Structure Learning in Coupled Dynamical Systems and Dynamic Causal Modelling

Identifying a coupled dynamical system out of many plausible candidates, each of which could serve as the underlying generator of some observed measurements, is a profoundly ill posed problem that commonly arises when modelling real world phenomena. In this review, we detail a set of statistical procedures for inferring the structure of nonlinear coupled dynamical systems (structure learning), which has proved useful in neuroscience research. A key focus here is the comparison of competing models of (ie, hypotheses about) network architectures and implicit coupling functions in terms of their Bayesian model evidence. These methods are collectively referred to as dynamical casual modelling (DCM). We focus on a relatively new approach that is proving remarkably useful; namely, Bayesian model reduction (BMR), which enables rapid evaluation and comparison of models that differ in their network architecture. We illustrate the usefulness of these techniques through modelling neurovascular coupling (cellular pathways linking neuronal and vascular systems), whose function is an active focus of research in neurobiology and the imaging of coupled neuronal systems

Whole Brain Network Dynamics of Epileptic Seizures at Single Cell Resolution

Epileptic seizures are characterised by abnormal brain dynamics at multiple scales, engaging single neurons, neuronal ensembles and coarse brain regions. Key to understanding the cause of such emergent population dynamics, is capturing the collective behaviour of neuronal activity at multiple brain scales. In this thesis I make use of the larval zebrafish to capture single cell neuronal activity across the whole brain during epileptic seizures. Firstly, I make use of statistical physics methods to quantify the collective behaviour of single neuron dynamics during epileptic seizures. Here, I demonstrate a population mechanism through which single neuron dynamics organise into seizures: brain dynamics deviate from a phase transition. Secondly, I make use of single neuron network models to identify the synaptic mechanisms that actually cause this shift to occur. Here, I show that the density of neuronal connections in the network is key for driving generalised seizure dynamics. Interestingly, such changes also disrupt network response properties and flexible dynamics in brain networks, thus linking microscale neuronal changes with emergent brain dysfunction during seizures. Thirdly, I make use of non-linear causal inference methods to study the nature of the underlying neuronal interactions that enable seizures to occur. Here I show that seizures are driven by high synchrony but also by highly non-linear interactions between neurons. Interestingly, these non-linear signatures are filtered out at the macroscale, and therefore may represent a neuronal signature that could be used for microscale interventional strategies. This thesis demonstrates the utility of studying multi-scale dynamics in the larval zebrafish, to link neuronal activity at the microscale with emergent properties during seizures

