Dynamic activation model for a glutamatergic neurovascular unit

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.

Advances in Astrocyte Computational Models: From Metabolic Reconstructions to Multi-omic Approaches

The growing importance of astrocytes in the field of neuroscience has led to a greater number of computational models devoted to the study of astrocytic functions and their metabolic interactions with neurons. The modeling of these interactions demands a combined understanding of brain physiology and the development of computational frameworks based on genomic-scale reconstructions, system biology, and dynamic models. These computational approaches have helped to highlight the neuroprotective mechanisms triggered by astrocytes and other glial cells, both under normal conditions and during neurodegenerative processes. In the present review, we evaluate some of the most relevant models of astrocyte metabolism, including genome-scale reconstructions and astrocyte-neuron interactions developed in the last few years. Additionally, we discuss novel strategies from the multi-omics perspective and computational models of other glial cell types that will increase our knowledge in brain metabolism and its association with neurodegenerative diseases.

Metabolism plays a central role in the cortical spreading depression: Evidence from a mathematical model

The slow propagating waves of strong depolarization of neural cells characterizing cortical spreading depression, or depolarization, (SD) are known to break cerebral homeostasis and induce significant hemodynamic and electro-metabolic alterations. Mathematical models of cortical spreading depression found in the literature tend to focus on the changes occurring at the electrophysiological level rather than on the ensuing metabolic changes. In this paper, we propose a novel mathematical model which is able to simulate the coupled electrophysiology and metabolism dynamics of SD events, including the swelling of neurons and astrocytes and the concomitant shrinkage of extracellular space. The simulations show that the metabolic coupling leads to spontaneous repetitions of the SD events, which the electrophysiological model alone is not capable to produce. The model predictions, which corroborate experimental findings from the literature, show a strong disruption in metabolism accompanying each wave of spreading depression in the form of a sharp decrease of glucose and oxygen concentrations, with a simultaneous increase in lactate concentration which, in turn, delays the clearing of excess potassium in extracellular space. Our model suggests that the depletion of glucose and oxygen concentration is more pronounced in astrocyte than neuron, in line with the partitioning of the energetic cost of potassium clearing. The model suggests that the repeated SD events are electro-metabolic oscillations that cannot be explained by the electrophysiology alone. The model highlights the crucial role of astrocytes in cleaning the excess potassium flooding extracellular space during a spreading depression event: further, if the ratio of glial/neuron density increases, the frequency of cortical SD events decreases, and the peak potassium concentration in extracellular space is lower than with equal volume fractions.

Effects of Xiaoyaosan on Depressive-Like Behaviors in Rats With Chronic Unpredictable Mild Stress Through HPA Axis Induced Astrocytic Activities

ABSTRACT

Astrocytes in the hippocampus are immediately relevant to depressive-like behavior. By regulating their activities, Xiaoyaosan (XYS), a traditional Chinese medicine compound, works in the treatment of depression.

OBJECTIVE

Chronic unpredictable mild stress (CUMS) rat model was established to observe the regulation of XYS. We investigated the behavioral changes of CUMS, the expression of corticosterone (CORT) of the hypothalamo-pituitary-adrenal (HPA) axis, the expression of Glu-NMDA receptor and astrocytes glial fibrillary acidic protein (GFAP) in the hippocampus. We also investigated whether these changes were linked to XYS.

METHODS

80 adult SD rats were randomly divided into four groups, control group, CUMS group, XYS group, and fluoxetine group. The rats in the control group and the CUMS group received 0.5 ml of deionized water once a day by intragastrically administration. Rats in the two treatment groups received XYS (2.224g/kg/d) and fluoxetine (2.0mg/kg/d) once a day, respectively. Rat hippocampus GFAP and Glu-NMDA receptor were respectively detected by real-time fluorescent quantitative PCR and western blot. The CORT of HPA axis was detected by Elisa. Body weight, food intake, and behavioral tests, such as open field tests, the sucrose preference test, and exhaustive swimming test, were used to assess depressive-like behavior in rats.

RESULTS

In this work, significant behavioral changes and differences in expression of the CORT of HPA axis and hippocampal GFAP and Glu-NMDA receptor were presented in CUMS-exposed rats. Like fluoxetine, XYS improved CUMS-induced rat's body weight, food intake, and depressive-like behavior. The study also proved that XYS could reverse the CUMS-induced changes of the CORT of HPA axis and affect the astrocytic activities and down-regulate the NR2B subunit of NMDA receptor (NR2B) level in the hippocampus.

CONCLUSION

Changes in the hippocampus GFAP and Glu-NMDA receptor may be an essential mechanism of depression. Besides, XYS may be critical to the treatment of depression by intervention the HPA axis, GFAP and Glu-NMDA receptor.

