Is gamma band EEG synchronization reduced during auditory driving in schizophrenia patients with auditory verbal hallucinations?

Auditory verbal hallucinations (AVH) in schizophrenia patients assumingly result from a state inadequate activation of the primary auditory system. We tested brain responsiveness to auditory stimulation in healthy controls (n=26), and in schizophrenia patients that frequently (n=18) or never (n=11) experienced AVH. Responsiveness was assessed by driving the EEG with click-tones at 20, 30 and 40Hz. We compared stimulus induced EEG changes between groups using spectral amplitude maps and a global measure of phase-locking (GFS). As expected, the 40Hz stimulation elicited the strongest changes. However, while controls and non-hallucinators increased 40Hz EEG activity during stimulation, a left-lateralized decrease was observed in the hallucinators. These differences were significant (p=.02). As expected, GFS increased during stimulation in controls (p=.08) and non-hallucinating patients (p=.06), which was significant when combining the two groups (p=.01). In contrast, GFS decreased with stimulation in hallucinating patients (p=0.13), resulting in a significantly different GFS response when comparing subjects with and without AVH (p<.01). Our data suggests that normally, 40Hz stimulation leads to the activation of a synchronized network representing the sensory input, but in hallucinating patients, the same stimulation partly disrupts ongoing activity in this network.

Functional and effective connectivity in MEG. Application to the study of epilepsy

