J Comput Neurosci (2017) 43:65–79DOI 10.1007/s10827-017-0647-7

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

Christopher M. Kim1,2 ·Duane Q. Nykamp1

Received: 3 November 2015 / Revised: 11 March 2017 / Accepted: 26 April 2017 / Published online: 20 May 2017© Springer Science+Business Media New York 2017

Abstract The inhibitory restraint necessary to suppressaberrant activity can fail when inhibitory neurons cease togenerate action potentials as they enter depolarization block.We investigate possible bifurcation structures that arise atthe onset of seizure-like activity resulting from depolariza-tion block in inhibitory neurons. Networks of conductance-based excitatory and inhibitory neurons are simulated tocharacterize different types of transitions to the seizurestate, and a mean field model is developed to verify the gen-erality of the observed phenomena of excitatory-inhibitorydynamics. Specifically, the inhibitory population’s activa-tion function in the Wilson-Cowan model is modified tobe non-monotonic to reflect that inhibitory neurons enterdepolarization block given strong input. We find that a phys-iological state and a seizure state can coexist, where theseizure state is characterized by high excitatory and lowinhibitory firing rate. Bifurcation analysis of the mean fieldmodel reveals that a transition to the seizure state may occurvia a saddle-node bifurcation or a homoclinic bifurcation.We explain the hysteresis observed in network simula-tions using these two bifurcation types. We also demon-strate that extracellular potassium concentration affects the

Action Editor: Steven J. Schiff

� Christopher M. Kimchristopher.kim3@nih.gov

1 School of Mathematics, University of Minnesota,Minneapolis, MN, USA

2 Present address: Laboratory of Biological Modeling, NIDDK,National Institute of Health, Bethesda, MD, USA

depolarization block threshold; the consequent changes inbifurcation structure enable the network to produce the tonicto clonic phase transition observed in biological epilepticnetworks.

Keywords Depolarization block · Seizures ·Excitatory-inhibitory network · Wilson-Cowan model

1 Introduction

An epileptic seizure is a neurological disorder that arisesfrom a hyper-excited neuronal ensemble. The focal epilepticactivity in particular is initiated within a spatially localizedregion and may propagate to other areas of the brain whenit develops into a full ictal event (Gastaut and Broughton1972).

The cortical circuit, however, has a spatial arrangementof inhibition that can suppress the activity in the terri-tory surrounding the focal region (Prince and Wilder 1967;Dichter and Spencer 1969a, b; Schwartz and Bonhoeffer2001). This surround inhibition can act as a mechanismto prevent focal activity from spreading into neighboringregions (Trevelyan and Schevon 2013). Recent studies ofin vitro models of epilepsy and human patients show thatwhen strong inhibitory barrages suppress intense excita-tory synaptic drive that would normally induce paroxysmaldischarges, excitatory cells can be prevented from beingrecruited to ictal events (Schevon et al. 2012; Trevelyanet al. 2006). When an ictal discharge succeeds in propagat-ing, inhibitory restraint is crucial in controlling the spreadof seizure activity (Trevelyan et al. 2007).

Various mechanisms can lead to failure of inhibitoryrestraint, e.g. synaptic depression, depletion of vesicles or

66 J Comput Neurosci (2017) 43:65–79

shift in GABAA reversal potential (Trevelyan et al. 2006;Trevelyan and Schevon 2013). A number of recent stud-ies in brain slices (Ziburkus et al. 2006; Cammarota et al.2013; Yi et al. 2015; Karlocai et al. 2014), rodents (Toyodaet al. 2015), and human patients (Ahmed et al. 2014) suggestthat inhibitory neurons enter depolarization block before orduring seizure activity. Excitatory and inhibitory neuronsexhibit intricate interplay as a result of depolarization block,which is a possible motif for generation and termination ofseizure-like events (Ziburkus et al. 2006). Excessive excita-tion can develop into a seizure-like activity if the inhibitorybarrage onto excitatory neurons is reduced due to depolar-ization block in inhibitory neurons (Cammarota et al. 2013),and seizure-like events terminate when inhibitory neuronsemerge from depolarization block (Ziburkus et al. 2006).Recordings from human patients exhibit similar dynamics:seizure activity is obstructed while fast-spiking inhibitoryneurons are active, but subsequent depolarization block ininhibitory neurons results in a large seizure amplitude andpropagation of epileptic waves (Ahmed et al. 2014).

Although it is not fully understood what causes inhibitoryneurons to become susceptible to depolarization block, anincrease in extracellular potassium beyond a physiologi-cal level as a result of excessive activity (Moody et al.1974; Somjen and Giacchino 1985; Sypert and Ward 1974;Heinemann and Lux 1977) is an important factor that couldpotentially lead to depolarization block. In vitro studiesshow that elevated extracellular potassium can trigger aber-rant activity in single neurons and in the network (Yaariet al. 1986; LeBeau et al. 2002; Korn et al. 1987; Traynelisand Dingledine 1988; Jensen et al. 1994). Inhibitory neu-rons, in particular, enter depolarization block if the extra-cellular potassium is sufficiently elevated (Shin et al. 2010).Computational modeling studies show that single neuronsexhibit various bursting patterns, episodes of alternatingtonic-clonic activity, and depolarization block in an envi-ronment where extracellular potassium is not well regulatedby glial cells or Na+/K+ pumps (Barreto and Cressman2011; Frohlich et al. 2006; Kager et al. 2007; Øyehaug et al.2012).

In this study, we characterize the transition dynamicsfrom a physiological to a pathological network state wheninhibitory restraint fails due to depolarization block. Weconsider the impact of extracellular potassium and blockadeof potassium channels on inhibitory restraint and the con-sequences on network dynamics. Networks of conductance-based excitatory and inhibitory neurons, where inhibitoryneurons are more susceptible to depolarization block thanexcitatory neurons, are simulated to investigate differenttypes of bifurcations to seizure-like activity. Furthermore, aphenomenological mean field model is developed to verifythat the dynamics found in simulations is a general propertyof excitatory-inhibitory network dynamics.

2 Methods

2.1 Network simulations

We used Morris-Lecar (M-L) (Morris and Lecar 1981) andHodgkin-Huxley (H-H) (Barreto and Cressman 2011) neu-ron models to simulate networks of excitatory and inhibitoryneurons. A M-L neuron can be considered as a reducedH-H neuron, where explicit variables modeling openingand closing of voltage-gated ion channels are replaced byvoltage-dependent functions. We modified the M-L neuronparameters such that excitatory neurons are regular-spiking,inhibitory neurons are fast-spiking, and inhibitory neuronsare more susceptible to depolarization block than excitatoryneurons. For the H-H neurons, we reduced the potassiumconductance of inhibitory neurons such that depolarizationblock can be induced more easily in inhibitory neurons.M-L neurons were used to simulate the effects of patho-logically high extracellular potassium concentration on net-work dynamics, which may result from excessive networkactivity. On the other hand, H-H neurons were used todemonstrate qualitatively similar network dynamics wheninhibitory neurons’ potassium channels are blocked prefer-entially, for example, by applying 4-aminopyridine (4-AP)to brain slices (Ziburkus et al. 2006).

