Oxygen and seizure dynamics: II. Computational modeling

The metabolic costs of cognition

Cognition and behavior are emergent properties of brain systems that seek to maximize complex and adaptive behaviors while minimizing energy utilization. Different species reconcile this trade-off in different ways, but in humans the outcome is biased towards complex behaviors and hence relatively high energy use. However, even in energy-intensive brains, numerous parsimonious processes operate to optimize energy use. We review how this balance manifests in both homeostatic processes and task-associated cognition. We also consider the perturbations and disruptions of metabolism in neurocognitive diseases.

Dynamical modeling and analysis of epileptic discharges transition caused by glutamate release with metabolism processes regulation from astrocyte

Glutamate (Glu) is a crucial excitatory neurotransmitter in the central nervous system that transmits brain information by activating excitatory receptors on neuronal membranes. Physiological studies have demonstrated that abnormal Glu metabolism in astrocytes is closely related to the pathogenesis of epilepsy. The astrocyte metabolism processes mainly involve the Glu uptake through astrocyte EAAT2, the Glu-glutamine (Gln) conversion, and the Glu release. However, the relationship between these Glu metabolism processes and epileptic discharges remains unclear. In this paper, we propose a novel neuron-astrocyte model by integrating the dynamical modeling of astrocyte Glu metabolism processes, which include Glu metabolism in astrocytes consisting of the Glu uptake, Glu-Gln conversion, Glu diffusion, and the resulting Glu release as well as Glu-mediated bidirectional communication between neuron and astrocyte. Furthermore, the influences of astrocyte multiple Glu metabolism processes on the Glu release and dynamics transition of neuronal epileptic discharges are verified through numerical experiments and dynamical analyses from various nonlinear dynamics perspectives, such as time series, phase plane trajectories, interspike intervals, and bifurcation diagrams. Our results suggest that the downregulation expression of EAAT2 uptake, the slowdown of the Glu-Gln conversion rate, and excessively elevated Glu equilibrium concentration in astrocytes can cause an increase in Glu released from astrocytes, which results in the aggravation of epileptic seizures. Meanwhile, neuronal epileptic discharge states transition from bursting to mixed-mode spiking and tonic firing induced by the combination of these abnormal metabolism processes. This study provides a theoretical foundation and dynamical analysis methodology for further exploring the dynamics evolution and physiopathological mechanisms of epilepsy.

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.

Multi-scale modelling of the epileptic brain: advantages of computational therapy exploration

Objective: Epilepsy is a complex disease spanning across multiple scales, from ion channels in neurons to neuronal circuits across the entire brain. Over the past decades, computational models have been used to describe the pathophysiological activity of the epileptic brain from different aspects. Traditionally, each computational model can aid in optimizing therapeutic interventions, therefore, providing a particular view to design strategies for treating epilepsy. As a result, most studies are concerned with generating specific models of the epileptic brain that can help us understand the certain machinery of the pathological state. Those specific models vary in complexity and biological accuracy, with system-level models often lacking biological details.Approach: Here, we review various types of computational model of epilepsy and discuss their potential for different therapeutic approaches and scenarios, including drug discovery, surgical strategies, brain stimulation, and seizure prediction. We propose that we need to consider an integrated approach with a unified modelling framework across multiple scales to understand the epileptic brain. Our proposal is based on the recent increase in computational power, which has opened up the possibility of unifying those specific epileptic models into simulations with an unprecedented level of detail.Main results: A multi-scale epilepsy model can bridge the gap between biologically detailed models, used to address molecular and cellular questions, and brain-wide models based on abstract models which can account for complex neurological and behavioural observations.Significance: With these efforts, we move toward the next generation of epileptic brain models capable of connecting cellular features, such as ion channel properties, with standard clinical measures such as seizure severity.

Dynamical modulation of hypersynchronous seizure onset with transcranial magneto-acoustic stimulation in a hippocampal computational model

Hypersynchronous (HYP) seizure onset is one of the frequently observed seizure-onset patterns in temporal lobe epileptic animals and patients, often accompanied by hippocampal sclerosis. However, the exact mechanisms and ion dynamics of the transition to HYP seizures remain unclear. Transcranial magneto-acoustic stimulation (TMAS) has recently been proposed as a novel non-invasive brain therapy method to modulate neurological disorders. Therefore, we propose a biophysical computational hippocampal network model to explore the evolution of HYP seizure caused by changes in crucial physiological parameters and design an effective TMAS strategy to modulate HYP seizure onset. We find that the cooperative effects of abnormal glial uptake strength of potassium and excessive bath potassium concentration could produce multiple discharge patterns and result in transitions from the normal state to the HYP seizure state and ultimately to the depolarization block state. Moreover, we find that the pyramidal neuron and the PV+ interneuron in HYP seizure-onset state exhibit saddle-node-on-invariant-circle/saddle homoclinic (SH) and saddle-node/SH at onset/offset bifurcation pairs, respectively. Furthermore, the response of neuronal activities to TMAS of different ultrasonic waveforms revealed that lower sine wave stimulation can increase the latency of HYP seizures and even completely suppress seizures. More importantly, we propose an ultrasonic parameter area that not only effectively regulates epileptic rhythms but also is within the safety limits of ultrasound neuromodulation therapy. Our results may offer a more comprehensive understanding of the mechanisms of HYP seizure and provide a theoretical basis for the application of TMAS in treating specific types of seizures.

The role of extracellular glutamate homeostasis dysregulated by astrocyte in epileptic discharges: a model evidence

Glutamate (Glu) is a predominant excitatory neurotransmitter that acts on glutamate receptors to transfer signals in the central nervous system. Abnormally elevated extracellular glutamate levels is closely related to the generation and transition of epileptic seizures. However, there lacks of investigation regarding the role of extracellular glutamate homeostasis dysregulated by astrocyte in neuronal epileptic discharges. According to this, we propose a novel neuron-astrocyte computational model (NAG) by incorporating extracellular Glu concentration dynamics from three aspects of regulatory mechanisms: (1) the Glu uptake through astrocyte EAAT2; (2) the binding and release Glu via activating astrocyte mGluRs; and (3) the Glu free diffusion in the extracellular space. Then the proposed model NAG is analyzed theoretically and numerically to verify the effect of extracellular Glu homeostasis dysregulated by such three regulatory mechanisms on neuronal epileptic discharges. Our results demonstrate that the neuronal epileptic discharges can be aggravated by the downregulation expression of EAAT2, the aberrant activation of mGluRs, and the elevated Glu levels in extracellular micro-environment; as well as various discharge states (including bursting, mixed-mode spiking, and tonic firing) can be transited by their combination. Furthermore, we find that such factors can also alter the bifurcation threshold for the generation and transition of epileptic discharges. The results in this paper can be helpful for researchers to understand the astrocyte role in modulating extracellular Glu homeostasis, and provide theoretical basis for future related experimental studies.

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.

