Minimal model of interictal and ictal discharges “Epileptor-2”
[['A simple canonical mathematical model of epileptic discharges has been proposed by V . Jirsa et al . [1] based on an analysis of experimental recordings of epileptic discharges in humans and animal models .', 'A comparison of the characteristic features of the canonical bifurcations of simple dynamical systems with the elements of the experimental signals revealed the most typical type of transitions to and from the ictal state .', 'A model called Epileptor , which reproduces this behavior , has been proposed .', 'The model can be expressed in a set of six ordinary differential equations [2] .', 'This model led to a series of predictions and applications in physiology , neuroinformatics , and translational medicine [2–4] .', 'A disadvantage of the Epileptor is a lack of physical meaning for its governing variables .', 'A biophysical interpretation would help clarify the experimental validation of the model’s predictions .', 'In the present work , an alternative model with a similar mathematical simplicity but that operates with physically meaningful variables is proposed .', 'The model is referred to as the Epileptor-2 .', 'The set of variables chosen to describe the epileptic state transitions along with the interpretation of the terms included in the equations constitute the primary model predictions .', 'Namely , the model distinguishes between main variables: two ionic concentrations of the extracellular potassium and intracellular sodium , a membrane potential , and a synaptic resource .', 'The selection of these variables as well as the forms of the governing equations with experimental observations taken from both the experiments and the literature are justified and described in detail .', 'While formulating the model assumptions , special attention was focused on two specific experimental observations .', 'First , in a number of experiments , IDs were formed from a clustered number of short discharges [5–8] .', 'These elementary discharges are similar to IIDs .', 'During each IID-like event , a principal neuron typically generated a burst of spikes .', 'Thus , the ID represents a burst of bursts of spikes .', 'Previous biophysical seizure-modeling attempts that considered the ionic dynamics in a single neuron model [9 , 10] or a network [11–13] did not show this feature .', 'The second observation suggested a crucial role of the Na-K pump in the termination of each ID .', 'In fact , the extracellular potassium concentration increased during the initial phase of the ID and began to decrease before the end of the ID [4 , 8 , 14] , whereas the intracellular sodium concentration increased until the actual end of the ID [15] .', 'Some simulations showed a similar behavior of ionic concentrations [9 , 10] .', 'Therefore , the ionic dynamics may be explained by the activation of the Na-K pump , and the consequent decrease in the extracellular potassium concentration may lead to ID termination [12] .', 'The two experimental observations described have not been modeled together .', 'The Results section of the paper is organized as follows .', 'First , the proposed model is presented , which is then justified .', 'The simulations of the interictal and ictal discharges are described and compared with experimental data in order to validate the main assumptions of the model .', 'Then , the main component of the Epileptor-2’s dynamics is discussed , which is an interictal-like discharge observed as a short burst of spikes .', 'To study these discharges , the case of constant ionic conductance was considered , the Epileptor-2 model was reduced to a second-order system of stochastic differential equations , and its mathematical analysis was carried out .', 'It is shown that the deterministic system of the reduced model has the only stable equilibrium as an attractor; however , under noise , the model exhibits a rather complex behavior of the stochastic generation of bursting , similar to the behavior of the stochastic Hindmarsh-Rose model [16] .', 'For weak noise , random states concentrate near the equilibrium .', 'With an increase of noise intensity , random trajectories can travel far from the stable equilibrium , and along with small-amplitude oscillations around the equilibrium , bursts are observed .', 'These bursts constitute the interictal and ictal events in the simulations of Epileptor-2 .', 'The slow oscillations of the potassium and sodium ionic concentrations were then analyzed , which determine the origination and shape of seizures .'], ['The proposed model consists of three subsystems that describe:', '( i ) the ionic dynamics ,', '( ii ) the neuronal excitability , and', '( iii ) a neuron-observer ( Fig 1A ) .', 'The proposed population model consists of the following equations:', 'd[K]odt=[K]bath−[K]oτK−2γIpump+δ[K]oν ( t )', '( 1 )', 'd[Na]idt=[Na]i0−[Na]iτNa−3Ipump+δ[Na]iν ( t )', '( 2 )', 'CdVdt=−gLV+u ( t )', '( 3 )', 'dxDdt=1−xDτD−δxDxDν ( t )', '( 4 )', 'All the terms of these equations are explained in the next section .', 'Here , only the main variables are introduced , which are as follows: [K]o and [Na]i represent extracellular potassium and intraneuronal sodium concentrations , respectively; V ( t ) is the membrane depolarization; xD ( t ) is the synaptic resource; ν ( t ) is the firing rate of an excitatory population; and an inhibitory population firing rate is assumed to be proportional to ν ( t ) .', 'The dynamics described by these equations is driven ν ( t ) , which is calculated with a sigmoidal input-output function:', 'ν ( V ( t ) ) =νmax[21+exp[−2 ( V ( t ) −Vth ) /kν]−1]+', '( 5 )', 'where [x]+ is equal to x for the positive argument and 0 otherwise .', 'The input current u ( t ) includes the potassium depolarizing current , the synaptic drive , and the noise ξ ( t ) , respectively:', 'u ( t ) =gK , leak ( VK ( t ) −VK0 ) +Gsynν ( t ) ( xD ( t ) −0 . 5 ) +σξ ( t )', '( 6 )', 'The potassium reversal potential is obtained from the ion concentrations via the Nernst equation:', 'VK=26 . 6mVln ( [K]o/130mM )', '( 7 )', 'The Na+/K+ pump current is taken from Cressman et al . [9] in the form:', 'Ipump=ρ ( 1+exp ( 3 . 5−[K]o ) ) ( 1+exp ( ( 25−[Na]i ) /3 ) )', '( 8 )', 'The basic set of parameters is as follows .', "The time constants for ionic dynamics , membrane polarization , and synaptic depression are τK = 100s , τNa = 20s , τm = C/gL = 10ms , and τD = 2s; the concentration increments at spike are δ[K]o = 0 . 02mM , δ[Na]i = 0 . 03mM , δxD = 0 . 01; the Gaussian white noise ξ ( t ) has zero mean and unity dispersion , ⟨ξ ( t ) ξ ( t' ) ⟩ = τm δ ( t−t' ) ; the noise amplitude is σ/gL = 25mV; the maximum pump flux is ρ = 0 . 2mM/s; the volume ration is γ = 10 ( close to 7 given in [17] ) ; the postsynaptic charge is Gsyn/gL = 5mV ∙ s; the potassium leak conductance is gK , leak/gL = 0 . 5; the initial extracellular and elevated bath potassium concentrations are [K]O0=3mM and [K]bath = 8 . 5mM; VK0=26 . 6mVln ( [K]O0/130mM ) ; initial and resting intracellular sodium concentrations are the same , [Na]i0=10mM .", 'The parameters of the input-output function are the maximal rate νmax = 100Hz , the threshold potential Vth = 25mV , and the gain kν = 20mV .', 'A representative neuron was modeled with a quadratic integrate-and-fire neuron [18 , 19] .', 'The equations for the membrane potential U ( t ) are as follows:', 'CUdUdt=gU ( U−U1 ) ( U−U2 ) +u', '( 9 )', 'ifU>VTthenU=Vreset', '( 10 )', 'with the parameters: gU = 0 . 4nS/mV , CU = 200pF , VT = 25mV , Vreset = −50mV , U1 = −60mV , U2 = −40mV; an initial condition is U = −70mV .', 'Eq 1 . The model states that the elevation of the extracellular potassium concentration [K]o plays a primary role in self-regenerating IDs .', 'Before each ID , the extracellular potassium concentration accumulates after a series of IIDs .', 'Each IID involves the activation of interneurons [5 , 7] , which results in opening GABAergic channels that are permeable for chloride and bicarbonate [20] and chloride flux into neurons ( note that due to bicarbonate participation , the reversal potential of the GABAergic current ( VGABA ) is more positive than that for chloride; thus , even if the membrane voltage converges to VGABA , the chloride is still transported into the neurons ) .', 'The increase in