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Neurobiology of Disease

Epilepsy in Small-World Networks

Theoden I. Netoff,1,3 RobertClewley,2,3 Scott Arno,1,3 Tara Keck,1,3 and JohnA.White1,31Department of Biomedical Engineering, 2Department of Mathematics and 3Center for BioDynamics and Center for Memory and Brain, Boston University,

Boston, Massachusetts 02215

In hippocampal slice models of epilepsy, two behaviors are seen: short bursts of electrical activity lasting 100 msec and seizure-like

electrical activity lasting seconds. The burstsoriginate from the CA3 region, where there is a high degree of recurrent excitatory connec-tions.Seizuresoriginate from theCA1, where there arefewerrecurrent connections.In attempting to explain this behavior,we simulated

model networks of excitatory neurons using several types of model neurons. The model neurons were connected in a ring containingpredominantly localconnections and some long-distance random connections, resulting in a small-world network connectivity pattern.

By changing parameters such as the synaptic strengths, number of synapses per neuron, proportion of local versus long-distanceconnections, we induced normal, seizing, and bursting behaviors. Based on these simulations, we made a simple mathematical

description of these networks under well-defined assumptions. This mathematical description explains how specific changes in thetopology or synaptic strength in the model cause transitions from normal to seizing and then to bursting. These behaviors appear to be

general properties of excitatory networks.

Key words:epilepsy; networks; small-world networks; seizures; computational modeling; interictal burst

IntroductionEpilepsy is characterized by two electrographic behaviors: inter-ictal bursts of activity that last100 msec and seizures that lastfrom seconds to minutes (Steriade, 2003). In slice models of ep-

ilepsy, bursts and seizures can be elicited in different regions ofthe hippocampus bathed in 4-aminopyridine (4-AP). Burstsoriginate in region CA3 of hippocampus (Chesnut and Swann,1988), whereas seizures originate in region CA1 (Netoff andSchiff, 2002). 4-AP increases excitability and effective synapticstrength (Perreault and Avoli, 1989); however, epileptiform be-havior can be induced in slices through a variety of methods(Traub and Miles, 1991), suggesting that the cause of these be-haviors is a general property of the network.

Traditionally, epilepsy is viewed as a disease of hypersyn-chronous neuronal activity (Penfield and Jasper, 1954; Steriade,2003). Evidence from hippocampal slices shows that bursts inCA3 are caused by neuronal activity that is synchronous on a fine

time scale (10 msec); however, neuronal activity during sliceseizures in CA1 is not synchronous (Netoff and Schiff, 2002; VanDrongelen et al., 2003). The most notable difference between thehippocampal regions is that CA3 has more recurrent synapticconnections than CA1. Staley et al. (1998) hypothesized thatbursts originate in region CA3 because the network activates

quickly, via recurrent excitation, depletingthe primary glutamatestores of the neurons and thus shutting down the network.

Our goal in this study was to use computational models toexplore how epileptiform behaviors relate to the connectivity of

the underlying networks. Our operating hypothesis for CA3bursts is similar to that of Staley et al. (1998), except that ourmodels rely on generally defined neuronal refractoriness toterminate burst activity. Refractoriness may arise via a number ofmechanisms, including synaptic depletion, inhibition, orvoltage-dependent properties in postsynaptic cells. In the CA1,which has less recurrent excitation, the activity spreads slower.Thus, an excitable pool of CA1 neurons is always available, lead-ing to sustainable seizure-like activity. To test these hypotheses,we simulated networks intended to mimic regions CA3and CA1.We used small-world network topologies, in which the major-ity of connections between cells are local, but a few cells havelong-distance connections (Watts and Strogatz, 1998; Watts,

1999). Small-world networks were used because they are simple,flexible, and reminiscent of the connectivity patterns of networksin the brain. As connectivity in the networks was changed, weobserved activity resembling epileptiform behaviors seen in slicemodels. Networks with large numbers of long-distance connec-tions were more prone to generating self-terminating bursts. Ad-ditionally, our results suggest that the greater level of intercon-nectivity in CA3 may be responsible for its tendency to burstrather than seize. Results were independent of the specific neu-ronal model (Poisson, integrate-and-fire, Hodgkin-Huxley)usedin the simulations. We derived a reduced mathematical descrip-tion of the networks that helped us to describe the conditionsunder which networks transition from normal to seizing to

bursting. Although this model provides only a heuristic de-scription of the slice behavior, it demonstrates how epileptiformbehaviors may depend on specific physical parameters.

Received Dec. 18, 2003; revised July 29, 2004; accepted July 30, 2004.

This work was supported by National Institutes of Health Grants R01-NS34425 and R01-MH43510 and National

Science Foundation Grant DMS-0211505. We thank Nancy Kopell, Sid Redner, Steven J. Schiff, Corey Acker, Alan

Dorval,andG.BardErmentroutforvaluablediscussionsinthedevelopmentofthiswork.Thereviewersofthispaper

made a number of suggestions that improved it dramatically.

