Seizure-like afterdischarges simulated in a model neuron

Mathematical modeling of intracellular osmolarity and cell volume stabilization: The Donnan effect and ion transport

The presence of impermeant molecules within a cell can lead to an increase in cell volume through the influx of water driven by osmosis. This phenomenon is known as the Donnan (or Gibbs-Donnan) effect. Animal cells actively transport ions to counteract the Donnan effect and regulate their volume, actively pumping Na+ out and K+ into their cytosol using the Na+/K+ ATPase (NKA) pump. The pump-leak equations (PLEs) are a system of algebraic-differential equations to model the membrane potential, ion (Na+, K+, and Cl-), and water flux across the cell membrane, which provide insight into how the combination of passive ions fluxes and active transport contribute to stabilizing cell volume. Our broad objective is to provide analytical insight into the PLEs through three lines of investigation: (1) we show that the provision of impermeant extracellular molecules can stabilize the volume of a passive cell; (2) we demonstrate that the mathematical form of the NKA pump is not as important as the stoichiometry for cell stabilization; and (3) we investigate the interaction between the NKA pump and cation-chloride co-transporters (CCCs) on cell stabilization, showing that NCC can destabilize a cell while NKCC and KCC can stabilize it. We incorporate extracellular impermeant molecules, NKA pump, and CCCs into the PLEs and derive the exact formula for the steady states in terms of all the parameters. This analytical expression enables us to easily explore the effect of each of the system parameters on the existence and stability of the steady states.

Metabolic implications of axonal demyelination and its consequences for synchronized network activity: An in silico and in vitro study

Myelination enhances the conduction velocity of action potentials (AP) and increases energy efficiency. Thick myelin sheaths are typically found on large-distance axonal connections or in fast-spiking interneurons, which are critical for synchronizing neuronal networks during gamma-band oscillations. Loss of myelin sheath is associated with multiple alterations in axonal architecture leading to impaired AP propagation. While numerous studies are devoted to the effects of demyelination on conduction velocity, the metabolic effects and the consequences for network synchronization have not been investigated. Here we present a unifying computational model for electrophysiology and metabolism of the myelinated axon. The computational model suggested that demyelination not only decreases the AP speed but AP propagation in demyelinated axons requires compensatory processes like mitochondrial mass increase and a switch from saltatory to continuous propagation to rescue axon functionality at the cost of reduced AP propagation speed and increased energy expenditure. Indeed, these predictions were proven to be true in a culture model of demyelination where the pharmacologically-induced loss of myelin was associated with increased oxygen consumption rates, and a significant broadening of bandwidth as well as a decrease in the power of gamma oscillations.

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.

Initiation and entrainment of multicellular automaticity via diffusion limited extracellular domains

Electrically excitable cells often spontaneously and synchronously depolarize in vitro and in vivo preparations. It remains unclear how cells entrain and autorhythmically activate above the intrinsic mean activation frequency of isolated cells with or without pacemaking mechanisms. Recent studies suggest that cyclic ion accumulation and depletion in diffusion-limited extracellular volumes modulate electrophysiology by ephaptic mechanisms (nongap junction or synaptic coupling). This report explores how potassium accumulation and depletion in a restricted extracellular domain induces spontaneous action potentials in two different computational models of excitable cells without gap junctional coupling: Hodgkin-Huxley and Luo-Rudy. Importantly, neither model will spontaneously activate on its own without external stimuli. Simulations demonstrate that cells sharing a diffusion-limited extracellular compartment can become autorhythmic and entrained despite intercellular electrical heterogeneity. Autorhythmic frequency is modulated by the cleft volume and potassium fluxes through the cleft. Additionally, inexcitable cells can suppress or induce autorhythmic activity in an excitable cell via a shared cleft. Diffusion-limited shared clefts can also entrain repolarization. Critically, this model predicts a mechanism by which diffusion-limited shared clefts can initiate, entrain, and modulate multicellular automaticity in the absence of gap junctions.

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.

