Osmotic forces and gap junctions in spreading depression: a computational model

A spatially distributed model of brain metabolism highlights the role of diffusion in brain energy metabolism

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

A model of neurovascular coupling and its application to cortical spreading depolarization

Cortical spreading depolarization (CSD) is a neuropathological condition involving propagating waves of neuronal silence, and is related to multiple diseases, such as migraine aura, traumatic brain injury (TBI), stroke, and cardiac arrest, as well as poor outcome of patients. While CSDs of different severity share similar roots on the ion exchange level, they can lead to different vascular responses (namely spreading hyperemia and spreading ischemia). In this paper, we propose a mathematical model relating neuronal activities to predict vascular changes as measured with near-infrared spectroscopy (NIRS) and fMRI recordings, and apply it to the extreme case of CSD, where sustained near-complete neuronal depolarization is seen. We utilize three serially connected models (namely, ion exchange, neurovascular coupling, and hemodynamic model) which are described by differential equations. Propagating waves of ion concentrations, as well as the associated vasodynamics and hemodynamics, are simulated by solving these equations. Our proposed model predicts vasodynamics and hemodynamics that agree both qualitatively and quantitatively with experimental literature. Mathematical modeling and simulation offer a powerful tool to help understand the underlying mechanisms of CSD and help interpret the data. In addition, it helps develop novel monitoring techniques prior to data collection. Our simulated results strongly suggest that fMRI is unable to reliably distinguish between spreading hyperemia and spreading ischemia, while NIRS signals are substantially distinct in the two cases.

Mathematical Psychiatry: On Cortical Spreading Depression-A Review

The concept of migraine with aura (MwA) is a widespread condition that can affect up to 30% of migraine patients and manifests itself as a temporary visual illusion followed by a prolonged headache. It was initially pitched as a neurological disease, and observed that the spread of accompanying electrophysiological waves as part of the condition, which came to be known as cortical spreading depression (CSD). A strong theoretical basis for a link between MwA and CSD has eventually led to knowledge of the dynamics between the pair. In addition to experiment-based observations, mathematical models make an important contribution towards a numerical means of expressing codependent neural-scale manifestations. This provides alternate means of understanding and observing the phenomena while helping to visualize the links between the variables and their magnitude in contributing towards the emanation and dynamic pulsing of the condition. A number of biophysical mechanisms are believed to contribute to the MwA-CSD, spanning ion diffusion, ionic currents of membranes, osmosis, spatial buffering, neurotransmission, gap junctions, metabolic pumping, and synapse connections. As part of this review study, the various mathematical models for the description of the condition are expressed, reviewed, and contrasted, all of which vary in their depth, perspective, and level of information presented. Subsequent to this, the review looked into links between electrophysiological data-driven manifestations from measurements such as EEG and fMRI. While concluding remarks forged a structured pathway in the area on sub-themes that need to be investigated in order to strengthen and robustify the existing models, they include an accounting for inter-personal variability in models, sex and hormonal factors, and age groups, i.e., pediatrics vs. adults.

Validating a Computational Framework for Ionic Electrodiffusion with Cortical Spreading Depression as a Case Study

Cortical spreading depression (CSD) is a wave of pronounced depolarization of brain tissue accompanied by substantial shifts in ionic concentrations and cellular swelling. Here, we validate a computational framework for modeling electrical potentials, ionic movement, and cellular swelling in brain tissue during CSD. We consider different model variations representing wild-type (WT) or knock-out/knock-down mice and systematically compare the numerical results with reports from a selection of experimental studies. We find that the data for several CSD hallmarks obtained computationally, including wave propagation speed, direct current shift duration, peak in extracellular K⁺ concentration as well as a pronounced shrinkage of extracellular space (ECS) are well in line with what has previously been observed experimentally. Further, we assess how key model parameters including cellular diffusivity, structural ratios, membrane water and/or K⁺ permeabilities affect the set of CSD characteristics.

The Critical Role of Spreading Depolarizations in Early Brain Injury: Consensus and Contention

BACKGROUND

When a patient arrives in the emergency department following a stroke, a traumatic brain injury, or sudden cardiac arrest, there is no therapeutic drug available to help protect their jeopardized neurons. One crucial reason is that we have not identified the molecular mechanisms leading to electrical failure, neuronal swelling, and blood vessel constriction in newly injured gray matter. All three result from a process termed spreading depolarization (SD). Because we only partially understand SD, we lack molecular targets and biomarkers to help neurons survive after losing their blood flow and then undergoing recurrent SD.

METHODS

In this review, we introduce SD as a single or recurring event, generated in gray matter following lost blood flow, which compromises the Na⁺/K⁺ pump. Electrical recovery from each SD event requires so much energy that neurons often die over minutes and hours following initial injury, independent of extracellular glutamate.

RESULTS

We discuss how SD has been investigated with various pitfalls in numerous experimental preparations, how overtaxing the Na⁺/K⁺ ATPase elicits SD. Elevated K⁺ or glutamate are unlikely natural activators of SD. We then turn to the properties of SD itself, focusing on its initiation and propagation as well as on computer modeling.

CONCLUSIONS

Finally, we summarize points of consensus and contention among the authors as well as where SD research may be heading. In an accompanying review, we critique the role of the glutamate excitotoxicity theory, how it has shaped SD research, and its questionable importance to the study of early brain injury as compared with SD theory.

