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Gamma Rhythms and Beta Rhythms Have Different Synchronization PropertiesAuthor(s): N. Kopell, G. B. Ermentrout, M. A. Whittington, R. D. TraubReviewed work(s):Source: Proceedings of the National Academy of Sciences of the United States of America,Vol. 97, No. 4 (Feb. 15, 2000), pp. 1867-1872Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/121568 .

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G a m m a r h y t h m s a n d b e t a r h y th m s h a v e dif f erentsynchronizationroper t i e sN. Kopelltt, G. B. Ermentrout?,M. A. Whittingtonml,nd R. D.TraublltDepartment of Mathematics and Center for BioDynamics, Boston University, Boston MA 02215; ?Department of Mathematics, University of Pittsburgh,Pittsburgh PA 15260; RSchool of Biomedical Sciences, Worsley Building, University of Leeds, Leeds LS29NL, United Kingdom; and blDivisionof Neuroscience,The Medical School, University of Birmingham, Birmingham B15 2TT, United KingdomContributed by Nancy J. Kopell, December 3, 1999Experimental nd modelingefforts suggest that rhythms n the CAlregion of the hippocampus hat are inthe beta range (12-29 Hz)havea different dynamical tructure han that of gamma (30-70 Hz).Weuse a simplified model to show that the different rhythms employdifferent dynamicalmechanisms o synchronize,based on differentionic currents.The beta frequency is able to synchronizeover longconduction delays (corresponding o signals traveling a significantdistance in the brain) hat apparentlycannot be tolerated by gammarhythms. The synchronizationproperties are consistent with datasuggesting that gamma rhythmsare used for relatively ocalcompu-tations whereas beta rhythmsare used for higher level interactionsinvolving more distant structures.

hythms in the gamma range (30--80 1Hz) and the beta range(12-30 Hz) are found in many parts of the nervous systemandare associated with attention. perception, and cognition (1-3). Ithas been noted in electroencephalogram (:EGG) signals thatrhythmsof different frequencies arefoundsimultaneously(4) Betaoscillations arereadilyobservable immediatelyafter evokedgam.maoscillations in sensory evoked potentia.l. ecordings (5). This betaactivityhas been correlated with the longrange synchronousac-tivityof neocorticalregionsduringvisuomotorreflex activation(6).This paperconcerns the correlation between the frequencybandof coherent oscillations and conduction delays between- the sitesparticipatingin the coherent rhythm. It has been. noted (7) inhuman EEG subjects that gamm.arhythmsare prevalent in local.visual response synchronization,but more distant coherence oc-curring duringmultimodal integrationbetween parietal and tem-poralcortices uses rhythmsatfrequenciesof 12--20Hz the so-calledbeta 1 range).We shall.use data from the CAl region of the hippocampus(8 10) as a paradigmto addressthe q-uestions f how l.ong-distancesynchrony is achieved and why there is a correlation betweenoscillationfrequency and the temporal distances betweeri partici-pating sites. The data available from the rat hippocampus slicepreparationgive clues about details of dynam;icshat areimportantto the synchronizationprocess.TFhe ork builds on earlierwork (l 11421) escribingandanalyzingthe role of doublet spikes in interneurons in producing synchronywhen there are significant conduction delays, Earlier work (13)using rate models showed,via simulations,that longer conductiondelayscould be tolerated and stillproduce synchrony f the carrierrhythm had lower frequencies. However, a rate model is notconsistent with the situation in which.excitatorycells fire at mrostone spike per cycle. and with high precisioninphase. An alternative

solution wassuggested bydata andlarge-scalemodels of the gammarhythmin the hippocampus (8, 9). In both data and models, theability to synchronize happened in those parameter regimes inwhich interneuronsproduced a spikedoublet in manyof the cycles.This mechanism was analyzed by Ermentrout and Kopell (11),where it was shown how the doublet provides a feedback.mecha-nism for the timing.The analysisgiventhere predictedthat,for longconduction delays (above 8-10 ms, depending on networkparam.-eters), synchronization n the gammafrequency bandis not robust.Although conduction delays in the neocortex are variable,there is