Ion dynamics at the energy-deprived tripartite synapse

The anatomical and functional organization of neurons and astrocytes at 'tripartite synapses' is essential for reliable neurotransmission, which critically depends on ATP. In low energy conditions, synaptic transmission fails, accompanied by a breakdown of ion gradients, changes in membrane potentials and cell swelling. The resulting cellular damage and cell death are causal to the often devastating consequences of an ischemic stroke. The severity of ischemic damage depends on the age and the brain region in which a stroke occurs, but the reasons for this differential vulnerability are far from understood. In the present study, we address this question by developing a comprehensive biophysical model of a glutamatergic synapse to identify key determinants of synaptic failure during energy deprivation. Our model is based on fundamental biophysical principles, includes dynamics of the most relevant ions, i.e., Na+, K+, Ca2+, Cl- and glutamate, and is calibrated with experimental data. It confirms the critical role of the Na+/K+-ATPase in maintaining ion gradients, membrane potentials and cell volumes. Our simulations demonstrate that the system exhibits two stable states, one physiological and one pathological. During energy deprivation, the physiological state may disappear, forcing a transit to the pathological state, which can be reverted when blocking voltage-gated Na+ and K+ channels. Our model predicts that the transition to the pathological state is favoured if the extracellular space fraction is small. A reduction in the extracellular space volume fraction, as, e.g. observed with ageing, will thus promote the brain's susceptibility to ischemic damage. Our work provides new insights into the brain's ability to recover from energy deprivation, with translational relevance for diagnosis and treatment of ischemic strokes.

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.

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.

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.

Evolution of our understanding of cell volume regulation by the pump-leak mechanism

Kay and Blaustein recount the history of the pump-leak mechanism, which stabilizes cell volume by pumping sodium ions out of cells.

All animal cells are surrounded by a flexible plasma membrane that is permeable to water and to small ions. Cells thus face a fundamental problem: the considerable tension that their membranes would experience if the osmotic influx of water, driven by the presence of impermeant intracellular ions, was left unopposed. The pivotal study that described the cell’s remedy for this impending osmotic catastrophe—the “pump-leak mechanism” (PLM)—was published in the Journal of General Physiology by Tosteson and Hoffman in 1960. Their work revealed how the sodium pump stabilizes cell volume by eliminating the osmotic gradient. Here we describe the mechanistic basis of the PLM, trace the history of its discovery, and place it into the context of our current understanding.

Slow moving neural source in the epileptic hippocampus can mimic progression of human seizures

Fast and slow neural waves have been observed to propagate in the human brain during seizures. Yet the nature of these waves is difficult to study in a surgical setting. Here, we report an observation of two different traveling waves propagating in the in-vitro epileptic hippocampus at speeds similar to those in the human brain. A fast traveling spike and a slow moving wave were recorded simultaneously with a genetically encoded voltage sensitive fluorescent protein (VSFP Butterfly 1.2) and a high speed camera. The results of this study indicate that the fast traveling spike is NMDA-sensitive but the slow moving wave is not. Image analysis and model simulation demonstrate that the slow moving wave is moving slowly, generating the fast traveling spike and is, therefore, a moving source of the epileptiform activity. This slow moving wave is associated with a propagating neural calcium wave detected with calcium dye (OGB-1) but is independent of NMDA receptors, not related to ATP release, and much faster than those previously recorded potassium waves. Computer modeling suggests that the slow moving wave can propagate by the ephaptic effect like epileptiform activity. These findings provide an alternative explanation for slow propagation seizure wavefronts associated with fast propagating spikes.

