A model of the coupling between brain electrical activity, metabolism, and hemodynamics: application to the interpretation of functional neuroimaging

Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism

Understanding cerebral energy metabolism in neurons and astrocytes is necessary for the interpretation of functional brain imaging data. It has been suggested that astrocytes can provide lactate as an energy fuel to neurons, a process referred to as astrocyte-neuron lactate shuttle (ANLS). Some authors challenged this hypothesis, defending the classical view that glucose is the major energy substrate of neurons, at rest as well as in response to a stimulation. To test the ANLS hypothesis from a theoretical point of view, we developed a mathematical model of compartmentalized energy metabolism between neurons and astrocytes, adopting hypotheses highly unfavorable to ANLS. Simulation results can be divided between two groups, depending on the relative neuron versus astrocyte stimulation. If this ratio is low, ANLS is observed during all the stimulus and poststimulus periods (continuous ANLS), but a high ratio induces ANLS only at the beginning of the stimulus and during the poststimulus period (triphasic behavior). Finally, our results show that current experimental data on lactate kinetics are compatible with the ANLS hypothesis, and that it is essential to assess the neuronal and astrocytic NADH/NAD+ ratio changes to test the ANLS hypothesis.

Brain lactate kinetics: Modeling evidence for neuronal lactate uptake upon activation

A critical issue in brain energy metabolism is whether lactate produced within the brain by astrocytes is taken up and metabolized by neurons upon activation. Although there is ample evidence that neurons can efficiently use lactate as an energy substrate, at least in vitro, few experimental data exist to indicate that it is indeed the case in vivo. To address this question, we used a modeling approach to determine which mechanisms are necessary to explain typical brain lactate kinetics observed upon activation. On the basis of a previously validated model that takes into account the compartmentalization of energy metabolism, we developed a mathematical model of brain lactate kinetics, which was applied to published data describing the changes in extracellular lactate levels upon activation. Results show that the initial dip in the extracellular lactate concentration observed at the onset of stimulation can only be satisfactorily explained by a rapid uptake within an intraparenchymal cellular compartment. In contrast, neither blood flow increase, nor extracellular pH variation can be major causes of the lactate initial dip, whereas tissue lactate diffusion only tends to reduce its amplitude. The kinetic properties of monocarboxylate transporter isoforms strongly suggest that neurons represent the most likely compartment for activation-induced lactate uptake and that neuronal lactate utilization occurring early after activation onset is responsible for the initial dip in brain lactate levels observed in both animals and humans.

Investigating the post-stimulus undershoot of the BOLD signal--a simultaneous fMRI and fNIRS study

Measuring the hemodynamic response with functional magnetic resonance imaging (fMRI) together with functional near-infrared spectroscopy (fNIRS) may overcome limitations of single-method approaches. Accordingly, we measured the event-related hemodynamic response with both imaging methods simultaneously in young subjects during visual stimulation. An intertrial interval of 60 s was chosen to include the prolonged post-stimulus undershoot of the blood oxygenation level dependent (BOLD) signal. During visual stimulation, the BOLD signal, oxy-, and total hemoglobin (Hb) increased, whereas deoxy-Hb decreased. The post-stimulus period was characterized by an undershoot of the BOLD signal, oxy-Hb, and an overshoot of deoxy-Hb. Total Hb as measured by fNIRS returned to baseline immediately after the end of stimulation. Results suggest that the post-stimulus events as measured by fNIRS are dominated by a prolonged high-level oxygen consumption in the microvasculature. The contribution of a delayed return of blood volume to the BOLD post-stimulus undershoot in post-capillary veins as suggested by the Balloon and Windkessel models remains ambiguous. Temporal changes in the BOLD signal were highly correlated with deoxy-Hb, with lower correlation values for oxy- and total Hb. Furthermore, data show that fNIRS covers the outer 1 cm of the brain cortex. These results were confirmed by simultaneous fMRI/fNIRS measurements during rest. In conclusion, multimodal imaging approaches may contribute to the understanding of neurovascular coupling.

