Weak connections form an infinite number of patterns in the brain

This work is supporting in part by NSFC (61172070), Innovation Research Team of Shaanxi Province (2013KCT-04), Key Program of Nature science Foundation of Shaanxi Province (20162DJC-01) and EPSRC (EP/I032606/1).Peer reviewedPublisher PD

Mathematical modelling and brain dynamical networks

In this thesis, we study the dynamics of the Hindmarsh-Rose (HR) model which studies the spike-bursting behaviour of the membrane potential of a single neuron. We study the stability of the HR system and compute its Lyapunov exponents (LEs). We consider coupled general sections of the HR system to create an undirected brain dynamical network (BDN) of Nn neurons. Then, we study the concepts of upper bound of mutual information rate (MIR) and synchronisation measure and their dependence on the values of electrical and chemical couplings. We analyse the dynamics of neurons in various regions of parameter space plots for two elementary examples of 3 neurons with two different types of electrical and chemical couplings. We plot the upper bound Ic and the order parameter rho (the measure of synchronisation) and the two largest Lyapunov exponents LE1 and LE2 versus the chemical coupling gn and electrical coupling gl. We show that, even for small number of neurons, the dynamics of the system depends on the number of neurons and the type of coupling strength between them. Finally, we evolve a network of Hindmarsh-Rose neurons by increasing the entropy of the system. In particular, we choose the Kolmogorov-Sinai entropy: HKS (Pesin identity) as the evolution rule. First, we compute the HKS for a network of 4 HR neurons connected simultaneously by two undirected electrical and two undirected chemical links. We get different entropies with the use of different values for both the chemical and electrical couplings. If the entropy of the system is positive, the dynamics of the system is chaotic and if it is close to zero, the trajectory of the system converges to one of the fixed points and loses energy. Then, we evolve a network of 6 clusters of 10 neurons each. Neurons in each cluster are connected only by electrical links and their connections form small-world networks. The six clusters connect to each other only by chemical links. We compare between the combined effect of chemical and electrical couplings with the two concepts, the information flow capacity Ic and HKS in evolving the BDNs and show results that the brain networks might evolve based on the principle of the maximisation of their entropies

Emergence of local synchronization in neuronal networks with adaptive couplings

Local synchronization, both prolonged and transient, of oscillatory neuronal behavior in cortical networks plays a fundamental role in many aspects of perception and cognition. Here we study networks of Hindmarsh-Rose neurons with a new type of adaptive coupling, and show that these networks naturally produce both permanent and transient synchronization of local clusters of neurons. These deterministic systems exhibit complex dynamics with 1/fη power spectra, which appears to be a consequence of a novel form of self-organized criticality

Control strategies of 3-cell Central Pattern Generator via global stimuli

The study of the synchronization patterns of small neuron networks that control several biological processes has become an interesting growing discipline. Some of these synchronization patterns of individual neurons are related to some undesirable neurological diseases, and they are believed to play a crucial role in the emergence of pathological rhythmic brain activity in different diseases, like Parkinson''s disease. We show how, with a suitable combination of short and weak global inhibitory and excitatory stimuli over the whole network, we can switch between different stable bursting patterns in small neuron networks (in our case a 3-neuron network). We develop a systematic study showing and explaining the effects of applying the pulses at different moments. Moreover, we compare the technique on a completely symmetric network and on a slightly perturbed one (a much more realistic situation). The present approach of using global stimuli may allow to avoid undesirable synchronization patterns with nonaggressive stimuli

Existence of Mutual Stabilization in Chaotic Neural Models

Recent work has demonstrated that interacting chaotic systems can establish persistent, periodic behavior, called mutual stabilization, when certain information is passed through interaction functions. In particular, this was first shown with two interacting cupolets (Chaotic Unstable Periodic Orbit-lets) of the double scroll oscillator. Cupolets are highly accurate approximations of unstable periodic orbits of a chaotic attractor that can be generated through a control scheme that repeatedly applies perturbations along Poincaré sections. The decision to perturb or not to perturb the trajectory is determined by a bit in a binary control sequence. One interaction function used in the original cupolet research was based on integrate-and-fire dynamics that are often seen in neural and laser systems and was used to demonstrate mutual stabilization between two double scroll oscillators. This result provided the motivation for this thesis where the stabilization of chaos in mathematical models of communicating neurons is investigated. This thesis begins by introducing mathematical models of neurons and discusses the biological realism of the models. Then, we consider the two-dimensional FitzHugh-Nagumo (FHN) neural model and we show how two FHN neurons can exhibit chaotic behavior when communication is mediated by a coupling constant, g, representative of the synaptic strength between the neurons. Through a bifurcation analysis, where the synaptic strength is the bifurcation parameter, we analyze the space of possible long-term behaviors of this model. After identifying regions of periodic and chaotic behavior, we show how a synaptic sigmoidal learning rule transitions the chaotic dynamics of the system to periodic dynamics in the presence of an external signal. After the signal passes through the synapse, synaptic learning alters the synaptic strength and the two neurons remain in a persistent, mutually stabilized periodic state even after the signal is removed. This result provides a proof-of-concept for chaotic stabilization in communicating neurons. Next, we focus on the 3-dimensional Hindmarsh-Rose (HR) neural model that is known to exhibit chaotic behavior and bursting neural firing. Using this model, we create a control scheme using two Poincaré sections in a manner similar to the control scheme for the double scroll system. Using the control scheme we establish that it is possible to generate cupolets in the HR model. We use the HR model to create neural networks where the communication between neurons is mediated by an integrate-and-fire interaction function. With this interaction, we show how a signal can propagate down a unidirectional chain of chaotic neurons. We further show how mutual stabilization can occur if two neurons communicate through this interaction function. Lastly, we expand the investigation to more complicated networks including a feedback network and a chain of neurons that ends in a feedback loop between the two terminal neurons. Mutual stabilization is found to exist in all cases. At each stage, we comment on the potential biological implications and extensions of these results