Network Analysis of Calcium Activity

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AVANCERAD NIVÅ, 30 HP STOCKHOLM, SVERIGE 2019

Network Analysis of Calcium Activity

Nätverksanalys av calciumaktivitet HELENA ENGSTRÖM

KTH

SKOLAN FÖR TEKNIKVETENSKAP

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Abstract

Networks exist in very different areas such as transport, social interaction, and biological systems. In multicellular organisms, communication between cells is absolutely essential.

Cells form communication networks that can transmit important information such as neuronal signals and immune responses. In this thesis, we have used network and graph theories to investigate and characterize calcium signaling networks that occur in arterial endothelial cells and brain neurons. Cell activity in endothelial cells was induced by four different agonists: adenosine triphosphate (ATP), adenosine diphosphate (ADP),

acetylcholine (Ach) and histamine (Hist), while the activity of neurons occurs

spontaneously without treatment. Our analysis shows that the network in endothelial cells was of Small-World type for all four agonists whereas the network in neurons were of Small-World type for developmental stage P8 and of Random type for E12.5, E16.5, E18.5 and P6.

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Sammanfattning

Nätverk förekommer inom vitt skilda områden såsom transport, social interaktion och cell biologi. I multicellulära organismer är kommunikation mellan celler absolut nödvändig.

Celler bildar kommunikationsnätverk som kan överföra viktig information såsom neuronsignaler och immunförsvar. I det här mastersarbetet har nät- och grafteorier använts för att undersöka och karakterisera kalciumsignalnätverk som förekommer i arteriella endotelceller och hjärnneuroner. Cellaktivitet i endotelceller inducerades av fyra olika agonister: adenosintrifosfat (ATP), adenosindifosfat (ADP), acetylkolin (Ach) och histamin (Hist), medan aktiviteten hos neuroner sker spontant utan behandling. Analysen visar att nätverket i endotelceller var av Small-World-typ för alla fyra agonister medan nätverken i neuroner var av Small-World-typ för utvecklingssteg P8 och av Random typ för E12.5, E16.5, E18.5 och P6.

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Acknowledgements

I would like to first of all thanks to Professor Per Uhlén for giving me this opportunity and for his continuous support during this thesis project. I would also like to thank Dr. Lauri Louhivuori for his constructive criticism, discussions and encouragement. Thanks to Dr. Erik Smedler for his previous work in the network theory field. Thanks to the Karolinska Institute and the Department of Medical Biochemistry and Biophysics for accepting me as a master thesis student. Thanks to Professor Hjalmar Brismar for agreeing to be my examiner. Also thanks to Royal Institute of Technology for all my acquired knowledge during the years at the engineering physics master’s program and to Professor Ulrich Vogt, teacher, course responsible and examiner for the degree project course, for his help with the practicalities around this thesis project. Last but not least thank you to my family and friends for their continuous support and encouragement.

Helena Engström Stockholm, Sweden June, 2019

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List of Figures

1.1 (A) A Small-World network, (B) Regular network (C) Random network and

(D) Scale-free network. . . 2 1.2 Three graphs where a and b are correlated while c isn’t correlated to

neither a nor b. . . 6 2.1 A visual representation of cleaning up the signal before analysis. (A) Removing

underlying trends caused by for ex. bleaching i.e. trend correction. This is done by fitting the signals to a polynomial function. (B) A specific threshold value is set, where only signals fulfilling the criteria are picked out as active cells to be further analyzed. (C) Signal magnitude is normalized to remove artefacts such as amplitude fluctuations and zero values . . . 9 2.2 Visual representation of the process to scramble the signal. (A) The original data

with start and end point. (B) By cutting the signal at a random time point and put it in opposite order a scrambled signal is created, maintaining the total activity of the signal. The scrambled data can then be used to calculate the cutoff value in an unbiased way . . . 10 2.3 The cylindrical model of an intact artery with the coordinate system as it was

during the analysis. The circumference of the cylinder corresponds to the image size on the x-axis and the length of the cylinder corresponds to the image size on the y-axis. . . 13 2.4 Representation of the recorded cell image where each cell (red circle) has an x

and y-coordinate. The distances between the cells were calculated using the Pythagorean Theorem . . . 14 2.5 Calculating the distances between cells in the cylindrical configuration for the

endothelial cell data. Adding the size of the image (in this case 1024 𝜇𝑚) the edges of the images simulated as being placed next to each other in a cylindrical shape. The y coordinate for all the cells stay the same, only the x- coordinate changes by adding the image size to the rightmost of the two cells in the image. The distance in both flat and cylindrical configuration is calculated and the shorter one of them is saved for further analysis . . . 15 3.1 A bar plot showing the Small-World parameter of the 4 agonists with their

resp. standard error of mean for the cylindrical (leftmost of the same colored bars) resp. flat (right most of the same colored bar)

configuration . . . 19

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3.2 Comparison of network plots for the 4 different agonists ATP, ACh, ADP and Hist. plots look similar in regards to where the connectivity is the highest which is mostly at the center of the plot and the distribution of colors which represents the value of connectivity. The network plot which differs the most from the rest is the Ach network plot, with less connections compared to the other plots. A few clusters are visible, which is a signature of the Small-World network . . . .. . . 20 3.3 Degree distribution of the networks for the different agonists, ATP, Ach, ADP

and Histamine . . . 21 3.4 The networks of the different stages E12.5, E16.5, 𝐸18.5, P6, and P8. . . 23 3.5 The degree distribution P(k) for the networks of the different stages. . . 24 3.6 Different percentiles of the connectivity of the active (spontaneous) cells for

the stages E12.5, E16.5, E18.5, P6 and P8. Blue indicates 0^th-20^th percentile, green is 20^th-40^th percentile, yellow is 40^th-60^th percentile, red 60^th-80^th percentile and black the 90^th percentile . . . 25 3.7 Boxplots of the average spike frequency distance for cells of different

connectivity groups. The green line represents the median, the red crosses shows the outliers, the blue upper and lower lines show the upper and lower quartile of the distribution and the vertical line shows the range of frequencies of the data points . . . 26 3.8 Histogram for each development stage E12.5, E16.5, E18.5, P6 and P8 showing

the number of cells for each k-value. . . 27 A1 n = 20 vertices are connected to k = 4 nearest neighbors in the Regular

network on the left hand side of the figure. The rewiring is done along the ring with increasing probability p. For p in between 0 and 1 the