A three-compartment model of the hemodynamic response and oxygen delivery to brain

We describe a mathematical model linking changes in cerebral blood flow, blood volume and the blood oxygenation state in response to stimulation. The model has three compartments to take into account the fact that the cerebral blood flow and volume as measured concurrently using laser Doppler flowmetry and optical imaging spectroscopy have contributions from the arterial, capillary as well as the venous compartments of the vasculature. It is an extension to previous one-compartment hemodynamic models which assume that the measured blood volume changes are from the venous compartment only. An important assumption of the model is that the tissue oxygen concentration is a time varying state variable of the system and is driven by the changes in metabolic demand resulting from changes in neural activity. The model takes into account the pre-capillary oxygen diffusion by flexibly allowing the saturation of the arterial compartment to be less than unity. Simulations are used to explore the sensitivity of the model and to optimise the parameters for experimental data. We conclude that the three-compartment model was better than the one-compartment model at capturing the hemodynamics of the response to changes in neural activation following stimulation.

Modelling of calcium dynamics in brain energy metabolism and Alzheimer's disease

Functional imaging techniques play a major role in the study of brain activation by monitoring the changes in blood flow and energy metabolism. In order to interpret functional neuroimaging data better, the existing mathematical models describing the links that may exist between electrical activity, energy metabolism and hemodynamics in literature are thoroughly analyzed for their advantages and disadvantages in terms of their prediction of available experimental data. Then, these models are combined within a single model that includes membrane ionic currents, glycolysis, mitochondrial activity, exchanges through the blood-brain barrier, as well as brain hemodynamics. Particular attention is paid to the transport and storage of calcium ions in neurons since calcium is not only an important molecule for signalling in neurons, but it is also essential for memory storage. Multiple efforts have underlined the importance of calcium dependent cellular processes in the biochemical characterization of Alzheimer's disease (AD), suggesting that abnormalities in calcium homeostasis might be involved in the pathophysiology of the disease. The ultimate goal of this study is to investigate the hypotheses about the physiological or biochemical changes in health and disease and to correlate them to measurable physiological parameters obtained from functional neuroimaging data as in the time course of blood oxygenation level dependent (BOLD) signal. When calcium dynamics are included in the model, both BOLD signal and metabolite concentration profiles are shown to exhibit temporal behaviour consistent with the experimental data found in literature. In the case of Alzheimer's disease, the effect of halved cerebral blood flow increase results in a negative BOLD signal implying suppressed neural activity.

Functional 1H-MRS findings in migraine patients with and without aura assessed interictally

The present study was aimed at investigating changes in brain metabolites due to visual cortex activation in migraineurs and normal subjects by (1)H-magnetic resonance spectroscopy (MRS). Twenty-two migraine patients with aura, 22 migraine patients without aura, and 10 control subjects were assessed. The volume of interest (about 8 cm(3)) was placed on the visual cortex area and the visual stimulus was applied using MR-compatible goggles with a flashing red light at a frequency of 8 Hz and an intensity of 14 lx. Data were acquired over 36'40". The experimental time course was: baseline phase, from 0 to 3'40" (1 spectrum); on phase (flashing light condition), from 3'40" to 29'20" (1540") (7 spectra), and off phase, from 29'20" to the end of the experiment at 36'40" (2 spectra). The main result of photic stimulation in patients with migraine with aura is the more consistent decrease (-14.61%) of the N-acetylaspartate (NAA) signal, which is significantly greater than that found in migraine patients without aura and control subjects. A parallel slight increase in the lactate peak was also detected. The above findings support little differences in brain metabolites between the two patient groups assessed in interictal periods, which suggests a less efficient mitochondrial functioning in migraine with aura patients.

Lateral modulation of BOLD activation in unstimulated regions of the human visual cortex

After staring at a blank region surrounded by a dynamic background for a few seconds, observers report a twinkle aftereffect in the unstimulated blank region. The significance of this twinkle aftereffect is that it occurs at a location that received no stimulation, and therefore reflects a rebound from lateral inhibition within the dynamic processing system. To study this inhibitory rebound effect, the blood oxygenation level dependent (BOLD) activation in the visual cortex was measured while the observers were viewing a flickering pin-wheel pattern alternating with a blank test. Retinotopic regions corresponding to the inter-wedge regions in the pin-wheel pattern showed activation negatively correlated with the test sequence. While the BOLD activation in the visual cortex is generally considered to be retinotopically driven by the visual stimuli, we were able to show a sustained negative activation in the unstimulated regions, with properties that correspond to those of the inhibitory rebound of the perceived aftereffect.