intracellular chloride activates the KCC2 cotransporter , which carries the potassium out of the cells [21] .', 'Thus , the extracellular potassium concentration increases at each IID , which was proven experimentally ( see Fig 12 from [5] for example ) .', 'Different types of IIDs were reported in the literature .', 'Some types are based on pure recurrent GABAergic interactions between interneurons [7 , 22] .', 'In this case , interneurons excite each other due to the depolarizing GABA effect; however , they evoke only subthreshold voltage fluctuations in the principal cells .', 'Other types of IIDs indicate that the IIDs recruit both GABAergic and glutamatergic synaptic interactions [5] .', 'In some cases , interneurons excite the glutamatergic principal neurons due to the depolarizing GABA effect , and then the principal neurons excite both populations [7] .', 'In other cases , the glutamatergic neurons begin the discharges , and interneurons are involved afterward [23] .', 'It was noted that in a rough consideration of the time scale of one-hundred milliseconds , the spiking activity of interneurons is proportional to that of pyramidal cells .', 'It is denoted as ν ( t ) in the model equations .', 'Thus , the chloride accumulation and subsequent potassium elevation by the potassium-chloride cotransporters is roughly proportional to ν ( t ) , as expressed by the final term in Eq 1 . Another contribution to the final term in Eq 1 is the potassium voltage-gated channels .', 'The total current is also roughly proportional to the spiking activity .', 'Also , glutamatergic currents contribute to the potassium flux [24] .', 'Again , they are roughly proportional to the presynaptic firing rate and thus to ν ( t ) .', 'All these contributions are efficiently taken into account by the final term in Eq 1 . Between IIDs , the potassium concentration [K]o tends to return to the bath concentration due to diffusion and glial buffering .', 'These processes are expressed by the first term in the right-hand side ( r . h . s ) of Eq 1 , and the characteristic time is τK .', 'The potassium is pumped into the neurons by the ATP-dependent Na-K pump .', 'In Eq 1 , this outflux affects the extracellular potassium concentration [K]o through the second term in the r . h . s . Eq 2 . The intracellular sodium concentration [Na]i increases due to the firing activity ν ( t ) , as expressed by the final term in Eq 1 . The same term implies a sodium flux through glutamatergic synapses [25] , whose activation is roughly proportional to the firing rate ν ( t ) .', 'The sodium is pumped from the neurons by the ATP-dependent Na-K pump .', 'In Eq 2 , this outflux affects [Na]i through the second term in the r . h . s . The leakage and intracellular diffusion are taken into account by the first term in the r . h . s . of Eq 2 . The characteristic time of the joint effect is evaluated by τNa .', 'Eq 3 . Describes a mean membrane depolarization due to the input u ( t ) using a single-compartment leaky neuron model .', 'Not taking into account a spike generation , the voltage V ( t ) reflects a nominal , extreme level of membrane polarization .', 'Together with Eq 6 , it is a stochastic ordinary differential equation , which corresponds to the Ornstein-Uhlenbeck process that provides colored Gaussian noise with the temporal correlation scale τm [26] .', 'Eq 4 . Describes the short-term synaptic depression according to the Tsodyks-Markram model [27] written in the rate-dependent form [28] .', 'Eq 5 . Describes the sigmoid-shaped dependence of the firing rate on depolarization V ( t ) ( Fig 2 ) .', 'The dependence is not strictly valid for the transient states .', 'Thus , it does not describe synchronization of neurons within a population and therefore not reproduce sharp peaks of the firing rate that are caused by this effect; however , it satisfactorily approximates activity averaged over a few tens of milliseconds [29] .', 'A synaptic conductance is also missing in the relation .', 'Thus its shunting effect is neglected .', 'Thus , inhibition is taken into account implicitly as a negative term in the expression for u ( t ) ( Eq 6 ) .', 'Eq 6 . For