Correspondence should be addressed to Theoden I. Netoff, Boston University, Department of Biomedical Engi-

neering, 44 Cummington Street, Boston, MA 02215. E-mail: [email protected]:10.1523/JNEUROSCI.1509-04.2004

Copyright 2004 Society for Neuroscience 0270-6474/04/248075-09$15.00/0

The Journal of Neuroscience, September 15, 2004 24(37):80758083 8075


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Materials andMethodsStructure of the network and connectivity.We generated simple networkmodels of excitatory neurons in hippocampus. To keep the number offreeparameters manageable, to moreeasily constrain activity to spread ina controlled manner, and to eliminatethe effects of boundaryconditions,we restricted our analyses to one-dimensional rings of neurons. Be-cause organization of the synaptic connections within cortical regions isneither a lattice of nearest-neighbor connections nor completely ran-domly connected (Mountcastle, 1997; Gonzalez-Burgos et al., 2000; Mc-Cormick and Contreras, 2001), we used a simple method to constructnetworks that lie between these two extremes. We began with a model inwhich each neuron is connected to a specific number k of its nearestneighbors and then randomly disconnected a proportionof the synap-tic connections and reconnected these synapses to a randomly chosenpostsynaptic cell. This method of network construction leads to small-world networks (Watts and Strogatz, 1998; Watts, 1999), in which mostconnections made by a given presynaptic neuronare local,but an impor-tant few can spread activity over long distances. An illustration of net-works with varying amount of long-distance connections is given inFigure 1. In the network in Figure 1a, eachnoderepresents a neuron.The ring of neurons is connected in a perfectlattice( 0), with eachcell connected to its four nearest neighbors. Figure 1,bandc, represents

networks increasing values of. These are small-world networks becausethey include a preponderance of local, regular connections but a smallnumber of long-distance connections, which greatly reduce the numberof synaptic steps required to connect any pair of neurons in the network.

The anatomy of our small-world networks is characterized by threefree parameters: network size (N), the number of synaptic connectionsper neuron (k), and the proportion of randomly made long-distanceconnections (). For most simulations, we used N 3000, approxi-mately corresponding to the smallest population size within which epi-leptiform activity is seen in the hippocampus (Fox et al., 2001). In somesimulations, we usedN 24,000to examine thegenerality of ourresults.We examined many values ofk but focused our attention on networkswith1% totalconnectivity(k 30forN 3000) to represent region CA1and networks with 3% total connectivity (k 90 for N 3000) torepresent region CA3. Bothkvalues are overestimated as follows: excita-

tory to excitatory coupling in region CA3 is closer to 2% (MacVicar andDudek, 1980) and for region CA1, it is 1%. Because the numbers oflocal versus long-distance connections are unknown in hippocampus, wetreated(the proportion of random connections in the network) as anexplicit free parameter that we varied from 0 to 1. is a particularlyimportant parameter for these models, because it controls the rate atwhich localwavesof activity give rise to new waves at distant locationsin the network.

Model neurons and synapses.We ran simulations using three differenttypes of model neurons: noisy and leaky integrate-and-fire neurons, sto-chastic HodgkinHuxley cells (Chow and White, 1996), and a Poissonspike-train cell model that is equivalent to the other models in terms of

first-order interspike interval statistics. The integrate-and-fire model is aregular leaky integrate-and-fire model with a stochastic component,given by

dV

dt Vleak V Isyn ,

where Vindicatesthe membranepotential, Vleakis theresting potentialof

the neuron,Isyn is the synaptic input (described in a subsequent para-graph), andis the stochastic component (white noise of sufficient vari-ance to generate spontaneous activity at a target rate). The neuron fireswhen Vreaches a threshold, resulting from noise or when a synapticcurrent is injected into postsynaptic neurons. Theneuron is then reset tozero, and all synaptic inputs are blocked for an absolute refractoryperiod of time R (see below for additional information onrefractoriness).

The stochasticHodgkinHuxley model is a conductance-basedmodel,which is describedin detailby Chow andWhite (1996). Parameters wereas in the original 1952 study, except that sodium channels were modeledas discrete, stochastic elements. The Langevin method was used to de-scribe the effects of channel noise (Chow and White, 1996). In thismodel, membrane noise causes the membrane potential to fluctuate and

occasionally causes the neuron to fire spontaneously. The number ofsodium channels wastunedto 3375 to match a target average sponta-neous firing rate (see below).

Synaptic currents for the integrate-and-fire and stochastic HodgkinHuxley models were calculated using a double exponential functionIsynA(e

t/ s et/ f) (VsynV), whereA is the synaptic amplitude,tis the time since synaptic input occurred,sandfare the slow and fastdecay rates, respectively, andVsynis the reversal potential of the synapse(Bower and Beeman, 1995).

To construct the Poisson model, we usedMatlab (MathWorks,Natick,MA) to selectspiketimesfrom a Poissonprocess. Synaptic inputs forthismodel were simulatedin thefollowing way: in response to the arrival of apresynaptic spike, the postsynaptic cell has a probability of immediatelyfiring, where the probability for a single synaptic input was set to 2.5%.We set the probability of firing after two or more simultaneous synaptic

inputs to 1. If the cell does not immediately fire in response to an input,then it uses the spike time previously drawn from the Poissondistribution.

All models were adjusted to have an absolute refractory period, R(after spiking), 10 times longer than the synaptic delay,d. This allowstheactive population to travel 10 steps before the neurons recover. For thePoisson model, the synaptic delay was set at 3.7 msec and a refractorytime of 36 msec, whereas in the integrate-and-fire model, the synapticdelay was 2.8 msec and the refractory time was 28 msec. The Hodgkin Huxley model had a synaptic delay dependent on the distance betweenthe neurons with a 1 msec delay between the local cells and up to 5 msecfor long-distance connections. An absolute refractory time was 36 msecandwas set to behavequalitatively similarly to the other models. Theoneexception is the network simulation with 24,000 neurons where the re-fractory time was 28 msec and the synaptic delay was 2 msec, which wasnecessary to allow enough time for the entire network to be activatedbefore the first cells recovered. Therefractory periodcan be generated bymany mechanisms, either presynaptic or postsynaptic, and we leave thisundetermined. The important factor is that the refractory time is on theorder of thetime it takes activity to spreadthroughout thenetwork in thebursting regimen; otherwise, the activity will re-enter and a clean transi-tion to bursting will not occur. We also matched all models so that theyhad an approximately exponential interspike interval distribution andfiring probability of 0.0315 spikes/sec. In most simulations, synaptic ef-ficacies were set in all models such that single inputs caused postsynapticaction potentials 2.5% of the time, and two simultaneous inputs led topostsynaptic firing with probability of firing approximately equal to one(assumed to be exactly one in our later theoretical analysis). For the24,000 cell simulations (see Fig. 3), we compensated for the eightfold

increase in network size with a 20-fold decrease in synaptic weight. Assuccessfully predicted by our reduced theoretical model, this change keptthe same overall rate of wave generation, leading to results (see Fig. 3).

Figure 1. Small-world network. a, Networks of neurons are generated in which all cells areonly coupled to their nearest neighbors (4 in this case).b, To generate small-world networks,small numbers of connections are broken and rewired to make long-distance connections at

randomlocations.Long-distanceconnectionsreducethenumberofsynapsesbetweenanypairof neurons in the network.c, As more long-distance connections are added, the network losesthe property that most connections are local, and the network looks much more random. We

find a range ofnormalandepileptiformbehaviors inthe small-world network regimen,wherefew connections are necessary to connect any pair of the neurons, but local connections stillpredominate.

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ResultsBasic properties of propagating activityIn all three types of networks (Poisson, integrate-and-fire, andstochastic HodgkinHuxley), we observed qualitatively simi-lar behaviors. Networks are quiescent at first, but eventually aspontaneous action potential in one neuron initiates activityin two neurons with common local postsynaptic targets. Be-cause convergence of two simultaneous inputs fires postsyn-aptic cells in these networks with probability near one, thisevent generates two waves that travel in opposite directions

around the ring. With few long-distance connections, a smallnumber of waves sweeps across the network. This results in asmall, stable amount of activity. With more long-distance

connections, existing waves frequently give rise to new wavesin distant locations, and network activity transitions to sus-tained high activity, which we liken to seizures. Figure 2cillus-trates still frames from a movie of an ongoing seizure (see Fig.2, a and b, for an explanation of the display method). With alarge enough number of long-distance connections (Fig. 2d),we observed bursts in which the majority of the network firedsynchronously and then became refractory. Movies of seizuresand bursts can be seen at http://www.bu.edu/ndl/people/net-off/SWN/JNeurosciSupplement.html. As we show, transition

points between normal, bursting, and seizing vary accordingto the number of long-distance connections, network size,synaptic strength, and number of synapses per neuron.

Figure2. Burstingandseizingbehaviorsasthenumberoflong-distanceconnectionsarechanged. a,TheringcontainsNneurons,eachofwhichareconnectedto k,mostlylocalneighbors(left).To visualizethe activityof this large network, we color coded each pointaccordingto thestate of theneuron andpulled every kthpointin theringtoward thecenter to make a spoke.Therefore, aneuronin thecenter ofa spoke isconnected toall theneuronsin thespoke, assumingthatall synaptic connectionsare local.A neuronat theendof thespoke isconnected to half of theneuronsonthespokeand half of theneuronson theoppositeend of thenextspoke.Thisresultsin a plotof theringthatresemblesa slinky. b, An illustrative temporal snapshot of network activity,with N3000, k30synapsesperneuron(i.e.,1%networkconnectivity),and 0.1.Lightgraydotsrepresentexcitableneurons,blackdotsarefiringneurons,anddarkgraydotsarerefractoryneurons.Thewavefrontsizestabilizestoapproximatelyhalfthesizeofthelocalneighborhood kandis followedby a refractory tail.Thistailis determined byhowmanystepsthewavefront cantravelbeforethe neurons begin to recover.c, Successive frames from a movie of seizing activity, with N 3000,k 30, and 0.1 (i.e., that 10% of synapses have been rewired). The frame rate is 250 Hz,corresponding to approximately two synaptic timedelays; therefore,the activewaves appeartwice as large(in space)as theiractual size.Spontaneous background activity generates a cascade ofactivity, which stabilizes into two traveling waves (frames 525). These traveling waves generate other waves in the network through the long-distance connections (e.g., frames 26, 31, 34).

