Journalof ComputationalNeuroscience9, 5–30,2000c© 2000Kluwer AcademicPublishers.Manufacturedin TheNetherlands.

A Temporal Mechanismfor Generating the PhasePrecessionof Hippocampal PlaceCells

AMITABHA BOSE,VICTORIA BOOTH AND MICHAEL RECCEDepartmentof MathematicalSciences,Centerfor AppliedMathematicsandStatistics,New Jersey Instituteof

Technology, Newark,NJ, 07102-1982bose@nimbu.njit.edu

ReceivedDecember1, 1998;RevisedJune21,1999;AcceptedJune25,1999

Action Editor: JohnRinzel

Abstract. The phaserelationshipbetweenthe activity of hippocampalplacecells and the hippocampalthetarhythm systematicallyprecessesas the animal runs throughthe region in an environmentcalled the placefieldof thecell. We presenta minimal biophysicalmodelof thephaseprecessionof placecells in region CA3 of thehippocampus.Themodeldescribesthedynamicsof two coupledpoint neurons—namely, a pyramidalcell andaninterneuron,thelatterof which is driven by a pacemaker input. Outsideof theplacefield, thenetwork displaysastable,backgroundfiring patternthat is locked to the thetarhythm. Thepacemaker input drives the interneuron,whichin turnactivatesthepyramidalcell. A singlestimulusto thepyramidalcell fromthedentategyrus,simulatingentranceinto theplacefield, reorganizesthefunctionalrolesof thecellsin thenetwork for anumberof cyclesof thethetarhythm. In thereorganizednetwork, thepyramidalcell drives theinterneuronat a higherfrequency thanthethetafrequency, thuscausingasystematicprecessionrelativeto thethetainput. Thefrequency of thepyramidalcellcanvary to accountfor changesin theanimal’s runningspeed.Thetransientdynamicsendafterup to 360degreesof phaseprecessionwhenthe pacemaker input to the interneuronoccursat a phaseto returnthe network to thestablebackgroundfiring pattern,thussignalingtheendof theplacefield. Our model,in contrastto others,reportsthatphaseprecessionis atemporally, andnotspatially, controlledprocess.Wealsopredictthatlikepyramidalcells,interneuronsphaseprecess.Ourmodelprovidesamechanismfor shuttingoff placecell firing aftertheanimalhascrossedtheplacefield, andit explainstheobservednearly360degreesof phaseprecession.Wealsodescribehowthismodelis consistentwith aproposedautoassociativememoryroleof theCA3 region.

Keywords: phaseprecession,minimal biophysicalmodel,placecells,thetarhythm

1. Intr oduction

Thereis considerablecurrentinterestin the ways inwhich thetemporalfiring patternof neuronsmaypro-videadditionalinformationthatis notconveyedby theaveragedfiring rate alone.This interesthasled to asearchfor ways in which temporalfiring propertiesof neuronsare generatedand detectedin the centralnervoussystem(for a review, seeRieke et al., 1997;Recce,1999).The main goal of this article is to useminimal biophysicalmodelingto demonstratehow a

groupof neuronsin region CA3 of the hippocampusof freelymoving ratscangenerateaspatiallyencodedoutput,calledphaseprecession, usingonly limitedandspatiallynonspecificinputs.

It hasbeenproposedthat hippocampalplacecellsprovide informationfor downstreamneuronsthroughthe phaserelationshipbetweenneuronalactivity andthe hippocampalEEG (O’Keefe and Recce,1993).Placecells werefirst describedin the CA1 region offreely moving rats (O’Keefeand Dostrovsky, 1971),andtheactivity of theseputative pyramidalcells(Fox

6 Boseetal.

Figure1. Extractionof thefiring phaseshift for eachspikeduringasinglerun throughtheplacefield of aplacecell on thelinearrunway. A:Eachactionpotentialfrom cell 3 duringtheonesecondof datashown in thefigureis markedwith a vertical line. B: Thephaseof eachspikerelative to thehippocampalthetarhythm. C: Hippocampalthetaactivity recordedat thesametime asthehippocampalunit. D: Half sinewavefit to the thetarhythm that was usedto find the beginning of eachthetacycle (shown with vertical ticks above andbelow the thetarhythm).Reprintedfrom O’KeefeandRecce(1993).

andRanck,1975)is highly correlatedwith therat’s lo-cationin anenvironment(O’Keefe,1976;Muller etal.,1987).The preferredfiring locationof a placecell iscalledits placefield.

During locomotor activity, the hippocampalEEGhasacharacteristic6 to12Hertzsinusoidalformcalledthe thetarhythm (GreenandArduini, 1954),andthephaseandfrequency of thisrhythmishighlycorrelatedacrossthe CA1 region of the hippocampus(Bullocketal.,1990).As therat runsthroughaplacefield, onalinear runway, thephaseof the thetarhythmat whicha placecell firessystematicallyprecesses.Eachtimethe animalentersthe placefield, the firing begins atthe samephase,and over the next five to ten cyclesof the thetarhythmit undergoesup to 360degreesofphaseprecession(O’KeefeandRecce,1993; Skaggset al., 1996).The maximumobserved phasepreces-sionwas355degrees(O’KeefeandRecce,1993).So,by thephrase“up to 360degreesof phaseprecession”we meanstrictly lessthan but in a neighborhoodof360degrees.An exampleof thephaseprecessionof aplacecell is shown in Fig. 1. During therunalongthetrack,thephaseof hippocampalplacecellfiring ismorecorrelatedwith theanimal’s locationwithin theplacefield thanwith thetimethathaspassedsinceit enteredtheplacefield (O’KeefeandRecce,1993). This sug-geststhatthephaseof placecell activity providesmoreinformationon the locationthanis availablefrom the

firing rateof thecell alone.A downstreamsystemthatmeasuresthe phaseof placecell activity would thenhave more information aboutthe location of the an-imal in the environmentand may be able to ignoreout of placefield firing that occurspreferentiallyata phasethat is different from the rangefound in theplacefield. Placecells in both the CA1 region andtheupstreamCA3 region arefound to undergo phaseprecession(O’KeefeandRecce,1993).

Skaggsandcoworkers (1996)confirmedthis find-ing andadditionally found that the initial phasewasconsistentamonga largenumberof placecells in theCA1 region. They alsofoundthatdentategranulecellsthat project to CA3 undergo a small numberof cy-clesof phaseprecessionandthereforeprovide a syn-chronized,timedexcitationtotheCA3pyramidalcells.Marr (1971)proposedthat this input from thedentategranulecells providesa seedinput for a memoryre-trieval and patterncompletionprocessthat is drivenby the excitatory feedbackamongpyramidalcells inthe CA3 region (Treves andRolls, 1994;GibsonandRobinson,1992;HiraseandRecce,1996). Thephaseprecessionof placecells may be an essentialpart ofthispatterncompletionprocess.

The hippocampalregions also contain a varietyof interneuronsthat contribute to the generationofhippocampaloscillations.Theseinterneuronsprojectto placecellsaswell asto otherinterneurons(Freund

PhasePrecessionof HippocampalPlaceCells 7

and Buzsaki, 1996). Skaggsand coworkers (1996)measuredthephaseof interneuronfiring in theCA1re-gionandfoundthatonaveragetheinterneuronsdonotphaseshift. On theotherhand,Csicsvari andcowork-ers(1998)found a large,significantcross-correlationbetweenpyramidalcell and interneuronpairs,wherethepyramidalcell firing precedesinterneuronfiring bytensof milliseconds.This implied high synapticcon-ductancefrompyramidalcellsto interneuronssuggeststhat interneuronscould phaseprecess.This sugges-tion may not be inconsistentwith the experimentallyobserved averagepropertiesof interneurons(Skaggset al., 1996) if only a subsetis shown to transientlyphaseprecess.

Themedialseptumisalsoinvolvedin modulatingthetemporalfiring patternsof hippocampalplacecellsandinterneurons.Theprojectionfromthemedialseptumtothehippocampusis bothGABAergic andCholinergic(FreundandAntal, 1988),andit providesapacemakerto drive the thetarhythm (GreenandArduini, 1954).Wehaveshownthattheinterburstfrequency of someoftheneuronsin themedialseptumisalinearfunctionof arat’srunningspeedonalineartrack(King etal.,1998).

In summary, the observed propertiesof the phaseprecessionphenomenoninclude the following: (1)placecells fire in only one direction of motion dur-ing runningon a linear track; (2) all placecells startfiring at thesameinitial phase;(3) the initial phaseisthesameoneachentryof theratinto theplacefieldof aplacecell; (4)thetotalamountof phaseprecessionisal-wayslessthan360degrees;(5) thecellsin thedentategyrus that project to CA3 undergo a smallernumberof cyclesof phaseprecession;(6) in one-dimensionalenvironments,thephaseplusfiring rateprovide moreinformationof therat’s locationthanthefiring rateofa placecell; (7) phaseis morecorrelatedwith theani-mal’s locationin a placefield thanwith thetime sincetheanimalenteredtheplacefield; and(8) backgroundfiring of placecells outsideof the placefield occursat a fixed phasethat is closestto the initial phasethatoccurswhentheanimalenterstheplacefield.

Several modelshave beenproposedto explain theneuralbasisof thephaseprecessionphenomenon.ThemodelproposedbyTsodyksandcoworkers(1996)pro-videsanenvironment-driven phaseprecession,whichiscertaintobemorecorrelatedwith positionthantime.Other models of environment-driven phasepreces-sionhavebeenproposedby WallensteinandHasselmo(1997)andJensenandLisman(1996). In thesemod-els,thephasecorrelationin CA1pyramidalcellsresultsfromCA3 inputthatisphaseprecessing.Kamondiand

coworkers(1998)haveproposedamodelfor phasepre-cessionin theCA1 regionthatdoesnotdependonpre-cisely timed inputs, but insteadthe phaseis a resultof thetotalamountof depolarizationof theplacecells.In this model,thetotal amountof phaseshift is muchlessthan360degrees.

In this article, we describea minimal biophysicaltemporalmodelfor phaseprecessionin theCA3regionof the hippocampus.It differs from prior modelsinthat(1) it generatesthespatialcorrelationof thephaseprecessionfrom a singlespatialinput andfrom infor-mationaboutthe animal’s runningspeed;(2) it doesnot requirephaseprecessionin the upstreamprojec-tion cells;(3) it doesnot requirea spatiallydependentdepolarizationof placecells; (4) it providesa mecha-nismfor determiningtheendof theplacefield; (5) itprovidesa mechanismfor up to 360degreesof phaseshift; and(6) it is consistentwith associative memorymodelsfor theCA3 region.

Alternatively, our modeldoesnot includea mecha-nismto accountfor thefiring rateof placecells. Theactivity of placecellsincludesbothrateandtiming in-formation.In thepresentmodel,weareonlyconcernedwith thetiming properties.

