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

Uncertainty quantification and sensitivity analysis of neuron models with ion concentration dynamics

This paper provides a comprehensive and computationally efficient case study for uncertainty quantification (UQ) and global sensitivity analysis (GSA) in a neuron model incorporating ion concentration dynamics. We address how challenges with UQ and GSA in this context can be approached and solved, including challenges related to computational cost, parameters affecting the system's resting state, and the presence of both fast and slow dynamics. Specifically, we analyze the electrodiffusive neuron-extracellular-glia (edNEG) model, which captures electrical potentials, ion concentrations (Na+, K+, Ca2+, and Cl-), and volume changes across six compartments. Our methodology includes a UQ procedure assessing the model's reliability and susceptibility to input uncertainty and a variance-based GSA identifying the most influential input parameters. To mitigate computational costs, we employ surrogate modeling techniques, optimized using efficient numerical integration methods. We propose a strategy for isolating parameters affecting the resting state and analyze the edNEG model dynamics under both physiological and pathological conditions. The influence of uncertain parameters on model outputs, particularly during spiking dynamics, is systematically explored. Rapid dynamics of membrane potentials necessitate a focus on informative spiking features, while slower variations in ion concentrations allow a meaningful study at each time point. Our study offers valuable guidelines for future UQ and GSA investigations on neuron models with ion concentration dynamics, contributing to the broader application of such models in computational neuroscience.

In silico study of the mechanisms of hypoxia and contractile dysfunction during ischemia and reperfusion of hiPSC cardiomyocytes

Interconnected mechanisms of ischemia and reperfusion (IR) has increased the interest in IR in vitro experiments using human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs). We developed a whole-cell computational model of hiPSC-CMs including the electromechanics, a metabolite-sensitive sarcoplasmic reticulum Ca2+-ATPase (SERCA) and an oxygen dynamics formulation to investigate IR mechanisms. Moreover, we simulated the effect and action mechanism of levosimendan, which recently showed promising anti-arrhythmic effects in hiPSC-CMs in hypoxia. The model was validated using hiPSC-CM and in vitro animal data. The role of SERCA in causing relaxation dysfunction in IR was anticipated to be comparable to its function in sepsis-induced heart failure. Drug simulations showed that levosimendan counteracts the relaxation dysfunction by utilizing a particular Ca2+-sensitizing mechanism involving Ca2+-bound troponin C and Ca2+ flux to the myofilament, rather than inhibiting SERCA phosphorylation. The model demonstrates extensive characterization and promise for drug development, making it suitable for evaluating IR therapy strategies based on the changing levels of cardiac metabolites, oxygen and molecular pathways.

Cellular communication among smooth muscle cells: The role of membrane potential via connexins

Communication via action potentials among neurons has been extensively studied. However, effective communication without action potentials is ubiquitous in biological systems, yet it has received much less attention in comparison. Multi-cellular communication among smooth muscles is crucial for regulating blood flow, for example. Understanding the mechanism of this non-action potential communication is critical in many cases, like synchronization of cellular activity, under normal and pathological conditions. In this paper, we employ a multi-scale asymptotic method to derive a macroscopic homogenized bidomain model from the microscopic electro-neutral (EN) model. This is achieved by considering different diffusion coefficients and incorporating nonlinear interface conditions. Subsequently, the homogenized macroscopic model is used to investigate communication in multi-cellular tissues. Our computational simulations reveal that the membrane potential of syncytia, formed by interconnected cells via connexins, plays a crucial role in propagating oscillations from one region to another, providing an effective means for fast cellular communication. Statement of Significance: In this study, we investigated cellular communication and ion transport in vascular smooth muscle cells, shedding light on their mechanisms under normal and abnormal conditions. Our research highlights the potential of mathematical models in understanding complex biological systems. We developed effective macroscale electro-neutral bi-domain ion transport models and examined their behavior in response to different stimuli. Our findings revealed the crucial role of connexinmediated membrane potential changes and demonstrated the effectiveness of cellular communication through syncytium membranes. Despite some limitations, our study provides valuable insights into these processes and emphasizes the importance of mathematical modeling in unraveling the complexities of cellular communication and ion transport.