Nuestro cerebro contiene cerca de 1014 sinapsis neuronales. Esta enorme cantidad de conexiones proporciona un entorno ideal donde distintos grupos de neuronas se sincronizan transitoriamente para provocar la aparición de funciones cognitivas, como la percepción, el aprendizaje o el pensamiento. Comprender la organización de esta compleja red cerebral en base a datos neurofisiológicos, representa uno de los desafíos más importantes y emocionantes en el campo de la neurociencia. Se han propuesto recientemente varias medidas para evaluar cómo se comunican las diferentes partes del cerebro a diversas escalas (células individuales, columnas corticales, o áreas cerebrales). Podemos clasificarlos, según su simetría, en dos grupos: por una parte, la medidas simétricas, como la correlación, la coherencia o la sincronización de fase, que evalúan la conectividad funcional (FC); mientras que las medidas asimétricas, como la causalidad de Granger o transferencia de entropía, son capaces de detectar la dirección de la interacción, lo que denominamos conectividad efectiva (EC). En la neurociencia moderna ha aumentado el interés por el estudio de las redes funcionales cerebrales, en gran medida debido a la aparición de estos nuevos algoritmos que permiten analizar la interdependencia entre señales temporales, además de la emergente teoría de redes complejas y la introducción de técnicas novedosas, como la magnetoencefalografía (MEG), para registrar datos neurofisiológicos con gran resolución. Sin embargo, nos hallamos ante un campo novedoso que presenta aun varias cuestiones metodológicas sin resolver, algunas de las cuales trataran de abordarse en esta tesis. En primer lugar, el creciente número de aproximaciones para determinar la existencia de FC/EC entre dos o más señales temporales, junto con la complejidad matemática de las herramientas de análisis, hacen deseable organizarlas todas en un paquete software intuitivo y fácil de usar. Aquí presento HERMES (http://hermes.ctb.upm.es), una toolbox en MatlabR, diseñada precisamente con este fin. Creo que esta herramienta será de gran ayuda para todos aquellos investigadores que trabajen en el campo emergente del análisis de conectividad cerebral y supondrá un gran valor para la comunidad científica. La segunda cuestión practica que se aborda es el estudio de la sensibilidad a las fuentes cerebrales profundas a través de dos tipos de sensores MEG: gradiómetros planares y magnetómetros, esta aproximación además se combina con un enfoque metodológico, utilizando dos índices de sincronización de fase: phase locking value (PLV) y phase lag index (PLI), este ultimo menos sensible a efecto la conducción volumen. Por lo tanto, se compara su comportamiento al estudiar las redes cerebrales, obteniendo que magnetómetros y PLV presentan, respectivamente, redes más densamente conectadas que gradiómetros planares y PLI, por los valores artificiales que crea el problema de la conducción de volumen. Sin embargo, cuando se trata de caracterizar redes epilépticas, el PLV ofrece mejores resultados, debido a la gran dispersión de las redes obtenidas con PLI. El análisis de redes complejas ha proporcionado nuevos conceptos que mejoran caracterización de la interacción de sistemas dinámicos. Se considera que una red está compuesta por nodos, que simbolizan sistemas, cuyas interacciones se representan por enlaces, y su comportamiento y topología puede caracterizarse por un elevado número de medidas. Existe evidencia teórica y empírica de que muchas de ellas están fuertemente correlacionadas entre sí. Por lo tanto, se ha conseguido seleccionar un pequeño grupo que caracteriza eficazmente estas redes, y condensa la información redundante. Para el análisis de redes funcionales, la selección de un umbral adecuado para decidir si un determinado valor de conectividad de la matriz de FC es significativo y debe ser incluido para un análisis posterior, se convierte en un paso crucial. En esta tesis, se han obtenido resultados más precisos al utilizar un test de subrogadas, basado en los datos, para evaluar individualmente cada uno de los enlaces, que al establecer a priori un umbral fijo para la densidad de conexiones. Finalmente, todas estas cuestiones se han aplicado al estudio de la epilepsia, caso práctico en el que se analizan las redes funcionales MEG, en estado de reposo, de dos grupos de pacientes epilépticos (generalizada idiopática y focal frontal) en comparación con sujetos control sanos. La epilepsia es uno de los trastornos neurológicos más comunes, con más de 55 millones de afectados en el mundo. Esta enfermedad se caracteriza por la predisposición a generar ataques epilépticos de actividad neuronal anormal y excesiva o bien síncrona, y por tanto, es el escenario perfecto para este tipo de análisis al tiempo que presenta un gran interés tanto desde el punto de vista clínico como de investigación. Los resultados manifiestan alteraciones especificas en la conectividad y un cambio en la topología de las redes en cerebros epilépticos, desplazando la importancia del ‘foco’ a la ‘red’, enfoque que va adquiriendo relevancia en las investigaciones recientes sobre epilepsia. ABSTRACT There are about 1014 neuronal synapses in the human brain. This huge number of connections provides the substrate for neuronal ensembles to become transiently synchronized, producing the emergence of cognitive functions such as perception, learning or thinking. Understanding the complex brain network organization on the basis of neuroimaging data represents one of the most important and exciting challenges for systems neuroscience. Several measures have been recently proposed to evaluate at various scales (single cells, cortical columns, or brain areas) how the different parts of the brain communicate. We can classify them, according to their symmetry, into two groups: symmetric measures, such as correlation, coherence or phase synchronization indexes, evaluate functional connectivity (FC); and on the other hand, the asymmetric ones, such as Granger causality or transfer entropy, are able to detect effective connectivity (EC) revealing the direction of the interaction. In modern neurosciences, the interest in functional brain networks has increased strongly with the onset of new algorithms to study interdependence between time series, the advent of modern complex network theory and the introduction of powerful techniques to record neurophysiological data, such as magnetoencephalography (MEG). However, when analyzing neurophysiological data with this approach several questions arise. In this thesis, I intend to tackle some of the practical open problems in the field. First of all, the increase in the number of time series analysis algorithms to study brain FC/EC, along with their mathematical complexity, creates the necessity of arranging them into a single, unified toolbox that allow neuroscientists, neurophysiologists and researchers from related fields to easily access and make use of them. I developed such a toolbox for this aim, it is named HERMES (http://hermes.ctb.upm.es), and encompasses several of the most common indexes for the assessment of FC and EC running for MatlabR environment. I believe that this toolbox will be very helpful to all the researchers working in the emerging field of brain connectivity analysis and will entail a great value for the scientific community. The second important practical issue tackled in this thesis is the evaluation of the sensitivity to deep brain sources of two different MEG sensors: planar gradiometers and magnetometers, in combination with the related methodological approach, using two phase synchronization indexes: phase locking value (PLV) y phase lag index (PLI), the latter one being less sensitive to volume conduction effect. Thus, I compared their performance when studying brain networks, obtaining that magnetometer sensors and PLV presented higher artificial values as compared with planar gradiometers and PLI respectively. However, when it came to characterize epileptic networks it was the PLV which gives better results, as PLI FC networks where very sparse. Complex network analysis has provided new concepts which improved characterization of interacting dynamical systems. With this background, networks could be considered composed of nodes, symbolizing systems, whose interactions with each other are represented by edges. A growing number of network measures is been applied in network analysis. However, there is theoretical and empirical evidence that many of these indexes are strongly correlated with each other. Therefore, in this thesis I reduced them to a small set, which could more efficiently characterize networks. Within this framework, selecting an appropriate threshold to decide whether a certain connectivity value of the FC matrix is significant and should be included in the network analysis becomes a crucial step, in this thesis, I used the surrogate data tests to make an individual data-driven evaluation of each of the edges significance and confirmed more accurate results than when just setting to a fixed value the density of connections. All these methodologies were applied to the study of epilepsy, analysing resting state MEG functional networks, in two groups of epileptic patients (generalized and focal epilepsy) that were compared to matching control subjects. Epilepsy is one of the most common neurological disorders, with more than 55 million people affected worldwide, characterized by its predisposition to generate epileptic seizures of abnormal excessive or synchronous neuronal activity, and thus, this scenario and analysis, present a great interest from both the clinical and the research perspective. Results revealed specific disruptions in connectivity and network topology and evidenced that networks’ topology is changed in epileptic brains, supporting the shift from ‘focus’ to ‘networks’ which is gaining importance in modern epilepsy research.