evidence that the delays between association areas could be sig-nificantly larger than 10 ms (see Discussion).In this paper, we show that the beta rhythm observed in thehippocampal slices is not merely a slowerversion of gamma, but hasa distinct dynamicalstructureandmakes use of intrinsicmembranecurrents not expressed during gamma. Furthermore, the betarhythm is much better adapted to synchronization n the presenceof long conduction delays. Via a very reduced model, we analyzewhythis is so. Predictionsfrom the analysisare shown to hold in thelarge-scale models.Backgroundon productionof beta and gamma rhythmsin thehippocampal slice can be found in ref. 3. In the tetanic stimulationparadigm(9, 10), with sufficiently strongstimuli, the hippocampalslice produces both gamma and beta, with a transition betweenthem. In intracellular ecordingsof beta in pyramidalcells, gamma-frequency oscillationscontinue between beta-frequency populationspikes, suggestingthat the interneuron networkcontinues to oscil-late at gamma frequency, which the pyramidal cells cannot follow(Fig. 1A). Two system parameters alter in time before and duringthe transition to beta: the strengthof recurrentexcitatory synapsesand the amplitude of one or more slow K conductances. Both ofthese parameters ncrease andthen level off, andexperimental dataand large-scalesimulationssuggest that evolution of both param-eters is necessary for the switch to beta to occur (10, 14). Betaoscillationsare synchronizedbetween the two sites when both sitesare stimulated together intensely (10, 14).In human EEG, occurring spontaneously or evoked by auditorystimulationby novel sounds, power in the gamma range coexistswithbeta, consistent with the beat-skippingstructure[C. Haenscheland J5 Gruzelier, personal communication; also see the work byTallon-Baudryet al. (15)].Local nhibition-Based hythms.The data andlarge scale simulationscited above all concern the behavior of the networkwhen two sitesare intensely stimulatedtogether. To better understand the mech-anismbehind the networkbehavior,we firstconsider the behaviorat one site. We show, via very reduced models, that the transitionfrom gamma to beta can be understood as a consequence of thechanges in recurrent excitatory synapses and expressions of K-conductances, Although this had previously been documented inlarge-scale simulations (14), the ability of the small network toreproduce this creates an excellent model within which to under-stand more deeply the long-distance synchronizationproperties ofbeta and gamma.We use models that are much reduced from the large scalesimulations in two ways.The network is pared down to a minimalnumber of cells and connections. We work with a local networkoftwo pyramidalcells (excitatory,or E-cells) and two interneurons(inhibitory,or I-cells). All cells are coupled to one another,exceptAbbreviation EEG,lectroencephalogram;AHP,after-hyperpolarization.tTowhomreprint equests hould be addressed.E-mail: k@bu.edu.Thepublicationcostsof this articlewere defrayedin part by page charge payment.Thisarticle musttherefore be hereby marked "advertisement"n accordancewith 18 U.S.C.?1734 solelyto indicate his fact.

0-o

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Fig.1 Gamma ndbeta oscillationsn vitro. ntracellularecordings f gammaandbetaoscillationsn a CAl hippocampal yramidal euron.Oscillations ereinducedby brief etanicstimulationseeWhittington tal. (9) ormethods].Theinitial posttetanic response s a gamma oscillationwith action potentials(fre-quency38 Hz) eparatedby a periodof hyperpolarizationmadeup of both AHPand inhibitory ynapticactivity.After the transition o beta activity, he under-lyinggammamembranepotentialoscillation s stillapparent frequency 2 Hz),but spikingoccurson everysecond or third period (frequency18 Hz). Actionpotentialsareseparatedbythe initialAHP/IPSPyperpolarizationndadditionalIPSPs.Bar= 1 mV,100ms.)

possiblyfor couplingbetween the excitatorycell (dotted lines in Fig.2A). E-E coupling,although sparsein the CAI (16), turnsout to beimportantfor the beta rhythm (10, 14); gamma rhythms, however,can be simulatedwithout E-E coupling (11, 17).The gammarhythmcorresponds to one in which all of the cells fire synchronouslyat30-70 Hz whereas in the beta rhythm,the I-cells synchronizeat agamma frequency and the E-cells synchroniizeat a frequency half

y-rhythm l g =1 (popy) l ge = 0.15(p-rhythm)50 _0VI(mV)

-10050

V2-(mV)-50

-100

-50

Vi (mV) 3E0 200 400 600time (msec)

Fig.2. (A)Minimal etwork or investigatingocalsynchronizationf gammaand beta rhythms.For he gammarhythms,he E-E onnectionsareabsent; orthe beta rhythms,hey are a necessarypartof the circuit. B)Gamma-to-betatransitionof local rhythmsoccurs as the AHP s turned on and the local E-Econnections restrengthened.Parameters re as in he appendix.For he gammarhythm,ee = 0andgm= 0. Atthe firstarrow,gm sset to 1,switching he rhythmfromgammato a rhythm nwhich the E-cellsmiss beats and fires nonsynchro-nously.Att = 400, gee = 0.15,andthe networkquickly uppresseshe nonsyn-chronous solution, leaving only the synchronousocal state. Throughout hetransitions, he I cells shown below exhibit only minorchanges, slowingdownslightlybecauseof the decreasedexcitation excitation veryother cycle).