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.

Quantifying the relationship between spreading depolarization and perivascular cerebrospinal fluid flow

Recent studies have linked spreading depolarization (SD, an electro-chemical wave in the brain following stroke, migraine, traumatic brain injury, and more) with increase in cerebrospinal fluid (CSF) flow through the perivascular spaces (PVSs, annular channels lining the brain vasculature). We develop a novel computational model that couples SD and CSF flow. We first use high order numerical simulations to solve a system of physiologically realistic reaction-diffusion equations which govern the spatiotemporal dynamics of ions in the extracellular and intracellular spaces of the brain cortex during SD. We then couple the SD wave with a 1D CSF flow model that captures the change in cross-sectional area, pressure, and volume flow rate through the PVSs. The coupling is modelled using an empirical relationship between the excess potassium ion concentration in the extracellular space following SD and the vessel radius. We find that the CSF volumetric flow rate depends intricately on the length and width of the PVS, as well as the vessel radius and the angle of incidence of the SD wave. We derive analytical expressions for pressure and volumetric flow rates of CSF through the PVS for a given SD wave and quantify CSF flow variations when two SD waves collide. Our numerical approach is very general and could be extended in the future to obtain novel, quantitative insights into how CSF flow in the brain couples with slow waves, functional hyperemia, seizures, or externally applied neural stimulations.

Inverse neurovascular coupling and associated spreading depolarization models for traumatic brain injury

The paper presents the mathematical model of cortical spreading depolarisation and its effect on inverse neurovascular coupling. The paper considers the potassium ion channels present in the neuron-astrocyte blood vascular network to access the role of potassium ions during spreading depolarisation and associated inverse neurovascular coupling. Simulation of our proposed mathematical model confirms the experimental results that an increase in concentration of potassium ions beyond 20mM in the perivascular space essentially leads to vasoconstriction and hence inverse neurovascular coupling. The propagatory nature of depolarizing potassium waves has been unraveled though our proposed mathematical model.

Grey-box modeling and hypothesis testing of functional near-infrared spectroscopy-based cerebrovascular reactivity to anodal high-definition tDCS in healthy humans

Transcranial direct current stimulation (tDCS) has been shown to evoke hemodynamics response; however, the mechanisms have not been investigated systematically using systems biology approaches. Our study presents a grey-box linear model that was developed from a physiologically detailed multi-compartmental neurovascular unit model consisting of the vascular smooth muscle, perivascular space, synaptic space, and astrocyte glial cell. Then, model linearization was performed on the physiologically detailed nonlinear model to find appropriate complexity (Akaike information criterion) to fit functional near-infrared spectroscopy (fNIRS) based measure of blood volume changes, called cerebrovascular reactivity (CVR), to high-definition (HD) tDCS. The grey-box linear model was applied on the fNIRS-based CVR during the first 150 seconds of anodal HD-tDCS in eleven healthy humans. The grey-box linear models for each of the four nested pathways starting from tDCS scalp current density that perturbed synaptic potassium released from active neurons for Pathway 1, astrocytic transmembrane current for Pathway 2, perivascular potassium concentration for Pathway 3, and voltage-gated ion channel current on the smooth muscle cell for Pathway 4 were fitted to the total hemoglobin concentration (tHb) changes from optodes in the vicinity of 4x1 HD-tDCS electrodes as well as on the contralateral sensorimotor cortex. We found that the tDCS perturbation Pathway 3 presented the least mean square error (MSE, median <2.5%) and the lowest Akaike information criterion (AIC, median -1.726) from the individual grey-box linear model fitting at the targeted-region. Then, minimal realization transfer function with reduced-order approximations of the grey-box model pathways was fitted to the ensemble average tHb time series. Again, Pathway 3 with nine poles and two zeros (all free parameters), provided the best Goodness of Fit of 0.0078 for Chi-Square difference test of nested pathways. Therefore, our study provided a systems biology approach to investigate the initial transient hemodynamic response to tDCS based on fNIRS tHb data. Future studies need to investigate the steady-state responses, including steady-state oscillations found to be driven by calcium dynamics, where transcranial alternating current stimulation may provide frequency-dependent physiological entrainment for system identification. We postulate that such a mechanistic understanding from system identification of the hemodynamics response to transcranial electrical stimulation can facilitate adequate delivery of the current density to the neurovascular tissue under simultaneous portable imaging in various cerebrovascular diseases.