The triterpenoid ursolic acid ameliorates stress in Caenorhabditis elegans by affecting the depression-associated genes skn-1 and prdx2

INTRODUCTION

Depression is one of the leading causes of death worldwide. Lower antioxidant concentrations and increased oxidative stress levels contribute to the development of depression. Effective and tolerable medications are urgently needed. Nrf2 and PRDX2 are promising targets in the treatment of oxidative stress and, therefore, promising for the development of novel antidepressants. Ursolic acid (UA), a natural triterpenoid found in various plants is known to exert neuroprotective and antioxidant effects. Skn-1 (which corresponds to human Nrf2) and prdx2 deficient mutants of the nematode Caenorhabditis elegans are suitable models to study the effect of UA on these targets. Additionally, stress assays are used to mimic stress or depressed state.

METHODS

We examined the antioxidant activity of UA in Caenorhabditis elegans wildtype and skn-1- and prdx2-deficient strains by H2DCF-DA and juglone assays as well as osmotic and heat stress assays. Additionally, we analyzed the binding of UA to human PRDX2 and Skn-1 proteins by molecular docking and microscale thermophoresis.

RESULTS

UA exerted strong antioxidant activities. Additionally, induction of stress resistance towards osmotic and heat stress was observed. qRT-PCR revealed that UA upregulated the gene expression of skn-1 and prdx2. Molecular docking studies supported these findings.

CONCLUSION

Our findings implicate that the strong antioxidant activity of UA may exert anti-depressive effects by its interaction with the Skn-1 transcription factor, which is part of a detoxification network, and the antioxidant PRDX2 protein, which protects the organism from the detrimental effects of radical oxygen species.

An electrodiffusive neuron-extracellular-glia model for exploring the genesis of slow potentials in the brain

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.

On the Effects of Mechanical Stress of Biological Membranes in Modeling of Swelling Dynamics of Biological Systems

We highlight mechanical stretching and bending of membranes and the importance of membrane deformations in the analysis of swelling dynamics of biological systems, including cells and subcellular organelles. Membrane deformation upon swelling generates tensile stress and internal pressure, contributing to volume changes in biological systems. Therefore, in addition to physical (internal/external) and chemical factors, mechanical properties of the membranes should be considered in modeling analysis of cellular swelling. Here we describe an approach that considers mechanical properties of the membranes in the analysis of swelling dynamics of biological systems. This approach includes membrane bending and stretching deformations into the model, producing a more realistic description of swelling. We also discuss the effects of membrane stretching on swelling dynamics. We report that additional pressure generated by membrane bending is negligible, compared to pressures generated by membrane stretching, when both membrane surface area and volume are variable parameters. Note that bending deformations are reversible, while stretching deformation may be irreversible, leading to membrane disruption when they exceed a certain threshold level. Therefore, bending deformations need only be considered in reversible physiological swelling, whereas stretching deformations should also be considered in pathological irreversible swelling. Thus, the currently proposed approach may be used to develop a detailed biophysical model describing the transition from physiological to pathological swelling mode.

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.

Systematic review of the pharmacological agents that have been tested against spreading depolarizations

Spreading depolarization (SD) occurs alongside brain injuries and it can lead to neuronal damage. Therefore, pharmacological modulation of SD can constitute a therapeutic approach to reduce its detrimental effects and to improve the clinical outcome of patients. The major objective of this article was to produce a systematic review of all the drugs that have been tested against SD. Of the substances that have been examined, most have been shown to modulate certain SD characteristics. Only a few have succeeded in significantly inhibiting SD. We present a variety of strategies that have been proposed to overcome the notorious harmfulness and pharmacoresistance of SD. Information on clinically used anesthetic, sedative, hypnotic agents, anti-migraine drugs, anticonvulsants and various other substances have been compiled and reviewed with respect to the efficacy against SD, in order to answer the question of whether a drug at safe doses could be of therapeutic use against SD in humans.

Turing-like structures in a functional model of cortical spreading depression

Cortical spreading depression (CSD) along with migraine waves and spreading depolarization events with stroke or injures are the front-line examples of extreme physiological behaviors of the brain cortex which manifest themselves via the onset and spreading of localized areas of neuronal hyperactivity followed by their depression. While much is known about the physiological pathways involved, the dynamical mechanisms of the formation and evolution of complex spatiotemporal patterns during CSD are still poorly understood, in spite of the number of modeling studies that have been already performed. Recently we have proposed a relatively simple mathematical model of cortical spreading depression which counts the effects of neurovascular coupling and cerebral blood flow redistribution during CSD. In the present study, we address the main dynamical consequences of newly included pathways, namely, the changes in the formation and propagation speed of the CSD front and the pattern formation features in two dimensions. Our most notable finding is that the combination of vascular-mediated spatial coupling with local regulatory mechanisms results in the formation of stationary Turing-like patterns during a CSD event.

A mathematical model of recurrent spreading depolarizations

A detailed biophysical model for a neuron/astrocyte network is developed in order to explore mechanisms responsible for the initiation and propagation of recurrent cortical spreading depolarizations. The model incorporates biophysical processes not considered in the earlier models. This includes a model for the Na⁺-glutamate transporter, which allows for a detailed description of reverse glutamate uptake. In particular, we consider the specific roles of elevated extracellular glutamate and K⁺ in the initiation, propagation and recurrence of spreading depolarizations.