Small-World network is shown and to the far right of the figure is the

random network . . . 39

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List of Tables

2.2 The network parameters for the four agonists; the shortest path length

λ, clustering coefficient σ and Small-World parameter ^σ_λ . . . 17

2.3 The connectivity, edge density and mean of all correlations for all four agonists . . . 17

2.4 The scale factor γ, mean of all correlations above cutoff and the 99^th percentile of all correlations for all four agonists in both flat and cylindrical configurations . . . .. . . 18

2.5 Network types for all agonists in flat and cylindrical configuration . . . 18

2.6 Neural brain development network parameters . . . 22

2.7 Network parameters for the neural brain network . . . 22

2.8 Network types for the different brain development stages . . . 22

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Abbreviations

ER Endoplasmic Reticulum ROI Region Of Interest SD Standard Deviation Ach Acetylcholine

ADP Adenosine DiPhospate ATP Adenosine TriPhosphate

Hist Histamine

Norm. Normalized

F Flat

C Cylindrical

Config. Configration

E Embryonic

P Post natal

TPLSM Two Photon Laser Scanning Microscope LSFM Light Sheet Fluorescence Microscopy

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1. Introduction

1.1 Background

1.1.1 Network theory and networks in life

Complex networks are the base for many different processes, from social networks to transport and even in biology. Studying their structure, underlying mechanism and behavior can reveal important knowledge about the system itself. [1]

Stanley Milgram introduced the Small-World network in order to model social

networks, which connected the network topology to human behavior. [2] In the 1990’s scientists started using network models in biological and physical contexts. [3] Many real world networks have turned out to be Small-World networks, including social sciences, the internet and transportation systems. [4]

Communication between cells is crucial for multicellular organisms to function. In the human brain cells form neural network circuits which is the base for the brain activity.

Network and topology, as well as their relationship are studied in order to expand the knowledge of biological network systems. [5]

1.1.2 Network topologies

The four network topologies that are discussed in this thesis are the Regular, Small- World, Random and Scale-Free network (see fig.1.1).

In Regular networks each node is connected to k other nodes while for Random networks nodes and links are connected stochastically. Small-World networks contain features of both Regular- and Random networks, such as high clustering like Regular networks and short internodal distances like Random networks. [6]

The clustering coefficient, often denoted 𝜎, is defined as the number of connections of a node divided by the possible number of connections between the nearest neighbors of the node. Averaging the clustering coefficients for all nodes of a network gives the clustering coefficient of the network. [7]

Random networks are created when m links are connected to N nodes randomly, which gives the probability for a node to have k =^𝑚_𝑁 connections, governed by the Poisson distribution. This type of network has fewer local connections which gives a small clustering coefficient, whereas for regular networks this number is higher. [8]

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Small-World networks are suggested to have an evolutionary advantage due to their robustness. Both Small-World and Scale-Free networks are found in widely different contexts such as transportation systems, the internet and biology. The characteristics of a Small-World network are high clustering and short path lengths, which is a combination of a Random and a Regular network. For a Scale-Free network hubs are characteristic. Hubs are nodes that are highly connected. [9]

If a node is connected to many other nodes, the probability for this node to connect with more nodes is higher compared to less connected nodes. This is called

preferential attachment. Eventually the highly connected node turns into a hub. [10]

Scale-free networks are for ex. online communities (i.e. Facebook) and follows the power law P(k) ∝ k^−γ, where 𝛾 is the degree exponent [11] and P(k) is the degree distribution giving the probability for the node to have k connections, also called connectivity. [12]

The network has Scale-Free properties if 1 < γ < 3. [13] The smaller the value of 𝛾 the more important the hubs are in the network. For γ > 3 the hubs are not important, and the network isn’t Scale-Free. [14]

Fig. 1.1 (A) A Small-World network, (B) Regular network (C) Random network and (D) Scale-Free network.

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1.1.3 Origin of the 𝐂𝐚

^𝟐+

signals and connection to disease

The Ca²⁺ ion is very well suited for signal transduction and is the most common cation [15] in the human body. The concentration of Ca²⁺ within the cell is ~10⁻⁷ M and outside the cell ~10⁻³ M. This large difference between the intra- and extracellular concentrations makes the cell sensitive to small fluctuations of Ca²⁺ added to the cell’s cytoplasm. High levels of Ca²⁺ in the cell can be lethal and lead to both necrosis [16]

and apoptosis [17]. Due to this fact, the cell has a system of transporters that rapidly remove increases of Ca²⁺ in the cytoplasm. The excess Ca²⁺ is mainly transported to the endoplasmic reticulum (ER); or out of the cell via the plasma membrane. The influx and efflux of Ca²⁺ keeps the concentration of Ca²⁺ balanced. The cell always tries to keep the concentration of Ca²⁺ low but there are many situations that can disrupt this balance. Sometimes the cell can only partly restore the Ca²⁺ balance. This may give rise to periodic Ca²⁺ oscillations. An example is the fertilization of the egg that is controlled by Ca²⁺ oscillations. When the sperm interacts with the egg Ca²⁺

oscillations are activated that continues for hours. The mechanism of Ca²⁺ oscillations although extensively studied, has not yet been fully explained. [18]

Ca²⁺ enters the cell either through the plasma membrane or is released by the ER giving rise to a Ca²⁺ signal. The ER is the most important intracellular storage of Ca²⁺. The influx of Ca²⁺ from the extracellular space occurs through Ca²⁺ channels located on the plasma membrane. When Ca²⁺ is released from the ER it occurs through ion channels located on the ER membrane, which are activated by different intracellular messengers. A few examples of diseases in which studies have shown malfunctioning in the Ca²⁺ signaling handling are diabetes, cancer, psoriasis and Alzheimer’s disease.