Dynamics of biologically informed neural mass models of the brain

This book contributes to the development and analysis of computational models that help brain function to be understood. The mean activity of a brain area is mathematically modeled in such a way as to strike a balance between tractability and biological plausibility. Neural mass models (NMM) are used to describe switching between qualitatively different regimes (such as those due to pharmacological interventions, epilepsy, sleep, or context-induced state changes), and to explain resonance phenomena in a photic driving experiment. The description of varying states in an ordered sequence gives a principle scheme for the modeling of complex phenomena on multiple time scales. The NMM is matched to the photic driving experiment routinely applied in the diagnosis of such diseases as epilepsy, migraine, schizophrenia and depression. The model reproduces the clinically relevant entrainment effect and predictions are made for improving the experimental setting.Die vorliegende Arbeit stellt einen Beitrag zur Entwicklung und Analyse von Computermodellen zum Verständnis von Hirnfunktionen dar. Es wird die mittlere Aktivität eines Hirnareals analytisch einfach und dabei biologisch plausibel modelliert. Auf Grundlage eines Neuronalen Massenmodells (NMM) werden die Wechsel zwischen Oszillationsregimen (z.B. durch pharmakologisch, epilepsie-, schlaf- oder kontextbedingte Zustandsänderungen) als geordnete Folge beschrieben und Resonanzphänomene in einem Photic-Driving-Experiment erklärt. Dieses NMM kann sehr komplexe Dynamiken (z.B. Chaos) innerhalb biologisch plausibler Parameterbereiche hervorbringen. Um das Verhalten abzuschätzen, wird das NMM als Funktion konstanter Eingangsgrößen und charakteristischer Zeitenkonstanten vollständig auf Bifurkationen untersucht und klassifiziert. Dies ermöglicht die Beschreibung wechselnder Regime als geordnete Folge durch spezifische Eingangstrajektorien. Es wird ein Prinzip vorgestellt, um komplexe Phänomene durch Prozesse verschiedener Zeitskalen darzustellen. Da aufgrund rhythmischer Stimuli und der intrinsischen Rhythmen von Neuronenverbänden die Eingangsgrößen häufig periodisch sind, wird das Verhalten des NMM als Funktion der Intensität und Frequenz einer periodischen Stimulation mittels der zugehörigen Lyapunov-Spektren und der Zeitreihen charakterisiert. Auf der Basis der größten Lyapunov-Exponenten wird das NMM mit dem Photic-Driving-Experiment überein gebracht. Dieses Experiment findet routinemäßige Anwendung in der Diagnostik verschiedener Erkrankungen wie Epilepsie, Migräne, Schizophrenie und Depression. Durch die Anwendung des vorgestellten NMM wird der für die Diagnostik entscheidende Mitnahmeeffekt reproduziert und es werden Vorhersagen für eine Verbesserung der Indikation getroffen

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

Dynamical mean field modelling and estimation of neuronal oscillations

Oscillations in neural activity are a ubiquitous phenomenon in the brain. They span multiple timescales and correlate with a myriad of physiological and pathological conditions. Given their intrinsic dynamical nature, mathematical and computational modelling tools have proven to be indispensible in order to interpret and formalize the mechanisms through which these oscillations arise. In this Thesis, I developed a new methodological framework that allows the assimilation of experimental data into biophysically plausible models of neural oscillations. Motivated by the fast oscillatory activity (30 ~ 130 Hz) at the onset of focal epileptic seizures, I started by investigating, via means of bifurcation analyses, whether such fast oscillations can be plausibly described by conductance-based neural mass models. Neural mass models have enjoyed success in describing several forms of epileptiform activity (e.g. spike-and-wave seizures and interictal spikes), but I found that, in order to generate such fast oscillations, the parameters of this family of models would have to depart significantly from biophysical plausibility. These results motivated the exploration of full mean-field models of spiking neurons to characterise this type of dynamics. I hence proposed a variant of a mean-field neural population model based on the Fokker-Planck equation of conductance-based, stochastic, leaky integrate-and-fire neurons. This modelling approach was chosen for its capacity to describe arbitrary network configurations and predict firing rates, trans-membrane currents and local field potentials. I introduced a new numerical scheme that makes the computational cost of integrating the ensuing partial differential equations scale linearly with the number of nodes of the networks. These advances are crucial for the practical implementation of model inversion schemes. I then built upon the literature of Dynamic Causal Modelling to develop a Bayesian model inversion algorithm applicable to dynamical systems in limit cycle regimes. I applied the scheme to the mean-field models described above, using experimental data recordings of carbachol-induced gamma oscillations, in the CA1 region of mice hippocampal slice preparations. The estimated model was able to make accurate predictions about independent data features; namely inter-spike-interval distributions. Also, the inverted models were qualitatively compatible with the observation that excitatory pyramidal cells and inhibitory interneurons play equally important roles in the dynamics of these oscillations (as opposed to interneuron-dominated gamma oscillations). I also explored the applicability of this inversion scheme to neural mass models of electroencephalographically recorded spike-and-wave seizures in humans. In conclusion, the work presented in this thesis provides significant new contributions to model based analyses of neuronal oscillatory data, and helps to bridge single-neuron measurements to network-level interactions