Neurobarrier coupling in the brain: a partner of neurovascular and neurometabolic coupling?

Neurovascular and neurometabolic coupling help the brain to maintain an appropriate energy flow to the neural tissue under conditions of increased neuronal activity. Both coupling phenomena provide us, in addition, with two macroscopically measurable parameters, blood flow and intermediate metabolite fluxes, that are used to dynamically image the functioning brain. The main energy substrate for the brain is glucose, which is metabolized by glycolysis and oxidative breakdown in both astrocytes and neurons. Neuronal activation triggers increased glucose consumption and glucose demand, with new glucose being brought in by stimulated blood flow and glucose transport over the blood-brain barrier. Glucose is shuttled over the barrier by the GLUT-1 transporter, which, like all transporter proteins, has a ceiling above which no further stimulation of the transport is possible. Blood-brain barrier glucose transport is generally accepted as a nonrate-limiting step but to prevent it from becoming rate-limiting under conditions of neuronal activation, it might be necessary for the transport parameters to be adapted to the increased glucose demand. It is proposed that the blood-brain barrier glucose transport parameters are dynamically adapted to the increased glucose needs of the neural tissue after activation according to a neurobarrier coupling scheme. This review presents neurobarrier coupling within the current knowledge on neurovascular and neurometabolic coupling, and considers arguments and evidence in support of this hypothesis.

Estimation of the Hemodynamic Response Function in event-related functional MRI: directed acyclic graphs for a general Bayesian inference framework

A convenient way to analyze BOLD fMRI data consists of modeling the whole brain as a stationary, linear system characterized by its transfer function: the Hemodynamic Response Function (HRF). HRF estimation, though of the greatest interest, is still under investigation, for the problem is ill-conditioned. In this paper, we recall the most general Bayesian model for HRF estimation and show how it can beneficially be translated in terms of graphical models, leading to (i) a clear and efficient representation of all structural and functional relationships entailed by the model, and (ii) a straightforward numerical scheme to approximate the joint posterior distribution, allowing for estimation of the HRF, as well as all other model parameters. We finally apply this novel technique on both simulations and real data.

Characterizing function-structure relationships in the human visual system with functional MRI and diffusion tensor imaging

A key objective in neuroscience is to improve our understanding of the relationship between brain function and structure. We investigated this in the posterior visual pathways of healthy volunteers by applying functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) with tractography. The optic radiations were segmented using the Probabilistic Index of Connectivity (PICo) tractography algorithm and extracted at several thresholds of connection confidence. The mean fractional anisotropy (FA) of the estimated tracts was found to correlate significantly with fMRI measures of visual cortex activity (induced by a photic stimulation paradigm). The results support the hypothesis that the visual cortical fMRI response is constrained by the external anatomical connections of the subserving optic radiations.

Estimation of the hemodynamic response in event-related functional MRI: Bayesian networks as a framework for efficient Bayesian modeling and inference

A convenient way to analyze blood-oxygen-level-dependent functional magnetic resonance imaging data consists of modeling the whole brain as a stationary, linear system characterized by its transfer function: the hemodynamic response function (HRF). HRF estimation, though of the greatest interest, is still under investigation, for the problem is ill-conditioned. In this paper, we recall the most general Bayesian model for HRF estimation and show how it can beneficially be translated in terms of Bayesian graphical models, leading to 1) a clear and efficient representation of all structural and functional relationships entailed by the model, and 2) a straightforward numerical scheme to approximate the joint posterior distribution, allowing for estimation of the HRF, as well as all other model parameters. We finally apply this novel technique on both simulations and real data.

The physiology and metabolism of neuronal activation: in vivo studies by NMR and other methods

In this article, a review is made of the current knowledge concerning the physiology and metabolism of neuronal activity, as provided by the application of NMR approaches in vivo. The evidence furnished by other functional spectroscopic and imaging techniques, such as PET and optical methods, are also discussed. In spite of considerable amounts of studies presented in the literature, several controversies concerning the mechanisms underlying brain function still remain, mainly due to the difficult assessment of the single vascular and metabolic dynamics which generally influence the functional signals. In this framework, methodological and technical improvements are required to provide new and reliable experimental elements, which can support or eventually modify the current models of activation.