Finite-element modeling of neuromodulation via controlled delivery of potassium ions using conductive polymer-coated microelectrodes

Objective. The controlled delivery of potassium is an interesting neuromodulation modality, being potassium ions involved in shaping neuron excitability, synaptic transmission, network synchronization, and playing a key role in pathological conditions like epilepsy and spreading depression. Despite many successful examples of pre-clinical devices able to influence the extracellular potassium concentration, computational frameworks capturing the corresponding impact on neuronal activity are still missing.Approach. We present a finite-element model describing a PEDOT:PSS-coated microelectrode (herein, simplyionic actuator) able to release potassium and thus modulate the activity of a cortical neuron in anin-vitro-like setting. The dynamics of ions in the ionic actuator, the neural membrane, and the cellular fluids are solved self-consistently.Main results. We showcase the capability of the model to describe on a physical basis the modulation of the intrinsic excitability of the cell and of the synaptic transmission following the electro-ionic stimulation produced by the actuator. We consider three case studies for the ionic actuator with different levels of selectivity to potassium: ideal selectivity, no selectivity, and selectivity achieved by embedding ionophores in the polymer.Significance. This work is the first step toward a comprehensive computational framework aimed to investigate novel neuromodulation devices targeting specific ionic species, as well as to optimize their design and performance, in terms of the induced modulation of neural activity.

Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection

For single neuron models, reproducing characteristics of neuronal activity such as the firing rate, amplitude of spikes, and threshold potentials as functions of both synaptic current and conductance is a challenging task. In the present work, we measure these characteristics of regular spiking cortical neurons using the dynamic patch-clamp technique, compare the data with predictions from the standard Hodgkin-Huxley and Izhikevich models, and propose a relatively simple five-dimensional dynamical system model, based on threshold criteria. The model contains a single sodium channel with slow inactivation, fast activation and moderate deactivation, as well as, two fast repolarizing and slow shunting potassium channels. The model quantitatively reproduces characteristics of steady-state activity that are typical for a cortical pyramidal neuron, namely firing rate not exceeding 30 Hz; critical values of the stimulating current and conductance which induce the depolarization block not exceeding 80 mV and 3, respectively (both values are scaled by the resting input conductance); extremum of hyperpolarization close to the midpoint between spikes. The analysis of the model reveals that the spiking regime appears through a saddle-node-on-invariant-circle bifurcation, and the depolarization block is reached through a saddle-node bifurcation of cycles. The model can be used for realistic network simulations, and it can also be implemented within the so-called mean-field, refractory density framework.

Astrocytes: new evidence, new models, new roles

Astrocytes have been in the limelight of active research for about 3 decades now. Over this period, ideas about their function and role in the nervous system have evolved from simple assistance in energy supply and homeostasis maintenance to a complex informational and metabolic hub that integrates data on local neuronal activity, sensory and arousal context, and orchestrates many crucial processes in the brain. Rapid progress in experimental techniques and data analysis produces a growing body of data, which can be used as a foundation for formulation of new hypotheses, building new refined mathematical models, and ultimately should lead to a new level of understanding of the contribution of astrocytes to the cognitive tasks performed by the brain. Here, we highlight recent progress in astrocyte research, which we believe expands our understanding of how low-level signaling at a cellular level builds up to processes at the level of the whole brain and animal behavior. We start our review with revisiting data on the role of noradrenaline-mediated astrocytic signaling in locomotion, arousal, sensory integration, memory, and sleep. We then briefly review astrocyte contribution to the regulation of cerebral blood flow regulation, which is followed by a discussion of biophysical mechanisms underlying astrocyte effects on different brain processes. The experimental section is closed by an overview of recent experimental techniques available for modulation and visualization of astrocyte dynamics. We then evaluate how the new data can be potentially incorporated into the new mathematical models or where and how it already has been done. Finally, we discuss an interesting prospect that astrocytes may be key players in important processes such as the switching between sleep and wakefulness and the removal of toxic metabolites from the brain milieu.

Mechanisms underlying pathological cortical bursts during metabolic depletion

Cortical activity depends upon a continuous supply of oxygen and other metabolic resources. Perinatal disruption of oxygen availability is a common clinical scenario in neonatal intensive care units, and a leading cause of lifelong disability. Pathological patterns of brain activity including burst suppression and seizures are a hallmark of the recovery period, yet the mechanisms by which these patterns arise remain poorly understood. Here, we use computational modeling of coupled metabolic-neuronal activity to explore the mechanisms by which oxygen depletion generates pathological brain activity. We find that restricting oxygen supply drives transitions from normal activity to several pathological activity patterns (isoelectric, burst suppression, and seizures), depending on the potassium supply. Trajectories through parameter space track key features of clinical electrophysiology recordings and reveal how infants with good recovery outcomes track toward normal parameter values, whereas the parameter values for infants with poor outcomes dwell around the pathological values. These findings open avenues for studying and monitoring the metabolically challenged infant brain, and deepen our understanding of the link between neuronal and metabolic activity.

Optogenetic Low-Frequency Stimulation of Principal Neurons, but Not Parvalbumin-Positive Interneurons, Prevents Generation of Ictal Discharges in Rodent Entorhinal Cortex in an In Vitro 4-Aminopyridine Model

Low-frequency electrical stimulation is used to treat some drug-resistant forms of epilepsy. Despite the effectiveness of the method in suppressing seizures, there is a considerable risk of side effects. An optogenetic approach allows the targeting of specific populations of neurons, which can increase the effectiveness and safety of low-frequency stimulation. In our study, we tested the efficacy of the suppression of ictal activity in entorhinal cortex slices in a 4-aminopyridine model with three variants of low-frequency light stimulation (LFLS): (1) activation of excitatory and inhibitory neurons (on Thy1-ChR2-YFP mice), (2) activation of inhibitory interneurons only (on PV-Cre mice after virus injection with channelrhodopsin2 gene), and (3) hyperpolarization of excitatory neurons (on Wistar rats after virus injection with archaerhodopsin gene). Only in the first variant did simultaneous LFLS of excitatory and inhibitory neurons replace ictal activity with interictal activity. We suggest that LFLS caused changes in the concentration gradients of K⁺ and Na⁺ cations across the neuron membrane, which activated Na-K pumping. According to the mathematical modeling, the increase in Na-K pump activity in neurons induced by LFLS led to an antiepileptic effect. Thus, a less specific and generalized optogenetic effect on entorhinal cortex neurons was more effective in suppressing ictal activity in the 4-aminopyridine model.

A mechanistic model of calcium homeostasis leading to occurrence and propagation of secondary brain injury

Secondary brain injury (SBI) refers to new or worsening brain insult after primary brain injury (PBI). Neurophysiological experiments show that calcium (Ca2+) is one of the major culprits that contribute to neuronal damage and death following PBI. However, mechanistic details about how alterations of Ca2+ levels contribute to SBI are not well characterized. In this paper, we first build a biophysical model for SBI related to calcium homeostasis (SBI-CH) to study the mechanistic details of PBI-induced disruption of CH, and how these disruptions affect the occurrence of SBI. Then, we construct a coupled SBI-CH model by formulating synaptic interactions to investigate how disruption of CH affects synaptic function and further promotes the propagation of SBI between neurons. Our model shows how the opening of voltage-gated calcium channels (VGCCs), decreasing of plasma membrane calcium pump (PMCA), and reversal of the Na+/Ca2+ exchanger (NCX) during and following PBI, could induce disruption of CH and further promote SBI. We also show that disruption of CH causes synaptic dysfunction, which further induces loss of excitatory-inhibitory balance in the system, and this might promote the propagation of SBI and cause neighboring tissue to be injured. Our findings offer a more comprehensive understanding of the complex interrelationship between CH and SBI.NEW & NOTEWORTHY We build a mechanistic model SBI-CH for calcium homeostasis (CH) to study how alterations of Ca2+ levels following PBI affect the occurrence and propagation of SBI. Specifically, we investigate how the opening of VGCCs, decreasing of PMCA, and reversal of NCX disrupt CH, and further induce the occurrence of SBI. We also present a coupled SBI-CH model to show how disrupted CH causes synaptic dysfunction, and further promotes the propagation of SBI between neurons.