Thisminimalmodel,whichiscomposedof twoneu-rons (onepyramidal cell andone interneuron)andapacemaker input, can explain the onset,occurrence,andend-of-phaseprecession.Thenetwork hastwo im-portantdynamicpatterns.Thefirst is astableattractingstate,whichmimicsthebehavior of CA3outsideof theplacefield. Theseconddynamicpatternis a transientstatethatencodesthebehavior of CA3within theplacefield. Themaindifferencebetweenthetwostatesis theinput thatcontrolsthefiring patternof theinterneuron.In the stablestate,the pacemaker input controls in-terneuronfiring, while in thetransientstate,excitationfrom thepyramidalcell does.Theseedfrom theden-tategyrusswitchescontrol from thepacemaker to theplacecell to initiatephaseprecession,andthedurationof phaseprecessionisdeterminedbythedurationof thetransientdynamicsasthenetwork returnsto thestablestate.

Thetransientplacecell firing in theplacefield is atemporalprocessin whichthephaseof firing of thecellin eachcycle of the thetarhythm strictly dependsonthephasein theprior cycle of the thetarhythm. Thisis in contrastto all othermodelsof phaseprecessionin which thephaseof firing of placecellsdependsonthe external inputs arriving at that cycle and not onthephasein othercyclesof thethetarhythm. For thisreason,to accountfor thespatialcorrelationof phase

8 Bose et al.

precession, the present model requires that the amountof phase change during a theta cycle depends linearlyon changes in the animal’s running speed. Neurons inthe medial septum have been found to have an inter-burst frequency that is a linear function of the runningspeed of a rat (King et al., 1998). Also, the theta fre-quency in the hippocampus of rats running on a lineartrack has been found to be highly correlated with therat’s speed (Recce, 1994). Since the phase-precessiondata includes a range of running speeds (O’Keefe andRecce, 1993), this implies the interburst frequency ofplace cells is also a function of the animal’s runningspeed. These data provide a mechanism to maintain aspatially correlated phase precession in the proposedtemporal model.

The model generates two different types of spatialinformation. First, it determines the length of the placefield. The place field ends when phase precession ends,which occurs when the pacemaker regains control ofthe interneuron from the place cell. Second, the modeldetermines the location of the animal within the placefield. Note that these two types of spatial informationare generated from the seed, a single location specificinput, and the velocity of the animal, a nonspatiallyspecific input.

Our model supports two themes that have arisen inother neural modeling studies. The first is that the con-trol of the dynamics in a network may switch amongdifferent neurons over time. These switches can occurin the absence of any synaptic changes. Instead, dy-namics are governed by the timing of events relativeto one another (Ermentrout and Kopell, 1998; Nadimet al., 1998). The second theme concerns the role ofinhibition in networks of neurons. In recent studies(Terman et al., 1998; Rubin and Terman, 1999; vanVreeswijk et al., 1994), it is shown that inhibition mayhave counterintuitive effects on the network, such assynchronizing mutually coupled inhibitory cells. In thepresent study, we use inhibition tospeed upthe oscil-lations of the pyramidal cell.

The primary prediction of this model is that phaseprecession is a spatially specific output that results froman integration of a spatial signal from the dentate gyrusat a single point and a nonspatial velocity coded signaloriginating from the medial septum. We predict that noupstream physiological signal exists that contains asmuch information about the animal’s spatial locationas the phase of hippocampal place cell firing in CA3.This prediction differentiates our model from all othermodels of phase precession that require an input that

is as spatially precise as the phase precessing output.Another prediction, which is easier to address experi-mentally, is that a subset of the interneurons transientlyphase precesses. The model also strongly depends on,and thus predicts, continued evidence for a linear fre-quency dependence of pyramidal cells in CA3 and pro-jection cells in the medial septum on running speed ofthe animal.

2. Model and Methods

2.1. Model

Our minimal biophysical model for phase precessionin CA3 consists of a pyramidal or place cell, an in-terneuron, and a pacemaker input, denotedP, I , andT ,respectively. The pacemaker inputT may be thoughtof as an individual cell or perhaps a conglomerationof inputs that produce the theta rhythm. The synap-tic connections amongP, I , and T reflect the net-work architecture of CA3 (see Fig. 2). Specifically,the pyramidal cellP receives fast, GABA-mediatedinhibition from the interneuronI and projects toIvia fast, glutamatergic excitation. The pacemaker input

Figure 2. Two cell and pacemaker network model consisting of apyramidal or place cell (P), an interneuron (I), and a pacemaker input(T). Synaptic connections reflect CA3 network architecture (arrowindicates excitatory and filled circle indicates inhibitory synapticconnections).

PhasePrecessionof HippocampalPlaceCells 9

T projectsonly to I through fast, GABA-mediatedinhibition.

Ouranalysisdoesnotdependonaspecificmodelfortheindividualneurons;hence,weuseageneralformforthemodelequationsfor eachcell. Thefew restrictionswerequireonthemodelproperties(outlinedbelow) aresatisfiedbyalargeclassof neuraloscillators,includingtheMorris-Lecar(Morris andLecar, 1981)equations,which have beensuccessfullyusedin numerousmod-eling studies(Rinzel and Ermentrout,1997; Somersand Kopell, 1993; Termanet al., 1998; TermanandLee,1997),andwhichweusefor oursimulations(seeAppendixB).

TheP cell is ageneralneuraloscillator, modeledbyequationsof thefollowing form:

v′p = f p(vp, wp)

w′p = εgp(vp, wp),

(1)

where′ denotesthe derivative with respectto time t .The variablevp representsthe membranevoltageoftheplacecell andthe function f p is composedof thesumof theionic andleakagecurrentsfoundin thecellaswell asanappliedcurrent. We considerP to haveburstingbehavior andthevariablevp tomodeltheburstenvelopeandnot to accountfor individual, fastspikesoccurringduringtheactive phaseof theburst. In thisgeneralclassof neuralmodels,the variablewp mod-els theactivationgatingvariableof anoutward, ioniccurrent,usuallypotassium-mediated,andthefunctiongp governsthe activation gatedynamicsof this con-ductance.The parameterε controlsthe time scaleofthewp dynamics.Ouranalysisassumesthatε is small(ε ¿ 1), sothat this generalneuraloscillatorbehavesasa relaxationoscillator.

Themodelfor the interneuronI alsohasa generalform, but we considerI to beanexcitablecell, ratherthanoscillatory. Themodelequationsare

v′i = fi (vi , wi )

w′i = εgi (vi , wi ),

(2)

wherevi representsthemembranevoltageof I andwi

representstheactivationgatingvariableof anoutward,ionic current.As describedbelow in termsof its phaseplane,we requirethat it is ableto fire in responsetoeitherexcitationreceived from P or inhibitionreceivedfrom T .

Theequationsfor thepacemaker T arecompletelygeneral. In fact, we simply assumethat T oscillates

between0 and1, with a periodTT in the thetarange.Whenthe valueof T exceedsa prescribedthreshold,thenweassumethatT fires,andwemeasurethedura-tionof its “spike” asthetimespentabovethethreshold.

For thenetworkmodel,appropriatesynapticcurrentsareaddedto theindividual cell models.In thesynap-tic currentvariablesandparameters,thefirst subscriptdenotesthe presynapticcell andthe secondsubscriptdenotesthepostsynapticcell. Thenetwork equationsfor P are

v′p = f p(vp, wp) − gipsip(t − σi p)[vp − vi p],

w′p = εgp(vp, wp).

(3)

For I , thenetwork equationsare

v′i = fi (vi , wi ) − gpispi (t − σpi )[vi − vpi ]

− gti sti (vi − vt i ), (4)

w′i = εgi (vi , wi ).

This form of the synapticcurrentshasbeenusedinothermodelingstudies(ErmentroutandKopell,1998;WangandRinzel,1992).Thevariablessip, spi , andsti

governthedynamicsof thesynapticcurrents,andeachsatisfiesanequationof theform

s′ = αH∞(v − vθ )(1 − s) − βs, (5)

where α and β are the rise and decayratesof thesynapse,and vθ is the synapticactivation threshold.TheHeavisidestepfunction H∞ actsasanactivationswitchfor thesynapse;H∞(v − vθ ) is 0 if v < vθ andis 1 if v ≥ vθ . Theparametersgpi , gti , andgip arethemaximalconductancesof thesesynapticcurrents,andvpi , vt i , andvi p arethereversalpotentials.In particu-lar, vi p andvt i aresetto make theassociatedcurrentsoutward,while vpi is setto make thesynapticcurrentinward. Finally, theparametersσi p andσpi representa synapticdelayfrom I to P andP to I , respectively.Withoutlossof generality, weassumethereis nodelaybetweenT and I .

2.2. PhasePlaneMethods

In this section,we describethe phaseplanemethodsweuseto analyzethemodel.Wefirst explainhow thebehavior of eachcell whenisolatedfrom thenetworkcan be studiedin the phaseplaneand then describetheeffectof network interactions.An importantaspect

10 Boseetal.

of our analysisthat is worth mentioningat this initialstageis thatwe oftencomparebehavior of uncoupledandcoupledversionsof P. By the former, we meanthe intrinsic P cell oscillation in isolation from thenetwork. By thelatter, we meantheoscillationof thesameP cell whenit mustnow interactwith T and Iin the network. Oneof the importantconclusionsofour analysisis that the network dynamicscanevolvein dramaticallydifferentways,dependingonthefiringsequenceof thecells.

Theoscillatorybehavior of theisolatedP cell canbeanalyzedin thevp − wp phaseplanewherethetrajec-tory of thecell is determinedby thenullclinesof eachequation(seeFig. 3). The vp-nullcline, denotedCP,is a cubic-shapedor N-shapedcurve foundby solvingf p(vp, wp) = 0 for wp. Thewp-nullcline,denotedDP,is a sigmoidal-shaped,nondecreasingcurve found bysolving gp(vp, wp) = 0. To ensurethat the isolatedPcell isoscillatory, parametersin thefunctions f p andgp

arechosensothatCP andDP intersectuniquelyalongthemiddlebranchof CP. As in prior models(SomersandKopell,1993;TermanandLee,1997;Termanetal.,1998),we requirethat the following conditionon gp

holdsneartheleft branchof CP:

∂gp

∂vp> 0. (6)

Followingstandardanalysisof relaxationoscillators,in thelimit ε = 0, thebursttrajectoryof the P cell canbe tracedin the phaseplane. As displayedin Fig. 3,the silent phaseof the burst occursas the trajectorytravelsdown theleft-handbranchof thecubic-shapednullclineCP, whileduringtheactivephaseof theburst,the trajectorytravelsup the right-handbranchof CP.Thetransitionsbetweentheactiveandsilentphasesofthe oscillation occur when the trajectoryreachesthekneesof the nullcline. One cycle of the burst oscil-lation is obtainedby tracingalong the four different“pieces”of thetrajectory. Specifically, settingε = 0 in(1), we obtainthefollowing, fastequationsgoverningthetransitionsof theP cellbetweenitsactiveandsilentstates:

v′p = f p(vp, wp)

w′p = 0.