A state-space model of the hemodynamic approach: nonlinear filtering of BOLD signals

In this paper, a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent (BOLD) responses. The proposed method constitutes an alternative to the recently proposed Friston [Neuroimage 16 (2002) 513] method and has some advantages over it. The procedure is based on recent groundbreaking time series analysis techniques that have been, in this case, adopted to characterize hemodynamic responses in functional magnetic resonance imaging (fMRI). This work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons. First, our model includes physiological noise. Previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation. Secondly, by using the innovation method and the local linearization filter, not only the parameters, but also the underlying states of the system generating responses can be estimated. These states can include things like a flow-inducing signal triggered by neuronal activation, de-oxyhemoglobine, cerebral blood flow and volume. Finally, radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach, which could be essential to understanding those brain areas indirectly related to the stimulus. Hence, thirdly, by inferring about the radial basis parameters, we are able to perform a blind deconvolution, which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also, to implicitly reconstruct the underlying synaptic dynamics, induced experimentally, which caused these states variations. From this study, we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps (SPM), nonlinear BOLD phenomena and unspecific input temporal sequences must be included in the fMRI analysis.

A neural mass model for MEG/EEG: coupling and neuronal dynamics

Although MEG/EEG signals are highly variable, systematic changes in distinct frequency bands are commonly encountered. These frequency-specific changes represent robust neural correlates of cognitive or perceptual processes (for example, alpha rhythms emerge on closing the eyes). However, their functional significance remains a matter of debate. Some of the mechanisms that generate these signals are known at the cellular level and rest on a balance of excitatory and inhibitory interactions within and between populations of neurons. The kinetics of the ensuing population dynamics determine the frequency of oscillations. In this work we extended the classical nonlinear lumped-parameter model of alpha rhythms, initially developed by Lopes da Silva and colleagues [Kybernetik 15 (1974) 27], to generate more complex dynamics. We show that the whole spectrum of MEG/EEG signals can be reproduced within the oscillatory regime of this model by simply changing the population kinetics. We used the model to examine the influence of coupling strength and propagation delay on the rhythms generated by coupled cortical areas. The main findings were that (1) coupling induces phase-locked activity, with a phase shift of 0 or pi when the coupling is bidirectional, and (2) both coupling and propagation delay are critical determinants of the MEG/EEG spectrum. In forthcoming articles, we will use this model to (1) estimate how neuronal interactions are expressed in MEG/EEG oscillations and establish the construct validity of various indices of nonlinear coupling, and (2) generate event-related transients to derive physiologically informed basis functions for statistical modelling of average evoked responses.

Modeling the hemodynamic response to brain activation

Neural activity in the brain is accompanied by changes in cerebral blood flow (CBF) and blood oxygenation that are detectable with functional magnetic resonance imaging (fMRI) techniques. In this paper, recent mathematical models of this hemodynamic response are reviewed and integrated. Models are described for: (1) the blood oxygenation level dependent (BOLD) signal as a function of changes in cerebral oxygen extraction fraction (E) and cerebral blood volume (CBV); (2) the balloon model, proposed to describe the transient dynamics of CBV and deoxy-hemoglobin (Hb) and how they affect the BOLD signal; (3) neurovascular coupling, relating the responses in CBF and cerebral metabolic rate of oxygen (CMRO(2)) to the neural activity response; and (4) a simple model for the temporal nonlinearity of the neural response itself. These models are integrated into a mathematical framework describing the steps linking a stimulus to the measured BOLD and CBF responses. Experimental results examining transient features of the BOLD response (post-stimulus undershoot and initial dip), nonlinearities of the hemodynamic response, and the role of the physiologic baseline state in altering the BOLD signal are discussed in the context of the proposed models. Quantitative modeling of the hemodynamic response, when combined with experimental data measuring both the BOLD and CBF responses, makes possible a more specific and quantitative assessment of brain physiology than is possible with standard BOLD imaging alone. This approach has the potential to enhance numerous studies of brain function in development, health, and disease.

