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In this thesis we use tools provided by dynamical systems theory to analyse transient behaviour of neural cells. We study models in Hodgkin-Huxley formalism that have a natural time-scale separation, which allows us to apply concepts of geometrical singular perturbation theory. The analysis presented in this thesis is applied to physiologically-realistic models represented by high-dimensional systems with multiple time scales. We construct a model of hippocampal pyramidal neurons and study after-depolarisation (ADP), which is a hallmark of excitability and precursor of transient bursting. Through careful analysis of the model we investigate the contribution of particular ionic currents to the excitability behaviour of the model. Furthermore using model simulations and experimental data we define ADP mathematically. Based on the essence of ADP and transient bursting we perform a reduction of the model, which enables an in-depth study of these phenomena. To understand spike adding during transient bursting we use a two-point boundary value formulation of the model, inspired by the experimental protocol. We show that the spikes are added through a canard-like transition, during which the orbit segment traces unstable sheets of a critical manifold up to a jump point. Our analysis suggests at least two mechanisms of spike adding: one is organised by a fold of the critical manifold and the other due to the presence of an additional unstable equilibria of the full system. The results of this study extend the definition of an excitability threshold and show that spikes can be added' through a continuous deformation of an orbit segment, not by a discontinuous abrupt process. We also apply the ideas of geometrical singular perturbation theory to study periodic bursting in a pituitary cell model. We use nullclines to investigate the nature of plateau bursting . taking place below the branch of attracting equilibria in the fast subsystem. Moreover, we continue orbit segments in order to compute the stable manifolds of the branch of saddle equilibria, which plays a role of separatrix in this system. We show that seemingly premature termination of the active phase of the plateau burst is related to the orbit crossing this separatrix before reaching the end of the stable equilibrium branch. Finally, we use numerical continuation to compute onsets of ADP and a spike in transient bursting. We show that these on sets correspond to extrema of slow variables of the full system. In our boundary value problem formation the on sets are detected as folds, which allows further continuation in two parameters to establish the boundaries of different model behaviours. This new technique is a form of parameter sensitivity analysis and, in principle, could be applied to other models.

The timely and appropriate trafficking of neurotransmitter receptors is fundamentally important to brain function, including synaptic plasticity. AMPA receptors mediate most of the fast excitatory neurotransmission in the central nervous system and are highly mobile proteins that undergo constitutive and activity-dependent insertion into and removal from synapses, which is thought to be the principal mechanism of synaptic plasticity. Meanwhile, the trafficking of inhibitory GABAB receptors is also involved in regulating synaptic plasticity, but the mechanisms underlying this regulation remain to be revealed. Receptor trafficking is heavily regulated by neuronal activation induced by neurotransmitters or other stimulations. We find that in primary cultured cortical neuron, elevated neuronal activity by activating NMDA receptors in different ways (chem-LTD, chem-LTP, Oxygen/Glucose deprivation) caused changed surface expression of GABABRs or AMPARs by cell surface biotinylation, indicating that both GABABRs and AMPARs undergo activation-dependent trafficking. In addition, activating AMPARs with agonist led to lysosomal-dependent internalization and degradation, providing a potential mechanism of loss of AMPARs during synaptic plasticity. Ubiquitination is a posttranslational modification, which involves conjugation of a 76-amino-acid protein ubiquitin to a lysine residue of the substrate protein. Ubiquitination mediates distinct cellular functions, including degradation. In the present study, ubiquitination of both GluA 1 and GluA2 was evaluated by immunoprecipitation. Interestingly, activation of NMDAR or AMPAR induced GluA2 ubiquitination in primary cultured cortical neuron; at the same time, ubiquitination of overexpressed GluA2 in HEK293 cell model can be accumulated through inhibiting protein degradation. These results suggest activation-dependent GluA2 ubiquitination is a key event in ligand binding-caused internalization and degradation. Ubiquitin-tagged protein cargoes in endosome membrane are sorted to lysosome for degradation by ESCRT (endosomal sorting complex required for transport) machinery. Here we also show that in HEK293 cell model, inhibition of TSG1 01, a member of ESCRT, by siRNA leads to an increased expression of AMPARs and accumulation of ubiquitinated GluA2. This observation infers that ESCRT plays a role in ubiquitination-regulated AMPAR degradation. The present study confirms the importance of ubiquitination to AMPAR trafficking. Ubiquitination-regulated receptordegradation through ESCRT machinery provides a mechanism for activation-dependent receptor trafficking.