Epilepsy

With the vision of including authors from different parts of the world, different educational backgrounds, and offering open-access to their published work, InTech proudly presents the latest edited book in epilepsy research, Epilepsy: Histological, electroencephalographic, and psychological aspects. Here are twelve interesting and inspiring chapters dealing with basic molecular and cellular mechanisms underlying epileptic seizures, electroencephalographic findings, and neuropsychological, psychological, and psychiatric aspects of epileptic seizures, but non-epileptic as well

Coupling and stochasticity in mesoscopic brain dynamics

The brain is known to operate under the constant influence of noise arising from a variety of sources. It also organises its activity into rhythms spanning multiple frequency bands. These rhythms originate from neuronal oscillations which can be detected via measurements such as electroen-cephalography (EEG) and functional magnetic resonance (fMRI). Experimental evidence suggests that interactions between rhythms from distinct frequency bands play a key role in brain processing, but the dynamical mechanisms underlying this cross-frequency interactions are still under investigation. Some rhythms are pathological and harmful to brain function. Such is the case of epileptiform rhythms characterising epileptic seizures. Much has been learnt about the dynamics of the brain from computational modelling. Particularly relevant is mesoscopic scale modelling, which is concerned with spatial scales exceeding those of individual neurons and corresponding to processes and structures underlying the generation of signals registered in the EEG and fMRI recordings. Such modelling usually involves assumptions regarding the characteristics of the background noise, which represents afferents from remote, non-modelled brain areas. To this end, Gaussian white noise, characterised by a flat power spectrum, is often used. In contrast, macroscopic fluctuations in the brain typically follow a `1/f b ¿ spectrum, which is a characteristic feature of temporally correlated, coloured noise. In Chapters 3-5 of this Thesis we address by means of a stochastically driven mesoscopic neuronal model, the three following questions. First, in Chapter 3 we ask about the significance of deviations from the assumption of white noise in the context of brain dynamics, and in particular we study the role that temporally correlated noise plays in eliciting aberrant rhythms in the model of an epileptic brain. We find that the generation of epileptiform dynamics in the model depends non-monotonically on the noise correlation time. We show that this is due to the maximisation of the spectral content of epileptogenic rhythms in the noise. These rhythms fall into frequency bands that indeed were experimentally shown to increase in power prior to epileptic seizures. We explain these effects in terms of the interplay between specific driving frequencies and bifurcation structure of the model. Second, in Chapter 4 we show how coupling between cortical modules leads to complex activity patterns and to the emergence of a phenomenon that we term collective excitability. Temporal patterns generated by this model bear resemblance to clinically observed characteristics of epileptic seizures. In that chapter we also introduce a fast method of tracking a loss of stability caused by excessive inter-modular coupling in a neuronal network. Third, in Chapter 5 we focus on cross-frequency interactions occurring in a network of cortical modules, in the presence of coloured noise. We suggest a mechanism that underlies the increase of power in a fast rhythm due to driving with a slow rhythm, and we find the noise parameters that best recapitulate experimental power spectra. Finally, in Chapter 6, we examine models of haemodynamic and metabolic brain processes, we test them on experimental data, and we consider the consequences of coupling them with mesoscopic neuronal models. Taken together, our results show the combined influence of noise and coupling in computational models of neuronal activity. Moreover, they demonstrate the relevance of dynamical properties of neuronal systems to specific physiological phenomena, in particular related to cross-frequency interactions and epilepsy. Insights from this Thesis could in the future empower studies of epilepsy as a dynamic disease, and could contribute to the development of treatment methods applicable to drug-resistant epileptic patients.El cervell opera sota la influència de sorolls amb diversos orígens. També organitza la seva activitat