Biology, Biostatistics

The connectivity underlying a complex system determines its global dynamics and its observable functional patterns. Examples are found in a variety of disciplines such as social networks, the Internet, the central nervous system including the cortex, as well as electronic circuits. Novel computational methods from fractal mathematics and "small world" networks provide an entry point to the understanding of the connectiyity and the interaction of its microscopic components from the study of the observable variables on the macroscopic system level. As an example of such an approach, we try to understand the underlying connectivity of the genome by analyzing the observable patterns of gene expression profiles made available by cDNA microarrays technology. We start by formulating different models of genetic interactions on a genomic scale and then we compute the statistics of gene expression levels produced from each model. By these means tire obtain a dictionary relating different connection topologies on the microscopic level to corresponding gene expression profiles on the macroscopic system level. To allow for comparison between theory and experiment, we compute the equivalent statistics of experimental cDNA microarrays data obtained from the public domain. Reading the theoretical dictionary backwards and applying it to the statistics of the experimental data, we are able to rule out improbable genetic connectivity patterns and identify the most promising candidates of genetic networks. Our results show that the most promising candidate of genetic network is the "small world" heterogeneous network where the value of the scaling exponent in g(k) = Ak-a is between three halves and six, 3/2 < a < 6. This conclusion is quantitatively supported by the measures of goodness of fit of the models to the experimental data. This would imply that some genes are regulated by the input from a few other genes, while some genes are regulated by the input from many other genes. However, all the genes have a similar pattern of regulatory output onto other genes. We also find that in our genetic interaction models the clustering of the input pattern of the structural connectivity matrices is reflected in the correlation pattern of the functional connectivity matrices. Hence, the model predicts a direct connection between the regulatory links among genes and the co-expression of these genes.

This dissertation is an investigation of the sources of commonly observed event-related transients in statistical measures of interdependence: variance, cross-correlation, power spectrum density and coherence spectrum density time functions. These measures are often employed in the analysis of spatio-temporal interdependence patterns in neural activity. In order to understand the phenomenon, the origins of the variability of event-related responses are revisited. The time series of single trial cortical event-related potentials typically have a random appearance, and their trial-to-trial variability is commonly explained by the classic signal-plus-noise model, in which random ongoing background noise activity is linearly combined with a stereotyped evoked response. Here, we demonstrate that more realistic models, challenging both the linear superposition and the trial-to-trial stationarity of the event-related responses, can account for such event-related transients. In particular, two effects are considered: the nonlinear gain modulation in neural networks coupled through sigmoid functions and the trial-to-trial variability in amplitude and latency of the event phase-locked responses. An extensive analysis and characterization of both effects in interdependence measures is carried out through both analytical and numerical simulations in Chapter 2. Chapter 3 presents the outcome of testing the predicted effects on UP data recorded from implanted intracortical electrodes in monkeys performing a visuo-motor pattern discrimination task. Overall, the results point to a large contribution of the trial-to-trial variability of event phase-locked responses on the observed event-related transient in statistical interdependence measures. Because variability of the event-related responses is commonly ignored, event-related modulations in power spectral density, cross-correlation, and spectral coherence are often attributed to dynamic changes in functional connectivity within and among neural populations. It becomes then crucial the separation or removal of the trial-to-trial amplitude and latency variability effect from the statistical measures. In order to achieve this goal, the reconstruction of the single trial event phase-locked potentials is required. In Chapter 4, we approach this problem from a Bayesian inference perspective. The posterior probability density is derived for a specified number of event phase-locked components using data from single or multiple sensors. The Maximum A Posteriori solution is used to obtain the phase-locked component waveforms and their single trial parameters. The outcome is a further and definitive support for predominance of the effect of the nonstationarity of the phase-locked responses on the statistical quantities. Based on the theoretical and experimental analysis conducted in Chapters 2, 3 and 4, a framework for the statistical analysis of dynamic spatio-temporal interdependence patterns in Local Field Potential data is articulated.