as fast. Compared with the detailed biophysical models (14), thecells are also simple: theyare modeled as single compartmentcellswith fast spiking currents for the gamma rhythm; for the betarhythm, an extra after-hyperpolarization AHP) current (slow Kconductance) is added to the E-cells (see Appendix).Although weuse a specific current (M-current) n the simulations,the analysis nthe mathematics given below will work for any AHP current withthe appropriate decay time.Increases in K-conductances, plus increases in the strength ofsynapses between excitatory cells, can transformthe output of thenetwork of E and I cells from gamma to beta. This transformationwas documented in the detailed biophysical model (14) with twoconnected sites. In Fig. 2B, we show that the transition is repro-duced in the reduced model, even without the synapticconnectionsbetween the sites.The first partof Fig. 2B displays hevoltage tracesof the two E-cells and one of the I-cells (they are synchronous)withparameters that elicit a gamma rhythm. In the middle section, aslower K-conductance hasbeen added to the model E-cells; now theE-cells, slowed downby the K-conductance, each fire on half of thegamma cycles. Note that they fire on opposite cycles. For the thirdpart of Fig. 2B, the parameterswere further changed by addingsynaptic (AMPA-mediated) connections between the E-cells; thenetworkis now as in Fig. 2A with the dotted lines. Now the E-cellsstill fire every other cycle, but this time on the same cycle; that is,they produce beta.

To understandwhy the E-cells miss opposite cyclesin the absenceof the E-E coupling, we note that the firingof one E-cell effectivelysilences the other in a given cycle through feedback inhibition,unless the laggingcell is so close that it fires before the onset of thefeedback inhibition.A major effect of the mutual excitatory con-nections is to increase the range of initial conditions under whichthe second cell can fire before receiving inhibition; in a mannergraded with the size of the excitatory conductance, the excitationadvances the firingof the second E-cell, preventingsuppressioninthat cycle.With some E-E coupling,there canbe other initial conditionsforthe same parametersforwhich the E-cellsdo fireon opposite cyclesthroughout the trajectory.However, if the E-E coupling is suffi-ciently large, that solution does not stably exist. With enoughexcitation from the cell that fires in a given cycle, the other cell isforcedto fire in the same cycle, rulingout the solution in which cellsfire on opposite cycles.We also note that there are many different ways to changeparametersto producethe gamma-to-betatransition.In additiontothe new excitatoryconnections, the essential change is to lower theexcitabilityof the E-cells relativeto the I-cells, by changingrelativedrives or intrinsic conductances.As we will see in the next section,synaptic input from distant sources can also change the balance ofexcitability.Long-Distance Synchronization in Inhibition-Based Rhythms. Strategyand basicdynamicalproperties.The different dynamicalstructuresand currents associated with gamma and beta lead to differentresults when these rhythms are used to coordinate dynamicalactivityof loci at a distance from one another. To show this, ourstrategyis to look at the dynamicsnear the gamma or beta rhythmand create a map (a functionrelatingthe timingof one cycle to thatof the next) containing information about whether, and in whatparameter ranges, that rhythmis dynamicallystable.There are two principlesthat govern the behavior of the maps.The first is that E-cells are able to fire when inhibition, eithersynapticor intrinsic(from AHP currents),has worn off sufficiently.This situation obtains when the effective membranetime constantof the excitatorycells is small comparedwith the decaytime of thesynapticcurrentand/or the AHP current;the voltage then tracksthe time course of the synapsesor AHP currents.Second, the I-cells have an extra property, associated withrelative refractoryperiod. Suppose a cell fires at t = 0 and rcceives

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DELAY50 (.

40.tB *-- T I - 0.2530 |R -1 -

-0.5 R-.20.

10 -0.75 R=O. .. ..-.-.- . .--** -

0 5 10 15 20 4 8 12 16 20

Fig.3. (A)Minimal etwork orinvestigating ynchronization ith conductiondelays.The long E-E onnectionsareessential or coherenceof the beta rhythmacrossdistances,but not for the gamma rhythm. B)The maps T1and Tpas afunctionof the delay8 forgm= 1 and all others parameters s inthe Appendix.Allaxes have units of milliseconds. C)The linearized calingfactorD for thegammarhythm dashed) ndforthe betarhythm t various atiosofthe effectiveAHP onductance ndthe effective nhibitory ynaptic onductance.ThecloserDisto zero, the faster he convergence o the synchronizedtate and the strongerisrobustnesso heterogeneity.