A mathematical model of recurrent spreading depolarizations

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

Computational model of interictal discharges triggered by interneurons

Interictal discharges (IIDs) are abnormal waveforms registered in the periods before or between seizures. IIDs that are initiated by GABAergic interneurons have not been mathematically modeled yet. In the present study, a mathematical model that describes the mechanisms of these discharges is proposed. The model is based on the experimental recordings of IIDs in pyramidal neurons of the rat entorhinal cortex and estimations of synaptic conductances during IIDs. IIDs were induced in cortico-hippocampal slices by applying an extracellular solution with 4-aminopyridine, high potassium, and low magnesium concentrations. Two different types of IIDs initiated by interneurons were observed. The first type of IID (IID1) was pure GABAergic. The second type of IID (IID2) was induced by GABAergic excitation and maintained by recurrent interactions of both GABA- and glutamatergic neuronal populations. The model employed the conductance-based refractory density (CBRD) approach, which accurately approximates the firing rate of a population of similar Hodgkin-Huxley-like neurons. The model of coupled excitatory and inhibitory populations includes AMPA, NMDA, and GABA-receptor-mediated synapses and gap junctions. These neurons receive both arbitrary deterministic input and individual colored Gaussian noise. Both types of IIDs were successfully reproduced in the model by setting two different depolarized levels for GABA-mediated current reversal potential. It was revealed that short-term synaptic depression is a crucial factor in ceasing each of the discharges, and it also determines their durations and frequencies.

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.

Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model

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

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.

The influence of depolarization block on seizure-like activity in networks of excitatory and inhibitory neurons

The inhibitory restraint necessary to suppress aberrant activity can fail when inhibitory neurons cease to generate action potentials as they enter depolarization block. We investigate possible bifurcation structures that arise at the onset of seizure-like activity resulting from depolarization block in inhibitory neurons. Networks of conductance-based excitatory and inhibitory neurons are simulated to characterize different types of transitions to the seizure state, and a mean field model is developed to verify the generality of the observed phenomena of excitatory-inhibitory dynamics. Specifically, the inhibitory population's activation function in the Wilson-Cowan model is modified to be non-monotonic to reflect that inhibitory neurons enter depolarization block given strong input. We find that a physiological state and a seizure state can coexist, where the seizure state is characterized by high excitatory and low inhibitory firing rate. Bifurcation analysis of the mean field model reveals that a transition to the seizure state may occur via a saddle-node bifurcation or a homoclinic bifurcation. We explain the hysteresis observed in network simulations using these two bifurcation types. We also demonstrate that extracellular potassium concentration affects the depolarization block threshold; the consequent changes in bifurcation structure enable the network to produce the tonic to clonic phase transition observed in biological epileptic networks.

From in silico astrocyte cell models to neuron-astrocyte network models: A review

The idea that astrocytes may be active partners in synaptic information processing has recently emerged from abundant experimental reports. Because of their spatial proximity to neurons and their bidirectional communication with them, astrocytes are now considered as an important third element of the synapse. Astrocytes integrate and process synaptic information and by doing so generate cytosolic calcium signals that are believed to reflect neuronal transmitter release. Moreover, they regulate neuronal information transmission by releasing gliotransmitters into the synaptic cleft affecting both pre- and postsynaptic receptors. Concurrent with the first experimental reports of the astrocytic impact on neural network dynamics, computational models describing astrocytic functions have been developed. In this review, we give an overview over the published computational models of astrocytic functions, from single-cell dynamics to the tripartite synapse level and network models of astrocytes and neurons.

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.

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.

How does spreading depression spread? Physiology and modeling

Spreading depression (SD) is a wave phenomenon in gray matter tissue. Locally, it is characterized by massive redistribution of ions across cell membranes. As a consequence, there is sustained membrane depolarization and tissue polarization that depress any normal electrical activity. Despite these dramatic events, SD remains difficult to observe in humans noninvasively, which, for long, has slowed advances in this field. The growing appreciation of its clinical importance in migraine and stroke is therefore consistent with an increasing need for computational methods that tackle the complexity of the problem at multiple levels. In this review, we focus on mathematical tools to investigate the question of spread and its two complementary aspects: What are the physiological mechanisms and what is the spatial extent of SD in the cortex? This review discusses two types of models used to study these two questions, namely, Hodgkin-Huxley type and generic activator-inhibitor models, and the recent advances in techniques to link them.