Here, we give detailed description of the modified M-L neurons. A comprehensive study of the H-H neuron wasconducted previously (Barreto and Cressman 2011). Wereplaced the Ca2+ channel with a Na+ channel, whichdid not change the essential dynamics of the M-L neuron.Ordinary differential equations consisting of two dynamicvariables, membrane potential V and potassium channelactivation w, describe its dynamics:

CdVi

dt= −gNam(Vi)(Vi − ENa) − gKw(t)(Vi − EK)

−gCl(Vi − ECl) + I exti (t) + I

syni (t)

dw

dt= φ

w∞(Vj ) − w

τ∞where m(V ) = (1+ tanh((V −V1)/V2))/2, w∞(V ) = (1+tanh((V − V3)/V4))/2, τ∞(V ) = 1/ cosh((V − V5/(2V6)),I exti (t) is an external input from other parts of the brain, and

Isyni (t) is a local synaptic input. See Table 1 for the neuron

parameters.Figure 1a and b show the bifurcation diagrams of exci-

tatory and inhibitory M-L neurons, respectively, where thepotassium reversal potential (EK ) is set to a physiological(−90 mV) or a pathological value (−70 mV). A neuroncan be in three different states depending on the strengthof external input Iext: rest state (solid black at low Iext),limit cycle (red), and depolarization block (solid blackat high Iext). When the rest state transitions to the limitcycle at a critical value Iext = Ic via a SNIC bifurcation

J Comput Neurosci (2017) 43:65–79 67

Table 1 Network simulation and mean field parameters

Morris-Lecar Excitatory Inhibitory

C 5 nF 1 nF

gK 22 μS 7 μS

gNa 20 μS 16 μS

gCl 2 μS 7 μS

φ 0.55 1

V1 −7 mV −7.35 mV

V2 15 mV 23 mV

V3 −8 mV −12 mV

V4 15 mV 12 mV

V5 0 mV −15 mV

V6 15 mV 10 mV

Hodgkin-Huxley Excitatory Inhibitory∗gK 40 μS 10 ∼ 20 μS

For other parameters, see Barreto and Cressman (2011)

Synaptic parameters Morris-Lecar Hodgkin-Huxley

τe 3 ms 3 ms

τi 7 ms 10 ms∗Jee 70 ∼ 90 nS 20 ∼ 35 nS

Jie 190 nS 40 nS

Jei 50 nS 70 nS

Jii 15 nS 5 nS

Eexcsyn 25 mV 55 mV

Einhsyn −70 mV −90 mV

∗EK −60 ∼ −90 mV −70 mV

ENa 60 mV 50 mV

ECl −60 mV −82 mV

�exc,inhth 15 mV, 10 mV 10 mV, 10 mV

Network parameters

N 3000

inhibitory fraction 0.25

p 0.01

Mean field parameters

τ 0.8∗Jee 8 ∼ 12

Jei 10

Jie 16

Jii 4∗θ 7 ∼ 10k 0.7

*These parameters are varied between simulations. The changedparameter values are given in the text

(saddle-node on invariant circle), the oscillation frequency ofmembrane potential increases monotonically as a function

of√

Iext − Ic (Rinzel and Ermentrout 1989). However, assoon as depolarization block is reached via a supercriticalHopf bifurcation, the membrane potential remains fixed athigh voltage, and the M-L neuron loses its spike-generatingmechanism.

If EK increases toward a pathological value as a resultof increased extracellular potassium concentration, both theexcitatory and inhibitory M-L neurons enter depolarizationblock at a reduced external input. This is consistent with aprevious study where increasing EK facilitated depolariza-tion block in the H-H neuron model (Barreto and Cressman2011). However, for the chosen M-L neuron parameters, thecritical external input that puts an excitatory M-L neuroninto depolarization block is much larger than the corre-sponding critical input for an inhibitory neuron, whichleads to qualitatively different shape of f-I curves for theexcitatory and inhibitory populations (Fig. 2b and c).

We performed network simulations with N excitatoryand N/4 inhibitory neurons where N = 3000 with con-nection probability p = 0.01. A neuron generates a spikewhen its membrane potential crosses a spike-threshold �th,and the δ-spikes from presynaptic neurons produce synapticcurrent in postsynaptic neurons by activating synaptic con-ductances. The total synaptic current neuron j receives isgiven by

Isynj (t) =

∑

k∈exc

Jjkgexcjk (t)(Vj − Eexc

syn)

+∑

k∈inh

Jjkginhjk (t)(Vj − Einh

syn)

where the synaptic conductance gjk(t) of postsynaptic neu-ron j and presynaptic neuron k rises instantaneously anddecays exponentially upon receiving a spike,

dgajk

dt= −ga

jk/τa +∑

t sk≤t

δ(t − t sk ), a ∈ {exc, inh}.

Here, Easyn is the synaptic reversal potential, τa is the synap-

tic time constant, and t sk represents the spike times ofpresynaptic neuron k. The synaptic coupling strength, Jjk ,from a neuron k in population b to a neuron j in populationa is equal to Jab if a connection exists and 0 otherwise fora, b ∈ {exc, inh}.

The network simulation code written with Brian simula-tor (Goodman and Brette 2008) is available from ModelDBwebsite (http://senselab.med.yale.edu/modeldb).

2.2 Mean field model

To characterize the network dynamics observed in simula-tions, we modified the Wilson-Cowan model (Wilson andCowan 1972) to account for the population-level effectsof depolarization block in inhibitory neurons. We left the

68 J Comput Neurosci (2017) 43:65–79

Fig. 1 Bifurcation diagrams of Morris-Lecar neurons. a ExcitatoryM-L neurons with (i) physiological EK = −90 mV and (ii) pathologi-cal EK = −70 mV. The rest state (solid black at low Iext) transitions toa limit cycle (red line) via a Hopf bifurcation (a.i) or a SNIC bifurca-tion (a.ii), which subsequently enters depolarization block (solid blackat high Iext) via a supercritical Hopf bifurcation. Solid and dashedblack lines denote stable and unstable fixed points, respectively, and

red lines indicate the max and min value of a limit cycle. b InhibitoryM-L neurons with (i) physiological EK = −90 mV and (ii) patho-logical EK = −70 mV. The rest state transitions to a limit cycle viaa SNIC bifurcation. For both excitatory and inhibitory M-L neurons,depolarization block can be reached at a reduced external input if EK

increases to a pathological value. However, the critical external inputfor inhibitory neurons is much lower than that for excitatory neurons

excitatory population’s activation function unchanged sinceexcitatory neurons did not enter depolarization block inthe parameter regime considered in simulations. On theother hand, we made the inhibitory population’s activationfunction be non-monotonic to reflect that the inhibitory pop-ulation’s output rate decreases when the external input isstrong enough to drive inhibitory neurons into depolariza-tion block.

The mean firing rates of excitatory and inhibitory pop-ulation, re and ri , evolved according to the differentialequations

dre

dt= −re + φe(Jeere − Jeiri + IE

ext)

τdri

dt= −ri + φi(Jiere − Jiiri + I I

ext) (1)

where Jab for a, b ∈ {e, i} denotes the coupling strengthfrom population b to population a, IE

ext and I Iext are exter-

nal inputs, and τ is a time constant. We used the activationfunctions

φe(x) = 1

1 + e−x, φi(x) = 1

1 + e−x· 1

1 + ek(x−θ)(2)

where the second factor of φi is monotonically decreasingif k > 0, and θ models the inhibitory population’s depo-larization block threshold. Low θ means that the inhibitoryneurons on average are more susceptible to depolarizationblock. Figure 2a shows the plots of φe and φi for different θ

values.Changing θ has the same effect as changing potassium

reversal potential EK of M-L neurons or inhibitory neuron’s

Fig. 2 f-I curves. a Activation functions of the mean field model.The excitatory population has a standard sigmoidal activation func-tion (black), while the inhibitory activation function is a productof a sigmoidal function and a monotonically decreasing function.See the definition in Eqs. (1) and (2). The maximum value andcenter of the inhibitory activation functions decrease as the depo-larization block threshold, θ , is reduced from 10 (blue), 8 (green)to 6 (red). b Population responses of unconnected excitatory M-L neurons. Each neuron receives external spikes generated by anindependent Poisson process at the given rate. The firing rate of

the excitatory population increases monotonically for pathological(−60 mV) and physiological (−90 mV) EK . The firing rate of neu-rons with pathological EK decreases substantially when the externalinput becomes strong (∼8 kHz). c Population responses of uncon-nected inhibitory M-L neurons. The firing rate of the inhibitorypopulation has a Gaussian shape since the firing rate decreasesas neurons enter depolarization block at relatively low input (∼4kHz). The maximum value and center of the inhibitory f-I curvesdecrease significantly as EK increases toward a pathological value(−60 mV)

J Comput Neurosci (2017) 43:65–79 69

potassium conductance ginhK of H-H neurons in the network

simulations.