(7)

In this case,wp actsas a parameterin the vp equa-tion, which is simply a scalarequationandthuseasyto analyze.Theequationsgoverningtheslow variationof vp during its active and silent statesare obtained

by rescalingtime usingτ = εt in (1) andthensettingε = 0:

0 = f p(vp, wp)

wp = gp(vp, wp),(8)

where˙ denotesthederivativewith respectto theslowtime variableτ . To mathematicallyconstructthe os-cillation cycle, we denotethe left andright kneesofthenullclineCP by (vLK, wLK) and(vRK, wRK) respec-tively. ResultsbyMishchenkoandRozov (1980)implythatfor ε sufficientlysmall,anactualperiodicorbit liesO(ε) closeto a so-calledsingularperiodicorbit. Thesingularperiodic orbit consistsof four pieces,all ofwhichareconstructedwith ε = 0. Thefirst isasolutionof (7) thatconnects(vLK, wLK) tosomepoint(vR, wLK)

on the right branchof CP (rising edgeof burst). Thenext pieceis a solutionof (8) thatconnects(vR, wLK)

to (vRK, wRK) (activephaseof burst). Thethirdpieceisasolutionof (7) thatconnects(vRK, wRK) to (vL , wRK)

(falling edgeof burst).Thefinal pieceis a solutionof(8) thatconnects(vL , wRK) backto (vLK, wLK) (silentphase).Note that with respectto the slow time scaleτ , thejumpsbetweentheactiveandsilentstatesof theburst are instantaneous.We denotethe periodof thesingularperiodicorbit by TP.

The nullclines for the model interneuronwith ex-citable behavior have the samequalitative shapeasthosefor the P cell, exceptmodelparametersaread-justedso that the cubic-shapedvi -nullcline, denotedCI , andthesigmoidalwi -nullcline, DI , intersectalongtheleft branchof CI . Theintersectionpointof thenull-clinesis astablefixedpointalongtheleft-handrestingbranchof CI , that ensuresthat the isolated I cell isunableto oscillateon its own. Thischangein thenull-clines can be obtained,for example,by shifting thehalf-activationvoltageof theoutwardcurrentto lowervoltagelevels. A propertyweassumefor theintersec-tion point of the nullclines is that this point be closeto the left kneeof the nullcline CI , thusallowing theinterneuronto fire eitherin responseto excitationfromtheP cell or via reboundfollowing inhibition from thepacemaker T . Aswith P, weassumethatneartheleftbranchof CI ,

∂gi

∂vi> 0. (9)

In our geometricanalysisin the phaseplane, thegeneraleffect of introducinga synapticcurrentis toeither raise or lower the cubic-shapednullcline of

Phase Precession of Hippocampal Place Cells 11

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12 Boseetal.

the postsynapticcell without qualitatively changingits shape,as long as the maximumsynapticconduc-tanceis not too large relative to the maximumionicconductances.Since(5) containsno factor of ε, thesynapsesturn on andoff on the fast time scalegov-ernedby t ; thus they are instantaneouswith respectto the slow flow of the trajectoriesgovernedby τ .To examinetheeffectsof thesynapticcurrentson theP cell in our network model,we denotethe solutioncurve of f p(vp, wp) − sip(t − σi p)gip[vp − vi p] = 0by CPI . Since P receives inhibition, CPI lies belowCP (seeFig. 3). We assumethatthenullcline DP andCPI intersectsomewherealongtheleft branchof CPI .This intersectionpoint will bea stablefixedpoint fortheinhibitedP cell,with theresultthattheinhibitedPcell will not spontaneouslyoscillate.Furthernotethatthisintersectionpointwill lie below theleft kneeof CP

becauseof our restriction(6) ∂gp

∂vp> 0 (seeSomersand

Kopell,1993).The effect on I of the excitatory synapticcurrent

from P andthe inhibitory synapticcurrentfrom T isthatits cubic-shapednullclinecanberaisedor lowereddependingontheactivationof spi andsti . For any pos-siblecase,weassumethatthewi -nullcline DI contin-uesto intersectthesynapticallyperturbedvi -nullclinesalongtheir left branches,thusensuringthat I remainsexcitable. As notedpreviously, we requirethat I beable to fire due to either excitation received from Por via reboundfrom inhibition received from T . Forexample,assumethat I issufficientlycloseto theinter-sectionpoint of thenullclineswhenit receives synap-tic input. If the input is excitatory from P, thecubic-shapednullclineof the I cell is instantaneouslyraisedreleasingthe trajectoryfrom its left branch. The tra-jectoryis thenattractedto therightbranchof theraisednullcline,thusinitiatinganactionpotential.If theinputis inhibitory from T , thecubic-shapednullclineis low-eredandthetrajectorymovesleft towardthenew stablefixedpointattheintersectionof theperturbednullclineandDI . For sufficiently stronginhibition, becauseofour restriction(9), the new fixed point will lie belowtheleft kneeof CI . Whenthesynapticinhibition shutsoff, the nullcline risesbackagaintowardCI , andthetrajectoryis attractedto theright branch,initiating anactionpotentialviapostinhibitoryrebound.Theabilityof the I cell to fire dueto thesetwo differentstimuli iscritical to ouranalysis.

The mathematicalanalysisfor this article, aswellasthe specificequationsusedin the simulations,canbe found in AppendicesA and B, respectively. We

work throughoutin theappropriateε = 0 limit andthenuseresultsof Mischenko andRozov (1980)to obtaintheε small result. Our proofsdependon theuseof atimemetricthatallowsustodefinetimesbetweencellsandalsotimesover which relevantbehavior occurs.Itturnsout thatwe canrelatemany of thetimesassoci-atedwith thenetwork modelbackto timesassociatedwith theisolatedcells.

3. Results

3.1. SystemProperties

Wefirstpresentanoverview of ourresultsthatprovidesa qualitative descriptionof how and why the modelgeneratesphaseprecession.In thesecondpartof thissection,weprovide theanalysisof ourmodelandspe-cific analyticalresults.

Thereexist two different, importantfiring patternsfor thesimplenetwork. Thefirst firing patternrepre-sentsnetwork behavior whenthe rat is outsideof theplacefield. The secondrepresentsnetwork behaviorwhenthe rat is insidethe placefield. The differencebetweenthesetwo behaviors is directly correlatedtowhetherT or P controlsthefiring of I . An importantaspectof ouranalysisis to identifymechanismsthatal-low thecontrolof firing tobeshiftedbetweenT andP.As describedbelow, thefirst changein controlfrom Tto P requiresa brief externalinput to thenetwork oc-curring whenthe rat first entersthe placefield. Thesecondchangein control from P backto T is deter-mined internally by the network andsignalsthat theplacefield hasended.

Thefiring patternrepresentingout-of-placefield be-havior is a stableperiodic statethat we call the TIPorbit. In this stablepattern,thepacemaker T controlsthefiring of I , which firesvia postinhibitoryrebound.In turn, I modulatesthe firing of P, so that it firesphase-locked to theunderlyingthetarhythm. Theef-fect of the inhibition from I is to eitherdelayor ad-vancethefiring of P, dependingonwherein P’scyclethe inhibition occurs. Thus, P is phase-locked to thethetarhythm anddoesnot phaseprecesswhetheritsintrinsic frequency is higheror lower thantheta(seeSection3.2.1). Moreover, P can only fire when re-leasedfrom inhibition from I .

Simulationresultsfrom the simplenetwork whereeachcell is modeledwith Morris-Lecarequationsareshown in Fig.4. Duringthefirst fiveburstsof thePcell(top trace),thenetwork is in thestableTIP orbit. The

PhasePrecessionof HippocampalPlaceCells 13

Figure 4. TIP andPIT orbits for the two cell andpacemaker net-workwith eachcellmodeledbyMorris-Lecarequations.TheT spikeis not explicitly shown andis denotedby dashedvertical lines. Thenetwork oscillatesin the TIP orbit for the first 5 cycles.The PITorbit is initiatedby thearrival of thememoryseed(heavy arrow at535msec,which lastsfor 3 msec)andphaseprecessionoccursoverthenext 7cycles(underheavy bar). Thenetwork returnsto thestableTIP orbit for theremainderof thesimulation.Themodelequationsandparametervaluesaregiven in AppendixB.

sequenceof cell firing with T firing first (not explic-itly shown; time of peakindicatedby dottedverticallines), followedby I (lower trace)andthen P is evi-dent.During theT spike, I is inhibitedasseenby thehyperpolarizationin the I voltagetraceimmediatelyfollowing thepeakof T . During the I spike (after itsreleasefrom inhibition from T), P is inhibitedasseenby thehyperpolarizationin the P voltagetraceimme-diatelyprecedingthespike.

Clearly, theout-of-placefield firing ratein theTIPorbit ishigherthanexperimentallyobserved. Thishighout-of-placefield firing leadsto a lossof spatialspeci-ficity to a downstreamdetectorof firing ratebut hasminimal effect on a downstreamdetectorthatusesthephaseof firing to determinethespatiallocationof theanimal.Thesimplestway to reduceor removetheout-of-placefieldfiring is tohaveaspatialinhibitorysignalthatprovidesinhibitionatall pointsoutsideof theplacefield. This approachis not takenherebecauseit reliesonspatialinformationupstreamfromthehippocampusaboutthesizeandextentof theplacefield. Thegoalof

this modelis to generatethespatialphasecorrelationfrom lessspatiallyspecificupstreaminformation.Fol-lowing thedescriptionof thepropertiesof thesimplemodel,we show in Section3.2.7how to completelysuppresstheout-of-placefield firing, withoutresortingto additionalupstreamspatialinformation(seeFig.9).

The secondfiring pattern, representingbehaviorwithin the placefield, is a transientstatethat we callthePIT orbit.ThePIT orbit is initiatedby abrief, soli-tary doseof excitation to P, representinga memoryseedexternallyprovided from thedentategyrus. TheseedcausesP to fire earlierthanit would have in theTIP orbit andtherebyallows P to seizecontrolof I ’sfiring. The P cell is ableto fire via its intrinsicoscilla-tory mechanismsfor aslongasit controlsI . Thus,theseedhastheeffectof switchingcontrolof I from T toP. During thePIT orbit, theperiodof P firing, TP, islessthantheperiodof T , TT , regardlessof theintrinsicperiodof P, providedthat TP andTT arenot too dis-parate.Thereasonphaseprecessionoccursin thePITorbit issomewhatsubtleanddependsbothontheintrin-sicperiodsof P andT andalsoonthetimedurationofinhibitionfrom I to P. If thisinhibitionisshort-lasting,thenduringPIT, P firesatits intrinsicfrequency. Thusif TP < TT , it is clearthatprecessionwill occur. Al-ternatively, if theinhibition is long-lasting,thenduringPIT, P fires at a substantiallyfasterrate than its in-trinsic frequency. Counterintuitively, this increaseinfiring rateis moststronglydependenton the strengthof theinhibitory synapticconductancefrom I to P. InSection3.2.2,weclarify how inhibition mayspeedupthe firing rateof P. For this case,precessionoccurswhethertheintrinsicperiodTP is lessthan,greaterthanor equalto TT . In eithercase,sinceP controlsthefir-ing of I in thePIT orbit, forcing I to fire with every Pburst, I alsophaseprecesses.

