Call number: LD2668 .T4 1963 W78 / Master of Science

LD2668 .T4 1966 W635 / Master of Science

published_or_final_version / Business Administration / Master / Master of Business Administration

published_or_final_version / abstract / Chemistry / Master / Master of Philosophy

published_or_final_version / abstract / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Radial glial cells play an important role during embryonic development in mammals. They are not only important for neural production but help to organise the architecture of the neocortex. Glial cells proliferate during the development of the brain in the embryo, before differentiating to produce neurons at a rate which increases towards the end of embryonic brain development. Glial cells communicate via Adenosine tri-phosphate (ATP) mediated calcium waves, which may have the effect of locally synchronising cell cycles, so that clusters of cells proliferate together, shedding cells in uniform sheets. Hence radial glial cells are not only responsible for the production of most neocortical neurons but also contribute to the architecture of the brain. It has been argued that human developmental disorders which are associated with cortical malfunctions such as infantile epilepsies and mental retardation may involve defects in neuronal production and/or architecture and mathematical modelling may shed some light upon these disorders. This thesis investigates, among other things, the conditions under which radial glial cells' cell cycles become `phase locked', radial glia proliferation and stochastic effects. There are various models for the cell cycle and for intracellular calcium dynamics. As part of our work, we marry two such models to form a model which incorporates the effect of calcium on the cell cycle of a single radial glial cell. Furthermore, with this achieved we consider populations of cells which communicate with each other via the secretion of ATP. Through bifurcation analysis, direct numerical simulation and the application of the theory of weakly coupled oscillators, we investigate and compare the behaviour of two models which differ from each other in the time during the cell cycle at which ATP is released. Our results from this suggest that cell cycle synchronisation is highly dependent upon the timing of ATP release. This in turn suggests that a malfunction in the timing of ATP release may be responsible for some cortical development disorders. We also show how the increase in radial glia proliferation may mostly be down to radial glial cells' ability to recruit quiescent cells onto the cell cycle. Furthermore, we consider models with an additive noise term and through the application of numerical techniques show that noise acts to advance the onset of oscillatory type solutions in both models. We build upon these results and show as a proof of concept how noise may act to enhance radial glia proliferation.

Modelling and reconstruction of genetic regulatory networks has developed in a wide field of study in the past few decades, with the application of ever sophisticated techniques. This thesis looks at how models for genetic networks have been developed from simple Boolean representations to more complicated models that take into account the inherent stochasticity of the biological system they are modelling. Statistical techniques are used to help predict the interaction between genes from microarray data in order to recover genetic regulatory networks and provide likely candidates for interactions that can be experimentally verified. The use of Granger causality is applied to statistically assess the effect of one gene upon another and modifications to this are presented, with bootstrapping used to understand the variability present within the parameters. Given the large amounts of data to be analysed from microarray experiments, clustering techniques are used to help reduce the computational burden and novel algorithms are developed to make use of such clustered data. Variability within clusters is also considered, by developing a novel approach with the use of principal component analysis. These algorithms that are developed are implemented with an observed dataset from Xenopus Laevis that has many genes but few timepoints in order to assess their effectiveness under such limited data. Predictions of likely interactions between genes are provided from the algorithms developed and their limitations discussed. Using extra information is considered, where a further dataset of gene knockout data is used to verify the predictions made for one particular gene.

