Phase-Dependent Response to Electrical Stimulation of Cortical Networks during Recurrent Epileptiform Short Discharge Generation In Vitro.

The closed-loop control of pathological brain activity is a challenging task. In this study, we investigated the sensitivity of continuous epileptiform short discharge generation to electrical stimulation applied at different phases between the discharges using an in vitro 4-AP-based model of epilepsy in rat hippocampal slices. As a measure of stimulation effectiveness, we introduced a sensitivity function, which we then measured in experiments and analyzed with different biophysical and abstract mathematical models, namely, (i) the two-order subsystem of our previous Epileptor-2 model, describing short discharge generation governed by synaptic resource dynamics; (ii) a similar model governed by shunting conductance dynamics (Epileptor-2B); (iii) the stochastic leaky integrate-and-fire (LIF)-like model applied for the network; (iv) the LIF model with potassium M-channels (LIF+KM), belonging to Class II of excitability; and (v) the Epileptor-2B model with after-spike depolarization.

Light-Driven Sodium Pump as a Potential Tool for the Control of Seizures in Epilepsy.

The marine flavobacterium Krokinobactereikastus light-driven sodium pump (KR2) generates an outward sodium ion current under 530 nm light stimulation, representing a promising optogenetic tool for seizure control. However, the specifics of KR2 application to suppress epileptic activity have not yet been addressed. In the present study, we investigated the possibility of KR2 photostimulation to suppress epileptiform activity in mouse brain slices using the 4-aminopyrindine (4-AP) model.

Response retention and apparent motion effect in visual cortex models.

Apparent motion is a visual illusion in which stationary stimuli, flashing in distinct spatial locations at certain time intervals, are perceived as one stimulus moving between these locations. In the primary visual cortex, apparent-motion stimuli produce smooth spatio-temporal patterns of activity similar to those produced by continuously moving stimuli. An important prerequisite for producing such activity patterns is prolongation of responses to brief stimuli.

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.

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.

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.

Hyperbolic equations for neuronal membrane deformation waves accompanying an action potential.

Experiments show that the propagation of an action potential along an axon is accompanied by mechanical deformations. We describe the mechanisms of the effect using fluid dynamic equations, Laplace's and Hook's laws for surface tension, and Lippmann's law, which relates membrane tension to membrane potential. We derived a minimal, 1-D model, which is a hyperbolic system of equations.

Insertion of Calcium-Permeable AMPA Receptors during Epileptiform Activity In Vitro Modulates Excitability of Principal Neurons in the Rat Entorhinal Cortex.

Epileptic activity leads to rapid insertion of calcium-permeable α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (CP-AMPARs) into the synapses of cortical and hippocampal glutamatergic neurons, which generally do not express them. The physiological significance of this process is not yet fully understood; however, it is usually assumed to be a pathological process that augments epileptic activity. Using whole-cell patch-clamp recordings in rat entorhinal cortex slices, we demonstrate that the timing of epileptiform discharges, induced by 4-aminopyridine and gabazine, is determined by the shunting effect of Ca-dependent slow conductance, mediated predominantly by K-channels.

A strategy for mapping biophysical to abstract neuronal network models applied to primary visual cortex.

A fundamental challenge for the theoretical study of neuronal networks is to make the link between complex biophysical models based directly on experimental data, to progressively simpler mathematical models that allow the derivation of general operating principles. We present a strategy that successively maps a relatively detailed biophysical population model, comprising conductance-based Hodgkin-Huxley type neuron models with connectivity rules derived from anatomical data, to various representations with fewer parameters, finishing with a firing rate network model that permits analysis. We apply this methodology to primary visual cortex of higher mammals, focusing on the functional property of stimulus orientation selectivity of receptive fields of individual neurons.

Molecular mechanisms of action determine inhibition of paroxysmal depolarizing shifts by NMDA receptor antagonists in rat cortical neurons.

N-methyl-d-aspartate glutamate receptors (NMDARs) are involved in numerous central nervous system (CNS) processes, including epileptiform activity. We used a picrotoxin-induced epileptiform activity model to compare the action of different types of NMDAR antagonists in rat brain slices. Paroxysmal depolarizing shifts (PDS) were evoked by external stimulation in the medial prefrontal cortex (mPFC) slices and recorded in pyramidal cells (PC) and in fast-spiking interneurons (FSI).

Presynaptic GABA receptors underlie the antiepileptic effect of low-frequency electrical stimulation in the 4-aminopyridine model of epilepsy in brain slices of young rats.

Low-frequency electrical stimulation (LFES) of the brain is one of the promising methods for helping patients with pharmacoresistant epilepsy. However, the mechanism of the antiepileptic effect of LFES is still unclear. We applied electrophysiological and pharmacological tools and mathematical modeling to investigate it.

A simple model of epileptic seizure propagation: Potassium diffusion versus axo-dendritic spread.