the neuronal input current u ( t ) , Eq 6 takes the depolarizing effect of the potassium concentration increase relative to its initial concentration ( term 1 ) , the total synaptic current ( term 2 ) , and the noise ( term 3 ) into account .', 'The current u ( t ) that determines the firing rate by Eq 5 does not contain intrinsic voltage-gated currents but reflects the indirect effects of the ionic concentrations on membrane polarization through the voltage-gated and leak channels .', 'Term 1 is a linear approximation of the dependence of a sum of potassium channels on their reversal potential deflection VK ( t ) −VK0 .', 'The potassium leak current provides the main contribution to this term .', 'Thus , the coefficient of proportionality gK , leak is called the potassium leak conductance .', 'At the same time , the chloride leak current , which is normally smaller than the potassium leak current [30] , also contributes to this term .', 'This is in accordance with the previous assumption on the cooperative change of intracellular chloride and extracellular potassium concentrations .', 'The change in the chloride leak current depends on the change in the chloride reversal potential , which is roughly proportional to VK ( t ) −VK0 , thus contributing to term 1 . In comparison with term 1 , which evaluates the effects of potassium and chloride ions , the effect of the sodium concentration change on membrane polarization is small because the dependence of the resting potential on the sodium concentration ( through the leak sodium current ) is at least twice lower [30] .', 'Therefore , the effect of sodium on u ( t ) was not considered .', 'The expression for the synaptic current ( term 2 ) implies additional assumptions .', 'It consists of two parts , a positive Gsyn ν ( t ) xD ( t ) and negative −0 . 5 Gsyn ν ( t ) .', 'The positive component of term 2 is an excitatory current; its approximation accounts for the synaptic resource xD and its dependence on the firing rate ν ( t ) , which implies that synaptic kinetics are instantaneous and assumes the driving force to be constant for the sake of simplicity .', 'Hence , the coefficient Gsyn is the maximum postsynaptic charge in response to a single presynaptic spike .', 'The critical role of the short-term depression of excitatory synapses in the termination of IIDs has been revealed for glutamate-based [23] and depolarized GABA-based IIDs [31] .', 'The role of short-term plasticity in inhibitory synapses does not seem to be essential .', 'Thus , the negative component of term 2 is inhibition , which is non-depressing and proportional to the excitation with a coefficient 0 . 5 .', 'The value of the factor is roughly estimated to be less than 1 to provide self-sustaining recurrent excitation at the beginning of discharges when the synaptic resource is full .', 'Below , the proportionality and the estimation of the factor are validated based on the experimental results .', 'The remainder of the equations , Eqs 7 and 8 , are canonical [9] .', 'Following the works by Bazhenov et al . [30] and Krishnan and Bazhenov [32] , we assume that the sodium pump approximation elaborated for a single neuron [9] is valid for the description of the population level ion dynamics .', 'After applying the basic set of parameter values and manipulating one of the parameters of the Epileptor-2 model , two regimes of activity with IIDs and IDs were stimulated .', 'As explained in this section , in both regimes , the discharges are composed of spontaneous short pulses that last a fraction of a second and correspond to short bursts ( SBs ) of spikes in single neurons .', 'Regime with interictal discharges .', 'In the case of a fast relaxation of the potassium concentration ( the parameter τK = 10 s was different from the basic parameter values ) , the model presents a regime with spontaneous short bursts ( SBs ) , which appear shortly after the onset of pro-epileptic bath solution with [K]bath = 8 . 