In most neuronal models, ion concentrations are assumed to be constant, and effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials are not accounted for. Here, we present the electrodiffusive Pinsky-Rinzel (edPR) model, which we believe is the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra-and extracellular space. The edPR model is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron. Unlike the PR model, the edPR model includes homeostatic mechanisms and ion-specific leakage currents, and keeps track of all ion concentrations (Na + , K + , Ca 2+ , and Cl − ), electrical potentials, and electrical conductivities in the intra-and extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity. For higher activity levels, or when homeostatic mechanisms are impaired, the homeostatic mechanisms fail in maintaining ion concentrations close to baseline, and the edPR model diverges from the PR model as it accounts for effects of concentration changes on neuronal firing. We envision that the edPR model will be useful for the field in three main ways. Firstly, as it relaxes commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model should supplement the PR model when simulating scenarios where ion concentrations are expected to vary over time. Thirdly, being applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.

Within the computational neuroscience community, there has been a focus on simulating the electrical activity of neurons, while other components of brain tissue, such as glia cells and the extracellular space, are often neglected. Standard models of extracellular potentials are based on a combination of multicompartmental models describing neural electrodynamics and volume conductor theory. Such models cannot be used to simulate the slow components of extracellular potentials, which depend on ion concentration dynamics, and the effect that this has on extracellular diffusion potentials and glial buffering currents. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine compartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for action potentials and dendritic calcium spikes in neurons, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. The edNEG model accounts for the concentration-dependent effects on ECS potentials that the standard models neglect. Using the edNEG model, we analyze these effects by splitting the extracellular potential into three components: one due to neural sink/source configurations, one due to glial sink/source configurations, and one due to extracellular diffusive currents. Through a series of simulations, we analyze the roles played by the various components and how they interact in generating the total slow potential. We conclude that the three components are of comparable magnitude and that the stimulus conditions determine which of the components that dominate.

The cerebral cortex is organized in cortical layers that differ in their cellular density, composition, and wiring. Cortical laminar architecture is also readily revealed by staining for cytochrome oxidase—the last enzyme in the respiratory electron transport chain located in the inner mitochondrial membrane. It has been hypothesized that a high-density band of cytochrome oxidase in cortical layer IV reflects higher oxygen consumption under baseline (unstimulated) conditions. Here, we tested the above hypothesis using direct measurements of the partial pressure of O2 (pO2) in cortical tissue by means of 2-photon phosphorescence lifetime microscopy (2PLM). We revisited our previously developed method for extraction of the cerebral metabolic rate of O2 (CMRO2) based on 2-photon pO2 measurements around diving arterioles and applied this method to estimate baseline CMRO2 in awake mice across cortical layers. To our surprise, our results revealed a decrease in baseline CMRO2 from layer I to layer IV. This decrease of CMRO2 with cortical depth was paralleled by an increase in tissue oxygenation. Higher baseline oxygenation and cytochrome density in layer IV may serve as an O2 reserve during surges of neuronal activity or certain metabolically active brain states rather than reflecting baseline energy needs. Our study provides to our knowledge the first quantification of microscopically resolved CMRO2 across cortical layers as a step towards better understanding of brain energy metabolism.

AbstractMost neuronal models are based on the assumption that ion concentrations remain constant during the simulated period, and do not account for possible effects of concentration variations on ionic reversal potentials, or of ionic diffusion on electrical potentials. Here, we present what is, to our knowledge, the first multicompartmental neuron model that accounts for electrodiffusive ion concentration dynamics in a way that ensures a biophysically consistent relationship between ion concentrations, electrical charge, and electrical potentials in both the intra- and extracellular space. The model, which we refer to as the electrodiffusive Pinsky-Rinzel (edPR) model, is an expanded version of the two-compartment Pinsky-Rinzel (PR) model of a hippocampal CA3 neuron, where we have included homeostatic mechanisms and ion-specific leakage currents. Whereas the main dynamical variable in the original PR model is the transmembrane potential, the edPR model in addition keeps track of all ion concentrations (Na+, K+, Ca2+, and Cl−), electrical potentials, and the electrical conductivities in the intra- as well as extracellular space. The edPR model reproduces the membrane potential dynamics of the PR model for moderate firing activity, when the homeostatic mechanisms succeed in maintaining ion concentrations close to baseline. For higher activity levels, homeostasis becomes incomplete, and the edPR model diverges from the PR model, as it accounts for changes in neuronal firing properties due to deviations from baseline ion concentrations. Whereas the focus of this work is to present and analyze the edPR model, we envision that it will become useful for the field in two main ways. Firstly, as it relaxes a set of commonly made modeling assumptions, the edPR model can be used to test the validity of these assumptions under various firing conditions, as we show here for a few selected cases. Secondly, the edPR model is a supplement to the PR model and should replace it in simulations of scenarios in which ion concentrations vary over time. As it is applicable to conditions with failed homeostasis, the edPR model opens up for simulating a range of pathological conditions, such as spreading depression or epilepsy.Author summaryNeurons generate their electrical signals by letting ions pass through their membranes. Despite this fact, most models of neurons apply the simplifying assumption that ion concentrations remain effectively constant during neural activity. This assumption is often quite good, as neurons contain a set of homeostatic mechanisms that make sure that ion concentrations vary quite little under normal circumstances. However, under some conditions, these mechanisms can fail, and ion concentrations can vary quite dramatically. Standard models are thus not able to simulate such conditions. Here, we present what to our knowledge is the first multicompartmental neuron model that in a biophysically consistent way does account for the effects of ion concentration variations. We here use the model to explore under which activity conditions the ion concentration variations become important for predicting the neurodynamics. We expect the model to be of great use for simulating a range of pathological conditions, such as spreading depression or epilepsy, which are associated with large changes in extracellular ion concentrations.

Significance: The cerebral metabolic rate of oxygen (CMRO 2) is an important indicator of brain function and pathology. Knowledge about its magnitude is also required for proper interpretation of the blood oxygenation level-dependent (BOLD) signal measured with functional MRI. Despite the need for estimating CMRO 2 , no gold standard exists. Traditionally, the estimation of CMRO 2 has been pursued with somewhat indirect approaches combining several different types of measurements with mathematical modeling of the underlying physiological processes. The recent ability to measure the level of oxygen (pO 2) in cortex with two-photon resolution in in vivo conditions has provided a more direct way for estimating CMRO 2 , but has so far only been used to estimate the average CMRO 2 close to cortical penetrating arterioles in rats. Aim: The aim of this study was to propose a method to provide spatial maps of CMRO 2 based on two-photon pO 2 measurements. Approach: The method has two key steps. First, the pO 2 maps are spatially smoothed to reduce the effects of noise in the measurements. Next, the Laplace operator (a double spatial derivative) in two spatial dimensions is applied on the smoothed pO 2 maps to obtain spatially resolved CMRO 2 estimates. Result: The smoothing introduces a bias, and a balance must be found where the effects of the noise are sufficiently reduced without introducing too much bias. In this model-based study, we explored this balance using synthetic model-based data, that is, data where the spatial maps of CMRO 2 were preset and thus known. The corresponding pO 2 maps were found by solving the Poisson equation, which relates CMRO 2 and pO 2. MATLAB code for using the method is provided. Conclusion: Through this model-based study, we propose a method for estimating CMRO 2 with high spatial resolution based on measurements of pO 2 in cerebral cortex.