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1.1.4 The cell

The smallest functional component of a living organism is the cell. Cells are performing multiple vital functions including metabolism, growth, reproduction, movement and hereditary transmission. All cells have a cell membrane, cytoplasm (or cytosol) and organelles. The different organelles each have specific functions. Examples of organelles are the nucleus where the genetic information is stored (in the DNA), mitochondria, which is responsible for energy production and cell metabolism, the lysosome, that breaks down macromolecules and the endoplasmic reticulum, which synthesizes proteins and stores Ca²⁺. [20]

1.1.5 𝐂𝐚

^𝟐+

Signaling

Due to the gradient between the intra- and extracellular concentrations of Ca²⁺, Ca²⁺ is a good signal messenger for many cellular processes, due to its frequency, amplitude and spatio-temporal characteristics in relation to other constituents. The regulation of the Ca²⁺ signal has two phases, the up- and the down-slope which are

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meant to increase and decrease the concentration of Ca²⁺ in the cell. The up-slope is caused by an influx of Ca²⁺ into the cytosol controlled by channels in the

membrane and the down-slope is controlled by an efflux of Ca²⁺ from the cytoplasm, regulated by pumps and exchangers. Efflux starts when the

concentration of Ca²⁺ becomes too high. The mitochondria stores Ca²⁺ when the concentration is too high (up-slope), then slowly releases it during the down-slope.

This is what causes the amplitude and the oscillatory behavior of the Ca²⁺ signal.

The oscillations can have different frequencies, from seconds to several days, where the variation of frequency is a response to changes in stimulus intensity. [21] The local maxima between an up- and a down-slope is what will later be referred to as a spike.

1.1.6 Robustness of a network

Robustness is a measure of how well the system can handle changes of external conditions and internal organization while still maintaining normal performance.

Some types of networks are known to be more robust than others, for example Scale-Free and Small-World networks. However, due to their hubs the Scale-Free network are more vulnerable to attacks in which the hubs are removed, since they are crucial to the function of the network. [22]

1.1.7 Endothelial cell network of the vascular system

The cells lining the innermost layer of the vascular system are called endothelial cells. The endothelium (the entire collective of the endothelial cells) can detect and generate several kinds of input and output unlike any other sensory system. This is what regulates blood pressure and distribution of blood flow in the vascular system, modulates blood clotting, determines when to form new blood cells, etc. In an adult human there are ~10¹³ endothelial cells, and coordinating these cells to regulate output even on a local scale is difficult. The network of cells is therefore studied in order to understand the behavior of individual cells. Each cell has its own sensing capacity to signals and by exchanging information between neighbors the sensing is distributed among the cells. This cell network in the endothelium makes it possible to solve sensory problems more complex than any single cell could. [23]

1.1.8 Neural networks

In mammalian neurons calcium signaling is an important intracellular messenger with a concentration of ~50 − 100 nM at rest and about 10 − 100 times more during electrical activity. [24]

A way to understand the complexity of neural networks is to study how the connectivity develops during brain development. In neuroscience studies have shown that stimulation of a single neuron can affect activity of the population both in vitro and in vivo. This shows that single neurons have a direct impact on the network, and behavior output are important to the underlying structures’ functional

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and structural organization. In vivo experiments have shown that the brain function is encoded by specific neuronal activity patterns; the information about the

anatomical structure of the microcircuits that is activated during behavioral tasks is often deficient. [25]

Since different functional microcircuits tend to develop successively, studying network development over time gives the opportunity to observe the impact of different microcircuits as they develop. Graph theory can help to link function and structure for these microcircuits during their development. The undeveloped networks often predict the end result of the adult circuit and ultimately spot and predict future dysfunction and diseases. [26]

1.1.9 Calcium imaging

Imaging calcium ions (Ca²⁺) in neurons is important since this signaling messenger is vital in all mammalian cells. Single cell Ca²⁺ imaging is often used to analyze signal events in neurons, dendrites and spines as well as interconnected local populations of neurons. An example of an application of Ca²⁺ imaging is the analysis of

pathological network activity, for example epileptiform events. [27]

1.1.10 Cross correlation analysis

Cross correlation analysis is used to mathematically quantify linear similarity between two waves where one of them is shifted in time. When used in signal processing, the waves are usually time series of discrete data point sets. The cross- covariance (normalized cross correlation function) is often used in image processing where the brightness of the image is measured. [28]

𝑛𝑜𝑟𝑚. 𝑐𝑟𝑜𝑠𝑠 𝑐𝑜𝑟𝑟(𝑥, 𝑦) = ∑^𝑛−1_𝑛=0𝑥(𝑛) ∙ 𝑦(𝑛)

√∑^𝑛−1_𝑛=0(𝑥(𝑛))²∙ ∑^𝑛−1_𝑛=0(𝑦(𝑛))² (1)

The higher the cross correlation value the more correlated the two series are. Two time series can be correlated even with one of them shifted in time. To account for this shift the correlation is calculated as a function of lag. Correction is when underlying trends are subtracted to filter the signals. This is done to ensure that gradual decay caused by for ex. bleaching doesn’t add to the actual signal. It’s done by fitting the signals to a polynomial function and then subtract the trend line. [29]

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1.1.11 Graph theory

In order to characterize the topology of networks, graph theory is used by

calculating a number of network parameters which determines the type of network.

These networks consist of nodes (also called vertices) and links (or edges) that connect them. Within the field of biology, when studying biological networks, cells, proteins or a specific part of the brain, can be represented as nodes. The links between them could represent interactions for example via ion fluxes, synapses or connections by bundles of axons. By comparing the network structure and activity in subjects of different ages and stages of development, physiology and

pathophysiology can be associated with the changes in the network structure. The values of the network parameters are what determines the type of network, for example the Small-World, Random, Scale-Free or Regular network. [30]

Fig. 1.2 Three graphs where a and b are correlated while c isn’t correlated to neither a nor b.

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1.2 Goal of thesis

The goal of this thesis is to study different kinds of cell networks such as the endothelial cell network and the neural cell network in the brain during the brain development using graph and network theory, in order to further the understanding of biological network activity on a cellular level.

