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

Children are more likely to develop epileptic seizures (ictal discharges lasting tens of seconds) than adults. A higher predisposition of the juvenile brain to the generation of epileptiform activity is thought to be due to a prevalence of inhibition over excitation at the early stage of brain development. However, the molecular and physiological mechanisms underlying these age-related differences are yet to be clarified. We compared the ictal activity induced by a convulsant 4-aminopyridine (4-AP) in the horizontal slices of the entorhinal cortex and hippocampus of 3- and 8-week-old Wistar rats. In 3-week-old rats, the ictal discharge was always preceded by a detectable preictal activity, as manifested in one or several 3–4-s GABA-glutamate events, whereas in 8-week-old rats, such events were typically absent or very rare (no more than one occasional short event). The ictal activity resistance to external exposures was also age-dependent. In 8-week-old rats, by contrast to 3-week-old animals, ictal discharge generation in the entorhinal cortex was blocked completely and replaced by 0.2–0.3 Hz interictal activity (simultaneous 1–3-s burst discharges) by a partial blockade of KCC2 cotransporter or Na+–K+-pump, as well as by low-frequency electric stimulation. Thus, our data indicate that ictal discharges in the immature (3-week-old) brain are more resistant to external exposures than in the brain of adult rats. Interictal and ictal epileptiform activities are antagonistic in 8-week-old animals. In contrast, the appearance of interictal activity interrupts the generation of ictal discharges completely. It can therefore be considered as one of the putative antiepileptic mechanisms in the mature rat brain.

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. Using the 4-aminopyridine (4-AP) model of epileptiform activity in juvenile rat brain slices, we found that LFES increased the interval between ictal discharges (IDs) in the entorhinal cortex. The blockade of GABAA, GABAB, AMPA, or NMDA synaptic receptors strongly affected the characteristics of epileptiform discharges in slices. However, only under the blockade of GABAB receptors, LFES becomes entirely ineffective, indicating that the activation of GABAB receptors underlies the main LFES antiepileptic effect. Further experiments allowed us to suggest that LFES activates mostly presynaptic GABAB receptors, which decrease the probability of glutamate release. In line with this hypothesis is the following data: 1) LFES reduces the short-term synaptic depression of excitatory postsynaptic currents similar to the agonist of GABAB receptors SKF-97541; 2) the blockade of excitatory amino acid transporters diminishes the antiepileptic effect of LFES; 3) modeling of the effects of LFES on the probability of glutamate release with a previously proposed mathematical model of epileptiform activity Epileptor-2 also shows the increase of the interval between IDs. Our findings point out a crucial role of presynaptic GABAB receptors in the antiepileptic effect of LFES in the 4-AP model in juvenile rat brain slices.

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. Our model qualitatively reproduces the membrane's mechanical deformation evoked by either the propagation of an action potential or the stepwise change of membrane potential. The understanding of the relationship between electrical activity and mechanical deformation provides guidance toward non-invasive imaging of neuronal activity.

Neuronal membrane undergoes mechanical deformations in response to electrical pulse or mechanical influence. We propose a one-dimensional mathematical minimal model that describes propagation and dissipation of the deformations. The model is based on fluid dynamics equations and uses the Lippmann equation to connect electrical effects with membrane elasticity. The effects of internal heterogeneity of neuron, e.g. cytoskeleton, on mechanical deformations and pressure waves is investigated and specified as relaxation. The pressure in response to the mechanical disturbance of the neuron is compared to known experiment.

In response to a short stimulus, the cortical areas of the brain retain the activity during a few hundred milliseconds. Our previous study has shown the major role of strong excitatory recurrent connections and slow NMDA receptor kinetics into the retention effect in case of single layer of recurrently-connected neuronal populations. In the present work, we investigate two multilayer models of an orientational hypercolumn as a functional unit of the visual cortex to study the role of interactions between the layers. Simulations with the simple firing rate-based multilayered ring-structured model, taking into account recurrent connections, synaptic depression and slow NMDA receptor currents, show that strong feed-forward connections between the layers could compensate for weak lateral recurrent connections and support the retention in the subsequent layers. In case of developed recurrent both lateral and feedforward connections, a similar response of populations in all layers is observed. Complex biologically detailed 2-d conductance-based refractory density model confirms the results. The retention effect can play an important role in such cognition processes as continuous motion perception.

