From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

Stochastic neural network dynamics: synchronisation and control

Biological brains exhibit many interesting and complex behaviours. Understanding of the mechanisms behind brain behaviours is critical for continuing advancement in fields of research such as artificial intelligence and medicine. In particular, synchronisation of neuronal firing is associated with both improvements to and degeneration of the brain’s performance; increased synchronisation can lead to enhanced information-processing or neurological disorders such as epilepsy and Parkinson’s disease. As a result, it is desirable to research under which conditions synchronisation arises in neural networks and the possibility of controlling its prevalence. Stochastic ensembles of FitzHugh-Nagumo elements are used to model neural networks for numerical simulations and bifurcation analysis. The FitzHugh-Nagumo model is employed because of its realistic representation of the flow of sodium and potassium ions in addition to its advantageous property of allowing phase plane dynamics to be observed. Network characteristics such as connectivity, configuration and size are explored to determine their influences on global synchronisation generation in their respective systems. Oscillations in the mean-field are used to detect the presence of synchronisation over a range of coupling strength values. To ensure simulation efficiency, coupling strengths between neurons that are identical and fixed with time are investigated initially. Such networks where the interaction strengths are fixed are referred to as homogeneously coupled. The capacity of controlling and altering behaviours produced by homogeneously coupled networks is assessed through the application of weak and strong delayed feedback independently with various time delays. To imitate learning, the coupling strengths later deviate from one another and evolve with time in networks that are referred to as heterogeneously coupled. The intensity of coupling strength fluctuations and the rate at which coupling strengths converge to a desired mean value are studied to determine their impact upon synchronisation performance. The stochastic delay differential equations governing the numerically simulated networks are then converted into a finite set of deterministic cumulant equations by virtue of the Gaussian approximation method. Cumulant equations for maximal and sub-maximal connectivity are used to generate two-parameter bifurcation diagrams on the noise intensity and coupling strength plane, which provides qualitative agreement with numerical simulations. Analysis of artificial brain networks, in respect to biological brain networks, are discussed in light of recent research in sleep theor

Discerning nonlinear brain dynamics from EEG:an application to autistic spectrum disorder in young children

A challenging goal in neuroscience is that of identifying specific brain patterns characterising autistic spectrum disorder (ASD). Genetic studies, together with investigations based on magnetic resonance imaging (MRI) and functional MRI, support the idea that distinctive structural features could exist in the ASD brain. In the developing brains of babies and small children, structural differences could provide the basis for different brain connectivity, giving rise to macroscopic effects detectable by e.g. electroencephalography (EEG). A significant body of research has already been conducted in this direction, mainly computing spectral power and coherence. Perhaps due to methodological limitations, together with high variability within and between the cohorts investigated, results have not been in complete agreement, and it is therefore still the case that the diagnosis of ASD is based on behavioural tests and interviews. This thesis describes a step-by-step characterisation and comparison of brain dynamics from ASD and neurotypical subjects, based on the analysis of multi-probe EEG time-series from male children aged 3-5 years. The methods applied are all ones that take explicit account of the intrinsically non-linear, open, and time-variable nature of the system. Time-frequency representations were first computed from the time-series to evaluate the spectral power and to categorise the ranges encompassing different activities as low-frequency (LF, 0.8-3.5 Hz), mid-range-frequency (MF, 3.5-12 Hz) or high-frequency (HF, 12-48 Hz). The spatial pathways for the propagation of neuronal activity were then investigated by calculation of wavelet phase coherence. Finally, deeper insight into brain connectivity was achieved by computation of the dynamical cross-frequency coupling between triplets of spatially distributed phases. In doing so, dynamical Bayesian inference was used to find the coupling parameters between the oscillators in the spatially-distributed network. The sets of parameters extracted by this means allowed evaluation of the strength of particular coupling components of the triplet LF, MF→HF, and enabled reconstruction of the coupling functions. By investigation of the form of the coupling functions, the thesis goes beyond conventional measures like the directionality and strength of an interaction, and reveals subtler features of the underlying mechanism. The measured power distributions highlight differences between ASD and typically developing children in the preferential frequency range for local synchronisation of neuronal activity: the relative power is generally higher at LF and HF, and lower at MF, in the ASD case. The phase coherence maps from ASD subjects also exhibited differences, with lower connectivity at LF and MF in the frontal and fronto-occipital pairs, and higher coherence at high frequencies for central links. There was higher inter-subject variability in a comparison of the forms of coupling functions in the ASD group; and a weaker coupling in their theta-gamma range, which can be linked with the cognitive features of the disorder. In conclusion, the approach developed in this thesis gave promising preliminary results, suggesting that a biomarker for ASD could be defined in terms of the described patterns of functional and effective connectivity computed from EEG measurements

Imaging the spatial-temporal neuronal dynamics using dynamic causal modelling

Oscillatory brain activity is a ubiquitous feature of neuronal dynamics and the synchronous discharge of neurons is believed to facilitate integration both within functionally segregated brain areas and between areas engaged by the same task. There is growing interest in investigating the neural oscillatory networks in vivo. The aims of this thesis are to (1) develop an advanced method, Dynamic Causal Modelling for Induced Responses (DCM for IR), for modelling the brain network functions and (2) apply it to exploit the nonlinear coupling in the motor system during hand grips and the functional asymmetries during face perception. DCM for IR models the time-varying power over a range of frequencies of coupled electromagnetic sources. The model parameters encode coupling strength among areas and allows the differentiations between linear (within frequency) and nonlinear (between-frequency) coupling. I applied DCM for IR to show that, during hand grips, the nonlinear interactions among neuronal sources in motor system are essential while intrinsic coupling (within source) is very likely to be linear. Furthermore, the normal aging process alters both the network architecture and the frequency contents in the motor network. I then use the bilinear form of DCM for IR to model the experimental manipulations as the modulatory effects. I use MEG data to demonstrate functional asymmetries between forward and backward connections during face perception: Specifically, high (gamma) frequencies in higher cortical areas suppressed low (alpha) frequencies in lower areas. This finding provides direct evidence for functional asymmetries that is consistent with anatomical and physiological evidence from animal studies. Lastly, I generalize the bilinear form of DCM for IR to dissociate the induced responses from evoked ones in terms of their functional role. The backward modulatory effect is expressed as induced, but not evoked responses

