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Proc Natl Acad Sci U S A. Mar 6, 2007; 104(10): 4188–4193.
Published online Feb 28, 2007. doi:  10.1073/pnas.0605864104
PMCID: PMC1820730
Neuroscience

A coherent neurobiological framework for functional neuroimaging provided by a model integrating compartmentalized energy metabolism

Abstract

Functional neuroimaging has undergone spectacular developments in recent years. Paradoxically, its neurobiological bases have remained elusive, resulting in an intense debate around the cellular mechanisms taking place upon activation that could contribute to the signals measured. Taking advantage of a modeling approach, we propose here a coherent neurobiological framework that not only explains several in vitro and in vivo observations but also provides a physiological basis to interpret imaging signals. First, based on a model of compartmentalized energy metabolism, we show that complex kinetics of NADH changes observed in vitro can be accounted for by distinct metabolic responses in two cell populations reminiscent of neurons and astrocytes. Second, extended application of the model to an in vivo situation allowed us to reproduce the evolution of intraparenchymal oxygen levels upon activation as measured experimentally without substantially altering the initial parameter values. Finally, applying the same model to functional neuroimaging in humans, we were able to determine that the early negative component of the blood oxygenation level-dependent response recorded with functional MRI, known as the initial dip, critically depends on the oxidative response of neurons, whereas the late aspects of the signal correspond to a combination of responses from cell types with two distinct metabolic profiles that could be neurons and astrocytes. In summary, our results, obtained with such a modeling approach, support the concept that both neuronal and glial metabolic responses form essential components of neuroimaging signals.

Keywords: astrocyte–neuron interactions, blood oxygenation level-dependent signal, lactate, mathematical model, NADH

Functional imaging methods that include functional MRI (fMRI), magnetic resonance spectroscopy (MRS), positron emission tomography, and optical imaging are widely used to explore brain activity (1). Signals recorded with these techniques originate from hemodynamic and metabolic changes occurring upon activation. Understanding the precise nature of these changes has led to renewed interest in cellular and molecular investigations in the regulation of brain energy metabolism and blood flow. Presently, a major challenge is to propose a coherent framework to link (i) cellular data obtained in vitro, e.g., in slices or cell cultures; (ii) data obtained in animal in vivo using, for instance, electrophysiological methods, biosensors, or fluorescence microscopy; and (iii) functional neuroimaging data obtained in humans.

In this context, several experimental data have challenged the classical view of brain energy metabolism, that complete oxidation of glucose in neurons is the sole process relevant to brain function. Fox and Raichle (2) reported, on the basis of their positron emission tomography data, that during a stimulation, the increased fraction of glucose consumption can be greater than the concomitant increase fraction of oxygen consumption. This observation is supported by MRS recordings that showed an increase in tissue lactate (LAC) concentration upon activation (35). On the other hand, Pellerin and Magistretti (6) demonstrated that astrocytes in cell cultures produce LAC in response to glutamate uptake. On this basis, Pellerin and Magistretti proposed that LAC released by astrocytes can be an additional energy fuel for neurons, a process referred to as the astrocyte–neuron LAC shuttle (ANLS). More recently, it was shown either in vitro or in vivo that several metabolic components, in response to a stimulation, display complex time courses, sometimes involving distinct cellular elements. First, an initial transient decrease in brain LAC concentration was observed in rat by using biosensors (7) and in human by using 1H-MRS (8). Second, an initial increase in deoxyhemoglobin concentration was observed by Malonek and Grinvald (9) using optical imaging in cat and an early negative phase of the blood oxygenation level-dependent (BOLD) signal in fMRI, referred to as an initial dip, was reported by Kim et al. (10), whereas it was already known that BOLD signal displays a poststimulus undershoot (11). Third, Ances et al. (12), using an oxygen microelectrode in vivo in rat, highlighted an initial decrease and poststimulus undershoot in tissue oxygen concentration upon activation, which are not present in concomitantly measured cerebral blood flow (CBF). Finally, Kasischke et al. (13), who used two-photon fluorescence imaging of NADH in rat hippocampal slices, showed that, in response to an activation, dendritic mitochondria undergo an early oxidation (early transient decrease in NADH), followed by a significant reduction in astrocytic cytosol (overshoot of NADH).

