A spatially distributed model of brain metabolism highlights the role of diffusion in brain energy metabolism.

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

The IAS-MEEG Package: A Flexible Inverse Source Reconstruction Platform for Reconstruction and Visualization of Brain Activity from M/EEG Data.

We present a standalone Matlab software platform complete with visualization for the reconstruction of the neural activity in the brain from MEG or EEG data. The underlying inversion combines hierarchical Bayesian models and Krylov subspace iterative least squares solvers. The Bayesian framework of the underlying inversion algorithm allows to account for anatomical information and possible a priori belief about the focality of the reconstruction. The computational efficiency makes the software suitable for the reconstruction of lengthy time series on standard computing equipment. The algorithm requires minimal user provided input parameters, although the user can express the desired focality and accuracy of the solution. The code has been designed so as to favor the parallelization performed automatically by Matlab, according to the resources of the host computer. We demonstrate the flexibility of the platform by reconstructing activity patterns with supports of different sizes from MEG and EEG data. Moreover, we show that the software reconstructs well activity patches located either in the subcortical brain structures or on the cortex. The inverse solver and visualization modules can be used either individually or in combination. We also provide a version of the inverse solver that can be used within Brainstorm toolbox. All the software is available online by Github, including the Brainstorm plugin, with accompanying documentation and test data.

Mining the Mind: Linear Discriminant Analysis of MEG Source Reconstruction Time Series Supports Dynamic Changes in Deep Brain Regions During Meditation Sessions.

Meditation practices have been claimed to have a positive effect on the regulation of mood and emotions for quite some time by practitioners, and in recent times there has been a sustained effort to provide a more precise description of the influence of meditation on the human brain. Longitudinal studies have reported morphological changes in cortical thickness and volume in selected brain regions due to meditation practice, which is interpreted as an evidence its effectiveness beyond the subjective self reporting. Using magnetoencephalography (MEG) or electroencephalography to quantify the changes in brain activity during meditation practice represents a challenge, as no clear hypothesis about the spatial or temporal pattern of such changes is available to date. In this article we consider MEG data collected during meditation sessions of experienced Buddhist monks practicing focused attention (Samatha) and open monitoring (Vipassana) meditation, contrasted by resting state with eyes closed. The MEG data are first mapped to time series of brain activity averaged over brain regions corresponding to a standard Destrieux brain atlas. Next, by bootstrapping and spectral analysis, the data are mapped to matrices representing random samples of power spectral densities in [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] frequency bands. We use linear discriminant analysis to demonstrate that the samples corresponding to different meditative or resting states contain enough fingerprints of the brain state to allow a separation between different states, and we identify the brain regions that appear to contribute to the separation. Our findings suggest that the cingulate cortex, insular cortex and some of the internal structures, most notably the accumbens, the caudate and the putamen nuclei, the thalamus and the amygdalae stand out as separating regions, which seems to correlate well with earlier findings based on longitudinal studies.

Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting.

A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation.

Uncertainty quantification in flux balance analysis of spatially lumped and distributed models of neuron-astrocyte metabolism.

Identifying feasible steady state solutions of a brain energy metabolism model is an inverse problem that allows infinitely many solutions. The characterization of the non-uniqueness, or the uncertainty quantification of the flux balance analysis, is tantamount to identifying the degrees of freedom of the solution. The degrees of freedom of multi-compartment mathematical models for energy metabolism of a neuron-astrocyte complex may offer a key to understand the different ways in which the energetic needs of the brain are met. In this paper we study the uncertainty in the solution, using techniques of linear algebra to identify the degrees of freedom in a lumped model, and Markov chain Monte Carlo methods in its extension to a spatially distributed case. The interpretation of the degrees of freedom in metabolic terms, more specifically, glucose and oxygen partitioning, is then leveraged to derive constraints on the free parameters to guarantee that the model is energetically feasible. We demonstrate how the model can be used to estimate the stoichiometric energy needs of the cells as well as the household energy based on the measured oxidative cerebral metabolic rate of glucose and glutamate cycling. Moreover, our analysis shows that in the lumped model the net direction of lactate dehydrogenase (LDH) in the cells can be deduced from the glucose partitioning between the compartments. The extension of the lumped model to a spatially distributed multi-compartment setting that includes diffusion fluxes from capillary to tissue increases the number of degrees of freedom, requiring the use of statistical sampling techniques. The analysis of the distributed model reveals that some of the conclusions valid for the spatially lumped model, e.g., concerning the LDH activity and glucose partitioning, may no longer hold.