5mM ( Fig 3 ) .', 'In a regime of non-clustered bursts , these SBs represent IIDs .', 'The decreased τK may correspond , for example , to the case of a more efficient extracellular diffusion of potassium ions .', 'This diffusion equilibrates the level of the potassium concentration between IIDs , thus preventing a cluster formation .', 'The mechanism of the IIDs is shown in Fig 1B .', 'The increase of [K]o from the initial level towards the bath concentration due to diffusion leads to a u ( t ) increase .', 'The generation of SB begins when the input current u ( t ) leads the depolarization V ( t ) close to the threshold Vth gL .', 'According to Eq 5 , the firing rate increases and recurrently affects u ( t ) in accordance to Eq 6 . The synaptic resource xD ( t ) begins to vanish , the current u ( t ) decreases , and SB terminates .', 'During each SB , a representative neuron generates a few spikes ( Fig 3F ) .', 'The spike generation terminates due to the decrease of u ( t ) .', 'The ionic concentrations [K]o and [Na]i increase at each SB and then relax during the interburst intervals ( Fig 3G ) .', 'The mean levels of the ion concentrations remain approximately constant .', 'Note that the regime with only IIDs is obtained with relatively fast [K]o relaxation ( τK = 10s ) compared with the characteristic interburst interval .', 'This fast relaxation prevents any significant potassium accumulation during the train of SBs .', 'Regime with ictal discharges .', 'The simulations of the model based on Eqs ( 1–8 ) with the basic parameter set show a population activity with recursive spontaneous IDs ( Fig 4A ) , shortly after the onset of pro-epileptic bath solution with [K]bath = 8 . 5mM .', 'Each ID is characterized by a high rate of activity ( black line ) for 30 seconds .', 'It consists of SBs resembling IIDs ( Fig 4B and Fig 5 ) , and each lasts hundreds of milliseconds .', 'The intervals between IDs are about two minutes .', 'The mechanism of IDs is given schematically in Fig 1B .', 'The concentration [K]o increases towards the level [K]bath .', 'At some critical level of [K]o , the network begins to generate SBs .', '[K]o , the extracellular potassium concentration , increases during each SB and partially relaxes between SBs .', 'The extracellular potassium accumulates ( Fig 4A and 4C , blue line ) .', 'The SB frequency rapidly grows with increasing [K]o , forming a cluster ( Fig 4B and 4C ) .', 'The intensified firing leads to an accumulation of sodium inside neurons ( Fig 4A and 4C , red line ) .', 'The Na-K-pump function strongly depends on [Na]i; therefore , the pump activity rapidly increases ( Fig 4A and 4C , orange line ) .', 'At some point , the pump dominates the outflux of potassium; thus , [K]o begins to decrease .', 'The decline of [K]o decreases the frequency of SBs .', 'The sodium concentration reaches its maximum and begins to restore .', 'While [Na]i is high , the Na-K-pump is highly active , which maintains the reduction of potassium concentration .', 'At some level of [K]o , the SB generation terminates .', 'During the next cycle , towards the next ID , [K]o begins to increase ( Fig 4A ) due to the diffusion with the bath solution , which dominates at this stage .', 'The regime with IDs is obtained with relatively slow [K]o relaxation ( τK = 100s , in accordance with the basic parameter set ) compared with the characteristic interburst interval .', 'This slow relaxation leads to potassium accumulation at some point of the train of SBs .', 'At some critical level of [K]o , depolarization due to the shift of VK ( t ) ( term 1 in Eq 6 ) overcomes the decrease of the excitation due to the decrease of the synaptic resource xD ( t ) .', 'It begins an ID via the positive feedback provided by the depolarizing effect of the potassium accumulation .', 'The results of this subsection lead to the conclusion that Epileptor-2 is able to describe both regimes with interictal and ictal discharges .', 'The mechanisms of the discharge generation correspond to those shown in the schematic in Fig 1B .', 'Simulations illustrate crucial roles of each of the processes included in the schematic ( S1 Text , section 1 ) .', 'Modifications of Epileptor-2 , given in S1 Text , sections 2–4 , extend the model to a case of epileptiform activity in a disinhibited network , a case of prominent depolarization block