The role of extracellular potassium dynamics in the different stages of ictal bursting and spreading depression: a computational study

An electrodiffusive network model with multicompartmental neurons and synaptic connections

Most computational models of neurons assume constant ion concentrations, disregarding the effects of changing ion concentrations on neuronal activity. Among the models that do incorporate ion concentration dynamics, simplifications are often made that sacrifice biophysical consistency, such as neglecting the effects of ionic diffusion on electrical potentials or the effects of electric drift on ion concentrations. A subset of models with ion concentration dynamics, often referred to as electrodiffusive models, account for ion concentration dynamics in a way that ensures a biophysical consistent relationship between ion concentrations, electric charge, and electrical potentials. These models include compartmental single-cell models, geometrically explicit models, and domain-type models, but none that model neuronal network dynamics. To address this gap, we present an electrodiffusive network model with multicompartmental neurons and synaptic connections, which we believe is the first compartmentalized network model to account for intra- and extracellular ion concentration dynamics in a biophysically consistent way. The model comprises an arbitrary number of "units," each divided into three domains representing a neuron, glia, and extracellular space. Each domain is further subdivided into a somatic and dendritic layer. Unlike conventional models which focus primarily on neuronal spiking patterns, our model predicts intra- and extracellular ion concentrations (Na+, K+, Cl-, and Ca2+), electrical potentials, and volume fractions. A unique feature of the model is that it captures ephaptic effects, both electric and ionic. In this paper, we show how this leads to interesting behavior in the network. First, we demonstrate how changing ion concentrations can affect the synaptic strengths. Then, we show how ionic ephaptic coupling can lead to spontaneous firing in neurons that do not receive any synaptic or external input. Lastly, we explore the effects of having glia in the network and demonstrate how a strongly coupled glial syncytium can prevent neuronal depolarization blocks.

Computational neurosciences and quantitative systems pharmacology: a powerful combination for supporting drug development in neurodegenerative diseases

Successful clinical development of new therapeutic interventions is notoriously difficult, especially in neurodegenerative diseases, where predictive biomarkers are scarce and functional improvement is often based on patient's perception, captured by structured interviews. As a consequence, mechanistic modeling of the processes relevant to therapeutic interventions in CNS disorders has been lagging behind other disease indications, probably because of the perceived complexity of the brain. However in this report, we develop the argument that a combination of Computational Neurosciences and Quantitative Systems Pharmacology (QSP) modeling of molecular pathways is a powerful simulation tool to enhance the probability of successful drug development for neurodegenerative diseases. Computational Neurosciences aims to predict action potential dynamics and neuronal circuit activation that are ultimately linked to behavioral changes and clinically relevant functional outcomes. These processes can not only be affected by the disease state, but also by common genotype variants on neurotransmitter-related proteins and the psycho-active medications often prescribed in these patient populations. Quantitative Systems Pharmacology (QSP) modeling of molecular pathways allows to simulate key pathological drivers of dementia, such as protein aggregation and neuroinflammatory responses. They often impact neurotransmitter homeostasis and voltage-gated ion-channels or lead to mitochondrial dysfunction, ultimately leading to changes in action potential dynamics and clinical readouts. Combining these two modeling approaches can lead to better actionable understanding of the many non-linear pharmacodynamic processes active in the human diseased brain. Practical applications include a rational selection of the optimal doses in combination therapies, identification of subjects more likely to respond to treatment, a more balanced stratification of treatment arms in terms of comedications, disease status and common genotype variants and re-analysis of small clinical trials to uncover a possible clinical signal. Ultimately this will lead to a higher success rate of bringing new therapeutics to the right patient populations.

The Na+/K+-ATPase generically enables deterministic bursting in class I neurons by shearing the spike-onset bifurcation structure

Slow brain rhythms, for example during slow-wave sleep or pathological conditions like seizures and spreading depolarization, can be accompanied by oscillations in extracellular potassium concentration. Such slow brain rhythms typically have a lower frequency than tonic action-potential firing. They are assumed to arise from network-level mechanisms, involving synaptic interactions and delays, or from intrinsically bursting neurons. Neuronal burst generation is commonly attributed to ion channels with slow kinetics. Here, we explore an alternative mechanism generically available to all neurons with class I excitability. It is based on the interplay of fast-spiking voltage dynamics with a one-dimensional slow dynamics of the extracellular potassium concentration, mediated by the activity of the Na+/K+-ATPase. We use bifurcation analysis of the complete system as well as the slow-fast method to reveal that this coupling suffices to generate a hysteresis loop organized around a bistable region that emerges from a saddle-node loop bifurcation-a common feature of class I excitable neurons. Depending on the strength of the Na+/K+-ATPase, bursts are generated from pump-induced shearing the bifurcation structure, spiking is tonic, or cells are silenced via depolarization block. We suggest that transitions between these dynamics can result from disturbances in extracellular potassium regulation, such as glial malfunction or hypoxia affecting the Na+/K+-ATPase activity. The identified minimal mechanistic model outlining the sodium-potassium pump's generic contribution to burst dynamics can, therefore, contribute to a better mechanistic understanding of pathologies such as epilepsy syndromes and, potentially, inform therapeutic strategies.

Inhibition of astroglial hemichannels prevents synaptic transmission decline during spreading depression

BACKGROUND

Spreading depression (SD) is an intriguing phenomenon characterized by massive slow brain depolarizations that affect neurons and glial cells. This phenomenon is repetitive and produces a metabolic overload that increases secondary damage. However, the mechanisms associated with the initiation and propagation of SD are unknown. Multiple lines of evidence indicate that persistent and uncontrolled opening of hemichannels could participate in the pathogenesis and progression of several neurological disorders including acute brain injuries. Here, we explored the contribution of astroglial hemichannels composed of connexin-43 (Cx43) or pannexin-1 (Panx1) to SD evoked by high-K+ stimulation in brain slices.