Figure4 illustratesthePIT orbit andphasepreces-sion for the Morris-Lecarmodelnetwork. Arrival ofthe memoryseedto P is modeledby a brief, excita-tory, appliedcurrentpulse(heavy arrow) that causesthe early firing of P and the interruptionof the TIPorbit. Due to theexcitation from P, I overcomestheinhibition from T andfireswith P, asseenby theslighthyperpolarizationin thepeakof the P burst. Over thenext six cycles(underthe heavy bar), P and I phaseprecessrelative to T . Thefiring of P dueto intrinsicoscillatorymechanismsisevidencedbythesmoothriseto thresholddisplayedby its voltagetrace.

Thephaseprecessionof I providesaninternalmech-anismthatreturnsthenetworktotheTIPorbit. Namely,

14 Boseetal.

in PIT, I continuesto receive inhibition with eachTspike,but thetimingorphasingof theinhibition issuchthatpostinhibitoryrebounddoesnot occurat eachcy-cle. For example,duringthefirst two spikesof phaseprecession,the inhibition from T arrives during the Ispike,asseenby thelowerspikeheights,andrebounddoesnot occur. Over thenext threecycles,T inhibitsI during early portions of its afterhyperpolarizationwhenreboundis not possible. But by the last spikeof phaseprecession,I is inhibited sufficiently late inits afterhyperpolarization,resultingin areboundspikethat interruptsthePIT orbit, returningthenetwork tothestableTIP orbit. The recaptureof I by T signalstheendof theplacefieldandoccurswhenP andI haveprecessedthroughupto 360degreesof phase,whichisconsistentwith experimentaldata(O’KeefeandRecce,1993). Thus, the precessionof I is necessaryfordeterminingtheendof theplacefield.

Thebrief doseof excitationin theform of themem-ory seedreorganizesthefunctionalrolesof thecellsintheTIPandPIT orbits.Namely, theexcitationswitchesthecontrolof I from T to P. T isableto regaincontrolof I only whenP and I have precessedthroughup to360 degrees.Note that this functionalreorganizationoccurswithout any changesin thecouplingor intrin-sic propertiesof the cells. We emphasizethat, in themodel,phaseprecessionis a transientphenomenathatceaseswith no furtherinput to thenetwork. Thus,thelengthof timeover whichthepyramidalcell precesses,or alternatively, thelengthin spaceover whichpreces-sion occurs,completelydeterminesthe spatialextentof the placefield. In this way, the network generatesthetemporalcodeembodiedin phaseprecession.

In thefollowing analyticresultssection,weprovidemathematicaljustificationfor thequalitative observa-tionsdescribedhere(seeTable1 for a list of relevantsymbols).In particular, inSection3.2.1,wediscusstheexistenceof theTIP orbit. In Sections3.2.2to 3.2.5,we focuson the PIT orbit anddetermineparameters

Table1. List of relevantsymbols.

TT = intrinsicperiodof thepacemaker input CP = intrinsic P cell cubic

TP = intrinsicperiodof the P cell CI = intrinsic I cell cubic

Ä = TT − TP CPI = P cell cubicwith sip = 1

TPC = periodof P cell in TIP network CI P = I cell cubicwith spi = 1

TP = periodof P cell in PIT network CIT = I cell cubicwith sti = 1.

τTon = durationof theinhibition from T to I τIon = durationof inhibition from I to P

τP A = durationof theactivestateof theintrinsic P cell τP R = durationof thesilentphaseof theintrinsic P cell

σi p = synapticdelayfrom I to P σpi = synapticdelayfrom P to I

Note:All timesaremeasuredat ε = 0 in slow timescaleτ .

that most strongly affect this network behavior. InSection3.2.6,weshow how experimentallysupporteddatarelatingrunningspeedto interburst frequency ofthepyramidalcellscanbeusedtoachieveaspatialcor-relationin thistemporalmodel.Finallyin Section3.2.7,weintroduceamodificationto thesimplenetwork thatsuppressesout-of-placefield firing.

3.2. Analysis

3.2.1. The Network Oscillatesat Theta FrequencyOutsideof thePlaceField. Thetrajectoriesof P andI in theTIP orbit canbetracedalongnullclinesin theirphaseplanes.We mayassumethatat τ = 0, T fires,the trajectoryof P lies on the left branchof CP withwp(0) = w? and the trajectoryof I lies somewherealongtheleft branchof CI . In responseto theinhibi-tion received from T , I fallsbacktoCIT andis releasedfrom inhibition at τ = τTon. We assumethatwi (0) issuchthatwi (τTon) liesbelow theleft kneeof CI . Thusat τ = τTon, I firesby postinhibitoryrebound.Allow-ing for synapticdelayfrom I to P, at τ = τTon +σi p, Pfeelsinhibition from I andasaresultfallsbackto CPI .At τ = τTon + σi p + τIon, P is releasedfrom inhibition.We show in AppendixA thatw? canbe chosensuchthatwp(τTon + σi p + τIon) < wLK . With thiscondition,when P is releasedfrom inhibition, it will fire. De-pendingon the positionof the trajectoryof P on theleft branchof CP whentheinhibition from I arrives,atτ = τTon + σi p, thefiring of P haseitherbeendelayedor advancedrelative to its intrinsic frequency. Wealsoshow inAppendixA thatwhenT firesagainatτ = TT ,wp(TT ) = w?, thusensuringthat this cycle of firingrepeats,andweshow thattheTIP orbit is stable.Notethat for P to fire, it mustbereleasedfrom inhibition;it is unableto take advantageof the fact that it is anoscillatorycell.

In Fig. 5, we superimposethephaseportrait of thecoupledP cell onto that of the uncoupledP cell to

PhasePrecessionof HippocampalPlaceCells 15

Fig

ure

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16 Boseetal.

clarify thedifferencesin their trajectories.In thefig-ure, we have usedlowercaseletters to denotetimesthatthesecellsspendin partsof theirorbit wheretheirtrajectoriesdiffer. Theletterswith barsuperscriptscor-respondto thecoupledP cell. NotethatTPC − TP =c + d − (b − b). SinceTPC = TT , a necessarycon-dition for theTIP orbit to exist is thatthedifferenceinintrinsicperiodsof P andT ,Ä = TT −TP, mustsatisfy

Ä = c + d − (b − b). (10)

Thedifferenceb− b> 0 isrelatedto themagnitudeof∂gp

∂vp. Since ∂gp

∂vp> 0, cellsevolve throughthesameEu-

clideandistanceonCP moreslowly thanonCPI . Thusb − b measurestheadditionaltime thattheuncoupledP cell must spendto cover the sameEuclideandis-tancealongthe left branchof CP that the coupledPcell coversonCPI .

In thecasewhenÄ > 0,astableTIPorbit isobtainedif c+ d > b− b. This can be achieved by choosingwp(τTon + σi p), andhencew?, suchthat the trajectoryof P entersa neighborhoodof the stablefixed pointat the intersectionof CPI and DP within thedurationof τIon. Thetrajectoryof P will remainnearthefixedpoint until P is releasedfrom inhibition. In this way,thefiring of P is delayedand P canbephase-lockedto thethetarhythm.

When Ä < 0, a stable TIP orbit is achieved ifb − b > c + d. In thiscase,wp(τTon + σi p), andhencew?, arechosensothatthetrajectoryof P remainssuf-ficiently far from thefixedpoint at the intersectionofCPI and DP during τIon. The trajectorydisplayedinFig. 5 is representativeof this case.Sincecellsevolvefasteron CPI thanon CP, the time b + c + d canbeshorterthanthetimeb. Thisshows thatinhibition canbeusedto speedup P. Thus,P canbephase-lockedto thethetarhythmif TP is greaterthanTT , as long astheperiodsarenot toodisparate.

Observe that(10)canalsoberewrittenas

Ä = τIon + d − b. (11)

For fixedÄ, bothbandd mayincreaseasτIon increases.An increasein b meansthat P mustfeel inhibition athighervaluesofwP alongtheleft branchof CP. Inturn,thismeansthatin TIP thephasedifferencebetweenTandP will begreater. Hence,thedurationτIon controlsthephasedifferencebetweenT andP in theTIP orbit.

In summary, theTIP orbit canbeobtainedwhetherthe differenceÄ = TT − TP is positive, negative, orzero.Theeffectof theinhibition from I to P depends

notonitsdurationτIon but on its timingduringthecycleof P. Wenotethatwhile theTIP orbit canbeachievedwith eitherpositive or negative Ä, independentof thedurationτIon, otherresultsof our model—namely, ob-taining the PIT orbit—requirethat for a short-lastingτIon, Ä mustbepositive.

3.2.2.Within the Place Field, Both P and I PhasePrecess. Similar to above, we analyzethePIT orbitby describingthetrajectoriesof P andI in theirphaseplanes.In thefollowing analysis,weassumethatneartherightbranchof CP, ∂gp

∂vp= 0.Thusthespeedatwhich

P evolvesalongtheright-handbranchesof CP andCPI

will be the same. Our analysiscontinuesto hold for∂gp

∂vp> 0butsufficientlysmall.Wealsoassumethatnear

theright branchof CI ,∂gi

∂vp= 0.

For simplicity, we assumethat the seedarrives atτ = 0, so thatboth T and P fire simultaneously. Thetrajectoryof P jumpsto theright branchof CP. The Icell receivesinhibition from T duringthetimeinterval(0, τTon), but, at τ = σpi , I firesdueto excitationfromP. At τ = σpi + σi p, P receives inhibition from I andfalls to the right branchof CPI . What happensnextis most strongly determinedby τIon, the time lengthover which P feels inhibition from I . The time τIon,however, ishardtodefinepreciselyasit changesduringeachcycleof thephaseprecession(seeAppendixA forclarificationsof this definition). We describehow thePIT orbit is obtainedin thecaseswhenτIon is shortandwhenit is longrelativeto thedurationof theactivestateof P.

First considerthecasewhenτIon is short,in partic-ular, τIon < τP A, thedurationof theactive stateof theuncoupledP cell. At τ = σpi + σi p, let the distancealongCPI from the positionof the trajectoryof P tothe right kneebe greaterthan τIon. Then the inhibi-tion to P shutsoff beforeP jumpsdown from therightkneeof CPI . Soat τ = σpi + σi p + τIon, P returnsto theright branchof CP andleaves theactive statethroughtheright kneeof its intrinsiccubicCP. In Fig. 6A, wehave drawn thetrajectorythat P follows onecycle af-ter theseedis given. It follows this trajectoryuntil thetimeatwhich thenetwork returnsto theTIP orbit. Wedenoteby TP theperiodof P duringphaseprecession.In this case,TP = TP. Thus,a necessaryconditionforphaseprecessionin this scenariois TP < TT or Ä > 0.Here, the solefactor that drives the precessionis thedifferenceÄ in theintrinsicperiodsof P andT .

It is instructivetocontrastthephaseportraitof I withthat of P duringPIT. Asopposedto the cleanand

Phase Precession of Hippocampal Place Cells 17

Fig

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Phase Precession of Hippocampal Place Cells 19

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20 Boseetal.

unchangingtrajectoryof P, thetrajectoryof I changeswith eachcycle of the precession.As we noticedinFig.4,theinhibitionfromT to I occursatprogressivelyearlierphasesduringPIT. Intermsof thephaseportrait(seeFig.6B), thismeansthattheinhibition from T to Ioccursatdifferentplacesin phasespace.In particular,during the first cycle, T inhibits I when I is on theright branchof CI P andstill far away from the rightknee. At the next cycle, T inhibits I while I is stillon the right branchbut now closerto the right kneeat a higherwi value. Progressingin this manner, theinhibition from T eventuallyoccurswhenI is backontheleft branchbut still refractory.Ultimately, I receivesinhibition whenit is closeenoughto the left kneeofCI to becapturedinto TIP, thusendingprecessionandsignalingtheendof theplacefield.

Phaseprecessionoccursfor amorecomplicatedrea-sonwhenτIon > τP A. In this case,we assumethat Preachestherightkneeof CPI beforetheinhibition fromI shutsoff. Notethattheright kneeof CPI lies belowtherightkneeof CP. Sothelengthof time P staysin itsactivestateduringthePIT orbit is lessthanthelengthoftimeit staysin itsactivestateduringtheTIPorbit. Thisdecreasein timeis theprimaryreasonwhy, in thiscase,phaseprecessionoccurs.This canbeseenby consid-eringthephaseplaneof P. Thereareseveralsubcasesdependingon thetime lengthof τIon (seeAppendixAfor details). In Fig. 6C, we have drawn the trajectorythat P follows one cycle after the seedis given foroneof thesesubcases.Let τ = τ1 bethetime P hitstherightkneeof CPI . ThenP jumpsbacktotheleft branchof CPI andevolvesdown CPI until τ = τ1 + σpi + σi p.Wehavedenotedby lowercaselettersvarioustimesofevolutionfor theuncoupledP cellandthecoupledpre-cessingP cell. NotethatTP − TP = kr + kl + m− m.Therearetwo sourcesof timereduction.Thelargerofthe two is the time kr + kl . This time is relatedto themaximumconductancefor thesynapticcurrentfrom Ito P, gip. If gip is small, thenthecubicCPI will notbetoofar below CP, andthusthetimekr + kl is small.As gip increases,sodoeskr + kl . Thesecondsourceoftime reduction,andthusphaseprecession,arisesdueto thedifferencem− m. This differenceis controlledby themagnitudeof ∂gP

∂vpneartheleft branchof CP. For

thiscase,phaseprecessionin thePIT orbit is achievedwhetherÄ is positive, negative, or zero.

3.2.3.The MemorySeedInitiates PhasePrecession.Granulecells in the dentategyrus make excitatorysynapseson CA3 pyramidal cells, and granulecells

haveaspatiallyspecific(JungandMcNaughton,1993)andthetarhythm phasespecific(Skaggset al., 1996)firing pattern.In our model, the arrival time of theexcitatory input, representingthe memoryseedfromthesegranulecells, neednot be so precise. In fact,thereexistsanopeninterval of potentialseedtimings(or phases)that result in phaseprecession.The seedmustbe timed to arrive when P is sufficiently closeto the left kneeof CP. To determinean expressionfor the appropriateseedarrival time τseed, let T fireat τ = 0 and assumethat the durationof the seedissmall. The seedexcitation momentarilyaddsa cur-rentof magnitudegdg[vp − vdg] to thecurrentbalanceequationfor the P cell, wheregdg is themaximalcon-ductance.Phaseprecessionoccursif theseedarrivesatτseed∈ (−δ, τTon + σi p) whereδ > 0. Thelowerbound−δ is relatedto theparametergdg. For largervaluesofgdg, −δ canbelarger(in magnitude).Theupperboundon τseed is a sufficient conditionfor phaseprecessionsincethe P cell will certainlyprecessif it is excitedbeforeit receives inhibition from I . As this intervalfor τseedgivesonly asufficientcondition,seedsthatar-riveslightlyafterτTon + σi p mayactuallyproducephaseprecession.

3.2.4.TheTotalAmountof PhasePrecessionDependson the Phaseof Memory SeedArrival. In additionto determiningwhetherphaseprecessionis initiated,the timing or phaseof the memoryseedalsoaffectsthe total amountof phasechangein the P cell dur-ing its precession.Sinceboth P and I phaseprecessandsince,in the TIP orbit, both cells fire at differentfixedphasesafter T , we needto considertheamountof precessionfor eachof thesecellsseparately.

First considerP. If T fires at τ = 0, thenduringthe TIP orbit, P fires at τ = τTon + σi p + τIon. Wecan translatethesetimesof firing to phasesof firingby multiplying thetimesby 360◦/TT . Theearliertheseedarrives,with respectto the time of P firing, thesmallertheamountthatP precesses.Let τadv = τTon +σi p + τIon − τseedbetheamountof time (or phase)thefiring of P is advancedby thememoryseed.Thetotalamountof phaseprecessionis a decreasingfunctionof τadv. In particular, if theseedarrives in theinterval(−δ, 0) (that is, if τadv is large), P precessesthroughlessthan360degrees.Whereasif theseedarrivesclosetoτ = τTon + σi p (τadvsmall),P precessesthroughcloseto 360degrees,providedthatthedurationof inhibitionfrom I to P, τIon, issufficientlyshort.Theoretically, theP cellcanonlyphaseprecessthroughafull 360degrees

PhasePrecessionof HippocampalPlaceCells 21

Figure 7. Amount of phasetraversedby the P cell during phaseprecessionin the Morris-Lecar-basednetwork model. Whenzerophaseis definedby thepeakof the T cell spike, P cell fires (bursttimes indicatedby circles, filled circles indicateFig. 4 results)at152degreesduringTIP (first fivebursts).Seedsarriving earlier(firstseedat500msec)resultin lessphaseprecessionthanseedsarrivinglater(lastseedat 550msec).Notealsothatthenumberof cyclesofphaseprecessionincreaseswith laterseedarrival.

in the limit τseed→ τTon + σi p + τIon from below. ForτIon large, sucha seedwould fall well outsideof theinterval describedin Section3.2.3andthuswould beunlikely to producephaseprecession.Thus,a lowerboundon themaximalamountof phaseprecessionis360◦(1− τIon

TT).

An exampleof theeffect of theseedtiming on theamountof phaseprecessionis shown in Fig. 7 for theMorris-Lecar-basednetwork model. In thefigure,thefilled circles indicatethe phaseof eachP cell burstduringtheTIP andPIT orbitsdisplayedin Fig.4. Dur-ing TIP, P firesatapproximately152degrees(dashedhorizontallines) when0 degreesis definedby the Tspike. ThememoryseedadvancesP cell firing to ap-proximately77 degreesandthenP precessesthrough285 degreesuntil returning to the TIP orbit and re-sumingfiring at152degrees.Theopencirclesindicatethe phaseof P firing whenthe seedarrives at differ-ent times. For the earliestseedarrival, P precessesthroughapproximately170degrees,andfor the latestseed,P precessesthroughroughly340degrees.

In contrast to the restriction on the amount ofphaseprecessionof the P cell, the I cell can pre-cessthroughthe full 360 degrees.This is possibleifτseed∈ (τTon − σpi , τTon + σi p). Thelowerboundoccursif theseedarrives at sucha time sothat theexcitationfrom P to I arrives exactly at themomentT releasesI from inhibition.

3.2.5.TheNumberof Cyclesof PhasePrecessionDe-pendson thePhasingof theMemorySeedandon theDuration of the Inhibition from I to P. Therearetwo possibleformulaefor thenumberof cyclesof pre-cession.Thedurationof inhibition from I to P, τIon,determineswhichof thetwoformulaeappliestoagivensituation. In thecaseÄ > 0 (TP < TT ), eitherformulamayapplywith τIon actuallydeterminingwhichonetouse.If Ä < 0 (TP > TT ), thenonly thesecondformulacanbeused.

If τIon < τP A, recallfrom Section3.2.2,thatthePITorbit exists only for Ä > 0. The numberof cycles isgiven simply by

n = TT − τadv

Ä, (12)

wheren is thenearestintegerto theratio value.Here,it is the differencebetweenthe intrinsic periodsof Tand P, togetherwith τadv, that determinesthe num-ber of cyclesof precession.The network model thatdisplayedtheresultsin Figs.4 and7 generatesa PITorbit by satisfyingtheseconditionson Ä andτIon andthenumberof cyclesof phaseprecessionis accuratelypredictedby (12), as shown in Table 2. The valuesTT = 100.3 ms and TP = 87.4 ms wereused.

If Ä is negative with τIon < τP A, thenaftertheseed,(12) predictsandsimulationscorroborate(not shown)that the P and I cells phaserecessback to the TIPperiodicorbit.

If τIon > τP A, as describedin Section3.2.2,thereisno requirementon the sign of Ä = TT − TP to obtainthe PIT orbit. In this case,the numberof cyclesofphaseprecessionis given by thefollowing expressionwherethetimesaredefinedin Fig. 6C:

n = TT − τadv

Ä + kr + kl + m − m. (13)

Recall that the quantity kr + kl + m− m is positive,which compensatesfor a potentiallynegative Ä. The

Table 2. Numberof cyclesof phaseprecessionpredictedby Eq. (12) andobserved in simulationsof Morris-Lecar-basednetwork(parametersasin Figs. 4 and 7) for differentarrival timesof thememoryseed.

τadv (ms) 54 39 29 19 14 9 3

npredicted 4 5 5 6 7 7 8

nobserved 4 5 6 7 7 8 8

22 Boseetal.

time m− m is relatedto the magnitudeof ∂gp

∂vpnear

the left branchof CP. The magnitudeof the timekr + kl isof theorderof gip andarisesbecausetheburstwidth andinterburstinterval arelongerfor theisolatedP cell than in the PIT orbit. We ran several simula-tions (not shown) of the Morris-Lecar-basednetworkmodelwith TP > TT (specifically, TT = 100.3 ms andTP = 102 ms). With gip = 1.0, we found n = 18, andfor gip = 1.5, n = 9, thuscorroboratingthestrongde-pendenceof thenumberof cyclesof phaseprecessionon gip.

Reiterating,if Ä > 0, either(12) or (13) mayholddependingonthedurationof theinhibitionfrom I to P.By continuousdependenceon parameters,thereex-istsanintermediatevaluefor τIon atwhichthenetworkswitchesfrom (12) to (13). Notethatnearthisswitch-ing point, small changesin the active durationof Ihavelargeramificationswith respectto theoccurrenceof precessionandtothenumberof cyclesof precession.