Cognitive reserve is associated with the functional organization of brain networks in healthy aging: a MEG study

The proportion of elderly people in the population has increased rapidly in the last century and consequently "healthy aging" is expected to become a critical area of research in neuroscience. Evidence reveals how healthy aging depends on three main behavioral factors: social lifestyle, cognitive activity and physical activity. In this study, we focused on the role of cognitive activity, concentrating specifically on educational and occupational attainment factors, which were considered two of the main pillars of cognitive reserve. 21 subjects with similar rates of social lifestyle, physical and cognitive activity were selected from a sample of 55 healthy adults. These subjects were divided into two groups according to their level of cognitive reserve; one group comprised subjects with high cognitive reserve (9 members) and the other contained those with low cognitive reserve (12 members). To evaluate the cortical brain connectivity network, all participants were recorded by Magnetoencephalography (MEG) while they performed a memory task (modified version of the Sternberg¿s Task). We then applied two algorithms (Phase Locking Value & Phase-Lag Index) to study the dynamics of functional connectivity. In response to the same task, the subjects with lower cognitive reserve presented higher functional connectivity than those with higher cognitive reserve. These results may indicate that participants with low cognitive reserve needed a greater 'effort' than those with high cognitive reserve to achieve the same level of cognitive performance. Therefore, we conclude that cognitive reserve contributes to the modulation of the functional connectivity patterns of the aging brain.

Diffractive optics-based heterodyne-detected four-wave mixing signals of protein motion: From “protein quakes” to ligand escape for myoglobin

Ligand transport through myoglobin (Mb) has been observed by using optically heterodyne-detected transient grating spectroscopy. Experimental implementation using diffractive optics has provided unprecedented sensitivity for the study of protein motions by enabling the passive phase locking of the four beams that constitute the experiment, and an unambiguous separation of the Real and Imaginary parts of the signal. Ligand photodissociation of carboxymyoglobin (MbCO) induces a sequence of events involving the relaxation of the protein structure to accommodate ligand escape. These motions show up in the Real part of the signal. The ligand (CO) transport process involves an initial, small amplitude, change in volume, reflecting the transit time of the ligand through the protein, followed by a significantly larger volume change with ligand escape to the surrounding water. The latter process is well described by a single exponential process of 725 ± 15 ns at room temperature. The overall dynamics provide a distinctive signature that can be understood in the context of segmental protein fluctuations that aid ligand escape via a few specific cavities, and they suggest the existence of discrete escape pathways.

Prediction of hidden multiferroic order in graphene zigzag ribbons

An electronic phase with coexisting magnetic and ferroelectric order is predicted for graphene ribbons with zigzag edges. The electronic structure of the system is described with a mean-field Hubbard model that yields results very similar to those of density functional calculations. Without further approximations, the mean-field theory is recasted in terms of a BCS wave function for electron-hole pairs in the edge bands. The BCS coherence present in each spin channel is related to spin-resolved electric polarization. Although the total electric polarization vanishes, due to an internal phase locking of the BCS state, strong magnetoelectric effects are expected in this system. The formulation naturally accounts for the two gaps in the quasiparticle spectrun, Δ0 and Δ1, and relates them to the intraband and interband self-energies.

Homodyne measurement of the average photon number

We describe a scheme for measurement of the mean photon flux at an arbitrary optical sideband frequency using homodyne detection. Experimental implementation of the technique requires an acousto-optic modulator in addition to the homodyne detector, and does not require phase locking. The technique exhibits polarization and frequency and spatial mode selectivity, as well as much improved speed, resolution, and dynamic range when compared to linear photodetectors and avalanche photodiodes, with potential application to quantum-state tomography and information encoding using an optical frequency basis. Experimental data also support a quantum-mechanical description of vacuum noise.