RESULTS

Focal high-K+ stimulation rapidly evoked a wave of SD linked to increased activity of the Cx43 and Panx1 hemichannels in the brain cortex, as measured by light transmittance and dye uptake analysis, respectively. The activation of these channels occurs mainly in astrocytes but also in neurons. More importantly, the inhibition of both the Cx43 and Panx1 hemichannels completely prevented high K+-induced SD in the brain cortex. Electrophysiological recordings also revealed that Cx43 and Panx1 hemichannels critically contribute to the SD-induced decrease in synaptic transmission in the brain cortex and hippocampus.

CONCLUSIONS

Targeting Cx43 and Panx1 hemichannels could serve as a new therapeutic strategy to prevent the initiation and propagation of SD in several acute brain injuries.

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.

Temporal dynamics of Na/K pump mediated memory traces: insights from conductance-based models of Drosophila neurons

Sodium potassium ATPases (Na/K pumps) mediate long-lasting, dynamic cellular memories that can last tens of seconds. The mechanisms controlling the dynamics of this type of cellular memory are not well understood and can be counterintuitive. Here, we use computational modeling to examine how Na/K pumps and the ion concentration dynamics they influence shape cellular excitability. In a Drosophila larval motor neuron model, we incorporate a Na/K pump, a dynamic intracellular Na⁺ concentration, and a dynamic Na⁺ reversal potential. We probe neuronal excitability with a variety of stimuli, including step currents, ramp currents, and zap currents, then monitor the sub- and suprathreshold voltage responses on a range of time scales. We find that the interactions of a Na⁺-dependent pump current with a dynamic Na⁺ concentration and reversal potential endow the neuron with rich response properties that are absent when the role of the pump is reduced to the maintenance of constant ion concentration gradients. In particular, these dynamic pump-Na⁺ interactions contribute to spike rate adaptation and result in long-lasting excitability changes after spiking and even after sub-threshold voltage fluctuations on multiple time scales. We further show that modulation of pump properties can profoundly alter a neuron's spontaneous activity and response to stimuli by providing a mechanism for bursting oscillations. Our work has implications for experimental studies and computational modeling of the role of Na/K pumps in neuronal activity, information processing in neural circuits, and the neural control of animal behavior.

Slow ion concentration oscillations and multiple states in neuron-glia interaction-insights gained from reduced mathematical models

When potassium in the extracellular space separating neurons and glia reaches sufficient levels, neurons may fire spontaneous action potentials or even become inactivated due to membrane depolarisation, which, in turn, may lead to increased extracellular potassium levels. Under certain circumstances, this chain of events may trigger periodic bursts of neuronal activity. In the present study, reduced neuron-glia models are applied to explore the relationship between bursting behaviour and ion concentration dynamics. These reduced models are built based on a previously developed neuron-glia model, in which channel-mediated neuronal sodium and potassium currents are replaced by a function of neuronal sodium and extracellular potassium concentrations. Simulated dynamics of the resulting two reduced models display features that are qualitatively similar to those of the existing neuron-glia model. Bifurcation analyses of the reduced models show rich and interesting dynamics that include the existence of Hopf bifurcations between which the models exhibit slow ion concentration oscillations for a wide range of parameter values. The study demonstrates that even very simple models can provide insights of possible relevance to complex phenomena.

Microscale Neuronal Activity Collectively Drives Chaotic and Inflexible Dynamics at the Macroscale in Seizures

Neuronal activity propagates through the network during seizures, engaging brain dynamics at multiple scales. Such propagating events can be described through the avalanches framework, which can relate spatiotemporal activity at the microscale with global network properties. Interestingly, propagating avalanches in healthy networks are indicative of critical dynamics, where the network is organized to a phase transition, which optimizes certain computational properties. Some have hypothesized that the pathologic brain dynamics of epileptic seizures are an emergent property of microscale neuronal networks collectively driving the brain away from criticality. Demonstrating this would provide a unifying mechanism linking microscale spatiotemporal activity with emergent brain dysfunction during seizures. Here, we investigated the effect of drug-induced seizures on critical avalanche dynamics, using in vivo whole-brain two-photon imaging of GCaMP6s larval zebrafish (males and females) at single neuron resolution. We demonstrate that single neuron activity across the whole brain exhibits a loss of critical statistics during seizures, suggesting that microscale activity collectively drives macroscale dynamics away from criticality. We also construct spiking network models at the scale of the larval zebrafish brain, to demonstrate that only densely connected networks can drive brain-wide seizure dynamics away from criticality. Importantly, such dense networks also disrupt the optimal computational capacities of critical networks, leading to chaotic dynamics, impaired network response properties and sticky states, thus helping to explain functional impairments during seizures. This study bridges the gap between microscale neuronal activity and emergent macroscale dynamics and cognitive dysfunction during seizures.SIGNIFICANCE STATEMENT Epileptic seizures are debilitating and impair normal brain function. It is unclear how the coordinated behavior of neurons collectively impairs brain function during seizures. To investigate this we perform fluorescence microscopy in larval zebrafish, which allows for the recording of whole-brain activity at single-neuron resolution. Using techniques from physics, we show that neuronal activity during seizures drives the brain away from criticality, a regime that enables both high and low activity states, into an inflexible regime that drives high activity states. Importantly, this change is caused by more connections in the network, which we show disrupts the ability of the brain to respond appropriately to its environment. Therefore, we identify key neuronal network mechanisms driving seizures and concurrent cognitive dysfunction.