2.3 Population response curves

The mean field model is an ad hoc characterization ofthe basic dynamics of interacting neuronal populations. Assuch, there is no systematic link between mean field param-eters and those from the network simulations. As a basiccalibration between the two distinct model classes, we setparameters of the mean field activation functions to mimicthe population response of unconnected M-L neurons.

Figure 2 demonstrates the qualitative agreement thatwe achieved between network simulations and the meanfield description of their population activity, comparing thesteady state responses of unconnected M-L neurons to theactivation functions of the mean field equation. Figure 2band c show the population response of unconnected excita-tory and inhibitory M-L neurons, respectively, when eachneuron is stimulated by independent Poisson spikes.

The excitatory population has a monotonically increasingpopulation response, which is in qualitative agreement withthe sigmoidal function φe of the mean field model (Fig. 2a,black). For the parameters chosen for excitatory M-L neu-rons, increasing EK to a pathological level has a relativelysmall impact on the firing rate (compare three populationresponses in Fig. 2b), although the f-I curves start diverg-ing if the neurons are stimulated at a high rate (∼8 kHz).For this reason, we used the same sigmoidal function φe todescribe the population responses of excitatory neurons withphysiological and pathological EK .

The firing rate of the inhibitory population, on the otherhand, is a non-monotonic function of the input rate sincethe fraction of inhibitory neurons driven into depolarizationblock increases with the external input. The non-monotonicactivation function of the mean field model shown in Fig. 2a(blue, green, red) captures the rising and decaying phase ofthe population response of inhibitory M-L neurons. WhenEK is increased to a pathological value, a relatively low rateof Poisson spikes is enough to drive the inhibitory popu-lation activity to its maximum value (Fig. 2c, green, red),after which the firing rate declines monotonically. This isa consequence of the fact that individual inhibitory M-Lneurons enter depolarization block at low input if EK isincreased. Moreover, the maximal inhibitory activity pro-duced with pathological EK is significantly lower comparedto that of a physiological EK . In the mean field model,the parameter θ of the inhibitory activation function repro-duces the effects of EK in network simulations. In Fig. 2a,the input rate required to reach the maximal value of theactivation function and the maximum output rate are bothreduced when θ decreases from θ = 10 (blue), 8 (green) to6 (red).

Similarly, the population response of unconnected H-Hneurons (Barreto and Cressman 2011) exhibited the rise anddecay phase when the neurons were stimulated with Poissonspikes. When inhibitory neuron’s potassium conductanceginh

K was reduced to mimic the effects of preferential block-ade of inhibitory neuron’s potassium channels by, for exam-ple, application of 4-AP, we found that the critical externalinput that puts inhibitory neurons into depolarization blockwas significantly less than that of excitatory neurons (datanot shown).

3 Results

The simplicity of our mean field model facilitated its anal-ysis. However, since its connection to the neuronal net-works is strictly phenomenological, the extent to which theresults of its analysis would apply to the network modelswas unclear. In this section, we analyze the dynamics ofthe mean field model and then demonstrate that the basicconclusions do hold for our network models.

We first examine under what conditions depolariza-tion can create a bistable network, where a normal stateand a seizure state (characterized by depolarization in theinhibitory neurons) are both supported by the network. Wethen outline a bifurcation analysis of the network that showstwo distinct transitions from the normal state to the seizurestate: a non-oscillatory transition characterized by a saddle-node bifurcation and an oscillatory transition characterizedby a homoclinic bifurcation. Finally, we demonstrate howa tonic to clonic phase transition, observed in grand malseizures, can arise naturally from the bifurcation structureof the model.

3.1 Depolarization block creates bistable network

We performed phase plane analysis of the mean fieldmodel to identify different network states arising fromdepolarization block in inhibitory neurons and verified thatcorresponding network states exist in M-L neuron network.

Figure 3a.i shows that two stable fixed points, which weterm the normal state and the seizure state, coexist if theinhibitory population is susceptible to depolarization blockand the excitatory population receives sufficiently strongexternal input. In the normal state, both the excitatory andinhibitory populations fire at low rate. The seizure state ischaracterized by excessive excitatory activity during whichthe inhibitory population is in depolarization block andhence fires at a low rate. The stable manifold of a saddlepoint forms a separatrix dividing the basins of attraction ofthe two stable network states.

We demonstrated such network bistability in simula-tions of M-L neuron network. As suggested by the mean

70 J Comput Neurosci (2017) 43:65–79

Fig. 3 Depolarization block creates a bistable network. The left col-umn shows phase planes of the mean field model, and the right columnshows the response of M-L neuron networks to a brief barrage of Pois-son spikes to the excitatory neurons. a (i) Coexistence of the normalstate (b1) and the seizure state (b2), separated by the stable mani-fold (inward arrow) of a saddle point (open black circle). The seizurestate is characterized by re at its maximum and ri near zero (whichwe characterize as depolarization block). θ = 7, IE

ext = −2, I Iext =

−4, Jee = 8. Blue: re-nullcline; red: ri -nullcline; solid green: sta-ble manifold; dashed green: unstable manifold. (ii) The upper panelshows the mean firing rate in response to an additional short (20 ms)1 kHz Poisson input applied to the excitatory neurons (black bar).The input drives the inhibitory population into depolarization block,allowing the excitatory population to generate action potentials at a

high rate. The lower panel shows voltage traces of sample excitatoryand inhibitory neurons. EK = −60 mV, IE

ext = 0.8 kHz, I Iext = 0.3

kHz, Jee = 70 nS. b (i) The mean field model has one stable fixedpoint (m1) if the external input to excitatory population is reduced(IE

ext = −7). (ii) The firing rate of Poisson spikes stimulating theexcitatory neurons is decreased to IE

ext = 0.35 kHz. In this case, thesame short input as in a.ii. evokes a transient increase in the excitatoryrate and briefly drives the inhibitory M-L neurons into depolariza-tion block. Subsequently, the network returns to the normal state. c (i)Increasing the depolarization block threshold to θ = 10 removes theseizure state. (ii) The same network simulation as in A.ii, but with aphysiological EK = −80 mV. The perturbation does not elicit depo-larization block in inhibitory neurons. Other parameters are given inTable 1

field analysis, we realized bistability by setting the EK ofexcitatory and inhibitory neurons to a pathological value(−60 mV), which made the inhibitory neurons prone todepolarization block, and providing strong baseline inputto the excitatory neurons. Figure 3a.ii demonstrates how abrief barrage of external spikes to the excitatory neuronsswitches the network immediately to the seizure state, as the

input effectively pushes the network state across the sep-aratrix into the basin of attraction of the seizure state. Asin the mean field network, the inhibitory population firesat a significantly reduced rate and the excitatory popula-tion sustains a high firing rate during the seizure state. Themembrane potential of a sample inhibitory neuron showsthat it was highly depolarized but did not generate any action

J Comput Neurosci (2017) 43:65–79 71

potentials, indicating that inhibitory neurons were in depo-larization block. The excitatory neuron, on the other hand,spiked continuously at high rate.

Phase plane analysis of the mean field model sug-gests that the seizure state of a bistable network can beremoved by either reducing the input to excitatory neurons(Fig. 3b.i) or increasing the depolarization block threshold

of inhibitory neurons (Fig. 3c.i)). After either manipulation,the re-nullcline did not intersect with the ri-nullcline at aseizure state.

