Synchronisation in signed complex networks motivated by neural population networks

© 2017 Dr Saeed AhmadizadehSynchronisation is a ubiquitous phenomenon which is observed in a wide range of network applications. The absence or presence of synchronisation depends on network features which are characterised by two factors: (i)~dynamics of each subsystem and (ii) network topology. This thesis develops tools for analysing synchronisation via the aforementioned two factors. Our results concerning the network topology are quite general, and, the rest of results are tailored for networks of neural populations. Nevertheless, they can be adopted for other classes of networks. Neural mass populations describe the averaged activity of cortical ensembles which, from the modelling perspective, is particularly important for epilepsy, as this is the scale observed through clinical electroencephalographic (EEG) and intracranial EEG recordings. From the analysis point of view, these networks pose new challenges in studying synchronisation, as their models are non-linear and their interconnections are directed with both positive and negative weights which, in network science, are known as directed signed graph networks. First, bifurcation analyses are conducted to explore various ranges of behaviours that can be generated by a simple network of neural populations by initialising it around equilibrium points. The underlying network contains two neural populations; each of them is modelled by a well-known Jansen-Rit model. For different values of interconnection gains, bifurcation diagrams are presented, where the input to the neural regions is the bifurcation parameter. The bifurcation analyses reveal that this network can generate various unexpected oscillatory activities, such as delta wave. Surprisingly, by changing the value of the input in one neural population, another neural population shows spike-wave output observed during epilepsy. Using human clinical data, we also investigated the suitability of these models for monitoring seizures. To analyse a larger network of neural populations, we use mathematical tools from control theory and graph theory to study synchronisation in Wilson-Cowan oscillators. This class of oscillators describes the activity of both excitatory and inhibitory populations of neurons and reproduces self-sustained oscillations observed in EEG signals. Since the interest is to investigate synchronisation with respect to a specific limit-cycle, that is responsible for producing oscillatory output, we linearise the model of the whole network around that limit-cycle, leading to a periodic linear time-varying system. By applying an appropriate change of coordinates, we transform the linearised system to a new system that itself contains a number of subsystems. We then observe that stability~(instability) of those subsystems, excluding one of them, implies the presence~(absence) of synchronous behaviour in the original network. The stability/instability of aforementioned subsystems depends on parameters of the models, as well as eigenvalues of a so-called ``Laplacian matrix" which solely depends on interconnection gains~(network structure). We propose a combination of numerical and theoretical approaches to check the stability/instability of those subsystems in terms of their eigenvalues. To prove the stability and instability results, we use Floquet theory along with Lyapunov stability and Chetaev instability theorems. For the numerical approach, we use a tool from robust control theory. We also investigate how changes in the network can affect the presence/absence of synchronisation. These results are developed using Weyl's inequality and results from the first part of analysis. Eigenvalues of the Laplacian matrix play a crucial role in reaching objectives such as synchronisation, formation control and etc in all network applications. For directed signed networks, it is still challenging to characterise eigenvalues of Laplacian matrix in terms of network structure. To this end, we consider a class of directed signed networks that have a specific property : the real parts of the non-zero eigenvalues of its Laplacian matrix are positive. This property, called a ``nice property", and ensures that this class of network achieves synchronisation for applications such as the decision-making in social networks. The absence of this nice property leads to the absence of synchronous behaviour in applications such as in neural populations. Under a mild assumption, we provide a necessary and sufficient condition under which the following statement holds: ``if edges between an arbitrary pair of nodes are perturbed with negative weights that satisfy an easily computed bound, the perturbed network has also the nice property. Furthermore, under certain conditions, we identify ``sensitive pairs of nodes", which if connected by infinitesimal negative weights, the network cannot have the nice property. Our proof techniques for these results are novel and based on tools from linear algebra