Effects of astrocytic mechanisms on neuronal hyperexcitability

While originally astrocytes have been thought to only act as support to neurons, recent studies have implicated them in multiple active roles, including the ability to moderate or alter neuronal firing patterns and to possibly be involved in both the prevention and propagation of epileptic seizures. In this study we propose a new model to incorporate pyramidal cells and interneurons (a common neural circuit in CA3 hippocampal slices) as well as a model of astrocyte. As both potassium and calcium ions have been shown to potentially affect neuronal hyperexcitability, the astrocytic model has both mechanisms--the clearance of potassium through potassium channels (such as KIR, KDR and sodium-potassium pump), and the influence of astrocyte in the synapse (forming the tripartite synapse with calcium-glutamate interactions). Preliminary findings of the model results show that when potassium conductances in the astrocyte are decreased, it results in the accumulation of extracellular potassium, leading to both spontaneous discharges and depolarization block, while the alteration of normal calcium response in the astrocyte can lead to just hyperexcitable conditions without the depolarization block.

Myelinating satellite oligodendrocytes are integrated in a glial syncytium constraining neuronal high-frequency activity

Satellite oligodendrocytes (s-OLs) are closely apposed to the soma of neocortical layer 5 pyramidal neurons but their properties and functional roles remain unresolved. Here we show that s-OLs form compact myelin and action potentials of the host neuron evoke precisely timed Ba²⁺-sensitive K⁺ inward rectifying (Kir) currents in the s-OL. Unexpectedly, the glial K⁺ inward current does not require oligodendrocytic Kir4.1. Action potential-evoked Kir currents are in part mediated by gap–junction coupling with neighbouring OLs and astrocytes that form a syncytium around the pyramidal cell body. Computational modelling predicts that glial Kir constrains the perisomatic [K⁺]_o increase most importantly during high-frequency action potentials. Consistent with these predictions neurons with s-OLs showed a reduced probability for action potential burst firing during [K⁺]_o elevations. These data suggest that s-OLs are integrated into a glial syncytium for the millisecond rapid K⁺ uptake limiting activity-dependent [K⁺]_o increase in the perisomatic neuron domain.

Satellite oligodendrocytes (s-OLs) are characterised by their close proximity to neocortical pyramidal cells. Here, the authors find that s-OLs myelinate axons and activity of host neurons evokes inward K⁺ currents in s-OLs which may work to modulate action potential burst firing by buffering extracellular K⁺ levels.

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.

Balance of synaptic and electrotonic connections controls the excitability of networks in biophysical model of epilepsy

Recent studies have implicated astrocytes in multiple active roles in neuronal networks. In particular they have been shown to be able to moderate and alter neural firing patterns both in normal and epileptic conditions. In addition, it has been proposed that one of the roles of gap junctions between astrocytes, as well as neurons is in increasing synchronization of neuronal firing and potential epileptogenic effect. In this study we build upon a model of a network that incorporates both pyramidal cells and interneurons as well as astrocytes with potassium clearance mechanisms and basic calcium dynamics. We include electrotonic connections between cells to be able to separate the effects of synaptic connections and gap junctions on neuronal hyperexcitability. Preliminary findings of this model show that under normal conditions, when gap junctions are blocked the network exists in an interictal-like state. When the system is put in a zero calcium environment (i.e. synaptic connections are disabled), the network enters spontaneous rhythmic bursting with very regular spiking. This suggests that electrotonic connections play a crucial role in the epileptogenesis within the neuronal network.

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.

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.

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

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

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.

The sodium-potassium pump controls the intrinsic firing of the cerebellar Purkinje neuron

In vitro, cerebellar Purkinje cells can intrinsically fire action potentials in a repeating trimodal or bimodal pattern. The trimodal pattern consists of tonic spiking, bursting, and quiescence. The bimodal pattern consists of tonic spiking and quiescence. It is unclear how these firing patterns are generated and what determines which firing pattern is selected. We have constructed a realistic biophysical Purkinje cell model that can replicate these patterns. In this model, Na(+)/K(+) pump activity sets the Purkinje cell's operating mode. From rat cerebellar slices we present Purkinje whole cell recordings in the presence of ouabain, which irreversibly blocks the Na(+)/K(+) pump. The model can replicate these recordings. We propose that Na(+)/K(+) pump activity controls the intrinsic firing mode of cerbellar Purkinje cells.