To verify these predictions, we used the same networkparameters as in Fig. 3a.ii but with two separate changesthat corresponded to the manipulations we made to themean field model. First, we reduced the baseline input to

Fig. 4 Saddle-node bifurcation at the onset of seizure-like activity. aBifurcation diagrams and a phase plane of the mean field model. (i)Depolarization block threshold, θ = 7. Starting at the monostable state(m1), the network stays at the lower branch of steady states (the normalstate) as the external input to excitatory population is increased until itjumps to the upper branch (the seizure state) via a saddle-node bifur-cation (SN). The bistable states b1 and b2 correspond to Fig. 3a, andthe monostable state m1 corresponds to Fig. 3b. The arrows show thehysteresis loop that starts and ends at the monostable state. Solid line:stable equilibrium; dashed line: unstable equilibrium; SN: saddle-nodebifurcation. Jee = 8, I I

ext = −4. (ii) Depolarization block thresh-old, θ = 10. The network is bistable over a narrow and increasedinput range if the depolarization block threshold is increased. Themonostable state m2 corresponds to Fig. 3c. (iii) The phase plane at asaddle-node bifurcation point when θ = 7 (black dot). b Simulations

of M-L neuron networks. (i) Hysteresis loops observed in network sim-ulations with EK = −90 mV, −80 mV, −70 mV. In agreement with themean field analysis, if EK is increased toward a physiological value,the transition to seizure state occurs at a higher input, and the networkreturns to normal state at a higher input. (ii) Time progression of exci-tatory and inhibitory firing rates used for constructing the hysteresisloops. The excitatory neurons are stimulated by Poisson spikes at a rateshown in the bottom panel. I I

ext = 0.6 kHz, Jee = 80 nS. c Simulationsof H-H neuron networks. The transition to the seizure state occurs at ahigher input if ginh

K is increased. The inhibitory neurons do not recoverfrom depolarization block even when the external input to the exci-tatory population is completely removed. The hysteresis shown in c.icorresponds to the portion of hysteresis loop in a.i that starts at b1 andends at b2. EK = −70 mV, I I

ext = 0.05 kHz, Jee = 20 nS. Othersimulation parameters are given in Table 1

72 J Comput Neurosci (2017) 43:65–79

excitatory neurons in Fig. 3b.ii. As before, a brief bar-rage of external spikes to the excitatory neurons drivesinhibitory neurons into depolarization block temporarily.This time, however, the network cannot sustain this statesince the excitatory drive onto inhibitory neurons is notstrong enough. For the second manipulation, we set theEK of excitatory and inhibitory neurons close to a phys-iological value (−80 mV), increasing inhibitory neuron’sresistance to depolarization block by effectively increas-ing the depolarization block threshold. The barrage of inputthat previously drove inhibitory neurons into depolarizationblock could no longer elicit such a response (Fig. 3c.ii).

3.2 Saddle-node bifurcation at the onset of seizure-likeactivity

Our primary motivation for developing the mean fieldmodel was to facilitate analysis of transitions between theseizure state and other states, such as the low rate normalstate. We begin our analysis of these transitions by examin-ing a non-oscillatory transition from the normal state to theseizure state via a saddle-node bifurcation and exploring theinfluence of EK and ginh

K on the bifurcation structure.Figure 4a.i shows a bifurcation diagram of the mean

field model for low θ , with the external input to excitatorypopulation as a bifurcation parameter. The network is mono-stable at low input (as in Fig. 3b) and becomes bistable foran increased input (as in Fig. 3a). The normal state tran-sitions to the seizure state via a saddle-node bifurcation ifthe external input became sufficiently large. Figure 4a.iiishows the phase plane at the saddle-node bifurcation point.Figure 4a.ii, on the other hand, shows the bifurcation dia-gram when θ was increased so that the inhibitory populationbecomes resistant to depolarization block. The bifurcationdiagram indicates that the bistable regime was reduced sig-nificantly, and the bistable state that existed at low θ turnedinto a monostable state, whose phase plane is illustrated inFig. 3c.

To verify that a similar bifurcation structure is presentin networks of realistic neurons, we demonstrated that hys-teresis loops, indicated by arrows in Fig. 4a.i and a.ii,can be observed in network simulations. Figure 4b showssimulation results with M-L neurons, where we gradu-ally increased the external input to excitatory neuronsuntil the network transitioned to the seizure state and thendecreased the input back to the original value. The result-ing hysteresis loops are consistent with the saddle-nodebifurcations suggested by the mean field analysis. Oncethe external input exceeds a critical value, the inhibitoryneurons enter depolarization block and the excitatory neu-rons exhibit runaway excitation. The network does notimmediately return to the normal state even when the exter-nal input is reduced below the critical value; instead, it

returns to the normal state only at a substantially decreasedinput.

Next, we investigated whether the depolarization blockthreshold θ of the mean field model can capture theeffects of extracellular potassium concentration on the non-oscillatory transition to seizure state in the simulations ofM-L neuron networks. When EK of M-L neurons wereshifted toward physiological values, we found that (1) astrong external input is required for the network to transi-tion from normal state to seizure state, and (2) the networkcan return to the normal state at a higher external input.These simulation results were predicted by the mean fieldanalysis; compare the IE

ext values in Fig. 4a.i and a.ii atwhich the arrows jump from the low to high branch andfrom the high to low branch. However, the mean field pre-diction, that the hysteresis loop becomes narrower when θ

increases, does not hold for network simulations since theincreased firing rate of excitatory neurons at a physiologi-cal EK (−90 mV) enables the network to sustain the seizurestate over a wide input range. We observed that the hys-teresis loop can become narrower if we shift Einh

K alonetoward a physiological value while keeping Eexc

K fixed,which corresponds more accurately to the mean field modelbut is less likely to occur in biological networks (data notshown).

For the simulations of H-H neuron networks (Fig. 4c),hysteresis did not form closed loops since the networkremained bistable even when the external input was com-pletely removed, due to a strong persistent excitatory drivepresent in the model of Barreto and Cressman (2011). Theresults were still consistent with the mean field analysis; thesimulation results corresponded to the portion of hysteresisloop in Fig. 4a.i that starts at b1 and ends at b2.

We systematically varied inhibitory H-H neuron’s potas-sium conductance ginh

K to mimic the blockade of inhibitoryneuron’s potassium channels. Reducing ginh

K had qualita-tively similar impact on the transition dynamics as in shift-ing EK to a pathological value for the M-L neuron network;it was possible to reach the seizure state by a reduced inputsince depolarization block in inhibitory neurons could beinduced more easily if ginh

K is reduced.During the course of hysteresis, we observed an intricate

interplay of excitatory and inhibitory population activity inM-L neuron networks (Figs. 4b(ii)), as reported in intracel-lular recordings of hippocampal cells (Ziburkus et al. 2006).Prior to the onset of seizure-like activity, the inhibitoryfiring rate increased gradually and reached its maximumimmediately before entering depolarization block. The exci-tatory firing rate, on the other hand, stayed relatively lowwhen the inhibition was strong. At the bifurcation point,where inhibitory neurons entered depolarization block, theexcitatory neurons were released from the inhibition andgenerated action potentials at their maximum rate. The

J Comput Neurosci (2017) 43:65–79 73

strong excitatory drive onto the inhibitory neurons keptthem in the depolarization block even when we reducedthe external input below the critical value. This hysteresismay explain why inhibitory neurons’ depolarization blockis long lasting as reported in (Ziburkus et al. 2006). Thenetwork finally exits the seizure state when we reduce exci-tatory drive to the point where it is too weak to sustain theseizure state. At the moment of exiting the seizure state, theinhibitory firing rate showed strong rebound as it emergedfrom depolarization block, and the excitatory firing ratedropped abruptly from its maximum rate to a normal rate.

3.3 Homoclinic bifurcation at the onset of seizure-likeactivity

The non-oscillatory saddle-node transition from the normalstate to the seizure state of the mean field model representeda transition between asynchronous states in the networkmodel. Next, we explore a transition from the resting statethrough a Hopf bifurcation to an oscillatory state, followedby a homoclinic bifurcation to the steady seizure state. Thisoscillatory state of the mean field model corresponds tosynchronous oscillations in the network models. Motivated

Fig. 5 Homoclinic bifurcation at the onset of seizure-like activ-ity. Increasing the excitatory-excitatory coupling strength producesan oscillatory transition to the seizure state. a A bifurcation dia-gram and phase planes of the mean field model. (i) Increasedexcitatory-excitatory coupling strength, Jee = 12. Limit cycles (reddots) emerge via a supercritical Hopf bifurcation and transition toseizure state via a homoclinic bifurcation. HB: Hopf bifurcation,HC: homoclinic bifurcation. θ = 7, I I

ext = −4. (ii) The phaseplane of a network state, where a limit cycle (b1) and seizure state(b2) coexist. (iii) A homoclinic orbit at the homoclinic bifurca-tion point. b Simulation of a M-L neuron network with increasedJee = 90 nS. EK = −70 mV, I I

ext = 0.6 kHz. (i) The hysteresis

loop corresponds to the loop indicated by arrows in the bifurca-tion diagram, a.i. (ii) Time progression of excitatory and inhibitoryfiring rates used for constructing the hysteresis loop. The bottompanel shows the firing rate of Poisson spikes each excitatory neu-ron receives. c Simulation of a H-H neuron network with increasedJee = 35 nS. EK = −70 mV, I I

ext = 0.05 kHz, ginhK =

15 nS. Network dynamics are qualitatively equivalent to the M-L neuron network. However, the inhibitory neurons do not emergefrom depolarization block even when the external input is completelyremoved. The excitatory neurons receive Poisson spikes at a rateshown in the bottom panel. Other simulation parameters are given inTable 1

74 J Comput Neurosci (2017) 43:65–79

by experimental findings suggesting that strong excitatorypostsynaptic potentials and increased recurrent excitatoryconnections in epileptic brain regions may contribute toseizure activity (Dudek and Sutula 2007; Wuarin and Dudek2001; Scharfman et al. 2003; Buckmaster et al. 2002), weinvestigated synchronized network activity that arises in ourmodels when the recurrent excitatory coupling strength isincreased.

For the mean field model, we increased Jee while keep-ing other parameters unchanged. A limit cycle emergedvia a supercritical Hopf bifurcation since the increased Jee

facililated achieving the Hopf bifurcation criterion, trL =−1 + Jeeφ

′e − 1

τ(1 + Jiiφ

′i ) = 0, where L is the linearized

dynamics of the mean field model. The bifurcation diagram(Fig. 5a(i)) had a new type of bistable state where a limitcycle and the seizure state coexisted. Figure 5a(ii) showsthe phase plane of this bistable network, in which the limitcycle and the seizure state are separated by the stable mani-fold of a saddle point. When the excitatory input was furtherincreased, the limit cycle coalesced with the stable andunstable manifolds of the saddle point and formed a homo-clinic orbit at a homoclinic bifurcation point (Fig. 5a(iii)).The limit cycle disappeared beyond the homoclinic bifur-cation point, and the seizure state became the only stablestate.

Figure 5b(i) shows hysteresis observed in simulations ofthe M-L neuronal network with increase excitatory cou-pling, which is in good agreement with the hysteresis loopshown in the bifurcation diagram, Fig. 5a(i). The gradualdevelopment of synchronous oscillations at low excitatoryinput in the simulations corresponds to the limit cycleappearing via a supercritical Hopf bifurcation, and the sharptransition from large amplitude oscillations to the maximumfiring rate corresponds to the homoclinic bifurcation. Whenwe reduced the excitatory input while the network was in theseizure state, the network bypassed the oscillatory state andcontinued to stay in the seizure state, as suggested from themean field model’s bifurcation diagram, until the input wastoo weak to sustain the seizure state. The asymmetric natureof hysteresis is well exhibited in Fig. 5b(ii), which showsthe time progression of excitatory and inhibitory firing ratesused for constructing the hysteresis loop in Fig. 5b(i). A richset of network dynamics appeared along the path towardsthe seizure state due to the emergence of synchronous oscil-lations from the Hopf bifurcation and their transition to theasynchronous seizure state via the homoclinic bifurcation.On the other hand, the dynamics along the path to the nor-mal state was rather simple: the seizure state transitioned tothe normal state via a saddle-node bifurcation.

The simulations of H-H neurons with stronger exci-tatory coupling also exhibited hysteresis and oscillations(Fig. 5c(i)) consistent with the bifurcation structure of themean field model. However, as in the simulations of the

H-H neuron networks from the previous section (Fig. 4c(i)),the excitatory activity did not form a closed hysteresis loop;the persistent excitatory activity in the seizure state wasstrong enough to keep the inhibitory neurons in depolar-ization block even when the external input was completelyremoved.

A two-parameter bifurcation diagram (Fig. 6) summa-rizes the bifurcations present in the mean field model andobserved in the network models. The saddle-node, Hopf,and homoclinic bifurcation lines divide the parameter spaceinto the three possible network states (normal, oscillatoryand seizure). For low values of Jee, the network passesthrough a saddle-node bifurcation to the seizure state. Forhigh values of Jee, the network passes through a homo-clinc bifurcation to the seizure state after passing through aHopf bifurcation to the oscillatory state. For high Jee, thesaddle-node bifurcation occurs between two unstable statesand doesn’t play a large role in the dynamics. The saddle-node, Hopf, and homoclinic bifurcation lines merge at aBogdanov-Takens bifurcation.

3.4 Tonic to clonic phase transition

A tonic to clonic phase transition is a hallmark of a grandmal seizure (Quian Quiroga et al. 1997) and is also observedin vitro when extracellular potassium concentration is mod-ulated (Jensen and Yaari 1997; Sypert and Ward 1974).

Fig. 6 Two-parameter bifurcation diagram. The bottom (top) hori-zontal arrow shows a pathway to the seizure state via a saddle-node(homoclinic) bifurcation as discussed in Fig. 4a.i (Fig. 5a.i). Saddle-node (SN), Hopf (HB) and homoclinic bifurcation (HC) lines merge ata Bogdanov-Takens bifurcation (BT). Other parameters are identicalto Fig. 4a.i

J Comput Neurosci (2017) 43:65–79 75

The tonic phase is characterized by a high frequency asyn-chronous spike discharge occurring in the early stages of aseizure. As the firing rate slows down for various reasons(e.g. synaptic depression, vesicle depletion, or reintroduc-tion of inhibition), the neurons begin to synchronize theirspiking activity and generate high amplitude oscillations,which is referred to as the clonic phase. We investigated thepossibility that the depolarization block in inhibitory neu-rons could be a mechanism that produces the tonic to clonicphase transition.

The mean field model analysis revealed a bifurcationstructure that could underlie a tonic to clonic transition. Toset up the transition, we configure the parameters to cre-ate a bistable network where the seizure state and a limitcycle coexist, and initialize the network to be in the seizurestate, as shown in Fig. 7a. The seizure state corresponds tothe tonic phase since excitatory neurons fire asynchronouslyat their maximum rate. As the limit cycle corresponds tosynchronous oscillations in the network models, it couldpotentially represent a clonic phase. One way to create atransition from the tonic phase to the clonic phase is toincrease the depolarization block threshold of the inhibitorypopulation, which moves the saddle-node bifurcation tohigher input rates, deforming the bifurcation structure toreduce the range of input rates where the tonic seizure stateexists (Fig. 7b). Since the limit cycle becomes the onlystable state, the network enters the clonic state, where theinhibition returns and participates in the oscillations.

To implement this mechanism behind a tonic to clonictransition in a M-L network model, we initialized the net-work into the seizure, or tonic, state by applying a strongexcitatory barrage of input to the excitatory neurons (notshown). Hence, the simulation began with the excitatoryneurons firing asynchronously at a high rate and inhibitoryneurons in depolarization block, as shown in Fig. 7c. To trig-ger the transition to the clonic stage, we slowly decreasedEK of excitatory and inhibitory neurons toward physio-logical values. As a result, the inhibitory neurons slowlyemerged from depolarization block and began to fire. Thenetwork then entered synchronous oscillations correspond-ing to the limit cycle of the mean field model, transitioningfrom the tonic phase into the clonic phase.

