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.

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.

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.

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.

Scenario and modelling uncertainty in global mean temperature change derived from emission-driven global climate models

We compare future changes in global mean temperature in response to different future scenarios which, for the first time, arise from emission-driven rather than concentration-driven perturbed parameter ensemble of a global climate model (GCM). These new GCM simulations sample uncertainties in atmospheric feedbacks, land carbon cycle, ocean physics and aerosol sulphur cycle processes. We find broader ranges of projected temperature responses arising when considering emission rather than concentration-driven simulations (with 10–90th percentile ranges of 1.7 K for the aggressive mitigation scenario, up to 3.9 K for the high-end, business as usual scenario). A small minority of simulations resulting from combinations of strong atmospheric feedbacks and carbon cycle responses show temperature increases in excess of 9 K (RCP8.5) and even under aggressive mitigation (RCP2.6) temperatures in excess of 4 K. While the simulations point to much larger temperature ranges for emission-driven experiments, they do not change existing expectations (based on previous concentration-driven experiments) on the timescales over which different sources of uncertainty are important. The new simulations sample a range of future atmospheric concentrations for each emission scenario. Both in the case of SRES A1B and the Representative Concentration Pathways (RCPs), the concentration scenarios used to drive GCM ensembles, lies towards the lower end of our simulated distribution. This design decision (a legacy of previous assessments) is likely to lead concentration-driven experiments to under-sample strong feedback responses in future projections. Our ensemble of emission-driven simulations span the global temperature response of the CMIP5 emission-driven simulations, except at the low end. Combinations of low climate sensitivity and low carbon cycle feedbacks lead to a number of CMIP5 responses to lie below our ensemble range. The ensemble simulates a number of high-end responses which lie above the CMIP5 carbon cycle range. These high-end simulations can be linked to sampling a number of stronger carbon cycle feedbacks and to sampling climate sensitivities above 4.5 K. This latter aspect highlights the priority in identifying real-world climate-sensitivity constraints which, if achieved, would lead to reductions on the upper bound of projected global mean temperature change. The ensembles of simulations presented here provides a framework to explore relationships between present-day observables and future changes, while the large spread of future-projected changes highlights the ongoing need for such work.

On the 'downward control' of extratropical diabatic circulations by eddy-induced mean zonal forces

The situation considered is that of a zonally symmetric model of the middle atmosphere subject to a given quasi-steady zonal force F̄, conceived to be the result of irreversible angular momentum transfer due to the upward propagation and breaking of Rossby and gravity waves together with any other dissipative eddy effects that may be relevant. The model's diabatic heating is assumed to have the qualitative character of a relaxation toward some radiatively determined temperature field. To the extent that the force F̄ may be regarded as given, and the extratropical angular momentum distribution is realistic, the extratropical diabatic mass flow across a given isentropic surface may be regarded as controlled exclusively by the F̄ distribution above that surface (implying control by the eddy dissipation above that surface and not, for instance, by the frequency of tropopause folding below). This “downward control” principle expresses a critical part of the dynamical chain of cause and effect governing the average rate at which photochemical products like ozone become available for folding into, or otherwise descending into, the extratropical troposphere. The dynamical facts expressed by the principle are also relevant, for instance, to understanding the seasonal-mean rate of upwelling of water vapor to the summer mesopause, and the interhemispheric differences in stratospheric tracer transport. The robustness of the principle is examined when F̄ is time-dependent. For a global-scale, zonally symmetric diabatic circulation with a Brewer-Dobson-like horizontal structure given by the second zonally symmetric Hough mode, with Rossby height HR = 13 km in an isothermal atmosphere with density scale height H = 7 km, the vertical partitioning of the unsteady part of the mass circulation caused by fluctuations in F̄ confined to a shallow layer LF̄ is always at least 84% downward. It is 90% downward when the force fluctuates sinusoidally on twice the radiative relaxation timescale and 95% if five times slower. The time-dependent adjustment when F̄ is changed suddenly is elucidated, extending the work of Dickinson (1968), when the atmosphere is unbounded above and below. Above the forcing, the adjustment is characterized by decay of the meridional mass circulation cell at a rate proportional to the radiative relaxation rate τr−1 divided by {1 + (4H2/HR2)}. This decay is related to the boundedness of the angular momentum that can be taken up by the finite mass of air above LF̄ without causing an ever-increasing departure from thermal wind balance. Below the forcing, the meridional mass circulation cell penetrates downward at a speed τr−1 HR2/H. For the second Hough mode, the time for downward penetration through one density scale height is about 6 days if the radiative relaxation time is 20 days, the latter being representative of the lower stratosphere. At any given altitude, a steady state is approached. The effect of a rigid lower boundary on the time-dependent adjustment is also considered. If a frictional planetary boundary layer is present then a steady state is ultimately approached everywhere, with the mass circulation extending downward from LF̄ and closing via the boundary layer. Satellite observations of temperature and ozone are used in conjunction with a radiative transfer scheme to estimate the altitudes from which the lower stratospheric diabatic vertical velocity is controlled by the effective F̄ in the real atmosphere. The data appear to indicate that about 80% of the effective control is usually exerted from below 40 km but with significant exceptions up to 70 km (in the high latitude southern hemispheric winter). The implications for numerical modelling of chemical transport are noted.

Mean motions induced by baroclinic instability in a jet

A study is made of the zonal-mean motions induced by a growing baroclinic wave in several contexts, under the framework of three different analysis schemes: the conventional Eulerian mean (EM), the transformed Eulerian mean (TEM), and the generalized Lagrangian mean (GLM). The effect of meridional shear in the initial jet on these induced mean motions is considered by treating the instability problem in the context of the two-layer model. The conceptual simplicity of the TEM formulation is shown to be useful in diagnosing the dynamics of instability, much as it has been found helpful in many problems of wave, mean-flow interaction. In addition, it is found that the TEM vertical velocity is a very good indicator of the GLM vertical velocity. However, the GLM meridional velocity is always convergent towards the centre of instability activity, and is not at all well represented by the nondivergent TEM meridional velocity. In comparing the results with Uryu's (1979) calculation of the GLM circulation induced by a growing Eady wave, it is found that the inclusion of meridional jet shear in the present work leads to some strikingly different effects in the GLM zonal wind acceleration. In the case of pure baroclinic instability treated by Uryu, the Eulerian and Stokes accelerations nearly cancel each other in the centre of the channel, leaving a weak Lagrangian acceleration opposed to the Eulerian one. In the more general case of mixed baroclinic-barotropic instability, however, the Eulerian and Stokes accelerations can reinforce one another, leading to a very strong Lagrangian zonal wind

The effect of non-uniform radiative damping on the zonal-mean dynamics of the extratropical middle atmosphere

The effect of spatial and temporal variations in the radiative damping rate on the response to an imposed forcing or diabatic heating is examined in a zonal-mean model of the middle atmosphere. Attention is restricted to the extratropics, where a linear approach is viable. It is found that regions with weak radiative damping rates are more sensitive in terms of temperature to the remote influence of the diabatic circulation. The delay in the response in such regions can mean that ‘downward’ control is not achieved on seasonal time-scales. A seasonal variation in the radiative damping rate modulates the evolution of the response and leaves a transient-like signature in the annual mean temperature field. Several idealized examples are considered, motivated by topical questions. It is found that wave drag outside the polar vortex can significantly affect the temperatures in its interior, so that high-latitude, high-altitude gravity-wave drag is not the only mechanism for warming the southern hemisphere polar vortex. Diabatic mass transport through the 100 hPa surface is found to lag the seasonal evolution of the wave drag that drives the transport, and thus cannot be considered to be in the downward control regime. On the other hand, the seasonal variation of the radiative damping rate is found to make only a weak contribution to the annual mean temperature increase that has been observed above the ozone hole. Copyright © 2002 Royal Meteorological Society.

The ocean’s role in setting the mean position of the Inter-Tropical Convergence Zone

Through study of observations and coupled climate simulations, it is argued that the mean position of the Inter-Tropical Convergence Zone (ITCZ) north of the equator is a consequence of a northwards heat transport across the equator by ocean circulation. Observations suggest that the hemispheric net radiative forcing of climate at the top of the atmosphere is almost perfectly symmetric about the equator, and so the total (atmosphere plus ocean) heat transport across the equator is small (order 0.2 PW northwards). Due to the Atlantic ocean’s meridional overturning circulation, however, the ocean carries significantly more heat northwards across the equator (order 0.4 PW) than does the coupled system. There are two primary consequences. First, atmospheric heat transport is southwards across the equator to compensate (0.2 PW southwards), resulting in the ITCZ being displaced north of the equator. Second, the atmosphere, and indeed the ocean, is slightly warmer (by perhaps 2 °C) in the northern hemisphere than in the southern hemisphere. This leads to the northern hemisphere emitting slightly more outgoing longwave radiation than the southern hemisphere by virtue of its relative warmth, supporting the small northward heat transport by the coupled system across the equator. To conclude, the coupled nature of the problem is illustrated through study of atmosphere–ocean–ice simulations in the idealized setting of an aquaplanet, resolving the key processes at work.

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.

Surface water temperature observations of large lakes by optimal estimation

Optimal estimation (OE) and probabilistic cloud screening were developed to provide lake surface water temperature (LSWT) estimates from the series of (advanced) along-track scanning radiometers (ATSRs). Variations in physical properties such as elevation, salinity, and atmospheric conditions are accounted for through the forward modelling of observed radiances. Therefore, the OE retrieval scheme developed is generic (i.e., applicable to all lakes). LSWTs were obtained for 258 of Earth's largest lakes from ATSR-2 and AATSR imagery from 1995 to 2009. Comparison to in situ observations from several lakes yields satellite in situ differences of −0.2 ± 0.7 K for daytime and −0.1 ± 0.5 K for nighttime observations (mean ± standard deviation). This compares with −0.05 ± 0.8 K for daytime and −0.1 ± 0.9 K for nighttime observations for previous methods based on operational sea surface temperature algorithms. The new approach also increases coverage (reducing misclassification of clear sky as cloud) and exhibits greater consistency between retrievals using different channel–view combinations. Empirical orthogonal function (EOF) techniques were applied to the LSWT retrievals (which contain gaps due to cloud cover) to reconstruct spatially and temporally complete time series of LSWT. The new LSWT observations and the EOF-based reconstructions offer benefits to numerical weather prediction, lake model validation, and improve our knowledge of the climatology of lakes globally. Both observations and reconstructions are publically available from http://hdl.handle.net/10283/88.

A statistical model for sea surface diurnal warming driven by numercial weather predictions fluxes and winds

A statistical model is derived relating the diurnal variation of sea surface temperature (SST) to the net surface heat flux and surface wind speed from a numerical weather prediction (NWP) model. The model is derived using fluxes and winds from the European Centre for Medium-Range Weather Forecasting (ECMWF) NWP model and SSTs from the Spinning Enhanced Visible and Infrared Imager (SEVIRI). In the model, diurnal warming has a linear dependence on the net surface heat flux integrated since (approximately) dawn and an inverse quadratic dependence on the maximum of the surface wind speed in the same period. The model coefficients are found by matching, for a given integrated heat flux, the frequency distributions of the maximum wind speed and the observed warming. Diurnal cooling, where it occurs, is modelled as proportional to the integrated heat flux divided by the heat capacity of the seasonal mixed layer. The model reproduces the statistics (mean, standard deviation, and 95-percentile) of the diurnal variation of SST seen by SEVIRI and reproduces the geographical pattern of mean warming seen by the Advanced Microwave Scanning Radiometer (AMSR-E). We use the functional dependencies in the statistical model to test the behaviour of two physical model of diurnal warming that display contrasting systematic errors.

Estimation of sea surface temperature from the Spinning Enhanced Visible and Infra Red Imager, improved using numerical weather prediction

Most of the operational Sea Surface Temperature (SST) products derived from satellite infrared radiometry use multi-spectral algorithms. They show, in general, reasonable performances with root mean square (RMS) residuals around 0.5 K when validated against buoy measurements, but have limitations, particularly a component of the retrieval error that relates to such algorithms' limited ability to cope with the full variability of atmospheric absorption and emission. We propose to use forecast atmospheric profiles and a radiative transfer model to simulate the algorithmic errors of multi-spectral algorithms. In the practical case of SST derived from the Spinning Enhanced Visible and Infrared Imager (SEVIRI) onboard Meteosat Second Generation (MSG), we demonstrate that simulated algorithmic errors do explain a significant component of the actual errors observed for the non linear (NL) split window algorithm in operational use at the Centre de Météorologie Spatiale (CMS). The simulated errors, used as correction terms, reduce significantly the regional biases of the NL algorithm as well as the standard deviation of the differences with drifting buoy measurements. The availability of atmospheric profiles associated with observed satellite-buoy differences allows us to analyze the origins of the main algorithmic errors observed in the SEVIRI field of view: a negative bias in the inter-tropical zone, and a mid-latitude positive bias. We demonstrate how these errors are explained by the sensitivity of observed brightness temperatures to the vertical distribution of water vapour, propagated through the SST retrieval algorithm.

Sea surface temperature from a geostationary satellite by optimal estimation

Optimal estimation (OE) is applied as a technique for retrieving sea surface temperature (SST) from thermal imagery obtained by the Spinning Enhanced Visible and Infra-Red Imager (SEVIRI) on Meteosat 9. OE requires simulation of observations as part of the retrieval process, and this is done here using numerical weather prediction fields and a fast radiative transfer model. Bias correction of the simulated brightness temperatures (BTs) is found to be a necessary step before retrieval, and is achieved by filtered averaging of simulations minus observations over a time period of 20 days and spatial scale of 2.5° in latitude and longitude. Throughout this study, BT observations are clear-sky averages over cells of size 0.5° in latitude and longitude. Results for the OE SST are compared to results using a traditional non-linear retrieval algorithm (“NLSST”), both validated against a set of 30108 night-time matches with drifting buoy observations. For the OE SST the mean difference with respect to drifter SSTs is − 0.01 K and the standard deviation is 0.47 K, compared to − 0.38 K and 0.70 K respectively for the NLSST algorithm. Perhaps more importantly, systematic biases in NLSST with respect to geographical location, atmospheric water vapour and satellite zenith angle are greatly reduced for the OE SST. However, the OE SST is calculated to have a lower sensitivity of retrieved SST to true SST variations than the NLSST. This feature would be a disadvantage for observing SST fronts and diurnal variability, and raises questions as to how best to exploit OE techniques at SEVIRI's full spatial resolution.