Studying the neurons in the brain during embryonic development is important to understand neurological development diseases such as epilepsy. Different neural signaling during embryonic development as well as postnatal defects of the neural network have been linked to epeleptogenesis. [31]

The goal is to be able to see how networks develop and ultimately being able to detect where in the development disease occur, as well as how the neural cell network is malfunctioning causing or contributing to this disease. This is important in order to understand how diseases like schizophrenia and epilepsy arise. Studying network development over time gives the opportunity to observe the impact of different microcircuits as they develop.

This will be done by analyzing endothelial cell data and neural brain development data. The goal is to determine the network type through network analysis for both of the data sets. Then further investigation will be done to the neuron data in the form of signal analysis by investigating the distribution of cell connectivity and spiking frequency.

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2. Method and material

2.1 Data analysis

Prior to the data analysis done in this thesis, the mean fluorescence value for each cell ROI is calculated and saved together with the x and y coordinates for each ROI is saved in a data file. [32]

The data analysis that was carried out during this thesis is described below and was done using Matlab R2018b and Excel 2013.

2.1.1 Graph theory and cross correlation analysis

Cross correlation analysis is used to study all the cell pair combinations. The correlation value lies between 0 and 1. This value corresponds to how strongly the cells interact with each other, i.e. how well their signal spikes resemble each other.

A cut-off value removes all the false correlations. To reduce insignificant network connections the cutoff value is calculated from a set of scrambled data. The scrambled data is created by mixing up the time series and assign them to random starting points. Each time series is divided into two at an arbitrary position and put together again in opposite order (see fig. 2.2). In this way the cell activity of both the scrambled and the original data are conserved. The 99^th percentile of all correlations of the scrambled data set is used as the cutoff value. If a cell pair has a correlation value higher than the cut-off value, they are considered to be interconnected. [33]

Fig. 2.1 A visual representation of cleaning up the signal before analysis. (A) Removing underlying trends caused by for ex. bleaching i.e. trend correction. This is done by fitting the signals to a polynomial function. (B) A specific threshold value is set, where only signals fulfilling the criteria are picked out as active cells to be further

analyzed. (C) Signal magnitude is normalized to remove artefacts such as amplitude fluctuations and zero values.

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Using this method gives a set of parameters describing the network properties;

o Mean value of all correlations

o Mean value of all correlations above cutoff o Connectivity C

o Edge density

o 99^th percentile of all correlations o Shortest path length λ

o Clustering coefficient 𝜎 o Small-World parameter ^𝜎

λ

o Scale factor 𝛾

Shortest path length 𝜆 = ^𝐿

𝐿_{𝑟𝑎𝑛𝑑𝑜𝑚} (2) Clustering coefficient

σ =

^C

C_random (3) Small-World parameter ^𝜎_𝜆 (4)

Fig. 2.2 Visual representation of the process to scramble the signal. (A) The original data with start and end point. (B) By cutting the signal at a random

time point and put it in opposite order a scrambled signal is created, maintaining the total activity of the signal. The scrambled data can then be

used to calculate the cutoff value in an unbiased way.

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maximum possible # edges in G_i=# edges in G_i 𝑘_𝑖(𝑘_𝑖− 1)/2 (5) G_i is the subgraph of 𝑖’s neighbors , ki is the number of neighbors of node i.

The clustering coefficient C is the average of Ci

C =1

N∑ C_i (6)

i∈G

L and

L

_random are the mean shortest path lengths of the network and the Random network respectively. The shortest path length 𝐿 is the minimum number of nodes that need to be passed in order to go from one node to another.

L(G) = 1

N(N − 1) ∑ d_ij (7)

i≠j∈G

Where 𝐺 is the graph with N nodes and d_ij shortest distance between i and j. [34]

In order for the network to be a small world network, the following criteria must be met;

C ≫ C_random and L ≈ L_random, i.e.

C

C_random≫ 1 (8) L

L_random≈ 1 (9)

This gives the condition for the small world parameter to be

σ

λ

> 1 (10) [35]

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2.1.2 Network analysis of cell data from mesenteric artery endothelial cells

The artery was cut open and loaded with florescent indicator. It’s then washed and mounted under the microscope. The fluorescent indicator is excited and the artery is imaged with one of the four agonists; (an agonist is a substance that binds to a receptor, causing the receptor to get activated) [36] Ach, ADP, ATP and Hist, one at a time to picture the Ca²⁺cell signaling. Software was used to extract the signals and ROIs were drawn around every cell where the average time-dependent florescence intensity is extracted as the Ca²⁺ signal. The signals were then normalized to their corresponding baseline. [37]

2.1.3 Data analysis endothelial cells

The signal is first corrected for underlying trends, i.e. trend line correction. Different artefacts can lead to gradual increase or decrease superimposed on the signal. This is corrected by fitting the signal to a polynomial function, for ex. a linear function. In order to reduce the likelihood of false positives in the cross correlation analysis, only the active cells were to be studied. An active cell was defined as having at least two transient intensity peaks, more than 3 standard deviations (standard deviation is calculated from each cell signal) (see eq. 11) above the noise, where the noise is defined as the fluctuating baseline of the non-active cells.

standard dev. = √∑ (x_i _𝑖− x̅)² n − 1 (11)

The formula for the standard deviation where n is the number of data points, x̅ and x𝑖

are the mean and the individual data point respectively.

The data points were then weighed (penalized) for distance in order to grade the importance of the correlations, by the distance between the correlating cells. The further they were from each other, the less likely it is that the correlation is real. This was done by normalizing all the distances in the distance matrix (i.e. the distances between each cell pair) using eq. 12:

d_norm=d − dmax

d_max (12)

Where 𝑑_𝑚𝑎𝑥 the maximum distance two cells could be separated by (in this case the maximum distance of the image). d is the actual distance between the two cells and dnorm is the normalized distance between them. dnorm Is then multiplied with the corresponding correlation for the cell pair.

The distances between all the cells were calculated in two ways, both in a flat configuration where the assumption is that the artery was cut open before recording the data, and a cylindrical configuration assuming the artery is completely intact

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without any cuts. The cylindrical configuration assumes that the artery was completely intact during the data recording, which makes the image form a cylinder where there are no edges to the image. Both configurations were used to do the analysis.

The distances between cell 𝑖 and 𝑗 were calculated using the Pythagorean Theorem.

d_ij= √(x_i− x_j)²+ (y_i− y_j)² (13)

Fig. 2.3 The intact artery as a cylindrical model with the coordinate system as it was during the analysis. The circumference of the cylinder corresponds to the image size on

the x-axis and the length of the cylinder corresponds to the image size on the y-axis.

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The difference between the cylindrical and flat configuration is that in the cylindrical configuration the edges of the image on the x-axis are considered to be connected resulting in a continuous surface of the image. This was done by not only calculating the distances between all cells on the image like in the flat configuration but also calculate the distance between cells across the edges of the image (see fig. 2.5) equivalent to a cylinder model. This was done by adding the size of the image to the left most of the two cells while the right most stay the same, mimicking that first cell’s position to be on the right side (in relation to the coordinate system) of the other cell.

Then the distance between these two cells are calculated. For this configuration both the distances in the image and the distances between cells on the edges were calculated. The shorter of the two was the distance which was more relevant and therefore the distance that was saved and used in the further analysis.

The network analysis is first done on the scrambled data where the 99^th percentile of all correlations value is set as a cutoff value for the correlations. All the parameters are calculated for the data (see tables under results) then the distance distribution, degree distribution and the signal cell network are plotted (see fig.3.1 and 3.2 under results).

Fig. 2.4 Representation of the recorded cell image where each cell (red circle) has an x and y- coordinate. The distances between the cells were calculated using the Pythagorean

Theorem.

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2.5 Calculating the distances between cells in the cylindrical configuration for the endothelial cell data. Adding the size of the image (in this case 1024 𝜇𝑚) the edges of the images simulated

as being placed next to each other in a cylindrical shape. The y coordinate for all the cells stay the same, only the x-coordinate changes by adding the image size to the rightmost of the two cells in the image. The distance in both flat and cylindrical configuration is calculated and the

shorter one of them is saved for further analysis.

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2.1.4 Network analysis of cell data from neural brain development

The measurements were carried out on mice embryos and newborn mice where the brain was extracted, cut in slices and injected with fluorescent dye in order to image the neural brain network in the brain.

2.1.5 Data analysis neural cell network

The difference between the analysis of the endothelial cell network and the neural cell network is the noise reduction before the cross correlation analysis for the neural cell data, since the neural cell data contains more noise. For the neural cells the

spontaneous activity of the cells was measured.

The different stages of the brain development is denoted as follows: E12,5, E14,5, E16,5,E18,5, P6 and P8. For ex. E12,5 indicates that the mouse is still in the embryonic state 12,5 days after fertilization when the data was recorded, while P6 indicates 6 days after birth.

The cell signals are first spline smoothed in order to estimate the underlying regression function. [38]

The signal is then subtracted by the spline smoothed function. The standard deviation is calculated from all values < 0 (negative deflections) and is then multiplied by 2. The baseline is calculated by subtracting the 2 times standard deviation from the spline smoothed function. The data is then subtracted and divided by the baseline.

The deviations of the positive deflections (noise > 0) are also multiplied by 2. The noise is normalized to the positive deflection by division. The data is then smoothed again by a moving point average and the standard deviation of this signal is multiplied by 3 to later be used as the cutoff for what counts as active cells.

The active cells have to pass a few criteria in order to be further analyzed. They need to have at least 2 peaks above the cutoff value of 3 times the standard deviation of the positive deflections. The width of the spikes (peaks) need to be between 1 − 12 s. A peak is defined as local maxima above the cutoff value. These active cells are then analyzed further in the network and signal analysis described in section 2.1.1.

In order for a network to be concluded as a Small-World network, it has to fulfill both the criteria σ ≥ 4 and λ~1 (see section 4.2 Limitations). This applies to both the endothelial cell network and neural cell network. If a network fails to fulfill the two criteria, it’s concluded to be a Random network.

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3. Results

3.1 Endothelial cell network

Table 2.2 The network parameters for the four agonists; the shortest path length ^𝜆, clustering coefficient 𝜎 and Small-World parameter ^𝜎

λ .

Table 2.3 The connectivity, edge density and mean of all correlations for all four agonists.

Agonist Config. λ 𝜎 𝜎

λ

Ach F 0,88019 ± 0,11082 6,03973 ± 1,32686 6,80588 ± 0,81162

C 0,89223 ± 0,10881 6,34793 ± 2,27849 7,02928 ± 1,97916

ADP F 0,89717 ± 0,25626 7,94385 ± 1,51244 9,17078 ± 1,28010

C 0,89368 ± 0,26389 7,07823 ± 1,20432 8,27545 ± 1,24499

ATP F 0,97647 ± 0,36882 7,71685 ± 1,54139 8,91378 ± 3,15772

C 0,97917 ± 0,43302 7,94408 ± 2,07323 9,40033 ± 3,74766

Hist F 0,86002 ± 0,10245 7,38823 ± 0,67891 8,73503 ± 1,38780

C 0,85695 ± 0,11247 7,22425 ± 0,64143 8,56655 ± 1,26436

Agonist Config. Connectivity Edge density Mean of all

correlations

Ach F 0,74220 ± 0,02937 0,02506 ± 0,01125 0,116258 ± 0,04509

C 0,75123 ± 0,05028 0,02563 ± 0,01146 0,12201 ± 0,04702

ADP F 0,64105 ± 0,10439 0,02010 ± 0,00103 0,158507 ± 0,06248

C 0,65068 ± 0,11697 0,02179 ± 0,00092 0,16728 ± 0,06631

ATP F 0,64327 ± 0,05287 0,01855 ± 0,00709 0,146183 ± 0,07796

C 0,63358 ± 0,06937 0,01851 ± 0,00683 0,15475 ± 0,08307

Hist F 0,63612 ± 0,08993 0,01908 ± 0,00618 0,144146 ± 0,04532

C 0,6142 ± 0,0613 0,01714 ± 0,00204 0,15237 ± 0,04874

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Table 2.4 The scale factor 𝛾, mean of all correlations above cutoff and the 99^th percentile of all correlations for all four agonists in both flat and cylindrical configurations.

Table 2.5 Network types for all agonists in flat and cylindrical configuration.

Agonist Config. 𝛾 Mean of all corr.

above cutoff

99^{t h} percentile of all corr.

Ach F − 1,18196 ± 0,28134 0,40851 ± 0,09125 0,40078 ± 0,10988

C −1,18479 ± 0,32967 0,41313 ± 0,08790 0,40628 ± 0,10862

ADP F −0,9141 ± 0,20056 0,57934 ± 0,12003 0,56333 ± 0,12697

C −0,89857 ± 0,22544 0,57952 ± 0,11922 0,56981 ± 0,12508

ATP F −0,99911 ± 0,14454 0,52122 ± 0,16147 0,48735 ± 0,19301

C −0,95084 ± 0,09698 0,52677 ± 0,15871 0,49346 ± 0,19204

Hist F −1,00798 ± 0,22224 0,54590 ± 0,10288 0,52252 ± 0,11381

C −1,09319 ± 0,14801 0,55863 ± 0,10710 0,53056 ± 0,11309

Agonist Config. Network type

Ach F Small-World

C

ADP F Small-World

C

ATP F Small-World

C

Hist F Small-World

C

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The results of the network analysis of the endothelial cell network are shown in table 2.5 above, showing the network type for all four agonists for both configurations.

Table 2.2, 2.3 and 2.4 show the mean of all the network parameters from all the data with their calculated standard deviations for all four agonists. The values are all very similar for all four agonists showing that the endothelial cell network in the mesenteric artery is a Small-World network ( 𝜎 ≫ 1 and λ~1).

Fig. 3.1 (A) A bar plot showing the Small-World parameter of the 4 agonists with their resp. standard error of mean for the cylindrical (leftmost of the same colored bars) resp. flat (right most of the same

colored bar) configuration.

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Fig. 3.2 Comparison of network plots for the 4 different agonists ATP, Ach, ADP and Hist. The plots look similar in regards to where the connectivity is the highest which is mostly at the center of the plot and the distribution of colors which represents the value of connectivity. The network plot which differs the most from the rest is the Ach network plot, with less connections compared to the other plots. A few

clusters are visible, which is a signature of the Small-World network.

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Fig. 3.3 Degree distribution of the networks for the different agonists, ATP, Ach, ADP and Histamine

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3.2 Neural cell network during brain development

Stage λ 𝜎 𝜎

λ

𝛾

E12.5 0,76908 ± 0,17983 2,80518 ± 0,48692 3,72538 ± 0,46440 −1,09280 ± 0,51075

E14.5 − − − −

E16.5 0,48308 ± 0,28083 4,7748 ± 0,47445 12,5668 ± 4,70285 −1,4974 ± 0,32428 E18.5 0,88201 ± 0,05268 3,56780 ± 0,80138 4,07160 ± 0,98749 −1,08760 ± 0,10278

PN6 0,49028 ± 0,06477 2,79413 ± 0,79227 5,93113 ± 2,09067 −1,12621 ± 0,32245 PN8 0,89736 ± 0,13444 4,84885 ± 1,24985 5,7409 ± 2,253 −1,1922 ± 0,0814

Table 2.6 Neural brain development network parameters.

Stage Mean corr. above cutoff

99^th perc. Of all correlations

Connectivity Edge density

E12.5 0,35752 ± 0,08574 0,3756 ± 0,1054 0,7240 ± 0,1119 0,03839 ± 0,01442

E14.5 −¹ − − −

E16.5 0,35867 ± 0,01288 0,34071 ± 0,00604 0,61523 ± 0,15005 0,01860 ± 0,00309 E18.5 0,32326 ± 0,02810 0,3386 ± 0,03976 0,81334 ± 0,05154 0,03431 ± 0,00814 PN6 0,39792 ± 0,01303 0,4036 ± 0,03504 0,56433 ± 0,03166 0,03792 ± 0,01907 PN8 0,34368 ± 0,00709 0,32132 ± 0,02897 0,61795 ± 0,09872 0,021303 ± 0,004686

Table 2.7 Network parameters for the neural brain network.

Stage Spontaneous cells

%

Network type E12.5 48,4 ± 14,1 Random

E14.5 ⁻ ⁻

E16.5 36,7 ± 5,2 Random E18.5 41,7 ± 8,4 Random

PN6 ^{30 ± 5} Random

PN8 41 ± 5 Small-world

Table 2.8 Network types for the different brain development stages.

1− Means no activity was discovered.

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Fig. 3.4 The network plots of the different stages E12.5, E16.5, E18.5, P6, and P8.

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Fig. 3.5. The degree distribution P(k) for the networks of the different stages.

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Fig. 3.6 Different percentiles of the connectivity of the active (spontaneous) cells for the stages E1.5, E16.5, E18.5, P6 and P8. Blue indicates 0^th-20^th percentile, green is 20^th-40^th percentile, yellow is 40^th-60^th

percentile, red 60^th-80^th percentile and black the 90^th percentile.

Fig. 3.5. The degree distribution of the different stages.

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Fig. 3.7 Boxplots of the average spike frequency distance for cells of different connectivity groups. The green line represents the median, the red crosses shows the outliers, the blue upper and lower lines show the upper and lower quartile of the distribution and the vertical line shows the range of frequencies of the

data points.

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Fig. 3.8 Histogram for each development stage E12.5, E16.5, E18.5, P6 and P8 showing the number of cells for each k-value.

