.

Once all selection probabilities are calculated, a parent sample can be picked from the population. A

cumulative probability distribution is created. The parents are selected at equal distances (d =

_N

1

p

) from

the cumulative probability distribution, where N

p

is the number of parents. The distance d is multiplied

by a uniform random number U [0, 1] to choose the first parent. Figure 5 and Table 1 illustrate this

principle for an example population with six individuals, with probabilities P

i

, as shown in the table.

Even with these small sample sizes, the samples roughly correspond to the scaled fitness distribution in

the population.

Figure 5: Stochastic Universal Sampling. On top, the cumulative

proba-bility distribution is shown. Based on a random uniform number, the first

position is chosen. Consequently, 12 random samples are drawn.

Table 1: Shows the probabilities used

in Figure 5, as well as the resulting

sam-ple rates.

Rank

F

P

Sample

1

1

∼0.27

0.3

2

∼0.71

∼0.19

0.2

3

∼0.58

∼0.16

0.2

4

0.5

∼0.14

0.2

5

∼0.45

∼0.12

0.1

6

∼0.41

∼0.11

0.2

2.5.3

Offspring creation

Offspring were created through three different methods: elitism, mutation and crossover.

First, the elitism parameter determined the amount of best individuals that were directly copied to the

next generation. Elitism is meant to ensure that the fit-test individual in the population is always kept.

The remaining population was created through either recombination or mutation. The percentage of the

population that was created by a crossover in proportion to mutation was determined by a parameter

called crossoverF raction.

Secondly, a uniform mutation algorithm was used. A random sample of edges was chosen that were

mutated. This sample was created based on the mutationRate: the probability that each of the alleles

is mutated. Each of the chosen edges was replaced by a random uniform number between 0 and the

individual’s maximum value.

Finally, intermediate crossover was implemented as a recombination method (Equation 7). An essential

parameter in this formula is ratio. The ratio determines the range around the parent values that a child

can adopt. When the ratio is 1, the child’s connection weight always lies between the parents’ weights.

When the ratio is 1.5, the probabilities of the child’s weight falling within or outside the parents’ range

are equal(Figure 6). A ratio within this range ([1,1.5]) is frequently used since this allows exploration

of new possible solutions.

child = p

1

+ U [0, 1] ∗ ratio ∗ (p

2

− p

1

)

(7)

Figure 6: Intermediate crossover.

2.5.4

Termination conditions

Evolutionary Algorithm runs were terminated when there had been no improvement for 100 generations,

or the maximum amount of generations (maxGenerations) was reached.

Table 2: Evolutionary Algorithm parameter tuning settings. Shows the initial values as well as the bounds of the search

space.

Initial value

Lower bound

Upper bound

grid search

populationSize

-

75

150

maxGenerations

-

100

600

eliteCount

-

1

5

SA

mutationRate

0.01

0.0005

0.02

crossoverF raction

0.8

0.6

1.0

2.6

Parameter tuning

As previously discussed, two algorithms used in this study required parameter tuning: the SAR model

and the Evolutionary Algorithm. First, the SAR parameter was optimised to maximise the correlation

between the average individual SC and FC. As there was only one parameter, a simple grid search was

performed. Initial broad tuning revealed the optimal value should be between 0.9 and 0.99. Within

that range, all values were checked with a step-size of 0.005. The Evolutionary Algorithm, on the other

hand, provided six parameters to be tuned: the population size, the maximum amount of generations

and the number of elite individuals, and also the crossover fraction, mutation rate and ratio. The first

three, all categorical, were explored using a grid search. The last three parameters were tuned using

a Simulated Annealing algorithm. Table 2 shows the different parameters and their respective ranges.

Final parameter settings and SA results are shown in Appendix A.

2.6.1

Simulated Annealing (SA)

A simulated annealing algorithm was implemented to find the optimal parameter settings for the

evolu-tionary algorithm. Simulated annealing is a method inspired by the process of annealing: the controlled

heating and cooling of metal to create the optimal density and strength. The algorithm is mainly used

to find approximate global optima in vast, multidimensional search spaces. Considering the running time

of the EA and the number of continuous parameters to be tuned, this specific algorithm was chosen.