sufficiently large excitationagain at t = at > 0. The cell fires againat t = T1(4i),where T1decays with fi.That is, the more recoveredthe cell, the shorterthe time to fire aftera givenfixed excitation. Forapery short, the cell may not fire at all. T1 s related to the intrinsicrefractoryperiod of the I-cells but is influenced by the rest of thenetwork. For example, increasing the self-inhibition in the localnetwork increases the values of TI. Standard integrate and fireneurons do not have this property, but it can be shown fromsimulations that this propertydoes hold for biophysicalmodels ofthe interneurons (11, 18). It also holds for more elaborateversionsof integrate and fire equations (12).An architecturallyminimal model combines all local cells of agiven type if they are synchronousin that rhythm.Thus, in bothgamma and beta, a minimal model of the local circuit uses only oneE cell and one I cell. In each of these, the model for long-distanceinteraction consists of two local circuits,with connections from Ecells to the distantI cells (Fig.3A, without the dotted lines). We alsoexplore the effects of addinglong connections between the E cells(Fig. 3A, dotted lines).Gammarhythms,beta rhythm,and feedback loops for two-sitesynchronization.In previous papers (8, 11, 12), we introduced amechanism for long distance synchronizationin gamma. In itssimplest form, a network embodying this mechanism has twominimal local circuits, as in Fig. 3A without the dotted lines. Thesynchronizationdepends on operating in a parameter range inwhich the I-cells produce double spikes per cycle ("doublets").Thefirst spike of the doublet is temporally tied to excitationfrom thelocal E-cell; the second is caused by excitation from the distantE-cell. The timingbetween the doublets in agiven cycle includesnotonly the difference in firing time between the two E-cells and theconduction delay, but a nonlinearpropertyof the I-cells associatedwith relative refractory period. It is this nonlinearitythat providesthe feedback responsible for the synchronizing properties of thedoublet configurationin the gamma rhythm.For the beta rhythm,there is an additionalnonlinearityattributable o the AHP current.We show below how the extra current and the beat-skippingstructurechangethe maps, andwhythis leads to tolerance of longerconduction delays for the beta rhythm.Let t1andt2 be the timesof firingof cells E1andE2 on some cycle,with A - t2. Starting with the gamma rhythm, if the initialconditionsare close enough to synchrony A 0), we can constructa map that gives the times ti and t2 of firing in the next cycle. The

time at which the inhibitionfor an E-cell wears off enough for it tofire is approximated by the time at which the inhibitory conduc-tance had decayed enough. [That this isan excellent approximationis easily checked by comparing simulations from the biophysicalmodels with predictions using that approximation (11).] The latterthreshold depends on the drive to the E-cells. We assume that thereis saturation of the synapse from the I to the E cell, so that it is onlythe timing of the second spike of the doublet that determines whenthe postsynaptic E-cell will fire.Cell I, is essentially recovered from firing in the previous cyclewhen it receives excitation from E1 at tl. (We assume no localconduction delay.) Hence, it fires a short (history independent)time, tj1, ater. (This time can be arbitrarily mall, depending on thedrive to the I-cells, but the latter drive maynot be so large that theI-cells fire before the E-cells.) Cell I1 receives excitation from E2 attime t2 + 6, where 6 is the delay time between the circuits. Cell Iithen fires the,second spike of its doublet at t2+ 8 + TI (t2 + 8 -t - tei) = TI. El fires in the next cycle when the inhibitoryconductanceequals some thresholdlevelg*, i.e., at a time ti definedby

gieexp[-(il - =gT , [1]where Tis the time constant for decay of inhibition.g. is related tothe drive to the cell via the intrinsicperiodpy induced by that drivein an uncoupled cell, namelygi exp[(-py + tei)/T] = g. From theabove,we can compute the time t1 n terms of t1andt2andsimilarlyfor t2.The map for analysisof the dynamicsnear the beta rhythmis avariationof Eq. 1, with two modifications. The cell I1now firesthreespikes per beta cycle, two d-uring he gamma cycle in which theE-cells fire, and one during the cycle in which the E-cells are silent.The excitation from the distant cell is, as before, received by anI-cell after it has fired its first spike of the period. The map TI forthe gamma rhythm is replaced by T1, which is defined to be theinterval between the time cell I, receives excitation from cell 2 andthe time it fires its third (not second, as before) spike; T, dependson the timest1, t2,defined as above.Thus,the time at which the thirdspike of the I cell fires is t2 + 8 + T(t2 + 8 - tl - tei) e T. Fig.3B shows TI,Tp.Note that, in thisparameterrange, Tli s almost TIplus a constant.As shown in ref. 11, it is the slope of the map T1 orTp)that matters in determining whether synchronizationwill takeplace; thus, this is not the key alteration that changes the synchro-nization properties.The second modification introducesan intrinsicallybased sourceof inhibition, namely the slowly decaying AHP current of theE-cells, with time constant Tahp, triggeredby a spike of that cell. Thetime t, of the next E1 spike, determined by the time of the lastinhibitorypulse received from cell I1andthe previous E1 spike time,is then defined implicitly bygieexp[ -( ?/i)/T] + gahp(xp[ (tl - tl)lTahp] = g*93 [21Here gahp is an "effective conductance" that takes into account theactual maximal conductance gahp but scales it using the maximalvalue of the gating variable for that current and the ratio of thedrivingforce of the AHP current to that of the synapticcurrent.Toderive the threshold value g*-, suppose an uncoupled local circuitdisplaysa beat-skippingbeta rhythm,and let tHbe the difference intime between the first and second I-spikes in a cycle. (Recall that,for the uncoupledbeta rhythm,there areonly two spikes per cycle.)Then,gf3 = gieexp[- (pp - tII)/T] + gahpexp[-pf31/Tahp], wherep, isthe period of the uncoupled beta rhythm.To understand the implication of these maps for stability,we doa stability analysiswith the implicit formula Eq. 3 by linearizingaround the synchronous solution;we will show that the map of Eq.1 can be considered a special case when there is no AUP. To do this,we define Pi ti -ti - p,f, the variationin a given cycle from the