These data not only prompt a reevaluation of our understanding of metabolic events taking place upon activation; they also indicate that to propose a coherent view of processes underlying neuroimaging techniques, one should further consider two particular points: (i) energetics can significantly differ between cell types and (ii) LAC, oxygen, and NADH concentrations display complex transient kinetics upon activation, notably an early and transient decrease. In the present study, on the basis of previous models (1416), we propose a more comprehensive mathematical model that takes into account both the compartmentalization of energy metabolism between two populations of cells with distinct metabolic behaviors and subcellular compartmentalization of energy metabolism between cytosol and mitochondria. We distinguish two kinds of cells with different levels of aerobic metabolism (previously identified as red and white cells, respectively; refs. 1719) that we assimilate here to neurons and astrocytes. Using this model, we can account now for several crucial in vitro and in vivo experimental data. Moreover, on the basis of physiological mechanisms suggested by this modeling, we propose a unified cellular and metabolic interpretation of some aspects of BOLD signal transients in fMRI.

Results

We present typical results of the model schematically illustrated in Fig. 1. This versatile model is designed to describe in vitro situations, taking into account neuronal cytosol (index n), neuronal mitochondria (nm), astrocytic cytosol (g), astrocytic mitochondria (gm), and extracellular (interstitial) space (e), all of which are displayed in solid lines in Fig. 1. The model can also account for in vivo situations, by adding capillaries (c) and small veins or venous balloons (v), which are represented in dashed lines. The input functions are JATPasesn and JATPasesg, the ATP consumption rates by neurons and astrocytes respectively, in the in vitro case, plus the CBF, in the in vivo case. Because possible differences between JATPasesn and JATPasesg are not well documented in the literature, we simply assumed that the kinetics of JATPases are similar between neurons and astrocytes, with equal time constants and equal increase fractions (peak of 134% of the baseline value). A study of the effect of varying these input terms can be found in supporting information (SI). The baseline level of glucose consumption (per unit tissue volume) is divided between neurons (JGlycon = 3.1 μM·s−1) and astrocytes (JGlycog = 8.9 μM·s−1) according to the MRS data collected by Hyder et al. (20). This results in an ANLS at resting state; however, the main results yielded by the model remains unchanged if we assume there is no ANLS at rest. Furthermore, we assume that metabolic regulations are the same in neurons and astrocytes, except that mitochondrial shuttles for NADH are less active in astrocytes (see SI).

Fig. 1.
The proposed model of compartmentalized energy metabolism. This model can describe in vitro situations by including five compartments (solid lines): neuronal cytosol (index n), neuronal mitochondria (index nm), glial or astrocytic cytosol (index g), astrocytic ...

Model in Vitro: Comparison with NADH Data in Slices.