The role of glutamate in neuronal ion homeostasis: A case study of spreading depolarization

Simultaneous changes in ion concentrations, glutamate, and cell volume together with exchange of matter between cell network and vasculature are ubiquitous in numerous brain pathologies. A complete understanding of pathological conditions as well as normal brain function, therefore, hinges on elucidating the molecular and cellular pathways involved in these mostly interdependent variations. In this paper, we develop the first computational framework that combines the Hodgkin-Huxley type spiking dynamics, dynamic ion concentrations and glutamate homeostasis, neuronal and astroglial volume changes, and ion exchange with vasculature into a comprehensive model to elucidate the role of glutamate uptake in the dynamics of spreading depolarization (SD)-the electrophysiological event underlying numerous pathologies including migraine, ischemic stroke, aneurysmal subarachnoid hemorrhage, intracerebral hematoma, and trauma. We are particularly interested in investigating the role of glutamate in the duration and termination of SD caused by K+ perfusion and oxygen-glucose deprivation. Our results demonstrate that glutamate signaling plays a key role in the dynamics of SD, and that impaired glutamate uptake leads to recovery failure of neurons from SD. We confirm predictions from our model experimentally by showing that inhibiting astrocytic glutamate uptake using TFB-TBOA nearly quadruples the duration of SD in layers 2-3 of visual cortical slices from juvenile rats. The model equations are either derived purely from first physical principles of electroneutrality, osmosis, and conservation of particles or a combination of these principles and known physiological facts. Accordingly, we claim that our approach can be used as a future guide to investigate the role of glutamate, ion concentrations, and dynamics cell volume in other brain pathologies and normal brain function.

Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model

A detailed biophysical model for a neuron/astrocyte network is developed to explore mechanisms responsible for the initiation and propagation of cortical spreading depolarizations and the role of astrocytes in maintaining ion homeostasis, thereby preventing these pathological waves. Simulations of the model illustrate how properties of spreading depolarizations, such as wave speed and duration of depolarization, depend on several factors, including the neuron and astrocyte Na(+)-K(+) ATPase pump strengths. In particular, we consider the neuroprotective role of astrocyte gap junction coupling. The model demonstrates that a syncytium of electrically coupled astrocytes can maintain a physiological membrane potential in the presence of an elevated extracellular K(+) concentration and efficiently distribute the excess K(+) across the syncytium. This provides an effective neuroprotective mechanism for delaying or preventing the initiation of spreading depolarizations.

Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression

Cortical spreading depression (SD) is a spreading disruption in brain ionic homeostasis during which neurons experience complete and prolonged depolarizations. SD is generally believed to be the physiological substrate of migraine aura and is associated with many other brain pathologies. Here, we perform simulations with a model of SD treating brain tissue as a triphasic continuum of neurons, glia and the extracellular space. A thermodynamically consistent incorporation of the major biophysical effects, including ionic electrodiffusion and osmotic water flow, allows for the computation of important physiological variables including the extracellular voltage (DC) shift. A systematic parameter study reveals that glia can act as both a disperser and buffer of potassium in SD propagation. Furthermore, we show that the timing of the DC shift with respect to extracellular [Formula: see text] rise is highly dependent on glial parameters, a result with implications for the identification of the propagating mechanism of SD.

Anions Govern Cell Volume: A Case Study of Relative Astrocytic and Neuronal Swelling in Spreading Depolarization

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little about the dynamics of cell and tissue swelling, and the differential changes in the volumes of neurons and glia during pathological states such as spreading depolarizations (SD) under ischemic and non–ischemic conditions, and epileptic seizures. By combining the Hodgkin–Huxley type spiking dynamics, dynamic ion concentrations, and simultaneous neuronal and astroglial volume changes into a comprehensive model, we elucidate why glial cells swell more than neurons in SD and the special case of anoxic depolarization (AD), and explore the relative contributions of the two cell types to tissue swelling. Our results demonstrate that anion channels, particularly Cl⁻, are intrinsically connected to cell swelling and blocking these currents prevents changes in cell volume. The model is based on a simple and physiologically realistic description. We introduce model extensions that are either derived purely from first physical principles of electroneutrality, osmosis, and conservation of particles, or by a phenomenological combination of these principles and known physiological facts. This work provides insights into numerous studies related to neuronal and glial volume changes in SD that otherwise seem contradictory, and is broadly applicable to swelling in other cell types and conditions.

Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs

The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

Radial, spiral and reverberating waves of spreading depolarization occur in the gyrencephalic brain

OBJECTIVES

The detection of the hemodynamic and propagation patterns of spreading depolarizations (SDs) in the gyrencephalic brain using intrinsic optical signal imaging (IOS).

METHODS

The convexity of the brain surface was surgically exposed in fourteen male swine. Within the boundaries of this window, brains were immersed and preconditioned with an elevated K(+) concentration (7 mmol/l) in the standard Ringer lactate solution for 30-40 min. SDs were triggered using 3-5 μl of 1 mol/l KCl solution. Changes in tissue absorbency or reflection were registered with a CCD camera at a wavelength of 564 nm (14 nm FWHM), which was mounted 25 cm above the exposed cortex. Additional monitoring by electrocorticography and laser-Doppler was used in a subset of animals (n=7) to validate the detection of SD.

RESULTS

Of 198 SDs quantified in all of the experiments, 187 SDs appeared as radial waves that developed semi-planar fronts. The morphology was affected by the surface of the gyri, the sulci and the pial vessels. Other SD patterns such as spirals and reverberating waves, which have not been described before in gyrencephalic brains, were also observed. Diffusion gradients created in the cortex surface (i.e., KCl concentrations), sulci, vessels and SD-SD interactions make the gyrencephalic brain prone to the appearance of irregular SD waves.