A continuum PNP model of double species ion transport between graphene

In this paper, we employ Poisson-Nernst-Planck equations in conjunction with the Navier-Stokes equations and the mean-field theory to investigate the charge transport of double species monovalent ions in double-layered graphene sheets, driven by an external electric field. Unlike most classical models, we develop a simple mechanical model by incorporating the microscopic effects of the physical systems so that the ionic interactions between ions and the host material, and the steric repulsions among ions are considered. Taking Li⁺-PF6^-1monovalent ions as an example, we find that the transport pattern for the present double species ionic systems and that for the pure lithium ions are dramatically different. Due to the mutual attractive ionic forces between ions and their counter-ions, such double species systems turn out to be more stable than that of the single species systems. In addition, the storage patterns of the former systems are richer, which depend heavily on initial ionic states, however less on the strength of applied electric fields and external temperatures when the graphene separation is not too large. The fluid flows, the electric conductivities and the stability of such double species systems are subsequently scrutinized. The current study could form a theoretical basis to help designing high-performance lithium batteries and explaining ion transport in biological channels.

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.

A tridomain model for potassium clearance in optic nerve of Necturus

Complex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system are important in a wide range of biological and medical processes. However, the exact driving mechanisms are often not known. In this work, we investigate flows induced by action potentials in an optic nerve as a prototype of the central nervous system. Different from traditional fluid dynamics problems, flows in biological tissues such as the central nervous system are coupled with ion transport. They are driven by osmosis created by concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group Orkand et al. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.

A mathematical model for persistent post-CSD vasoconstriction

Cortical spreading depression (CSD) is the propagation of a relatively slow wave in cortical brain tissue that is linked to a number of pathological conditions such as stroke and migraine. Most of the existing literature investigates the dynamics of short term phenomena such as the depolarization and repolarization of membrane potentials or large ion shifts. Here, we focus on the clinically-relevant hour-long state of neurovascular malfunction in the wake of CSDs. This dysfunctional state involves widespread vasoconstriction and a general disruption of neurovascular coupling. We demonstrate, using a mathematical model, that dissolution of calcium that has aggregated within the mitochondria of vascular smooth muscle cells can drive an hour-long disruption. We model the rate of calcium clearance as well as the dynamical implications on overall blood flow. Based on reaction stoichiometry, we quantify a possible impact of calcium phosphate dissolution on the maintenance of F0F1-ATP synthase activity.

An electrodiffusive, ion conserving Pinsky-Rinzel model with homeostatic mechanisms

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⁺, Ca²⁺, 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.

Neurons 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 accounts for ion concentration variations in a way that ensures complete and consistent ion concentration and charge conservation. In this work, we 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 value for the field of neuroscience, as it can be used to simulate a range of pathological conditions, such as spreading depression or epilepsy, which are associated with large changes in extracellular ion concentrations.

Altered Macular Vasculature in Migraine Patients without Aura: Is It Associated with Ocular Vasculature and White Matter Hyperintensities?

Aim

We aimed to determine the alterations in macular and optic nerve vasculature in patients with migraine without aura using optical coherence tomography-angiography (OCTA). We also aimed to determine whether there were clinical differences and alterations in ocular structures in migraine cases with white matter hyperintensities (WMH) using magnetic resonance imaging (MRI). Materials and Methods. The study group comprised patients with migraine without aura and age- and sex-matched healthy controls. Detailed histories of the patients with migraine were recorded including the disease duration, number of attacks in the last month, and attack durations. Visual evoked potentials (VEP) were recorded in all migraine patients. The migraine disability assessment (MIDAS) questionnaire was administered to all patients. The patients were divided into two groups as migraine with WMHs and migraine without WMHs. All subjects underwent a complete neurological and ophthalmological examination. Only the right eyes of the patients were included in the study. Retinal imaging was performed using OCT and OCTA.