Eventually, waves start to meet and annihilate each other (e.g., frames 4, 33, 43). This network attains equilibrium when the new waves are generated at the same rate that the waves annihilateeachother.d,Stillframesfromamovieofburstingactivity(N 3000; k 90; 0.1).Inthisnetwork,thenumberoflong-distanceconnectionscauseswavestogeneratenewwavesfasterthanthewavesannihilateeachother.Thisresultsinalloftheneuronsfiringinthenetwork,alloftheneuronsbecomingrefractory,andtheactivityinthenetworkshuttingoff.Moviesofnetworkactivity

can be seen at: http://www.bu.edu/ndl/people/netoff/SWN/JNeurosciSupplement.html.

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mean path length is low. The regimen of seizing activity in theCA1 model corresponds with this small-world regimen. The be-ginning of the bursting regimen corresponds with the drop inclustering coefficient, which signals the transition of the networkfrom a small world to a random graph. As indicated by compar-ing the right and left columns in Figure 3, having a larger numberof connections leads to bursting at lower values of.

Scaling relationships and the reduced model ofpropagating activityPropagating activity in simulated networks has a number of ste-reotypical characteristics that allow us to represent it accuratelyusing a reduced model. First, because the time needed for supra-thresholdinputs to activate the model neurons is small comparedwith thesynaptic delay time, thesynapticdelaysets theamount oftime required for wave fronts to propagate from one group ofactive neurons to another. Consequently, the network evolves inapproximately discrete time steps, corresponding to the synapticdelay. Second, assuming a nearly regular lattice (i.e., that thenumber of long-distance connections is small relative toN), thewave front size,, is just less than half the neighborhoodsize:k/2 1. The division by 2 comes from the fact that the wavespreads in both directions (unless one set of cells is refractory);subtracting 1 from the total removes the neuron that only re-ceives a single synaptic input from the active neurons. Third,following each wave is a wake of refractory neurons, approxi-mately the sizeR R/d, whereRdenotes the refractorytime of each neuron, and ddenotes the synaptic delay. Thisequation defines the number of time steps that a neuron remainsrefractory, which we have denotedR. Finally, knowing the num-berof waves,their size, andtheir wake size, it is straightforwardtocalculate the characteristics of propagation and the probability ofemergence of a new wave per time step, as well as the probabilityper unit time that two waves will collide and thus annihilate.

Thus, we should be able to construct a discrete time, birth-and-death process describing activity in the ring network.

Letwidenote the number of active waves at time step i. Wedefine the number of new waves that will begin spontaneously inone time step to be Si seip2. Here,s is the probability that anygiven neuron will fire in that time, which we set ats 0.0315d, where 0.0315 is the spontaneous spike rate per second. ei is thenumber of excitable neurons present in the network at time i.p2is the probability that two or more neurons fire in response to thefiring of the same presynaptic neuron. The dependent variablep2can be calculated from the binomial theorem using synapticstrengthp1 (which we define as the probability that firing of asingle presynaptic cell induces an action potential in a given

postsynaptic cell) and the number of synapses per neuron k:p21 (1 p1)

k kp1(1 p1)

k1. In our reduced model, if twosuch neurons in a neighborhood fire, we will assume that a trav-eling wave will be initiated with probability of exactly one (therelative frequency of such an event was observed to be 1 in thesimulations), with wave front size exactly equal to . This as-sumption holds true for our 3000 cell simulations but not for the24,000 cell simulations. Interestingly, behavior of the network isnot sensitive to violations of this assumption (Fig. 3).

Once a wave has been initiated, it can start new waves ofactivity in other regions of the network through long-distanceconnections. The rate at which new waves are generated dependson the current number of waves,wi, which determines the num-

berof activeneurons,wi. These activeneurons havewiklong-distance connections on average. The probability that synapses attheend of these connections start a wave at a postsynapticneuron

isp1p2ei/N, whereeiis the number of excitable cells (see below).The approximate symmetry in the connectivity of the ring (be-cause is assumed to be much less than 1) means that a new wavewill propagate in two directions as two separate wave fronts. Thisis reflected in theformula for the newwave rate by multiplying bya factor of two. This approximation is valid, provided that thenetwork is far from being saturated with activity (i.e., when the

network size N is much larger than the number of active andrefractory neurons) and the resulting wave fronts develop intofull waves with refractory wakes, an assumption that is true aslong as the waves emerge in a region of nonrefractory cells. Thenumber of excitable cells, ei, is thetotal numberof neurons minusthe number that are active or refractory:

ei N wi t1

R

wit.

Thelast term in this equation for ei accounts forthe recenthistoryof activity, by summing that past activity up to the refractory

time. This sum gives the total number of refractory neurons attime i. Initially, we will further simplify this equation by assumingthat the number of refractory cells is simply proportional to thenumber of active waves:eiN wi(1 R). This approxima-tion is also valid only when the network activity is far from satu-ration. Accordingto this model,the numberof newwavesborn attime step i is approximated byn i (2wik)(p1p2ei/N) S i.This function, a quadratic function ofwi, is plotted in the toppanelsof Figure4 (dashed lines)for three differentvalues of,theproportion of long-distance connections.