Focal seizures are organized by feedback between neural activity and ion concentration changes

Human and animal EEG data demonstrate that focal seizures start with low-voltage fast activity, evolve into rhythmic burst discharges and are followed by a period of suppressed background activity. This suggests that processes with dynamics in the range of tens of seconds govern focal seizure evolution. We investigate the processes associated with seizure dynamics by complementing the Hodgkin-Huxley mathematical model with the physical laws that dictate ion movement and maintain ionic gradients. Our biophysically realistic computational model closely replicates the electrographic pattern of a typical human focal seizure characterized by low voltage fast activity onset, tonic phase, clonic phase and postictal suppression. Our study demonstrates, for the first time in silico, the potential mechanism of seizure initiation by inhibitory interneurons via the initial build-up of extracellular K+ due to intense interneuronal spiking. The model also identifies ionic mechanisms that may underlie a key feature in seizure dynamics, i.e., progressive slowing down of ictal discharges towards the end of seizure. Our model prediction of specific scaling of inter-burst intervals is confirmed by seizure data recorded in the whole guinea pig brain in vitro and in humans, suggesting that the observed termination pattern may hold across different species. Our results emphasize ionic dynamics as elementary processes behind seizure generation and indicate targets for new therapeutic strategies.

Modulation of seizure-like events by the small conductance and ATP-sensitive potassium ion channels

Potassium ion channels are extensively involved in the regulation of epileptic seizures. The small conductance calcium-sensitive potassium channels (SK channels) and ATP-sensitive potassium (K_ATP) channels are activated by calcium ion entry and decrease ATP levels, respectively. These channels can underlie the post-burst afterhyperpolarization and be upregulated during seizures, providing negative feedback during epileptic activity. Using the whole-cell patch-clamp method in rat brain slices, we investigated the effect of SK- and K_ATP-affecting drugs on seizure-like events (SLEs) in the 4-aminopyridine model of epileptic seizures in vitro. We demonstrate that SK and K_ATP channels contribute to sustaining the high-frequency firing of the principal neurons in the deep layers of the entorhinal cortex during injections of depolarizing current and epileptiform discharges. Neither the pharmacological blockade nor the activation of these channels was able to prevent the epileptiform activity in brain slices. However, the blockade of K_ATP channels increases the SLE duration, suggesting that these channels may contribute to the termination of SLEs. Thus, K_ATP channels can be considered a promising target for pharmacological interventions for the treatment of epilepsy.

Ictal wavefront propagation in slices and simulations with conductance-based refractory density model

The mechanisms determining ictal discharge (ID) propagation are still not clear. In the present study, we aimed to examine these mechanisms in animal and mathematical models of epileptiform activity. Using double-patch and extracellular potassium ion concentration recordings in rat hippocampal-cortical slices, we observed that IDs moved at a speed of about 1 mm/s or less. The mechanisms of such slow propagation have been studied with a mathematical, conductance-based refractory density (CBRD) model that describes the GABA- and glutamatergic neuronal populations’ interactions and ion dynamics in brain tissue. The modeling study reveals two main factors triggerring IDs: (i) increased interneuronal activity leading to chloride ion accumulation and a consequent depolarizing GABAergic effect and (ii) the elevation of extracellular potassium ion concentration. The local synaptic transmission followed by local potassium ion extrusion and GABA receptor-mediated chloride ion accumulation underlies the ID wavefront’s propagation. In contrast, potassium ion diffusion in the extracellular space is slower and does not affect ID’s speed. The short discharges, constituting the ID, propagate much faster than the ID front. The accumulation of sodium ions inside neurons due to their hyperactivity and glutamatergic currents boosts the Na⁺/K⁺ pump, which terminates the ID. Knowledge of the mechanism of ID generation and propagation contributes to the development of new treatments against epilepsy.

During an epileptic seizure, neuronal excitation spreads across the brain tissue and is accompanied by significant changes in ionic concentrations. Ictal discharge front spreads at low speeds, less than 1 mm/s. Mechanisms underlying this phenomenon are not yet well understood. We study these mechanisms using electrophysiological recordings in brain slices and computer simulations. Our detailed biophysical model describing neuronal populations’ interaction, spatial propagation, and ionic dynamics reproduces the generation and propagation of spontaneously repeating ictal discharges. The simulations are consistent with our recordings of the electrical activity and the extracellular potassium ion concentration. We distinguished between the two alternative mechanisms of the ictal wavefront propagation: (i) the diffusion of potassium ions released from excited neurons, which depolarizes distant neurons and thus supports excitation, and (ii) the axonal spread of excitation followed by the local extracellular potassium ion accumulation that supports the excitation. Our simulations provide evidence in favor of the latter mechanism. Our experiment-based modeling contributes to a mathematical description of brain tissue functioning and potentially contributes to developing new treatments against epilepsy.

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.

A model of metabolic supply-demand mismatch leading to secondary brain injury

Secondary brain injury (SBI) is defined as new or worsening injury to the brain after an initial neurologic insult, such as hemorrhage, trauma, ischemic stroke, or infection. It is a common and potentially preventable complication following many types of primary brain injury (PBI). However, mechanistic details about how PBI leads to additional brain injury and evolves into SBI are poorly characterized. In this work, we propose a mechanistic model for the metabolic supply demand mismatch hypothesis (MSDMH) of SBI. Our model, based on the Hodgkin-Huxley model, supplemented with additional dynamics for extracellular potassium, oxygen concentration, and excitotoxity, provides a high-level unified explanation for why patients with acute brain injury frequently develop SBI. We investigate how decreased oxygen, increased extracellular potassium, excitotoxicity, and seizures can induce SBI and suggest three underlying paths for how events following PBI may lead to SBI. The proposed model also helps explain several important empirical observations, including the common association of acute brain injury with seizures, the association of seizures with tissue hypoxia and so on. In contrast to current practices which assume that ischemia plays the predominant role in SBI, our model suggests that metabolic crisis involved in SBI can also be nonischemic. Our findings offer a more comprehensive understanding of the complex interrelationship among potassium, oxygen, excitotoxicity, seizures, and SBI.NEW & NOTEWORTHY We present a novel mechanistic model for the metabolic supply demand mismatch hypothesis (MSDMH), which attempts to explain why patients with acute brain injury frequently develop seizure activity and secondary brain injury (SBI). Specifically, we investigate how decreased oxygen, increased extracellular potassium, excitotoxicity, seizures, all common sequalae of primary brain injury (PBI), can induce SBI and suggest three underlying paths for how events following PBI may lead to SBI.

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.