Results

A total of 66 migraine patients (29 with WMH and 37 without WMH) and 43 healthy controls were included in this study. Among the migraine patients, disease duration, attack frequency in the last month, attack durations, and the visual analogue scale (VAS), MIDAS, and VEP scores were all similar between those with and without WMHs. There was no significant difference between the groups regarding the ganglion cell complex, foveal, and retinal nerve fiber layer thicknesses. The superficial or deep vascular perfusion densities of the optic disc were also similar between the groups. The foveal avascular zone (FAZ) was significantly larger (P=0.034), and both superficial and deep macular vascular densities were significantly lower in the migraine groups compared with the healthy controls (P=0.001). There was no significant difference concerning the FAZ size or vascular densities between the migraine groups with and without WMHs. In the correlation analysis performed between the migraine patients, the FAZ size was correlated with age and VAS and MIDAS scores while both superficial and deep macular vascular densities were negatively correlated with age and VAS and MIDAS scores.

Conclusion

We suggest that for not only migraine with aura but also migraine without aura, neurovascular structures play an important role in pathogenesis, and novel studies are warranted to elucidate the alterations in these and determine the significance of WMHs in these patient groups.

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.

Metabolic basis of brain-like electrical signalling in bacterial communities

Information processing in the mammalian brain relies on a careful regulation of the membrane potential dynamics of its constituent neurons, which propagates across the neuronal tissue via electrical signalling. We recently reported the existence of electrical signalling in a much simpler organism, the bacterium Bacillus subtilis. In dense bacterial communities known as biofilms, nutrient-deprived B. subtilis cells in the interior of the colony use electrical communication to transmit stress signals to the periphery, which interfere with the growth of peripheral cells and reduce nutrient consumption, thereby relieving stress from the interior. Here, we explicitly address the interplay between metabolism and electrophysiology in bacterial biofilms, by introducing a spatially extended mathematical model that combines the metabolic and electrical components of the phenomenon in a discretized reaction-diffusion scheme. The model is experimentally validated by environmental and genetic perturbations, and confirms that metabolic stress is transmitted through the bacterial population via a potassium wave. Interestingly, this behaviour is reminiscent of cortical spreading depression in the brain, characterized by a wave of electrical activity mediated by potassium diffusion that has been linked to various neurological disorders, calling for future studies on the evolutionary link between the two phenomena. This article is part of the theme issue 'Liquid brains, solid brains: How distributed cognitive architectures process information'.

Global Sensitivity Analysis of High-Dimensional Neuroscience Models: An Example of Neurovascular Coupling

The complexity and size of state-of-the-art cell models have significantly increased in part due to the requirement that these models possess complex cellular functions which are thought-but not necessarily proven-to be important. Modern cell models often involve hundreds of parameters; the values of these parameters come, more often than not, from animal experiments whose relationship to the human physiology is weak with very little information on the errors in these measurements. The concomitant uncertainties in parameter values result in uncertainties in the model outputs or quantities of interest (QoIs). Global sensitivity analysis (GSA) aims at apportioning to individual parameters (or sets of parameters) their relative contribution to output uncertainty thereby introducing a measure of influence or importance of said parameters. New GSA approaches are required to deal with increased model size and complexity; a three-stage methodology consisting of screening (dimension reduction), surrogate modeling, and computing Sobol' indices, is presented. The methodology is used to analyze a physiologically validated numerical model of neurovascular coupling which possess 160 uncertain parameters. The sensitivity analysis investigates three quantities of interest, the average value of [Formula: see text] in the extracellular space, the average volumetric flow rate through the perfusing vessel, and the minimum value of the actin/myosin complex in the smooth muscle cell. GSA provides a measure of the influence of each parameter, for each of the three QoIs, giving insight into areas of possible physiological dysfunction and areas of further investigation.

