Multi-timescale modeling of activity-dependent metabolic coupling in the neuron-glia-vasculature ensemble

Abstract

Glucose is the main energy substrate in the adult brain under normal conditions. Accumulating evidence, however, indicates that lactate produced in astrocytes (a type of glial cell) can also fuel neuronal activity. The quantitative aspects of this so-called astrocyte-neuron lactate shuttle (ANLS) are still debated. To address this question, we developed a detailed biophysical model of the brain's metabolic interactions. Our model integrates three modeling approaches, the Buxton-Wang model of vascular dynamics, the Hodgkin-Huxley formulation of neuronal membrane excitability and a biophysical model of metabolic pathways. This approach provides a template for large-scale simulations of the neuron-glia-vasculature (NGV) ensemble, and for the first time integrates the respective timescales at which energy metabolism and neuronal excitability occur. The model is constrained by relative neuronal and astrocytic oxygen and glucose utilization, by the concentration of metabolites at rest and by the temporal dynamics of NADH upon activation. These constraints produced four observations. First, a transfer of lactate from astrocytes to neurons emerged in response to activity. Second, constrained by activity-dependent NADH transients, neuronal oxidative metabolism increased first upon activation with a subsequent delayed astrocytic glycolysis increase. Third, the model correctly predicted the dynamics of extracellular lactate and oxygen as observed in vivo in rats. Fourth, the model correctly predicted the temporal dynamics of tissue lactate, of tissue glucose and oxygen consumption, and of the BOLD signal as reported in human studies. These findings not only support the ANLS hypothesis but also provide a quantitative mathematical description of the metabolic activation in neurons and glial cells, as well as of the macroscopic measurements obtained during brain imaging.

Figure 1. Model structure.

The model is divided in four main compartments: a neuronal compartment, an astrocytic compartment, the extracellular space and a vascular compartment. Neurons and astrocytes are further divided between a cytosolic sub-compartment and a mitochondrial sub-compartment to account for the compartmentalization of oxidative and glycolytic metabolisms. Both neurons and astrocytes contain a metabolic network including glycolytic enzymes, lactate dehydrogenase, glucose and lactate transporters, NADH shuttles, oxidative metabolism, phosphocreatine and the Na, K-ATPase electrogenic pump. Additionally, the neuronal compartment contains voltage- and calcium-gated ion channels following the Hodgkin-Huxley formalism. The system as a whole is driven by two independent inputs (red). First, a glutamatergic presynaptic population activates AMPA receptors on the neuronal membrane and excitatory amino-acid transporters (EAATs) on the astrocytic membrane. The activation of both AMPA receptors and EAATs leads to an increase of the intracellular sodium concentration which activates the energy consuming Na, K-ATPase pump and subsequent metabolic processes. Activation of AMPA receptors also depolarizes neurons and might lead to the generation of action potentials, which will also lead to an increase in intracellular sodium in the neuronal compartment via opening of voltage-gated sodium channels. Second, the cerebral blood flow (CBF) is modulated as a separate input. For comparison to in vitro experiments using acute brain slices, the only input is the presynaptic neuronal population while the supply of oxygen and glucose that normally comes from the CBF is held constant as a proxy for the laminar flow of a controlled perfusing solution. Finally, the extracellular space is the place where cells and capillaries exchange metabolites

Figure 2. Simulated dynamics of Hodgkin-Huxley equations and of intracellular sodium concentrations during an in vitro 20 sec stimulation episode.

A. Time course of the global excitatory conductance simulating the activation of AMPA receptors by the presynaptic population. The gray area denotes the stimulation period. B. Neuronal membrane voltage recorded in response to the excitatory stimulation plotted in A and instantaneous firing rate (inset). C. Sodium concentration in the neuronal cytosol (blue) and in the astrocytic cytosol (red) recorded in response to the excitatory stimulation plotted in A

Figure 3. Simulated dynamics of ATP and PCr concentrations during an in vitro 20 sec stimulation episode.

A. Concentration of phosphocreatine (PCr; upper lines at ~5.0 mM) and adenosine triphosphate (ATP; lower lines at ~2.2 mM) in the neuronal (blue) and astrocytic compartments (red) during a 20 sec in vitro stimulation episode (same simulation as in Fig. 2). B. Zoom-in re-scaling of the area of interest in panel A for PCr in neuronal (blue) and astrocytic compartments (red). C. Zoom-in re-scaling of the area of interest in panel A for ATP in neuronal (blue) and astrocytic compartments (red). In panels A, B and C, the gray area denotes the stimulation period

Figure 4. Simulated dynamics of NADH concentrations, of lactate concentrations and of glucose and oxygen consumption during an in vitro 20 sec stimulation episode.