A spatially distributed computational model of brain cellular metabolism.

This paper develops a three-dimensional spatially distributed model of brain cellular metabolism and investigates how the locus of the synaptic activity in reference to the capillaries and diffusion affects the behavior of the model, a type of analysis which is impossible to carry out in spatially lumped models, which are shown to be consistent spatially averaged approximations of the distributed model. To avoid a geometrically detailed modeling of the complex structure of the tissue consisting of different cell types and the extracellular space, the distributed model is based on a novel multi-domain formulation of reaction-diffusion equations, accounting also for separate mitochondria. The model reduction relating the spatially distributed model and lower dimensional reduced models, including the well-mixed spatially lumped compartment model, is carefully explained. We illustrate the effects of losing the spatial resolution with a computed example which is based on a reduced one-dimensional distributed radial model, and look into how the model behaves when the locus of the synaptic activity in reference to the capillaries is changed. By averaging the fluxes and concentrations in the distributed radial model to correspond to quantities in a lumped model, and further by estimating the parameters in the lumped, we conclude that varying the locus of the synaptic activity in reference to the capillaries alters significantly the lumped model parameters. This observation seems to be consequential for parameter estimation procedures from data when the spatial resolution is insufficient to determine the locus of the activity, indicating that the model uncertainty is an inherent feature of lumped models.

Modeling HIV-1 dynamics and fitness in cell culture across scales.

A common approach to understand and analyze complex biological systems is to describe the dynamics in terms of a system of ordinary differential equations (ODE) depending on numerous biologically meaningful and descriptive parameters that are estimated using observed data. The ODE models are often based on the implicit assumption of well-mixed dynamics, i.e., the delay of interaction due to spatial distribution is not included in the model. In this article, we address the question how the heterogeneity of the underlying system affects the estimated parameter values of the ODE model, and on the other hand, what information about the microscopic system can be drawn from these values. The system we are considering is a pairwise growth competition assay used to quantify ex vivo replicative fitness of different HIV-1 isolates. To overcome the lack of ground truth, we generate data using a detailed microscopic spatially distributed hybrid stochastic-deterministic (HSD) infection model in which the dynamics is controlled by parameters directly related to cell level infection, virus production processes, and diffusion of virus particles. The synthetic data sets are then analyzed using an ODE based well-mixed model, in which the corresponding macroscopic parameter distributions are estimated using Markov chain Monte Carlo (MCMC) methods. This approach provides a comprehensive picture of the statistical dependencies of the model parameter across different scales.

A reaction-diffusion model of CO2 influx into an oocyte.

We have developed and implemented a novel mathematical model for simulating transients in surface pH (pH(S)) and intracellular pH (pH(i)) caused by the influx of carbon dioxide (CO(2)) into a Xenopus oocyte. These transients are important tools for studying gas channels. We assume that the oocyte is a sphere surrounded by a thin layer of unstirred fluid, the extracellular unconvected fluid (EUF), which is in turn surrounded by the well-stirred bulk extracellular fluid (BECF) that represents an infinite reservoir for all solutes. Here, we assume that the oocyte plasma membrane is permeable only to CO(2). In both the EUF and intracellular space, solute concentrations can change because of diffusion and reactions. The reactions are the slow equilibration of the CO(2) hydration-dehydration reactions and competing equilibria among carbonic acid (H(2)CO(3))/bicarbonate (HCO(3)(-)) and a multitude of non-CO(2)/HCO(3)(-) buffers. Mathematically, the model is described by a coupled system of reaction-diffusion equations that-assuming spherical radial symmetry-we solved using the method of lines with appropriate stiff solvers. In agreement with experimental data [Musa-Aziz et al. 2009, PNAS 106 5406-5411], the model predicts that exposing the cell to extracellular 1.5% CO(2)/10 mM HCO(3)(-) (pH 7.50) causes pH(i) to fall and pH(S) to rise rapidly to a peak and then decay. Moreover, the model provides insights into the competition between diffusion and reaction processes when we change the width of the EUF, membrane permeability to CO(2), native extra- and intracellular carbonic anhydrase-like activities, the non-CO(2)/HCO(3)(-) (intrinsic) intracellular buffering power, or mobility of intrinsic intracellular buffers.

A hybrid stochastic-deterministic computational model accurately describes spatial dynamics and virus diffusion in HIV-1 growth competition assay.