The average number of wave collisions per time step dependson how many waves are present in the network. The more wavesin the network, the more likelycollisions will occur.The expected

number of dying waves in a time step can be approximated by thetime it takes the currentlyactive waves, evenlydistributed aroundthe ring, to collide. We estimate the death rate to bedi 2wi/ei.In this equation, the term 2 reflects the fact that two wave frontspropagate through an excitable region toward each other at a rateof 2 neurons per time step; the term wi reflects the fact thatevenlydistributedwaves grow, on average, closer as their numbergrows; and the termeireflects the fact that the average distancebetween two waves is proportional to the number of remainingexcitable cells. As for ei, the approximation in di is most validwhen the network activity is far from saturation. The death rate diis plotted (solid lines) in the top panels in Figure 4. Its valueincreases without bound as the number of active and refractory

neurons approachesN, when the denominator approaches zero(at the point where the network reaches saturation). The netgrowth in the number of waves at time i isni di, and thus thewave birth death process can be written as a one-dimensionalmap,wi1f(wi) wi ni di, in which we have defined thefunctionf(wi) for notational convenience. This one-dimensionalmap is only valid under the assumption that the number of re-fractory cells is directly proportional to the current number ofactive wave (see above) (Table 1).

Activity in thenetwork reaches an equilibrium state when newwaves are generated at the same rate that they annihilate eachother, i.e., when there is a steady state number of wavesw*thatsolves the equationf(w*)w*. A stable equilibrium corresponds

to the slope of the functionf(wi) at theequilibriumbetween 1 and1; otherwise, it is unstable and the new wave rate is faster thanthe dying wave rate. Thestrength of attraction of this equilibrium



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affects the spread of the distribution in thenumber of neurons firing at any giventime in a stochastic network.

The equilibrium level of activity and itsstability depend on the parameters in thenew wave rate ni and dying wave rate di. Aswe vary parameters such as the proportion

of long-distance connections (), we caninvestigate how changes in the equilib-rium cause the network to switch fromnormal to seizing to bursting. In the bot-tom panels in Figure 4, we plot the num-ber of waves on the next time step wi1versus the number of waves on this timestep wi (solid line) as well as the line ofidentity (y x; dotted line). In theseplots, equilibrium points f(w*) are indi-cated by intersections of the solid and dot-ted lines. Stability of equilibrium points isgiven by the slope off(w*). At low long-distance connectivity ( 0.001), bothnianddiare small, resulting in a weakly at-tracting equilibrium (with a slope just lessthan one). As indicated by the fact thatwi1 wi, waves are rarely born or die,implyingthat thenetwork is dominatedbythe low rate of spontaneous wave genera-tion. As more long-distance connectionsare introduced ( 0.01), the rate ofemergence of new waves increases. Thistrend increases the number of waves thatwe expect to see at the equilibrium andalso makes the equilibrium more stronglyattracting (the slope at the equilibrium

point is near zero). As even more long-distance connections are introduced ( 0.05), the slope offatthe equilibrium point is less than 1, implying that the equilib-rium is unstable. [This is known as aflip bifurcation,or some-times as a pitchfork bifurcationfor maps (Baker and Gollub,1996).] At this point in the full network model, new waves aregenerated rapidly, giving rise to a burst (i.e., a large, synchronousincrease in the amount of activity). The burst (when wi 6)would lead to a system that is dominated by wave death on sub-sequent time steps. Then, almost the entire network will becomerefractory, implying that the system will be quiescent untilenough cells have returned to the excitable state and the networkcan burst again. This detail of bursting behavior is not captured

by our reduced model, which was derived under the assumptionthat the number of active and refractory cells at any given time issmall compared with the network sizeN. Instead, asis increasedabove a critical point at 0.05, the reduced system continues tooscillate between increasingly higher and lower states (aroundthe unstable equilibrium) until the state fluctuates chaoticallywith a very large amplitude. This describes a well-understoodperiod-doubling route to chaosfor discrete-time maps such asthat described byf(which resembles the classic logisticmap)(Baker and Gollub, 1996). The amplitudes are large enough thatthe activity of a full network model would reach saturation at thispoint and therefore enter our bursting regimen. Because thelarge-amplitude chaotic behavior is reached for values very

close to that which first caused the equilibrium to become unsta-ble, the onset of instability is a reasonable indicator for the onsetof bursting in the full network model. Results from this reduced

model and those from simulations are compared in Figure 5(discussed below).

For both CA1 and CA3 network simulations, the transitionfrom normalactivity to seizures corresponds to an increase in thepercentage of active cells and the onset of a sustained oscillationin the population activity. Because analysis of the one-dimensional map does not provide us with an explicit conditionfor this transition, we used the full refractory dynamics in thedefinition ofei, making the map (1R)-dimensional, where R isthe length of the refractory period. In a similar way to the one-dimensional map, the (1R)-dimensional map can be analyzedto predict qualitative changes in the dynamics. For few long-

distance connections, both maps exhibit the same equilibria andstability properties. As more long-distance connections areadded, the (1 R)-dimensional map exhibits oscillations notpresent in the one-dimensional map, after aHopf bifurcationfor mapsoccurs as is increased (Agarwal et al., 2000). Thesesmall-amplitude oscillations resemble those in the full networksimulations during seizures (data not shown) and also have aperiod approximately equal to the refractory time of the neurons.The transition to oscillation occurs at a value ofthat can becomputed analytically and compared with results from compu-tational simulations (see below). Numerical implementations ofthe two maps both transition to bursting for similar values ofmodel parameters. However, the (1 R)-dimensional map does

notprovide an explicit analytic conditionfor theburstonset. Thisis because the transition occurs in the full map when the oscilla-tions grow so large that the activity saturates. The saturation of