From phenomenological to biophysical models of seizures

Epilepsy is a complex disease that requires various approaches for its study. In this short review, we discuss the contribution of theoretical and computational models. The review presents theoretical frameworks that underlie the understanding of certain seizure properties and their classification based on their dynamical properties at the onset and offset of seizures. Dynamical system tools are valuable resources in the study of seizures. By analyzing the complex, dynamic behavior of seizures, these tools can provide insights into seizure mechanisms and offer a framework for their classification. Additionally, computational models have high potential for clinical applications, as they can be used to develop more accurate diagnostic and personalized medicine tools. We discuss various modeling approaches that span different scales and levels, while also questioning the neurocentric view, and emphasize the importance of considering glial cells. Finally, we explore the epistemic value provided by this type of approach.

The role of selective SATB1 deletion in somatostatin expressing interneurons on endogenous network activity and the transition to epilepsy

Somatostatin (SST) expressing interneurons are the second most abundant group of inhibitory neurons in the neocortex. They mainly target the apical dendrites of excitatory pyramidal cells and are implicated in feedforward and feedback inhibition. In the present study, we employ a conditional knockout mouse, in which the transcription factor Satb1 is selectively deleted in SST-expressing interneurons resulting to the reduction of their number across the somatosensory barrel field. Our goal was to investigate the effect of the reduced number of Satb1 mutant SST-interneurons on (i) the endogenous cortical network activity (spontaneously recurring Up/Down states), and (ii) the transition to epileptiform activity. By conducting LFP recordings in acute brain slices from young male and female mice, we demonstrate that mutant animals exhibit significant changes in network excitability, reflected in increased Up state occurrence, decreased Up state duration and higher levels of extracellular spiking activity. Epileptiform activity was induced through two distinct and widely used in vitro protocols: the low magnesium and the 4-Aminopyridine (4-AP) model. In the former, slices from mutant animals manifested shorter latency for the expression of stable seizure-like events. In contrast, when epilepsy was induced by 4-AP, no significant differences were reported. We conclude that normal SST-interneuron function has a significant role both in the regulation of the endogenous network activity, and in the transition to seizure-like discharges in a context-dependent manner.

Effects of central nervous system electrical stimulation on non-neuronal cells

Over the past few decades, much progress has been made in the clinical use of electrical stimulation of the central nervous system (CNS) to treat an ever-growing number of conditions from Parkinson's disease (PD) to epilepsy as well as for sensory restoration and many other applications. However, little is known about the effects of microstimulation at the cellular level. Most of the existing research focuses on the effects of electrical stimulation on neurons. Other cells of the CNS such as microglia, astrocytes, oligodendrocytes, and vascular endothelial cells have been understudied in terms of their response to stimulation. The varied and critical functions of these cell types are now beginning to be better understood, and their vital roles in brain function in both health and disease are becoming better appreciated. To shed light on the importance of the way electrical stimulation as distinct from device implantation impacts non-neuronal cell types, this review will first summarize common stimulation modalities from the perspective of device design and stimulation parameters and how these different parameters have an impact on the physiological response. Following this, what is known about the responses of different cell types to different stimulation modalities will be summarized, drawing on findings from both clinical studies as well as clinically relevant animal models and in vitro systems.

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.

A unified physiological framework of transitions between seizures, sustained ictal activity and depolarization block at the single neuron level

The majority of seizures recorded in humans and experimental animal models can be described by a generic phenomenological mathematical model, the Epileptor. In this model, seizure-like events (SLEs) are driven by a slow variable and occur via saddle node (SN) and homoclinic bifurcations at seizure onset and offset, respectively. Here we investigated SLEs at the single cell level using a biophysically relevant neuron model including a slow/fast system of four equations. The two equations for the slow subsystem describe ion concentration variations and the two equations of the fast subsystem delineate the electrophysiological activities of the neuron. Using extracellular K⁺ as a slow variable, we report that SLEs with SN/homoclinic bifurcations can readily occur at the single cell level when extracellular K⁺ reaches a critical value. In patients and experimental models, seizures can also evolve into sustained ictal activity (SIA) and depolarization block (DB), activities which are also parts of the dynamic repertoire of the Epileptor. Increasing extracellular concentration of K⁺ in the model to values found during experimental status epilepticus and DB, we show that SIA and DB can also occur at the single cell level. Thus, seizures, SIA, and DB, which have been first identified as network events, can exist in a unified framework of a biophysical model at the single neuron level and exhibit similar dynamics as observed in the Epileptor.Author Summary: Epilepsy is a neurological disorder characterized by the occurrence of seizures. Seizures have been characterized in patients in experimental models at both macroscopic and microscopic scales using electrophysiological recordings. Experimental works allowed the establishment of a detailed taxonomy of seizures, which can be described by mathematical models. We can distinguish two main types of models. Phenomenological (generic) models have few parameters and variables and permit detailed dynamical studies often capturing a majority of activities observed in experimental conditions. But they also have abstract parameters, making biological interpretation difficult. Biophysical models, on the other hand, use a large number of variables and parameters due to the complexity of the biological systems they represent. Because of the multiplicity of solutions, it is difficult to extract general dynamical rules. In the present work, we integrate both approaches and reduce a detailed biophysical model to sufficiently low-dimensional equations, and thus maintaining the advantages of a generic model. We propose, at the single cell level, a unified framework of different pathological activities that are seizures, depolarization block, and sustained ictal activity.

