Random geometric graphs and their applications in neuronal modelling

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.