Computational singular perturbation analysis of brain lactate metabolism

Lactate in the brain is considered an important fuel and signalling molecule for neuronal activity, especially during neuronal activation. Whether lactate is shuttled from astrocytes to neurons or from neurons to astrocytes leads to the contradictory Astrocyte to Neuron Lactate Shuttle (ANLS) or Neuron to Astrocyte Lactate Shuttle (NALS) hypotheses, both of which are supported by extensive, but indirect, experimental evidence. This work explores the conditions favouring development of ANLS or NALS phenomenon on the basis of a model that can simulate both by employing the two parameter sets proposed by Simpson et al. (J Cereb. Blood Flow Metab., 27:1766, 2007) and Mangia et al. (J of Neurochemistry, 109:55, 2009). As most mathematical models governing brain metabolism processes, this model is multi-scale in character due to the wide range of time scales characterizing its dynamics. Therefore, we utilize the Computational Singular Perturbation (CSP) algorithm, which has been used extensively in multi-scale systems of reactive flows and biological systems, to identify components of the system that (i) generate the characteristic time scale and the fast/slow dynamics, (ii) participate to the expressions that approximate the surfaces of equilibria that develop in phase space and (iii) control the evolution of the process within the established surfaces of equilibria. It is shown that a decisive factor on whether the ANLS or NALS configuration will develop during neuronal activation is whether the lactate transport between astrocytes and interstitium contributes to the fast dynamics or not. When it does, lactate is mainly generated in astrocytes and the ANLS hypothesis is realised, while when it doesn't, lactate is mainly generated in neurons and the NALS hypothesis is realised. This scenario was tested in exercise conditions.

A computational model integrating brain electrophysiology and metabolism highlights the key role of extracellular potassium and oxygen

The human brain is a small organ which uses a disproportionate amount of the total metabolic energy production in the body. While it is well understood that the most significant energy sink is the maintenance of the neuronal membrane potential during the brain signaling activity, the role of astrocytes in the energy balance continues to be the topic of a lot of research. A key function of astrocytes, besides clearing glutamate from the synaptic clefts, is the potassium clearing after neuronal activation. Extracellular potassium plays a significant role in triggering neuronal firing, and elevated concentration of potassium may lead to abnormal firing patterns, e.g., seizures, thus emphasizing the importance of the glial K⁺ buffering role. The predictive mathematical model proposed in this paper elucidates the role of glial potassium clearing in brain energy metabolism, integrating a detailed model of the ion dynamics which regulates neuronal firing with a four compartment metabolic model. Because of the very different characteristic time scales of electrophysiology and metabolism, care must be taken when coupling the two models to ensure that the predictions, e.g., neuronal firing frequencies and the oxygen-glucose index (OGI) of the brain during activation and rest, are in agreement with empirical observations. The temporal multi-scale nature of the problem requires the design of new computational tools to ensure a stable and accurate numerical treatment. The model predictions for different protocols, including combinations of elevated activation and ischemic episodes, are in good agreement with experimental observations reported in the literature.

Combination Therapy for Multi-Target Manipulation of Secondary Brain Injury Mechanisms

Traumatic brain injury (TBI) is a major healthcare problem that affects millions of people worldwide. Despite advances in understanding and developing preventative and treatment strategies using preclinical animal models, clinical trials to date have failed, and a 'magic bullet' for effectively treating TBI-induced damage does not exist. Thus, novel pharmacological strategies to effectively manipulate the complex and heterogeneous pathophysiology of secondary injury mechanisms are needed. Given that goal, this paper discusses the relevance and advantages of combination therapies (COMTs) for 'multi-target manipulation' of the secondary injury cascade by administering multiple drugs to achieve an optimal therapeutic window of opportunity (e.g., temporally broad window) and compares these regimens to monotherapies that manipulate a single target with a single drug at a given time. Furthermore, we posit that integrated mechanistic multiscale models that combine primary injury biomechanics, secondary injury mechanobiology/neurobiology, physiology, pharmacology and mathematical programming techniques could account for vast differences in the biological space and time scales and help to accelerate drug development, to optimize pharmacological COMT protocols and to improve treatment outcomes.