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.

Heterogeneous firing behavior during ictal-like epileptiform activity in vitro

Seizure activity in vivo is caused by populations of neurons displaying a high degree of variability in activity pattern during the attack. The reason for this variability is not well understood. Here we show in an in vitro preparation that hippocampal CA1 pyramidal cells display four types of afterdischarge behavior during stimulus-induced ictal-like events in the presence of Cs(+) (5 mM): type I (43.7%) consisting of high-frequency firing riding on a plateau potential; type II (28.2%) consisting of low-frequency firing with no plateau potential; type III (18.3%) consisting of high-frequency firing with each action potential preceded by a transient hyperpolarization and time-locked to population activity, no plateau potential; "passive" (9.9%) typified by no afterdischarge. Type I behavior was blocked by TTX (0.2 μM) and intracellular injection of QX314 (12.5-25 mM). TTX (0.2 μM) or phenytoin (50 μM) terminated ictal-like events, suggesting that the persistent Na(+) current (I(NaP)) is pivotal for type I behavior. Type I behavior was not correlated to intrinsic bursting capability. Blockade of the M current (I(M)) with linopirdine (10 μM) increased the ratio of type I neurons to 100%, whereas enhancing I(M) with retigabine (50-100 μM) greatly reduced the epileptiform activity. These results suggest an important role of I(M) in determining afterdischarge behavior through control of I(NaP) expression. We propose that type I neurons act as pacemakers, which, through synchronization, leads to recruitment of type III neurons. Together, they provide the "critical mass" necessary for ictogenesis to become regenerative.

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.

The influence of the chloride currents on action potential firing and volume regulation of excitable cells studied by a kinetic model

In excitable cells, the generation of an action potential (AP) is associated with transient changes of the intra- and extracellular concentrations of small ions such as Na(+), K(+) and Cl(-). If these changes cannot be fully reversed between successive APs cumulative changes of trans-membrane ion gradients will occur, impinging on the cell volume and the duration, amplitude and frequency of APs. Previous computational studies focused on effects associated with excitation-induced changes of potassium and sodium. Here we present a model based study on the influence of chloride on the fidelity of AP firing and cellular volume regulation during excitation. Our simulations show that depending on the magnitude of the basal chloride permeability two complementary types of responsiveness and volume variability exist: (i) At high chloride permeability (typical for muscle cells), large excitatory stimuli are required to elicit APs; repetitive stimuli of equal strength result in almost identical spike train patterns (Markovian behavior), however, long excitation may lead to after discharges due to an outward directed current of intracellular chloride ions which accumulate during excitation; cell volume changes are large. (ii) At low chloride permeability (e.g., neurons), small excitatory stimuli are sufficient to elicit APs, repetitive stimuli of equal strength produce spike trains with progressively changing amplitude, frequency and duration (short-term memory effects or non-Markovian behavior); cell volume changes are small. We hypothesize that variation of the basal chloride permeability could be an important mechanism of neuronal cells to adapt their responsiveness to external stimuli during learning and memory processes.

Ion concentration dynamics as a mechanism for neuronal bursting

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.

Imbalance of ionic conductances contributes to diverse symptoms of demyelination

Fast axonal conduction of action potentials in mammals relies on myelin insulation. Demyelination can cause slowed, blocked, desynchronized, or paradoxically excessive spiking that underlies the symptoms observed in demyelination diseases. The diversity and timing of such symptoms are poorly understood, often intermittent, and uncorrelated with disease progress. We modeled the effects of demyelination (and secondary remodeling) on intrinsic axonal excitability using Hodgkin-Huxley and reduced Morris-Lecar models. Simulations and analysis suggested a simple explanation for the breadth of symptoms and revealed that the ratio of sodium to leak conductance, g(Na)/g(L), acted as a four-way switch controlling excitability patterns that included spike failure, single spike transmission, afterdischarge, and spontaneous spiking. Failure occurred when this ratio fell below a threshold value. Afterdischarge occurred at g(Na)/g(L) just below the threshold for spontaneous spiking and required a slow inward current that allowed for two stable attractor states, one corresponding to quiescence and the other to repetitive spiking. A neuron prone to afterdischarge could function normally unless it was switched to its "pathological" attractor state; thus, although the underlying pathology may develop slowly by continuous changes in membrane conductances, a discontinuous change in axonal excitability can occur and lead to paroxysmal symptoms. We conclude that tonic and paroxysmal positive symptoms as well as negative symptoms may be a consequence of varying degrees of imbalance between g(Na) and g(L) after demyelination. The KCNK family of g(L) potassium channels may be an important target for new drugs to treat the symptoms of demyelination.