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periodic rhythm with period pp. A is defined as before. We canrewriteEq. 2 as. gi,eexp[-(pp + Pi - A - 8- Tp(A + - tei))/T]

+ gaphexp[ - (p, + PI)/Tahp] [3]To lowest order in the small quantities A, P1, P2, the above can beexpressed as

Ap1 + B(p, - A(1 + T'(8- tei)/T)) = 0,whereA = (gahp/ahp)exp(-p1/Tahp) andB = (gie/T)exp(-(pp -, -T(6 - tei))/T). (See Appendix for more details about A and B.) Asimilar formula holds for p2, with A replaced by -A. Subtractingthese two formulae, and noting that p2 - PI = A - A, we have

(A + B)(A -A) -2BA(1 + T (8- tei)/T))-A -B - 2BT'(5 -t,I/TA-- BT -_'A DA [4]

Synchrony s stable if the coefficient D of A in Eq. 4 lies between-1 and + 1.The closer Eq. 4 is to the stabilityboundary, the slowerthe synchronization, o the fastestsynchronizationoccurs whenD =0; similarly,when the systemis close to its stabilityboundary,smalldifferences in local circuits can produce significant phase lagsbetween the circuits.Fig. 3C shows D as a function of 5 for otherparameters as used in the simulations.Eq. 4 and Fig. 3B revealthat there are two effects associated withsynchronization.For short conduction delays, T, is significantandaffects synchronization.For largerconduction delays (e.g., 5 > 10ms), T, is essentially zero, and the synchronizationcomes from abalance between the decay of the inhibition (encoded in B, anddependent on 5) and the decay of the AHP current.Fig. 3C showsthat increasing the amount of the AHP currentbringsD signifi-cantlycloser to zero, increasingthe robustness and rate of synchro-nization. See Appendix for information about the calculation of Dandestimation of the quantities n Eq. 4. The analysis or the gammarhythm gives rise to a similar relationship as Eq. 4, in which theexpression Tpis replaced by TI and the coefficient A is set to zerobecause there is no AHP current.In this case, there is no balancebetween the decay of inhibition and the AHP current, so allsynchronizationeffects come from the non-zero slope of TI; thelatter function is essentiallyflat by 8-10 ms,which gives the criticalconstraint on long-distancesynchronization n the gamma rhythm.As shownin ref. 11, changesinparametersdo not significantlyaffectthis conclusion.Fig. 4 shows that the conclusions of the analysis hold for largescale networks as well as small ones. Fig. 4A shows a large-scalerealistic model (14), showing the gamma-to-beta transitionfor Eand I cells. The lower panels shows that there is synchronization nboth frequency regimes.The right-handpanels of Fig. 4A show thesame quantitieswhen an extra 10-ms decay has been added to theconduction times between the two sites (in Fig. 4A, the cells aredistributed,with a smaller maximal conduction delay across thetotal array of -4 ms). Note that the beta rhythm synchronizesacross the two sites whereas the gamma rhythms n the two sites arein antiphase.Effects of mutual excitation. The analysis we did above forsynchronization between two distant sites used as signals onlyexcitation to distant I-cells. Because there can also be mutuallyexcitatory connections (as in Fig. 3A with the dotted lines), it isimportantto know the effects of these. We show in this section thatlong-range E-E connections are not able to stabilize the synchro-nous solution if that solution is unstable in the absence of thoseconnections. Nevertheless, especially in the beta rhythm, they canbe important in producing the appropriate network output byeliminating the possibilityof other unwanted solutions.

A Control +10 ms delaye-cel!