A Bidomain Model for Lens Microcirculation

There exists a large body of research on the lens of the mammalian eye over the past several decades. The objective of this work is to provide a link between the most recent computational models and some of the pioneering work in the 1970s and 80s. We introduce a general nonelectroneutral model to study the microcirculation in the lens of the eye. It describes the steady-state relationships among ion fluxes, between water flow and electric field inside cells, and in the narrow extracellular spaces between cells in the lens. Using asymptotic analysis, we derive a simplified model based on physiological data and compare our results with those in the literature. We show that our simplified model can be reduced further to the first-generation models, whereas our full model is consistent with the most recent computational models. In addition, our simplified model captures in its equations the main features of the full computational models. Our results serve as a useful link intermediate between the computational models and the first-generation analytical models. Simplified models of this sort may be particularly helpful as the roles of similar osmotic pumps of microcirculation are examined in other tissues with narrow extracellular spaces, such as cardiac and skeletal muscle, liver, kidney, epithelia in general, and the narrow extracellular spaces of the central nervous system, the "brain." Simplified models may reveal the general functional plan of these systems before full computational models become feasible and specific.

Inadequate brain glycogen or sleep increases spreading depression susceptibility

OBJECTIVE

Glycogen in astrocyte processes contributes to maintenance of low extracellular glutamate and K⁺ concentrations around excitatory synapses. Sleep deprivation (SD), a common migraine trigger, induces transcriptional changes in astrocytes, reducing glycogen breakdown. We hypothesize that when glycogen utilization cannot match synaptic energy demand, extracellular K⁺ can rise to levels that activate neuronal pannexin-1 channels and downstream inflammatory pathway, which might be one of the mechanisms initiating migraine headaches.

METHODS

We suppressed glycogen breakdown by inhibiting glycogen phosphorylation with 1,4-dideoxy-1,4-imino-D-arabinitol (DAB) and by SD.

RESULTS

DAB caused neuronal pannexin-1 large pore opening and activation of the downstream inflammatory pathway as shown by procaspase-1 cleavage and HMGB1 release from neurons. Six-hour SD induced pannexin-1 mRNA. DAB and SD also lowered the cortical spreading depression (CSD) induction threshold, which was reversed by glucose or lactate supplement, suggesting that glycogen-derived energy substrates are needed to prevent CSD generation. Supporting this, knocking down the neuronal lactate transporter MCT2 with an antisense oligonucleotide or inhibiting glucose transport from vessels to astrocytes with intracerebroventricularly delivered phloretin reduced the CSD threshold. In vivo recordings with a K⁺ -sensitive/selective fluoroprobe, Asante Potassium Green-4, revealed that DAB treatment or SD caused a significant rise in extracellular K⁺ during whisker stimulation, illustrating the critical role of glycogen in extracellular K⁺ clearance.

INTERPRETATION

Synaptic metabolic stress caused by insufficient glycogen-derived energy substrate supply can activate neuronal pannexin-1 channels as well as lower the CSD threshold. Therefore, conditions that limit energy supply to synapses (eg, SD) may predispose to migraine attacks, as suggested by genetic studies associating glucose or lactate transporter deficiency with migraine. Ann Neurol 2018;83:61-73.

Eye movements and imaging in vestibular migraine

BACKGROUND AND OBJECTIVE

Migrainous dizziness is one of the most frequent complaints. Dizziness associated with migraine may be the result of abnormal eye movements. Brain imaging and changes in eye movements may explain the dizziness and highlight possible pathophysiological substrates in migraine dizziness. Our aim is to evaluate eye movement using videonystagmography (VNG) and video head impulse test (vHIT) and to study the occipital lobe metabolic profile in vestibular migraine patients (VM).

MATERIALS AND METHODS

There were 2 groups enrolled in the study; the first group consisted of 25 vestibular migraine patients (VM) according to the recent criteria of Barany society. The second group consisted of 20 age matched healthy subjects. Both groups underwent the following: (1) A detailed history, VNG test protocol, vHIT in three planes. (2) Magnetic resonance imaging (MRI) for the brain and inner ear using 1.5T magnet and proton magnetic resonance spectroscopy (H1-MRS).

RESULTS

Sixty eight percent of the patients complained of spontaneous vertigo and 28% complained of positional vertigo. Non-paroxysmal positional nystagmus was recorded in 92% during their dizzy spell. The brain MRI was unremarkable in 72% of the cases. Chemical shift in the occipital lobe was found in 92% of VM. Lactate peaks were statistically significant related with the presence of non-paroxysmal positional nystagmus.

CONCLUSIONS

A statistically significant relationship exists between non-paroxysmal positional nystagmus and presence of lactate peaks in the occipital lobe in VM patients.

Treatment of acute migraine by a partial rebreathing device: A randomized controlled pilot study

Background Impaired brain oxygen delivery can trigger and exacerbate migraine attacks. Normoxic hypercapnia increases brain oxygen delivery markedly by vasodilation of the cerebral vasculature, and hypercapnia has been shown to abort migraine attacks. Stable normoxic hypercapnia can be induced by a compact partial rebreathing device. This pilot study aimed to provide initial data on the device's efficacy and safety. Methods Using a double-blinded, randomized, cross-over study design, adult migraine-with-aura patients self-administered the partial rebreathing device or a sham device for 20 minutes at the onset of aura symptoms. Results Eleven participants (mean age 35.5, three men) self-treated 41 migraine attacks (20 with the partial rebreathing device, 21 with sham). The partial rebreathing device increased mean End Tidal CO₂ by 24%, while retaining mean oxygen saturation above 97%. The primary end point (headache intensity difference between first aura symptoms and two hours after treatment (0-3 scale) - active/sham difference) did not reach statistical significance (-0.55 (95% CI: -1.13-0.04), p = 0.096), whereas the difference in percentage of attacks with pain relief at two hours was significant ( p = 0.043), as was user satisfaction ( p = 0.022). A marked efficacy increase was seen from first to second time use of the partial rebreathing device. No adverse events occurred, and side effects were absent or mild. Conclusion Normoxic hypercapnia shows promise as an adjunctive/alternative migraine treatment, meriting further investigation in a larger population. Clinical study registered at ClinicalTrials.gov with identifier NCT03472417.

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.

Evaluation of retinal nerve fiber layer, ganglion cell layer and macular changes in patients with migraine

The aim of this study was to investigate retinal nerve fiber layer (RNFL), ganglion cell layer (GCL) thickness, macular changes (central subfield thickness (CST), cube average thickness (CAT), cube volume (CV) in patients with migraine using spectral-domain optical coherence tomography (OCT) and to assess if there was any correlation with white matter lesions (WML). In this prospective case-control study, RNFL, GCL thickness and macular changes of 19 migraine patients with aura (MA), 41 migraine without aura (MO) and 60 age- and gender-matched healthy subjects were measured using OCT device. OCT measurements were taken at the same time of the day to minimize the effects of diurnal variation. The average, inferior and superior quadrant RNFL thickness were significantly thinner in patients with migraine (p = 0.017, p = 0.010, p = 0.048). There was also a significant difference between patients with and without aura in the mean and superior quadrant RNFL thickness (p = 0.02, p = 0.043).While there was a significant thinning in CST and CAT in patients with migraine (p = 0.020), there were no significant difference in GCL measurements (p = 0.184). When the groups were compared to the control group, there were significant differences between MA and the control group regarding average, superior and inferior quadrant RNLF thickness (p < 0.001, p = 0.025, p < 0.001). On the other hand, there were significant differences between MO and the control group regarding average and inferior faces (p = 0.037, p = 0.04). When OCT measurements were evaluated according to the frequency of attacks, CST and GCL thickness were significantly thinner in patients who had more than four attacks a month (p = 0.024, p = 0.014). In patients with WML, only CV measurements were significantly thinner than migraine patients without WML (p = 0.014). The decreased RNFL, CST, CAT and CV of the migraine patients might be related to the vascular pathology of the disease. Because WML was not correlated with the same measurements except CV, we think that further studies are needed to evaluate the etiopathologic relationship between OCT measurements and WML in migraine patients.