Simulated annealing is an iterative algorithm, starting with an initial set of parameter values. At each

iteration, the fitness value is calculated. The fitness, in this case, is the best fitness reached in an

Evolu-tionary Algorithm run with that set of parameters. Then, a new set of parameter values is chosen from

a ‘neighbouring’ range. The size of this range is determined by the temperature (T ). At each iteration,

a step of length T is taken in a random uniform direction. If this new set of parameters has a better

fitness than the previous solution, it is selected as the current solution. If the fitness is low, there is

still a small probability the parameter set is accepted (Equation 8). The lower the temperature, and the

more substantial the decrease in fitness, the smaller the likelihood of acceptance. The selection of lower

fitness values, albeit with a lower probability, prevents the model from getting stuck in local optima.

The temperature gradually decreases for a set amount of generations (reannealInterval), after which

it ‘re-anneals’: the temperature is reset to a higher value. The temperature, aside from re-annealing

steps, is determined by Equation 9. In this equation, g is the number of generations since (re-)annealing.

The initial parameter settings, as well as the possible ranges that were used, are shown in Table 2.

P

acceptance

=

1

1 − exp(

_{max(T )}

∆

)

(8)

T = T

0

∗ 0.95

g

(9)

2.7

Experimental design

The Evolutionary Algorithm was used in 4 different conditions (Table 3). This study aimed to evolve

the controls’ individual SC matrices towards an SC that allowed the predicted FC to match the real

patients’ FC. This was the experimental condition. Moreover, three control conditions were used. Most

importantly, the individual control SC was evolved towards the control functional connectivity, to be able

to differentiate between changes that emerged from optimally fitting to the SAR model and changes

that directly stemmed from the schizophrenia related alterations in patient FC. Then, both of these

experiments were repeated starting with the individual patient SC instead of the control SC.

For each of these conditions, the EA was executed 100 times. The simulated SC and fitness of all

individuals were collected for each generation in every run.

Table 3: Shows the different test conditions that were used to run the Evolutionary Algorithm.

Initial population

Reference FC

Experimental condition

Control SC

Patient FC

Control condition 1

Control SC

Control FC

Control condition 2

Patient SC

Patient FC

Control condition 3

Patient SC

Control FC

2.8

Data analysis

2.8.1

Real structural connectivity

The real structural connectivity matrices of the control and patient group were compared, based on

a set of graph theoretical metrics: the strength, mean shortest path length, betweenness centrality,

clustering coefficient, and local- and global efficiency. Differences between SC of patients and controls

were compared using t-tests, with False Discovery Rate (FDR) correction.

Graph theoretical measures

For each node i, the strength was calculated: the sum of weights of all edges connecting to node i.

Furthermore, the shortest path between node i and every other node was measured. The betweenness

centrality represents the amount of times node i is present on the shortest path between any two other

nodes in the network. The clustering coefficient, which describes the probability that any two neighbours

of node i are interconnected, was calculated by dividing the number of edges in the neighbourhood of

node i by the maximum possible amount of edges (

_d

1

i

(d

−1)

, where d

i

is the degree of node i). Finally,

the local efficiency is the average inverse shortest path length between the neighbours of node i. The

global efficiency is calculated in the same manner, but in this case the entire network is defined as the

neighbourhood.

2.8.2

Simulated structural connectivity

Fitness

Fitness values over time were compared between all experimental conditions (Table 3).

Simulated SC

Before analysis, a threshold was applied to the EA output matrix, as the simulated network tends to

become fully connected over time. The threshold percentages were based on the actual SC matrices

(µ = 0.474, minimum = 0.374, maximum = 0.563). Density thresholds of 37%, 47% and 57% were

chosen in order to keep equal distances between thresholds. After thresholds were applied, the simulated

SC was analysed using the graph theoretical measures described in Section 2.8.1. Differences between

simulated SC of patients and controls were compared using t-tests, with False Discovery Rate (FDR)

correction.

Rich club

For each individual, the mean value for each graph theory measure was calculated. Then, the ratio of

the rich club value compared to the local (non-rich club) value was calculated (Equation 10). This ratio

was compared between the two groups using a t-test for each measure. The rich club region was defined

based on earlier research [33] and included the bilateral superior frontal gyrus, superior parietal gyrus,

precuneus and precentral gyrus.

richRatio =

µ

rich

µ

local

Atrophy

Finally, the graph theory findings were compared to the atrophy pattern associated with schizophrenia,

using Pearson’s correlation.