We present a new hybrid stochastic-deterministic, spatially distributed computational model to simulate growth competition assays on a relatively immobile monolayer of peripheral blood mononuclear cells (PBMCs), commonly used for determining ex vivo fitness of human immunodeficiency virus type-1 (HIV-1). The novel features of our approach include incorporation of viral diffusion through a deterministic diffusion model while simulating cellular dynamics via a stochastic Markov chain model. The model accounts for multiple infections of target cells, CD4-downregulation, and the delay between the infection of a cell and the production of new virus particles. The minimum threshold level of infection induced by a virus inoculum is determined via a series of dilution experiments, and is used to determine the probability of infection of a susceptible cell as a function of local virus density. We illustrate how this model can be used for estimating the distribution of cells infected by either a single virus type or two competing viruses. Our model captures experimentally observed variation in the fitness difference between two virus strains, and suggests a way to minimize variation and dual infection in experiments.

Dynamic activation model for a glutamatergic neurovascular unit.

This article considers a dynamic spatially lumped model for brain energy metabolism and proposes to use the results of a Markov chain Monte Carlo (MCMC) based flux balance analysis to estimate the kinetic model parameters. By treating steady state reaction fluxes and transport rates as random variables we are able to propagate the uncertainty in the steady state configurations to the predictions of the dynamic model, whose responses are no longer individual but ensembles of time courses. The kinetic model consists of five compartments and is governed by kinetic mass balance equations with Michaelis-Menten type expressions for reaction rates and transports between the compartments. The neuronal activation is implemented in terms of the effect of neuronal activity on parameters controlling the blood flow and neurotransmitter transport, and a feedback mechanism coupling the glutamate concentration in the synaptic cleft and the ATP hydrolysis, thus accounting for the energetic cost of the membrane potential restoration in the postsynaptic neurons. The changes in capillary volume follow the balloon model developed for BOLD MRI. The model follows the time course of the saturation levels of the blood hemoglobin, which link metabolism and BOLD FMRI signal. Analysis of the model predictions suggest that stoichiometry alone is not enough to determine glucose partitioning between neuron and astrocyte. Lactate exchange between neuron and astrocyte is supported by the model predictions, but the uncertainty on the direction and rate is rather elevated. By and large, the model suggests that astrocyte produces and effluxes lactate, while neuron may switch from using to producing lactate. The level of ATP hydrolysis in astrocyte is substantially higher than strictly required for neurotransmitter cycling, in agreement with the literature.

Interpretation of NMR spectroscopy human brain data with a multi-compartment computational model of cerebral metabolism.

One of the main difficulties in studying human brain metabolism at rest and during neuronal stimulation is that direct quantitative information of metabolite and intermediate concentrations in real time from in vivo and in situ brain cells is extremely difficult to obtain. We present a new six compartment dynamic computational model of the astrocyte-glutamatergic neuron cellular complex, previously used and validated for steady state investigations [1],which utilizes Michaelis-Menten type kinetic expressions for the reaction fluxes and transport rates. The model is employed to interpret experimental data (total tissue concentrations of glucose, lactate, aspartate, and glutamate) collected via NMR spectroscopy [2] in terms of compartmentalized metabolism. By integrating numerical methods with Bayesian statistics, we obtain an ensemble of models in statistical agreement with the data. Moreover, our preliminary results seem to suggest that NMR spectroscopy detects the time profile of the concentrations of glucose, lactate and aspartate in glutamatergic neuron.

An efficient deconvolution algorithm for estimating oxygen consumption during muscle activities.

The reconstruction of an unknown input function from noisy measurements in a biological system is an ill-posed inverse problem. Any computational algorithm for its solution must use some kind of regularization technique to neutralize the disastrous effects of amplified noise components on the computed solution. In this paper, following a hierarchical Bayesian statistical inversion approach, we seek estimates for the input function and regularization parameter (hyperparameter) that maximize the posterior probability density function. We solve the maximization problem simultaneously for all unknowns, hyperparameter included, by a suitably chosen quasi-Newton method. The optimization approach is compared to the sampling-based Bayesian approach. We demonstrate the efficiency and robustness of the deconvolution algorithm by applying it to reconstructing the time courses of mitochondrial oxygen consumption during muscle state transitions (e.g., from resting state to contraction and recovery), from the simulated noisy output of oxygen concentration dynamics on the muscle surface. The model of oxygen transport and metabolism in skeletal muscle assumes an in vitro cylindrical structure of the muscle in which the oxygen from the surrounding oxygenated solution diffuses into the muscle and is then consumed by the muscle mitochondria. The algorithm can be applied to other deconvolution problems by suitably replacing the forward model of the system.