Typical results from the model applied in vitro are displayed in Fig. 2. The stimulation time was 20 s, to compare our results with the temporal evolution of the kinetics of NADH published by Kasischke et al. (13). Neuronal glycolysis, JGlycon, undergoes an early increase with a fast decay toward baseline. In contrast, astrocytic glycolysis, JGlycog returns to baseline with a longer time constant (Fig. 2A). Furthermore, the extra glucose consumption in neurons (0.55 μM·s−1 at t = 8 s) corresponds to an extra oxygen consumption of 6 × 0.55 = 3.3 μM·s−1, whereas the increase in neuronal oxygen consumption is 8 μM·s−1 at t = 8 s (Fig. 2A Inset). The difference requires an extra substrate supply, which is provided by extracellular LAC to neurons; the change in the rate of the LAC dehydrogenase (LDH)-catalyzed reaction in neurons is ΔJLDHn = −1.5 μM·s−1 at t = 8 s (Fig. 2B), which indicates an extra pyruvate (PYR) production from LAC and corresponds to an extra oxygen consumption of 3 × 1.5 = 4.5 μM·s−1. On the contrary, ΔJLDHg equals 1.3 μM·s−1 at t = 10 s, meaning that astrocytes produce more LAC from PYR (Fig. 2B). The maximal changes in fluxes of monocarboxylate transporters (MCTs) are ΔJMCTn = −1.0 μM·s−1 at t = 14 s, ΔJMCTg = 0.9 μM·s−1 at t = 21 s for neurons and astrocytes, respectively. This clearly indicates that an extra ANLS occurs both at the MCT and LDH levels. More precisely, neurons consume LAC from the extracellular space, and extracellular LAC is partially replenished by the astrocytic production. This results in a decrease in extracellular LAC concentration (Fig. 2D), which is followed by a reincrease in LAC level. The neuronal mitochondrial NADH first decreases, then increases, with a minimum at t = 10 s, namely before the end of the stimulation. The amplitude of this minimum is 9.1% of the baseline value (Fig. 2C), which is fully consistent with the kinetics of dendritic mitochondrial NADH of Kasischke et al. (13), showing a minimum of ≈10% at t = 10 s. Astrocytic cytosolic NADH begins to increase slowly, then more rapidly, and is increased by 12% above its baseline value at about t = 40 s, which is close to the 7–9% increase reported experimentally. Astrocytic mitochondrial NADH modifications are much less prominent than in neurons, whereas neuronal cytosolic NADH changes are moderate, results that are fully consistent again with the data of Kasischke et al. (13) (see SI, which also displays other simulations where different JATPases in neurons and in astrocytes can result in an even better fit with experimental data, especially for extracellular LAC and astrocytic NADH).

Fig. 2.
Dynamics of the main variables of the model in vitro upon stimulation. Shaded area indicates the stimulation duration (20 s). Red, blue, green, and violet lines relate to neuronal, astrocytic, tissue, and extracellular responses, respectively. Δ ...

Model in Vivo.

Comparison with oxygen and LAC recordings in animal.

Typical results from the model applied in vivo are displayed in Fig. 3. Parameter values are nearly the same as in Fig. 2. The stimulation time is 60 s, to compare the temporal evolution of intraparenchymal oxygen (O2i) with the PO2 measurements of Ances et al. (12). As in the in vitro case, we obtained an early stimulation of JO2n (Fig. 3A Inset), followed by a slightly delayed increase of JGlycog (Fig. 3A). This is accompanied by a significant oxidation of neuronal mitochondria and a slower reduction of astrocytic cytosol (Fig. 3C). Furthermore, changes in ANLS level and the biphasic behavior of LAC kinetics still occur as in vitro (Fig. 3 B and D). The resulting cerebral metabolic rate of oxygen consumption (CMRO2), combined with a CBF time course (Fig. 3E) based on the data recorded by Ances et al. (12), results in a temporal evolution of O2i that is consistent with the measurements of Ances et al. (12). Notably, O2i displays an initial transient decrease followed by a peak, and a significant poststimulus undershoot; most interestingly, during the stimulation, O2i reaches a minimum around the middle of the stimulation time, then further increases (Fig. 3E). It can be noted that such a behavior is due to the persistence of a relatively high oxygen consumption (Fig. 3A Inset) with respect to CBF and can be observed whether the input term JATPases has a similar time course in both cell types, as shown in Fig. 3, or has slower kinetics in astrocytes than in neurons (as shown in SI). Finally, the calculated BOLD signal also displays a significant poststimulus undershoot but, because of the CBF time course, the initial dip of the BOLD signal is very slight (Fig. 3F). The positive part of the BOLD signal tends to increase again at the end of the stimulation, so there is an overshoot at both the beginning and end of the stimulation. This aspect is not present for CBF but is correlated to the intraparenchymal oxygen time course. This peculiar time course of the simulated BOLD signal has some common features with the BOLD signal recorded in human by Nakai et al. (21), who also found two overshoots but with almost equal amplitude.

Fig. 3.
Dynamics of the main variables of the model in vivo upon stimulation. Shaded area indicates the stimulation duration (60 s). (A–D) Red, blue, green, and violet lines relate to neuronal, astrocytic, tissue, and extracellular responses, respectively. ...