Modeling NaV1.1/SCN1A sodium channel mutations in a microcircuit with realistic ion concentration dynamics suggests differential GABAergic mechanisms leading to hyperexcitability in epilepsy and hemiplegic migraine

Loss of function mutations of SCN1A, the gene coding for the voltage-gated sodium channel Na_V1.1, cause different types of epilepsy, whereas gain of function mutations cause sporadic and familial hemiplegic migraine type 3 (FHM-3). However, it is not clear yet how these opposite effects can induce paroxysmal pathological activities involving neuronal networks’ hyperexcitability that are specific of epilepsy (seizures) or migraine (cortical spreading depolarization, CSD). To better understand differential mechanisms leading to the initiation of these pathological activities, we used a two-neuron conductance-based model of interconnected GABAergic and pyramidal glutamatergic neurons, in which we incorporated ionic concentration dynamics in both neurons. We modeled FHM-3 mutations by increasing the persistent sodium current in the interneuron and epileptogenic mutations by decreasing the sodium conductance in the interneuron. Therefore, we studied both FHM-3 and epileptogenic mutations within the same framework, modifying only two parameters. In our model, the key effect of gain of function FHM-3 mutations is ion fluxes modification at each action potential (in particular the larger activation of voltage-gated potassium channels induced by the Na_V1.1 gain of function), and the resulting CSD-triggering extracellular potassium accumulation, which is not caused only by modifications of firing frequency. Loss of function epileptogenic mutations, on the other hand, increase GABAergic neurons’ susceptibility to depolarization block, without major modifications of firing frequency before it. Our modeling results connect qualitatively to experimental data: potassium accumulation in the case of FHM-3 mutations and facilitated depolarization block of the GABAergic neuron in the case of epileptogenic mutations. Both these effects can lead to pyramidal neuron hyperexcitability, inducing in the migraine condition depolarization block of both the GABAergic and the pyramidal neuron. Overall, our findings suggest different mechanisms of network hyperexcitability for migraine and epileptogenic Na_V1.1 mutations, implying that the modifications of firing frequency may not be the only relevant pathological mechanism.

The voltage-gated sodium channel Na_V1.1 is a major target of human mutations implicated in different pathologies. In particular, mutations identified in certain types of epilepsy cause loss of function of the channel, whereas mutations identified in certain types of migraine (in which spreading depolarizations of the cortical circuits of the brain are involved) cause instead gain of function. Here, we study dysfunctions induced by these differential effects in a two-neuron (GABAergic and pyramidal) conductance-based model with dynamic ion concentrations. We obtain results that can be related to experimental findings in both situations. Namely, extracellular potassium accumulation induced by the activity of the GABAergic neuron in the case of CSD, and higher propensity of the GABAergic neuron to depolarization block in the epileptogenic scenario, without significant modifications of its firing frequency prior to it. Both scenarios can induce hyperexcitability of the pyramidal neuron, leading in the migraine condition to depolarization block of both the GABAergic and the pyramidal neuron. Our results are successfully confronted to experimental data and suggest that modification of firing frequency is not the only key mechanism in these pathologies of neuronal excitability.

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.

GABA receptor inhibition and severe hypoxia induce a paroxysmal depolarization shift in goldfish neurons

Mammalian neurons undergo rapid excitotoxic cell death when deprived of oxygen; however, the common goldfish (Carassius auratus) has the unique ability of surviving in oxygen-free waters, under anoxia. This organism utilizes γ-amino butyric acid (GABA) signaling to suppress excitatory glutamatergic activity during anoxic periods. Although GABA_A receptor antagonists are not deleterious to the cellular survival, coinhibition of GABA_A and GABA_B receptors is detrimental by abolishing anoxia-induced neuroprotective mechanisms. Here we show that blocking the anoxic GABAergic neurotransmission induces seizure-like activity (SLA) analogous to a paroxysmal depolarization shift (PDS), with hyperpolarization of action potential (AP) threshold and elevation of threshold currents. The observed PDS was attributed to an increase in excitatory postsynaptic currents (EPSCs) that are normally attenuated with decreasing oxygen levels. Furthermore, for the first time, we show that in addition to PDS, some neurons undergo depolarization block and do not generate AP despite a suprathreshold membrane potential. In conclusion, our results indicate that with severe hypoxia and absence of GABA receptor activity, telencephalic neurons of C. auratus manifest a paroxysmal depolarization shift, a key feature of epileptic discharge.NEW & NOTEWORTHY This work shows that the combination of anoxia and inhibition of GABA receptors induces seizure-like activities in goldfish telencephalic pyramidal and stellate neurons. Importantly, to prevent seizure-like activity, an intact GABA-mediated inhibitory pathway is required.

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.

