Random geometric graphs and their applications in neuronal modelling

University dissertation from Lund University, Faculty of Science, Centre for Mathematical Sciences

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 con-nections (edges) in the brain using inhomogeneous random distance graph models. We presentfour models which have in common the characteristic of having a probability of connectionsbetween the nodes dependent on the distance between the nodes. In Paper I it is described aone-dimensional inhomogeneous random graph which introduce this connectivity dependenceon the distance, then the degree distribution and some clustering properties are studied. PaperII extend the model in the two-dimensional case scaling the probability of the connection bothwith the distance and the dimension of the network. The threshold of the giant componentis analysed. In Paper III and Paper IV the model describes in simplied way the growth ofpotential synapses between the nodes and describe the probability of connection with respectto distance and time of growth. Many observations on the behaviour of the brain connectivityand functionality indicate that the brain network has the capacity of being both functionalsegregated and functional integrated. This means that the structure has both densely inter-connected clusters of neurons and robust number of intermediate links which connect thoseclusters. The models presented in the thesis are meant to be a tool where the parametersinvolved can be chosen in order to mimic biological characteristics.

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