Effect of stimulus features on the BOLD signal in human.

We further studied possible links between compartmentalized energy metabolism and BOLD signal features, as displayed in Fig. 4. The parameters are the same as in Fig. 3, except that the CBF time course is modeled by using a trapezoidal function, which is commonly used to account for fMRI data in humans (22). Our results show that separate components of the BOLD response depend, in a specific manner, on to what extent each cell type is stimulated. Increasing neuronal stimulation enhances the initial dip amplitude (Fig. 4A). On the other hand, stimulation of astrocytes has no effect on the initial dip (Fig. 4B). Increasing neuronal or astrocytic stimulation tends to decrease the BOLD value at the end of the stimulation, but neuronal stimulation has a dramatic effect on the beginning of the positive part of the BOLD signal, which is less modified by astrocytic stimulation. The amplitude of the poststimulus undershoot increases both for an activation of neurons and astrocytes, a neuronal stimulation being more efficient than an astrocytic stimulation. In summary, increasing either neuronal or astrocytic stimulation affects the BOLD signal for the most part of the stimulation and during the poststimulus undershoot, but only a neuronal stimulation is responsible for the initial dip.

Fig. 4.
Influence of distinct metabolic activations on the BOLD signal in human. Shaded area indicates the duration of the stimulation (40 s). Effect of increasing neuronal (A) or astrocytic (B) stimulation levels on the BOLD signal evolution: Σn and ...

Discussion

Previous modeling efforts to provide a physiological interpretation of neuroimaging data were intended to resolve either hemodynamic aspects (22, 23), oxygen exchanges (18), or relationships between brain electrical activity, metabolism, and hemodynamics (14). Recently, we developed models to investigate compartmentalized energy metabolism between astrocytes and neurons (15) and identify mechanisms necessary to explain brain LAC kinetics (16). Using these models, we showed that current experimental data on LAC and NADH kinetics upon various kinds of activation are compatible with the ANLS hypothesis. Recent experimental data, while confirming the main conclusions of our models, emphasized the importance of the subcellular compartmentalization of energy metabolism between the cytosol and mitochondria (13). Meanwhile, advances in neuroimaging underlined that changes of the metabolic state of the brain strongly affect the fMRI signal (24), whereas recent in vivo MRS data on glucose fluxes (20) gave insights on the astrocyte–neuron interactions. In an attempt to integrate these experimental data, obtained at the cellular and regional levels, as well as identify specific cellular mechanisms underlying functional neuroimaging, we developed a versatile model designed to account for both in vitro and in vivo experimental data. This model takes into account (i) the compartmentalization between neurons and astrocytes; (ii) the compartmentalization between the cytosol and mitochondria; and (iii) in the in vivo version of the model, the exchanges through the blood–brain barrier, hemodynamics, and the relationship to the BOLD signal.

Distinct Metabolic Responses of Neurons and Astrocytes Support the LAC Shuttle Concept.