en una sèrie de ritmes que s'expandeixen en diverses bandes de freqüència. Aquests ritmes tenen el seu origen en les osci∙lacions neuronals i poden detectar-se via mesures com les electroencefalogràfiques (EEG) o la ressonància magnètica funcional (fMRI). Les evidències experimentals suggereixen que les interaccions entre ritmes operant en bandes de freqüència diferents juguen un paper central en el processat cerebral però els mecanismes dinàmics subjacents a les interaccions inter-freqüència encara estan investigant-se. Alguns ritmes són patològics i fan malbé el funcionament cerebral. És el cas dels ritmes epileptiformes que caracteritzen les convulsions epilèptiques. Fent servir el modelatge computacional s'ha après molt sobre la dinàmica del cervell. Especialment rellevant és el modelatge a l’escala mesoscòpica, que té a veure amb les escales espacials superiors a les de les neurones individuals i que correspon als processos que generen EEG i fMRI. Tal modelatge, en general, implica supòsits relatius a les característiques del soroll de fons que representa zones remotes del cervell no modelades. Amb aquesta finalitat s'utilitza sovint el soroll blanc gaussià, que es caracteritza per un espectre de potència pla. Les fluctuacions macroscòpiques en el cervell, però, normalment segueixen un espectre '1/fb', que és un tret característic de les correlacions temporals i el soroll de color. Als Capítols 3-5 d'aquesta tesi abordem mitjançant un model neuronal mesoscòpic forçat estocàsticament, les tres preguntes següents. En primer lloc, en el Capítol 3 ens preguntem sobre la importància de les desviacions de l'assumpció de soroll blanc en el context de la dinàmica del cervell i, en particular, estudiem el paper que juga el soroll amb correlació temporal en l'obtenció de ritmes aberrants d'un cervell epilèptic. Trobem que la generació de les dinàmiques epileptiformes depèn de forma monòtona del temps de correlació del soroll. Aquests ritmes es divideixen en bandes de freqüència que, segons, s'ha mostrat experimentalment, augmenten la seva potència espectral abans de les crisis epilèptiques. Expliquem aquests efectes en termes de la interacció entre les freqüències específiques del forçament i l'estructura de bifurcació del model. En segon lloc, en el Capítol 4 es mostra com l'acoblament entre mòduls corticals condueix a patrons d'activitat complexes i a l'aparició d'un fenomen que anomenem excitabilitat col∙lectiva. Els patrons temporals generats per aquest model s'assemblen a les observacions clíniques de les convulsions epilèptiques. En aquest capítol també introduïm un mètode d'anàlisi de la pèrdua d'estabilitat causada per l'acoblament inter-modular excessiu en les xarxes neuronals. En tercer lloc, en el Capítol 5 ens centrem en les interaccions inter-freqüència que es produeixen en una xarxa de mòduls corticals en presència de soroll de color. Suggerim un mecanisme subjacent a l'augment de la potència spectral de ritmes ràpids a causa del forçament amb un ritme lent, i veiem quins paràmetres del soroll descriuen millor els espectres de potència experimental. Finalment, en el Capítol 6, estudiem models dels processos hemodinàmics i metabòlics del cervell, els comparem amb dades experimentals i considerem les conseqüències del seu acoblament amb models neuronals mesoscopics. En conjunt, els nostres resultats mostren la influència combinada del soroll i l'acoblament en models computacionals de l'activitat neuronal. D'altra banda, també demostren la importància de les propietats dinàmiques dels sistemes neuronals en fenòmens fisiològics específics com les interaccions inter-frequència i l'epilèpsia. Els resultats d'aquesta Tesi contribueixen a potenciar l’estudi de l'epilèpsia com una malaltia dinàmica, i el desenvolupament de mètodes de tractament aplicables a pacients epilèptics resistents als fàrmacs.Postprint (published version

The Dynamic Brain: From Spiking Neurons to Neural Masses and Cortical Fields

The cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. In this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. Computational models at different space–time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. Modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. Macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. Each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fMRI), electroencephalogram (EEG), and magnetoencephalogram (MEG). Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. This makes dynamic models critical in integrating theory and experiments. We argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences