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.

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons

Two subpopulations of midbrain dopamine (DA) neurons are known to have different dynamic firing ranges in vitro that correspond to distinct projection targets: the originally identified conventional DA neurons project to the dorsal striatum and the lateral shell of the nucleus accumbens, whereas an atypical DA population with higher maximum firing frequencies projects to prefrontal regions and other limbic regions including the medial shell of nucleus accumbens. Using a computational model, we show that previously identified differences in biophysical properties do not fully account for the larger dynamic range of the atypical population and predict that the major difference is that originally identified conventional cells have larger occupancy of voltage-gated sodium channels in a long-term inactivated state that recovers slowly; stronger sodium and potassium conductances during action potential firing are also predicted for the conventional compared to the atypical DA population. These differences in sodium channel gating imply that longer intervals between spikes are required in the conventional population for full recovery from long-term inactivation induced by the preceding spike, hence the lower maximum frequency. These same differences can also change the bifurcation structure to account for distinct modes of entry into depolarization block: abrupt versus gradual. The model predicted that in cells that have entered depolarization block, it is much more likely that an additional depolarization can evoke an action potential in conventional DA population. New experiments comparing lateral to medial shell projecting neurons confirmed this model prediction, with implications for differential synaptic integration in the two populations.

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

Loss of function mutations of SCN1A, the gene coding for the voltage-gated sodium channel NaV1.1, cause different types of epilepsy, whereas gain of function mutations cause sporadic and familial hemiplegic migraine type 3 (FHM-3). However, it is not clear yet how these opposite effects can induce paroxysmal pathological activities involving neuronal networks’ hyperexcitability that are specific of epilepsy (seizures) or migraine (cortical spreading depolarization, CSD). To better understand differential mechanisms leading to the initiation of these pathological activities, we used a two-neuron conductance-based model of interconnected GABAergic and pyramidal glutamatergic neurons, in which we incorporated ionic concentration dynamics in both neurons. We modeled FHM-3 mutations by increasing the persistent sodium current in the interneuron and epileptogenic mutations by decreasing the sodium conductance in the interneuron. Therefore, we studied both FHM-3 and epileptogenic mutations within the same framework, modifying only two parameters. In our model, the key effect of gain of function FHM-3 mutations is ion fluxes modification at each action potential (in particular the larger activation of voltage-gated potassium channels induced by the NaV1.1 gain of function), and the resulting CSD-triggering extracellular potassium accumulation, which is not caused only by modifications of firing frequency. Loss of function epileptogenic mutations, on the other hand, increase GABAergic neurons’ susceptibility to depolarization block, without major modifications of firing frequency before it. Our modeling results connect qualitatively to experimental data: potassium accumulation in the case of FHM-3 mutations and facilitated depolarization block of the GABAergic neuron in the case of epileptogenic mutations. Both these effects can lead to pyramidal neuron hyperexcitability, inducing in the migraine condition depolarization block of both the GABAergic and the pyramidal neuron. Overall, our findings suggest different mechanisms of network hyperexcitability for migraine and epileptogenic NaV1.1 mutations, implying that the modifications of firing frequency may not be the only relevant pathological mechanism.

Fast–Slow Variable Dissection with Two Slow Variables: A Case Study on Bifurcations Underlying Bursting for Seizure and Spreading Depression

The identification of nonlinear dynamics of bursting patterns related to multiple time scales and pathology of brain tissues is still an open problem. In the present paper, representative cases of bursting related to seizure (SZ) and spreading depression (SD) simulated in a theoretical model are analyzed. When the fast–slow variable dissection method with only one slow variable (extracellular potassium concentration, [Formula: see text]) taken as the bifurcation parameter of the fast subsystem is used, the mismatch between bifurcation points of the fast subsystem and the beginning and ending phases of burst appears. To overcome this problem, both slow variables [Formula: see text] and [Formula: see text] (intracellular sodium concentration) are regarded as bifurcation parameters of the fast subsystem, which exhibits three codimension-2 bifurcation points and multiple codimension-1 bifurcation curves containing the saddle-node bifurcation on an invariant cycle (SNIC), the supercritical Hopf bifurcation (the border between spiking and the depolarization block), and the saddle homoclinic (HC) bifurcation. The bursting patterns for SD are related to the Hopf bifurcation and the depolarization block while for SZ to SNIC. Furthermore, at the intersection points between the bursting trajectory and the bifurcation curves in plane ([Formula: see text], [Formula: see text]), the initial or termination phases of burst match the SNIC or HC point well or the Hopf point to a certain extent due to the slow passage effect, showing that the fast–slow variable dissection method with suitable process is still effective to analyze bursting activities. The results present the complex bifurcations underlying the bursting patterns and a proper performing process for the fast–slow variable dissection with two slow variables, which are helpful for modulation to bursting patterns related to brain disfunction.

Somatostatin-positive Interneurons Contribute to Seizures in SCN8A Epileptic Encephalopathy

SCN8A epileptic encephalopathy is a devastating epilepsy syndrome caused by mutant SCN8A which encodes the voltage-gated sodium channel NaV1.6. To date, it is unclear if and how inhibitory interneurons, which express NaV1.6, influence disease pathology. We found that selective expression of the R1872W mutation in somatostatin (SST) interneurons was sufficient to convey susceptibility to audiogenic seizures. SST interneurons from mutant mice were hyperexcitable but hypersensitive to action potential failure via depolarization block under normal and seizure-like conditions. Remarkably, GqDREADD-mediated activation of wild-type SST interneurons resulted in prolonged electrographic seizures and was accompanied by SST hyperexcitability and depolarization block. Aberrantly large persistent sodium currents, a hallmark of SCN8A mutations, were observed and were found to contribute directly to aberrant SST physiology in computational and pharmacological experiments. These novel findings demonstrate a critical and previously unidentified contribution of SST interneurons to seizure generation not only in SCN8A encephalopathy, but epilepsy in general.

Depolarization Block in the Endocannabinoid System of the Hippocampus

Depolarization block is such a mechanism that the firing activity of a neuronal system is stopped for particular values of the input current. It is important to block epilepsy or unpleasant firing rates. We investigate this property for a non-linear model of CA3 hippocampal neurons under the action of endocannabinoid transmitters. The aim is to discover if they induce depolarization block, a property already seen in other neuronal models and observed in some experiments, signifying that the neural population increases its spiking frequency as some main parameter changes until reaching a situation of no firing. The results is theoretical and it could be useful for investigating real system of neurons of the hippocampus. In some papers it has been shown that this property is connected with bistability, which means that the system has two equilibrium states for some ranges of its parameters. Endocannabinoids influence the learning and memory process and so we concentrate our attention on the CA3 neurons of the hippocampus. We find bistability and depolarization block for the considered model, which is a generalization of the Wilson-Cowan model. The model describes average properties of neurons divided in three classes: the excitatory neuronal population (CA3 neurons) and two types of inhibitory neuron populations (basket cells). The exogenous concentration of cannabinoids is the parameter that controls bistability. This result can be used for an experiment that could give information for medical therapy. We study the time evolution of the synapses connecting the excitatory population with two types of basket cells. The evolution of synaptic weights is considered to be a toy model of the learning process. But this model cannot encompass the complexity and diversity of exogenous and endogenous endocannabinoids effects in vivo.

A unified physiological framework of transitions between seizures, status epilepticus 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 status epilepticus (SE) 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 SE and DB, we show that SE-like events and DB can also occur at the single cell level. Thus, seizures, SE 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 SummaryEpilepsy 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 status epilepticus.