Metabolism plays a central role in the cortical spreading depression: Evidence from a mathematical model

The slow propagating waves of strong depolarization of neural cells characterizing cortical spreading depression, or depolarization, (SD) are known to break cerebral homeostasis and induce significant hemodynamic and electro-metabolic alterations. Mathematical models of cortical spreading depression found in the literature tend to focus on the changes occurring at the electrophysiological level rather than on the ensuing metabolic changes. In this paper, we propose a novel mathematical model which is able to simulate the coupled electrophysiology and metabolism dynamics of SD events, including the swelling of neurons and astrocytes and the concomitant shrinkage of extracellular space. The simulations show that the metabolic coupling leads to spontaneous repetitions of the SD events, which the electrophysiological model alone is not capable to produce. The model predictions, which corroborate experimental findings from the literature, show a strong disruption in metabolism accompanying each wave of spreading depression in the form of a sharp decrease of glucose and oxygen concentrations, with a simultaneous increase in lactate concentration which, in turn, delays the clearing of excess potassium in extracellular space. Our model suggests that the depletion of glucose and oxygen concentration is more pronounced in astrocyte than neuron, in line with the partitioning of the energetic cost of potassium clearing. The model suggests that the repeated SD events are electro-metabolic oscillations that cannot be explained by the electrophysiology alone. The model highlights the crucial role of astrocytes in cleaning the excess potassium flooding extracellular space during a spreading depression event: further, if the ratio of glial/neuron density increases, the frequency of cortical SD events decreases, and the peak potassium concentration in extracellular space is lower than with equal volume fractions.

Cellular, molecular, and clinical mechanisms of action of deep brain stimulation-a systematic review on established indications and outlook on future developments

Deep brain stimulation (DBS) has been successfully used to treat movement disorders, such as Parkinson's disease, for more than 25 years and heralded the advent of electrical neuromodulation to treat diseases with dysregulated neuronal circuits. DBS is now superseding ablative techniques, such as stereotactic radiofrequency lesions. While serendipity has played a role in developing DBS as a therapy, research during the past two decades has shown that electrical neuromodulation is far more than a functional lesion that can be switched on and off. This understanding broadens the field to enable new types of stimulation, clinical indications, and research. This review highlights the complex effects of DBS from the single cell to the neuronal network. Specifically, we examine the electrical, cellular, molecular, and neurochemical mechanisms of DBS as applied to Parkinson's disease and other emerging applications.

A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons

Many pathological conditions, such as seizures, stroke, and spreading depression, are associated with substantial changes in ion concentrations in the extracellular space (ECS) of the brain. An understanding of the mechanisms that govern ECS concentration dynamics may be a prerequisite for understanding such pathologies. To estimate the transport of ions due to electrodiffusive effects, one must keep track of both the ion concentrations and the electric potential simultaneously in the relevant regions of the brain. Although this is currently unfeasible experimentally, it is in principle achievable with computational models based on biophysical principles and constraints. Previous computational models of extracellular ion-concentration dynamics have required extensive computing power, and therefore have been limited to either phenomena on very small spatiotemporal scales (micrometers and milliseconds), or simplified and idealized 1-dimensional (1-D) transport processes on a larger scale. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. By assuming electroneutrality, the KNP-framework circumvents charge-relaxation processes on the spatiotemporal scales of nanometers and nanoseconds, and makes it feasible to run simulations on the spatiotemporal scales of millimeters and seconds on a standard desktop computer. In the present work, we use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell. In addition to elucidating the single neuron contribution to electrodiffusive effects in the ECS, the simulation demonstrates the efficiency of the 3-D KNP framework. We envision that future applications of the framework to more complex and biologically realistic systems will be useful in exploring pathological conditions associated with large concentration variations in the ECS.

Adaptation and Inhibition Control Pathological Synchronization in a Model of Focal Epileptic Seizure

Pharmacoresistant epilepsy is a common neurological disorder in which increased neuronal intrinsic excitability and synaptic excitation lead to pathologically synchronous behavior in the brain. In the majority of experimental and theoretical epilepsy models, epilepsy is associated with reduced inhibition in the pathological neural circuits, yet effects of intrinsic excitability are usually not explicitly analyzed. Here we present a novel neural mass model that includes intrinsic excitability in the form of spike-frequency adaptation in the excitatory population. We validated our model using local field potential (LFP) data recorded from human hippocampal/subicular slices. We found that synaptic conductances and slow adaptation in the excitatory population both play essential roles for generating seizures and pre-ictal oscillations. Using bifurcation analysis, we found that transitions towards seizure and back to the resting state take place via Andronov-Hopf bifurcations. These simulations therefore suggest that single neuron adaptation as well as synaptic inhibition are responsible for orchestrating seizure dynamics and transition towards the epileptic state.

Role of astrocyte connexin hemichannels in cortical spreading depression

Cortical spreading depression (CSD) is an intriguing phenomenon consisting of massive slow brain depolarizations that affects neurons and glial cells. It has been recognized since 1944, but its pathogenesis has only been uncovered during the last decade. Acute brain injuries can be further complicated by CSD in >50% of severe cases. This phenomenon is repetitive and produces a metabolic overload that increments secondary damage. Propagation of CSD is known to be linked to excitotoxicity, but the mechanisms associated with its initiation remain less understood. It has been shown that CSD can be initiated by increases in extracellular [K⁺] ([K⁺]_e), and animal models use high [K⁺]_e to promote CSD. Connexin hemichannel activity increases due to high [K⁺]_e and low extracellular [Ca²⁺], conditions that occur after brain injury. Moreover, glial cell gap junction channels are fundamental in controlling extracellular medium composition, particularly in maintaining normal extracellular glutamate and K⁺ concentrations through "spatial buffering". However, the role of astrocytic gap junctions under tissue stress can change to damage spread in the acute damage zone whereas the reduced communication in adjacent zone would reduce cell dead propagation. Here, we review the main findings associated with CSD, and discuss the possible involvement of astrocytic connexin-based channels in secondary damage propagation. This article is part of a Special Issue entitled: Gap Junction Proteins edited by Jean Claude Herve.

