A mathematical model of recurrent spreading depolarizations

Computational modeling of neuron-glia signaling interactions to unravel cellular and neural circuit functioning

Glial cells have been shown to be vital for various brain functions, including homeostasis, information processing, and cognition. Over the past 30 years, various signaling interactions between neuronal and glial cells have been shown to underlie these functions. This review summarizes the interactions, particularly between neurons and astrocytes, which are types of glial cells. Some of the interactions remain controversial in part due to the nature of experimental methods and preparations used. Based on the accumulated data, computational models of the neuron-astrocyte interactions have been developed to explain the complex functions of astrocytes in neural circuits and to test conflicting hypotheses. This review presents the most significant recent models, modeling methods and simulation tools for neuron-astrocyte interactions. In the future, we will especially need more experimental research on awake animals in vivo and new computational models of neuron-glia interactions to advance our understanding of cellular dynamics and the functioning of neural circuits in different brain regions.

Analysis of Network Models with Neuron-Astrocyte Interactions

Neural networks, composed of many neurons and governed by complex interactions between them, are a widely accepted formalism for modeling and exploring global dynamics and emergent properties in brain systems. In the past decades, experimental evidence of computationally relevant neuron-astrocyte interactions, as well as the astrocytic modulation of global neural dynamics, have accumulated. These findings motivated advances in computational glioscience and inspired several models integrating mechanisms of neuron-astrocyte interactions into the standard neural network formalism. These models were developed to study, for example, synchronization, information transfer, synaptic plasticity, and hyperexcitability, as well as classification tasks and hardware implementations. We here focus on network models of at least two neurons interacting bidirectionally with at least two astrocytes that include explicitly modeled astrocytic calcium dynamics. In this study, we analyze the evolution of these models and the biophysical, biochemical, cellular, and network mechanisms used to construct them. Based on our analysis, we propose how to systematically describe and categorize interaction schemes between cells in neuron-astrocyte networks. We additionally study the models in view of the existing experimental data and present future perspectives. Our analysis is an important first step towards understanding astrocytic contribution to brain functions. However, more advances are needed to collect comprehensive data about astrocyte morphology and physiology in vivo and to better integrate them in data-driven computational models. Broadening the discussion about theoretical approaches and expanding the computational tools is necessary to better understand astrocytes' roles in brain functions.

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.

A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model

Cortical spreading depression (SD) is a spreading disruption of ionic homeostasis in the brain during which neurons experience complete and prolonged depolarizations. SD is the basis of migraine aura and is increasingly associated with many other brain pathologies. Here, we study the role of glutamate and NMDA receptor dynamics in the context of an ionic electrodiffusion model. We perform simulations in one (1D) and two (2D) spatial dimension. Our 1D simulations reproduce the "inverted saddle" shape of the extracellular voltage signal for the first time. Our simulations suggest that SD propagation depends on two overlapping mechanisms; one dependent on extracellular glutamate diffusion and NMDA receptors and the other dependent on extracellular potassium diffusion and persistent sodium channel conductance. In 2D simulations, we study the dynamics of spiral waves. We study the properties of the spiral waves in relation to the planar 1D wave, and also compute the energy expenditure associated with the recurrent SD spirals.

Mitochondrial dysfunction and role in spreading depolarization and seizure

The effect of pathological phenomena such as epileptic seizures and spreading depolarization (SD) on mitochondria and the potential feedback of mitochondrial dysfunction into the dynamics of those phenomena are complex and difficult to study experimentally due to the simultaneous changes in many variables governing neuronal behavior. By combining a model that accounts for a wide range of neuronal behaviors including seizures, normoxic SD, and hypoxic SD (HSD), together with a detailed model of mitochondrial function and intracellular Ca²⁺ dynamics, we investigate mitochondrial dysfunction and its potential role in recovery of the neuron from seizures, HSD, and SD. Our results demonstrate that HSD leads to the collapse of mitochondrial membrane potential and cellular ATP levels that recover only when normal oxygen supply is restored. Mitochondrial organic phosphate and pH gradients determine the strength of the depolarization block during HSD and SD, how quickly the cell enters the depolarization block when the oxygen supply is disrupted or potassium in the bath solution is raised beyond the physiological value, and how fast the cell recovers from SD and HSD when normal potassium concentration and oxygen supply are restored. Although not as dramatic as phosphate and pH gradients, mitochondrial Ca²⁺ uptake has a similar effect on neuronal behavior during these conditions.

Using NEURON for Reaction-Diffusion Modeling of Extracellular Dynamics

Development of credible clinically-relevant brain simulations has been slowed due to a focus on electrophysiology in computational neuroscience, neglecting the multiscale whole-tissue modeling approach used for simulation in most other organ systems. We have now begun to extend the NEURON simulation platform in this direction by adding extracellular modeling. The extracellular medium of neural tissue is an active medium of neuromodulators, ions, inflammatory cells, oxygen, NO and other gases, with additional physiological, pharmacological and pathological agents. These extracellular agents influence, and are influenced by, cellular electrophysiology, and cellular chemophysiology-the complex internal cellular milieu of second-messenger signaling and cascades. NEURON's extracellular reaction-diffusion is supported by an intuitive Python-based where/who/what command sequence, derived from that used for intracellular reaction diffusion, to support coarse-grained macroscopic extracellular models. This simulation specification separates the expression of the conceptual model and parameters from the underlying numerical methods. In the volume-averaging approach used, the macroscopic model of tissue is characterized by free volume fraction-the proportion of space in which species are able to diffuse, and tortuosity-the average increase in path length due to obstacles. These tissue characteristics can be defined within particular spatial regions, enabling the modeler to account for regional differences, due either to intrinsic organization, particularly gray vs. white matter, or to pathology such as edema. We illustrate simulation development using spreading depression, a pathological phenomenon thought to play roles in migraine, epilepsy and stroke. Simulation results were verified against analytic results and against the extracellular portion of the simulation run under FiPy. The creation of this NEURON interface provides a pathway for interoperability that can be used to automatically export this class of models into complex intracellular/extracellular simulations and future cross-simulator standardization.