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The mammalian suprachiasmatic nuclei (SCN) are located in the ventral part of the hypothalamus and orchestrate circadian rhythms in physiology and behaviour. The ~20.000 neurones of the murine SCN express key molecular clock components including the Cryptochrome (Cry1/2) and Period (Per1/2/3) genes and their protein products CRY1/2 and PER1/2/3. Using different mouse models, this work demonstrates that with disrupted expression of CRY in the after-hours (Afh/Afh) mouse, cells of the ventral part of the SCN (vSCN) have a propensity to desynchronise. They receive increased GABAergic inputs and are less excitable during the projected night but not during the day compared to congenic wildtype (+/+). The linkage between CRY protein expression and the reduced excitability at night is supported by recordings from SCN cells of Cry2 deficient mice (Cry2-/-), which exhibit similar electrophysiological behaviour. Luminometrical recordings of single cell Per2 expression confirms the involvement of GABAergic signalling in both, maintaining a coherent rhythm in synchronised SCN cells from +/+ controls and the propensity of Afh/Afh SCN cells to desynchronise.A mechanism by which neuronal excitability is regulated in mammals, is the modulation of activity of small-conductance Ca2+-activated K+ (SK) channels. Western blot analysis demonstrates the expression of SK2 and SK3 channel protein in SCN neurones. Functionally, we show with whole cell electrophysiology, calcium imaging and luminometry how SK channels regulate the levels of intracellular calcium ([Ca2+]i) from day to night. In the more hyperpolarised SCN network of the Afh/Afh genotype at night, SK channel activity is altered and contributes to the lower single cell excitability.Vasoactive intestinal polypeptide (VIP) and its cognate receptor, VPAC2, are synthesised by SCN neurones and this intercellular signal facilitates coordination of suprachiasmatic neuronal activity. How the loss of VPAC2 receptor signalling affects the electrophysiology of SCN neurones and their response to excitatory inputs is unclear. Here we made patch clamp recordings of SCN neurones in brain slices prepared from animals that do not express VPAC2 receptors (Vipr2-/- mice) as well as non-transgenic animals (Vipr2+/+ mice). While Vipr2+/+ SCN neurones exhibit coordinated day-night variation in their electrical state, Vipr2-/- neurones do not and instead manifest a range of states during both day and night. We find that Vipr2+/+ neurones vary the membrane threshold potential at which they start to fire actions potentials from day to night, while Vipr2-/- neurones lack this variation. This is due to Vipr2-/- neurones lacking a voltage-gated sodium current. Subsequently we determine that this aberrant temporal control of neuronal state and excitability alters appropriate neuronal responses to a neurochemical mimic of the light-input pathway to the SCN. Conclusively, these results highlight the critical role intercellular signalling plays in the activity of individual neuronal state and their response to neural input as well as ensemble activity and function of the suprachiasmatic neural network.

Daily rhythms of physiology and behaviour in mammals are orchestrated by a hierarchical network of cellular oscillators. The master pacemaker that defines local and systemic timing across the brain and body are the suprachiasmatic nuclei of the hypothalamus (SCN). Disruption to the timing of sleep and daily behavioural activity can manifest in a range of pathologies including neuropsychiatric disorders. Bipolar disorder (BPD) is once such neurological condition that exhibits profound associations with altered circadian rhythm generation and whose toolkit of pharmacological interventions impact upon circadian rhythm generation. Currently it is unclear exactly how changes to rhythmic physiology contribute to the aetiology and pathology of BPD. In recent years, rodent models possessing lesions within genes that make up the basic cellular oscillator are widely reported to exhibit concomitant changes in affective behaviours, namely mania-like phenotypes. Recently a mouse model possessing a mutation within the neuron-specific Na+/K+-ATPase (NKA) alpha3 subunit, known as Myshkin, was described as a model of the manic phase of BPD. The NKA alpha3 is not reported as a critical element of the circadian oscillator and we used this opportunity to characterise the behavioural and physiological circadian system of these animals. Under wheel-running paradigms Myk/+ animals exhibited a broad array of behavioural deficits including lengthened, low amplitude and labile free-running rhythms, altered phase re-setting and elevated metabolic activity. Physiological characterisation of the SCN revealed deficits in amplitude of electrical output and changes to post-synaptic signalling although the ex vivo molecular pacemaking of the SCN remained intact. Myshkin animals therefore represent a novel model in which changes to central output arise independently of changes to basic molecular pacemaking. Despite this seemingly distinct mechanism Myshkin animals share many mood and circadian phenotypes with other clock gene models of affective behaviours highlighting that changes to pacemaking output of the SCN may be a critical factor across animal models exhibiting circadian and mood deficits. In addition, the impact of the mood stabiliser lithium, commonly prescribed in BPD, on cellular pathways within the SCN was investigated. Lithium consistently lengthens the period of cellular and behavioural rhythms in mammals although the mechanism of this action is yet undefined. Glycogen synthase kinase 3β (GSK3β) and inositol monophosphatase (IMPase) are the major biochemical targets of lithium at therapeutic concentrations. GSK3β is known to shorten rhythms and this study targeted IMPase and inositol phosphate turnover in the period lengthening effects of lithium. We reveal that although inhibition of IMPase dampens SCN molecular rhythms, the period of oscillations remains unchanged and therefore lithium acts upon distinct cellular pathways within the SCN to exert effects on period.

