Random geometric graphs and their applications in neuronal modelling

Ajazi, Fioralba

Random geometric graphs and their applications in neuronal modelling

Mark

Ajazi, Fioralba ^LU (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
one-dimensional 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 two-dimensional 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
one-dimensional 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 two-dimensional 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
one-dimensional 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 two-dimensional 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
one-dimensional 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 two-dimensional 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/b68cb74f-3fa5-4260-a7c4-1d2a4bb33507

author

Ajazi, Fioralba ^LU

supervisor

Tatyana Turova ^LU
Valerie Chavez

opponent

Professor Britton, Tom, Stockholm University, Sweden

organization

Mathematical Statistics

publishing date

2018-09

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, Sölvegatan 18, Lund

defense date

2018-09-27 09:00:00

ISBN

9789177537984

9789177537991

language

English

LU publication?

yes

id

b68cb74f-3fa5-4260-a7c4-1d2a4bb33507

date added to LUP

2018-09-03 14:23:13

date last changed

2024-02-13 15:05:53

@phdthesis{b68cb74f-3fa5-4260-a7c4-1d2a4bb33507,
  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/>one-dimensional 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 two-dimensional 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}},
}

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Random geometric graphs and their applications in neuronal modelling