Switching and generation of ultrashort pulses using all-fibre devices

Serial and parallel interconnection of photonic devices is integral to the construction of any all-optical data processing system. This thesis presents results from a series of experiments centering on the use of the nonlinear-optical loop mirror (NOLM) switch in architectures for the manipulation and generation of ultrashort pulses. Detailed analysis of soliton switching in a single NOLM and cascade of two NOLM's is performed, centering on primary limitations to device operation, effect of cascading on amplitude response, and impact of switching on the characteristics of incident pulses. By using relatively long input pulses, device failure due to stimulated Raman generation is postponed to demonstrate multiple-peaked switching for the first time. It is found that while cascading leads to a sharpening of the overall switching characteristic, pulse spectral and temporal integrity is not significantly degraded, and emerging pulses retain their essential soliton character. In addition, by including an asymmetrically placed in-fibre Bragg reflector as a wavelength selective loss element in the basic NOLM configuration, both soliton self-switching and dual-wavelength control-pulse switching are spectrally quantised. Results are presented from a novel dual-wavelength laser configuration generating pulse trains with an ultra-low rms inter-pulse-stream timing jitter level of 630fs enabling application in ultrafast switching environments at data rates as high as 130GBits/s. In addition, the fibre NOLM is included in architectures for all-optical memory, demonstrating storage and logical inversion of a 0.5kByte random data sequence; and ultrafast phase-locking of a gain-switched distributed feedback laser at 1.062GHz, the fourteenth harmonic of the system baseband frequency. The stringent requirements for environmental robustness of these architectures highlight the primary weaknesses of the NOLM in its fibre form and recommendations to overcome its inherent drawbacks are presented.

Contradiction in universal and particular reasoning

A wide range of essential reasoning tasks rely on contradiction identification, a cornerstone of human rationality, communication and debate founded on the inversion of the logical operators "Every" and "Some." A high-density electroencephalographic (EEG) study was performed in 11 normal young adults. The cerebral network involved in the identification of contradiction included the orbito-frontal and anterior-cingulate cortices and the temporo-polar cortices. The event-related dynamic of this network showed an early negative deflection lasting 500 ms after sentence presentation. This was followed by a positive deflection lasting 1.5 s, which was different for the two logical operators. A lesser degree of network activation (either in neuron number or their level of phase locking or both) occurred while processing statements with "Some," suggesting that this was a relatively simpler scenario with one example to be figured out, instead of the many examples or the absence of a counterexample searched for while processing statements with "Every." A self-generated reward system seemed to resonate the recruited circuitry when the contradictory task is successfully completed.

Graphene based widely-tunable and singly-polarized pulse generation with random fiber lasers

Pulse generation often requires a stabilized cavity and its corresponding mode structure for initial phase-locking. Contrastingly, modeless cavity-free random lasers provide new possibilities for high quantum efficiency lasing that could potentially be widely tunable spectrally and temporally. Pulse generation in random lasers, however, has remained elusive since the discovery of modeless gain lasing. Here we report coherent pulse generation with modeless random lasers based on the unique polarization selectivity and broadband saturable absorption of monolayer graphene. Simultaneous temporal compression of cavity-free pulses are observed with such a polarization modulation, along with a broadly-tunable pulsewidth across two orders of magnitude down to 900 ps, a broadly-tunable repetition rate across three orders of magnitude up to 3 MHz, and a singly-polarized pulse train at 41 dB extinction ratio, about an order of magnitude larger than conventional pulsed fiber lasers. Moreover, our graphene-based pulse formation also demonstrates robust pulse-to-pulse stability and widewavelength operation due to the cavity-less feature. Such a graphene-based architecture not only provides a tunable pulsed random laser for fiber-optic sensing, speckle-free imaging, and laser-material processing, but also a new way for the non-random CW fiber lasers to generate widely tunable and singly-polarized pulses.

Mode-locked semiconductor lasers with optical injection

We perform characterization of the pulse shape and noise properties of quantum dot passively mode-locked lasers (PMLLs). We propose a novel method to determine the RF linewidth and timing jitter, applicable to high repetition rate PMLLs, through the dependence of modal linewidth on the mode number. Complex electric field measurements show asymmetric pulses with parabolic phase close to threshold, with the appearance of waveform instabilities at higher currents. We demonstrate that the waveform instabilities can be overcome through optical injection-locking to the continues wave (CW) master laser, leading to time-bandwidth product (TBP) improvement, spectral narrowing, and spectral tunability. We discuss the benefits of single- and dual-tone master sources and demonstrate that dual-tone optical injection can additionally improve the noise properties of the slave laser with RF linewidth reduction below instrument limits (1 kHz) and integrated timing jitter values below 300 fs. Dual-tone injection allowed slave laser repetition rate control over a 25 MHz range with reduction of all modal optical linewidths to the master source linewidth, demonstrating phase-locking of all slave modes and coherence improvement.