or consideration of more realistic , adaptive neuron-observer .', 'The IDs simulated using the proposed model are similar to the those registered in slices of a rat entorhinal cortex ( Fig 6 ) .', 'The ictal events occurred with a period of about three minutes ( compare Fig 6A with Fig 5A ) .', 'Each ID consisted of SBs ( Fig 6B with Fig 5B ) .', 'Each SB consisted of several spikes ( Fig 6C with Fig 5C ) .', 'The shift of the baseline of the potential during an ID was observed during the experiment ( Fig 6A ) and in the average potential V ( t ) in the model ( Fig 4A ) ; it is not present in U ( t ) due to the approximate nature of the single neuron model used .', 'Ionic dynamics .', 'The transmembrane concentration gradient of the six major ions ( K+ , Na+ , Cl- , Ca2+ , H+ , and HCO3- ) is altered during an epileptic seizure [15] .', 'The most well-studied are the levels of extracellular K+ and Ca2+ and intracellular Cl- , Na+ , and H+ concentrations [15] .', 'Changes in the concentration of extracellular K+ and intracellular Cl- and Na+ play the most critical roles in seizure induction and maintenance [15] .', 'The central assumptions of the Epileptor-2 assert that only extracellular K+ and intracellular Na+ concentrations govern the activity , thus excluding intracellular Cl- from consideration .', 'This assumption implies a proportionality between the changes of [K]o and [Cl]in .', 'Fig 7 was reproduced from the study by Raimondo [15] and supports this assumption .', 'The profiles of [K]o and [Cl]in during ID are similar .', 'Presumably , this proportionality is explained by the activity of K-Cl-cotransporters , which evoke a potassium outflux in response to chloride accumulation within the neurons .', 'The extracellular potassium concentration increases during a single SB up to 1–2 mM according to data from [33] ( see their Fig 7 or Fig 12 in [5] ) and [8 , 14] .', 'This potassium flux originates from both pyramidal cells and interneurons [8] .', 'The potassium voltage-gated channels , glutamatergic receptors , and K-Cl cotransporters contribute to the flux .', 'The increase of [K]o during a single SB results in moderate depolarization on about 1 mV according to Eq 7 . In the Epileptor-2 , the value of parameter δ[K]o provides the same level of potassium increase after each SB; the parameter τK defines the speed of potassium level relaxation , and the parameter gK , leak controls the level of polarization dependent on [K]o .', 'According to the experimental observations [5 , 8 , 14 , 15] , the extracellular potassium concentration has a peak in the middle of an ID , while the intracellular sodium concentration still gradually increases until the end of an ID [15] ( Fig 7 ) .', 'The model reproduces the ionic dynamics ( Fig 4A ) , employing the nonlinear activation of the Na-K-pump .', 'The pump is activated by a high sodium concentration ( Fig 4C ) according to the approximation ( Eq 8 ) .', 'The influx of potassium through the pump overcomes the outflux , and [K]o begins to decrease .', 'Note that the peak of [K]o precedes the decrease of [Na]i in the simulation , the mentioned experiment ( Fig 7 ) , and the previous modeling [9] .', 'The decrease of [K]o results in the termination of the ID , as shown by the experiments .', 'The model explains this ID termination by the decrease of the potassium-dependent current ( term 1 in the r . h . s . of Eq 6 ) .', 'This suggests a crucial role of the Na-K pump in ID termination .', 'Indirect support of the hypothesis of the crucial role of the ATP-dependent Na-K-pump in ID termination is provided by the evidence of a drop in the oxygen concentration during an ID , which reflects the activity of energy metabolism [1] .', 'Synaptic currents .', 'The second term in Eq 6 implies a proportionality between the excitatory and inhibitory currents and the firing rate .', 'The experiments support this approximation .', 'Assuming that two neighboring pyramidal neurons receive roughly identical synaptic input during epileptiform events [7] , the pairing recordings were made from the neighboring neurons during IDs and IIDs .', 'The excitatory and inhibitory currents from