Compensation for geometric mismodelling by anisotropies in optical tomography.

We propose an approach for the estimation of the optical absorption coefficient in medical optical tomography in the presence of geometric mismodelling. We focus on cases in which the boundaries of the measurement domain or the optode positions are not accurately known. In general, geometric distortion of the domain produces anisotropic changes for the material parameters in the model. Hence, geometric mismodelling in an isotropic case may correspond to an anisotropic model. We seek to approximate the errors due to geometric mismodelling as extraneous additive noise and to pose a simple anisotropic model for the diffusion coefficient. We show that while geometric mismodelling may deteriorate the estimates of the absorption coefficient significantly, using the proposed model enables the recovery of the main features.

Representation of bioelectric current sources using Whitney elements in the finite element method.

Bioelectric current sources of magneto- and electroencephalograms (MEG, EEG) are usually modelled with discrete delta-function type current dipoles, despite the fact that the currents in the brain are naturally continuous throughout the neuronal tissue. In this study, we represent bioelectric current sources in terms of Whitney-type elements in the finite element method (FEM) using a tetrahedral mesh. The aim is to study how well the Whitney elements can reproduce the potential and magnetic field patterns generated by a point current dipole in a homogeneous conducting sphere. The electric potential is solved for a unit sphere model with isotropic conductivity and magnetic fields are calculated for points located on a cap outside the sphere. The computed potential and magnetic field are compared with analytical solutions for a current dipole. Relative difference measures between the FEM and analytical solutions are less than 1%, suggesting that Whitney elements as bioelectric current sources are able to produce the same potential and magnetic field patterns as the point dipole sources.

Stochastic modelling of muscle recruitment during activity.

Muscle forces can be selected from a space of muscle recruitment strategies that produce stable motion and variable muscle and joint forces. However, current optimization methods provide only a single muscle recruitment strategy. We modelled the spectrum of muscle recruitment strategies while walking. The equilibrium equations at the joints, muscle constraints, static optimization solutions and 15-channel electromyography (EMG) recordings for seven walking cycles were taken from earlier studies. The spectrum of muscle forces was calculated using Bayesian statistics and Markov chain Monte Carlo (MCMC) methods, whereas EMG-driven muscle forces were calculated using EMG-driven modelling. We calculated the differences between the spectrum and EMG-driven muscle force for 1-15 input EMGs, and we identified the muscle strategy that best matched the recorded EMG pattern. The best-fit strategy, static optimization solution and EMG-driven force data were compared using correlation analysis. Possible and plausible muscle forces were defined as within physiological boundaries and within EMG boundaries. Possible muscle and joint forces were calculated by constraining the muscle forces between zero and the peak muscle force. Plausible muscle forces were constrained within six selected EMG boundaries. The spectrum to EMG-driven force difference increased from 40 to 108 N for 1-15 EMG inputs. The best-fit muscle strategy better described the EMG-driven pattern (R (2) = 0.94; RMSE = 19 N) than the static optimization solution (R (2) = 0.38; RMSE = 61 N). Possible forces for 27 of 34 muscles varied between zero and the peak muscle force, inducing a peak hip force of 11.3 body-weights. Plausible muscle forces closely matched the selected EMG patterns; no effect of the EMG constraint was observed on the remaining muscle force ranges. The model can be used to study alternative muscle recruitment strategies in both physiological and pathophysiological neuromotor conditions.

Astrocytic tracer dynamics estimated from [1-¹¹C]-acetate PET measurements.

We address the problem of estimating the unknown parameters of a model of tracer kinetics from sequences of positron emission tomography (PET) scan data using a statistical sequential algorithm for the inference of magnitudes of dynamic parameters. The method, based on Bayesian statistical inference, is a modification of a recently proposed particle filtering and sequential Monte Carlo algorithm, where instead of preassigning the accuracy in the propagation of each particle, we fix the time step and account for the numerical errors in the innovation term. We apply the algorithm to PET images of [1-¹¹C]-acetate-derived tracer accumulation, estimating the transport rates in a three-compartment model of astrocytic uptake and metabolism of the tracer for a cohort of 18 volunteers from 3 groups, corresponding to healthy control individuals, cirrhotic liver and hepatic encephalopathy patients. The distribution of the parameters for the individuals and for the groups presented within the Bayesian framework support the hypothesis that the parameters for the hepatic encephalopathy group follow a significantly different distribution than the other two groups. The biological implications of the findings are also discussed.