Assimilating seizure dynamics

Observability of a dynamical system requires an understanding of its state-the collective values of its variables. However, existing techniques are too limited to measure all but a small fraction of the physical variables and parameters of neuronal networks. We constructed models of the biophysical properties of neuronal membrane, synaptic, and microenvironment dynamics, and incorporated them into a model-based predictor-controller framework from modern control theory. We demonstrate that it is now possible to meaningfully estimate the dynamics of small neuronal networks using as few as a single measured variable. Specifically, we assimilate noisy membrane potential measurements from individual hippocampal neurons to reconstruct the dynamics of networks of these cells, their extracellular microenvironment, and the activities of different neuronal types during seizures. We use reconstruction to account for unmeasured parts of the neuronal system, relating micro-domain metabolic processes to cellular excitability, and validate the reconstruction of cellular dynamical interactions against actual measurements. Data assimilation, the fusing of measurement with computational models, has significant potential to improve the way we observe and understand brain dynamics.

Firing patterns and synchronization in nonsynaptic epileptiform activity: the effect of gap junctions modulated by potassium accumulation

Several lines of evidence point to the modification of firing patterns and of synchronization due to gap junctions (GJs) as having a role in the establishment of epileptiform activity (EA). However, previous studies consider GJs as ohmic resistors, ignoring the effects of intense variations in ionic concentration known to occur during seizures. In addition to GJs, extracellular potassium is regarded as a further important factor involved in seizure initiation and sustainment. To analyze how these two mechanisms act together to shape firing and synchronization, we use a detailed computational model for in vitro high-K(+) and low-Ca(2+) nonsynaptic EA. The model permits us to explore the modulation of electrotonic interactions under ionic concentration changes caused by electrodiffusion in the extracellular space, altered by tortuosity. In addition, we investigate the special case of null GJ current. Increased electrotonic interaction alters bursts and action potential frequencies, favoring synchronization. The particularities of pattern changes depend on the tortuosity and array size. Extracellular potassium accumulation alone modifies firing and synchronization when the GJ coupling is null.

Evidence of multistability in a realistic computer simulation of hippocampus subfield CA1

The manner in which hippocampus processes neural signals is thought to be central to the memory encoding process. A theoretically oriented literature has suggested that this is carried out via "attractors" or distinctive spatio-temporal patterns of activity. However, these ideas have not been thoroughly investigated using computational models featuring both realistic single-cell physiology and detailed cell-to-cell connectivity. Here we present a 452 cell simulation based on Traub et al.'s pyramidal cell [Traub RD, Jefferys JG, Miles R, Whittington MA, Toth K. A branching dendritic model of a rodent CA3 pyramidal neurone. J Physiol (Lond) 1994;481:79-95] and interneuron [Traub RD, Miles R, Pyramidal cell-to-inhibitory cell spike transduction explicable by active dendritic conductances in inhibitory cell. J Comput Neurosci 1995;2:291-8] models, incorporating patterns of synaptic connectivity based on an extensive review of the neuroanatomic literature. When stimulated with a one second physiologically realistic input, our simulated tissue shows the ability to hold activity on-line for several seconds; furthermore, its spiking activity, as measured by frequency and interspike interval (ISI) distributions, resembles that of in vivo hippocampus. An interesting emergent property of the system is its tendency to transition from stable state to stable state, a behavior consistent with recent experimental findings [Sasaki T, Matsuki N, Ikegaya Y. Metastability of active CA3 networks. J Neurosci 2007;27:517-28]. Inspection of spike trains and simulated blockade of K(AHP) channels suggest that this is mediated by spike frequency adaptation. This finding, in conjunction with studies showing that apamin, a K(AHP) channel blocker, enhances the memory consolidation process in laboratory animals, suggests the formation of stable attractor states is central to the process by which memories are encoded. Ways that this methodology could shed light on the etiology of mental illness, such as schizophrenia, are discussed.