Modeling seizures: From single neurons to networks

Dynamical system tools offer a complementary approach to detailed biophysical seizure modeling, with a high potential for clinical applications. This review describes the theoretical framework that provides a basis for theorizing certain properties of seizures and for their classification according to their dynamical properties at onset and offset. We describe various modeling approaches spanning different scales, from single neurons to large-scale networks. This narrative review provides an accessible overview of this field, including non-exhaustive examples of key recent works.

Activity-mediated accumulation of potassium induces a switch in firing pattern and neuronal excitability type

During normal neuronal activity, ionic concentration gradients across a neuron’s membrane are often assumed to be stable. Prolonged spiking activity, however, can reduce transmembrane gradients and affect voltage dynamics. Based on mathematical modeling, we investigated the impact of neuronal activity on ionic concentrations and, consequently, the dynamics of action potential generation. We find that intense spiking activity on the order of a second suffices to induce changes in ionic reversal potentials and to consistently induce a switch from a regular to an intermittent firing mode. This transition is caused by a qualitative alteration in the system’s voltage dynamics, mathematically corresponding to a co-dimension-two bifurcation from a saddle-node on invariant cycle (SNIC) to a homoclinic orbit bifurcation (HOM). Our electrophysiological recordings in mouse cortical pyramidal neurons confirm the changes in action potential dynamics predicted by the models: (i) activity-dependent increases in intracellular sodium concentration directly reduce action potential amplitudes, an effect typically attributed solely to sodium channel inactivation; (ii) extracellular potassium accumulation switches action potential generation from tonic firing to intermittently interrupted output. Thus, individual neurons may respond very differently to the same input stimuli, depending on their recent patterns of activity and/or the current brain-state.

Ionic concentrations in the brain are not constant. We show that during intense neuronal activity, they can change on the order of seconds and even switch neuronal spiking patterns under identical stimulation from a regular firing mode to an intermittently interrupted one. Triggered by an accumulation of extracellular potassium, such a transition is caused by a specific, qualitative change in of the neuronal voltage dynamics—a so-called bifurcation—which affects crucial features of action-potential generation and bears consequences for how information is encoded and how neurons behave together in the network. Also, changes in intracellular sodium can induce measurable effects, like a reduction of spike amplitude that occurs independently of the fast amplitude effects attributed to sodium channel inactivation. Taken together, our results demonstrate that a neuron can respond very differently to the same stimulus, depending on its previous activity or the current brain state. This finding may be particularly relevant when other regulatory mechanisms of ionic homeostasis are challenged, for example, during pathological states of glial impairment or oxygen deprivation. Finally, categorization of cortical neurons as intrinsically bursting or regular spiking may be biased by the ionic concentrations at the time of the observation, highlighting the non-static nature of neuronal dynamics.

An interictal measurement of cerebral oxygen extraction fraction in MRI-negative refractory epilepsy using quantitative susceptibility mapping

PURPOSE

Oxygen extraction fraction (OEF) can be a factor to identify brain tissue's disability in epileptic patients. This study aimed to assess the OEF's level measurement in refractory epileptic patients (REPs) using a quantitative susceptibility mapping (QSM) method and to determine whether the OEF parameters change.

METHODS

QSM-OEF maps of 26 REPs and 16 healthy subjects were acquired using 3T MRI with a 64-channel coil. Eighteen regions-of-interest (ROIs) were chosen around the cortex in one appropriate slice of the brain and the mean QSM-OEF for each ROI was obtained. The correlations of QSM-OEF among different clinical characteristics of the disease, as well as between the patients and normal subjects, were also investigated.

RESULTS

QSM-OEF was shown to be significantly higher in REPs (44.9 ± 5.8) than that in HS (41.9 ± 6.2) (p < 0.05). Mean QSM-OEF was statistically lower in the ipsilateral side (44.5 ± 6.6) compared to the contralateral side (46.4 ± 6.8) (P < 0.01). QSM-OEF was illustrated to have a strong positive correlation with the attack duration (r = 0.6), and a moderate negative correlation with the attack frequency (r = -0.3). Using an optimized support vector machine algorithm, we could predict the disease in subjects having abnormal OEF values in the brain-selected-ROIs with sensitivity, specificity, AUC, and the precision of 0.96, 1, 0.98, and 1, respectively.

CONCLUSIONS

The results of this study revealed that QSM-OEF of the REPs' brain is higher than that of HS, which indicates that QSM-OEF is associated with disease activity.

On the origin of ultraslow spontaneous Na+ fluctuations in neurons of the neonatal forebrain

Spontaneous neuronal and astrocytic activity in the neonate forebrain is believed to drive the maturation of individual cells and their integration into complex brain-region-specific networks. The previously reported forms include bursts of electrical activity and oscillations in intracellular Ca2+ concentration. Here, we use ratiometric Na+ imaging to demonstrate spontaneous fluctuations in the intracellular Na+ concentration of CA1 pyramidal neurons and astrocytes in tissue slices obtained from the hippocampus of mice at postnatal days 2-4 (P2-4). These occur at very low frequency (∼2/h), can last minutes with amplitudes up to several millimolar, and mostly disappear after the first postnatal week. To further investigate their mechanisms, we model a network consisting of pyramidal neurons and interneurons. Experimentally observed Na+ fluctuations are mimicked when GABAergic inhibition in the simulated network is made depolarizing. Both our experiments and computational model show that blocking voltage-gated Na+ channels or GABAergic signaling significantly diminish the neuronal Na+ fluctuations. On the other hand, blocking a variety of other ion channels, receptors, or transporters including glutamatergic pathways does not have significant effects. Our model also shows that the amplitude and duration of Na+ fluctuations decrease as we increase the strength of glial K+ uptake. Furthermore, neurons with smaller somatic volumes exhibit fluctuations with higher frequency and amplitude. As opposed to this, larger extracellular to intracellular volume ratio observed in neonatal brain exerts a dampening effect. Finally, our model predicts that these periods of spontaneous Na+ influx leave neonatal neuronal networks more vulnerable to seizure-like states when compared with mature brain.NEW & NOTEWORTHY Spontaneous activity in the neonate forebrain plays a key role in cell maturation and brain development. We report spontaneous, ultraslow, asynchronous fluctuations in the intracellular Na+ concentration of neurons and astrocytes. We show that this activity is not correlated with the previously reported synchronous neuronal population bursting or Ca2+ oscillations, both of which occur at much faster timescales. Furthermore, extracellular K+ concentration remains nearly constant. The spontaneous Na+ fluctuations disappear after the first postnatal week.

Glial Modulation of Electrical Rhythms in a Neuroglial Network Model of Epilepsy

OBJECTIVE

An important EEG-based biomarker for epilepsy is the phase-amplitude cross-frequency coupling (PAC) of electrical rhythms; however, the underlying pathways of these pathologic markers are not always clear. Since glial cells have been shown to play an active role in neuroglial networks, it is likely that some of these PAC markers are modulated via glial effects.

METHODS

We developed a 4-unit hybrid model of a neuroglial network, consisting of 16 sub-units, that combines a mechanistic representation of neurons with an oscillator-based Cognitive Rhythm Generator (CRG) representation of glial cells-astrocytes and microglia. The model output was compared with recorded generalized tonic-clonic patient data, both in terms of PAC features, and state classification using an unsupervised hidden Markov model (HMM).