i-cells

B 0s15 0.1Sy~~~~~~~~~~~

-75 -50 -25 0 25 So 75 ms -76 -50 -25 O 25 So 75 mes

Fig.4. Innetworkmodel,betaremains ynchronizedwith longeraxonaldelaysthan does gamma.The networkconsists f a 96x 32 array f pyramidal ells 1.92mmwide) and a superimposed96 x 4 arrayof interneurons, s in ref. 14. Incontrolconditions,he maximum xonconductiondelaywas 3.84 msacross hearray.Oscillationswere evoked by tonicdepolarizationof both pyramidal ells(E-cells)nd interneuronsI-cells), iththe gamma i beta transition ccurring spyramidal ell AHP onductances nd excitatorypostsynaptic otentialconduc-tancessimultaneouslyncrease 10, 18).In he case shown onthe right,allaxonalsignalscrossinghe midlineof the arraywere subjected o an additional10-msdelay. (A)Gamma, ollowed by beta, as plotted insimultaneous races of localaverage signals (224 nearby E-cells, 8 nearbyI-cells).The E-cell ignals appearsimilar,with and without the extra 10-msdelay. Duringbeta, both E-and I-celltracesrevealsan underlying scillation t gammafrequency. Bars= 20 mv,200ms.) B)Cross-correlationsf localaverageE-cellignals romoppositeends ofthearray,or 200 msof gamma (thin lines)and 800 msof beta (thick ines).Inthecontrolcase,bothgammaand beta havecross-correlationeakswithin 3 ms of0.With he extra 10-msconductiondelay, he gamma signal s almostanticorre--lated between the two sites whereas he beta signalcross-correlationeakis at- 1.4ms.

Fig. 5A illustratesthe key points. In the first part of Fig. 5A, weshow the behavior of local circuitsfor the network in Fig. 3A (nodotted lines),with the parameters adjustedso that the local cir-cuitsare each oscillatingin the gamma regime and the conduction delayis 13 ms. Note the lack of synchrony.With coupling between the Ecells (add dotted lines), the circuits change behavior but do notsynchronize,as shownin the second part.Thus,E-E connections donot overcome the inability of the gamma rhythm to synchronize inthe presence of substantialdelays. (This was true for all strengthsof E-E couplingwe tried.)The lastportionof Fig.5A shows that thesystem synchronizes f enough AHP current is added to the E-cellsto slow down the local rhythmto beta.To explain these results we make two points. First,we note thatthe E-E coupling does not significantlychange the dynamicsof thenetwork when the cells are close to synchrony.The reason is thatthe conduction delay insures that the effect is felt after the spike ofthe cell postsynapticto the excitatory postsynaptic potential, so isnot felt on the currentcycle.In addition,the excitatory postsynapticpotential (simulated as being AMPA receptor-mediated) decaysfast enough so that the effect is negligible by the next cycle. Thus,with no AHP current, E-E coupling cannot change dynamics inwhich the synchronous state is unstable to dynamics that havcsynchronyas a stable state.For the beta rhythm, B-B coupling is very important, as illus-trated by Fig.5B: The first portion shows the antiphase between thetwo sites that occur when each uncoupled site displays a beta

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fasty ceen and on(fe)

0v~'(mV)-50

-1000 200 400 600 800

c= 0 c=O0 C =050 - -

Vo''2(mV) A~ j A { {50

0 200 400 600 800 1000 1200tiine (msec)

Fig. 5. (A) Fast gamma will not synchronize with a delay of 13 ms. If the longdistance E-Ecoupling is turned on (cee = 0.5), synchrony still does not occur.However, if the AHP is then turned on (g, = 1), the network shifts to a betarhythm and synchronizes within two cycles. Parameters are as in the Appendix.Excitatory drive is 6 in one site and 6.5 in the other; inhibitory drive is 1.15 in bothsites. The voltages of the two E-cellsare represented by black and gray lines. (B)The main role played by the long-distance E-E connections is to prevent theanti-phase solution. For the first 400 ms, there is no long E-Econnection. At t =400, c, = 0.05, which induces synchronywithin a few cycles. Att = 400, cee is resetto zero, but synchrony remains robust. Drives and AHP are as in A.

rhythm and the only long connections are E to 1.Adding long E-Econnections synchronizesthe network.The last portion shows thatthis synchrony is maintained when the E-E connections are thenremoved, showing that network is bistable.The role of the E-E coupling in Fig. 5A is not to stabilizesynchrony,as shown above. Instead,the E-E coupling preventsthenonsynchronoussolutionthat is shown in Fig.5B. Near the solutionshown in Fig. 5B, the effect of an excitatory postsynaptic potentialon the postsynapticE-cell can cause the latter to spikewell beforeit would in the absence of E-E connections, preventingbistability.DiscussionModeling ssues.Thispaperaddresses questions of behaviorof largenetworks of neurons using small networks reduced to a minimalnumber of currents. The biophysical equations associated withthose small networksstillhave on the order of 20 equations,fartoomany for an analytical approach.Thus, the map analyses that wepresented represent a still furtherreduction of complexity.Never-theless, the predictions that beta rhythmswill synchronize at muchlarger delays than gamma was shown to hold for the very large,detailed, and realisticmodels. Indeed, the power of the mathemat-ics we used was its ability to explain why any equations with theessential structural eatures (in our case, inhibition and an intrinsicAHP with the appropriate time constants) should lead to theobserved outcomes.The success of this method raises the question of why the veryreduced descriptions are able to capture essential aspects of thenetwork behavior: n particular,how the low dimensionalmaps cancapture behavior of high dimensional differential equations. Thekey point we emphasize here is that the voltage-gatedconductanceequationshave a large range of time scales associated with intrinsicand synaptic properties. The action potentials occur at (approxi-mately) discrete times, initializing the time course of many of thecurrents; dependinlgon the frequency of the rhythm,most of thefastest currents have stopped changing by the next discrete event.All of these fast events are lumped in the maps, leaving only the