Functional connectivity networks describe how regions of the brain interact. The timing, location, and frequency of these interactions inform about memory, decision making, motor movement, affective states, and more. However, while these interactions are well described as networks, these networks, like many others throughout nature, are constantly changing. Complex network evolution poses a highly dimensional problem but also contains much information about the system in question. In this thesis, a recent class of evolving complex network models was explored and extended to capture the functional connectivity dynamics observed in neuronal networks. Functional connectivity was investigated through data- and model-driven techniques at the cellular level, with cultures of cortical neurones on multi-electrode arrays, and at the whole-brain level, with electroencephalography. At the neuronal level, complex spatial dependencies were identified in bursts of excitation and two novel network models, the Starburst model and the Excitation Flow model, are used to capture the resulting functional connectivity. At the whole-brain level, functional connectivity dynamics were used to perform single-trial classification of intentional motor movement. Again, spatiotemporal dependencies were identified and used to present three novel techniques for modelling the network dynamics. The first two techniques decomposed networks into network templates (one model-based and one spectral-based) and modelled the dynamics with hidden Markov models. The final technique was a generalised evolving version of the Starburst model. The hidden Markov model of spectrally decomposed networks was shown to classify motor intentions with an accuracy around 80%. Firstly, this thesis shows that time plays an important role in the production of the complex network topologies observed in functional connectivity, both at the cellular and whole-brain leve1. Further, it is shown that evolving complex network models are very useful tools for modelling these topologies and that the network dynamics can be used to uncover features that are crucial to identifying functional states.

Neural fields models have been developed. to emulate large scale brain dynamics. They exhibit similar types of patterns as observed in real cortical tissue, such as travelling waves and persistent localised activity. The study of neural field models is yet a growing field of research, and in this thesis we contribute by developing new approaches to the analysis of pattern formation. A particular focus is on interface methods in one and two spatial dimensions. In the first part of this thesis we Study the influence of inhomogeneities on the velocity of propagating waves. We examine periodically modulated connectivity functions as well as fluctuating firing thresholds. For strong inhomogeneities we observe wave propagation failure and the emergence of stable localised solutions that do not exist in the homogeneous model. In the second part we develop a method to approximate stationary localised solutions and travelling waves in neural field models with sigmoidal firing rates. In particular, we devise a scheme that approximates the slope of these solutions and yields refined results upon iteration. We calculate explicit solutions for piecewise linear and piecewise polynomial firing rates. In the third part we develop an interface approach for planar neural field models. We derive the equations of motion for a certain class of synaptic connectivity function. In the interface description the evolution of a contour, which is defined by a level set condition, is governed by the normal velocity which depends exclusively on the shape of the contour. We present results for the existence and stability of various types of patterns. The interface description is also incorporated into a numerical scheme which allows to investigate pattern formation beyond instabilities.

The monoamines play important roles as neurotransmitters both centrally and peripherally, and are believed to play a key role in a number of pathological conditions such as Parkinson's disease, schizophrenia, drug addiction, and mood disorders, which has driven many years of research, resulting in a relatively good understanding of the neurochemistry of this group of amines. [n contrast, relatively little is known about the role of a second class of endogenous amines, the 'trace amines, that share substantial overlap with the monoamines. For 30 years the trace amines, have also been thought to be associated with affective behaviour, paranoid chronic schizophrenia and depression. However, until the recent discovery of a novel family Gprotein- coupled receptors (GPCRs), with high affinity for the trace amines, research was restricted. One member of this family, trace amine 1 receptor (TA1) is functional in both rodents and man, and is expressed widely throughout the brain of the mouse, rat and human. The functional role of the TAl receptor in the mammalian brain was therefore the focus of the current project. In order to investigate the function of the TAl receptor, two specific TAl receptor agonists were used, kindly provided by Hoffman La Roche. As the TAl receptor has been implicated in the neuropathophysiology of several psychiatric disorders, such as ADHD and schizophrenia, which are characterised by substantial and debilitating deficits in cognitive performance, initial experiments aimed to assess for any effect of the TAl agonists on cognitive performance in laboratory rodents. In part, the association between the TAl receptor and mental disorders is based on evidence suggesting an interaction with monoamine neurotransmission. However, previously reported findings from in vitro studies do not support a mechanism of interaction comparable with results from in vivo studies. Therefore later experiments investigated the relationship between TAl and monoamine neurotransmission in vivo. Initial experiments focused on the interaction between TAl activation and behaviours induced by psychostimulants known to alter monoamine function, and subsequent investigation examined the possible mechanisms involved using in vivo microdialysis.

A wealth of recent findings has demonstrated the outmost importance of neural activity the establishment of neural networks. GABAergic activity plays a central role in this process by exerting the earliest excitatory drive onto developing neurons. Progenitor proliferation, neuronal migration, arbour elaboration and development are all influenced by GABA. This early excitatory effect is due to an elevated intracellular concentration of C1- ([C1-]in), resulting in an outwardly directed driving force for C1- that renders GABAa transmission excitatory. During late developmental stages, as a result of an ontogenic decrease in [C1-]in that switches the polarity of the driving force for C1-, GABA becomes inhibitory. The commonly accepted hypothesis concerning the mechanisms underlying the shift in [C1-]n stipulates that the Na+/K+/C1- cotransporter 1 (NKCC1) accumulates C1- in embryonic neurons and that an ontogenic increase in the expression of the K+/C1- cotransporter 2 (KCC2) that extrudes C1- from adult neurons, induces the shift. A concurrent downregulation of NKCCl is also thought to participate in the shift. The switch of GABAergic polarity constitutes a major step in the maturation of neural networks.