Making generalizations about seizure propagation

The Source of Afterdischarge Activity in Neocortical Tonic–Clonic Epilepsy. Trevelyan AJ, Baldeweg T, van Drongelen W, Yuste R, Whittington M. J Neurosci 2007;27(49):13513–13519. Tonic–clonic seizures represent a common pattern of epileptic discharges, yet the relationship between the various phases of the seizure remains obscure. Here we contrast propagation of the ictal wavefront with the propagation of individual discharges in the clonic phase of the event. In an in vitro model of tonic–clonic epilepsy, the after discharges (clonic phase) propagate with relative uniform speed and are independent of the speed of the ictal wavefront (tonic phase). For slowly propagating ictal wave fronts, the source of the afterdischarges, relative to a given recording electrode, switched as the wavefront passed by, indicating that afterdischarges are seeded from wavefront itself. In tissue that has experienced repeated ictal events, the wavefront generalizes rapidly, and the afterdischarges in this case show a different “flip-flop” pattern, with frequent switches in their direction of propagation. This same flip-flop pattern is also seen in subdural EEG recordings in patients suffering intractable focal seizures caused by cortical dysplasias. Thus, in both slowly and rapidly generalizing ictal events, there is not a single source of afterdischarge activity: rather, the source is continuously changing. Our data suggest a complex view of seizures in which the ictal event and its constituent discharges originate from distinct locations.

Tracking and control of neuronal Hodgkin-Huxley dynamics

Nonlinear ensemble state estimation offers a paradigm-shifting improvement in our ability to observe, predict, and control the state of spiking neuronal systems. We use an ensemble Kalman filter to predict hidden states and future trajectories in the Hodgkin-Huxley equations, reconstruct ion dynamics, control neuronal activity including a strategy for dynamic conductance clamping, and show the feasibility of controlling pathological cellular activity such as seizures.

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

Experimental evidences point out the participation of nonsynaptic mechanisms (e.g., fluctuations in extracellular ions) in epileptiform bursting and spreading depression (SD). During these abnormal oscillatory patterns, it is observed an increase of extracellular potassium concentration [K(+)](o) and a decrease of extracellular calcium concentration [Ca(2+)](o) which raises the neuronal excitability. However, whether the high [K(+)](o) triggers and propagates these abnormal neuronal activities or plays a secondary role into this process is unclear. To better understand the influence of extracellular potassium dynamics in these oscillatory patterns, the experimental conditions of high [K(+)](o) and zero [Ca(2+)](o) were replicated in an extended Golomb model where we added important regulatory mechanisms of ion concentration as Na(+)-K(+) pump, ion diffusion and glial buffering. Within these conditions, simulations of the cell model exhibit seizure-like discharges (ictal bursting). The SD was elicited by the interruption of the Na(+)-K(+) pump activity, mimicking the effect of cellular hypoxia (an experimental protocol to elicit SD, the hypoxia-induced SD). We used the bifurcation theory and the fast-slow method to analyze the interference of K(+) dynamics in the cellular excitability. This analysis indicates that the system loses its stability at a high [K(+)](o), transiting to an elevated state of neuronal excitability. Effects of high [K(+)](o) are observed in different stages of ictal bursting and SD. In the initial stage, the increase of [K(+)](o) creates favorable conditions to trigger both oscillatory patterns. During the neuronal activity, a continuous growth of [K(+)](o) by outward K(+) flow depresses K(+) currents in a positive feedback way. At the last stage, due to the depression of K(+) currents, the Na(+)-K(+) pump is the main mechanism in the end of neuronal activity. Thus, this work suggests that [K(+)](o) dynamics may play a fundamental role in these abnormal oscillatory patterns.