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

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

Neural Connectivity with Hidden Gaussian Graphical State-Model

The noninvasive procedures for neural connectivity are under questioning. Theoretical models sustain that the electromagnetic field registered at external sensors is elicited by currents at neural space. Nevertheless, what we observe at the sensor space is a superposition of projected fields, from the whole gray-matter. This is the reason for a major pitfall of noninvasive Electrophysiology methods: distorted reconstruction of neural activity and its connectivity or leakage. It has been proven that current methods produce incorrect connectomes. Somewhat related to the incorrect connectivity modelling, they disregard either Systems Theory and Bayesian Information Theory. We introduce a new formalism that attains for it, Hidden Gaussian Graphical State-Model (HIGGS). A neural Gaussian Graphical Model (GGM) hidden by the observation equation of Magneto-encephalographic (MEEG) signals. HIGGS is equivalent to a frequency domain Linear State Space Model (LSSM) but with sparse connectivity prior. The mathematical contribution here is the theory for high-dimensional and frequency-domain HIGGS solvers. We demonstrate that HIGGS can attenuate the leakage effect in the most critical case: the distortion EEG signal due to head volume conduction heterogeneities. Its application in EEG is illustrated with retrieved connectivity patterns from human Steady State Visual Evoked Potentials (SSVEP). We provide for the first time confirmatory evidence for noninvasive procedures of neural connectivity: concurrent EEG and Electrocorticography (ECoG) recordings on monkey. Open source packages are freely available online, to reproduce the results presented in this paper and to analyze external MEEG databases

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size

Neural population equations such as neural mass or field models are widely used to study brain activity on a large scale. However, the relation of these models to the properties of single neurons is unclear. Here we derive an equation for several interacting populations at the mesoscopic scale starting from a microscopic model of randomly connected generalized integrate-and-fire neuron models. Each population consists of 50 -- 2000 neurons of the same type but different populations account for different neuron types. The stochastic population equations that we find reveal how spike-history effects in single-neuron dynamics such as refractoriness and adaptation interact with finite-size fluctuations on the population level. Efficient integration of the stochastic mesoscopic equations reproduces the statistical behavior of the population activities obtained from microscopic simulations of a full spiking neural network model. The theory describes nonlinear emergent dynamics like finite-size-induced stochastic transitions in multistable networks and synchronization in balanced networks of excitatory and inhibitory neurons. The mesoscopic equations are employed to rapidly simulate a model of a local cortical microcircuit consisting of eight neuron types. Our theory establishes a general framework for modeling finite-size neural population dynamics based on single cell and synapse parameters and offers an efficient approach to analyzing cortical circuits and computations

Critical fluctuations in cortical models near instability

Australian Research Council, the National Health and Medical Research Council, the Brain Network Recovery Group Grant JSMF22002082, and Netherlands Organization for Scientific Research (NWO #451–10-030

Far from Equilibrium Percolation, Stochastic and Shape Resonances in the Physics of Life

Key physical concepts, relevant for the cross-fertilization between condensed matter physics and the physics of life seen as a collective phenomenon in a system out-of-equilibrium, are discussed. The onset of life can be driven by: (a) the critical fluctuations at the protonic percolation threshold in membrane transport; (b) the stochastic resonance in biological systems, a mechanism that can exploit external and self-generated noise in order to gain efficiency in signal processing; and (c) the shape resonance (or Fano resonance or Feshbach resonance) in the association and dissociation processes of bio-molecules (a quantum mechanism that could play a key role to establish a macroscopic quantum coherence in the cell)