Application of the model in vitro can satisfactorily account for NADH fluorescence data obtained in rat hippocampal slices by using two-photon microscopy by Kasischke et al. (13), as shown in Fig. 2. To obtain this concordance, it was necessary to assume that (i) in astrocytes, cytosolic NADH at resting state is higher than in neurons, this latter point being fully consistent with fluorescence data (13); (ii) mitochondrial shuttles for NADH are less active in astrocytes than in neurons (25); and (iii) stimulation is high at the beginning of the activation period (26). It must be emphasized that we assumed no difference between neurons and astrocytes for the regulation of glycolysis, phosphocreatine buffer, PYR dehydrogenase (PDH), and oxidative phosphorylation. Furthermore, to reduce the number of free parameters, we assumed the time courses of the increase fractions of ATP consumption are the same in neurons and astrocytes. In these conditions, the increase of oxygen consumption is much greater in neurons that in astrocytes. Glycolysis increases both in astrocytes and in neurons, but the increase in glycolysis is more prolonged in astrocytes. These time evolutions result in conditions favorable to the ANLS. However, as suggested (15, 27), ANLS does not require to be a time-invariant phenomenon. More precisely, our results suggest the following sequence of events: (i) an early onset of neuronal oxidative phosphorylation, resulting in the rapid decrease in mitochondrial NADH; and (ii) stimulation of neuronal PDH and Krebs cycle, resulting in a decrease in neuronal PYR and LAC, which induces thermodynamical conditions favorable to a conversion of LAC into PYR and LAC uptake by neurons, which can be reinforced by the favorable kinetic properties of neuronal MCTs (16). Thus, at the beginning of a stimulation, extracellular LAC is taken up by neurons and only partly replenished by astrocytes, which results in a decrease in extracellular LAC concentration, as was reported in vivo by Hu and Wilson (7). This is consistent with our previous analysis of in vivo LAC kinetics, which shows that the LAC initial dip has mainly a metabolic origin and is due to a consumption of LAC by neurons at the very start of a stimulation (16). Later, NADH and PYR production by astrocytic glycolysis is markedly enhanced, which results in a greater conversion of PYR into LAC and an increased secretion of LAC by these cells, such that extracellular LAC increases again. All these modifications favor the use of LAC produced by astrocytes in neuronal mitochondria via the ANLS. In addition, it can be noted that in our model, an even higher stimulation of neurons results in a marked overshoot of neuronal mitochondrial NADH following the initial decrease. In this case, neuronal NADH contributes to the tissue NADH increase together with astrocytic cytoplasmic NADH (data not shown), which can contribute to explain the results obtained by Brennan et al. (28) by using pharmacological inhibition of glycolysis in slices. Furthermore, our model does not address directly the question of which physiological processes contribute to the energetic demands in each cell type upon stimulation. The main contributor to JATPases in both cell types is assumed to be Na+ entry. Considering that glutamatergic activity accounts for the majority of energy expenditures in the central nervous system (29), it is likely that glutamate, by both activation of ionotropic glutamate receptors on neurons and its Na+-dependent uptake in astrocytes, is the main contributor of Na+ influx and enhanced JATPases in both cell types. It can be emphasized that we used identical time courses for JATPasesn or JATPasesg increase fractions. A somewhat better fit between simulations and experimental data was obtained when we assumed that ATP consumption kinetics is slower in astrocytes than in neurons (see SI). This prediction of the model that JATPasesg is slower than JATPasesn remains to be confirmed experimentally. It is noteworthy, however, that some data reporting changes in ATP levels in astrocytes upon exposure to glutamate indicate ATP consumption must be a relatively slow process in these cells (≈120 s to reach maximal effect for a stimulus of the same duration), which would be consistent with the model prediction (30). Nevertheless, our model emphasizes that comparative experimental measurements of astrocytic and neuronal ATP consumption kinetics would be of crucial importance for a more precise description of the system.

Consequences of the Cellular Metabolic Compartmentalization for Oxygen Concentration Changes in Vivo.

Our versatile model enabled us to address the possible consequences of cellular metabolic compartmentalization on the intraparenchymal oxygen time course in vivo, taking into account the effect of CBF variations and transport of oxygen between capillaries and parenchyma. We also simulated the BOLD signal, whose main determinants are CBF, cerebral metabolic rate of oxygen consumption (CMRO2), and cerebral blood volume, whereas intraparenchymal oxygen variations depend both on CMRO2 and CBF. It is well known that viscoelastic venous properties tend to enhance the first overshoot of the positive component of the BOLD signal and to increase the poststimulus undershoot, as demonstrated by Buxton et al. (22). We could reproduce intraparenchymal oxygen variations consistent with experimental in vivo data, for instance, those obtained by Ances et al. (12), displaying an initial transient decrease, an increase above the baseline value with overshoots of intraparenchymal oxygen at both the beginning and end of the stimulation, and a poststimulus undershoot. It can be noted that Offenhauser et al. (31) reported a decrease in oxygen concentration upon activation, but this result is not contradictory with our results, because in our model, higher neuronal stimulation implies a marked and longer decrease in oxygen concentration (data not shown). Our simulation results were obtained with parameter values nearly the same as in the in vitro case, this choice being validated by the simulated oxygen kinetics. The initial transient decrease in oxygen concentration is mainly due to the rapid oxygen consumption by neuronal mitochondria, but this phenomenon can be hindered by the oxygen supply because of the high level of CBF in rat; this can be a cause of the elusive character of the reported initial dip of the BOLD signal (32, 33). A second cause of the elusive character of the initial dip is the venous viscoelastic properties (22), as confirmed by Fig. 3 (compare intraparenchymal oxygen and BOLD time courses). According to our simulations, oxygen consumption decreases relatively more slowly than CBF, resulting in a decrease in tissue oxygen concentration occurring around the middle of the stimulation, as observed by Ances et al. (12). This result suggests a possible metabolic mechanism to explain the existence of two peaks within the positive component of the BOLD signal (21, 23). In the poststimulus period, the undershoot of intraparenchymal oxygen is caused by the oxygen debt created during the stimulation, which results in a continuation of oxygen consumption, notably by neurons but also by astrocytes. Thus, the model suggests that the initial transient decrease of intraparenchymal oxygen is almost exclusively due to neuronal stimulation of oxygen consumption, whereas later aspects of oxygen transients depend both on neuronal and astrocytic metabolism.

Contribution of Each Cellular Metabolic Phenotype to the Different Components of the BOLD Signal.

To make clearer the contribution of each cellular metabolic phenotype to the BOLD signal, we studied the effect of either predominant astrocytic or neuronal stimulation, which has been referred to as white or red activities (17, 19), on the BOLD signal time course (Fig. 4). Our results unambiguously show that, whereas the initial dip originates only from a neuronal stimulation, both the decrease of the positive part of the BOLD signal and the poststimulus undershoot can be caused by a neuronal as well as an astrocytic stimulation, although their contribution somewhat differs because of their distinct oxygen consumption kinetics. It can be noted that the contribution of astrocytes to the change in the positive part of the BOLD and poststimulus undershoot is enhanced if we assume that ATP consumption kinetics is slower in astrocytes than in neurons. These statements remain qualitatively valid whatever the venous viscoelastic properties may be, even if a high venous hysteresis tends to mask the initial dip and to increase the amplitude of the poststimulus undershoot (see SI). Hence, the model can account to some extent for the variety of BOLD signal time courses, especially the elusive character of the initial dip (10, 32), and the various aspects of the positive component (21, 22). Most interestingly, our results strongly suggest that a careful analysis of the shape of the BOLD signal could give new insights about the involvement of various cellular components, namely astrocytes or weakly oxidative neurons and highly oxidative neurons (17, 18). On the one hand, one can wonder whether interregional variations in BOLD signal could in part be explained by the ratio between highly and weakly oxidative neurons (or the number of astrocytes) within the voxel (34, 35). On the other hand, a stimulation for which presynaptic afferents and astrocytes are significantly stimulated but the stimulation of postsynaptic neurons remains relatively low, a situation that correlates with relatively low information treatment (17, 19), would affect the poststimulus undershoot and the positive component of the BOLD signal. Conversely, a stimulus for which postsynaptic activity is greatly enhanced, corresponding to a high degree of information processing within the voxel (17, 19), would result not only in an increase of the poststimulus undershoot and a decrease of the positive component but also in a marked increase in the amplitude of the initial dip. It should be noted that these putative rules can be modulated by the hemodynamic features of the explored brain region, for instance the venous viscoelastic properties. As a note of caution when interpreting functional imaging experiments, our modeling highlights the fact that metabolic time constants may be very long (up to tens of minutes), which means that if not enough time is allowed between two activations, the two situations may not be entirely comparable. Finally, a major advantage of the present model is that it should be possible to assess the validity of the proposed rules in the near future by comparing ATP consumption kinetics in neurons and astrocytes, and by recording in vivo the transients of NADH, LAC and oxygen in animals.