Astrocytic gap junction blockade markedly increases extracellular potassium without causing seizures in the mouse neocortex

Extracellular potassium concentration, [K⁺]_o, is a major determinant of neuronal excitability. In the healthy brain, [K⁺]_o levels are tightly controlled. During seizures, [K⁺]_o increases up to 15mM and is thought to cause seizures due to its depolarizing effect. Although astrocytes have been suggested to play a key role in the redistribution (or spatial buffering) of excess K⁺ through Connexin-43 (Cx43)-based Gap Junctions (GJs), the relation between this dynamic regulatory process and seizure generation remains unknown. Here we contrasted the role of astrocytic GJs and hemichannels by studying the effect of GJ and hemichannel blockers on [K⁺]_o regulation in vivo. [K⁺]_o was measured by K⁺-sensitive microelectrodes. Neuronal excitability was estimated by local field potential (LFP) responses to forepaw stimulation and changes in the power of resting state activity. Starting at the baseline [K⁺]_o level of 1.61±0.3mM, cortical microinjection of CBX, a broad spectrum connexin channel blocker, increased [K⁺]_o to 11±3mM, Cx43 GJ/hemichannel blocker Gap27 increased it from 1.9±0.7 to 9±1mM. At these [K⁺]_o levels, no seizures were observed. Cx43 hemichannel blockade with TAT-Gap19 increased [K⁺]_o by only ~1mM. Microinjection of 4-aminopyridine, a known convulsant, increased [K⁺]_o to ~10mM and induced spontaneously recurring seizures, whereas direct application of K⁺ did not trigger seizure activity. These findings are the first in vivo demonstration that astrocytic GJs are major determinants for the spatial buffering of [K⁺]_o and that an increase in [K⁺]_o alone does not trigger seizures in the neocortex.

Reduced Efficacy of the KCC2 Cotransporter Promotes Epileptic Oscillations in a Subiculum Network Model

Pharmacoresistant epilepsy is a chronic neurological condition in which a basal brain hyperexcitability results in paroxysmal hypersynchronous neuronal discharges. Human temporal lobe epilepsy has been associated with dysfunction or loss of the potassium-chloride cotransporter KCC2 in a subset of pyramidal cells in the subiculum, a key structure generating epileptic activities. KCC2 regulates intraneuronal chloride and extracellular potassium levels by extruding both ions. Absence of effective KCC2 may alter the dynamics of chloride and potassium levels during repeated activation of GABAergic synapses due to interneuron activity. In turn, such GABAergic stress may itself affect Cl- regulation. Such changes in ionic homeostasis may switch GABAergic signaling from inhibitory to excitatory in affected pyramidal cells and also increase neuronal excitability. Possibly these changes contribute to periodic bursting in pyramidal cells, an essential component in the onset of ictal epileptic events. We tested this hypothesis with a computational model of a subicular network with realistic connectivity. The pyramidal cell model explicitly incorporated the cotransporter KCC2 and its effects on the internal/external chloride and potassium levels. Our network model suggested the loss of KCC2 in a critical number of pyramidal cells increased external potassium and intracellular chloride concentrations leading to seizure-like field potential oscillations. These oscillations included transient discharges leading to ictal-like field events with frequency spectra as in vitro Restoration of KCC2 function suppressed seizure activity and thus may present a useful therapeutic option. These simulations therefore suggest that reduced KCC2 cotransporter activity alone may underlie the generation of ictal discharges.

SIGNIFICANCE STATEMENT

Ion regulation in the brain is a major determinant of neural excitability. Intracellular chloride in neurons, a partial determinant of the resting potential and the inhibitory reversal potentials, is regulated together with extracellular potassium via kation chloride cotransporters. During temporal lobe epilepsy, the homeostatic regulation of intracellular chloride is impaired in pyramidal cells, yet how this dysregulation may lead to seizures has not been explored. Using a realistic neural network model describing ion mechanisms, we show that chloride homeostasis pathology provokes seizure activity analogous to recordings from epileptogenic brain tissue. We show that there is a critical percentage of pathological cells required for seizure initiation. Our model predicts that restoration of the chloride homeostasis in pyramidal cells could be a viable antiepileptic strategy.

Effect of Ionic Diffusion on Extracellular Potentials in Neural Tissue

Recorded potentials in the extracellular space (ECS) of the brain is a standard measure of population activity in neural tissue. Computational models that simulate the relationship between the ECS potential and its underlying neurophysiological processes are commonly used in the interpretation of such measurements. Standard methods, such as volume-conductor theory and current-source density theory, assume that diffusion has a negligible effect on the ECS potential, at least in the range of frequencies picked up by most recording systems. This assumption remains to be verified. We here present a hybrid simulation framework that accounts for diffusive effects on the ECS potential. The framework uses (1) the NEURON simulator to compute the activity and ionic output currents from multicompartmental neuron models, and (2) the electrodiffusive Kirchhoff-Nernst-Planck framework to simulate the resulting dynamics of the potential and ion concentrations in the ECS, accounting for the effect of electrical migration as well as diffusion. Using this framework, we explore the effect that ECS diffusion has on the electrical potential surrounding a small population of 10 pyramidal neurons. The neural model was tuned so that simulations over ∼100 seconds of biological time led to shifts in ECS concentrations by a few millimolars, similar to what has been seen in experiments. By comparing simulations where ECS diffusion was absent with simulations where ECS diffusion was included, we made the following key findings: (i) ECS diffusion shifted the local potential by up to ∼0.2 mV. (ii) The power spectral density (PSD) of the diffusion-evoked potential shifts followed a 1/f2 power law. (iii) Diffusion effects dominated the PSD of the ECS potential for frequencies up to several hertz. In scenarios with large, but physiologically realistic ECS concentration gradients, diffusion was thus found to affect the ECS potential well within the frequency range picked up in experimental recordings.