one neuron in a voltage-clamp mode and detected spikes in another neuron in a cell-attached mode were recorded ( Fig 8A and 8B ) .', 'The holding voltages in the voltage clamp corresponded to the reversal potentials of GABAergic and glutamatergic currents , -50 mV and 0 mV , respectively .', 'The current profiles for several SBs were superimposed ( Fig 8C ) .', 'The mean currents were rescaled to correspond to the voltage level -40 mV , which is close to the spike threshold .', 'These mean inhibitory and inverted excitatory currents are shown in Fig 8D .', 'The instantaneous firing rate was calculated from cell-attached recordings as an average number of spikes per interval divided by the length of the interval ( 12 . 5 ms ) .', 'It was found that', '( i ) the shapes of the currents and the instantaneous firing rate are roughly similar;', '( ii ) at spike threshold potential , the excitatory current is bigger than the inhibitory one;', '( iii ) at the initial stage , the peak of the excitatory current is twice larger than that of the inhibitory current; and', '( iv ) the excitatory current decreases faster than the inhibitory one .', 'These findings support the assumptions of the proportionality of the currents to the firing rate , the effect of short-term synaptic depression , and the ratio of excitatory versus inhibitory currents before the action of depression .', 'Therefore , these findings ( observations ) validate term 2 of Eq 6 . Based on the simulations of the system of Eqs 1–8 ( Fig 4 ) , the variables xD ( t ) and V ( t ) oscillate faster than the dynamics of [K]O and [Na]i , producing the SBs .', 'Thus , the subsystem of Eqs 3–5 was studied under the conditions of fixed ionic concentrations , i . e . , with VK ( t ) =VK0 .', 'The system of Eqs 3–5 is planar and stochastic with non-smooth r . h . s . , as follows:', 'CdVdt=−gLV+Gsynν ( t ) ( xD ( t ) −0 . 5 ) +σξ ( t )', '( 3’ )', 'dxDdt=1−xDτD−δxDxDν ( t )', '( 4’ )', 'ν ( V ( t ) ) =νmax[21+exp[−2 ( V ( t ) −Vth ) /kν]−1]+', '( 5’ )', 'Nullclines of the correspondent deterministic system ( σ = 0 ) are shown in the phase space ( Fig 9A ) .', 'The deterministic system has a unique stable equilibrium point with V = 0 and xD = 1 . There are no other equilibrium points or limit circles .', 'Thus , all possible trajectories converge to this stable steady state .', 'Consider a trajectory with an initial position inside or to the left of the U-shaped nullcline .', 'The trajectory first approaches the U-shaped nullcline , then reaches the bottom point , separates towards the straight line nullcline , and proceeds towards the steady state .', 'Because a voltage trace that corresponds to this trajectory would be shaped as an impulse , the zones inside and to the left of the U-shaped nullcline could be referred to as zones of excitation .', 'Next , the effect of random disturbances on the system ( 3’-5’ ) that has the only attractor of the deterministic system was examined .', 'A small amplitude noise that leads to small voltage fluctuations near 0 cannot excite the system .', 'Indeed , small deviations from the equilibrium result in the monotonic convergence to the equilibrium; however , deviations larger than the threshold result in a large excursion before returning to the resting state .', 'This implies a stochastic excitability .', 'As the noise intensity increases , the time between two successive activations decreases .', 'A random trajectory beginning from the stable equilibrium and declined by the noise to the zone of excitation is plotted in the phase plane ( Fig 9A ) along with corresponding time series ( Fig 9B ) .', 'When the noise intensity is higher than some threshold value , random trajectories can move far from the stable equilibrium , and large-amplitude oscillations occur along with small-amplitude oscillations .', 'The intermittency of the oscillations forms the stochastic bursting .', 'Finally , the effect of the extracellular potassium concentration on the frequency and magnitude of SBs was examined .', 'The level of [K]O determines an additive current in Eq 3 according to Eqs 6 and 7 . An increase in [K]O leads to an inward current .', 