Cortical involvement in schizophrenia was assessed using the BrainMap database (http://www.brainmap.org)

[44]. The Sleuth toolbox was utilised to extract schizophrenia Voxel-Based Morphometry (VBM) and

meta-analyses were conducted using the GingerALE toolbox. This resulted in brain maps of Activation

Likelihood Estimation (ALE). Finally, the regional ALE was calculated by taking the mean of all voxels

within each region of the DK-68 atlas [40] using FreeSurfer. The z-score was calculated for each regional

averaged ALE. The average z-score was used in the cortical involvement map for schizophrenia.

3

Results

3.1

Real structural connectivity

Graph-theoretical measures of the structural connectivity matrices of schizophrenia patients and healthy

controls were compared. No significant differences in density, betweenness centrality, mean shortest path

length, or local efficiency were found between the groups. The left temporal pole showed a significant

decrease in strength in schizophrenia patients (µ ± SD = 3846 ± 2254) compared to healthy controls

(µ ± SD = 4452 ± 2800, FDR q = 0.011). The structural brain network in schizophrenia showed

a decrease in global efficiency (control µ ± SD = 219 ± 40.26, patient µ ± SD = 202.67 ± 43.01,

P = 0.0232).

3.2

Simulated structural connectivity

3.2.1

Fitness

Figure 7 shows the fitness over time during the simulation of structural connectivity using the EA. EA

runs starting from the healthy control (Figure 7A) and runs starting from the patient SC (Figure 7B)

both show a better fitness when evolving to controls as compared to evolving to patients. This finding

may suggest a disassociation between SC and FC in schizophrenia patients.

A.

B.

Figure 7: Fitness ± SD over time, for runs starting from the control SC (A) as well as the runs starting from the patient

SC (B). The green lines represent simulations evolving towards the patient FC, and blue lines represent those evolving

towards the control FC.

3.2.2

Simulated SC

The best individual of each simulation was selected and the simulated SC of patients and controls were

compared. The simulated SC matrix was thresholded using proportional thresholding, with the top 47%

strongest connectivity (the mean network density in real SC matrices) kept for further analysis. Results

using other thresholds (37% and 57%, the minimum and maximum density across subjects) are shown

in Appendix B.

Increased strength was found throughout the frontal lobe and most of the temporal lobe in the simulated

structural networks of patients (Figure 8A), with the largest increases observed in the bilateral rostral

anterior cingulate gyrus, entorhinal gyrus, frontal pole. The right supramarginal gyrus, right precentral

gyrus, right superior temporal gyrus, left transverse temporal gyrus and left fusiform gyrus showed

de-creased strength in the simulated structural network of patients.

Figure 8B shows the differences in betweenness centrality that were present between the simulated SC

of schizophrenia and controls. Areas with significantly decreased betweenness centrality in the simulated

schizophrenia SC included the bilateral superior frontal gyrus, precentral gyrus and fusiform gyrus. The

left rostral anterior cingulate cortex and the right pericalcarine cortex, entorhinal cortex and

parahip-pocampal gyrus showed a significantly higher betweenness centrality in the simulated structural networks

of patients.

A.

B.

C.

D.

E.

Figure 8: The final difference in graph theory measures between the runs evolving SC of healthy controls towards the

FC of controls and patients. The controls’ values were subtracted from the patients’ values, so positive values indicate

that the measure in the simulated patients was higher than the simulated control SC. Only the areas that are significantly

different after FDR correction (Q < 0.05) are displayed. A. Strength, B. Betweenness centrality, C. Clustering coefficient,

D. Local efficiency, E. Mean shortest path length.

The between-group differences in clustering coefficient are shown in Figure 8C. In general, a higher

clus-tering coefficient was found in the simulated SC of patients compared to that of controls. The largest

effect was observed in frontal- and temporal areas, such as the bilateral temporal pole and entorhinal

cortex, and the right fusiform gyrus. The bilateral transverse temporal cortex showed a decreased

clus-tering coefficient in the simulated structural network of patients compared to controls.

The local efficiency showed a very similar pattern to the clustering coefficients (Figure 8D). The largest

increase in local efficiency were found in the temporal pole and entorhinal cortex. The bilateral frontal

pole and inferior temporal cortex also showed an increased local efficiency in the simulated patient SC

compared to the simulated control SC.