A study has been made on the effects of the removal of the frontal ganglion from adult Locusta migratoria migratorioides R & F on various aspects of metabolism. A decrease in lipid and carbohydrate reserves was observed in the fat body of operated animals although this was less marked when compared with starved specimens. This decrease in fat body reserves was not due to an increased release into the haemolymph nor to increased utilisation. Indeed oxygen consumption was significantly lower than in operated controls. The fine structural appearance of the fat body cells indicated a cessation of the normal functional development. This tended to be confirmed by the observation that fat body from operated animals showed a reduced ability to incorporate C(^14) glucose into triglycerides in vitro. Six digestive enzymes have been characterised and the optimal conditions for assay employed in determining the effects of frontal ganglion removal and starvation on the activities of these enzymes in various regions of the gut. Both treatments resulted in a reduction in amount of enzyme activity. The distribution of enzyme activity was also affected such that the foregut of starved and frontal ganglionectomised treatments had a smaller proportion of the activity than in the control. The fact that frontal ganglion removal effects a dramatic reduction in haemolymph volume was confirmed. Ultrastructural changes in the Malpighian tubules adds support to the suggestion that this reduced blood volume vas a result of chronic diuresis. However, this was shown not to be due to an effect on the activity of the Na (^+)K (^+)ATPase exchange pump. Removal of the frontal ganglion was shown to result in a cessation of growth and to reduce or prevent altogether the release of neurosecretory material from the corpus cardiacum, confirming previous work on larvae and adults. The above observations are discussed in the light of endocrine control of metabolism and the proposal that food passage through the gut is reduced in frontal ganglionectomised animals.

This thesis examines the roles of inflammation and the chondroitin sulphate proteoglycan NG2 in regeneration of injured axons in the adult mammalian nervous system, against a background suggesting that inflammation around the cell bodies of axotomised neurons enhances axonal regeneration and that NG2 is a major inhibitor of CNS axonal regeneration. 1) Lipopolysaccharide was placed on / injected in motor cortex of rats, with or without concomitant injury of the cervical corticospinal tract (CST). The inflammatory response and expression of the growth-associated genes c-jun, ATF3, SCG10 and GAP-43 was investigated by immunohistochemistry or in situ hybridisation. Retrograde labelling identified CST neuron cell bodies, and anterograde tracing of CST axons identified axonal sprouting / regeneration. Lipopolysaccharide-induced inflammation promoted upregulation of GAP-43 (briefly), c-jun and SCG10 (for two weeks) in CST neurons, but did not enhance regeneration of injured CST axons. 2) Axonal regeneration was examined in the CNS and PNS of NG2 knockout mice. CNS regeneration was assessed following dorsal column injury in ascending axons with cholera toxin-conjugated horseradish peroxidase (CT-HRP) and in descending CST axons with anterograde labelling with biotinylated dextran amine (BDA), as well as transganglionic labelling of transected dorsal roots with CT-HRP. PNS regeneration after sciatic nerve crush was assessed anatomically by: retrograde labelling (from the hindpaw) of L4/5 dorsal root ganglion cells immunohistochemistry to detect sensory axons in hindpaw skin silver-cholinesterase staining of soleus motor axons / end plates EM counts of tibial and digital nerves. Functional recovery was assessed by the (motor) toe spreading reflex and (sensory) responses to von Frey hairs. There was neither anatomical nor functional evidence for significant effects on CNS or PNS axonal regeneration in the knockout mice. These findings suggest that NG2 is not a major inhibitory factor in the failure of CNS regeneration and is not important for successful axonal regeneration in the PNS.

The stereotyped motor behaviour of the Drosophila larva provides a useful model for the development and function of neural circuitry. However, the cellular and molecular basis of this behaviour is poorly understood. Reported in this thesis is an analysis of slowmo (slmo), a gene previously shown to be expressed in the developing embryonic central nervous system of Drosophila. Null mutants of slmo are able to hatch, but the resulting larvae exhibit a progressive phenotype of defective locomotor activity. An enhancer trap, P(GAL4)c682, which is inserted in the slmo locus, reports expression in a subset of neurons within the embryonic, larval and adult central nervous systems. Inactivation of marked neurons with tetanus toxin light chain leads to severely impaired motor function, but not total paralysis. Affected embryos are capable of some sporadic movements, but are unable to hatch and normal peristaltic contraction waves are largely absent. The slmo gene is shown to encode a member of a novel family of conserved proteins of unknown function. GFP fusions of Slmo are shown to localise to the mitochondria in a cultured cell line. Two novel slmo related genes, termed pre/i/ (pre!) and real-time (retm), are identified in Drosophila. The prel gene is expressed ubiquitously during embryonic development, and disruption of the gene by a P-insertion results in lethality during larval development. retm encodes a member of a novel subclass of larger Slmo related proteins which contain the conserved CRAL- TRIO domain thought to be involved in the transport of small hydrophobic ligands. GFP fusions of both Prel and Retm are also associated with the mitochondria in cell culture, suggesting this might be true all proteins of this family. Using a yeast-2-hybrid approach, the identification of candidate Slmo interacting proteins is described, providing a basis for future work on the function of Slmo.