ones capable of transmitting phasic signals. Which processes arelumped together can change with alterations of parameters.The simulationsshow that, for physiologicalparameter regimes,gamma rhythmssupport robust synchronizationbetween sites fordelays up to -8-10 ms. The beta rhythms (a subharmonic ofgamma in the beta 1 range) support synchronization in our simu-lations for 20+ ms. The maps explainhow this can be so and showthat the ability to synchronizewith long conduction delays dependson more than the frequency of the network: e.g., on the time scalesof the currents participating in the rhythm. For example, simula-tions (not given) show that the faster the two-site (beat-skipping)beta, the more delay can be tolerated and still get synchronization;the analysis predicts this because, at higher frequencies, there ismore of the AHP current left at the end of the cycle to provide asynchronization.Parameter ranges can also affect the outcome bychangingthe order of the spikes (excitatoryand inhibitory)in thesteady state configuration.The architecture of the local and long connections also haveeffects that are not intuitive.For example,the local E-E connectionsare critical for producing beta at a single site (no internal delay).However, synchronyof the excitatorycells is possiblebetween sitesusingE to I as the only long connections. The major role of the longB-B connections is to prevent a configurationin which the E-cellsskip every other cycle, but with the two sites having E-cells inantiphase instead of synchrony. We note that E-E connectionsalone are not sufficient to produce synchronybetween the E-cells(19, 20);that they aresynchronizing n thiscase is attributablepartlyto the adaptationcurrents 21) andpartly o the interactionwith theinhibition in the network.The analysis shows that coupling across distances lowers thefrequency of the coupled rhythm.This was shown explicitlyfor thegammarhythm (8, 11), and similar echniques provide aformula forthe frequency of the beta rhythm.The lower frequency is attrib-utable to two effects: First, the conduction delay itself lowers thefrequency. Second, when there is synchronizationacross distances,thereis an extra nhibitoryspikeassociatedwith the extraexcitationonto the I-cell;this extra nhibition slowsdownthe next firingof theE-cells.The addition of long-distancecoupling can change the structureof the rhythm as well as the frequency. In particular, f two localcircuitsdisplayingthe beta rlhythmare coupled with a conductiondelay, the resulting rhythmcan have the structureof either a slowgamma rhythm (B cells and I cells have same frequency) or a betarhythm (E cells skip beats); increasing the drive to the 1-cellschanges the coupled systemfrom slow gamma to beta (simulationnot shown.) The switch is accompanied by a drop in frequency,associatedwith the extra I-spikebetween pulses of the E-cells.Connections o ExperimentalObservations.The induction of excita-tory pyramidalcell firing as a subharmonic of a continuing gammafrequency subthresholdoscillationis a robustphenomenon in thehippocampus (10, 14) (Fig. 1A). Frequency analysisof oscillatorycomponents of corticalEEG responses to visualor auditorysensorystimuli also reveals a similarpattern. An initial gamma response(evoked) switches to a beta response superimposed on which is asecond gamma frequency component (C. Haenschel and J. Gru-zelier, personal communication; ref. 15).Further EEG studies have demonstrated that, in terms of co-herence, the above type of beta oscillations are associated withtemporalrelationshipsbetween more spatiallydistantregionsof thecentralnervous systemthan gamma oscillations. This suggeststhatthe anatomically hierarchicalpattern of processing sensory infor-mation (primary, supplemenitarysensory-associational areas) ismirroredbya hierarchicaluse of oscillationfrequencyused to bringabouttemporal correlations (7), at least within the gamma andbetabands.Physical separation of areatsof the central nervous system doesnot correlatewell with axonalconduction delaysbetween areas.For

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example, ipsilateral, intrahemisphericaxon conduction velocitieshave been quoted to be as low as 1 mm/ms for longer collaterals(22), whereas interhemispheric,callosal axon conductionvelocitiesappearto be 2-4 mm/ms, with some researchersobservingcallosalfiberconduction velocities of over 20 mm/ms (22). These estimatessuggestthat, at least for some conduction pathways, he conductiontime between associationalareasin the temporaland parietallobeswould be well over 10 ms.However, conduction velocities for axonal projections in thecentral nervous system are plastic, changing in a use-dependentmanner over time (23). Bilateral primary processing of sensoryinformation s common, andsynchronizationatgammafrequenciesacross the corpuscallosumhas been seen for visualevokedresponse(24). The brainmay therefore organizeaxonal connectionsbetweenareas in the temporal domain (25). We suggest that the enhancedcoherence afforded by beta frequency oscillations over gammaoscillationswith long conduction delaysmaybe used, in conjunctionwith modified signal velocities, to functionally delineate differentinterareal interactionsinvolved in sensory processing.AppendixSimulations.Each site hastwo excitatory andone inhibitoryneuron.Each excitatory neuron receives input from the local inhibitoryneuron and all of the other excitatory neurons; each inhibitoryneuron receives input from all of the excitatory neurons and alsohas self-inhibition.The units of conductance are mS/cm2, those ofcurrent are ptA/cm2, and capacitance is ,uF/cm2. The excitatoryneurons satisfy equations of the form:

CV' = -0.1(V+ 67) - lOOm3h(V- 50) - 80n4(V+ 100)gAHPw(V 100) IY + IePP [5]

where the variables m, h, n, and w satisfy0.32(54 + 1) 0.28(V + 27)

m=1 - exp (-(V+ 554)/4) exp((v + 27)/5) - 1h'= 0.128 exp(-(50 + V)/18)(1 - h)

41 + exp(-(v + 27)/5)0.032(v + 52)

1 - exp(v + 52)/5) (1 - n) - 0.5 exp(-(57 + v)/40)n,

w,T(V)-w

where wcO(V) 1 + exp(-(V + 35)/10) and400

TW(V)= 3.3 exp((V+ 35)/20) + exp(-(V+ 35)/20)

The inhibitory neurons have identical equations, but there is noAHP current. The capacitance is 1.Synaptic currents are l', = gisi(t)(V + 80) + ge-e(t)V + CeeSe(t 6)and lUsn = [ei(Sl(t) + S2(t)) + Cei(1(t - 6) + 32(t - 6)]V +giisi(t)(V+ 80). In all of the simulations,gie= 1, gei= 0.15,gii = 0.2,gee = 0.15, and Cei = 0.15. The barred variablescorrespondto otherexcitatorycells in the same column, and the hatted correspond toother excitatorycells from the distant column. The synapsessatisfyfirst order equations of the form: se = 5(1 + tanh(V/4))(1 - Se) -Se/2, s' = 2(1 + tanh(V/4))(1 - si) - sj/15. The current applied tothe excitatorycells was between 5.5 and 7 and acted as the sourcefor the heterogeneitybetween columns. The currentappliedto theinhibitorycells was 1.15. The main parameters that varied in thepaper are the cross EE coupling, Cee (0-0.05), the gAHP (0-1.25),and the delay, a.The equations are integratedbyusing modified Eulerwith a stepsize of 0.025 ms because of the fact that the extrapolationused forthe delay equations is only second order. The simulations werechecked by using a smaller step size with no difference found.Codefor the computationsis availablefrom G.B.E. in the form of an xppfile. xpp is a package for solving differential equations and isavailableat http://www.pitt.edu/-phase.Computationof D.The plot of D in Fig. 3C is done by using Eq. 4and the definitionsofA andB. To do this, a formula must be derivedfor the maps TI and Tp.Asymptoticssuggestsa form for the maps,and this formula was then hand-fitted to the numericalcomputa-tions of the maps. The period of the oscillation scaled linearlywiththe delay. For Fig. 3C, TI(5) = 3.2/[1 - 1/(1 + 0.5(5 + 0.2))],Tp(5) = 34 + 12/(1 + 1.3a2) -0.13a, and pp(5) = 60 + 0.75. Inapproximatingthe maps, we assume an exponential decay of theinhibitory synapses and the adaptation; this is not strictlycorrectbecause, in both cases, the equations are nonlinear. However,empirically,we can fit the parametersin Eqs. 2 and 4 by looking atthe magnitudes of the variables, w (for the AHP) and si for theinhibition. We multiplythese by their respectivemaximal conduc-tances and then multiply that by V,,, - V. Here, V is the meanpotential of the cell, and ,,ev s the reversalpotential of the current.For the AHP it is -100 mV, andfor the synapseit is -80 mV. Themean potential is -69 mV. Empirically,we find that TAHP= 50, T =20. From Eq. 4, it is clear that, because the quantitiesA, B appearvia a ratio, only the ratio r = gAHP1g'i7 atters. Our empiricalestimates of this areR 0.6, based on the above approximations[the maximumthat the synaptic gates get is -0.7, that of the AHPis -0.14, the maximal conductances are both -1, and (-100 +69)/(-80 + 69) 3 so that the ratio is -0.6]. With these threeparameters T, TAHP, and R, as well as the empiricallydeterminedfunctions Tpj1(5),we can plot the function D. Note that, to plot Dfor the gamma rhythm,the period is not needed and neither is thedecay of the inhibitory synapse;only the map TI is required.Thisworkwaspartially upported yNational nstitutes f HealthGrantROI MH47150 o N.K. and G.B.E., grants rom the NationalScienceFoundation to N.K. and G.B.E. and grants from the Wellcome Trust toM.A.W. and R.D.T., who is a Wellcome Trust PrincipalResearch Fellow.

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