A. Relative fluctuations of the NADH concentration in the astrocytic cytosol (red), in the neuronal mitochondria (blue) and averaged over the whole tissue (black) as evoked by a 20 sec stimulation episode in vitro (grey area; same simulation as in Fig. 2 and 3). The dotted lines indicate corresponding in vitro data reproduced from Kasischke et al. [23]. B. Oxygen utilization (thick lines) and glucose utilization (thin lines) by neurons (blue) and astrocytes (red) as evoked by the same 20 sec stimulation episode as in A. Glucose utilization is multiplied by 6 to allow a direct comparison to oxygen utilization. C. Net transport of lactate between the four compartments of the model during the same 20 sec stimulation episode as in A. Lactate is exported by the astrocyte to the extracellular space (thick red line) and imported by the neuron from the extracellular space (thick blue line; net import is negative by convention in this model). The thin red line denotes the activity of the lactate dehydrogenase converting pyruvate into lactate in the astrocytic cytosol while the thin blue line denotes the activity of the lactate dehydrogenase converting lactate into pyruvate in the neuronal cytosol (again, the negative sign is a convention). Note how increase in lactate-to-pyruvate conversion precedes the increase in net lactate import by neurons, while pyruvate-to-lactate conversion follows the increase in net lactate export by astrocytes. D. Relative fluctuations of tissue lactate (pink) and of extracellular lactate (black) during the same 20 sec stimulation episode as in A to C. Both lines are superimposed and barely distinguishable

Figure 5. Predicted evoked responses of tissue lactate and of tissue oxygen in vivo in rodents during a 60 sec stimulation episode.

A. Temporal evolution of the cerebral blood flow chosen as an input to the model during a simulated 60 sec stimulation episode in vivo (grey area). This specific time course closely matches measurements in rodents during functional forepaw or whisker stimulation [50, 51]. Note that the blood flow only significantly increases approximately 1 sec after the onset of activation (inset). B. Relative fluctuations of the intra-parenchymal oxygen concentration as evoked by a 60 sec stimulation episode with the blood flow as in A. These results closely match the experimental results of Ances et al. [50] (their Fig. 1). C. Relative fluctuations of tissue lactate (pink line) and of extracellular lactate (black line) during the same 60 sec stimulation episode as in A to B. These results closely match the experimental results of Hu and Wilson [16] (their Fig. 1). D. Net transport of lactate between the four compartments of the model during the same 60 sec stimulation episode as in A to C. Like in the in vitro case (Fig. 4), lactate is exported by the astrocyte to the extracellular space (thick red line) and imported by the neuron from the extracellular space (thick blue line; net import is negative by convention in this model). A small amount of lactate is exported from the extracellular space to the capillary at baseline and this export increases by 69% after the end of the stimulation (pink line). The thin red line denotes the activity of the lactate dehydrogenase converting pyruvate into lactate in the astrocytic cytosol while the thin blue line denotes the activity of the lactate dehydrogenase converting lactate into pyruvate in the neuronal cytosol (again, the negative sign is a convention)

Figure 6. Predicted evoked responses of tissue lactate, CMR_glc, CMR_O2, oxygen-glucose index (OGI) and BOLD in vivo in humans.

A. Temporal evolution of the cerebral blood flow chosen as an input to the model during a simulated 900 sec stimulation episode in vivo (grey area). This specific time course closely matches in vivo measurements in humans during imaging experiments [52, 60]. B. Relative fluctuations of tissue lactate concentration during the same stimulation episode as in A. The model predicts an initial lactate dip followed by a 60% increase sustained till the end of the stimulation. The presence of a dip matches experimental data from Mangia et al. [17]. C and D. Cerebral metabolic rate of glucose consumption (CMR_glc) and cerebral metabolic rate of oxygen consumption (CMR_O2) during the same 900 sec stimulation episode as in A to B. In both cases, the light area corresponds to the contribution of the astrocytic compartment towards the total tissue consumption, while the dark area corresponds to the contribution of the neuronal compartment. While glucose consumption increases by about 40%, the increase is mostly due to the astrocytic compartment (light red) with the neuronal glucose utilization even slightly decreasing at the onset of activation (dark red). On the contrary, while oxygen utilization increases by about 10%, most of this increase is due to the neuronal compartment (dark blue) with the astrocytic oxygen utilization being almost constant (light blue). E. The predicted ratio of CMR_O2 to CMR_glc or oxygen-glucose index (OGI) during the same 900 sec stimulation episode as in A to D. F. Predicted BOLD signal for the same 900 sec stimulation episode as in A to E. Like the tissue lactate concentration (B), the BOLD shows a clear dip at the onset of activation