RESULTS

The neuroglial model output showed PAC features similar to those observed in epileptic seizures. These generated PAC features were able to accurately identify spontaneous epileptiform discharges (SEDs) as seizure-like states, as well as a postictal-like state following the long-duration SED, when applied to the HMM machine learning algorithm trained on patient data. The evolution profile of the maximal PAC during the SED compared well with patient data, showing similar association with the duration of the postictal state.

CONCLUSION

The hybrid neuroglial network model was able to generate PAC features similar to those observed in ictal and postictal epileptic states, which has been used for state classification and postictal state duration prediction.

SIGNIFICANCE

Since PAC biomarkers are important for epilepsy research and postictal state duration has been linked with risk of sudden unexplained death in epilepsy, this model suggests glial synaptic effects as potential targets for further analysis and treatment.

Dynamics of a neuron-glia system: the occurrence of seizures and the influence of electroconvulsive stimuli : A mathematical and numerical study

In this paper, we investigate the dynamics of a neuron-glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro-scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single-cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters.

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.

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.

Ionic and synaptic mechanisms of seizure generation and epileptogenesis

The biophysical mechanisms underlying epileptogenesis and the generation of seizures remain to be better understood. Among many factors triggering epileptogenesis are traumatic brain injury breaking normal synaptic homeostasis and genetic mutations disrupting ionic concentration homeostasis. Impairments in these mechanisms, as seen in various brain diseases, may push the brain network to a pathological state characterized by increased susceptibility to unprovoked seizures. Here, we review recent computational studies exploring the roles of ionic concentration dynamics in the generation, maintenance, and termination of seizures. We further discuss how ionic and synaptic homeostatic mechanisms may give rise to conditions which prime brain networks to exhibit recurrent spontaneous seizures and epilepsy.

Mathematical model of Na-K-Cl homeostasis in ictal and interictal discharges

Despite big experimental data on the phenomena and mechanisms of the generation of ictal and interictal discharges (IDs and IIDs), mathematical models that can describe the synaptic interactions of neurons and the ionic dynamics in biophysical detail are not well-established. Based on experimental recordings of combined hippocampal-entorhinal cortex slices from rats in a high-potassium and a low-magnesium solution containing 4-aminopyridine as well as previous observations of similar experimental models, this type of mathematical model has been developed. The model describes neuronal excitation through the application of the conductance-based refractory density approach for three neuronal populations: two populations of glutamatergic neurons with hyperpolarizing and depolarizing GABAergic synapses and one GABAergic population. The ionic dynamics account for the contributions of voltage-gated and synaptic channels, active and passive transporters, and diffusion. The relatively slow dynamics of potassium, chloride, and sodium ion concentrations determine the transitions from pure GABAergic IIDs to IDs and GABA-glutamatergic IIDs. The model reproduces different types of IIDs, including those initiated by interneurons; repetitive IDs; tonic and bursting modes of an ID composed of clustered IID-like events. The simulations revealed contributions from different ionic channels to the ion concentration dynamics before and during ID generation. The proposed model is a step forward to an optimal mathematical description of the mechanisms of epileptic discharges.

Origin of slow spontaneous resting-state neuronal fluctuations in brain networks

Resting- or baseline-state low-frequency (0.01-0.2 Hz) brain activity is observed in fMRI, EEG, and local field potential recordings. These fluctuations were found to be correlated across brain regions and are thought to reflect neuronal activity fluctuations between functionally connected areas of the brain. However, the origin of these infra-slow resting-state fluctuations remains unknown. Here, using a detailed computational model of the brain network, we show that spontaneous infra-slow (<0.05 Hz) activity could originate due to the ion concentration dynamics. The computational model implemented dynamics for intra- and extracellular K⁺ and Na⁺ and intracellular Cl^- ions, Na⁺/K⁺ exchange pump, and KCC2 cotransporter. In the network model simulating resting awake-like brain state, we observed infra-slow fluctuations in the extracellular K⁺ concentration, Na⁺/K⁺ pump activation, firing rate of neurons, and local field potentials. Holding K⁺ concentration constant prevented generation of the infra-slow fluctuations. The amplitude and peak frequency of this activity were modulated by the Na⁺/K⁺ pump, AMPA/GABA synaptic currents, and glial properties. Further, in a large-scale network with long-range connections based on CoCoMac connectivity data, the infra-slow fluctuations became synchronized among remote clusters similar to the resting-state activity observed in vivo. Overall, our study proposes that ion concentration dynamics mediated by neuronal and glial activity may contribute to the generation of very slow spontaneous fluctuations of brain activity that are reported as the resting-state fluctuations in fMRI and EEG recordings.

Astrocytic Kir4.1 channels and gap junctions account for spontaneous epileptic seizure

Experimental recordings in hippocampal slices indicate that astrocytic dysfunction may cause neuronal hyper-excitation or seizures. Considering that astrocytes play important roles in mediating local uptake and spatial buffering of K⁺ in the extracellular space of the cortical circuit, we constructed a novel model of an astrocyte-neuron network module consisting of a single compartment neuron and 4 surrounding connected astrocytes and including extracellular potassium dynamics. Next, we developed a new model function for the astrocyte gap junctions, connecting two astrocyte-neuron network modules. The function form and parameters of the gap junction were based on nonlinear regression fitting of a set of experimental data published in previous studies. Moreover, we have created numerical simulations using the above single astrocyte-neuron network module and the coupled astrocyte-neuron network modules. Our model validates previous experimental observations that both Kir4.1 channels and gap junctions play important roles in regulating the concentration of extracellular potassium. In addition, we also observe that changes in Kir4.1 channel conductance and gap junction strength induce spontaneous epileptic activity in the absence of external stimuli.

Astrocytes are critical regulators of normal physiological activity in the central nervous system, and one of their key functions is removing extracellular K⁺. In recent years, numerous biological studies have shown that astrocytic Kir4.1 channels and gap junctions between astrocytes act as major K⁺ clearance mechanisms. Dysfunction of either of these regulatory mechanisms may cause generation of K⁺-induced seizures. However, it is unclear how and to what extent these two K⁺-regulating processes lead to spontaneous epileptic activity. These questions were addressed in the present study by constructing novel single astrocyte-neuron network models and a coupled astrocyte-neuron module network connected by an astrocyte gap junction based on existing experimental observations and previous theoretical reports. Simulation results first verified that either down-regulation of astrocytic Kir4.1 channels or a decrease of the gap junction strength between astrocytes causes neuropathological hyper-excitability and spontaneous epileptic activity. These results imply that dysfunctional astrocytes should be considered as targets for therapeutic strategies in epilepsy.

Predicting the spatiotemporal diversity of seizure propagation and termination in human focal epilepsy

Recent studies have shown that seizures can spread and terminate across brain areas via a rich diversity of spatiotemporal patterns. In particular, while the location of the seizure onset area is usually invariant across seizures in an individual patient, the source of traveling (2–3 Hz) spike-and-wave discharges during seizures can either move with the slower propagating ictal wavefront or remain stationary at the seizure onset area. Furthermore, although many focal seizures terminate synchronously across brain areas, some evolve into distinct ictal clusters and terminate asynchronously. Here, we introduce a unifying perspective based on a new neural field model of epileptic seizure dynamics. Two main mechanisms, the co-existence of wave propagation in excitable media and coupled-oscillator dynamics, together with the interaction of multiple time scales, account for the reported diversity. We confirm our predictions in seizures and tractography data obtained from patients with pharmacologically resistant epilepsy. Our results contribute toward patient-specific seizure modeling.