Computational Flux Balance Analysis Predicts that Stimulation of Energy Metabolism in Astrocytes and their Metabolic Interactions with Neurons Depend on Uptake of K+ Rather than Glutamate

Brain activity involves essential functional and metabolic interactions between neurons and astrocytes. The importance of astrocytic functions to neuronal signaling is supported by many experiments reporting high rates of energy consumption and oxidative metabolism in these glial cells. In the brain, almost all energy is consumed by the Na⁺/K⁺ ATPase, which hydrolyzes 1 ATP to move 3 Na⁺ outside and 2 K⁺ inside the cells. Astrocytes are commonly thought to be primarily involved in transmitter glutamate cycling, a mechanism that however only accounts for few % of brain energy utilization. In order to examine the participation of astrocytic energy metabolism in brain ion homeostasis, here we attempted to devise a simple stoichiometric relation linking glutamatergic neurotransmission to Na⁺ and K⁺ ionic currents. To this end, we took into account ion pumps and voltage/ligand-gated channels using the stoichiometry derived from available energy budget for neocortical signaling and incorporated this stoichiometric relation into a computational metabolic model of neuron-astrocyte interactions. We aimed at reproducing the experimental observations about rates of metabolic pathways obtained by ¹³C-NMR spectroscopy in rodent brain. When simulated data matched experiments as well as biophysical calculations, the stoichiometry for voltage/ligand-gated Na⁺ and K⁺ fluxes generated by neuronal activity was close to a 1:1 relationship, and specifically 63/58 Na⁺/K⁺ ions per glutamate released. We found that astrocytes are stimulated by the extracellular K⁺ exiting neurons in excess of the 3/2 Na⁺/K⁺ ratio underlying Na⁺/K⁺ ATPase-catalyzed reaction. Analysis of correlations between neuronal and astrocytic processes indicated that astrocytic K⁺ uptake, but not astrocytic Na⁺-coupled glutamate uptake, is instrumental for the establishment of neuron-astrocytic metabolic partnership. Our results emphasize the importance of K⁺ in stimulating the activation of astrocytes, which is relevant to the understanding of brain activity and energy metabolism at the cellular level.

A New Computational Model for Neuro-Glio-Vascular Coupling: Astrocyte Activation Can Explain Cerebral Blood Flow Nonlinear Response to Interictal Events

Developing a clear understanding of the relationship between cerebral blood flow (CBF) response and neuronal activity is of significant importance because CBF increase is essential to the health of neurons, for instance through oxygen supply. This relationship can be investigated by analyzing multimodal (fMRI, PET, laser Doppler…) recordings. However, the important number of intermediate (non-observable) variables involved in the underlying neurovascular coupling makes the discovery of mechanisms all the more difficult from the sole multimodal data. We present a new computational model developed at the population scale (voxel) with physiologically relevant but simple equations to facilitate the interpretation of regional multimodal recordings. This model links neuronal activity to regional CBF dynamics through neuro-glio-vascular coupling. This coupling involves a population of glial cells called astrocytes via their role in neurotransmitter (glutamate and GABA) recycling and their impact on neighboring vessels. In epilepsy, neuronal networks generate epileptiform discharges, leading to variations in astrocytic and CBF dynamics. In this study, we took advantage of these large variations in neuronal activity magnitude to test the capacity of our model to reproduce experimental data. We compared simulations from our model with isolated epileptiform events, which were obtained in vivo by simultaneous local field potential and laser Doppler recordings in rats after local bicuculline injection. We showed a predominant neuronal contribution for low level discharges and a significant astrocytic contribution for higher level discharges. Besides, neuronal contribution to CBF was linear while astrocytic contribution was nonlinear. Results thus indicate that the relationship between neuronal activity and CBF magnitudes can be nonlinear for isolated events and that this nonlinearity is due to astrocytic activity, highlighting the importance of astrocytes in the interpretation of regional recordings.

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.

Energy metabolism and glutamate-glutamine cycle in the brain: a stoichiometric modeling perspective

Background

The energetics of cerebral activity critically relies on the functional and metabolic interactions between neurons and astrocytes. Important open questions include the relation between neuronal versus astrocytic energy demand, glucose uptake and intercellular lactate transfer, as well as their dependence on the level of activity.

Results

We have developed a large-scale, constraint-based network model of the metabolic partnership between astrocytes and glutamatergic neurons that allows for a quantitative appraisal of the extent to which stoichiometry alone drives the energetics of the system. We find that the velocity of the glutamate-glutamine cycle (V_cyc) explains part of the uncoupling between glucose and oxygen utilization at increasing V_cyc levels. Thus, we are able to characterize different activation states in terms of the tissue oxygen-glucose index (OGI). Calculations show that glucose is taken up and metabolized according to cellular energy requirements, and that partitioning of the sugar between different cell types is not significantly affected by V_cyc. Furthermore, both the direction and magnitude of the lactate shuttle between neurons and astrocytes turn out to depend on the relative cell glucose uptake while being roughly independent of V_cyc.