Figure 8E shows the difference in mean shortest path length between the simulated structural networks

of patients and controls. An overall decreased mean shortest path was found, with the most significant

decreases at the bilateral rostral anterior cingulate cortex and frontal pole, and the left banks of the

superior temporal sulcus.

The group mean and standard deviation as well as P- and Q-values of the results above are shown in

Appendix C.

3.2.3

Rich club

The ratio of the above-mentioned graph theoretical measures between rich-club nodes and non-rich-club

nodes was calculated and shown in Table 4. The rich club region was defined based on earlier findings

[33] and included the bilateral superior frontal gyrus, superior parietal gyrus, precuneus and precentral

gyrus. In the simulated structural networks of schizophrenia patients, the relative rich club strength,

betweenness centrality, clustering coefficient, and local efficiency were decreased as compared to the

simulated structural networks of controls. The relative mean shortest path length of rich club areas was

longer in simulated structural networks of patients.

Table 4: The relative strength, betweenness centrality, mean shortest path length, clustering coefficient, and local efficiency

of the rich club areas compared to the local areas for the two groups.

Measure

µ control

SD control

µ patient

SD patient

P

Strength

1.3839

0.0397

1.2981

0.0337

6.30E-39

Betweenness centrality

1.9784

0.3639

1.6282

0.2866

1.47E-12

Mean shortest path length

0.8944

0.0182

0.9249

0.0192

7.69E-24

Clustering coefficient

1.0789

0.0439

1.0423

0.0406

4.95E-09

Local efficiency

1.0904

0.0366

1.0484

0.032

1.98E-15

3.2.4

Atrophy

Finally, Pearson’s correlation was performed comparing the graph theory findings to the pattern of brain

atrophy in schizophrenia (Figure 9). The atrophy map showed a positive correlation with the differences

in strength of the simulated SC (R=0.23, p=0.0496; Figure 10A), and a negative correlation with the

mean shortest path (R=-0.27, P=0.0214; Figure 10B). No significant correlations between atrophy and

betweenness centrality, clustering or local efficiency were found.

Figure 10: The correlation between schizophrenia-related brain atrophy and the schizophrenia-related alterations

(patients-controls) in strength (A) and mean shortest path length (B), respectively.

4

Discussion

The aim of this study was to simulate the schizophrenic brain using computational modelling techniques.

With the use of the SAR model and evolutionary algorithm, the individual control SC matrices were

evolved towards simulated SC matrices that resulted in simulated FC matrices resembling the real FC

matrix of schizophrenia patients. In general, an increase in strength, clustering coefficient and local

effi-ciency, and a relative decrease in mean shortest path length were found in the simulated SC of patients

compared to that of controls. The betweenness centrality showed a more heterogeneous pattern but

a predominantly decreased betweenness centrality was observed in the frontal cortex of the simulated

structural network of patients. The regions that showed the most significant alterations were the frontal

and temporal lobes.

Comparison of fitness over time showed that, regardless of whether the EA was started from the

con-trol or schizophrenia structural network, runs evolving towards the concon-trol FC showed better fitness

throughout the simulation. These findings may reflect a disassociation between structural and

func-tional connectivity in patients. Earlier research has consistently shown altered SC-FC coupling among

rich club regions in schizophrenia [7][20][21], probably enlarging the differentiation between whole-brain

SC and FC in schizophrenia patients.

Our findings revealed that the rich club showed lower strength, betweenness centrality, clustering

coef-ficient and local efficiency, and higher mean shortest path length in the simulated structural networks

of schizophrenia patients. These results were consistent with earlier studies that analysed the structural

connectivity in schizophrenia and showed reduced rich club connectivity [7][19].The rich club is thought

to play an essential role in the functional integration of the brain [45]. Our results further suggest a

reduction of the central role of rich club areas in the simulated patients’ structural network.

Over all, the simulated structural network in this study shows the most significantly altered connectivity

pattern in the frontal and temporal regions. This is consistent with previous studies, which have

sim-ilarly observed the most consistent changes in those regions [30][13][12]. However, this study showed

increased strength, clustering coefficient and simulated SC, whereas previous research has predominantly

shown decreased connectivity patterns [30][13][12]. Such discrepancies could be attributed to multiple

factors. Primarily, the evolutionary algorithm adapts the SC matrices to predict FC optimally, but in

reality, there is no one-to-one translation between SC and FC. At the end of an EA run, the correlation

between predicted and real FC is very high. As such, the simulated SC may be overfitting to the FC

matrix. Nonetheless, the simulated SC shows the theoretical SC matrix that could explain the functional

connectivity pattern that is observed.