Analyses at microscopic, mesoscopic, and mean-field scales

Die Aktivität des Hippocampus im Tiefschlaf ist geprägt durch sharp wave-ripple Komplexe (SPW-R): kurze (50–100 ms) Phasen mit erhöhter neuronaler Aktivität, moduliert durch eine schnelle “Ripple”-Oszillation (140–220 Hz). SPW-R werden mit Gedächtniskonsolidierung in Verbindung gebracht, aber ihr Ursprung ist unklar. Sowohl exzitatorische als auch inhibitorische Neuronpopulationen könnten die Oszillation generieren. Diese Arbeit analysiert Ripple-Oszillationen in inhibitorischen Netzwerkmodellen auf mikro-, meso- und makroskopischer Ebene und zeigt auf, wie die Ripple-Dynamik von exzitatorischem Input, inhibitorischer Kopplungsstärke und dem Rauschmodell abhängt. Zuerst wird ein stark getriebenes Interneuron-Netzwerk mit starker, verzögerter Kopplung analysiert. Es wird eine Theorie entwickelt, die die Drift-bedingte Feuerdynamik im Mean-field Grenzfall beschreibt. Die Ripple-Frequenz und die Dynamik der Membranpotentiale werden analytisch als Funktion des Inputs und der Netzwerkparameter angenähert. Die Theorie erklärt, warum die Ripple-Frequenz im Verlauf eines SPW-R-Ereignisses sinkt (intra-ripple frequency accommodation, IFA). Weiterhin zeigt eine numerische Analyse, dass ein alternatives Modell, basierend auf einem transienten Störungseffekt in einer schwach gekoppelten Interneuron-Population, unter biologisch plausiblen Annahmen keine IFA erzeugen kann. IFA kann somit zur Modellauswahl beitragen und deutet auf starke, verzögerte inhibitorische Kopplung als plausiblen Mechanismus hin. Schließlich wird die Anwendbarkeit eines kürzlich entwickelten mesoskopischen Ansatzes für die effiziente Simulation von Ripples in endlich großen Netzwerken geprüft. Dabei wird das Rauschen nicht im Input der Neurone beschrieben, sondern als stochastisches Feuern entsprechend einer Hazard-Rate. Es wird untersucht, wie die Wahl des Hazards die dynamische Suszeptibilität einzelner Neurone, und damit die Ripple-Dynamik in rekurrenten Interneuron-Netzwerken beeinflusst.Hippocampal activity during sleep or rest is characterized by sharp wave-ripples (SPW-Rs): transient (50–100 ms) periods of elevated neuronal activity modulated by a fast oscillation — the ripple (140–220 Hz). SPW-Rs have been linked to memory consolidation, but their generation mechanism remains unclear. Multiple potential mechanisms have been proposed, relying on excitation and/or inhibition as the main pacemaker. This thesis analyzes ripple oscillations in inhibitory network models at micro-, meso-, and macroscopic scales and elucidates how the ripple dynamics depends on the excitatory drive, inhibitory coupling strength, and the noise model. First, an interneuron network under strong drive and strong coupling with delay is analyzed. A theory is developed that captures the drift-mediated spiking dynamics in the mean-field limit. The ripple frequency as well as the underlying dynamics of the membrane potential distribution are approximated analytically as a function of the external drive and network parameters. The theory explains why the ripple frequency decreases over the course of an event (intra-ripple frequency accommodation, IFA). Furthermore, numerical analysis shows that an alternative inhibitory ripple model, based on a transient ringing effect in a weakly coupled interneuron population, cannot account for IFA under biologically realistic assumptions. IFA can thus guide model selection and provides new support for strong, delayed inhibitory coupling as a mechanism for ripple generation. Finally, a recently proposed mesoscopic integration scheme is tested as a potential tool for the efficient numerical simulation of ripple dynamics in networks of finite size. This approach requires a switch of the noise model, from noisy input to stochastic output spiking mediated by a hazard function. It is demonstrated how the choice of a hazard function affects the linear response of single neurons and therefore the ripple dynamics in a recurrent interneuron network

Vibrational and stochastic resonances in driven nonlinear systems:part two

Nonlinearity is ubiquitous in both natural and engineering systems. The resultant dynamics has emerged as a multidisciplinary field that has been very extensively investigated, due partly to the potential occurrence of nonlinear phenomena in all branches of sciences, engineering and medicine. Driving nonlinear systems with external excitations can yield a plethora of intriguing and important phenomena – one of the most prominent being that of resonance. In the presence of additional harmonic or stochastic excitation, two exotic forms of resonance can arise: vibrational resonance or stochastic resonance, respectively. Several promising state-of-the-art technologies that were not covered in Part One of this Theme Issue are discussed here. They include inter alia the improvement of image quality, the design of machines and devices that exert vibrations on materials, the harvesting of energy from various forms of ambient vibration, and control of aerodynamic instabilities. They form an important part of the Theme Issue as a whole, which is dedicated to an overview of vibrational and stochastic resonances in driven nonlinear systems. resonances in driven nonlinear systems

Taming neuronal noise with large networks

How does reliable computation emerge from networks of noisy neurons? While individual neurons are intrinsically noisy, the collective dynamics of populations of neurons taken as a whole can be almost deterministic, supporting the hypothesis that, in the brain, computation takes place at the level of neuronal populations. Mathematical models of networks of noisy spiking neurons allow us to study the effects of neuronal noise on the dynamics of large networks. Classical mean-field models, i.e., models where all neurons are identical and where each neuron receives the average spike activity of the other neurons, offer toy examples where neuronal noise is absorbed in large networks, that is, large networks behave like deterministic systems. In particular, the dynamics of these large networks can be described by deterministic neuronal population equations. In this thesis, I first generalize classical mean-field limit proofs to a broad class of spiking neuron models that can exhibit spike-frequency adaptation and short-term synaptic plasticity, in addition to refractoriness. The mean-field limit can be exactly described by a multidimensional partial differential equation; the long time behavior of which can be rigorously studied using deterministic methods. Then, we show that there is a conceptual link between mean-field models for networks of spiking neurons and latent variable models used for the analysis of multi-neuronal recordings. More specifically, we use a recently proposed finite-size neuronal population equation, which we first mathematically clarify, to design a tractable Expectation-Maximization-type algorithm capable of inferring the latent population activities of multi-population spiking neural networks from the spike activity of a few visible neurons only, illustrating the idea that latent variable models can be seen as partially observed mean-field models. In classical mean-field models, neurons in large networks behave like independent, identically distributed processes driven by the average population activity -- a deterministic quantity, by the law of large numbers. The fact the neurons are identically distributed processes implies a form of redundancy that has not been observed in the cortex and which seems biologically implausible. To show, numerically, that the redundancy present in classical mean-field models is unnecessary for neuronal noise absorption in large networks, I construct a disordered network model where networks of spiking neurons behave like deterministic rate networks, despite the absence of redundancy. This last result suggests that the concentration of measure phenomenon, which generalizes the ``law of large numbers'' of classical mean-field models, might be an instrumental principle for understanding the emergence of noise-robust population dynamics in large networks of noisy neurons