A modelling error approach for the estimation of optical absorption in the presence of anisotropies.

Optical tomography is an emerging method for non-invasive imaging of human tissues using near-infrared light. Generally, the tissue is assumed isotropic, but this may not always be true. In this paper, we present a method for the estimation of optical absorption coefficient allowing the background to be anisotropic. To solve the forward problem, we model the light propagation in tissue using an anisotropic diffusion equation. The inverse problem consists of the estimation of the absorption coefficient based on boundary measurements. Generally, the background anisotropy cannot be assumed to be known. We treat the uncertainties in the background anisotropy parameter values as modelling error, and include this in our model and reconstruction. We present numerical examples based on simulated data. For reference, examples using an isotropic inversion scheme are also included. The estimates are qualitatively different for the two methods.

Anisotropic effects in highly scattering media.

In this paper, we study anisotropic scattering and light propagation models applicable to diffuse optical tomography. We propose a model for anisotropic scattering in the radiative transfer framework and derive the corresponding anisotropic diffusion model. To verify the anisotropic diffusion model, we consider the case of a simple anisotropic scattering model also presentable within the diffusion approximation. For numerical computations, we present a three-dimensional (3D) anisotropic Monte Carlo model and 2D finite element and boundary element solutions of the anisotropic diffusion model, and compare the results of the simulations.

Quantitative in silico Analysis of Neurotransmitter Pathways Under Steady State Conditions.

The modeling of glutamate/GABA-glutamine cycling in the brain tissue involving astrocytes, glutamatergic and GABAergic neurons leads to a complex compartmentalized metabolic network that comprises neurotransmitter synthesis, shuttling, and degradation. Without advanced computational tools, it is difficult to quantitatively track possible scenarios and identify viable ones. In this article, we follow a sampling-based computational paradigm to analyze the biochemical network in a multi-compartment system modeling astrocytes, glutamatergic, and GABAergic neurons, and address some questions about the details of transmitter cycling, with particular emphasis on the ammonia shuttling between astrocytes and neurons, and the synthesis of transmitter GABA. More specifically, we consider the joint action of the alanine-lactate shuttle, the branched chain amino acid shuttle, and the glutamine-glutamate cycle, as well as the role of glutamate dehydrogenase (GDH) activity. When imposing a minimal amount of bound constraints on reaction and transport fluxes, a preferred stoichiometric steady state equilibrium requires an unrealistically high reductive GDH activity in neurons, indicating the need for additional bound constants which were included in subsequent computer simulations. The statistical flux balance analysis also suggests a stoichiometrically viable role for leucine transport as an alternative to glutamine for replenishing the glutamate pool in neurons.

Computational tools for calculating alternative muscle force patterns during motion: a comparison of possible solutions.

Comparing the available electromyography (EMG) and the related uncertainties with the space of muscle forces potentially driving the same motion can provide insights into understanding human motion in healthy and pathological neuromotor conditions. However, it is not clear how effective the available computational tools are in completely sample the possible muscle forces. In this study, we compared the effectiveness of Metabolica and the Null-Space algorithm at generating a comprehensive spectrum of possible muscle forces for a representative motion frame. The hip force peak during a selected walking trial was identified using a lower-limb musculoskeletal model. The joint moments, the muscle lever arms, and the muscle force constraints extracted from the model constituted the indeterminate equilibrium equation at the joints. Two spectra, each containing 200,000 muscle force samples, were calculated using Metabolica and the Null-Space algorithm. The full hip force range was calculated using optimization and compared with the hip force ranges derived from the Metabolica and the Null-Space spectra. The Metabolica spectrum spanned a much larger force range than the NS spectrum, reaching 811N difference for the gluteus maximus intermediate bundle. The Metabolica hip force range exhibited a 0.3-0.4 BW error on the upper and lower boundaries of the full hip force range (3.4-11.3 BW), whereas the full range was imposed in the NS spectrum. The results suggest that Metabolica is well suited for exhaustively sample the spectrum of possible muscle recruitment strategy. Future studies will investigate the muscle force range in healthy and pathological neuromotor conditions.