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 Ca2+-dependent slow conductance, mediated predominantly by K+-channels. The blockade of CP-AMPARs by IEM-1460 eliminates this extra conductance and consequently increases the rate of discharge generation. The blockade of NMDARs reduced the additional conductance to a lesser extent than the blockade of CP-AMPARs, indicating that CP-AMPARs are a more significant source of intracellular Ca2+. The study’s main findings were implemented in a mathematical model, which reproduces the shunting effect of activity-dependent conductance on the generation of discharges. The obtained results suggest that the expression of CP-AMPARs in principal neurons reduces the discharge generation rate and may be considered as a protective mechanism.

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. The mapping produces compact expressions for the parameters of the abstract model that clearly identify the impact of specific electrophysiological and anatomical parameters on the analytical results, in particular as manifested by specific functional signatures of visual cortex, including input-output sharpening, conductance invariance, virtual rotation and the tilt after effect. Importantly, qualitative differences between model behaviours point out consequences of various simplifications. The strategy may be applied to other neuronal systems with appropriate modifications., Author summary A hierarchy of theoretical approaches to study a neuronal network depends on a tradeoff between biological fidelity and mathematical tractibility. Biophysically-detailed models consider cellular mechanisms and anatomically defined synaptic circuits, but are often too complex to reveal insights into fundamental principles. In contrast, increasingly abstract reduced models facilitate analytical insights. To better ground the latter to the underlying biology, we describe a systematic procedure to move across the model hierarchy that allows understanding how changes in biological parameters—physiological, pathophysiological, or because of new data—impact the behaviour of the network. We apply this approach to mammalian primary visual cortex, and examine how the different models in the hierarchy reproduce functional signatures of this area, in particular the tuning of neurons to the orientation of a visual stimulus. Our work provides a navigation of the complex parameter space of neural network models faithful to biology, as well as highlighting how simplifications made for mathematical convenience can fundamentally change their behaviour.

Cortical neural networks in vivo are able to retain their activity evoked by a short stimulus. We compare and analyze two mathematical models of ring-structured networks of an orientational hypercolumn of the visual cortex in order to distinguish the contributions of recurrent connections, synaptic depression and slow synaptic kinetics into the retention effect. Comparison of a more elaborated model with the classical ring-model has helped to translate the mathematical analysis of the later model to the former one. As shown, the network with developed recurrent connections reproduces the retention effect compared to that in experiments. The synaptic depression prevents the effect, however the long-lasting excitatory synaptic current recovers the property. Accounting of the slow kinetics of the NMDA receptors, a characteristic post-peak plateau of activity is reproduced. The models show an invariance to contrast of visual stimuli. Simulations reveal a major role of strong excitatory recurrent connections in the retention effect.