Cortico-muscular coherence in sensorimotor synchronisation

This thesis sets out to investigate the neuro-muscular control mechanisms underlying the ubiquitous phenomenon of sensorimotor synchronisation (SMS). SMS is the coordination of movement to external rhythms, and is commonly observed in everyday life. A large body of research addresses the processes underlying SMS at the levels of behaviour and brain. Comparatively, little is known about the coupling between neural and behavioural processes, i.e. neuro-muscular processes. Here, the neuro-muscular processes underlying SMS were investigated in the form of cortico-muscular coherence measured based on Electroencephalography (EEG) and Electromyography (EMG) recorded in human healthy participants. These neuro-muscular processes were investigated at three levels of engagement: passive listening and observation of rhythms in the environment, imagined SMS, and executed SMS, which resulted in the testing of three hypotheses: (i) Rhythms in the environment, such as music, spontaneously modulate cortico-muscular coupling, (ii) Movement intention modulates cortico-muscular coupling, and (iii) Cortico-muscular coupling is dynamically modulated during SMS time-locked to the stimulus rhythm. These three hypotheses were tested through two studies that used Electroencephalography (EEG) and Electromyography (EMG) recordings to measure Cortico-muscular coherence (CMC). First, CMC was tested during passive music listening, to test whether temporal and spectral properties of music stimuli known to induce groove, i.e., the subjective experience of wanting to move, can spontaneously modulate the overall strength of the communication between the brain and the muscles. Second, imagined and executed movement synchronisation was used to investigate the role of movement intention and dynamics on CMC. The two studies indicate that both top-down, and somatosensory and/or proprioceptive processes modulate CMC during SMS tasks. Although CMC dynamics might be linked to movement dynamics, no direct correlation between movement performance and CMC was found. Furthermore, purely passive auditory or visual rhythmic stimulation did not affect CMC. Together, these findings thus indicate that movement intention and active engagement with rhythms in the environment might be critical in modulating CMC. Further investigations of the mechanisms and function of CMC are necessary, as they could have important implications for clinical and elderly populations, as well as athletes, where optimisation of motor control is necessary to compensate for impaired movement or to achieve elite performance

Complexity and Criticality in financial markets: systemic risk across frequencies and cross sections

Extreme market events and systemic collapses cause most of the popular attention to finance and financial markets. Extreme phenomena and the dynamics of con- nected/interacting systems have been the subject of financial modeling since early derivatives modeling, exposure risk modeling and portfolio construction. In the present work we discuss how traditional methods have for the most part failed to properly model the interconnected global financial and economic system. This led to systemic risk events and simplistic regulation which does not properly account for its implications. Analogously, we discuss how from as early as Mandelbrot’s works on financial prices and fat tails, academics, practitioners and regulators alike were warned of fat tails in financial modeling and in particular market making and derivatives pricing. The improper modeling or dismissal of these lies at the cen- tre of financial downturns ranging from LTCM’s collapse to the quant downturn of August 2007. The solution I promote in this thesis is that of complexity and criticality. In line with this we propose two lines of work. The former analyses markets as complex networks and their structure through to practical takeaways including a proof of concept for portfolio construction. The latter instead focuses on extreme events in high frequency markets with results for both tail modeling and systemic events and practical insights from those. Recent events have shown how retail investors and their savings are now heavily involved in financial markets. We hope that our contribution of methods of practical use for proper risk modeling will encourage their adoption by practitioners and regulators with the outcome of a more stable and efficient financial system

The laminar profile of spatial attention in macaque V1 and V4

PhD ThesisSpatial attention allows processing to be prioritised for one or more locations in the visual field, even in the presence of other distracting or irrelevant stimuli. Previous work has shown that attention modulates the activity of the brain at the level of spiking activity, local field potentials and coherence between and within neuronal groups. However currently little is known about how these attentional modulations differ between groups of neurons in different cortical layers and areas. We trained two adult male rhesus macaques to perform a covert visuospatial attention task whilst we recorded simultaneously from V1 and V4. Recordings were taken with multichannel laminar electrodes allowing recording from supragranular, granular and infragranular cells within the same cortical microcolumns. We used current source density analysis to align our recording contacts to the cortical laminar profile (layers). The receptive fields of the V1 and V4 cells we recorded from were overlapping which meant they could be driven by the same stimulus in the task. To measure the attentional modulation of information flow between different groups of neurons we calculated field coherence, Granger causality and spike-rate correlations. Attention increased firing rates for all of the cell types, layers and areas in our study. We also show that variability as measured by gain variance and noise correlations is reduced by attention. Although we find differences between the two monkeys regarding LFP power changes and regarding coherence measures within and between the areas investigated, we find that attention consistently increased the Granger causality in the gamma frequency band between V1 and V4. We demonstrate that the flow of information in the alpha/beta and gamma bands follows expected interareal feedback and feedforward patterns between V1 and V4. We also provide evidence that feedforward gamma oscillations are generated, contrary to expectations, in the infragranular layers of V1