3.2.6.The Spatial Correlation of PhasePrecessionIs a Speed-CorrectedTemporalPhenomena. Othermodelsof phaseprecessionrely explicitly on the as-sumptionthatphaseprecessionisaspatialphenomena.We show herethattheapparentspatialdependenceofphaseprecessioncanbeaccountedfor by our tempo-ral model. More precisely, if the interburst frequencyof the placecell andthe frequency of the underlyingthetarhythmarelinearfunctionsof theanimal’sspeed,thenphaseprecessioncanbea temporalphenomenonandalsobemorecorrelatedwith theanimal’s locationthanwith thetime thathaspassedsinceit enteredtheplacefield.

Assume the linear relationship fT = fB + γ1v,f P = fB + γ2v, where fT is the thetafrequency, f Pis the frequency of the precessingP cell andγ2 > γ1

arepositiveconstants.Thevelocityv is settov0 + 1v,wherev0 is theminimumvelocityneededfor thethetarhythmto exist, and1v is thepositive deviation fromthis baselinevelocity. Without lossof generality, thefrequency fB is a commonbaselinefor 1v = 0. Letγ = γ2 − γ1, then f P = fT + γ v, whereγ > 0. Thetimeof thenth thetacyclepeakis nTT , andthetimeofthenth P cell burst,τn, isnTP = nTT

1+ TT γ v. Theamount

of phaseshift after the nth burst φn is the differencenTT − nTP dividedby theperiodof thethetarhythm;thusφn = 360◦ nTT γ v

1+ TT γ v. The positionof the rat at the

time of the nth burst is xn = vτn = vnTT1+ TT γ v

. By in-spection,φn = 360◦γ xn, or equivalently, the phaseisspatiallycorrelated.

Figure 8. Phaseshift of P firing during eachcycle of precessionchangeslinearly with changesin animal’s runningspeed,modeledby changingintrinsic frequency of P. In Morris-Lecar-basednet-work model,phasedifferenceof P burstsduringprecession(circles,filled circle correspondsto Fig. 4 results)increasedlinearly asin-trinsicperiodof P wasdecreased,correspondingto anincreasein Ä

(appliedcurrentto P, I p = 95, 98, 100, 103and105µA/cm2). TheperiodTT = 100mswas fixed throughout.

The phaseshift at eachcycle is given by 1φ =360◦ Ä

TT. Thus the phaseshift is a linear function of

thedifferencein period,andfor fixed TT it is a linearfunction of TP. This linear relationshipholds in ourMorris-Lecar-basednetwork modelasshown in Fig.8.The filled circle showing the largestphaseshift dur-ing eachcycle of phaseprecessioncorrespondsto theparametervaluesin Figs. 4 and 7. To model a de-creasein runningspeed,theintrinsic frequency of theP cell was decreased.In responseto this increaseinTP, correspondingto a decreasein Ä, thephaseshift1φ decreaseslinearly.

3.2.7. Suppressionof Out-of-Place Field Firing.The primary objective of our model is to illustrateamechanismfor generatingthephaseprecessionof placecells. In themodel,phaseprecessiononlyoccursin theplacefield, but theout-of-placefield firing is toohigh.This implies that a downstreamphase-baseddetectorcould preciselydeterminethe animal’s locationat allphasesexceptfor thephase-lockedone.

Differentcomplicatedfeaturescouldbeaddedto themodel to reduceor remove the out-of-placefield fir-ing. For example,therearemany typesof interneu-rons in the CA3 region that have a diversepatternof connectionsand an incompletelydeterminedrolein network activity. In particular, someof the in-terneuronsmake synapticconnectionsonly on otherinterneurons(FreundandBuzsaki,1996).By introduc-ing oneadditionalinterneuroninto our currentmodel,

PhasePrecessionof HippocampalPlaceCells 23

Figure 9. Thevoltagetracesfor thethreecell andpacemaker net-work. Phaseprecessionbeginsat t roughlyequalto 500msecwiththearrival of thememoryseed.ThecellsP andI2 thenphaseprecessfor thenext 6 cycles. Outsideof theplacefield, the longerlastinginhibition from I1, whichis periodicallyreinforced,forcesP to staybelow threshold.

theout-of-placefieldfiring canbesuppressedasshownin Fig. 9. In this morecomplex modelT andP areasbefore,but now thereexist two interneurons,I1 andI2.Both interneuronsreceive fastGABA-mediatedinhi-bition from T andarecapableof firing reboundspikes.The interneuronI1 now provides a slowly decayingGABA-mediatedinhibitory currentto P andreceivesa similar currentfrom I2. TheinterneuronI2 receivesfast glutamatergic excitation from P. As before, Pmayalsohave a feedbackconnectionfrom I2 thatcaneitherbe a fastor slow inhibitory current. With thisnetwork architecture,we requirethat theslow inhibi-tion decayin roughlyonethetacycle—thatis, roughly100msec.

Outsideof the placefield, I1 and I2 fire by postin-hibitory reboundin responseto T . Notethatthesynap-tic delayfrom I2 to I1 allowsthelattertoreboundbefore

the inhibition from I2 cantake effect. Moreover, thisslow inhibition decaysaway over onethetacycle thusallowing I1 to againbein apositionto fire by rebound.Theinhibition from I1 to P, whichlaststheentirethetacycle is strongenoughto prevent P from firing duringthecycle. Sincethis inhibition is renewedat eachcy-cle, P never fires. Wecall this theT I1I2{P} orbit andnotethatT controlsthedynamicsof thenetwork.

As in the simple model, the externally providedmemoryseedfiresP andinterruptstheT I1I2{P} orbit.Excitationfrom P causesI2 to fire. The subsequentslowly decayinginhibition from I2 to I1 now comesbeforethe fastinhibition from T . Thus I1 is stronglyinhibited andinitially will be unableto respondto Tinput. In the reorganizedP I2{I1}T orbit, phasepre-cessionoccurswith P controlling I2 directly, causingit to fire with eachP burstandthusphaseprecess.Inthisorbit, P alsocontrolsI1, indirectly throughI2. Asin the simplemodel, the timing andeffect of the in-put from T to both I1 and I2 changesfrom cycle tocycle. The effect on I2 is similar to that discussedin Section3.2.2andshown in Fig. 6B. The inhibitionfrom T chasesI2 aroundits phaseplane.When I2 hasprecessedthroughup to 360degrees,it canberecap-turedby T . For precessionto end,however, I1 mustberecapturedby T .

BecauseP and I2 arephaseprecessing,the inhibi-tion from I2 to I1 comesatprogressivelyearlierphases.Thismeansthatateachcycleof theta,thereexistspro-gressively moretimefor thisinhibition to decaybeforethe T input to I1 arrives. Similar to the basicmodel,phaseprecessionendswhenT recapturescontrolof I1.This occurswhenthe input from I2 to I1 comesearlyenoughin the T cycle so that the inhibition decaysawaysufficiently to allow I1 to reboundin responsetoT input. Oncethis happens,theslow inhibition fromI1 to P is reinitiated,thussuppressingP. Moreover,since P hasbeenfunctionally removed from the cir-cuit, I2 mustnow wait for T inputto fire. As discussedabove, this inputoccursataphasethatallows I1 to fireperiodically.

Thecomplex modeldescribedhereoperatesin afun-damentallysimilarfashionto thebasicTIP/PIT model.As before,theroleof inhibition differsinsideandout-sideof theplacefield. Indeed,controlof thenetworkof interneuronsis again crucial for determiningthenetwork output. Moreover, our analytical resultsinSections3.2.2 through3.2.6carry over with little orno modification. In this model,it appearsthat I1 hasan “antiplacefield” in that it fires everywhereexcept

24 Boseetal.

within theplacefield. In a lessreducedandmorere-alistic network model,this maynot bethecase.As aresult,wedonot intendour resultsto beinterpretedaspredictingtheexistenceof antiplacefields.

4. Discussion

Wedescribedaminimalbiophysicalmodelof thephaseprecessionof hippocampalplacecells that is consis-tentwith theessentialempiricallydeterminedproper-ties of the phenomenon.We identified mechanismswherebytemporalcontrolof phaseprecessioncanoc-cur. Inparticular, weproposedamechanismfor chang-ing thefiring patternof placecellsastheanimalenters,crosses,andthenleaves the placefield. This mecha-nismof changingcontrolof theinterneuronfrom T toP switchesthe network from a stablestateto a tran-sient state. The precessionof the interneuronspro-videsasecondmechanismto switchthecontrolof theinterneuronbackto T andthusreturnthenetwork to astablestate.

Theinitial formof themodelaccountsfor mostof theempiricalobservationsthatwerelistedin theintroduc-tion. Namely, (1) sinceweprovidetheseedonly whentherat moves in onedirection,placecellsonly fire inonedirectionof motion; (2) all placecells, only thesingleP cell in ourcase,startfiring at thesameinitialphase;(3) theinitial phaseis thesameoneachentryofthe rat in theplacefield sincetheseedalwaysoccursat the samephase;(4) sincethe changein control inthenetwork occursbefore360degreesof precession,it is impossiblefor thereto bemorethanthis amountof phaseprecession;(5) while thecells in thedentategyrusmay undergo a small numberof cyclesof pre-cession,ourmodelrequiresonly onecycle; additionalcycleswouldnotchangetheresults;(6) themodeldoesnotaccountfor theincreasein firing rate(alessreducednetwork modelthataccountsfor individualspikesmaybe ableto includefiring rateinformation;we believethatthemechanismswehaveuncoveredfor phasepre-cessionin our idealizedburstingneuronswould carryover to neuronsthat insteadexhibit burstsof spikes);and (7) the linear dependenceon frequency of pyra-midal cell firing is usedhereto maintaina correlationwith locationratherthantimethathaspassedsincetheanimalenteredtheplacefield.

Theinitial model,however, hada high out-of-placefield firing rate. This out-of-placefield activity wouldresultin alossof spatialspecificityif adownstreamsys-temusesfiring rateto determinetheanimal’s location

in the environment. However, the phaserelationshipbetweenplacecell activity alsoprovidesinformationon the locationof theanimal,andthebackgroundfir-ing outsideof theplacefield maynot bea substantialproblemfor adownstreamsystemthatusesphasetode-terminetheanimal’s location.We presenteda method(not necessarilyunique),usinganadditionalinterneu-ron, to remove theout-of-placefield firing without re-quiringupstreaminformationonthesizeandextentoftheplacefield.

Fromamathematicalpointof view, thiswork showstheimportanceof determiningfunctionalityof thesub-piecesof themodel.Wehaveshownhow controlwithinthenetworkcanbeswitchedfromonenetworkmemberto anotherandthenbackagain.Wehavedemonstratedthat geometricanalysisis well suitedto identify themechanismsthatchangecontrol. Our work alsohigh-lights thechangingandperhapsnonintuitiveeffectsofinhibition in thenetwork. In particular, wehaveshownhow a fasteroscillatorcanentraina slower oneusingonly inhibition.Thisdependscritically onthetimingoftheinhibitory input from thefasteroscillator, which isconsistentwith thethemethattiming isof fundamentalimportancein temporalcodegeneration.Wehavealsoshown how thenetwork canuseinhibition to function-ally enhanceor removetheeffectsof achosenmember.