Mode-locked semiconductor lasers with optical injection

We perform characterization of the pulse shape and noise properties of quantum dot passively mode-locked lasers (PMLLs). We propose a novel method to determine the RF linewidth and timing jitter, applicable to high repetition rate PMLLs, through the dependence of modal linewidth on the mode number. Complex electric field measurements show asymmetric pulses with parabolic phase close to threshold, with the appearance of waveform instabilities at higher currents. We demonstrate that the waveform instabilities can be overcome through optical injection-locking to the continues wave (CW) master laser, leading to time-bandwidth product (TBP) improvement, spectral narrowing, and spectral tunability. We discuss the benefits of single- and dual-tone master sources and demonstrate that dual-tone optical injection can additionally improve the noise properties of the slave laser with RF linewidth reduction below instrument limits (1 kHz) and integrated timing jitter values below 300 fs. Dual-tone injection allowed slave laser repetition rate control over a 25 MHz range with reduction of all modal optical linewidths to the master source linewidth, demonstrating phase-locking of all slave modes and coherence improvement.

The effect of Gaussian noise on the threshold, dynamic range, and loudness of analogue cochlear implant stimuli

The deliberate addition of Gaussian noise to cochlear implant signals has previously been proposed to enhance the time coding of signals by the cochlear nerve. Potentially, the addition of an inaudible level of noise could also have secondary benefits: it could lower the threshold to the information-bearing signal, and by desynchronization of nerve discharges, it could increase the level at which the information-bearing signal becomes uncomfortable. Both these effects would lead to an increased dynamic range, which might be expected to enhance speech comprehension and make the choice of cochlear implant compression parameters less critical (as with a wider dynamic range, small changes in the parameters would have less effect on loudness). The hypothesized secondary effects were investigated with eight users of the Clarion cochlear implant; the stimulation was analogue and monopolar. For presentations in noise, noise at 95% of the threshold level was applied simultaneously and independently to all the electrodes. The noise was found in two-alternative forced-choice (2AFC) experiments to decrease the threshold to sinusoidal stimuli (100 Hz, 1 kHz, 5 kHz) by about 2.0 dB and increase the dynamic range by 0.7 dB. Furthermore, in 2AFC loudness balance experiments, noise was found to decrease the loudness of moderate to intense stimuli. This suggests that loudness is partially coded by the degree of phase-locking of cochlear nerve fibers. The overall gain in dynamic range was modest, and more complex noise strategies, for example, using inhibition between the noise sources, may be required to get a clinically useful benefit. © 2006 Association for Research in Otolaryngology.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder. © 2014 American Physical Society.

Designing Short Term Wave Traces to Assess Wave Power Devices

In recent years modern numerical methods have been employed in the design of Wave Energy Converters (WECs), however the high computational costs associated with their use makes it prohibitive to undertake simulations involving statistically relevant numbers of wave cycles. Experimental tests in wave tanks could also be performed more efficiently and economically if short time traces, consisting of only a few wave cycles, could be used to evaluate the hydrodynamic characteristics of a particular device or design modification. Ideally, accurate estimations of device performance could be made utilizing results obtained from investigations with a relatively small number of wave cycles. However the difficulty here is that many WECs, such as the Oscillating Wave Surge Converter (OWSC), exhibit significant non-linearity in their response. Thus it is challenging to make accurate predictions of annual energy yield for a given spectral sea state using short duration realisations of that sea. This is because the non-linear device response to particular phase couplings of sinusoidal components within those time traces might influence the estimate of mean power capture obtained. As a result it is generally accepted that the most appropriate estimate of mean power capture for a sea state be obtained over many hundreds (or thousands) of wave cycles. This ensures that the potential influence of phase locking is negligible in comparison to the predictions made. In this paper, potential methods of providing reasonable estimates of relative variations in device performance using short duration sea states are introduced. The aim of the work is to establish the shortness of sea state required to provide statistically significant estimations of the mean power capture of a particular type of Wave Energy Converter. The results show that carefully selected wave traces can be used to reliably assess variations in power output due to changes in the hydrodynamic design or wave climate. 

A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.