A major goal of epilepsy research is understanding the spatiotemporal dynamics of seizure. Here, the authors extend the Epileptor neural mass model into a neural field model, in order to provide a unified and patient-specific model of seizure initiation, propagation, and termination.

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.

Low-to-High Cross-Frequency Coupling in the Electrical Rhythms as Biomarker for Hyperexcitable Neuroglial Networks of the Brain

OBJECTIVE

One of the features used in the study of hyperexcitablility is high-frequency oscillations (HFOs, >80 Hz). HFOs have been reported in the electrical rhythms of the brain's neuroglial networks under physiological and pathological conditions. Cross-frequency coupling (CFC) of HFOs with low-frequency rhythms was used to identify pathologic HFOs in the epileptogenic zones of epileptic patients and as a biomarker for the severity of seizure-like events in genetically modified rodent models. We describe a model to replicate reported CFC features extracted from recorded local field potentials (LFPs) representing network properties.

METHODS

This study deals with a four-unit neuroglial cellular network model where each unit incorporates pyramidal cells, interneurons, and astrocytes. Three different pathways of hyperexcitability generation-Na - ATPase pump, glial potassium clearance, and potassium afterhyperpolarization channel-were used to generate LFPs. Changes in excitability, average spontaneous electrical discharge (SED) duration, and CFC were then measured and analyzed.

RESULTS

Each parameter caused an increase in network excitability and the consequent lengthening of the SED duration. Short SEDs showed CFC between HFOs and theta oscillations (4-8 Hz), but in longer SEDs the low frequency changed to the delta range (1-4 Hz).

CONCLUSION

Longer duration SEDs exhibit CFC features similar to those reported by our team.

SIGNIFICANCE

First, Identifying the exponential relationship between network excitability and SED durations; second, highlighting the importance of glia in hyperexcitability (as they relate to extracellular potassium); and third, elucidation of the biophysical basis for CFC coupling features.

Changes of Ionic Concentrations During Seizure Transitions - A Modeling Study

Traditionally, it is considered that neuronal synchronization in epilepsy is caused by a chain reaction of synaptic excitation. However, it has been shown that synchronous epileptiform activity may also arise without synaptic transmission. In order to investigate the respective roles of synaptic interactions and nonsynaptic mechanisms in seizure transitions, we developed a computational model of hippocampal cells, involving the extracellular space, realistic dynamics of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] ions, glial uptake and extracellular diffusion mechanisms. We show that the network behavior with fixed ionic concentrations may be quite different from the neurons' behavior when more detailed modeling of ionic dynamics is included. In particular, we show that in the extended model strong discharge of inhibitory interneurons may result in long lasting accumulation of extracellular [Formula: see text], which sustains the depolarization of the principal cells and causes their pathological discharges. This effect is not present in a reduced, purely synaptic network. These results point to the importance of nonsynaptic mechanisms in the transition to seizure.

Glycolytic inhibition by 2-deoxy-d-glucose abolishes both neuronal and network bursts in an in vitro seizure model

Neuronal activity is energy demanding and coupled to cellular metabolism. In this study, we investigated the effects of glycolytic inhibition with 2-deoxy-d-glucose (2-DG) on basal membrane properties, spontaneous neuronal firing, and epileptiform network bursts in hippocampal slices. The effect of glycolytic inhibition on basal membrane properties was examined in hippocampal CA1 neurons, which are not ordinarily active spontaneously. Intracellular application of 2-DG did not significantly alter the membrane input resistance, action-potential threshold, firing pattern, or input-output relationship of these neurons compared with simultaneously recorded neighboring neurons without intracellular 2-DG. The effect of glycolytic inhibition on neuronal firing was tested in spontaneously active hippocampal neurons (CA3) when synaptic transmission was left intact or blocked with AMPA, NMDA, and GABA_A receptor antagonists (DNQX, APV, and bicuculline, respectively). Under both conditions (synaptic activity intact or blocked), bath application of 2-DG (2 mM) blocked spontaneous firing in ~2/3 (67 and 71%, respectively) of CA3 pyramidal neurons. In contrast, neuronal firing of CA3 neurons persisted when 2-DG was applied intracellularly, suggesting that glycolytic inhibition of individual neurons is not sufficient to stop neuronal firing. The effects of 2-DG on epileptiform network bursts in area CA3 were tested in Mg²⁺-free medium containing 50 µM 4-aminopyridine. Bath application of 2-DG abolished these epileptiform bursts in a dose-dependent and all-or-none manner. Taken together, these data suggest that altered glucose metabolism profoundly affects cellular and network hyperexcitability and that glycolytic inhibition by 2-DG can effectively abrogate epileptiform activity.NEW & NOTEWORTHY Neuronal activity is highly energy demanding and coupled to cellular metabolism. In this study, we demonstrate that glycolytic inhibition with 2-deoxy-d-glucose (2-DG) effectively suppresses spontaneous neuronal firing and epileptiform bursts in hippocampal slices. These data suggest that an altered metabolic state can profoundly affect cellular and network excitability, and that the glycolytic inhibitor 2-DG may hold promise as a novel treatment of drug-resistant epilepsy.

Interaction between synaptic inhibition and glial-potassium dynamics leads to diverse seizure transition modes in biophysical models of human focal seizures

How focal seizures initiate and evolve in human neocortex remains a fundamental problem in neuroscience. Here, we use biophysical neuronal network models of neocortical patches to study how the interaction between inhibition and extracellular potassium ([K (+)] o ) dynamics may contribute to different types of focal seizures. Three main types of propagated focal seizures observed in recent intracortical microelectrode recordings in humans were modelled: seizures characterized by sustained (∼30-60 Hz) gamma local field potential (LFP) oscillations; seizures where the onset in the propagated site consisted of LFP spikes that later evolved into rhythmic (∼2-3 Hz) spike-wave complexes (SWCs); and seizures where a brief stage of low-amplitude fast-oscillation (∼10-20 Hz) LFPs preceded the SWC activity. Our findings are fourfold: (1) The interaction between elevated [K (+)] o (due to abnormal potassium buffering by glial cells) and the strength of synaptic inhibition plays a predominant role in shaping these three types of seizures. (2) Strengthening of inhibition leads to the onset of sustained narrowband gamma seizures. (3) Transition into SWC seizures is obtained either by the weakening of inhibitory synapses, or by a transient strengthening followed by an inhibitory breakdown (e.g. GABA depletion). This reduction or breakdown of inhibition among fast-spiking (FS) inhibitory interneurons increases their spiking activity and leads them eventually into depolarization block. Ictal spike-wave discharges in the model are then sustained solely by pyramidal neurons. (4) FS cell dynamics are also critical for seizures where the evolution into SWC activity is preceded by low-amplitude fast oscillations. Different levels of elevated [K (+)] o were important for transitions into and maintenance of sustained gamma oscillations and SWC discharges. Overall, our modelling study predicts that the interaction between inhibitory interneurons and [K (+)] o glial buffering under abnormal conditions may explain different types of ictal transitions and dynamics during propagated seizures in human focal epilepsy.