The mechanisms of epileptic discharge generation and spread are not yet fully known. A recently proposed simple biophysical model of interictal and ictal discharges, Epileptor-2, reproduces well the main features of neuronal excitation and ionic dynamics during discharge generation. In order to distinguish between two hypothesized mechanisms of discharge propagation, we extend the model to the case of two-dimensional propagation along the cortical neural tissue.

Mind the last spike - firing rate models for mesoscopic populations of spiking neurons.

The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level.

Correction: Minimal model of interictal and ictal discharges "Epileptor-2".

[This corrects the article DOI: 10.1371/journal.pcbi.

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.

Seizure-Induced Potentiation of AMPA Receptor-Mediated Synaptic Transmission in the Entorhinal Cortex.

Excessive excitation is considered one of the key mechanisms underlying epileptic seizures. We investigated changes in the evoked postsynaptic responses of medial entorhinal cortex (ERC) pyramidal neurons by seizure-like events (SLEs), using the modified 4-aminopyridine (4-AP) model of epileptiform activity. Rat brain slices were perfused with pro-epileptic solution contained 4-AP and elevated potassium and reduced magnesium concentration.

Spatial propagation of interictal discharges along the cortex.

Interictal discharges (IIDs) accompany epileptic seizures and highlight the mechanisms of pathological activity. The propagation of IIDs along the neural tissue is not well understood. To simulate IID propagation, this study proposes a new mathematical model that uses the conductance-based refractory density approach for glutamatergic and GABAergic neuronal populations.

Minimal model of interictal and ictal discharges "Epileptor-2".

Seizures occur in a recurrent manner with intermittent states of interictal and ictal discharges (IIDs and IDs). The transitions to and from IDs are determined by a set of processes, including synaptic interaction and ionic dynamics. Although mathematical models of separate types of epileptic discharges have been developed, modeling the transitions between states remains a challenge.

Acute Changes in Electrophysiological Properties of Cortical Regular-Spiking Cells Following Seizures in a Rat Lithium-Pilocarpine Model.

Profound alterations in both the synaptic and intrinsic membrane properties of neurons that increase the neuronal network excitability are found in epileptic tissue. However, there are still uncertainties regarding the kind of changes in the intrinsic membrane properties occurring during epileptogenesis. Epileptogenesis is typically triggered by the initial brain-damaging insult, and status epilepticus (SE) is one of such insults.

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.

Conductance-based refractory density approach: comparison with experimental data and generalization to lognormal distribution of input current.

The conductance-based refractory density (CBRD) approach is an efficient tool for modeling interacting neuronal populations. The model describes the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons, each receiving individual Gaussian noise and a common time-varying deterministic input. However, the approach requires experimental validation and extension to cases of distributed input signals (or input weights) among different neurons of such an ensemble.

Synaptic Conductances during Interictal Discharges in Pyramidal Neurons of Rat Entorhinal Cortex.

In epilepsy, the balance of excitation and inhibition underlying the basis of neural network activity shifts, resulting in neuronal network hyperexcitability and recurrent seizure-associated discharges. Mechanisms involved in ictal and interictal events are not fully understood, in particular, because of controversial data regarding the dynamics of excitatory and inhibitory synaptic conductances. In the present study, we estimated AMPAR-, NMDAR-, and GABA R-mediated conductances during two distinct types of interictal discharge (IID) in pyramidal neurons of rat entorhinal cortex in cortico-hippocampal slices.

Simplest relationship between local field potential and intracellular signals in layered neural tissue.

The relationship between the extracellularly measured electric field potential resulting from synaptic activity in an ensemble of neurons and intracellular signals in these neurons is an important but still open question. Based on a model neuron with a cylindrical dendrite and lumped soma, we derive a formula that substantiates a proportionality between the local field potential and the total somatic transmembrane current that emerges from the difference between the somatic and dendritic membrane potentials. The formula is tested by intra- and extracellular recordings of evoked synaptic responses in hippocampal slices.

A mathematical model of color and orientation processing in V1.

Orientation processing in the primary visual cortex (V1) has been experimentally investigated in detail and reproduced in models, while color processing remains unclear. Thus, we have constructed a mathematical model of color and orientation processing in V1. The model is mainly based on the following experimental evidence concerning color blobs: A blob contains overlapping neuronal patches activated by different hues, so that each blob represents a full gamut of hue and might be structured with a loop (Xiao et al.

Firing clamp: a novel method for single-trial estimation of excitatory and inhibitory synaptic neuronal conductances.

Understanding non-stationary neuronal activity as seen in vivo requires estimation of both excitatory and inhibitory synaptic conductances from a single trial of recording. For this purpose, we propose a new intracellular recording method, called "firing clamp." Synaptic conductances are estimated from the characteristics of artificially evoked probe spikes, namely the spike amplitude and the mean subthreshold potential, which are sensitive to both excitatory and inhibitory synaptic input signals.