Materials and Methods

The structure of the model has been outlined at the beginning of Results. As already mentioned, the input functions of the model in vitro are JATPases(t)n and JATPases(t)g, which are the ATP consumption by neurons and astrocytes, respectively. Plausible time evolutions of these functions were based on our previous models (14, 15), characterized by a peak followed by a plateau, which took into account literature data on neural stimulation (26). For in vivo stimulations, we took as a supplementary input function the regional CBF(t), based on the data of Ances et al. (12) in rat (Fig. 3) and Buxton et al. (22) in human (Fig. 4).

In neuronal and astrocytic cytosol, the model describes ATP, NADH, PYR, and LAC balance (see SI). ATP balance takes into account, for x = n or x = g, (i) ATP consumption JATPasesx, ATP net production by glycolysis (2 × JGlycox), ATP production by mitochondria, which is assumed to be equal to 5 × JO2x, JO2x being the oxygen consumption by mitochondria, which is related to oxidative phosphorylation processes. Moreover, ATP balance takes into account the reaction catalyzed by adenylate kinase, as previously described (14, 36, 37), and the effect of phosphocreatine buffering. For simplicity, we did not include the possible effects of a rise of intracellular calcium on ATP production. PYR balance is the result of JGlycox, JLDHx, the rate of the reaction catalyzed by LDH, and JPDH-Krebsx, the PYR consumption by mitochondria, which is assumed to be nearly equal to both the PDH rate, JPDHx, and the Krebs cycle rate, JKrebsx. In a similar way, NADH balance depends on JGlycox, JLDHx, and the rate of transfer of reducing equivalents from cytosol to mitochondria, JShuttlex. Note that, in our model, these shuttle rates need not be related to JPDH-Krebsx. LAC balance depends both on JLDHx and LAC transport toward the extracellular space by the MCTs, namely JMCTx. Variation in extracellular LAC concentration (LACe) depends on JMCTx. To relate theoretical results to subcellular NADH measurements, we took the intramitochondrial NADH concentration, namely NADHxm, as a state variable. Its balance is determined by JShuttlex, JPDH-Krebsx, and JO2x. The rate equation for JO2x is based on the model of Holzhütter et al. (38).

In the in vivo model, changes in JO2n and JO2g notably depend on the variations of intraparenchymal oxygen concentration, O2i, and conversely are determinants of O2i balance. Furthermore, we took into account the CBF(t), as an input term, and four more state variables were added, namely the mean concentrations of capillary LAC (LACc) and capillary oxygen (O2c), venous deoxyhemoglobin (dHb), and the venous balloon volume (Vv). O2c balance depends on the blood flow contribution, JO2Cap, and exchanges between capillary and parenchyma (JO2m), as described by Vafaee and Gjedde (39), and some of us (14, 15). LACc balance depends on the blood flow contribution JLACCap and the LAC efflux through the blood–brain barrier JLACm, which is determined by MCT1 properties, as described (16). The O2c value directly determines the deoxyhemoglobin concentration at the end of the capillary. Finally, using the balloon model of Buxton et al. (22), we can calculate both the dHb (per unit tissue volume) and Vv, which are the main determinants of the BOLD signal.

Supplementary Material

Supporting Information:

Acknowledgments

This work was supported by the Fondation pour la Recherche Médicale (A.A.), the Action Concertée Incitative “Neurosciences Intégratives et Computationnelles” (French Ministry of Research) (R.C.), and the Swiss Fonds National de la Recherche Scientifique [Grants 3100A0-100679 (to L.P.) and 31-56930-99 (to P.J.M.)].

Abbreviations

fMRI
functional MRI
MRS
magnetic resonance spectroscopy
LAC
lactate
ANLS
astrocyte–neuron LAC shuttle
BOLD
blood oxygenation level-dependent
CBF
cerebral blood flow
MCT
monocarboxylate transporter
PYR
pyruvate
PDH
PYR dehydrogenase
LDH
LAC dehydrogenase.

Footnotes

The authors declare no conflict of interest.

This article is a PNAS direct submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0605864104/DC1.

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