Conclusions

These findings suggest that, in absence of ad hoc activity-related constraints on neuronal and astrocytic metabolism, the glutamate-glutamine cycle does not control the relative energy demand of neurons and astrocytes, and hence their glucose uptake and lactate exchange.

Mathematical Models of Blast-Induced TBI: Current Status, Challenges, and Prospects

Blast-induced traumatic brain injury (TBI) has become a signature wound of recent military activities and is the leading cause of death and long-term disability among U.S. soldiers. The current limited understanding of brain injury mechanisms impedes the development of protection, diagnostic, and treatment strategies. We believe mathematical models of blast wave brain injury biomechanics and neurobiology, complemented with in vitro and in vivo experimental studies, will enable a better understanding of injury mechanisms and accelerate the development of both protective and treatment strategies. The goal of this paper is to review the current state of the art in mathematical and computational modeling of blast-induced TBI, identify research gaps, and recommend future developments. A brief overview of blast wave physics, injury biomechanics, and the neurobiology of brain injury is used as a foundation for a more detailed discussion of multiscale mathematical models of primary biomechanics and secondary injury and repair mechanisms. The paper also presents a discussion of model development strategies, experimental approaches to generate benchmark data for model validation, and potential applications of the model for prevention and protection against blast wave TBI.

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.

The metabolism of neurons and astrocytes through mathematical models

Mathematical modeling of the energy metabolism of brain cells plays a central role in understanding data collected with different imaging modalities, and in making predictions based on them. During the last decade, several sophisticated brain metabolism models have appeared. Unfortunately, the picture of the metabolic details that emerges from them is far from coherent: while each model has its justification and is in agreement with some experimental data, some of the predictions of different models can diverge from each other significantly. In this article, we review some of the recent published models, emphasizing similarities and differences between them to understand where the differences in predictions stem from. In that context we present a probabilistic approach, which rather than assigning fixed values to the model parameters, regard them as random variables whose distributions are inferred on in the light of stoichiometric information and different observations. The probabilistic approach reveals how much intrinsic variability a metabolic system may contain, which in turn may be a valid explanation of the different findings.

Ménage à trois: the role of neurotransmitters in the energy metabolism of astrocytes, glutamatergic, and GABAergic neurons

This work is a computational study based on a new detailed metabolic network model comprising well-mixed compartments representing separate cytosol and mitochondria of astrocytes, glutamatergic and gamma aminobutyric acid (GABA)ergic neurons, communicating through an extracellular space compartment and fed by arterial blood flow. Our steady-state analysis assumes statistical mass balance of both carbons and amino groups. The study is based on Bayesian flux balance analysis, which uses Markov chain Monte Carlo sampling techniques and provides a quantitative description of steady states when the two exchangers aspartate-glutamate carrier (AGC1) and oxoglutarate carrier (OGC) in the malate-aspartate shuttle in astrocyte are not in equilibrium, as recent studies suggest. It also highlights the importance of anaplerotic reactions, pyruvate carboxylase in astrocyte and malic enzyme in neurons, for neurotransmitter synthesis and recycling. The model is unbiased with respect to the glucose partitioning between cell types, and shows that determining the partitioning cannot be done by stoichiometric constraints alone. Furthermore, the intercellular lactate trafficking is found to depend directly on glucose partitioning, suggesting that a steady state may support different scenarios. At inhibitory steady state, characterized by high rate of GABA release, there is elevated oxidative activity in astrocyte, not in response to specific energetic needs.

Brain lactate metabolism: the discoveries and the controversies

Potential roles for lactate in the energetics of brain activation have changed radically during the past three decades, shifting from waste product to supplemental fuel and signaling molecule. Current models for lactate transport and metabolism involving cellular responses to excitatory neurotransmission are highly debated, owing, in part, to discordant results obtained in different experimental systems and conditions. Major conclusions drawn from tabular data summarizing results obtained in many laboratories are as follows: Glutamate-stimulated glycolysis is not an inherent property of all astrocyte cultures. Synaptosomes from the adult brain and many preparations of cultured neurons have high capacities to increase glucose transport, glycolysis, and glucose-supported respiration, and pathway rates are stimulated by glutamate and compounds that enhance metabolic demand. Lactate accumulation in activated tissue is a minor fraction of glucose metabolized and does not reflect pathway fluxes. Brain activation in subjects with low plasma lactate causes outward, brain-to-blood lactate gradients, and lactate is quickly released in substantial amounts. Lactate utilization by the adult brain increases during lactate infusions and strenuous exercise that markedly increase blood lactate levels. Lactate can be an 'opportunistic', glucose-sparing substrate when present in high amounts, but most evidence supports glucose as the major fuel for normal, activated brain.