This thesis examines various aspects of time series and their applications. In the rst part, we study numerical and asymptotic properties of Box-Pierce family of portmanteau tests. We compare size and power properties of time series model diagnostic tests using their asymptotic c2 distribution and bootstrap distribution (dynamic and fixed design) against various linear and non-linear alternatives. In general, our results show that dynamic bootstrapping provides a better approximation of the distribution underlying these statistics. Moreover, we find that Box-Pierce type tests are powerful against linear alternatives while the CvM due to Escanciano (2006b) test performs better against non linear alternative models. The most challenging scenario for these portmanteau tests is when the process is close to the stationary boundary and value of m, the maximum lag considered in the portmanteau test, is very small. In these situations, the c2 distribution is a poor approximation of the null asymptotic distribution. Katayama (2008) suggested a bias correction term to improve the approximation in these situations. We numerically study Katayama's bias correction in Ljung and Box (1978) test. Our results show that Katayama's correction works well and conrms the results as shown in Katayama (2008). We also provide a number of algorithms for performing the necessary calculations efciently. We notice that the bootstrap automatically does bias correction in Ljung-Box statistic. It motivates us to look at theoretical properties of the dynamic bootstrap in this context. Moreover, noticing the good performance of Katayama's correction, we suggest a bias correction term for the Monti (1994) test on the lines of Katayama's correction. We show that our suggestion improves Monti's statistic in a similar way to what Katayama's suggestion does for Ljung-Box test. We also make a novel suggestion of using the pivotal portmanteau test. Our suggestion is to use two separate values of m, one a large value for the calculation of the information matrix and a smaller choice for diagnostic purposes. This results in a pivotal statistic which automatically corrects the bias correction in Ljung-Box test. Our suggested novel algorithm efciently computes this novel portmanteau test. In the second part, we implement lasso-type shrinkage methods to linear regression and time series models. We look through simulations in various examples to study the oracle properties of these methods via the adaptive lasso due to Zou (2006). We study consistent variable selection by the lasso and adaptive lasso and consider a result in the literature which states that the lasso cannot be consistent in variable selection if a necessary condition does not hold for the model. We notice that lasso methods have nice theoretical properties but it is not very easy to achieve them in practice. The choice of tuning parameter is crucial for these methods. So far there is not any fully explicit way of choosing the appropriate value of tuning parameter, so it is hard to achieve the oracle properties in practice. In our numerical study, we compare the performance of k-fold cross-validation with the BIC method of Wang et al. (2007) for selecting the appropriate value of the tuning parameter. We show that k-fold crossvalidation is not a reliable method for choosing the value of the tuning parameter for consistent variable selection. We also look at ways to implement lasso-type methods time series models. In our numerical results we show that the oracle properties of lasso-type methods can also be achieved for time series models. We derive the necessary condition for consistent variable selection by lasso-type methods in the time series context. We also prove the oracle properties of the adaptive lasso for stationary time series.

The standard memory consolidation model assumes that new memories are temporarily stored in the hippocampus and later transferred to the neocortex, during deep sleep, for long-term storage, signifying the importance of studying functional and structural cortico-hippocampal interactions. Our work offers a thorough analysis on such interactions between neocortex and hippocampus, along with a detailed study of their intrinsic dynamics, from two complementary perspectives: statistical data analysis and computational modelling. The first part of this study reviews mathematical tools for assessing directional interactions in multivariate time series. We focus on the notion of Granger Causality and the related measure of generalised Partial Directed Coherence (gPDC) which we then apply, through a custom built numerical package, to electrophysiological data from the medial prefrontal cortex (mPFC) and hippocampus of anaesthetized rats. Our gPDC analysis reveals a clear lateral-to-medial hippocampus connectivity and suggests a reciprocal information flow between mPFC and hippocampus, altered during cortical activity. The second part deals with modelling sleep-related intrinsic rhythmic dynamics of the two areas, and examining their coupling. We first reproduce a computational model of the cortical slow oscillation, a periodic alteration between activated (UP) states and neuronal silence. We then develop a new spiking network model of hippocampal areas CA3 and CA1, reproducing many of their intrinsic dynamics and exhibiting sharp wave-ripple complexes, suggesting a novel mechanism for their generation based on CA1 interneuronal activity and recurrent inhibition. We finally couple the two models to study interactions between the slow oscillation and hippocampal activity. Our simulations propose a dependence of the correlation between UP states and hippocampal spiking on the excitation-to-inhibition ratio induced by the mossy fibre input to CA3 and by a combination of the Schaffer collateral and temporoammonic input to CA1. These inputs are shown to affect reported correlations between UP states and ripples.

Telomeres are repetitive elements of DNA which are located at the ends of chromosomes. During cell division, telomeres on daughter chromomeres shorten until the telomere length falls below a critical level. This shortening restricts the number of cell divisions. In this thesis, we use mathematical modelling to study dynamics of telomere length in a cell in order to understand normal ageing (telomere shortening),Werner’s syndrome (a disease of accelerated ageing) and the immortality of cells caused by telomerase (telomere constant length maintenance). In the mathematical models we compared four possible mechanisms for telomere shortening. The simplest model assumes that a fixed amount of telomere is lost on each replication; the second supposes that telomere loss depends on telomere length; for the third case the amount of telomeres loss per division is fixed but the probability of dividing depends on telomere length; the fourth cases has both telomere loss and the probability of division dependent on telomere length. We start by developing Monte Carlo simulations of normal ageing using these four cases. Then we generalize the Monte Carlo simulations to consider Werner’s syndrome, where the extra telomeres are lost during replication accelerate the ageing process. In order to investigate how the distribution of telomere length varies with time, we derive, from the discrete model, continuum models for the four different cases. Results from the Monte Carlo simulations and the deterministic models are shown to be in good agreement. In addition to telomere loss, we also consider increases in telomere length caused by the enzyme telomerase, by appropriately extending the earlier Monte Carlo simulations and continuum models. Results from the Monte Carlo simulations and the deterministic models are shown to be in good agreement. We also show that the concentration of telomerase in cells can control their proliferative potential.