Animal models of monogenic migraine

Migraine is a highly prevalent and disabling neurological disorder with a strong genetic component. Rare monogenic forms of migraine, or syndromes in which migraine frequently occurs, help scientists to unravel pathogenetic mechanisms of migraine and its comorbidities. Transgenic mouse models for rare monogenic mutations causing familial hemiplegic migraine (FHM), cerebral autosomal dominant arteriopathy with subcortical infarcts and leukoencephalopathy (CADASIL), and familial advanced sleep-phase syndrome (FASPS), have been created. Here, we review the current state of research using these mutant mice. We also discuss how currently available experimental approaches, including epigenetic studies, biomolecular analysis and optogenetic technologies, can be used for characterization of migraine genes to further unravel the functional and molecular pathways involved in migraine.

The neuroglial potassium cycle during neurotransmission: role of Kir4.1 channels

Neuronal excitability relies on inward sodium and outward potassium fluxes during action potentials. To prevent neuronal hyperexcitability, potassium ions have to be taken up quickly. However, the dynamics of the activity-dependent potassium fluxes and the molecular pathways underlying extracellular potassium homeostasis remain elusive. To decipher the specific and acute contribution of astroglial Kir4.1 channels in controlling potassium homeostasis and the moment to moment neurotransmission, we built a tri-compartment model accounting for potassium dynamics between neurons, astrocytes and the extracellular space. We here demonstrate that astroglial Kir4.1 channels are sufficient to account for the slow membrane depolarization of hippocampal astrocytes and crucially contribute to extracellular potassium clearance during basal and high activity. By quantifying the dynamics of potassium levels in neuron-glia-extracellular space compartments, we show that astrocytes buffer within 6 to 9 seconds more than 80% of the potassium released by neurons in response to basal, repetitive and tetanic stimulations. Astroglial Kir4.1 channels directly lead to recovery of basal extracellular potassium levels and neuronal excitability, especially during repetitive stimulation, thereby preventing the generation of epileptiform activity. Remarkably, we also show that Kir4.1 channels strongly regulate neuronal excitability for slow 3 to 10 Hz rhythmic activity resulting from probabilistic firing activity induced by sub-firing stimulation coupled to Brownian noise. Altogether, these data suggest that astroglial Kir4.1 channels are crucially involved in extracellular potassium homeostasis regulating theta rhythmic activity.

Effects of extracellular potassium diffusion on electrically coupled neuron networks

Potassium accumulation and diffusion during neuronal epileptiform activity have been observed experimentally, and potassium lateral diffusion has been suggested to play an important role in nonsynaptic neuron networks. We adopt a hippocampal CA1 pyramidal neuron network in a zero-calcium condition to better understand the influence of extracellular potassium dynamics on the stimulus-induced activity. The potassium concentration in the interstitial space for each neuron is regulated by potassium currents, Na(+)-K(+) pumps, glial buffering, and ion diffusion. In addition to potassium diffusion, nearby neurons are also coupled through gap junctions. Our results reveal that the latency of the first spike responding to stimulus monotonically decreases with increasing gap-junction conductance but is insensitive to potassium diffusive coupling. The duration of network oscillations shows a bell-like shape with increasing potassium diffusive coupling at weak gap-junction coupling. For modest electrical coupling, there is an optimal K(+) diffusion strength, at which the flow of potassium ions among the network neurons appropriately modulates interstitial potassium concentrations in a degree that provides the most favorable environment for the generation and continuance of the action potential waves in the network.

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.

Electrodiffusive model for astrocytic and neuronal ion concentration dynamics

The cable equation is a proper framework for modeling electrical neural signalling that takes place at a timescale at which the ionic concentrations vary little. However, in neural tissue there are also key dynamic processes that occur at longer timescales. For example, endured periods of intense neural signaling may cause the local extracellular K⁺-concentration to increase by several millimolars. The clearance of this excess K⁺ depends partly on diffusion in the extracellular space, partly on local uptake by astrocytes, and partly on intracellular transport (spatial buffering) within astrocytes. These processes, that take place at the time scale of seconds, demand a mathematical description able to account for the spatiotemporal variations in ion concentrations as well as the subsequent effects of these variations on the membrane potential. Here, we present a general electrodiffusive formalism for modeling of ion concentration dynamics in a one-dimensional geometry, including both the intra- and extracellular domains. Based on the Nernst-Planck equations, this formalism ensures that the membrane potential and ion concentrations are in consistency, it ensures global particle/charge conservation and it accounts for diffusion and concentration dependent variations in resistivity. We apply the formalism to a model of astrocytes exchanging ions with the extracellular space. The simulations show that K⁺-removal from high-concentration regions is driven by a local depolarization of the astrocyte membrane, which concertedly (i) increases the local astrocytic uptake of K⁺, (ii) suppresses extracellular transport of K⁺, (iii) increases axial transport of K⁺ within astrocytes, and (iv) facilitates astrocytic relase of K⁺ in regions where the extracellular concentration is low. Together, these mechanisms seem to provide a robust regulatory scheme for shielding the extracellular space from excess K⁺.