Coupling of changes in cerebral blood flow with neural activity: what must initially dip must come back up

Activation flow coupling, increases in neuronal activity leading to changes in cerebral blood flow (CBF), is the basis of many neuroimaging methods. An early rise in deoxygenation, the "initial dip," occurs before changes in CBF and cerebral blood volume (CBV) and may provide a better spatial localizer of early neuronal activity compared with subsequent increases in CBF. Imaging modality, anesthetic, degree of oxygenation, and species can influence the magnitude of this initial dip. The observed initial dip may reflect a depletion of mitochondrial oxygen (O(2)) buffers caused by increased neuronal activity. Changes in CBF mediated by nitric oxide (NO) or other metabolites and not caused by a lack of O(2) or energy depletion most likely lead to an increased delivery of capillary O(2) in an attempt to maintain intracellular O(2) buffers.

Pre-surgical mapping of primary motor cortex by functional MRI at 3 T: effects of intravenous administration of Gd-DTPA

The functional magnetic resonance imaging (fMRI) is often performed at the end of a routine MRI examination during which, dependent on the clinical indication, contrast agent has been administered; however, the effects of Gd-DTPA injection on the results of blood oxygenation level dependent (BOLD)-fMRI remain unknown. The present study was conducted to investigate the effects of the intravenous administration of Gd-DTPA on the results of pre-surgical localization of the primary motor cortex by BOLD-fMRI at 3 T. Eight normal subjects were included in this study. After the anatomical scans, pre- and post-contrast fMRI scanning was performed. The number of significantly activated voxels and the mean percentage signal change were compared. The mean number of significantly activated voxels was 115.0+/-27.0 in pre-contrast runs and 90.8+/-27.1 in post-contrast runs (mean value of all 8 volunteers+/-standard deviation; p<0.05). The mean mean percentage signal change was 4.07+/-0.39 in pre-contrast runs and 3.86+/-1.91 in post-contrast runs ( p=0.16). Pre-surgical localization of the motor area by BOLD-fMRI should be performed before the administration of Gd contrast material.

Neuroprotection in hypothermia linked to redistribution of oxygen in brain

Hypothermia improves the outcome of acute ischemic stroke, traumatic injury, and inflammation of brain tissue. We tested the hypothesis that hypothermia reduces the energy metabolism of brain tissue to a level that is commensurate with the prevailing blood flow and hence allows adequate distribution of oxygen to the entire tissue. To determine the effect of 32 degrees C hypothermia on brain tissue, we measured the sequential changes of physiological variables by means of PET in pigs. Cerebral blood flow and oxygen consumption (cerebral metabolic rate of oxygen) declined to 50% of the baseline in 3 and 5 h, respectively, thus elevating the oxygen extraction fraction to 140% of the baseline at 3 h. The results are consistent with the claim that cooling of the brain to 32 degrees C couples both energy metabolism and blood flow to a lower rate of work of the entire tissue.

Relation between cerebral blood flow and metabolism explained by a model of oxygen exchange

The cerebral blood flow (CBF) and cerebral metabolic rate of oxygen (CMRo(2)) are major determinants of the contrast in functional magnetic resonance imaging and optical imaging. However, the coupling between CBF and CMRo(2) during cerebral activation remains controversial. Whereas most of the previous models tend to show a nonlinear coupling, experimental studies have led to conflicting conclusions. A physiologic model was developed of oxygen transport through the blood-brain barrier (BBB) for dynamic and stationary states. Common model simplifications are proposed and their implications for the CBF/CMRo(2) relation are studied. The tissue oxygen pool, the BBB permeability, and the hemoglobin dissociation curve are physiologic parameters directly involved in the CBF/CMRo(2) relation. We have been shown that the hypothesis of a negligible tissue oxygen pool, which was admitted by most of the previous models, implies a tight coupling between CBF and CMRo(2). By relaxing this hypothesis, a real uncoupling was allowed that gives a more coherent view of the CBF/CMRo(2) relation, in better agreement with the hypercapnia data and with the variability reported in experimental works for the relative changes of those two variables. This also allows a temporal mismatch between CBF and CMRo(2), which influences the temporal shape of oxygenation at the capillary end.