CONCLUSION

The gyrencephalic brain is capable of irregular SD propagation patterns. The irregularities of the gyrencephalic brain cortex may promote the presence of re-entrance waves, such as spirals and reverberating waves.

Bistable dynamics underlying excitability of ion homeostasis in neuron models

When neurons fire action potentials, dissipation of free energy is usually not directly considered, because the change in free energy is often negligible compared to the immense reservoir stored in neural transmembrane ion gradients and the long-term energy requirements are met through chemical energy, i.e., metabolism. However, these gradients can temporarily nearly vanish in neurological diseases, such as migraine and stroke, and in traumatic brain injury from concussions to severe injuries. We study biophysical neuron models based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent ion concentrations inside and outside the cell and metabolic energy-driven pumps. We reveal the basic mechanism of a state of free energy-starvation (FES) with bifurcation analyses showing that ion dynamics is for a large range of pump rates bistable without contact to an ion bath. This is interpreted as a threshold reduction of a new fundamental mechanism of ionic excitability that causes a long-lasting but transient FES as observed in pathological states. We can in particular conclude that a coupling of extracellular ion concentrations to a large glial-vascular bath can take a role as an inhibitory mechanism crucial in ion homeostasis, while the Na⁺/K⁺ pumps alone are insufficient to recover from FES. Our results provide the missing link between the HH formalism and activator-inhibitor models that have been successfully used for modeling migraine phenotypes, and therefore will allow us to validate the hypothesis that migraine symptoms are explained by disturbed function in ion channel subunits, Na⁺/K⁺ pumps, and other proteins that regulate ion homeostasis.

Refractory period modulates the spatiotemporal evolution of cortical spreading depression: a computational study

Cortical spreading depression (CSD) is a pathophysiological phenomenon, which underlies some neurological disorders, such as migraine and stroke, but its mechanisms are still not completely understood. One of the striking facts is that the spatiotemporal evolution of CSD wave is varying. Observations in experiments reveal that a CSD wave may propagate through the entire cortex, or just bypass some areas of the cortex. In this paper, we have applied a 2D reaction-diffusion equation with recovery term to study the spatiotemporal evolution of CSD. By modulating the recovery rate from CSD in the modeled cortex, CSD waves with different spatiotemporal evolutions, either bypassing some areas or propagating slowly in these areas, were present. Moreover, spiral CSD waves could also be induced in case of the transiently altered recovery rate, i.e. block release from the absolute refractory period. These results suggest that the refractory period contributes to the different propagation patterns of CSD, which may help to interpret the mechanisms of CSD propagation.

Mathematical approaches to modeling of cortical spreading depression

Migraine with aura (MwA) is a debilitating disease that afflicts about 25%-30% of migraine sufferers. During MwA, a visual illusion propagates in the visual field, then disappears, and is followed by a sustained headache. MwA was conjectured by Lashley to be related to some neurological phenomenon. A few years later, Leão observed electrophysiological waves in the brain that are now known as cortical spreading depression (CSD). CSD waves were soon conjectured to be the neurological phenomenon underlying MwA that had been suggested by Lashley. However, the confirmation of the link between MwA and CSD was not made until 2001 by Hadjikhani et al. [Proc. Natl. Acad. Sci. U.S.A. 98, 4687-4692 (2001)] using functional MRI techniques. Despite the fact that CSD has been studied continuously since its discovery in 1944, our detailed understandings of the interactions between the mechanisms underlying CSD waves have remained elusive. The connection between MwA and CSD makes the understanding of CSD even more compelling and urgent. In addition to all of the information gleaned from the many experimental studies on CSD since its discovery, mathematical modeling studies provide a general and in some sense more precise alternative method for exploring a variety of mechanisms, which may be important to develop a comprehensive picture of the diverse mechanisms leading to CSD wave instigation and propagation. Some of the mechanisms that are believed to be important include ion diffusion, membrane ionic currents, osmotic effects, spatial buffering, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. Discrete and continuum models of CSD consist of coupled nonlinear differential equations for the ion concentrations. In this review of the current quantitative understanding of CSD, we focus on these modeling paradigms and various mechanisms that are felt to be important for CSD.

Towards dynamical network biomarkers in neuromodulation of episodic migraine

Computational methods have complemented experimental and clinical neurosciences and led to improvements in our understanding of the nervous systems in health and disease. In parallel, neuromodulation in form of electric and magnetic stimulation is gaining increasing acceptance in chronic and intractable diseases. In this paper, we firstly explore the relevant state of the art in fusion of both developments towards translational computational neuroscience. Then, we propose a strategy to employ the new theoretical concept of dynamical network biomarkers (DNB) in episodic manifestations of chronic disorders. In particular, as a first example, we introduce the use of computational models in migraine and illustrate on the basis of this example the potential of DNB as early-warning signals for neuromodulation in episodic migraine.

A mathematical model of the metabolic and perfusion effects on cortical spreading depression

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism-aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium-ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD--vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization--found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.