A biophysically detailed description of the mechanisms of the primary vision is still being developed. We have incorporated a simplified, filter-based description of retino-thalamic visual signal processing into the detailed, conductance-based refractory density description of the neuronal population activity of the primary visual cortex. We compared four mechanisms of the direction selectivity (DS), three of them being based on asymmetrical projections of different types of thalamic neurons to the cortex, distinguishing between (i) lagged and nonlagged, (ii) transient and sustained, and (iii) On and Off neurons. The fourth mechanism implies a lack of subcortical bias and is an epiphenomenon of intracortical interactions between orientation columns. The simulations of the cortical response to moving gratings have verified that first three mechanisms provide DS to an extent compared with experimental data and that the biophysical model realistically reproduces characteristics of the visual cortex activity, such as membrane potential, firing rate, and synaptic conductances. The proposed model reveals the difference between the mechanisms of both the intact and the silenced cortex, favoring the second mechanism. In the fourth case, DS is weaker but significant; it completely vanishes in the silenced cortex.DS in the On-Off mechanism derives from the nonlinear interactions within the orientation map. Results of simulations can help to identify a prevailing mechanism of DS in V1. This is a step towards a comprehensive biophysical modeling of the primary visual system in the frameworks of the population rate coding concept., Author summary A major mechanism that underlies tuning of cortical neurons to the direction of a moving stimulus is still debated. Considering the visual cortex structured with orientation-selective columns, we have realized and compared in our biophysically detailed mathematical model four hypothetical mechanisms of the direction selectivity (DS) known from experiments. The present model accomplishes our previous model that was tuned to experimental data on excitability in slices and reproduces orientation tuning effects in vivo. In simulations, we have found that the convergence of inputs from so-called transient and sustained (or lagged and nonlagged) thalamic neurons in the cortex provides an initial bias for DS, whereas cortical interactions amplify the tuning. In the absence of any bias, DS emerges as an epiphenomenon of the orientation map. In the case of a biased convergence of On- and Off- thalamic inputs, DS emerges with the help of the intracortical interactions on the orientation map, also. Thus, we have proposed a comprehensive description of the primary vision and revealed characteristic features of different mechanisms of DS in the visual cortex with columnar structure.

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. Using the 4-aminopyridine (4-AP) model of epileptiform activity in juvenile rat brain slices, we found that LFES increased the interval between ictal discharges (IDs) in the entorhinal cortex. The blockade of GABA

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. We review recent advances achieved by the RDM to obtain efficient population density equations for networks of generalized integrate-and-fire (GIF) neurons -- a class of neuron models that has been successfully fitted to various cell types. The theory not only predicts the nonstationary dynamics of large populations of neurons but also permits an extension to finite-size populations and a systematic reduction to low-dimensional rate dynamics. The new types of rate models will allow a re-examination of models of cortical computations under biological constraints.

Direction selectivity (DS) of visual cortex neurons is modelled with a filter-based description of retino-thalamic pathway and a conductance-based population model of the cortex as a 2-d continuum. The DS mechanism is based on a pinwheel-dependent asymmetry of projections from lagged and non-lagged thalamic neurons to the cortex. The model realistically reproduces responses to drifting gratings. The model reveals the role of the cortex in sharpening DS, keeping interneurons non-selective.

The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modeling interact- ing populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting popula- tion model makes use of slow-fast analysis, which leads to a novel method- ology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explo- rations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy). Russian Science Foundation grant (project 16-15- 10201) Spanish grant MINECO-FEDER-UE MTM-2015-71509-C2-2-R Catalan Grant number 2017SGR1049

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). The NMDAR antagonists APV and memantine reduced the duration of PDS. However, the competitive antagonist APV caused similar effects on the PC and FSI, while the open-channel blocker memantine had a much stronger effect on the PDS in the FSI than in the PC. This difference cannot be explained by a corresponding difference in NMDAR sensitivity to memantine because the drug inhibited the excitatory postsynaptic current (EPSC) similarly in both cell types. Importantly, the PDS were significantly longer in the FSI than in the PC. The degree of PDS inhibition by memantine correlated with individual PDS durations in each cell type. Computer modeling of a synaptic network in the mPFC suggests that the different effects of memantine on the PDS in the PC and FSI can be explained by use dependence of its action. An open-channel blocking mechanism and competition with Mg2+ ions for the binding site result in pronounced inhibition of the long PDS, whereas the short PDS are weakly sensitive. Our results show that peculiarities of kinetics and the mechanism of action largely determine the effects of NMDAR antagonists on physiological and/or pathological processes.

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. We demonstrated that 15-min robust epileptiform activity in slices leads to an increase in the amplitude of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR)-mediated component of the evoked response, as well as an increase in the polysynaptic γ-aminobutyric acid (GABA) and N-methyl-D-aspartate (NMDA) receptor-mediated components. The increase in AMPA-mediated postsynaptic conductance depends on NMDA receptor activation. It persists for at least 15 min after the cessation of SLEs and is partly attributed to the inclusion of calcium-permeable AMPA receptors in the postsynaptic membrane. The mathematical modeling of the evoked responses using the conductance-based refractory density (CBRD) approach indicated that such augmentation of the AMPA receptor function and depolarization by GABA receptors results in prolonged firing that explains the increase in polysynaptic components, which contribute to overall network excitability. Taken together, our data suggest that AMPA receptor enhancement could be a critical determinant of sustained status epilepticus (SE).

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. The mathematical model is found to be consistent with experimental double-patch registrations in the 4-aminopyridine in vitro model of epilepsy. In slices, the spontaneous activity of interneurons leads to their synchronization by means of the depolarizing GABAmediated response, thus initiating IIDs. Modeling reveals a clustering of interneuronal synchronization followed by IIDs with activity fronts that propagate along the cortex. The GABA-mediated depolarization either remains to be subthreshold for the principal neurons and thus results in pure GABAergic IIDs (IID1s) or leads to glutamatergic excitation, thus resulting in another type of IIDs (IID2s). In both the model and experiment, IIDs propagate as waves, with constant activity profiles and velocity. The speed of IIDs is of the order of tens of mm/s and is larger for IID2s than for IID1s (40 and 20 mm/s, respectively). The simulations, consistent with experimental observations, show that the wavelike propagation of IIDs initiated by interneurons is determined by local synaptic connectivity under the conditions of depolarizing GABA.

Mechanisms of epileptic discharge generation and spread are not well known. Interictal and ictal discharges (IIDs and IDs) are determined by neuronal interactions and ionic dynamics. In order to reproduce the discharges in a simplest way, we have recently proposed a minimal mathematical model that is alternative to the known model Epileptor. Our model is of similar complexity, but in contrast to the Epileptor formulated in terms of abstract variables, it attributes physical meaning to the main variables. The model is expressed in ordinary differential equations for four principal variables, extracellular potassium and intracellular sodium concentrations, a mean membrane potential and a short-term depressing synaptic resource. Our model reproduces IIDs as bursts of spikes, and IDs as clusters of spike bursts. Potassium accumulation governs the transition to IDs. Here we generalize the model to the case of spatial propagation. Diffusion of the extracellular potassium concentration is assumed to govern the spatial spread of spiking activity across cortical tissue. Simulations are consistent with experimental registrations of waves in pro-epileptic conditions, propagating at a speed of about 0.5 mm/s.

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. A simple generic mathematical model of seizure dynamics (Epileptor) has recently been proposed by Jirsa et al. (2014); however, it is formulated in terms of abstract variables. In this paper, a minimal population-type model of IIDs and IDs is proposed that is as simple to use as the Epileptor, but the suggested model attributes physical meaning to the variables. The model is expressed in ordinary differential equations for extracellular potassium and intracellular sodium concentrations, membrane potential, and short-term synaptic depression variables. A quadratic integrate-and-fire model driven by the population input current is used to reproduce spike trains in a representative neuron. In simulations, potassium accumulation governs the transition from the silent state to the state of an ID. Each ID is composed of clustered IID-like events. The sodium accumulates during discharge and activates the sodium-potassium pump, which terminates the ID by restoring the potassium gradient and thus polarizing the neuronal membranes. The whole-cell and cell-attached recordings of a 4-AP-based in vitro model of epilepsy confirmed the primary model assumptions and predictions. The mathematical analysis revealed that the IID-like events are large-amplitude stochastic oscillations, which in the case of ID generation are controlled by slow oscillations of ionic concentrations. The IDs originate in the conditions of elevated potassium concentrations in a bath solution via a saddle-node-on-invariant-circle-like bifurcation for a non-smooth dynamical system. By providing a minimal biophysical description of ionic dynamics and network interactions, the model may serve as a hierarchical base from a simple to more complex modeling of seizures., Author summary In pathological conditions of epilepsy, the functioning of the neural network crucially depends on the ionic concentrations inside and outside neurons. A number of factors that affect neuronal activity is large. That is why the development of a minimal model that reproduces typical seizures could structure further experimental and analytical studies of the pathological mechanisms. Here, on a base of known biophysical models, we present a simple population-type model that includes only four principal variables, the extracellular potassium concentration, the intracellular sodium concentration, the membrane potential and the synaptic resource diminishing due to short-term synaptic depression. A simple modeled neuron is used as an observer of the population activity. We validate the model assumptions with in vitro experiments. Our model reproduces ictal and interictal events, where the latter result in bursts of spikes in single neurons, and the former represent the cluster of spike bursts. Mathematical analysis reveals that the bursts are spontaneous large-amplitude oscillations, which may cluster after a saddle-node on invariant circle bifurcation in the pro-epileptic conditions. Our consideration has significant bearing in understanding pathological neuronal network dynamics.