Figure4. Thebirthdeathprocess(1-dimensionalmap)modelofwavegenerationandannihilation.Thetoppanelsshowhowmany new waves are generated (n

i; dashed lines) and annihilated (d

i; solid lines) per time step as functions of the number of

currently active waves, withk 90. They-intercept of the dashed lines indicates the spontaneous background rate of wavegeneration. An equilibrium point exists where the new wave rate is equal to the dying wave rate (indicated by the arrows). Thebottom panels map the number of waves on one time step to the average number expected on the next time step (solid lines).Equilibriaoccurwhenthenumber of waveson thenexttimestepis equalto that onthe current time step [i.e.,at theintersectionofthesolidlineoff(w

i) andthedottedlineof identity(alsoindicated byarrows)].For 0.0001,the equilibriumpoint at w

i

1.82 isonly weaklyattracting(theslopeof thesolidlineis approximately1),andthenumberofwaveschangesonlyslightlyateachtimestep.For 0.01, thesystemhas a stronglyattractingequilibrium(wheretheslopeof thesolid line isapproximatelyzero), corresponding to ongoing seizures characterized by an average of 3.87 existing simultaneous waves. For 0.05, theequilibriumisunstable,andthedynamicshasenteredachaoticregimen.Theonsetofchaosindicatesthattheentirenetworkwillrepeatedly fire brief synchronous bursts.



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activity breaks the assumption of the der-ivation of the map and is not a preciselydefinable transition. Thus, it cannot becaptured by an analytical expression in-volving a local change in stability. In con-trast, the one-dimensional map representsan averaged view of the full dynamics

(through itssimplifiedtreatment of there-fractory wake), for which transition tobursting occurs while the low-activity as-sumption still holds.

Figure 5ashows the boundaries of thedifferent behavior regimens predicted byboth of the analytical models as the num-ber of synapses per neuron (k) and pro-portion of long-distance connections ()is varied. The seizing-bursting border(solid black lines) was calculated from thesimplified one-dimensional map. Thenormal-seizing border (dotted blacklines) was derived from the (1 R)-dimensional map. The two horizontalgray lines in each panel correspond to theCA1 and CA3models fromFigure 3. afinFigure 5a correspond to the examplesfrom Figure 3, again as the proportion oflong-distance connections is changedover several orders of magnitude. Verticaltick marks in Figure 5acorrespond to theborders between the normal, bursting,and seizing regimens for the simulatedCA1 and CA3 models from Figure 3. Pre-dicted transitions from simplified mapsoccur near the transition points observed

in simulations for a wide range of param-eter values.

In the brain and in slice models, epi-lepsy is probably not caused by an increasein long-distance connections but rather achange in the synaptic strength resultingfrom drugs or changes in the ionic con-centrations of the fluid bathing the slice.Using our reduced map models, we canexplore the effects of changing the synap-tic efficacyp1(Fig. 5b), which we define asthe probability that a single postsynapticinput gives rise to a postsynaptic spike on

the next time step. We look for the regi-men transitionsasp1 is varied, keepingat0.01 (the same value used for panelsaande in Fig. 3). The transitions predicted bythe models are plotted in Figure 5b. Thetotal network activity measured from thenetwork simulations is plotted in the in-sets and shows a drop in activity coincid-ing with the burst transition for CA3 aspredicted(indicated by theline in theinsetand the dash in the full panel). The CA1network did not transition to bursts andhad a smooth transition to seizures with

an onset that is harder to define but ap-proximately coincides with the predictedvalues. Thus, the reduced model predicts

Figure 5. Change in network behavior as a function of number of synaptic connections per neuron and proportion of long-distance connections using the reduced model (for network size of 3000 neurons). The left panel shows curves delineating thenormal,seizing,andburstingregimensasthenumberofsynapsesperneuron,andtheproportionoflong-distanceconnectionsarechanged. Thesolidblackcurveis calculated from analysis of theone-dimensional mapand thedotted black curve from the(1R)-dimensional map. Tick marks indicate the boundaries of normal, seizing, and bursting behavior in network simulations fromFigure3.HorizontallinesindicatespecificparameterchoicesfortheCA3andtheCA1models.Pointslabeled afcorrespondtotheconditions simulated in Figure 3af. These plots imply that the CA3 network will transition from normal to bursting at a muchsmaller proportion of long-distance connections or smaller synaptic strength thanthe CA1.The rightpanel illustratesthe bound-

aries between the behavioral regimens as the number of synapses per neuron and the synaptic strength are varied (with theproportionoflong-distanceconnectionsfixedat0.01).Thecurveswerecalculatedinasimilarwaytothatintheleftpanel.Thetick mark in the line for the CA3 indicates the boundary between bursting and seizing observed in the integrate-and-fire model,

wherethe populationfiring rate asa functionof synapticstrengthare plotted inthe inset.Theresults of thesimulations correlatewell with theanalyses ofthe reduced mapmodels.No clear transition from seizing to burstingwas seen inthe full CA1model, aspredicted by the one-dimensional map model.