A model of neurovascular coupling and the BOLD response: PART I

The mechanisms with which neurons communicate with the vasculature to increase blood flow, termed neurovascular coupling is still unclear primarily due to the complex interactions between many parameters and the difficulty in accessing, monitoring and measuring them in the highly heterogeneous brain. Hence a solid theoretical framework based on existing experimental knowledge is necessary to study the relation between neural activity, the associated vasoactive factors released and their effects on the vasculature. Such a framework should also be related to experimental data so that it can be validated against repetitive experiments and generate verifiable hypothesis. We have developed a mathematical model which describes a signaling mechanism of neurovascular coupling with a model of pyramidal neuron and its corresponding fMRI BOLD response. In the first part of two papers we describe the integration of the neurovascular coupling unit extended to include a complex neuron model, which includes the important Na/K ATPase pump, with a model that provides a BOLD signal taking its input from the cerebral blood flow and the metabolic rate of oxygen consumption. We show that this produces a viable signal in terms of initial dip, positive and negative BOLD signals.

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.

ASICs as therapeutic targets for migraine

Migraine is the most common neurological disorder and one of the most common chronic pain conditions. Despite its prevalence, the pathophysiology leading to migraine is poorly understood and the identification of new therapeutic targets has been slow. Several processes are currently thought to contribute to migraine including altered activity in the hypothalamus, cortical-spreading depression (CSD), and afferent sensory input from the cranial meninges. Decreased extracellular pH and subsequent activation of acid-sensing ion channels (ASICs) may contribute to each of these processes and may thus play a role in migraine pathophysiology. Although few studies have directly examined a role of ASICs in migraine, studies directly examining a connection have generated promising results including efficacy of ASIC blockers in both preclinical migraine models and in human migraine patients. The purpose of this review is to discuss the pathophysiology thought to contribute to migraine and findings that implicate decreased pH and/or ASICs in these events, as well as propose issues to be resolved in future studies of ASICs and migraine. This article is part of the Special Issue entitled 'Acid-Sensing Ion Channels in the Nervous System'.

Microvascular blood flow monitoring with laser speckle contrast imaging using the generalized differences algorithm

Laser speckle contrast imaging (LSCI) is a full-field optical technique to monitor microvascular blood flow with high spatial and temporal resolutions. It is used in many medical fields such as dermatology, vascular medicine, or neurosciences. However, LSCI leads to a large amount of data: image sampling frequency is often of several Hz and recordings usually last several minutes. Therefore, clinicians often perform regions of interest in which a spatial averaging of blood flow is performed and the result is followed with time. Unfortunately, this leads to a poor spatial resolution for the analyzed data. At the same time, a higher spatial resolution for the perfusion maps is wanted. To get over this dilemma we propose a new post-acquisition visual representation for LSCI perfusion data using the so-called generalized differences (GD) algorithm. From a stack of perfusion images, the procedure leads to a new single image with the same spatial resolution as the original images and this new image reflects perfusion changes. The algorithm is herein applied on simulated stacks of images and on experimental LSCI perfusion data acquired in three different situations with a commercialized laser speckle contrast imager. The results show that the GD algorithm provides a new way of visualizing LSCI perfusion data.

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.

Neurovascular coupling and the influence of luminal agonists via the endothelium

A numerical model of neurovascular coupling (NVC) is presented based on neuronal activity coupled to vasodilation/contraction models via the astrocytic mediated perivascular K(+) and the smooth muscle cell Ca(2+) pathway. Luminal agonists acting on P2Y receptors on the endothelial cell surface provide a flux of IP3 into the endothelial cytosol. This concentration of IP3 is transported via gap junctions between endothelial and smooth muscle cells providing a source of sacroplasmic derived Ca(2+) in the smooth muscle cell. The model is able to relate a neuronal input signal to the corresponding vessel reaction. Results indicate that blood flow mediated IP3 production via the agonist ATP has a substantial effect on the contraction/dilation dynamics of the SMC. The resulting variation in cytosolic Ca(2+) can enhance and inhibit the flow of blood to the cortical tissue. IP3 coupling between endothelial and smooth muscle cells seems to be important in the dynamics of the smooth muscle cell. The VOCC channels are, due to the hyperpolarisation from K(+) SMC efflux, almost entirely closed and do not seem to play a significant role during neuronal activity. The current model shows that astrocytic Ca(2+) is not necessary for neurovascular coupling to occur in contrast to a number of experiments outlining the importance of astrocytic Ca(2+) in NVC, however this Ca(2+) pathway is not the only one mediating NVC. Importantly agonists in flowing blood have a significant influence on the endothelial and smooth muscle cell dynamics.

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.