In this thesis, our aim is to study a Cournot tatonnement system which exhibits destabilisation of the Cournot equilibrium as the number of firms increase. Our approach is to first consider the special case of firms behaving identically in a market share attraction model in two different adjustment process: Cournot tatonnement and bounded rationality adjustment. Results from the Cournot tatonnement system shows a superstable equilibrium in two firms model and an unstable equilibrium in a five firms model. In the five firms model, we show that introducing heterogeneity stabilises the Cournot equilibrium. For both two and five firms model, the differences of costs between firms are critical for the convergence of the system to the Cournot equilibrium. Lastly, we study the effect of entries and exits of firms on the number of active firms in the market. We discover that the market can sustain between two to four firms, and the factors are differences of costs and initial outputs between firms, and barriers to entries.

A key part of a fruit fly's development is the formation of segmentsin its body. These structures are built by the protein forms of so-calledsegment polarity (SP) genes. It is the asymmetric expression of SPgenes which creates the fruit-fly's segmental structure. The SP genesand their products (e.g. proteins) can be said to form a system which isself-regulating, i.e. genes are used to make proteins and, in turn,proteins are used to turn genes on or off.How this system achieves stable asymmetry of this kind ismathematically interesting as it can be thought of in a differentway - multiple symmetries in the same system. This is unusual and weattempt to explain how it is possible using a mathematical modelconstructed by von Dassow et al. When trying to understand abiological system of this kind, there are two main approaches -reductionist and holistic. We try to show that they are notmutually exclusive - we look at the whole system but reduce what ismeant by the whole.For example, von Dassow's model is large scale and, using it as atemplate, we show that a similar (but smaller) model inherits itsproperties. Smaller models can be made by short-handing thetranslation process (through which RNA is used to make protein)wherever an SP gene has a unique protein form.Our data indicates that the simultaneous wild-type expression ofkey SP genes (engrailed and wingless) takes places only whencumulative regulation of the wingless gene by two SP proteins isweak. The absence of this regulation would explain coexistence ofmultiple mathematical symmetries in one system (representative ofgenetic asymmetry) as it acts like a division between them. In thisway, the system itself can be thought to divide into two independentsub-systems which can be treated separately.

mRNA translation is one of the fundamental and well controlled cellular process requiringthe combined function of a large number of molecular components. The three main stagesof translation, initiation, elongation and termination are facilitated by more than 20 proteinsknown as translation factors. Translation is the final step in the flow of genetic information,and regulation at this level allows for an immediate and rapid response to changes inphysiological conditions. The control exerted at the systems level of translation has notprecisely been characterized. Three different techniques have been employed toquantitative the control exerted by the respective translation factors.In the first approach, employing the microscopic techniques, in vivo intra-cellulardistribution of translation elongation and release factors were analysed with TCM and GFPtags. The result indicates that the factors are cytoplasmically distributed which cannotinfluence the overall translational control. In the second approach, the protein expressionlevels of the elongation and release factors were titrated progressively to explore theircontrol effects on global translation regulation. The endogenous promoter of eachtranslation factor was substituted by the tetO7 synthetic promoter to regulate the expressionlevel in response to varying concentrations of doxycycline. Measurement of proteinsynthesis rate and the growth rate at different levels of the elongation and release factorsprovide insight to system-level control. The results indicate that the elongation factorseEF1A and eEF2 and the release factor eRF1 exert an unexpectedly high degree of controlover translation rate. Moreover, these factors, along with elongation factor eEF3 werefound to be functionally dedicated to translation, in contrast to eEF1B and eRF3, which isevidently multifunctional. In the third approach, a mathematical model has been developedto represent the control landscape of the translational machinery. This translation model is apowerful tool that will be used in the quantitative analysis of translation when two factorsare made limiting at a time. The extensive study carried out on the translational regulationof Saccharomyces cerevisiae reveals an interesting observation of the involvement of eachtranslation factors. For the first time, the quantitative measurement of the translationalregulation reveals the translational regulation exerted by individual translation factors.