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4. Discussion

4.1 Discussion of results

4.1.1 The endothelial cell network

The endothelium processes a large amount of information provided by the chemical environment that the vascular system is surrounded by. This information is what determines the physiological output. The endothelium controls nearly all

cardiovascular functions and the majority of cardiovascular diseases stems from dysfunction in the endothelium. The extracellular signals need to be detected and transferred correctly through Ca²⁺ signals and other signaling pathways in order to not lose information. It’s not yet fully understood how the endothelium processes the chemical composition it’s exposed to, at the same time as it transduces extracellular signals to stimulate cellular responses. [39]

Previous studies suggest that different stimuli induce characteristic signals in

endothelial cells, i.e. the input and output relationship for the cell and a given agonist isn’t fixed. When several signals are present, it appears that the cell acts as a

computational element. This computational ability is a collective feature of the entire endothelial network, making it possible to monitor the external environment through the sensing ability distributed across separate cells. This might be what gives the endothelium its various functions with its fixed network that communicates using only one signaling molecule Ca²⁺. [40]

The sensing and control of the endothelium are distributed over many endothelial cells. The spatial structure of the sensing cells and the temporal encoding of signals are vital for cell responses. This helps the endothelium to process several instructions as well as acting as a sensory system with much higher computational power than that of any single cell. [41]

The analysis in this thesis showed that the signaling cell networks for all the four agonists Ach, ATP, ADP and Hist were the Small-World network with very similar parameters for all the agonists. The topology of the signaling network was also very similar for the different agonists (see fig.3.2). The range of connectivity values seen in the network plots is also similar for all four agonists and the areas where the

connectivity is the highest is in the center of the plots. The characteristic clusters for Small-World networks are also visible. This indicates that the signaling network of the endothelium is robust with its high clustering and short path lengths and acts similarly regardless of any of the four agonists. Interestingly the difference between the results of the analysis done in the flat resp. cylindrical configuration was small.

Fig. 3.3 shows the degree distributions, P(k) for the agonists in log-log plots. The degree refers to the number of connections for each node (cell) and the plots show the probability distribution of the degrees for the entire network. It’s clear that the degree

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distribution is linear for all the agonists indicating that P(k) follows the power law, which means the networks are Scale-Free (see section 1.1.2).

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4.1.2 The neural cell network

In order for the network to be considered a Small-World network both of the λ and 𝜎 values need to indicate that (see section Method and Material), which they don’t for E12.5, E14.5, E16.5, E18.5 and P6.

At stage E12.5 the network is classified as a Random network, even though some of the E12.5 files had a λ value very close to 1. The rest of the E12.5 files had a very low λ value, reducing the mean of λ. Since 𝜎 < 4 for all of the E12.5 files, it is classified as a Random network. Since the line between Random and Small-World networks is so vague, this could possibly be a Small-World network. It does indicate that by the higher λ value but looking at the trend of this data, it points more strongly towards a Random network. The conclusion drawn about stage E12.5 is therefore that the network is a Random network but with Small-World tendencies. More measurements will be needed to decide with more certainty which network type it is.

For stage E14.5 there was only one file that generated a result, so it’s impossible to draw a conclusion for that stage, except that there seem to be no network activity in the analyzed data for stage E14.5. More tests and analyses need to be conducted in order to draw a more confident conclusion about the network activity for E14.5. One interesting parallel however, is at stage E14.5 the neurogenesis starts [42], which could affect the network topology.

At the later embryonic stage, E16.5 the gliogenesis starts [43] and continues into the postnatal stages [44], which is interesting to note since E16.5 seemed to point to a Random network and the gliogenesis could have an effect on the network topology at this stage.

The network parameters for E18.5 indicates a Random network with Small-World tendencies due to the sigma (𝜎) value being on the lower side (see table 2.6). At P6 the network is also a Random network and at P8 the network was found to be a Small- World network.

It’s interesting to see that the network topology seem to change from Random to Small-World during the development, especially if there’s a connection between network topology and neuro-and gliogenesis. This could indicate that the network topology is connected to these processes. If, how and why that is would be of interest to explore in the future.

The network plots in fig. 3.4 varies a lot in appearance between development stages.

This could be due to a few things. Generally, the more connections between cells the higher the network activity is. The color of the lines indicates the value of connectivity, where blue is the lowest and red the highest. The majority of the connectivity values between cells lies in the blue range, meaning that the connectivity is mostly quite low.

Looking at the degree distributions P(k) for the different stages in fig. 3.5 they are linear, like the degree distributions for the endothelial cell networks. Therefore they follow the power law and the networks for the development stages are thus Scale- Free.

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Fig. 3.7 shows the boxplots of the average spike frequency distance of the different connectivity groups for the different stages of development. The average spike frequency distance is the average distance in seconds between frequency spikes of active cells. There seem to be a trend where the majority of the distances lies around the same range regardless of their connectivity and stage of development, which means that the time between signals are in the same range for all the different stages.

In fig. 3.6 showing the spatial distribution of the different connectivity percentages of cells, there seem to be clusters of highly connected cells for the stages E12.5, E18.5 and P8. Stage E12.5, E18.5 were concluded to be Random networks and stage P8 Small-world network, but clusters of highly connected cells is a sign of a Small-World network. This indicates that the stages E12.5, E18.5 have some Small-World

characteristics, even though they were concluded to be Random networks. Observing the plots for E16.5 and P6 (which were concluded to be Random networks), the highly connected cells are more spread out.

E12.5 and E18.5 both had a large spread of k-values compared to the other stages, seen in fig. 3.7, which is also clearly visible in fig.3.2 showing more network activity.

There seem to be a direct link between network activity and the range of k-values.

There’s more activity and they also have more highly connected cells. The reason why there’s more activity for these two stages could have to do with the way the data is collected, or that the activity is generally higher for these stages. What determines the level of activity is not clear at this point. For the stages E16.5, P6 and P8 the k-values ranges from 0 − 5. For all stages the number of cells decrease from the smallest to the highest k-values (see fig. 3.8).

The motivation for doing the network analysis of the neural brain network

development was to understand the development of the neural cell network in the brain. The goal is to be able to see how the network develops over time and thereby being able to detect where in the development disease occur, as well as how the neural cell network is malfunctioning, causing or contributing to this disease. This is important in order to understand how diseases like schizophrenia and epilepsy occur and working towards finding cures for diseases connected to neural brain network development.

The conclusions that were drawn in this thesis were drawn based on the provided data. It’s important to validate the conclusions and improve the existing techniques used to reach these conclusions, especially since some of the definitions allow for a lot of interpretation of the results. It might be that the stages of the neural network development will be concluded to be of different network types in the future due to improved techniques, reduced noise etc. The analysis done in this thesis on this kind of data has never been done prior and it’s therefore somewhat expected that future results from similar methods, gives a different or more consistent result.

