Random geometric graphs and their applications in neuronal modelling
(2018) Abstract
 Random graph theory is an important tool to study different problems arising from real world.
In this thesis we study how to model connections between neurons (nodes) and synaptic connections (edges) in the brain using inhomogeneous random distance graph models. We present
four models which have in common the characteristic of having a probability of connections
between the nodes dependent on the distance between the nodes. In Paper I it is described a
onedimensional inhomogeneous random graph which introduce this connectivity dependence
on the distance, then the degree distribution and some clustering properties are studied. Paper
II extend the model in the twodimensional case scaling the probability of the... (More)  Random graph theory is an important tool to study different problems arising from real world.
In this thesis we study how to model connections between neurons (nodes) and synaptic connections (edges) in the brain using inhomogeneous random distance graph models. We present
four models which have in common the characteristic of having a probability of connections
between the nodes dependent on the distance between the nodes. In Paper I it is described a
onedimensional inhomogeneous random graph which introduce this connectivity dependence
on the distance, then the degree distribution and some clustering properties are studied. Paper
II extend the model in the twodimensional case scaling the probability of the connection both
with the distance and the dimension of the network. The threshold of the giant component
is analysed. In Paper III and Paper IV the model describes in simplied way the growth of
potential synapses between the nodes and describe the probability of connection with respect
to distance and time of growth. Many observations on the behaviour of the brain connectivity
and functionality indicate that the brain network has the capacity of being both functional
segregated and functional integrated. This means that the structure has both densely inter
connected clusters of neurons and robust number of intermediate links which connect those
clusters. The models presented in the thesis are meant to be a tool where the parameters
involved can be chosen in order to mimic biological characteristics. (Less)  Abstract (Swedish)
 Random graph theory is an important tool to study different problems arising from real world.
In this thesis we study how to model connections between neurons (nodes) and synaptic con
nections (edges) in the brain using inhomogeneous random distance graph models. We present
four models which have in common the characteristic of having a probability of connections
between the nodes dependent on the distance between the nodes. In Paper I it is described a
onedimensional inhomogeneous random graph which introduce this connectivity dependence
on the distance, then the degree distribution and some clustering properties are studied. Paper
II extend the model in the twodimensional case scaling the probability of the... (More)  Random graph theory is an important tool to study different problems arising from real world.
In this thesis we study how to model connections between neurons (nodes) and synaptic con
nections (edges) in the brain using inhomogeneous random distance graph models. We present
four models which have in common the characteristic of having a probability of connections
between the nodes dependent on the distance between the nodes. In Paper I it is described a
onedimensional inhomogeneous random graph which introduce this connectivity dependence
on the distance, then the degree distribution and some clustering properties are studied. Paper
II extend the model in the twodimensional case scaling the probability of the connection both
with the distance and the dimension of the network. The threshold of the giant component
is analysed. In Paper III and Paper IV the model describes in simplied way the growth of
potential synapses between the nodes and describe the probability of connection with respect
to distance and time of growth. Many observations on the behaviour of the brain connectivity
and functionality indicate that the brain network has the capacity of being both functional
segregated and functional integrated. This means that the structure has both densely inter
connected clusters of neurons and robust number of intermediate links which connect those
clusters. The models presented in the thesis are meant to be a tool where the parameters
involved can be chosen in order to mimic biological characteristics. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/b68cb74f3fa54260a7c41d2a4bb33507
 author
 Ajazi, Fioralba ^{LU}
 supervisor

 Tatyana Turova ^{LU}
 Valerie Chavez
 opponent

 Professor Britton, Tom, Stockholm University, Sweden
 organization
 publishing date
 201809
 type
 Thesis
 publication status
 published
 subject
 keywords
 random graph, Neural Network, Probability, Inhomogeneous random graph, random distance graph, random grown networks
 pages
 113 pages
 publisher
 Lund University, Faculty of Science, Centre for Mathematical Sciences
 defense location
 Lecture Hall MH:R, Matematikcentrum, SÃ¶lvegatan 18, Lund
 defense date
 20180927 09:00:00
 ISBN
 9789177537984
 9789177537991
 language
 English
 LU publication?
 yes
 id
 b68cb74f3fa54260a7c41d2a4bb33507
 date added to LUP
 20180903 14:23:13
 date last changed
 20240213 15:05:53
@phdthesis{b68cb74f3fa54260a7c41d2a4bb33507, abstract = {{Random graph theory is an important tool to study different problems arising from real world.<br/>In this thesis we study how to model connections between neurons (nodes) and synaptic connections (edges) in the brain using inhomogeneous random distance graph models. We present<br/>four models which have in common the characteristic of having a probability of connections<br/>between the nodes dependent on the distance between the nodes. In Paper I it is described a<br/>onedimensional inhomogeneous random graph which introduce this connectivity dependence<br/>on the distance, then the degree distribution and some clustering properties are studied. Paper<br/>II extend the model in the twodimensional case scaling the probability of the connection both<br/>with the distance and the dimension of the network. The threshold of the giant component<br/>is analysed. In Paper III and Paper IV the model describes in simplied way the growth of<br/>potential synapses between the nodes and describe the probability of connection with respect<br/>to distance and time of growth. Many observations on the behaviour of the brain connectivity<br/>and functionality indicate that the brain network has the capacity of being both functional<br/>segregated and functional integrated. This means that the structure has both densely inter<br/>connected clusters of neurons and robust number of intermediate links which connect those<br/>clusters. The models presented in the thesis are meant to be a tool where the parameters<br/>involved can be chosen in order to mimic biological characteristics.}}, author = {{Ajazi, Fioralba}}, isbn = {{9789177537984}}, keywords = {{random graph; Neural Network; Probability; Inhomogeneous random graph; random distance graph; random grown networks}}, language = {{eng}}, publisher = {{Lund University, Faculty of Science, Centre for Mathematical Sciences}}, school = {{Lund University}}, title = {{Random geometric graphs and their applications in neuronal modelling}}, url = {{https://lup.lub.lu.se/search/files/50587728/PhD_thesis_F.pdf}}, year = {{2018}}, }