4 Discussion

4.1 Insights from the mean field bifurcation structure

We have investigated a possible bifurcation structure ofexcitatory and inhibitory network dynamics near the onsetof seizure-like activity when it arises as a result of depo-larization block in inhibitory neurons. A simple mean fieldmodel of depolarization block enabled an analysis of thebifurcations underlying transitions among the normal state,the asynchronous depolarization block state, and the syn-chronous oscillation state. Given the loose relationship

Fig. 7 Tonic to clonic phase transition. a (i) Bifurcation diagram withincreased Jee = 12 and low depolarization block threshold, θ = 7.At the outset, we initialize the network in the seizure state, which cor-responds to the tonic phase. (ii) Corresponding network state in thephase plane (solid circle). b (i) The depolarization block threshold isincreased to θ = 10. The tonic phase no longer exists, thus the networkstate is shifted to the clonic phase. The amplitude of oscillations of the

clonic phase is indicated by black open circles. (ii) Corresponding net-work state in the phase plane (black ellipse). c Simulation of a M-Lneuron network with increased Jee = 90nS, exhibiting the tonic-clonicphase transition. The tonic phase switches to the clonic phase whenEK is gradually recovered from a pathological value −60 mV towardphysiological values, Eexc

K = −70 mV and EinhK = −80 mV, over the

period indicated by the black line. IEext = 0.4 kHz, I I

ext = 0.2 kHz

76 J Comput Neurosci (2017) 43:65–79

between the mean field model and neuronal network mod-els, we had no assurances that the network models wouldhave the same transitions or that the transitions’ dependenceon parameters would be predicted by the mean field model.In the mean field model, we simply modified the activa-tion function of the inhibitory neurons to be non-monotonic,which imitated how depolarization block decreases the fir-ing rate in response to a strong input. We included a singledepolarization block threshold parameter to control the levelof input that initiated the depolarization block.

Network simulations of M-L and H-H neurons demon-strated how well the mean field analysis captured the dif-ferent dynamical states induced by depolarization block aswell as the types of transitions among the states. In par-ticular, as predicted by the mean field model, the networksimulations demonstrated how the depolarization block ininhibitory neurons can induce bistability where a normalstate coexists with a seizure state in which the inhibitoryneurons are in depolarization block. We also observed inthe network models the two transition pathways to theseizure state that we encountered in the mean field bifurca-tion analysis. We found a non-oscillatory transition to theseizure state in network simulations that corresponded toa saddle-node bifurcation in the mean field model, and anoscillatory transition to the seizure state that correspondedto a homoclinic bifurcation. For both types of transitions,the mean field bifurcation analysis explained the hysteresisloops observed in the network simulations. Such a corre-spondance between the mean field model and the networksimulations demonstrated that the simplistic mean fieldmodel could capture important structural features of thedynamics produced by the depolarization block in inhibitoryneurons.

For networks exhibiting a homoclinic bifurcation, themean field model predicted a transition from asynchronous(or tonic) firing in the seizure state to synchronized (orclonic) activity if inhibitory neurons were pulled out ofdepolarization block. This mechanism for the transitionto the clonic state required an increase in the depolariza-tion block threshold of the inhibitory population, which weimplemented in the M-L neuron network by decreasing thepotassium reversal potential EK . Since we did not modelpotassium dynamics that could underlie such a decrease,this result simply represents a proof of principle that adecrease in EK could initiate the tonic to clonic transition.

A possible way to model changes in EK is to includesynaptic fatigue, e.g. synaptic depression or depletion ofvesicles, and reuptake of extracellular potassium by glialcells to the current network model. In such an extended net-work model, the recurrent excitatory syapses will becomeweaker due to synaptic fatigue in the tonic state, whichleads to gradual decrease of excitatory neuron’s firingrate. Consequently, less potassium will be pumped into the

extracellular space, allowing glial cells to clear the extracel-lular potassium and restore EK toward a normal level.

4.2 Relationship to experiment results

Network simulations exhibiting a saddle-node bifurcationwere characterized by strong inhibition before the seizureonset, runaway excitation during depolarization block andrecovery to normal state as inhibition emerged from thedepolarization block (Figs. 4b,c). Ziburkus et al. (2006)have observed this sequence of events in excitatory andinhibitory neurons in hippocampal slices, where excita-tion and inhibition activate at different stages of a seizureepisode. In addition, they reported that inhibitory neuronsexhibited long-lasting depolarization block (5–40 s) duringthe ictal discharges. This observation is consistent with ourresult that, once inhibitory neurons are driven into depo-larization block, recurrently connected excitatory neuronscan produce strong excitatory drive to sustain the seizurestate over a wide range of parameters; the external input hadto be decreased substantially for the network to return tothe normal state (Fig. 4b, also see Fig. 4c). Therefore, ourcomputational model suggests a robust excitatory-inhibitorydynamic that may underlie the partitioning of seizure-likeevents into different stages as reported by Ziburkus et al.(2006).

For network simulations exhibiting a homoclinic bifur-cation, we demonstrated that the asynchronous seizurestate transitions to synchronized activity when the potas-sium reversal potential is restored to a physiological value(Fig. 7c). The increase of network synchrony resembles thetonic to clonic phase transition observed in epilepsy modelsinduced by elevated extracellular potassium concentration([K+]o) or by repeated stimulations (Moody et al. 1974;Sypert and Ward 1974; Jensen and Yaari 1997). Measure-ments of [K+]o have shown that the level of extracellularpotassium increases at the outset of the tonic phase anddeclines gradually starting from the late tonic phase andthroughout the clonic phase (Jensen and Yaari 1997). Wefound that the direction of changes in EK and subsequentphase transitions observed in our computational model areconsistent with the experimental findings. Increasing EK atthe outset of the tonic phase lowered the bifurcation point toseizure state, thus facilitated the transition to the tonic phase(Fig. 4b), and decreasing EK during depolarization blockenabled the transition to the clonic phase (Fig. 7b).

We note that there are other possible ionic mechanismsby which increased [K+]o can lead to seizure-like activity.One important factor is chloride loading in the intracellu-lar space. Previous studies have shown that accumulationof extracellular potassium can increase the intracellularchloride concentration of the NKCC1 cotransporter equi-librium state (Dzhala et al. 2010) or block the efflux

J Comput Neurosci (2017) 43:65–79 77

of intracellular chloride through KCC2 cotransporters(Thompson and Gahwiler 1989). The increased intracellularchloride concentration depolarizes the reversal potential ofGABAA receptor-mediated postsynaptic current, which caninvert its effect from inhibition to excitation (Ben-Ari 2002).Consequently, networks of fast-spiking interneurons, with-out any contribution of pyramidal cell activity, can developprototypic epileptic-discharges (Fujiwara-Tsukamoto et al.2010), and activating PV+ interneurons during clonic phasecan increase the frequency of epileptic-discharges (Ellenderet al. 2014).

In our effort to model seizure-like activity, the goal wasto characterize the types of transitions to the seizure statewhen inhibition fails. For this purpose, we selected a set ofneuron parameters for which the inhibitory neurons but notthe excitatory neurons were susceptible to depolarizationblock. However, recordings from in vitro models of epilepsyreport that excitatory neurons are also prone to depolariza-tion block, although less frequently (Ziburkus et al. 2006).The connectivity structure of neurons may be one reasonwhy inhibitory neurons are more easily depolarized: Par-valbumin positive basket cells, which are the most effectiveinhibitory neurons, receive perisomatic and somatic synap-tic inputs, while pyramidal cells (excitatory) are less likelyto receive them (Karlocai et al. 2014; Gulyas et al. 1999;Megıas et al. 2001). It is possible that strong excitatoryinputs entering the soma can drive inhibitory neurons moreeasily to a depolarized state. Another reason may be thatslow adaption currents, e.g. Ca2+-induced potassium cur-rent, are present in regular-spiking excitatory neurons butnot in fast-spiking inhibitory neurons. The hyperpolarizingcurrent that activates during high frequency firing can pre-vent excitatory neurons from staying at a depolarized state.

4.3 Comparison to other computational models

Other computational studies investigated possible mecha-nisms of the tonic to clonic phase transition. Using pulsecoupled oscillator theory, Beverlin et al. (2012) showed thatrecurrently connected excitatory M-L neurons can synchro-nize their activity if the external input is reduced gradually,which mimicked the effect that inhibition is reintroducedto the system. Their result is in line with the mechanismproposed in our model, where recovery of extracellularpotassium concentration to a physiological level releasedinhibition from depolarization block and enabled the tran-sition from tonic to clonic phase. Frohlich et al. (2006)have demonstrated that a biophysically detailed single neu-ron model including various ion channels can exhibit tonicfiring and bursting if the extracellular potassium concen-tration is elevated. A network of these neurons showedalternating epochs of tonic firing and bursting with slowoscillations in the extracellular potassium concentration

(Frohlich et al. 2006, 2010). The main difference with ourresult is that our neuron models do not burst, therefore thesynchronized activity in the clonic phase is a networkphenomenon but not a consequence of single neurondynamics.

The mean field equation used in this study to describedifferent types of bifurcations at the seizure onset is a phe-nomenological model that accounts for the depolarizationblock of inhibitory neurons. Specifically, the sigmoidal acti-vation function of the Wilson-Cowan equation was modifiedto a non-monotonic function for the inhibitory populationsuch that a strong external input can decrease the outputfiring rate as the fraction of inhibitory neurons in depo-larization block increases. In a recent study, Meijer et al.(2015) developed a similar mean field model by modify-ing the Wilson-Cowan equation, where depolarization blockwas modeled with a Gaussian activation function. In agree-ment with our results, they showed that the network can bebistable, i.e. normal and seizure state coexist, and stronginhibition precedes an epileptic traveling wave propagatingacross a spatially continuous model.

These phenomenological mean field models lack a pre-cise mapping between single cell dynamics and populationdynamics; therefore, they cannot quantitatively explain howpathological changes in single neuron properties alter themacroscopic dynamics. A recent study by Zandt et al.(2014) showed that the f-I curve of a single neuron canbe used to quantitatively link neural mass models describ-ing population dynamics and pathological changes in singleneuron parameters. In particular, they simulated an increasein extracellular potassium concentration by varying thepotassium reversal potential and demonstrated that oscil-lations appearing in network simulations can be matchedquantitatively to limit cycles of the neural mass model.However, a bifurcation analysis of the neural mass modelwas not performed to unravel how pathological changesin single neurons altered the network dynamics. It wouldbe of interest to adapt their methods to, first, justifythe choice of non-monotonic activation function by deriv-ing it from single neuron’s f-I curve and, second, estab-lish quantitative mapping between changes in the meanfield model and pathological changes in single neuronparameters.

Acknowledgements This research was supported by the NationalScience Foundation grant DMS-0847749. We thank Tay Netoff forinsightful discussions and motivating our investigation. CMK thanksBernstein Center Freiburg for their support where part of this work wasconducted.

Compliance with Ethical Standards

Conflict of interest The authors declare that they have no conflictof interest.

78 J Comput Neurosci (2017) 43:65–79

References

Ahmed, O.J., Kramer, M.A., Truccolo, W., Naftulin, J.S., Potter, N.S.,Eskandar, E.N., Cosgrove, G.R., Blum, A.S., Hochberg, L.R., &Cash, S.S. (2014). Inhibitory single neuron control of seizures andepileptic traveling waves in humans. BMCNeuroscience, 15(Suppl1), F3.

Barreto, E., & Cressman, J.R. (2011). Ion concentration dynamics asa mechanism for neuronal bursting. Journal of Biological Physics,37(3), 361–373.

Ben-Ari, Y. (2002). Excitatory actions of gaba during development:the nature of the nurture. Nature Reviews Neuroscience, 3(9), 728–739.

Beverlin, B., Kakalios, J., Nykamp, D., & Netoff, T.I. (2012). Dynam-ical changes in neurons during seizures determine tonic to clonicshift. Journal of Computational Neuroscience, 33(1), 41–51.

Buckmaster, P.S., Zhang, G.F., & Yamawaki, R. (2002). Axon sprout-ing in a model of temporal lobe epilepsy creates a predominantlyexcitatory feedback circuit. The Journal of Neuroscience: TheOfficial Journal of the Society for Neuroscience, 22(15), 6650–6658.

Cammarota, M., Losi, G., Chiavegato, A., Zonta, M., & Carmignoto,G. (2013). Fast spiking interneuron control of seizure propaga-tion in a cortical slice model of focal epilepsy. The Journal ofPhysiology, 591(Pt 4), 807–822.

Dichter, M., & Spencer, W.A. (1969a). Penicillin-induced interic-tal discharges from the cat hippocampus. I. Characteristics andtopographical features. Journal of Neurophysiology, 32(5), 649–662.

Dichter, M., & Spencer, W.A. (1969b). Penicillin-induced interictaldischarges from the cat hippocampus. II. Mechanisms underlyingorigin and restriction. Journal of Neurophysiology, 32(5), 663–687.

Dudek, F.E., & Sutula, T.P. (2007). Epileptogenesis in the den-tate gyrus: a critical perspective. Progress in Brain Research,163(801), 755–773.

Dzhala, V.I., Kuchibhotla, K.V., Glykys, J.C., Kahle, K.T., Swiercz,W.B., Feng, G., Kuner, T., Augustine, G.J., Bacskai, B.J., & Staley,K.J. (2010). Progressive nkcc1-dependent neuronal chloride accu-mulation during neonatal seizures. The Journal of Neuroscience,30(35), 11745–11761.

Ellender, T.J., Raimondo, J.V., Irkle, A., Lamsa, K.P., & Akerman, C.J.(2014). Excitatory effects of parvalbumin-expressing interneu-rons maintain hippocampal epileptiform activity via synchronousafterdischarges. The Journal of Neuroscience, 34(46), 15208–15222.

Frohlich, F., Bazhenov, M., Timofeev, I., Steriade, M., & Sejnowski,T.J. (2006). Slow state transitions of sustained neural oscilla-tions by activity-dependent modulation of intrinsic excitability.The Journal of Neuroscience: The Official Journal of the Societyfor Neuroscience, 26(23), 6153–6162.

Frohlich, F., Sejnowski, T.J., & Bazhenov, M. (2010). Network bista-bility mediates spontaneous transitions between normal and patho-logical brain states. The Journal of Neuroscience: The OfficialJournal of the Society for Neuroscience, 30(32), 10734–10743.

Fujiwara-Tsukamoto, Y., Isomura, Y., Imanishi, M., Ninomiya, T.,Tsukada, M., Yanagawa, Y., Fukai, T., & Takada, M. (2010). Pro-totypic seizure activity driven by mature hippocampal fast-spikinginterneurons. The Journal of Neuroscience, 30(41), 13679–13689.

Gastaut, H., & Broughton, R.J. (1972). Epileptic seizures: clinicaland electrographic features, diagnosis and treatment. Springfield:Thomas. https://books.google.de/books/about/Epileptic Seizures.html?id=Xb1rAAAAMAAJ&pgis=1.

Goodman, D.F., & Brette, R. (2008). The brian simulator. Frontiers inNeuroscience, 3, 26.

Gulyas, A.I., Megıas, M., Emri, Z., & Freund, T.F. (1999). Totalnumber and ratio of excitatory and inhibitory synapses converg-ing onto single interneurons of different types in the CA1 areaof the rat hippocampus. The Journal of Neuroscience: The Offi-cial Journal of the Society for Neuroscience, 19(22), 10082–10097.