The influence of potassium concentration on epileptic seizures in a coupled neuronal model in the hippocampus

Experiments on hippocampal slices have recorded that a novel pattern of epileptic seizures with alternating excitatory and inhibitory activities in the CA1 region can be induced by an elevated potassium ion (K(+)) concentration in the extracellular space between neurons and astrocytes (ECS-NA). To explore the intrinsic effects of the factors (such as glial K(+) uptake, Na(+)-K(+)-ATPase, the K(+) concentration of the bath solution, and K(+) lateral diffusion) influencing K(+) concentration in the ECS-NA on the epileptic seizures recorded in previous experiments, we present a coupled model composed of excitatory and inhibitory neurons and glia in the CA1 region. Bifurcation diagrams showing the glial K(+) uptake strength with either the Na(+)-K(+)-ATPase pump strength or the bath solution K(+) concentration are obtained for neural epileptic seizures. The K(+) lateral diffusion leads to epileptic seizure in neurons only when the synaptic conductance values of the excitatory and inhibitory neurons are within an appropriate range. Finally, we propose an energy factor to measure the metabolic demand during neuron firing, and the results show that different energy demands for the normal discharges and the pathological epileptic seizures of the coupled neurons.

Dynamic transition of neuronal firing induced by abnormal astrocytic glutamate oscillation

The gliotransmitter glutamate released from astrocytes can modulate neuronal firing by activating neuronal N-methyl-D-aspartic acid (NMDA) receptors. This enables astrocytic glutamate(AG) to be involved in neuronal physiological and pathological functions. Based on empirical results and classical neuron-glial "tripartite synapse" model, we propose a practical model to describe extracellular AG oscillation, in which the fluctuation of AG depends on the threshold of calcium concentration, and the effect of AG degradation is considered as well. We predict the seizure-like discharges under the dysfunction of AG degradation duration. Consistent with our prediction, the suppression of AG uptake by astrocytic transporters, which operates by modulating the AG degradation process, can account for the emergence of epilepsy.

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.

Large extracellular space leads to neuronal susceptibility to ischemic injury in a Na+/K+ pumps-dependent manner

The extent of anoxic depolarization (AD), the initial electrophysiological event during ischemia, determines the degree of brain region-specific neuronal damage. Neurons in higher brain regions exhibiting nonreversible, strong AD are more susceptible to ischemic injury as compared to cells in lower brain regions that exhibit reversible, weak AD. While the contrasting ADs in different brain regions in response to oxygen-glucose deprivation (OGD) is well established, the mechanism leading to such differences is not clear. Here we use computational modeling to elucidate the mechanism behind the brain region-specific recovery from AD. Our extended Hodgkin-Huxley (HH) framework consisting of neural spiking dynamics, processes of ion accumulation, and ion homeostatic mechanisms unveils that glial-vascular K(+) clearance and Na(+)/K(+)-exchange pumps are key to the cell's recovery from AD. Our phase space analysis reveals that the large extracellular space in the upper brain regions leads to impaired Na(+)/K(+)-exchange pumps so that they function at lower than normal capacity and are unable to bring the cell out of AD after oxygen and glucose is restored.

The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little of how cell volume affects neuronal dynamics. We here performed the first detailed study of the effects of cell volume on neuronal dynamics. By incorporating cell swelling together with dynamic ion concentrations and oxygen supply into Hodgkin-Huxley type spiking dynamics, we demonstrate the spontaneous transition between epileptic seizure and spreading depression states as the cell swells and contracts in response to changes in osmotic pressure. Our use of volume as an order parameter further revealed a dynamical definition for the experimentally described physiological ceiling that separates seizure from spreading depression, as well as predicted a second ceiling that demarcates spreading depression from anoxic depolarization. Our model highlights the neuroprotective role of glial K buffering against seizures and spreading depression, and provides novel insights into anoxic depolarization and the relevant cell swelling during ischemia. We argue that the dynamics of seizures, spreading depression, and anoxic depolarization lie along a continuum of the repertoire of the neuron membrane that can be understood only when the dynamic ion concentrations, oxygen homeostasis,and cell swelling in response to osmotic pressure are taken into consideration. Our results demonstrate the feasibility of a unified framework for a wide range of neuronal behaviors that may be of substantial importance in the understanding of and potentially developing universal intervention strategies for these pathological states.

Role of multiple-scale modeling of epilepsy in seizure forecasting

Over the past three decades, a number of seizure prediction, or forecasting, methods have been developed. Although major achievements were accomplished regarding the statistical evaluation of proposed algorithms, it is recognized that further progress is still necessary for clinical application in patients. The lack of physiological motivation can partly explain this limitation. Therefore, a natural question is raised: can computational models of epilepsy be used to improve these methods? Here, we review the literature on the multiple-scale neural modeling of epilepsy and the use of such models to infer physiologic changes underlying epilepsy and epileptic seizures. The authors argue how these methods can be applied to advance the state-of-the-art in seizure forecasting.

Characterising seizures in anti-NMDA-receptor encephalitis with dynamic causal modelling

We characterised the pathophysiology of seizure onset in terms of slow fluctuations in synaptic efficacy using EEG in patients with anti-N-methyl-d-aspartate receptor (NMDA-R) encephalitis. EEG recordings were obtained from two female patients with anti-NMDA-R encephalitis with recurrent partial seizures (ages 19 and 31). Focal electrographic seizure activity was localised using an empirical Bayes beamformer. The spectral density of reconstructed source activity was then characterised with dynamic causal modelling (DCM). Eight models were compared for each patient, to evaluate the relative contribution of changes in intrinsic (excitatory and inhibitory) connectivity and endogenous afferent input. Bayesian model comparison established a role for changes in both excitatory and inhibitory connectivity during seizure activity (in addition to changes in the exogenous input). Seizures in both patients were associated with a sequence of changes in inhibitory and excitatory connectivity; a transient increase in inhibitory connectivity followed by a transient increase in excitatory connectivity and a final peak of excitatory–inhibitory balance at seizure offset. These systematic fluctuations in excitatory and inhibitory gain may be characteristic of (anti NMDA-R encephalitis) seizures. We present these results as a case study and replication to motivate analyses of larger patient cohorts, to see whether our findings generalise and further characterise the mechanisms of seizure activity in anti-NMDA-R encephalitis.

•

We characterised seizures in patient with anti-NMDA-R encephalitis using EEG.

•

Dynamic causal modelling was used to estimate causes of seizure activity.

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Characteristic variation of excitatory–inhibitory balance during seizure activity.

•

This variation was seen for seizures within and between patients.