Table 1. Principal definitions, symbols, and default parametervalues usedin equations

Symbol Identification Definition/values used

N Number of neurons in network 3000 (24,000 in some network simulations)k Number of synapses per neuron 30 (for CA1), 90 (for CA3)

s Spontaneous firing rate of a single neuron per time step of

size d

0.0315 d

R

Absolute refractory t ime of neuron 28 msec ( IF) , 36 msec (Poiss , HH)

d Synaptic time delay 2.8 msec (IF), 3.7 msec (Poiss), 15 msec (HH) Proportion of long-distance connections generated by

breaking a synapse and rewiring it to a randomly cho-

sen postsynaptic cell

Varied from 1.0 105 to 0.4

p1

Synaptic strength (i.e., probability that postsynaptic neu-

ron will fire given that a particular presynaptic neuron

fired)

0.025

p2

Probability that two postsynaptic neurons fire given the

presynaptic neuron fired (dependent onkandp1)

p2 1 (1p

1)k kp

1(1p

1)k1

Approximate number of neurons in wave front k/2 1R Number of time steps that a neuron remains refractory R

R/

d 10

wi

Number of waves present in network at timei wi1 f(w

i)

ei

Number of excitable neurons in the network at timeiei Nw

i

R

t1

wit

orei N

wi(1 R) (for one-dimensional map)ni

Number of new waves generated at timeiresulting from

long-distance connections

ni (2w

ik)(p

1p

2ei/N)S

i

di

Number of waves that die in time stepiresulting from

wave collision

di 2w

i/e

i

Si

Spontaneous wave generation resulting from spontane-

ous cellular activity

Sise

ip

2

w* Number of waves in network where new wave rate and

dying wave rate are equal (equilibrium point)

f(w*)w*

f (wi) Function describing number of waves on next time step

given number of waves on time step i

f(wi)w

i n

i d

i



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that as synaptic strengths are enhanced, CA3 is more likely totransition to bursting, but CA1 is more likely to transition to aseizing regimen, agreeing with experimental results.

DiscussionHere, we have introduced a simple small-world network represen-tation of excitatory neurons in the hippocampal slice. The network

was implemented using three different neuron models: noisy andleaky integrate-and-fire, stochastic HodgkinHuxley, and a Poissonspike-train model. We used these models to explain why seizuresmay not be hypersynchronous, but bursts are. In these simulations,seizure activity in the CA1-like networks is not fully synchronous,allowing the activity to be sustained. The same model supports thatbursts in CA3 are caused by synchronous rapid recruitment of neu-ronal activity. Simulation resultswereindependent of particular cel-lular models used, indicating that the proportion of long-distanceconnections is more important than the details of individual neu-rons in determining the epileptiform properties of the network. Wederived a reduced mathematical model of the average activity in thenetwork asa birth death process(map) in discretetime.Thismodelavoids a purely mean-field approximation (which would ignore thesmall-world properties of the network) and instead retains parame-ters directly related to the physiology and the connectivity of thenetwork. The one-dimensional map predicted the transition fromseizing to bursting found in the network simulations as the numberof synapses per neuron, proportion of long-distance connections,and synaptic strength were varied. These transitions were describedas a loss of stability of an equilibrium in the map. A more detaileddescription of the refractory dynamics gave rise to a higher-dimensional map. This map could be analyzed to predict the transi-tion fromnormalactivity to seizing.These reducedmodels highlightthe roles of physical parameters that could underlie the differentepileptiform behaviors observed in CA1 versus CA3.

Relation to previous modeling studiesA number of recent modeling studies have examined problemssimilar to those studied here. For example, Tsodyks et al. (2000)studied coherent activity in randomly connected networks withdepressing synapses. They observed coherent activity, somewhatsimilar to the bursting seen here. In their work and in relatedwork (Tabak et al., 2000; Wiedemann and Luthi, 2003), networksof neurons with depressing synapses initiate bursts from excita-tory neurons that had not fired for some time and thus gave riseto large postsynaptic effects. Synaptic depression shut downbursts in these studies. In contrast to these models, our model hasexplicit small-world connectivity that plays an important role inthe behavior of the network. We focused on how changes in the

number of long-distance connections, synaptic strength, andoverall connectivity led to dramatic changes in network activity.

Nishikawa et al. (2003) studied synchronization in small-world networks that ranged from our WattsStrogatz style tonetworks in which a few hyperconnectedhubneurons servedto reduced the mean path length. They found that hubbed net-works were less likely to synchronize than WattsStrogatz net-works, although the hubbed networks have smaller mean pathlengths. Superficially, this result seems to contrast with our resultthat the greatest synchronization was seen with small pathlengths. But this apparent discrepancy is an artifact of the way thenetworks were constructed in these two studies. Our results arecompatible with the general argument of Nishikawa and col-

leagues, in that network synchronization decreases with increas-ing heterogeneity in the number of connections per neuron. Ourbursting networks with largehave both small mean path and

low heterogeneity, because in a randomly connected network,each cell receives approximately the same number of connec-tions. Our seizing networks with smallare more heterogeneousand thus are expected to be less synchronized.