Heinemann, U., & Lux, H.D. (1977). Ceiling of stimulus induced risesin extracellular potassium concentration in the cerebral cortex ofcat. Brain Research, 120(2), 231–249.

Jensen, M.S., & Yaari, Y. (1997). Role of intrinsic burst firing, potas-sium accumulation, and electrical coupling in the elevated potas-sium model of hippocampal epilepsy. Journal of Neurophysiology,77(3), 1224–1233.

Jensen, M.S., Azouz, R., & Yaari, Y. (1994). Variant firing patternsin rat hippocampal pyramidal cells modulated by extracellularpotassium. Journal of Neurophysiology, 71(3), 831–839.

Kager, H., Wadman, W.J., & Somjen, G.G. (2007). Seizure-like after-discharges simulated in a model neuron. Journal of ComputationalNeuroscience, 22(2), 105–128.

Karlocai, M.R., Kohus, Z., Kali, S., Ulbert, I., Szabo, G., Mate, Z.,Freund, T.F., & Gulyas, A.I. (2014). Physiological sharp wave-ripples and interictal events in vitro: what’s the difference? Brain:A Journal of Neurology, 137(Pt 2), 463–485.

Korn, S.J., Giacchino, J.L., Chamberlin, N.L., & Dingledine, R.(1987). Epileptiform burst activity induced by potassium in thehippocampus and its regulation by GABA-mediated inhibition.Journal of Neurophysiology, 57(1), 325–340.

LeBeau, F.E.N., Towers, S.K., Traub, R.D., Whittington, M.A., &Buhl, E.H. (2002). Fast network oscillations induced by potas-sium transients in the rat hippocampus in vitro. The Journal ofPhysiology, 542(Pt 1), 167–179.

Megıas, M., Emri, Z., Freund, T.F., & Gulyas, A.I. (2001). Totalnumber and distribution of inhibitory and excitatory synapses onhippocampal CA1 pyramidal cells. Neuroscience, 102(3), 527–540.

Meijer, H.G.E., Eissa, T.L., Kiewiet, B., Neuman, J.F., Schevon, C.A.,Emerson, R.G., Goodman, R.R., McKhann, G.M., Marcuccilli,C.J., Tryba, A.K., Cowan, J.D., van Gils, S.A., & van Drongelen,W. (2015). Modeling focal epileptic activity in the Wilson-cowanmodel with depolarization block. Journal of Mathematical Neuro-science, 5, 7.

Moody, W.J., Futamachi, K.J., & Prince, D.A. (1974). Extracellularpotassium activity during epileptogenesis. Experimental Neurol-ogy, 42(2), 248–263.

Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnaclegiant muscle fiber. Biophysical Journal, 35(1), 193–213.

Øyehaug, L., Østby, I., Lloyd, C.M., Omholt, S.W., & Einevoll, G.T.(2012). Dependence of spontaneous neuronal firing and depo-larisation block on astroglial membrane transport mechanisms.Journal of Computational Neuroscience, 32(1), 147–165.

Prince, D.A., & Wilder, B.J. (1967). Control mechanisms in corticalepileptogenic foci. “Surround” inhibition. Archives of Neurology,16(2), 194–202.

Quian Quiroga, R., Blanco, S., Rosso, O.A., Garcia, H., & Rabinowicz,A. (1997). Searching for hidden information with Gabor Trans-form in generalized tonic-clonic seizures. Electroencephalogra-phy and Clinical Neurophysiology, 103(4), 434–439.

Rinzel, J., & Ermentrout, G.B. (1989). Analysis of neural excitabilityand oscillations. In C.I. Koch, & E. Segev (Eds.) Methods in neu-ronal modeling (pp. 135–169). Cambridge: MIT Press. http://dl.acm.org/citation.cfm?id=94605.94613.

Scharfman, H.E., Sollas, A.L., Berger, R.E., & Goodman, J.H.(2003). Electrophysiological evidence of monosynaptic excita-tory transmission between granule cells after seizure-inducedmossy fiber sprouting. Journal of Neurophysiology, 90(4), 2536–2547.

J Comput Neurosci (2017) 43:65–79 79

Schevon, C.A., Weiss, S.A., McKhann, G., Goodman, R.R., Yuste,R., Emerson, R.G., & Trevelyan, A.J. (2012). Evidence of aninhibitory restraint of seizure activity in humans. Nature Commu-nications, 3, 1060.

Schwartz, T.H., & Bonhoeffer, T. (2001). In vivo optical mapping ofepileptic foci and surround inhibition in ferret cerebral cortex.Nature Medicine, 7(9), 1063–107.

Shin, D.S.H., Yu, W., Fawcett, A., & Carlen, P.L. (2010). Char-acterizing the persistent CA3 interneuronal spiking activity inelevated extracellular potassium in the young rat hippocampus.Brain Research, 1331, 39–50.

Somjen, G.G., & Giacchino, J.L. (1985). Potassium and calcium con-centrations in interstitial fluid of hippocampal formation duringparoxysmal responses. Journal of Neurophysiology, 53(4), 1098–1108.

Sypert, G., & Ward, A. (1974). Changes in extracellular potassiumactivity during neocortical propagated seizures. ExperimentalNeurology, 45(1), 19–41.

Thompson, S.M., & Gahwiler, B.H. (1989). Activity-dependent dis-inhibition. ii. effects of extracellular potassium, furosemide, andmembrane potential on ecl-in hippocampal ca3 neurons. Journalof Neurophysiology, 61(3), 512–523.

Toyoda, I., Fujita, S., Thamattoor, A.K., & Buckmaster, P.S. (2015).Unit activity of hippocampal interneurons before spontaneousseizures in an animal model of temporal lobe epilepsy. Journal ofNeuroscience, 35(16), 6600–6618.

Traynelis, S.F., & Dingledine, R. (1988). Potassium-induced sponta-neous electrographic seizures in the rat hippocampal slice. Journalof Neurophysiology, 59(1), 259–276.

Trevelyan, A.J., & Schevon, C.A. (2013). How inhibition influencesseizure propagation. Neuropharmacology, 69, 45–54.

Trevelyan, A.J., Sussillo, D., Watson, B.O., & Yuste, R. (2006). Modu-lar propagation of epileptiform activity: evidence for an inhibitoryveto in neocortex. The Journal of Neuroscience: The OfficialJournal of the Society for Neuroscience, 26(48), 12447–12455.

Trevelyan, A.J., Sussillo, D., & Yuste, R. (2007). Feedforward inhibi-tion contributes to the control of epileptiform propagation speed.The Journal of Neuroscience: The Official Journal of the Societyfor Neuroscience, 27(13), 3383–3387.

Wilson, H.R., & Cowan, J.D. (1972). Excitatory and inhibitory inter-actions in localized populations of model neurons. BiophysicalJournal, 12(1), 1–24.

Wuarin, J.P., & Dudek, F.E. (2001). Excitatory synaptic input to gran-ule cells increases with time after kainate treatment. Journal ofNeurophysiology, 85(3), 1067–1077.

Yaari, Y., Konnerth, A., & Heinemann, U. (1986). Nonsynaptic epilep-togenesis in the mammalian hippocampus in vitro. II. Role of extra-cellular potassium. Journal of Neurophysiology, 56(2), 424–438.

Yi, F., DeCan, E., Stoll, K., Marceau, E., Deisseroth, K., &Lawrence, J.J. (2015). Muscarinic excitation of parvalbumin-positive interneurons contributes to the severity of pilocarpine-induced seizures. Epilepsia, 56(2), 297–309.

Zandt, B.J., Visser, S., van Putten, M.J.A.M., & Ten Haken, B. (2014).A neural mass model based on single cell dynamics to modelpathophysiology. Journal of Computational Neuroscience.

Ziburkus, J., Cressman, J.R., Barreto, E., & Schiff, S.J. (2006).Interneuron and pyramidal cell interplay during in vitro seizure-like events. Journal of Neurophysiology, 95(6), 3948–3954.