It might be that other conclusions could be drawn for the neural networks during brain development based on other data, the results here isn’t written in stone. These are the first steps on the way and opening up for discussion and further analysis about this subject, as well as working towards understanding the bigger picture of signaling networks in biology on a deeper level.

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4.2 Limitations

A big obstacle during this thesis project was how to successfully remove the noise while analyzing the neural brain network, since this data had quite a low signal-to- noise ratio. Therefore the threshold for active cells were set quite high, to reduce the risk of drawing incorrect conclusions.

Another limitation to this project was the limited time for all that could have been done with this data. There are a lot more information available to extract from these kinds of datasets beyond what was done here. This thesis’ main focus was the network activity and the network properties of the two datasets, as well as some signal analysis and hub cells (connectivity plots) for the neuronal brain network.

The Small-World coefficient has been found to be sensitive in classifying Small-World networks as well as having low precision. This can lead to miss-classification of networks, where they are classified as Small-World while in fact they are Random networks. It also doesn’t indicate where on the spectrum between Random and Regular network the Small-World network is. The clustering in C_random has a large impact on σ due to being in the denominator, which can cause large impact even for small changes of Crandom. The often large difference between C and Crandom also affects the Small-World coefficient. [45]

Regarding the definition of what a Small-World network is, it’s rather vague. There’s no exact definition of what counts as 𝜎 ≫ 1, which caused difficulty in trying to reach a definite answer for some of the files.

The vague definition of the clustering coefficient and the shortest path length (eq. 8 and 9) gives room for too much interpretation which in turn gives a large risk of error. Where is the line between Random and Small-World network for the clustering coefficient?

There’s no rigorous definition of 𝑎 ≫ 𝑏, it varies from case to case. So in this case, what does the definition of the clustering coefficient for a Small-World network σ ≫ 1 mean?

In this thesis σ ≥ 4 was considered to be a Small-World network iff λ~1, based on the article E. Smedler, S. Malmersjö & P. Uhlén, ”Network analysis of time-lapse microscopy recordings”, Frontiers in neural circuits.

Network theory is a rather static method of describing dynamic systems, which means that additional methods and tools are needed to extract as much and as deep

knowledge and understanding as possible. In order to further the understanding of networks in biology and to surpass the current limitations of trying to study networks over time, fluctuating data, noise etc., techniques to integrate data from different sources are needed. [46]

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4.3 Future work

There is a lot more information to extract from these types of datasets, especially regarding the neural brain development network. If there was more time during this project, the next step would’ve been to continue with the spatial analysis. A suggestion would be to look at the distance distribution of different k-values, distances between hub cells and trying to find a spatial pattern to how they are distributed, to further understand the network.

It would’ve been beneficial to have even more recorded data for each development stage, in order to draw conclusions with more certainty. This is also something that would be beneficial to do in the future.

In order to increase the reliability of deciding whether or not a network is a small- world network, a new definition of the small-world parameter ω (denoted ^𝜎_λin this thesis) has been suggested by Telesford, Qawi K. et al. (2011) as the following

ω =L_random

L − C

C_latt (14)

Where C of the network to be studied is compared to C_latt of an equivalent lattice network. By doing so, the Small-World parameter is less affected by fluctuations than it is when using eq. (3) since Crandom isn’t present in the denominator (for the explanation to why C_random in the denominator can lead to misclassification, see section 4.1 Limitations).

The values of ω in eq. 14 lies between −1 and 1, where −1 indicates a Regular network (L ≫ Lrandom, C ≈ C_latt), 𝜔 close to 0 indicates Small-World network (L ≈ L_random, C ≈ C_latt) and ω positive and close to 1 indicates Random network (L ≈ L_random, C ≪ C_latt). [47]

The field of network theory in cell biology is aiming to expand the knowledge about how the cells are the building blocks of the functional organism, especially within neuroscience, where circuits of neurons perform structure dependent computations [48] and there’s a lot more to learn in this field. In the future focus should be to further improve existing techniques for data collecting and cleaning the signal, as well as collecting more data and looking into this new equation for the Small-World parameter (eq. 14) in order to get more certain and reliable results.

Two photon laser scanning microscopes (TPLSM) and light sheet fluorescence microscopy (LSFM) are two methods that can image samples in 3D. This makes it possible to image the inside of a specimen without cutting. [49] Due to its low photo toxicity and the ability to image depth of the specimen, it’s a good technique to get live images of large biological samples, for example embryos. [50] This would improve the data recording and it could possibly take into account the network in 3D compared to in 2D as was done in this thesis. Keeping the specimen intact, without cutting (which obviously influences the specimen negatively, if the goal is to get as close as possible

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to the real life situation) while recording the data would give a result closer to the in vivo environment. The 2D picture that was obtained in this thesis doesn’t fully represent the in vivo circumstances.

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5. Conclusions

Complex networks are fundamental in widely different areas of life, and is an

important part of understanding how system works. In this thesis biological networks have been investigated with focus on the endothelial cell network of the mesenteric artery, and the neuron cell network in the developing brain. The ultimate goal of studying cell networks is to understand the complex system of how cells communicate, in order to further the understanding of how the body works and to be able to

discover ailments caused by deviations in the function of a network, to be able to find ways to treat these ailments.

The endothelial signal cell network was found to be Small-World network for all the four applied agonists and appeared to react similarly to all of them. Comparing the network plots of the endothelial cell networks they look similar with similar ranges of connectivity, where clusters are visible. The network parameters clearly indicates the Small-World network where the values of the parameters are very similar for all four agonists. The Small-World network has characteristics of both the Regular network and Random network, with short path lengths and high clustering. This makes the Small- World network a robust type of network and is found in many different contexts.

For the neural brain development networks, P8 was found to be a Small-World network, whereas the stages E12.5, E16.5, E18.5 and P6 were Random networks and for stage E14.5 the result was inconclusive. At stage E14.5 and E16.5 neuro- resp.

gliogenesis occur, which could have an effect on the network topology.

Network Analysis of Calcium Activity