A limitation of this study is that the EA only evolves towards a mean functional connectivity. In reality,

there are substantial individual differences in FC. For instance, studies have shown that specific

symp-toms can correspond with specific dysconnectivity patterns in both structural [12][30] and functional

networks [30][46]. By averaging a large group of patients, such effects may be lost. Therefore, in future

research, separate EA runs could be used to evolve the SC to individual FC patterns. This would allow

studying specific symptomatic patterns and disease stages, and could be used to study whether individual

differences in real SC are replicable in the simulated connectivity pattern.

Another limitation might be the simplicity of the SAR model. The SAR model is a simple, linear model.

As such, it has been argued that the SAR model cannot explain the complex underlying dynamics, and

the non-stationary elements of FC [22][47]. One of the advantages of the current experimental setup is

that the Evolutionary Algorithm can be used in combination with any other model for FC prediction. In

the future, it might be valuable to see whether results can be replicated with other models. That being

said, the SAR model has been shown to outperform many of the frequently used complex computational

models [22].

To our knowledge, this study is the first attempt to use an Evolutionary Algorithm in the

simula-tion of brain connectivity alterasimula-tions in schizophrenia. The mechanisms underlying the development of

schizophrenia and the link between structural and functional connectivity changes are still largely

un-known. The current study showed promising results, as the model was able to replicate earlier findings

from connectomics research, such as reduced rich club activity and alterations in SC-FC coupling. In

the future, we hope the Evolutionary algorithm could be used to provide insight into the structural

and functional connectivity alterations underlying the disease, and in understanding how the disease

progresses over time.

Acknowledgements

I would like to thank my daily supervisor, Yongbin Wei, who always made time for me. I am very

grateful for his feedback, ideas and encouragement. I would also like to thank Dr. Martijn van den

Heuvel, whose door was always open, for his guidance and enthusiasm. Finally, I’d like to thank all the

other members of the CTG connectomics lab at VU, who were willing to discuss my research and give

me advice whenever I needed it.

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Appendix A: Parameter tuning

Table 5: Parameter values used in this study

Parameter

Final value

Sar model

k

0.9870

Evolutionary algorithm

MaxGenerations

600

PopSize

125

Elitism

4

Ratio

1.1018

CrossoverFraction

0.6012

MutationRate

0.0005

A

B

C

D

Figure 11: Simulated annealing optimisation of three parameters, Ratio, CrossoverF raction and M utationRate. Shows

the tested parameter values and the corresponding fitness. Optimal values found were 1.1018, 0.6012 and 5.02e-4,

respec-tively. A. Fitness of data points in all three dimensions, B. Ratio vs CrossoverF raction, C. Ratio vs M utationRate,

D. CrossoverF raction vs M utationRate.

Appendix B: Simulated SC thresholds

A

B

C

Figure 12: Shows the alterations in strength in the simulated schizophrenia SC compared to the simulated control SC for

different thresholds. Positive values indicate that the strength is higher in the simulated patients. Only the significant

areas after FDR correction are shown (Q < 0.05). A. 37%, B. 47%, C. 57%.

A

B

C

Figure 13: Shows the alterations in clustering coefficient in the simulated schizophrenia SC compared to the simulated

control SC for different thresholds. Positive values indicate that the clustering coefficient is higher in the simulated patients.

Only the significant areas after FDR correction are shown (Q < 0.05). A. 37%, B. 47%, C. 57%.

A

B

C

Figure 14: Shows the alterations in local efficiency in the simulated schizophrenia SC compared to the simulated control

SC for different thresholds. Positive values indicate that the clustering coefficient is higher in the simulated patients. Only

the significant areas after FDR correction are shown (Q < 0.05). A. 37%, B. 47%, C. 57%.

Appendix C: Data tables

In this appendix, the statistical data is shown for each of the tested thresholds, for each of the

graph-theoretical measures.

• Table 5-7: Strength

• Table 8-10: Clustering coefficient

• Table 11: Mean shortest path length

• Table 12: Betweenness centrality

• Table 13-15: Local efficiency