'The inward current shifts the U-shaped nullcline left and down and shifts the straight line nullcline to the right ( dashed lines in Fig 9A ) .', 'Consequently , the probability of the excursion in the phase-space to cross the nullcline from left to right is higher .', 'The amplitude of oscillations decreases , and the frequency increases .', 'The noise brings a new quality to the system , which cannot be observed in the deterministic case .', 'This phenomenon , known as coherence resonance , is a generic feature of noise-driven excitable systems and is mostly independent of the particular details of the model [16 , 34 , 35] .', 'For epileptic discharges , the mechanism of stochastic large-amplitude oscillations underlies the SBs observed as interictal events and as primary components of IDs .', 'To analyze an appearance of IDs , the full model was reduced to a slow subsystem .', 'In the simulations ( Fig 4 ) , the dynamics of [K]o and [Na]i consisted of slow and low-amplitude fluctuations .', 'The former are much slower than the dynamics of xD , V , and ν .', 'Also , it was noted that Eqs 1 and 2 have slower time scales ( τK and τNa ) than Eqs 3 and 4 with the time scales τm and τD .', 'These observations allowed for reducing the full model to a slow subsystem that is based on Eqs 1 and 2 . The fast subsystem based on Eqs 3 and 4 and discussed in the previous subsection is controlled by the only slow variable [K]o .', 'On the other hand , Eqs 1 and 2 are affected by the fast subsystem only through ν ( t ) .', 'In the simulations , the slow high-amplitude oscillations of [K]o and [Na]i did not follow the fast fluctuations of ν but were controlled by a slow component of ν .', 'Therefore , to extract a slow subsystem , ν ( t ) was averaged , and the dependence of this average variable ν¯ on [K]o was evaluated .', 'This was done with the integration of Eqs 3–7 for the slow ramp of [K]bath ( from 3 to 22 mM during 200s ) with the other parameters taken from the basic parameter set .', 'The obtained dependence was approximated as follows:', 'ν¯ ( [K]o ) ={0 , if[K]o<4 . 5;otherwise−63 . 9093+20 . 0921[K]o−1 . 53505 ( [K]o ) 2+0 . 0533615 ( [K]o ) 3−0 . 000690027 ( [K]o ) 4', '( 5” )', 'which is valid for [K]o < 20 mM .', 'After the substitution of ν by ν¯ in Eqs 1 , 2 and 8 , the slow subsystem is obtained:', 'd[K]odt=[K]bath−[K]oτK−2γIpump+δ[K]oν¯ ( [K]o )', '( 1’ )', 'd[Na]idt=[Na]i0−[Na]iτNa−3Ipump+δ[Na]iν¯ ( [K]o )', '( 2’ )', 'Ipump=ρ ( 1+exp ( 3 . 5−[K]o ) ) ( 1+exp ( ( 25−[Na]i ) /3 ) )', '( 8’ )', 'where [K]bath = 8 . 5 mM , [K]o0=3mM , [Na]i0=10mM τK = 100 s , τNa = 20 s , δ[K]o = 0 . 02 mM , δ[Na]i = 0 . 03 mM , γ = 10 , ρ = 0 . 2 mM/s , as in the basic parameter set .', 'The reduced model based on Eqs 1’ , 2’ , 5” and 8’ behaves similarly to the full Epileptor-2 , as shown in Fig 10 .', 'The slow dynamical system was then analyzed based on Eqs 1’ , 2’ , 5” and 8’ .', 'Under “normal” conditions of a low potassium concentration in the bath solution , [K]bath = 3 mM , the system converges to a fixed point , which is a stable node .', 'The other two equilibrium points are an unstable focus and a saddle .', 'At some critical value of [K]bathcrit .', '=6 . 42017mM , the stable node couples with the saddle at the boundary that is given by a kink of the function ν¯ ( [K]o ) , Eq 5” .', 'This bifurcation is a codimension-two bifurcation in a piecewise-smooth system [36] .', 'To the best of the authors’ knowledge , a full classification of the codimension-two b
['Introduction', 'Results', 'Discussion', 'Methods']
['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 .']
['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 .']
['medicine and health sciences', 'action potentials', 'chemical compounds', 'depolarization', 'membrane potential', 'electrophysiology', 'neuroscience', 'systems science', 'mathematics', 'interneurons', 'potassium', 'computer and information sciences', 'animal cells', 'chemistry', 'dynamical systems', 'chlorides', 'cellular neuroscience', 'cell biology', 'physiology', 'neurons', 'biology and life sciences', 'cellular types', 'physical sciences', 'neurophysiology', 'chemical elements']