Transient localized wave patterns and their application to migraine

Transient dynamics is pervasive in the human brain and poses challenging problems both in mathematical tractability and clinical observability. We investigate statistical properties of transient cortical wave patterns with characteristic forms (shape, size, duration) in a canonical reaction-diffusion model with mean field inhibition. The patterns are formed by ghost behavior near a saddle-node bifurcation in which a stable traveling wave (node) collides with its critical nucleation mass (saddle). Similar patterns have been observed with fMRI in migraine. Our results support the controversial idea that waves of cortical spreading depression (SD) have a causal relationship with the headache phase in migraine and, therefore, occur not only in migraine with aura (MA), but also in migraine without aura (MO), i.e., in the two major migraine subtypes. We suggest a congruence between the prevalence of MO and MA with the statistical properties of the traveling waves' forms according to which two predictions follow: (i) the activation of nociceptive mechanisms relevant for headache is dependent upon a sufficiently large instantaneous affected cortical area; and (ii) the incidence of MA is reflected in the distance to the saddle-node bifurcation. We also observed that the maximal instantaneous affected cortical area is anticorrelated to both SD duration and total affected cortical area, which can explain why the headache is less severe in MA than in MO. Furthermore, the contested notion of MO attacks with silent aura is resolved. We briefly discuss model-based control and means by which neuromodulation techniques may affect pathways of pain formation.

Diffusing substances during spreading depolarization: analytical expressions for propagation speed, triggering, and concentration time courses

Spreading depolarization (SD) is an important phenomenon in stroke and migraine. However, the processes underlying the propagation of SD are still poorly understood, and an elementary model that is both physiological and quantitative is lacking. We show that, during the onset and propagation of SD, the concentration time courses of excitatory substances such as potassium and glutamate can be described with a reaction-diffusion equation. This equation contains four physiological parameters: (1) a concentration threshold for excitation; (2) a release rate; (3) a removal rate; and (4) an effective diffusion constant. Solving this equation yields expressions for the propagation velocity, concentration time courses, and the minimum stimulus that can trigger SD. This framework allows for analyzing experimental results in terms of these four parameters. The derived time courses are validated with measurements of potassium in rat brain tissue.

Connexin 36 promotes cortical spreading depolarization and ischemic brain damage

Cortical spreading depolarization (CSD) promotes the progression of neuronal injury after cerebral ischemia. However, the mechanisms of propagation of postischemic CSD events are still unclear. In this study we characterized the role of the main neuronal gap junction protein connexin 36 (Cx36) in generating postischemic CSDs. In Cx36-deficient mice and controls we occluded the distal middle cerebral artery. To detect CSD events we recorded the direct current and laser Doppler flow. In addition, locomotor function and the infarct size were determined. Cx36-deficient mice had significantly fewer and shorter CSD events than wild-type controls. Additionally, Cx36 deletion is neuroprotective, leading to a better functional outcome and decreased infarct size after ischemia. These results suggest a detrimental role for Cx36 after ischemia, possibly by promoting CSD.

Is spreading depolarization characterized by an abrupt, massive release of gibbs free energy from the human brain cortex?

In the evolution of the cerebral cortex, the sophisticated organization in a steady state far away from thermodynamic equilibrium has produced the side effect of two fundamental pathological network events: ictal epileptic activity and spreading depolarization. Ictal epileptic activity describes the partial disruption, and spreading depolarization describes the near-complete disruption of the physiological double Gibbs-Donnan steady state. The occurrence of ictal epileptic activity in patients has been known for decades. Recently, unequivocal electrophysiological evidence has been found in patients that spreading depolarizations occur abundantly in stroke and brain trauma. The authors propose that the ion changes can be taken to estimate relative changes in Gibbs free energy from state to state. The calculations suggest that in transitions from the physiological state to ictal epileptic activity to spreading depolarization to death, the cortex releases Gibbs free energy in a stepwise fashion. Spreading depolarization thus appears as a twilight state close to death. Consistently, electrocorticographic recordings in the core of focal ischemia or after cardiac arrest display a smooth transition from the initial spreading depolarization component to the later ultraslow negative potential, which is assumed to reflect processes in cellular death.

SPREADING DEPRESSION IN MAINLY LOCALLY CONNECTED CELLULAR AUTOMATON

A computational model based on cellular automaton (CA) is developed for studying the spreading depression (SD) phenomenon. CA cells correspond to neurons which are mainly locally connected. The influence of the parameter values in the spatiotemporal dynamics is investigated in order to obtain ways of reducing, or even preventing, occurrences of SD.

Modeling the slow wave shapes of spreading depression in a rat cortex: a methodology for seeking physiological parameters

Spreading depression (SD) consists of a transient significant suppression of the spontaneous neural electrical activity that spreads slowly across regions of the gray matter in a wave form. Nowadays, this phenomenon is being studied by means of mathematical and computational models to reproduce the main characteristics of SD. Given the high number of parameters and their unknown ranges of variation, the setting of parameters for current SD models is usually a hard task that must be addressed in order to make such models reproduce real data. In this paper, we present a 1-D model which is able to reproduce the most important characteristics of SD waves observed in laboratory experiments: the slow extracellular potential shift and extracellular ionic concentration variations regarding speed, shape, and amplitude. Such a reproduction is possible due to a methodology that we introduced to set the parameters of the SD models. The methodology allows the impact of each parameter on the results produced by the model and the range of parameters for which the model displays plausible behavior to be determined. The methodology also helps to identify features that the model cannot produce and it gives insights about what parts of the model should be modified to improve its capacities through the identification of parameters involved with each behavior.