Computational modeling of seizure dynamics using coupled neuronal networks: factors shaping epileptiform activity

Epileptic seizure dynamics span multiple scales in space and time. Understanding seizure mechanisms requires identifying the relations between seizure components within and across these scales, together with the analysis of their dynamical repertoire. Mathematical models have been developed to reproduce seizure dynamics across scales ranging from the single neuron to the neural population. In this study, we develop a network model of spiking neurons and systematically investigate the conditions, under which the network displays the emergent dynamic behaviors known from the Epileptor, which is a well-investigated abstract model of epileptic neural activity. This approach allows us to study the biophysical parameters and variables leading to epileptiform discharges at cellular and network levels. Our network model is composed of two neuronal populations, characterized by fast excitatory bursting neurons and regular spiking inhibitory neurons, embedded in a common extracellular environment represented by a slow variable. By systematically analyzing the parameter landscape offered by the simulation framework, we reproduce typical sequences of neural activity observed during status epilepticus. We find that exogenous fluctuations from extracellular environment and electro-tonic couplings play a major role in the progression of the seizure, which supports previous studies and further validates our model. We also investigate the influence of chemical synaptic coupling in the generation of spontaneous seizure-like events. Our results argue towards a temporal shift of typical spike waves with fast discharges as synaptic strengths are varied. We demonstrate that spike waves, including interictal spikes, are generated primarily by inhibitory neurons, whereas fast discharges during the wave part are due to excitatory neurons. Simulated traces are compared with in vivo experimental data from rodents at different stages of the disorder. We draw the conclusion that slow variations of global excitability, due to exogenous fluctuations from extracellular environment, and gap junction communication push the system into paroxysmal regimes. We discuss potential mechanisms underlying such machinery and the relevance of our approach, supporting previous detailed modeling studies and reflecting on the limitations of our methodology.

Neurons communicate via different types of synapses on very fast time scales. The combination of hundred thousand of such interconnected cells within a fluctuating extracellular environment forms a complex network that gives rise to function and behavior via the formation of dynamical patterns of activity. In the context of epilepsy, the functional properties of the network at the source of a seizure are disrupted by a possibly large set of factors at the cellular and molecular levels. It is therefore needed to sacrifice some biological accuracy to model seizure dynamics in favor of macroscopic realizations. Here, we present a neuronal network model that convenes both neuronal and network representations with the goal to describe brain dynamics involved in the development of epilepsy. We compare our modeling results with animal in vivo recordings to validate our approach in the context of seizures. Such system-level methodology has significant bearing in understanding neuronal network dynamics that entangle multiple synaptic and extracellular modalities.

Oxygen-sensing scaffolds for 3-dimensional cell and tissue culture

Porous membrane scaffolds are widely used materials for three-dimensional cell cultures and tissue models. Additional functional modification of such scaffolds can significantly extend their use and operational performance. Here we describe hybrid microporous polystyrene-based scaffolds impregnated with a phosphorescent O2-sensitive dye PtTFPP, optimized for live cell fluorescence microscopy and imaging of O2 distribution in cultured cells. Modified scaffolds possess high brightness, convenient spectral characteristics (534 nm excitation, 650 nm emission), stable and robust response to pO2 in phosphorescence intensity and lifetime imaging modes (>twofold response over 21/0% O2), such as confocal PLIM. They are suitable for prolonged use under standard culturing conditions without affecting cell viability, and for multi-parametric imaging analysis of cultured cells and tissue samples. We tested the O2 scaffolds with cultured cancer cells (HCT116), multicellular aggregates (PC12) and rat brain slices and showed that they can inform on tissue oxygenation at different depths and cell densities, changes in respiration activity, viability and responses to drug treatment. Using this method multiplexed with staining of dead cells (CellTox Green) and active mitochondria (TMRM), we demonstrated that decreased O2 (20-24 μM) in scaffold corresponds to highest expression of tyrosine hydroxylase in PC12 cells. Such hypoxia is also beneficial for action of hypoxia-specific anti-cancer drug tirapazamine (TPZ). Thus, O2 scaffolds allow for better control of conditions in 3D tissue cultures, and are useful for a broad range of biomaterials and physiological studies.

Modeling focal epileptic activity in the Wilson-cowan model with depolarization block

Measurements of neuronal signals during human seizure activity and evoked epileptic activity in experimental models suggest that, in these pathological states, the individual nerve cells experience an activity driven depolarization block, i.e. they saturate. We examined the effect of such a saturation in the Wilson-Cowan formalism by adapting the nonlinear activation function; we substituted the commonly applied sigmoid for a Gaussian function. We discuss experimental recordings during a seizure that support this substitution. Next we perform a bifurcation analysis on the Wilson-Cowan model with a Gaussian activation function. The main effect is an additional stable equilibrium with high excitatory and low inhibitory activity. Analysis of coupled local networks then shows that such high activity can stay localized or spread. Specifically, in a spatial continuum we show a wavefront with inhibition leading followed by excitatory activity. We relate our model simulations to observations of spreading activity during seizures.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1186/s13408-015-0019-4) contains supplementary material 1.

Unification of neuronal spikes, seizures, and spreading depression

The pathological phenomena of seizures and spreading depression have long been considered separate physiological events in the brain. By incorporating conservation of particles and charge, and accounting for the energy required to restore ionic gradients, we extend the classic Hodgkin-Huxley formalism to uncover a unification of neuronal membrane dynamics. By examining the dynamics as a function of potassium and oxygen, we now account for a wide range of neuronal activities, from spikes to seizures, spreading depression (whether high potassium or hypoxia induced), mixed seizure and spreading depression states, and the terminal anoxic "wave of death." Such a unified framework demonstrates that all of these dynamics lie along a continuum of the repertoire of the neuron membrane. Our results demonstrate that unified frameworks for neuronal dynamics are feasible, can be achieved using existing biological structures and universal physical conservation principles, and may be of substantial importance in enabling our understanding of brain activity and in the control of pathological states.

Critical role for resource constraints in neural models

Criticality has emerged as a leading dynamical candidate for healthy and pathological neuronal activity. At the heart of criticality in neural systems is the need for parameters to be tuned to specific values or for the existence of self-organizing mechanisms. Existing models lack precise physiological descriptions for how the brain maintains its tuning near a critical point. In this paper we argue that a key ingredient missing from the field is a formulation of reciprocal coupling between neural activity and metabolic resources. We propose that the constraint of optimizing the balance between energy use and activity plays a major role in tuning brain states to lie near criticality. Important recent findings aligned with our viewpoint have emerged from analyses of disorders that involve severe metabolic disturbances and alter scale-free properties of brain dynamics, including burst suppression. Moreover, we argue that average shapes of neuronal avalanches are a signature of scale-free activity that offers sharper insights into underlying mechanisms than afforded by traditional analyses of avalanche statistics.

Oxygen and seizure dynamics: I. Experiments

We utilized a novel ratiometric nanoquantum dot fluorescence resonance energy transfer (NQD-FRET) optical sensor to quantitatively measure oxygen dynamics from single cell microdomains during hypoxic episodes as well as during 4-aminopyridine (4-AP)-induced spontaneous seizure-like events in rat hippocampal slices. Coupling oxygen sensing with electrical recordings, we found the greatest reduction in the O2 concentration ([O2]) in the densely packed cell body stratum (st.) pyramidale layer of the CA1 and differential layer-specific O2 dynamics between the st. pyramidale and st. oriens layers. These hypoxic decrements occurred up to several seconds before seizure onset could be electrically measured extracellularly. Without 4-AP, we quantified a narrow range of [O2], similar to the endogenous hypoxia found before epileptiform activity, which permits a quiescent network to enter into a seizure-like state. We demonstrated layer-specific patterns of O2 utilization accompanying layer-specific neuronal interplay in seizure. None of the oxygen overshoot artifacts seen with polarographic measurement techniques were observed. We therefore conclude that endogenously generated hypoxia may be more than just a consequence of increased cellular excitability but an influential and critical factor for orchestrating network dynamics associated with epileptiform activity.