Two additional computational efforts have shown that small-world networks similar to ours can synchronize. Networks ofoscillatory elements synchronize when the network contains

enough long-distance connections of sufficient synaptic strength(Hong et al., 2002). Roxin and colleagues (2004) have showedthat adding long-distance connections makes small-world net-works of integrate-and-fire neurons transition from sustainedactivity to synchronous bursts of finite duration. Oscillation ofpopulation activity has been studied by Curtu and Ermentrout(2001) and Wilson and Cowan (1973) using differential equa-tions similar in form to ourdiscrete-time maps. Ourwork utilizesthese types of models to study epilepsy. First, we relate our mod-els to epilepsy in a specific brain region (hippocampus) and at-tempt to explain why regions CA1 and CA3 exhibit differentepileptiform behaviors in slice models. Second, we show that therelationship between the number of long-distance connectionsand seizing or bursting is remarkably independent of the neuro-nalmodel used. Third, ourderivation of the reduced mapmodelsretains important physiological parameters of which the effect onepileptiform behavior can be studied directly.

Relation to experimental resultsThis model helps to explain how a stable network may becomeunstable and prone to epileptiform behaviors. Many drugs anddisturbances to the cell equilibria can cause this. For example,4-AP induces epileptiform behaviors in slice models by blockingvoltage-gated K channels and thus indirectly enhancing EPSPamplitudes (Chesnut and Swann, 1988). It is shown in Figure 5that enhancing synaptic strength can transition a stable networkinto bursting or seizing. Epilepsy can also be induced following

cell death. Although decreasing the number of cells alone cannotinduce epilepsy in our networks, a concomitant increase in syn-aptic strength to compensate for the reduced synaptic activitymight. Although our simulations only included excitatory cells,normal network activity depends on a balance of excitation andinhibition for stability. The parameterp1could be interpreted asrepresenting the ratio of excitation to inhibition. In hippocampalslices, epileptiform activity can be induced by pharmacologicallyblocking inhibitory synaptic activity (Amitai et al., 1993). Thesedisinhibited slice models of epilepsy show three stages during theictal event (Borck and Jefferys, 1999). After the first stage of de-polarization, the second consists of high-frequency oscillationssimilar to our seizures. The third stage consists of postictal bursts

that are similar to our bursts. Our model suggests that the seiz-ing bursting transition in disinhibited slices may correspond toincreasing synaptic strength, connectivity, or cellular excitabilityduring the ictal event.

We predict from our model that networks with fewer recur-rent connections are more likely to seize than networks withmore recurrent connections. Our model is consistent with thetheory that epileptiform behaviors are generated by positive feed-back in excitatory network activity (Schwartzkroin, 1994), result-ing in runaway excitation. This may help to explain why someregions of the brain produce seizures, whereas others produceepileptiformbursts. Therange of synaptic strengths forwhich thenetwork will produce seizing behavior is smaller for networks

with more recurrent excitatory synapses. These networks aremore likely to transition from normal directly to bursting with-out observing seizures. Our model suggests that epileptiform



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bursts in CA1 may occur only when it receives strong synchro-nous synaptic input from theburst-proneCA3 region. If burstinginput to a seizure-producing region is stopped, the region may bereleased to produce its own seizing behavior, as observed exper-imentally by Barbarosie and Avoli (1997) and Bragdon et al.(1992).

Organotypic cultures of neocortex generate waves of activity

that are reminiscent of our normal and bursting activity (Beggsand Plenz, 2003). Normal activity in the organotypic cultures isscale-free, in that the probability distribution of sizes of wavesand the distribution of wave lifetimes both obey a power law,whereas picrotoxin-induced bursts were not. Beggs and Plenz(2003) replicated their scale-free behavior in a multilayer, feed-forward model. In our simulations, distributions of activity donot resemble power laws for normal, seizing, or bursting activity.We believe that our simulated activity is not scale-free primarilybecause waves of activity, once initiated, almost always propagateover long distances and thus have long lives. This behavior standsin contrast to that observed and modeled by Beggs and Plenz(2003), in which the most common events are small in spatialscale and short in duration.

The time scales that we observe for the bursting and seizing donot necessarily correspond to actual time scales observed exper-imentallyin vitro. Furthermore, the seizures in the model do notend as seizures do in thein vitromodels, because the small-worldnetwork model and our mathematical description of the modelare highly reduced compared with the hippocampal slice. Weexpect that fuller models, including inhibition and more spatialrealism, may address these discrepancies. The purpose of thismodel is to develop an understanding of how changes in thephysiology change the behavior from normal to bursts to sei-zures. Therefore, we simulated the networks with only excitatoryactivity and overestimated the number of recurrent excitatoryconnections in the CA1 and CA3. This model could be expanded

to include inhibitory neurons and more realistic connectionschemes, such as a two-dimensional lattice of neurons. The pur-pose of this model was not to calculate the exact transition of thenetwork behavior or to derive physiological values for these tran-sitions butto give an intuitivefeel forwhy these transitions occur.The exact parameter values where the networks transition be-tween behaviors may differ as we change the form and constitu-ents of the network, but we expect that these transitions willremain qualitatively the same. Nevertheless, we feel that our sim-ple network is useful as one approximation to the qualitativeproperties of collective behavior in the hippocampus.

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