Self-terminating wave patterns and self-organized pacemakers in a phenomenological model of spreading depression

A simple reaction-diffusion model of spreading depression (SD) is presented. Its local dynamics are governed by two activator and two inhibitor variables that provide an extremely simplified description of the mutual interaction between the neurons and extracellular space. This interaction is realized by the substances in the extracellular space that are increasing excitability of the neurons that have released them and are diffusing to the neighboring neurons, thereby spreading this excitation. Typical dynamic patterns of simulated activity are presented. The focus is laid on the case where response of the extracellular medium is relatively fast, and retracting waves, spiral-shaped waves, and autonomous pacemakers are observed, which is in good agreement with experimental observations. The underlying mechanisms are found to be related to switching between the local bi-stable, excitable, and self-sustained dynamics in the simulated medium. This article is part of a Special Issue entitled: Neural Coding.

Generation of Sustained Field Potentials by Gradients of Polarization Within Single Neurons: A Macroscopic Model of Spreading Depression

Spreading depression (SD) is a pathological wave of depolarization of the neural tissue producing a negative macroscopic field potential ( Vo), used as a marker for diagnostic purposes. The cellular basis of SD and neuronal mechanisms of generation of Vo at the microscopic level are poorly understood. Using a CA1 mathematical model and experimental verification, we examined how transmembrane currents in single cells scale up in the extracellular space shaping Vo. The model includes an array of 17,000 realistically modeled neurons (responsible for generating transmembrane currents) dynamically coupled to a virtual aggregate/extracellular space (responsible for Vo). The SD wave in different tissue bands is simulated by imposing membrane shunts in the corresponding dendritic elements as suggested by experimentally assessed drop in membrane resistance. We show that strong isopotential depolarization of wide domains (as in the main SD phase) produce broad central cancellation of axial and transmembrane currents in single cells. When depolarization is restricted to narrow dendritic domains (as in the late SD phase), the internal cancellation shrinks and the transmembrane current increases. This explains why in the laminated CA1 the Vo is smaller in the main phase of SD, when both dendritic layers are seized, than in the SD tail restricted to an apical band. Moreover, scattering of the neuronal somatas (as in cortical regions) further decreases the aggregate Vo due to the volume averaging. Although mechanistically the Vo associated to SD is similar to customary transient fields, its changes maybe related to spatial factors in single cells rather than cell number or depolarization strength.

Infant Brain Stem Is Prone to the Generation of Spreading Depression During Severe Hypoxia

Spreading depression (SD) resembles a concerted, massive neuronal/glial depolarization propagating within the gray matter. Being associated with cerebropathology, such as cerebral ischemia or hemorrhage, epileptic seizures, and migraine, it is well studied in cortex and hippocampus. We have now analyzed the susceptibility of rat brain stem to hypoxia-induced spreading depression-like depolarization (HSD), which could critically interfere with cardiorespiratory control. In rat brain stem slices, severe hypoxia (oxygen withdrawal) triggered HSD within minutes. The sudden extracellular DC potential shift of approximately −20 mV showed the typical profile known from other brain regions and was accompanied by an intrinsic optical signal (IOS). Spatiotemporal IOS analysis revealed that in infant brain stem, HSD was preferably ignited within the spinal trigeminal nucleus and then mostly spread out medially, invading the hypoglossal nucleus, the nucleus of the solitary tract (NTS), and the ventral respiratory group (VRG). The neuronal hypoxic depolarizations underlying the generation of HSD were massive, but incomplete. The propagation velocity of HSD and the associated extracellular K+ rise were also less marked than in other brain regions. In adult brain stem, HSD was mostly confined to the NTS and its occurrence was facilitated by hypotonic solutions, but not by glial poisoning or block of GABAergic and glycinergic synapses. In conclusion, brain stem tissue reliably generates propagating HSD episodes, which may be of interest for basilar-type migraine and brain stem infarcts. The preferred occurrence of HSD in the infant brain stem and its propagation into the VRG may be of importance for neonatal brain stem pathology such as sudden infant death syndrome.

Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback

The onset of pulse propagation is studied in a reaction-diffusion (RD) model with control by augmented transmission capability that is provided either along nonlocal spatial coupling or by time-delayed feedback. We show that traveling pulses occur primarily as solutions to the RD equations, while augmented transmission changes excitability. For certain ranges of the parameter settings, defined as weak susceptibility and moderate control, respectively, the hybrid model can be mapped to the original RD model. This results in an effective change in RD parameters controlled by augmented transmission. Outside moderate control parameter settings new patterns are obtained, for example, stepwise propagation due to delay-induced oscillations. Augmented transmission constitutes a signaling system complementary to the classical RD mechanism of pattern formation. Our hybrid model combines the two major signaling systems in the brain, namely, volume transmission and synaptic transmission. Our results provide insights into the spread and control of pathological pulses in the brain.

A quantitative model of cortical spreading depression due to purinergic and gap-junction transmission in astrocyte networks

Spreading depression (SD), a propagating wave of electrical silence in the cortex and archicortex, involves depolarization of neurons and astrocytes for approximately 1 min, due principally to a large increase in extracellular K+. SD is accompanied by large increases in extracellular ATP and is blocked by glutamate N-methyl-D-aspartate receptor antagonists. As a principal means of transmission between astrocytes is through their release of ATP, we have investigated if a model in which SD is driven by the effects of astrocyte waves of ATP interacting with waves of glutamate release from neurons and astrocytes can give a quantitative account of experimental observations on SD. We show that the characteristics of SD and the accompanying extracellular ionic changes can be accommodated by such a model-whether astrocyte transmission is principally through the release of ATP, as in archicortex (hippocampus) and spinal cord, or via gap junctions, as in the neocortex. Furthermore, these models give quantitative accounts of the effects on the characteristics of SD of agents toxic for astrocytes and of gap-junction blockers. Finally, an additional series of critical tests of the model is suggested.

Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke

The stability of cortical function depends critically on proper regulation. Under conditions of migraine and stroke a breakdown of transmembrane chemical gradients can spread through cortical tissue. A concomitant component of this emergent spatio-temporal pattern is a depolarization of cells detected as slow voltage variations. The propagation velocity of approximately 3 mm/min indicates a contribution of diffusion. We propose a mechanism for spreading depolarizations (SD) that rests upon a nonlocal or noninstantaneous feedback in a reaction-diffusion system. Depending upon the characteristic space and time scales of the feedback, the propagation of cortical SD can be suppressed by shifting the bifurcation line, which separates the parameter regime of pulse propagation from the regime where a local disturbance dies out. The optimization of this feedback is elaborated for different control schemes and ranges of control parameters.

Efficient control of transient wave forms to prevent spreading depolarizations

In various neurological disorders spatio-temporal excitation patterns constitute examples of excitable behavior emerging from pathological pathways. During migraine, seizure, and stroke an initially localized pathological state can temporarily spread indicating a transition from non-excitable to excitable behavior. We investigate these transient wave forms in the generic FitzHugh-Nagumo (FHN) system of excitable media. Our goal is to define an efficient control minimizing the volume of invaded tissue. The general point of such a therapeutic optimization is how to apply control theory in the framework of structures in differential geometry by regarding parameter plane M of the FHN system as a differentiable manifold endowed with a metric. We suggest to equip M with a metric given by pharmacokinetic-pharmacodynamic models of drug receptor interaction.

A global phenomenological model of ischemic stroke with stress on spreading depressions

In this paper, we establish a new global phenomenological model of ischemic stroke. It takes into account local ischemia, energy reduction, propagation of spreading depressions (SD), damages to the cells and cellular death by apoptosis or necrosis. The spatial diffusion of the ions in the extracellular space which triggers the propagation of SD is a central point here. First we expose the various biological hypotheses that we have made in this model, and then we explain how to determine the parameters and solve the system of equations that we obtain. Next we present some results of this model: we simulate a KCl injection and then a local ischemia. Finally we discuss results and propose some improvements for this model.

A steady-state model of spreading depression predicts the importance of an unknown conductance in specific dendritic domains

Spreading depression (SD) is a pathological wave of transient neuronal inactivation. We recently reported that the characteristic sustained complete depolarization is restricted to specific cell domains where the input resistance (R(in)) first becomes negligible before achieving partial recovery, whereas in adjacent, more polarized membranes it drops by much less. The experimental study of the participating membrane channels is hindered by their mixed contribution and heterogeneous distribution. Therefore, we derived a biophysical model to analyze the conductances that replicate the subcellular profile of R(in) during SD. Systematic variation of conductance densities far beyond the ranges reported failed to fit the experimental values. Besides standard potassium, sodium, and Glu-mediated conductances, the initial opening and gradual closing of an as yet undetermined large conductance is required to account for the evolution of R(in). Potassium conductances follow in the relative contribution and their closing during the late phase is also predicted. Large intracellular potential gradients from zero to rest are readily sustained between shunted and adjacent SD-spared membranes, which remain electroregenerative. The gradients are achieved by a combination of high-conductance subcellular domains and transmembrane ion redistribution in extended but discrete dendritic domains. We conclude that the heterogeneous subcellular behavior is due to local membrane properties, some of which may be specifically activated under extreme SD conditions.

Preseizure increased gamma electroencephalographic activity has no effect on extracellular potassium or calcium

Extracellular ion concentrations change during seizures in seizure models. [K(+)](o) increases and [Ca(2+)](o) decreases, resulting from population discharges, enhanced neuronal excitability, though not obviously before seizure onset. In acute pharmacological epilepsy models, there are striking increases in preictal high-frequency (gamma) electroencephalographic (EEG) activity. It is not known whether enhanced gamma EEG results in ionic changes, because gamma and ions have not been measured simultaneously. In this study, unanesthetized, paralyzed rats were given intravenous injections of kainic acid or picrotoxin to induce EEG discharges. Changes in EEG, [K(+)](o), and [Ca(2+)](o) in cortex and hippocampus were recorded. Kainic acid caused small [K(+)](o) fluctuations, without a temporal relationship of these with increased gamma EEG or with onset of discharges. Gamma EEG increases after picrotoxin also failed to affect [K(+)](o) and [Ca(2+)](o). Picrotoxin-induced electrical discharges led to [K(+)](o) rises of >9 mM and [Ca(2+)](o) falls of 0.1-0.2 mM. Kainic acid-induced discharges generated only moderate (2-3 mM) rises in [K(+)](o) and no changes in [Ca(2+)](o). In both models, there were large potassium rises (15-80 mM) and calcium falls (>0.5 mM), suggesting spreading depressions. Small [K(+)](o) fluctuations after kainic acid are consistent with disruption in potassium homeostasis, possibly because of depolarization of astrocytes. To reveal possible latent [K(+)](o) or [Ca(2+)](o) changes, we injected fluorocitrate intracortically to impair astrocytic function, before administering picrotoxin. Even fluorocitrate did not cause gamma-related ion changes but did cause low-magnitude, transient, potassium increases and slower potassium homeostasis during discharges, minor changes consistent with involvement of both astrocytes and neurons in [K(+)](o) regulation. (c) 2007 Wiley-Liss, Inc.