4.1. RelatedWork

Othermodelsof phaseprecessiondiffer from oursinimportantways. In prior work (Tsodykset al., 1996;JensenandLisman,1996;WallensteinandHasselmo,1997;Kamondietal.,1998),phaseprecessionis mod-eledasa spatialphenomena.Eachof thesemodelsinsomeway utilizesexternalinput thatalreadyencodesfor thephaseprecession.Thusnoneof themodelstrulyexplainsthegenesisof phaseprecession.Namely, themodelsdo not addresshow a single P cell behavesboth outsideandinsideits placefield or what causesthetransitionin behavior of thecell betweenthesetwoareas.

Theabove modelsrequireeithera network of exci-tatorycellsorahighlyparameterizeddescriptionof theneurontoachieveprecession.In theworkof JensenandLisman(1996),a one-dimensionalchainof pyramidalcells is considered,andphaseprecessionoccursdueto thelocalunidirectionalexcitatorysynapsesbetweenthesecells. Selective excitation to a specificmemberof the chain, occurringat eachthetacycle, providesphase-precessedinput to thenetwork. In themodelof

PhasePrecessionof HippocampalPlaceCells 25

Tsodykset al. (1996), strongexcitatory connectionsbetweenplace cells is also required.In this model,precessionis driven by asymmetricsynapticweightsbetweenthesecells. Thetotalamountof excitatoryin-putacell receives isat amaximumin thecenterof theplacefieldanddecreasesmonotonicallyasthedistancefrom the centerincreases.Tsodykset al. (1996)callthis “directional tuning.” A similar idea is employedby Kamondietal. (1998).Therearamplikedepolariz-ing currentis providedto theplacecell to mimic pas-sagethroughtheplacefield.Thusin thesetwo models,therunninganimalis given externalinformationaboutits position in spacethat is encodedin the spatiallydependentdepolarization.Kamondiet al. (1998)andWallensteinandHasselmo(1997)bothusemulticom-partmentdescriptionsof the placecells. Precessionin thesemodelsdependscritically on a more com-plex, multiparameterdescriptionof eachneuron.More-over, the modelof WallensteinandHasselmo(1997)also requiresphase-precessedexternal input at eachthetacycle asin JensenandLisman(1996)to achieveprecession.

The presentmodel takes an entirely different ap-proach. First, we believe phaseprecessionis a tem-poral mechanismthat, as demonstrated,can accountfor changesin therat’s runningspeed.Second,we donotrequireany externalprecessedinputto thenetwork.Instead,we simply needa onetime doseof excitationthatmimics a memoryseedandchangesthenetworkfrom the TIP to the PIT orbit. We do not requireanetwork of excitatorycellsor multicompartmentmod-elsfor thecells. Finally, our analysisgives a differentinterpretationof the significanceof phaseprecessionthanall of thepreviousstudies.Namely, in our modeltheendof phaseprecessionsignalstheendof theplacefield. In the otherstudies,the endof the placefieldsignalstheendof theprecession.Thusourmodelpro-videsaninternalmechanismfor determiningtheendoftheplacefield (via theendof precession),whereastheothermodelsrequireanotherexternal input to notifythepyramidalcell thatits placefield hasended.

4.2. ExperimentalSupport

Severalstudieshavedemonstratedthatthemeanphaseof activity of hippocampalinterneuronsis justprior tothe meanphaseof pyramidalcell activity (Fox et al.,1986;Buzsaki andEidelberg, 1982).This impliesthatduringthephaseprecession,theactivity of aplacecellmustpassthroughthephaseof thetarhythmat which

interneuronsare most strongly inhibiting pyramidalcells. In the presentmodel the pyramidal cell con-trols the firing of the interneuronduring the phaseprecessionprocess,and its activity is not adverselyinhibitedby theinterneuron.

In recentrecordingsof a larger numberof pyra-midal cells and interneurons(Csicsvari et al., 1998),pyramidalcellswerefound thathadlarge,significantcross-correlationpeaksat times on the scaleof tenmillisecondsthat precededthe activity of simultane-ously recordedinterneurons.Thesecorrelationsfindthatthereis a tight couplingin thetimeatwhichpyra-midal cells and interneuronsare active and suggeststhat theactivity of a subsetof the interneuronsmightprecessalongwith placecells. The experimentalev-idenceof the high level of correlationbetweenplacecells and interneuronsdemonstratesthat thereexistsa sufficiently strongconductancebetweensubsetsofthesetwo typesof neurons.Thesefindingsareconsis-tentwith thepredictionsin thepresentmodelin whichthephaseprecessionof interneuronsmayonlyoccurina subsetof interneuronsandonly whenthey arebeingdriven by aphaseprecessingplacecell.

4.3. Consistencyof BiophysicalandFunctionalModels

In a modelof phaseprecession,thereareseveral im-portantlargergoalsandanatomicalconsiderationsthatshouldbe taken into account.There is considerabledata that demonstratesthat the hippocampushas arole in episodicmemory. We have proposeda func-tional modelthatdescribeshow thespatialandmem-ory rolesof thehippocampuscanbecombined(RecceandHarris, 1996; Recce,1999). In this view, whena rat returnsto a previously experiencedenvironment,thesetof coactiveplacecellscorrespondsto therecallof anegocentricmapof theenvironment.Thememoryrecallisperformedbyapatterncompletionprocessthatis controlledthroughtheexcitatoryfeedbackpathwayin the CA3 region of the hippocampus.The seedforthis recall enterstheCA3 region throughtheentorhi-nal cortex to thedentategyrusandthenin themossyfiber projectionfrom the dentategranulecells to theCA3 pyramidalcellsandinterneurons.The temporalcontrolof the recall process,in this view, is providedby thepacemaker input from themedialseptum.

ThefeedbackconnectionsbetweenCA3 pyramidalcells are relatively sparse,which can limit the num-ber of egocentricmapsof spacethat canbe storedif

26 Boseetal.

the region is modeledasan autoassociative memory.However, Gardner-Medwin (1976)hasdemonstratedthat this limitation canbe overcomeif the recall pro-cessis performedover a sequenceof steps. HiraseandRecce(1996)foundthattheoptimalperformancein a multistepautoassociative memorymodel of theCA3 regioncanbeachieved if an interneuronnetworkcontrolsthe recall processby providing an inhibitoryinput to thepyramidalcellsthat is a linearfunctionofthe numberof simultaneouslyactive pyramidalcells.Further, it hasbeenproposedthatthephaseprecessioncorrespondsto themultiplestepsin this recallprocessand that the phaseprecessionsuggestsa methodforthe concurrentrecall of several egocentricmemories(Recce,1999).

In the presentwork, we have begun the processof systematicallyaddressingtheseissues.Thepresentmodelfor phaseprecession,while conceptuallyquitesimple,is notnecessarilytheonethatmosteasilypro-ducesoutputthatphaseprecesses.Instead,our mod-eling is basedon thelargerconsiderationsof identify-ing mechanismsthatcanallow multiplememoryrecallprocessesto besimultaneouslyperformed.Thebasicframework describedabove, which accountsfor pre-cession,is compatiblewith a functionalmodelof thespatialandepisodicmemoryrolesof CA3 placecells.

Appendix A. Analysis: Existenceand Stabilityof TIP Periodic Orbit

Outsidetheplacefield,thenetwork is in theTIPconfig-uration.Weshow thatthereis aunique,stableperiodicorbit to which the cells areattracted.Assumethat Tfires at τ = 0. We will locatean initial conditionforP andshow thatwhenT firesagainat τ = TT , P hasreturnedexactly to this initial condition. Theanalysiswill bein termsof a time metricthatwill measurethetimesof evolution of P on variouspartsof its orbit inphasespace.Thesetimesdependon variousbiophys-ical parametersthat are relatedto both intrinsic andsynapticpropertiesof thecellsandthenetwork.

Theorem1. Thereexistsa locally uniqueasymptoti-callystableTIPperiodicorbit withaperiodO(ε) closeto TT .

Proof Theanalysisbelow occursat ε = 0, but usingthework of Mischenko andRozov (1980),theresultsactuallyholdfor ε > 0andsufficientlysmall. Considerthe left branchof CP betweenwRK andwLK . Associ-atedtoeachpointonthiscurve,denotedwp, isthetime

1(wp) it takes P to reachwLK startingfrom wp. Inparticular, 1(wRK) = τP R and1(wLK) = 0. It is clearthat1 is a monotonedecreasingfunctionof wp. Thedescriptionof theTIP network given earliershowsthatthereis aone-dimensionalmap5 thattakestheinitialpositionof P at τ = 0 alongtheleft branchof CP andreturnsthepositionof P at τ = TT .

Proposition 1. The map 5 definesa uniform con-traction on a subinterval(wlo,whi ) of (wLK,wRK). Ifw? is theresultinglocally uniqueasymptoticallystablefixedpoint, thenthenetworkpossessesalocallyuniqueasymptoticallystablesingularTIP periodicorbit withperiodTT , wherewp(0) = w?.

Proof Weprovetheexistenceof afixedpointbycom-paringthebehavior of theuncoupledP cell to its cou-pledcounterpart.SinceTT = TP + Ä, wecancomparetheperiodof thecoupledP cell to TT . First we locatea suitablewhi . Associatedwith it is a time Thi fromwhi to wLK on CP. At τ = τTon + σi p, P receives inhi-bition andjumpsbackto CPI . At τ = τTon + σi p + τIon,theinhibition to P shutsoff. We choosewhi suchthatwp(τTon + σi p + τIon) = wLK . Next let us considertheevolutionof the P cell, solelyonCP in theabsenceofany synapticinput. Since ∂gp

∂vp> 0 neartheleft branch

of CP andCPI , the P cell movesdown CP ataslowerrate than on CPI . In other words, P takes a longertime to cover thesameEuclideandistanceonCP thanonCPI . Thusatτ = τTon + σi p + τIon, theuncoupledPcell will be above wLK . It is seenthat inhibition ac-tually shortensthe time distanceto the kneefor thisinitial condition. This is very similar to, but timewisetheoppositeof, theideaof virtual delayof Kopell andSomers(1995).Therethey showedthatexcitationhadtheoppositeeffect of increasingthe time to theknee.The coupledanduncoupledversionsof P follow thesametrajectoryin theactivestate.

AssumemomentarilythatÄ = 0. For theuncoupledP cell, 1(5(whi )) = Thi . But for thecoupledP cell,1(5(whi )) < Thi .Thusthetimeit takesfor thecoupledP cell to returnto its initial positionwhi is Tpc < TP. IfÄ is sufficiently small,thenTpc < TP + Ä = TT . Notethat Thi > τTon + σi p + τIon. How big the differencebetweenthesetimesneedsto bedependsonthesizeof∂gp

∂vp, whichdeterminesthedifferencein theratesalong

CP andCPI . Thuswehavelocatedaninitial conditionwhosetime distanceto thekneecompressesover oneoscillation.