When neurons generate electrical signals they release potassium ions (K⁺) into the extracellular space. During periods of intense neural activity, the local extracellular K⁺ may increase drastically. If it becomes too high, it can lead to neural dysfunction. Astrocytes (a kind of glial cells) are involved in preventing this from happening. Astrocytes can take up excess K⁺, transport it intracellularly, and release it in regions where the concentration is lower. This process is called spatial buffering, and a full mechanistic understanding of it is currently lacking. The aim of this work is twofold: First, we develop a formalism for modeling ion concentration dynamics in the intra- and extracellular space. The formalism is general, and could be used to simulate many cellular processes. It accounts for ion transports due to diffusion (along concentration gradients) as well as electrical migration (along voltage gradients). It extends previous, related formalisms, which have focused only on intracellular dynamics. Secondly, we apply the formalism to model how astrocytes exchange ions with the extracellular space. We conclude that the membrane mechanisms possessed by astrocytes seem optimal for shielding the extracellular space from excess K⁺, and provide a full mechanistic description of the spatial (K⁺) buffering process.

Computational modeling reveals dendritic origins of GABA(A)-mediated excitation in CA1 pyramidal neurons

GABA is the key inhibitory neurotransmitter in the adult central nervous system, but in some circumstances can lead to a paradoxical excitation that has been causally implicated in diverse pathologies from endocrine stress responses to diseases of excitability including neuropathic pain and temporal lobe epilepsy. We undertook a computational modeling approach to determine plausible ionic mechanisms of GABA(A)-dependent excitation in isolated post-synaptic CA1 hippocampal neurons because it may constitute a trigger for pathological synchronous epileptiform discharge. In particular, the interplay intracellular chloride accumulation via the GABA(A) receptor and extracellular potassium accumulation via the K/Cl co-transporter KCC2 in promoting GABA(A)-mediated excitation is complex. Experimentally it is difficult to determine the ionic mechanisms of depolarizing current since potassium transients are challenging to isolate pharmacologically and much GABA signaling occurs in small, difficult to measure, dendritic compartments. To address this problem and determine plausible ionic mechanisms of GABA(A)-mediated excitation, we built a detailed biophysically realistic model of the CA1 pyramidal neuron that includes processes critical for ion homeostasis. Our results suggest that in dendritic compartments, but not in the somatic compartments, chloride buildup is sufficient to cause dramatic depolarization of the GABA(A) reversal potential and dominating bicarbonate currents that provide a substantial current source to drive whole-cell depolarization. The model simulations predict that extracellular K(+) transients can augment GABA(A)-mediated excitation, but not cause it. Our model also suggests the potential for GABA(A)-mediated excitation to promote network synchrony depending on interneuron synapse location - excitatory positive-feedback can occur when interneurons synapse onto distal dendritic compartments, while interneurons projecting to the perisomatic region will cause inhibition.

Multistability in a neuron model with extracellular potassium dynamics

Experiments show a primary role of extracellular potassium concentrations in neuronal hyperexcitability and in the generation of epileptiform bursting and depolarization blocks without synaptic mechanisms. We adopt a physiologically relevant hippocampal CA1 neuron model in a zero-calcium condition to better understand the function of extracellular potassium in neuronal seizurelike activities. The model neuron is surrounded by interstitial space in which potassium ions are able to accumulate. Potassium currents, Na{+}-K{+} pumps, glial buffering, and ion diffusion are regulatory mechanisms of extracellular potassium. We also consider a reduced model with a fixed potassium concentration. The bifurcation structure and spiking frequency of the two models are studied. We show that, besides hyperexcitability and bursting pattern modulation, the potassium dynamics can induce not only bistability but also tristability of different firing patterns. Our results reveal the emergence of the complex behavior of multistability due to the dynamical [K{+}]{o} modulation on neuronal activities.

Dependence of spontaneous neuronal firing and depolarisation block on astroglial membrane transport mechanisms

Exposed to a sufficiently high extracellular potassium concentration ([K( + )]₀), the neuron can fire spontaneous discharges or even become inactivated due to membrane depolarisation ('depolarisation block'). Since these phenomena likely are related to the maintenance and propagation of seizure discharges, it is of considerable importance to understand the conditions under which excess [K( + )]₀ causes them. To address the putative effect of glial buffering on neuronal activity under elevated [K( + )](o) conditions, we combined a recently developed dynamical model of glial membrane ion and water transport with a Hodgkin-Huxley type neuron model. In this interconnected glia-neuron model we investigated the effects of natural heterogeneity or pathological changes in glial membrane transporter density by considering a large set of models with different, yet empirically plausible, sets of model parameters. We observed both the high [K( + )]₀-induced duration of spontaneous neuronal firing and the prevalence of depolarisation block to increase when reducing the magnitudes of the glial transport mechanisms. Further, in some parameter regions an oscillatory bursting spiking pattern due to the dynamical coupling of neurons and glia was observed. Bifurcation analyses of the neuron model and of a simplified version of the neuron-glia model revealed further insights about the underlying mechanism behind these phenomena. The above insights emphasise the importance of combining neuron models with detailed astroglial models when addressing phenomena suspected to be influenced by the astroglia-neuron interaction. To facilitate the use of our neuron-glia model, a CellML version of it is made publicly available.

Experimental observation of increased fluctuations in an order parameter before epochs of extended brain synchronization

The identification of epileptic seizure precursors has potential clinical relevance. It is conjectured that seizures may be represented by dynamical bifurcations and that an adequate order parameter to characterize brain dynamics is the phase difference in the oscillatory activity of neural systems. In this study, the critical point hypothesis that seizures, or more generally periods of widespread high synchronization, represent bifurcations is empirically tested by monitoring the growth of fluctuations in the putative order parameter of phase differences between magnetoencephalographic and electroencephalographic signals in nearby brain regions in patients with epilepsy and normal subjects during hyperventilation. Implications of the results with regard to epileptic phenomena are discussed.