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. In the present study, we explored the acute changes in the intrinsic membrane properties of pyramidal cells one day after SE in a rat lithium-pilocarpine model. Using whole-cell patch-clamp recording and the dynamic-clamp technique, we investigated the properties of regular-spiking neurons in the entorhinal cortex (EC) and the medial prefrontal cortex (PFC), two areas differentially affected by SE. We found that one day after SE: (1) the intrinsic membrane properties of EC neurons are significantly altered, while the properties of PFC neurons are mostly unchanged; (2) the input resistance and membrane time constant of regular-spiking neurons are reduced due to enhanced leak current; (3) the active membrane properties of neurons are mostly unaffected; and (4) changes in the passive membrane properties diminish the intrinsic neuronal excitability. Therefore, our results suggest that the acute changes in the intrinsic membrane properties of entorhinal neurons following pilocarpine-induced SE do not contribute to network hyperexcitability. In contrast, at the early stage of epileptogenesis, protective homeostatic plasticity of intrinsic membrane properties is observed in the EC; it reduces the neuronal excitability in response to increased network excitability.

In epilepsy, the number of persistent sodium (NaP) channels increases. To study their effects on neuronal excitability we applied dynamic-clamp (DC). We have revealed that NaP current decreases rheobase, promotes depolarization block (DB) and changes membrane potential between spikes. Bifurcation analysis of a Hodgkin-Huxley-like neuron reveals that NaP current shifts saddle-node and Hopf bifurcations which correspond to the rheobase and DB, in agreement with experiments. By shifting DB, NaP current can make an antiepileptic effect via excitatory neurons.

International audience; No abstract available

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.

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 GABAAR-mediated conductances during two distinct types of interictal discharge (IID) in pyramidal neurons of rat entorhinal cortex in cortico-hippocampal slices. Repetitively emerging seizure-like events and IIDs were recorded in high extracellular potassium, 4-aminopyridine, and reduced magnesium-containing solution. An original procedure for estimating synaptic conductance during IIDs was based on the differences among the current-voltage characteristics of the synaptic components. The synaptic conductance dynamics obtained revealed that the first type of IID is determined by activity of GABAAR channels with depolarized reversal potential. The second type of IID is determined by the interplay between excitation and inhibition, with prominent early AMPAR and prolonged depolarized GABAAR and NMDAR-mediated components. The study then validated the contribution of these components to IIDs by intracellular pharmacological isolation. These data provide new insights into the mechanisms of seizures generation, development, and cessation.

A hypercolumn of the visual cortex is a functional unit formed of neighboring columns whose neurons respond to a stimulus of particular orientation. The function of the hypercolumn is to amplify the orientation tuning of visually evoked responses. According to the conventional simple model of a hypercolumn, neuronal populations with different orientation preferences are distributed on a ring. Every population is described by a firing rate (FR) model. To determine the limitations of the FR-ring model, it was compared with a more detailed ring model, which takes into account the distribution of neurons of each population according to their voltage values. In the case of leaky integrate-and-fire neurons, every neuronal population is described by the Fokker-Planck equation (FPE). The mapping of parameters was obtained. The simulations revealed differences in the behavior of the two models. The FPE-based model reacts faster to a change in stimulus orientation. The FPE ring model gives a steady-state solution in the form of waves of activity traveling on the ring, whereas the FR ring model presents amplitude instability for the same parameter set. The FPE ring model reproduces the characteristic effects of the FR ring model: virtual rotation and symmetry breaking.