We next locatean initial conditionwhosetime dis-tanceto thekneeexpands.Now let wlo = wp(0) such

PhasePrecessionof HippocampalPlaceCells 27

that for the uncoupledP cell wP(τTon + σi p) = wLK .Assumethattheinitial conditionsfor T andI areiden-tical to the above case. Thusat the momentP wereto fire, it receives inhibition from I . It will thusfallbackto CPI andmovedown thiscubictowardthefixedpoint on CPI for a time τIon. At τ = τTon + σi p + τIon,P is releasedfrom inhibition and jumps to the rightbranch.Thew valueof thepoint to whereit jumpsliesbelow wLK . On the the right branch,let the time be-tweenthis point andwLK bedenotedby τD. OncePpassesthroughwp = wLK on theright branchof CP, itwill follow thesameorbit astheuncoupledP cell. ThecoupledP cell hasbeendelayedby a time τIon + τD.Soit will returnto its initial positionatatimeTpc > TP,if Ä = 0. Thusif Ä is sufficiently small,Tpc > TT . Forthis initial condition,thetime distanceto thekneehasbeenexpanded—thatis, 1(5(wlo)) > 1(wlo).

Therefore,wehavelocatedasetof initial conditionssuchthat the lower and upperboundariesof the setaremappedinto the interior of thesetby themap5.By continuousdependenceon initial conditions,thisimpliesthatthereexistsat leastoneelementof thesetthatremainsfixedunderthemap5. To provethatthispoint is uniqueandattracting,we needto show that5 is a contractionmapping. We follow two differentversionsof theP cell,denotedP1 andP2, andshow thatthedynamicsof theTIPnetworkbringthesecellsclosertogetherafter one iterateof 5. Let w1 < w2 denoteany two pointsin (wlo, whi ) suchthatwp1(0) = w1 andwp2(0) = w2. Thecells P1 andP2 do not interactwithoneanotherbut receive commoninhibition from thesameI cell.

Lemma 1. There existsλ ∈ (0,1) such that |5(w2)

− 5(w1)| < λ|w2 − w1|.Proof: Fromtheaboveanalysis,it is clearthatin theTIP network, P1 and P2 will jump to the active statefrom apointonCPI thatliesbelow theleft kneeof CP.This occursat τ = τTon + τIon + σi p. Let δ denotethetime betweenthe cells at τ = 0. As the cells evolvealongtheleft branchof CP andthenCPI , thetime be-tweenthemremainsinvariant.However, theEuclideandistancebetweenthecellsdecreases.Thisisastandardfactfor two cellsthatareevolving alongthesameone-dimensionalcurve toward a fixed point. The longerthecellsspendnearthecritical point formedby thein-tersectionof CPI andDP, themoretheEuclideandis-tanceis compressed.At τ = τTon + τIon + σi p, thecellsarereleasedfrom inhibition andjump horizontallytotheright branchof CP. Thesamemechanismthatfos-

terssynchrony due to fast thresholdmodulation(seeSomersandKopell, 1993) implies that thereis com-pressionacrossthe jump. The rational is the follow-ing. Acrossthejump, theEuclideandistancedoesnotchange.However, therateof evolution of thecellsontheleft branchis muchslowerdueto thefactthatthesecells areevolving neara critical point. On the rightbranchthecellsarevery far way from thew-nullclineDp, so the rateof evolution is muchfaster. Thusthecells travel throughthe sameEuclideandistancein amuchshortertime. Thusthe new time betweencellsis lessthanδ. Sincethecellsalwayssatisfythesamedifferentialequation,the time betweenthemremainsinvarianton the right branchof CP. Moreover, sincethey both jump down from the right kneeof the CP,the time is invariantacrossthedown jump. When P1

andP2 returnto theleft branchof CP, P2 will lie belowP1. So whenP2 returnsto theinitial positionof P1 atw1, the time betweenthecellswill be lessthanwhenthey started,so the Euclideandistancebetweenthemmusthave compressed.This proves that 5 is a con-traction.Theconstantλ canroughlybeapproximatedby the ratio of the speedat the jump on point on theright branchto thespeedat the jump off point on theleft branch(SomersandKopell,1993).

Wehavemadesomeimplicit assumptionsaboutthebehavior of I . Namely, we have not proved that I ac-tually returnsto its initial positionat τ = TT . We areassumingthat the time on the left branchof CI fromw = wRK of CI to a neighborhoodof the fixed pointon CI is boundedfrom below andabove. The upperboundarisesfrom thefactthatwewant I to beabletorespondto T at τ = TT . Thelower boundcomesfromthefactthatwedonotwant I to fire for thetimeperiodτP A + σi p. Providedtheseconditionsaremet, theex-istenceof anasymptoticallystablefixedpoint impliestheexistenceof anasymptoticallystablesingularperi-odicorbit. It isnow fairly standardtoshow thatthemap5 perturbssmoothlyfor ε > 0, but sufficiently small,thusyielding theactualTIP periodicorbit (MischenkoandRozov, 1980). 2

Let usnow addressmorecarefullywhatwemeanbytheactivestateof theinterneuron.In PIT, it isnotpos-sibleto locatethepreciseplacefromwhich I jumpsup.Thusthepreciselocationat which it reachestherightbranchis not determinable,and thereforewe cannotpreciselydefineits active duration. To give an esti-matefirst on thejump up point we needto definetwomorecubics. Oneis CI P , which occurswhensP I = 1

28 Boseetal.

andsT I = 0. Thesecondis CI PT , which occurswhenbothsP I = 1 andsT I = 1. NotethatCI PT lies betweenCI P andCIT . Dependingontheparametersgpi , gti , vpi ,andvt i , it maylie aboveorbelow CI . For thisargumentassumeit lies above CI . Thenthemaximumw valueto which I canjump is thew valueof theleft kneeofCI P , call it wi M . Theminimumvalueis thew valueoftheintersectionof CI andDI , call it wim. Since∂gi

∂vi= 0

neartheright branches,thespeedsof evolution alongall theright branchesareidentical. For this argumenttheorderingof theright kneesof therelevantcubicsisRIT < RI < RI PT < RI P , wherethenotationshouldbeclear. So,by alongactivephaseof theinterneuron,wemeanthetimefrom wi M to RIT is longerthenτP AI . Byashortactivephasewemeanthetimefromwim to RI P islessthanτP AI . Notethatthesetimesaresimplyboundsfor the lengthof theactive phase.They do not implythat I necessarilyjumps down from a specificknee.Indeed,I ’s jump down point may changeeachcycledependingonthetiming of inhibition from T . Finally,if the lengthof the active stateof I is too long—forexample, if the time wi M to RI is much longer thanτP AI + σpi , then the analysispresentedabove breaksdown. This is becauseI will not be in a positiontorespondto thenext boutof excitationfrom P.

Appendix B. Model Equationsand Parameter Values

We have numerically implementedour two cell andpacemaker network usingtheMorris-Lecarequations(1981)tomodeltheneurons.Thecurrentbalanceequa-tionsfor theP andI cellsare

Cmdvp

dt= −gCam∞(vp)(vp − vCa) − gK wp(vp − vK )

−gL(vp − vL) − sipgIP(vp − vIP) + I p,

Cmdvi

dt= −gCam∞(vi )(vi − vCa) − gK wi (vi − vK )

− gL(vi − vL) − spi gP I (vi − vPI)

− sti gTI(vi − vTI) + Ii ,

where vX (in mV) is the membranevoltage in thepyramidal cell (X = p) and the interneuron(X = i ).Themaximalconductancesfor thecalciumandpotas-sium currentsin the cells are the same(gCa = 4.4,gK = 8.0). The reversalpotentialsfor the ionic cur-rentsareVCa = 120 mV andVK = − 84mV. Theleakconductancedensityin eachcell isgL = 2mS/cm2,and

theleakreversalpotentialis VL = −60mV. Themem-branecapacitanceis Cm = 20 µF/cm2. The appliedcurrentvalues(in µA/cm2) areI p = 105andIi = 120.In our implementationof themodel,theT cell is mod-eledby thesameequationsasanisolatedP cell,exceptappliedcurrentis setto 92 µA/cm2. In orderto ob-tain frequenciesin thethetarange,timewas scaledby4.5. Themodelequationswerenumericallyintegratedusing XPP (information on the programavailable athttp://www.pitt.edu/∼phase).

The gatingkineticsof the potassiumconductancesin eachcell (X = p and i ) aregovernedby equationsof thefollowing form:

dwX

dt= φ

w∞,X(v) − wX

τw,X(v),

whereφ = 0.005correspondsto ε in thegeneralformof theequations.Thesteady-stateactivationandinac-tivationfunctionsandthevoltage-dependenttimecon-stantfunctionsfor thecalciumandpotassiumcurrentsin eachcell (X = p andi ) aregiven by

m∞(v) = 1

2

[1 + tanh

(v − v1

v2

)],

w∞,X(v) = 1

2

[1 + tanh

(v − v3,X

v4,X

)],

τw,X(v) = 1

sech[(v − v3,X)/2v4,X].

Thehalf-activationvoltagesfor thegatingfunctionsare(in mV) v1,p = v1,t = − 1.2, v3,p = 2, andv3,i = −25.Theactivationandinactivationsensitivitiesfor thegat-ing functionsare(in mV) v2,p = v2,i = 18, v4,p = 30,andv4,i = 10.

The synapticcurrentsin eachcell aregovernedbythevariablesXY whereX indicatesthepresynapticcellandY indicatesthepostsynapticcell. Thesevariablessatisfyequationsof theform

s′XY = α(1 − sXY)

1

2

(1 + tanh

(vX − v5

v6

))− β sXY,

wheretheHeavisidefunctionin thegeneralformof theequationsfor sXY is replacedby asigmoidlikefunctionthatdependsonthepresynapticvoltage(half-activationv5 = 0andactivationsensitivity v6 = 10). For thethreesynapticcurrentsin the model,governedby spi , sip,and sti , the rise and decayratesof the synapseare

PhasePrecessionof HippocampalPlaceCells 29

thesame(α = 2, β = 1). In thecurrentbalanceequa-tions, the effect of the synapseis determinedby thereversalpotentialsof the synapticcurrents(vpi = 80,vi p = vt i = −80), and the maximal conductancesofthesynapticcurrentsareall equal(gpi = gip = gti = 1).Wetookσi p = σpi = 0 in thesimulations,thusshowingthattheexistenceof delaysarenotcrucialfor theaboveresults.

For thesimulationin Fig. 9, the following parame-terswereused:I p = 118, Ii1 = 80, Ii2 = 80,gi2i1 = 0.5,αi2i1 = 5, βi2i1 = 0.005,vi2i1 = −95, gpi2 = 1, αpi2 = 2,βpi2 = 1, vpi2 = 80, gi1 p = 0.8, αi1 p = 3, βi1 p = 0.001,vi1 p = −95, gti1 = 2.5, αt i1 = 2, βt i1 = 2, vt i1 = −80,gti2 = 2.5, αt i2 = 2, βt i2 = 2, andvt i2 = −80.

Acknowledgments

This work was supported,in part, by the New JerseyInstitute of Technologyunder grants421540(AB),421590(VB), and 421920(MR). V. Booth also ac-knowledgessupportfrom theNationalScienceFoun-dation(grantIBN-9722946).

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