Neural Dynamics On Complex Networks

This thesis investigate the dynamics of neural networks. The main investigations are the influence of the topological properties of networks on the dynamics of neural networks, and the role of the sparse nature of complex networks in the functions of neural networks. Meanwhile, the dynamical origin of preventing the synchrony in neural networks is studied. Before the main investigations, from a nonlinear dynamics point of view, the features of the study on neural activities are introduced and the theory of complex networks invoked in this thesis is introduced. The studies in this thesis are as following:First, the effects of the degree distribution on mutual synchronization of two-layer neural networks are studied. In the study three coupling strategies are carried out: large-large coupling, random coupling, and small-small coupling. By computer simulations and analytical methods, it is find that couplings between nodes with large degree play an important role in the synchronization. For large-large coupling, less couplings are needed for inducing synchronization for both random and scale-free networks. In contrast, cutting couplings between nodes with large degree is very efficient for preventing neural systems from synchronization, especially when subnetworks are scale free. The analysis treatment shows that the degree distribution of subsystems affects the synchronization between two networks.Second, we investigate the influence of efficacy of synaptic interaction on firing synchronization in excitatory neuronal networks. We find spike death phenomena: namely, the state of neurons transits from the limit cycle to a fixed point or transient state. The phenomena occur under the perturbation of an excitatory synaptic interaction, which has a high efficacy. We show that the decrease of synaptic current results in spike death through depressing the feedback of the sodium ionic current. In the networks with the spike death property the degree of synchronization is lower and insensitive to the heterogeneity of neurons. The mechanism of the influence is that the transition of the neuron state disrupts the adjustment of the rhythm of the neurons oscillation and prevents a further increase of the firing synchronization.Third, the response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated networks. We find that the degree correlation of scale-free networks makes the dynamics of attractor systems different from uncorrelated ones. The-dynamics of correlated scale-free attractor networks results in the effects of degree correlation on the response to stimuli.Fourth, we use the Hopfield attractor networks as an example to study the role of sparse connection density in the difference of functional performance of complex networks. In simulations, we find that the stability of patterns between random and scale-free networks has a maximum difference with a specific sparse connection density. The result suggests that there exists a sparse density with which the dynamics of networks are affected most significantly by topological properties. Using the signal-to-noise-ratio analysis, we show that the non-monotonicity is induced by the competition between the distinction of degree distribution and the signal strength.