A firing rate (FR) model for a population of adaptive leaky integrate-and-fire neurons has been proposed. Unlike known FR models, it describes more precisely the unsteady firing regimes and takes into account the effect of slow potassium currents of spike adaptation. Approximations of the adaptive channel conductances are rewritten from voltage-dependent to spike-dependent and then to rate-dependent ones. The proposed FR model is compared with a very detailed population model, namely, the conductance-based Refractory Density model. This comparison shows the coincidence of the first peak of activity after the start of stimulation as well as of the stationary state. As an example of simulation of coupled adaptive neuronal populations, a ring model has been constructed, which reproduces a visual illusion known as tilt after-effect. The FR model is recommended for mathematical analysis of neuronal population activity as well as for computationally expensive large-scale network simulations.

Synchronization plays important role in generation of brain activity patterns. Experimental data show that neurons demonstrate more reproducible activity for noise-like input than for constant current injection, and that effect can not be reproduced by standard oversimplified Firing-Rate (FR) models. The paper proposes a modification of FR model which reproduces these kinds of activity. The FR model approximates the firing rate of an infinite number of leaky integrate-and-fire neurons, considered as a population, and in contrary to conventional models it accounts for not only a steady-state firing regime but a fast rising excitation as well. Comparison of our simulations with the experimental data shows that the synchronous firing of the neuronal population strongly depends on the synchrony of neuronal states just before spiking. This effect is reproduced by the proposed FR model in contrary to the conventional FR models and is in agreement with the direct Monte-Carlo simulation of individual neurons.

A method for estimating the synaptic current distribution is described, which is based on solving the inverse problem for a system of homogeneously distributed parallel cylinders as a model of dendrites of the pyramidal neurons. The estimates are compared with the data known from literature and those obtained in an independent 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 asin 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 STATEMENTIon 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.

We consider a firing-rate model, that may be used to model EEG, justifying its use by comparison with a conductance-based refractory density population model and a set of individual neurons. It is shown that stimulation of the system by applying a step-wise current, results in a sharp peak in the population activity that can be reproduced by the EEG-model. In addition the steady-state activity may also be reproduced. Similar comparisons are obtained for stimulation via oscillatory inputs.

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. Additionally, the contribution of different membrane currents to the field potential is demonstrated in a two-population mean-field model. Our formalism, which allows for a simple estimation of unknown dendritic currents directly from somatic measurements, provides an interpretation of the local field potential in terms of intracellularly measurable synaptic signals. It is also applicable to the study of cortical activity using two-compartment neuronal population models.

The processes arising when a high-velocity liquid nitrogen drop impacts a hot rigid wall have been studied by means of numerical simulation. An extremely thin near-wall layer of fluid undergoing heating and liquid–vapor phase transition is clearly distinguished in the droplet, the rest of the liquid remaining cold. A model for the layer, based on the reduced compressible Navier–Stokes equations with a two-phase equation of state, has been constructed and adjusted to the known model for the cold liquid part of the drop, based on the barotropic liquid Lagrangian equations with the Tait equation of state. A predictor-corrector finite-difference scheme for the thin layer equations has been devised and coupled with the finite-element method for the barotropic liquid. Drop shapes and flow field distributions in the layer have been obtained for the impacts of nitrogen droplets of different sizes with an initial velocity 186 m/s and different wall temperatures. The influence of the vapor layer on the drop has been analyzed. The heat flux on the wall has been calculated. An analytic formula has been derived for the heat flux, which is in agreement with the calculations for droplets up to 0.1 mm radius.