Role of astrocytes in grey matter during stroke: a modelling approach

The astrocytic response to stroke is extremely complex and incompletely understood. On the one hand, astrocytes are known to be neuroprotective when extracellular glutamate or potassium is slightly increased. But, on the other hand, they are considered to contribute to the extracellular glutamate increase during severe ischaemia. A mathematical model is used to reproduce the dynamics of the membrane potentials, intracellular and extracellular concentrations and volumes of neurons and astrocytes during ischaemia in order to study the role of astrocytes in grey matter during the first hour of a stroke. Under conditions of mild ischaemia, astrocytes are observed to take up glutamate via the glutamate transporter, and potassium via the Na/K/Cl cotransporter, which limits glutamate and potassium increase in the extracellular space. On the contrary, under conditions of severe ischaemia, astrocytes appear to be unable to maintain potassium homeostasis. Moreover, they are shown to contribute to the excitotoxicity process by expelling glutamate out of the cells via the reversed glutamate transporter. A detailed understanding of astrocytic function and influence on neuron survival during stroke is necessary to improve the neuroprotective strategies for stroke patients.

A role for glutamate and glia in the fast network oscillations preceding spreading depression

The mechanism of the propagation of spreading depression is unclear. Classical theories proposed a self-maintained cycle fed by elevated potassium and/or glutamate in the extracellular space. Earlier we found in vivo a characteristic oscillatory field activity that is synchronous in a strip of tissue ahead of the oncoming wave of neuron depolarization and that occurs before the extracellular potassium level begins to rise [Herreras O, Largo C, Ibarz JM, Somjen GG, Marrín del Río R (1994) Role of neuronal synchronizing mechanisms in the propagation of spreading depression in the in vivo hippocampus. J Neurosci 14:7087-7098]. We investigated here the possible participation of glutamate and the role of glia in the prodromal field oscillations using extra and intracellular recordings and pharmacological manipulations in rat hippocampal slices. As earlier shown in vivo, field oscillations propagated ahead of the negative potential shift covering distances of up to 1 mm. The oscillatory prodromals were initially subthreshold but then each wave became crowned by a population spike. The frequency of the oscillatory prodromals was variable among slices (80-115 Hz), but constant in individual slices. The blockade of ionotropic glutamate receptors decreased the frequency of prodromal oscillations, retarded spreading depression propagation, and shortened the duration of depolarization. Blocking the glutamate membrane transport increased the oscillatory frequency. The selective metabolic poisoning of astrocytes led to gradual disorganization of prodromal oscillations whose frequency first increased and then decreased. Also, the amplitude of the population spikes within the burst diminished as individual cells fired fewer action potentials, although still phase-locked with population spikes. The effects of glial metabolic impairment were observed within the period when neuron electrical properties were still normal, and were blocked by glutamate receptor antagonists. These data suggest that glutamate released from glial cells and possibly also from neurons has a role in the generation of oscillations and neuron firing synchronization that precede the spreading depression-related depolarization, but additional mechanisms are required to fully explain the onset and propagation of spreading depression.

A mathematical model of ion movements in grey matter during a stroke

The development of cytotoxic oedema during a stroke consists in cell swelling and shrinking of the extracellular space. This phenomenon is triggered by ion movements through voltage-gated channels, exchangers and pumps. During ischaemia, sodium, calcium and chloride enter the neurons whereas potassium and glutamate are expelled out of the cells. A mathematical model is proposed to represent the long-term dynamics of membrane potentials, cell volumes and ionic concentrations in intracellular and extracellular spaces during a stroke and to study the influence of each ionic current on cell swelling. The model relies on electrophysiological mechanisms and takes into account the behaviour of two types of cells: neurons and also astrocytes known to play a key role in the excitotoxic process in grey matter. The results obtained when a severe or a moderate ischaemia is simulated are consistent with those observed in the in vitro and in vivo experiments. As this model appears to be robust, it is used to perform illustrative simulations aimed at studying the effect of some channel blockers on cell swelling. This approach may help to explore new therapeutic strategies in order to reduce stroke damage.

A computational perspective on migraine aura

The classical visual aura of migraine is characterized by a unilateral hallucination, composed of a zigzag fortification pattern followed by a trailing scotoma. This pattern usually starts in central vision, expands and spreads to the periphery, and then disappears. We review a number of historical attempts to explain the migraine aura in terms of brain events, then summarize recent theories of the pathophysiology of the aura. We describe an approach to the computational modeling of migraine aura, based on the principles of (a) cortical organization, and (b) active wave propagation in an excitable medium. We demonstrate correspondences between properties of the model system and aspects of the pathophysiology of the aura. The simulations produced by the model are in agreement with descriptions and drawings of visual aura from migraine patients. We outline several testable predictions stemming from the implementation of the model, and explain how model-based empirical research has the capacity to (a) improve recording of the phenomena of the visual aura, (b) improve understanding of the spatio-temporal dynamics of other types of aura, in particular somatosensory and dysphasic aurae, and (c) clarify the theoretical requirements for the initiation of aura in the brain.