Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient, average features of neural masses in order to model activity at the population level, thereby linking microscopic physiology to macroscopic observations, e.g., with the electroencephalogram (EEG). One of the common aspects of MFM descriptions is the presence of a high-dimensional parameter space capturing neurobiological attributes deemed relevant to the brain dynamics of interest. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two archetypal categories or “families”. After investigating and characterizing them in depth, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli such as thalamic input, and distributions of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We here unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They instead emerge when the nonlinear structure of parameter space is partitioned according to bifurcation responses. We call this general method “metabifurcation analysis”. The partitioning cannot be achieved by the investigation of only a small number of parameter sets and is instead the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible parameter sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces.

Dynamical complexity in a mean-field model of human EEG

A recently proposed mean-field theory of mammalian cortex rhythmogenesis describes the salient features of electrical activity in the cerebral macrocolumn, with the use of inhibitory and excitatory neuronal populations (Liley et al 2002). This model is capable of producing a range of important human EEG (electroencephalogram) features such as the alpha rhythm, the 40 Hz activity thought to be associated with conscious awareness (Bojak & Liley 2007) and the changes in EEG spectral power associated with general anesthetic effect (Bojak & Liley 2005). From the point of view of nonlinear dynamics, the model entails a vast parameter space within which multistability, pseudoperiodic regimes, various routes to chaos, fat fractals and rich bifurcation scenarios occur for physiologically relevant parameter values (van Veen & Liley 2006). The origin and the character of this complex behaviour, and its relevance for EEG activity will be illustrated. The existence of short-lived unstable brain states will also be discussed in terms of the available theoretical and experimental results. A perspective on future analysis will conclude the presentation.

Connecting mean field models of neural activity to EEG and fMRI data

Progress in functional neuroimaging of the brain increasingly relies on the integration of data from complementary imaging modalities in order to improve spatiotemporal resolution and interpretability. However, the usefulness of merely statistical combinations is limited, since neural signal sources differ between modalities and are related non-trivially. We demonstrate here that a mean field model of brain activity can simultaneously predict EEG and fMRI BOLD with proper signal generation and expression. Simulations are shown using a realistic head model based on structural MRI, which includes both dense short-range background connectivity and long-range specific connectivity between brain regions. The distribution of modeled neural masses is comparable to the spatial resolution of fMRI BOLD, and the temporal resolution of the modeled dynamics, importantly including activity conduction, matches the fastest known EEG phenomena. The creation of a cortical mean field model with anatomically sound geometry, extensive connectivity, and proper signal expression is an important first step towards the model-based integration of multimodal neuroimages.

Ketamine, propofol, and the EEG: A neural field analysis of HCN1-mediated interactions

Ketamine and propofol are two well-known, powerful anesthetic agents, yet at first sight this appears to be their only commonality. Ketamine is a dissociative anesthetic agent, whose main mechanism of action is considered to be N-methyl-D-aspartate (NMDA) antagonism; whereas propofol is a general anesthetic agent, which is assumed to primarily potentiate currents gated by γ-aminobutyric acid type A (GABAA) receptors. However, several experimental observations suggest a closer relationship. First, the effect of ketamine on the electroencephalogram (EEG) is markedly changed in the presence of propofol: on its own ketamine increases θ (4–8 Hz) and decreases α (8–13 Hz) oscillations, whereas ketamine induces a significant shift to beta band frequencies (13–30 Hz) in the presence of propofol. Second, both ketamine and propofol cause inhibition of the inward pacemaker current Ih, by binding to the corresponding hyperpolarization-activated cyclic nucleotide-gated potassium channel 1 (HCN1) subunit. The resulting effect is a hyperpolarization of the neuron’s resting membrane potential. Third, the ability of both ketamine and propofol to induce hypnosis is reduced in HCN1-knockout mice. Here we show that one can theoretically understand the observed spectral changes of the EEG based on HCN1-mediated hyperpolarizations alone, without involving the supposed main mechanisms of action of these drugs through NMDA and GABAA, respectively. On the basis of our successful EEG model we conclude that ketamine and propofol should be antagonistic to each other in their interaction at HCN1 subunits. Such a prediction is in accord with the results of clinical experiment in which it is found that ketamine and propofol interact in an infra-additive manner with respect to the endpoints of hypnosis and immobility.

Realistic mean field forward predictions for the integration of co-registered EEG/fMRI

Brain activity can be measured non-invasively with functional imaging techniques. Each pixel in such an image represents a neural mass of about 105 to 107 neurons. Mean field models (MFMs) approximate their activity by averaging out neural variability while retaining salient underlying features, like neurotransmitter kinetics. However, MFMs incorporating the regional variability, realistic geometry and connectivity of cortex have so far appeared intractable. This lack of biological realism has led to a focus on gross temporal features of the EEG. We address these impediments and showcase a "proof of principle" forward prediction of co-registered EEG/fMRI for a full-size human cortex in a realistic head model with anatomical connectivity, see figure 1. MFMs usually assume homogeneous neural masses, isotropic long-range connectivity and simplistic signal expression to allow rapid computation with partial differential equations. But these approximations are insufficient in particular for the high spatial resolution obtained with fMRI, since different cortical areas vary in their architectonic and dynamical properties, have complex connectivity, and can contribute non-trivially to the measured signal. Our code instead supports the local variation of model parameters and freely chosen connectivity for many thousand triangulation nodes spanning a cortical surface extracted from structural MRI. This allows the introduction of realistic anatomical and physiological parameters for cortical areas and their connectivity, including both intra- and inter-area connections. Proper cortical folding and conduction through a realistic head model is then added to obtain accurate signal expression for a comparison to experimental data. To showcase the synergy of these computational developments, we predict simultaneously EEG and fMRI BOLD responses by adding an established model for neurovascular coupling and convolving "Balloon-Windkessel" hemodynamics. We also incorporate regional connectivity extracted from the CoCoMac database [1]. Importantly, these extensions can be easily adapted according to future insights and data. Furthermore, while our own simulation is based on one specific MFM [2], the computational framework is general and can be applied to models favored by the user. Finally, we provide a brief outlook on improving the integration of multi-modal imaging data through iterative fits of a single underlying MFM in this realistic simulation framework.

Electrorhythmogenesis and anaesthesia in a physiological mean field theory

We solve eight partial-differential, two-dimensional, nonlinear mean field equations, which describe the dynamics of large populations of cortical neurons. Linearized versions of these equations have been used to generate the strong resonances observed in the human EEG, in particular the α-rhythm (8–), with physiologically plausible parameters. We extend these results here by numerically solving the full equations on a cortex of realistic size, which receives appropriately “colored” noise as extra-cortical input. A brief summary of the numerical methods is provided. As an outlook to future applications, we explain how the effects of GABA-enhancing general anaesthetics can be simulated and present first results.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

The tensor-kinetic field in nuclear collisions

The role of the tensor terms in the Skyrme interaction is studied for their effect in dynamic calculations where non-zero contributions to the mean-field may arise, even when the starting nucleus, or nuclei are even-even and have no active time-odd potentials in the ground state. We study collisions in the test-bed 16O-16O system, and give a qualitative analysis of the behaviour of the time-odd tensor-kinetic density, which only appears in the mean field Hamiltonian in the presence of the tensor force. We find an axial excitation of this density is induced by a collision.

Assessment of the RE(OH)3 Ising magnetic materials as possible candidates for the study of transverse-field-induced quantum phase transitions

The LiHoxY1−xF4 Ising magnetic material subject to a magnetic field perpendicular to the Ho3+ Ising direction has shown over the past 20 years to be a host of very interesting thermodynamic and magnetic phenomena. Unfortunately, the availability of other magnetic materials other than LiHoxY1−xF4 that may be described by a transverse-field Ising model remains very much limited. It is in this context that we use here a mean-field theory to investigate the suitability of the Ho(OH)3, Dy(OH)3, and Tb(OH)3 insulating hexagonal dipolar Ising-type ferromagnets for the study of the quantum phase transition induced by a magnetic field, Bx, applied perpendicular to the Ising spin direction. Experimentally, the zero-field critical (Curie) temperatures are known to be Tc≈2.54, 3.48, and 3.72 K, for Ho(OH)3, Dy(OH)3, and Tb(OH)3, respectively. From our calculations we estimate the critical transverse field, Bxc, to destroy ferromagnetic order at zero temperature to be Bxc=4.35, 5.03, and 54.81 T for Ho(OH)3, Dy(OH)3, and Tb(OH)3, respectively. We find that Ho(OH)3, similarly to LiHoF4, can be quantitatively described by an effective S=1/2 transverse-field Ising model. This is not the case for Dy(OH)3 due to the strong admixing between the ground doublet and first excited doublet induced by the dipolar interactions. Furthermore, we find that the paramagnetic (PM) to ferromagnetic (FM) transition in Dy(OH)3 becomes first order for strong Bx and low temperatures. Hence, the PM to FM zero-temperature transition in Dy(OH)3 may be first order and not quantum critical. We investigate the effect of competing antiferromagnetic nearest-neighbor exchange and applied magnetic field, Bz, along the Ising spin direction ẑ on the first-order transition in Dy(OH)3. We conclude from these preliminary calculations that Ho(OH)3 and Dy(OH)3 and their Y3+ diamagnetically diluted variants, HoxY1−x(OH)3 and DyxY1−x(OH)3, are potentially interesting systems to study transverse-field-induced quantum fluctuations effects in hard axis (Ising-type) magnetic materials.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Rainfall estimation by rain gauge-radar combination: a concurrent multiplicative-additive approach

A procedure (concurrent multiplicative-additive objective analysis scheme [CMA-OAS]) is proposed for operational rainfall estimation using rain gauges and radar data. On the basis of a concurrent multiplicative-additive (CMA) decomposition of the spatially nonuniform radar bias, within-storm variability of rainfall and fractional coverage of rainfall are taken into account. Thus both spatially nonuniform radar bias, given that rainfall is detected, and bias in radar detection of rainfall are handled. The interpolation procedure of CMA-OAS is built on Barnes' objective analysis scheme (OAS), whose purpose is to estimate a filtered spatial field of the variable of interest through a successive correction of residuals resulting from a Gaussian kernel smoother applied on spatial samples. The CMA-OAS, first, poses an optimization problem at each gauge-radar support point to obtain both a local multiplicative-additive radar bias decomposition and a regionalization parameter. Second, local biases and regionalization parameters are integrated into an OAS to estimate the multisensor rainfall at the ground level. The procedure is suited to relatively sparse rain gauge networks. To show the procedure, six storms are analyzed at hourly steps over 10,663 km2. Results generally indicated an improved quality with respect to other methods evaluated: a standard mean-field bias adjustment, a spatially variable adjustment with multiplicative factors, and ordinary cokriging.

Neural population models and cortical field theory: overview

The term neural population models (NPMs) is used here as catchall for a wide range of approaches that have been variously called neural mass models, mean field models, neural field models, bulk models, and so forth. All NPMs attempt to describe the collective action of neural assemblies directly. Some NPMs treat the densely populated tissue of cortex as an excitable medium, leading to spatially continuous cortical field theories (CFTs). An indirect approach would start by modelling individual cells and then would explain the collective action of a group of cells by coupling many individual models together. In contrast, NPMs employ collective state variables, typically defined as averages over the group of cells, in order to describe the population activity directly in a single model. The strength and the weakness of his approach are hence one and the same: simplification by bulk. Is this justified and indeed useful, or does it lead to oversimplification which fails to capture the pheno ...

Terahertz analysis of stratified wall plaster at buildings of cultural importance across Europe

Terahertz (THz) radiation is being developed as a tool for the analysis of cultural heritage, and due to recent advances in technology is now available commercially in systems which can be deployed for field analysis. The radiation is capable of penetrating up to one centimetre of wall plaster and is delivered in ultrafast pulses which are reflected from layers within this region. The technique is non-contact, non-invasive and non-destructive. While sub-surface radar is able to penetrate over a metre of wall plaster, producing details of internal structures, infrared and ultraviolet techniques produce information about the surface layers of wall plaster. THz radiation is able to provide information about the interim region of up to approximately one centimetre into the wall surface. Data from Chartres Cathedral, France, Riga Dome Cathedral, Latvia, and Chartreuse du Val de Bénédiction, France is presented each with different research questions. The presence of sub-surface paint layers was expected from documentary evidence, dating to the 13th Century, at Chartres Cathedral. In contrast, at the Riga Dome Cathedral surface painting had been obscured as recently as 1941 during the Russian occupation of Latvia using white lead-based paint. In the 13th Century, wall paintings at the Chapel of the Frescos, Chartreuse du Val de Benediction in Villeneuve les Avignon were constructed using sinopia under-painting on plaster covering uneven stonework.. This paper compares and contrasts the ability of THz radiation to provide information about sub-surface features in churches and Cathedrals across Europe by analysing depth based profiles gained from the reflected signal. © (2013) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE).