Journal of Computational Neuroscience 9, 5–30, 2000c© 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

A Temporal Mechanism for Generating the Phase Precessionof Hippocampal Place Cells

AMITABHA BOSE, VICTORIA BOOTH AND MICHAEL RECCEDepartment of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of

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

Received December 1, 1998; Revised June 21, 1999; Accepted June 25, 1999

Action Editor: John Rinzel

Abstract. The phase relationship between the activity of hippocampal place cells and the hippocampal thetarhythm systematically precesses as the animal runs through the region in an environment called theplace fieldof the cell. We present a minimal biophysical model of the phase precession of place cells in region CA3 of thehippocampus. The model describes the dynamics of two coupled point neurons—namely, a pyramidal cell and aninterneuron, the latter of which is driven by a pacemaker input. Outside of the place field, the network displays astable, background firing pattern that is locked to the theta rhythm. The pacemaker input drives the interneuron,which in turn activates the pyramidal cell. A single stimulus to the pyramidal cell from the dentate gyrus, simulatingentrance into the place field, reorganizes the functional roles of the cells in the network for a number of cycles of thetheta rhythm. In the reorganized network, the pyramidal cell drives the interneuron at a higher frequency than thetheta frequency, thus causing a systematic precession relative to the theta input. The frequency of the pyramidal cellcan vary to account for changes in the animal’s running speed. The transient dynamics end after up to 360 degreesof phase precession when the pacemaker input to the interneuron occurs at a phase to return the network to thestable background firing pattern, thus signaling the end of the place field. Our model, in contrast to others, reportsthat phase precession is a temporally, and not spatially, controlled process. We also predict that like pyramidal cells,interneurons phase precess. Our model provides a mechanism for shutting off place cell firing after the animal hascrossed the place field, and it explains the observed nearly 360 degrees of phase precession. We also describe howthis model is consistent with a proposed autoassociative memory role of the CA3 region.

Keywords: phase precession, minimal biophysical model, place cells, theta rhythm

1. Introduction

There is considerable current interest in the ways inwhich the temporal firing pattern of neurons may pro-vide additional information that is not conveyed by theaveraged firing rate alone. This interest has led to asearch for ways in which temporal firing propertiesof neurons are generated and detected in the centralnervous system (for a review, see Rieke et al., 1997;Recce, 1999). The main goal of this article is to useminimal biophysical modeling to demonstrate how a

group of neurons in region CA3 of the hippocampusof freely moving rats can generate a spatially encodedoutput, calledphase precession, using only limited andspatially nonspecific inputs.

It has been proposed that hippocampal place cellsprovide information for downstream neurons throughthe phase relationship between neuronal activity andthe hippocampal EEG (O’Keefe and Recce, 1993).Place cells were first described in the CA1 region offreely moving rats (O’Keefe and Dostrovsky, 1971),and the activity of these putative pyramidal cells (Fox

6 Bose et al.

Figure 1. Extraction of the firing phase shift for each spike during a single run through the place field of a place cell on the linear runway. A:Each action potential from cell 3 during the one second of data shown in the figure is marked with a vertical line. B: The phase of each spikerelative to the hippocampal theta rhythm. C: Hippocampal theta activity recorded at the same time as the hippocampal unit. D: Half sine wavefit to the theta rhythm that was used to find the beginning of each theta cycle (shown with vertical ticks above and below the theta rhythm).Reprinted from O’Keefe and Recce (1993).

and Ranck, 1975) is highly correlated with the rat’s lo-cation in an environment (O’Keefe, 1976; Muller et al.,1987). The preferred firing location of a place cell iscalled itsplace field.

During locomotor activity, the hippocampal EEGhas a characteristic 6 to 12 Hertz sinusoidal form calledthe theta rhythm(Green and Arduini, 1954), and thephase and frequency of this rhythm is highly correlatedacross the CA1 region of the hippocampus (Bullocket al., 1990). As the rat runs through a place field, on alinear runway, the phase of the theta rhythm at whicha place cell fires systematically precesses. Each timethe animal enters the place field, the firing begins atthe same phase, and over the next five to ten cyclesof the theta rhythm it undergoes up to 360 degrees ofphase precession (O’Keefe and Recce, 1993; Skaggset al., 1996). The maximum observed phase preces-sion was 355 degrees (O’Keefe and Recce, 1993). So,by the phrase “up to 360 degrees of phase precession”we mean strictly less than but in a neighborhood of360 degrees. An example of the phase precession of aplace cell is shown in Fig. 1. During the run along thetrack, the phase of hippocampal place cell firing is morecorrelated with the animal’s location within the placefield than with the time that has passed since it enteredthe place field (O’Keefe and Recce, 1993). This sug-gests that the phase of place cell activity provides moreinformation on the location than is available from the

firing rate of the cell alone. A downstream system thatmeasures the phase of place cell activity would thenhave more information about the location of the an-imal in the environment and may be able to ignoreout of place field firing that occurs preferentially ata phase that is different from the range found in theplace field. Place cells in both the CA1 region andthe upstream CA3 region are found to undergo phaseprecession (O’Keefe and Recce, 1993).

Skaggs and coworkers (1996) confirmed this find-ing and additionally found that the initial phase wasconsistent among a large number of place cells in theCA1 region. They also found that dentate granule cellsthat project to CA3 undergo a small number of cy-cles of phase precession and therefore provide a syn-chronized, timed excitation to the CA3 pyramidal cells.Marr (1971) proposed that this input from the dentategranule cells provides a seed input for a memory re-trieval and pattern completion process that is drivenby the excitatory feedback among pyramidal cells inthe CA3 region (Treves and Rolls, 1994; Gibson andRobinson, 1992; Hirase and Recce, 1996). The phaseprecession of place cells may be an essential part ofthis pattern completion process.

The hippocampal regions also contain a varietyof interneurons that contribute to the generation ofhippocampal oscillations. These interneurons projectto place cells as well as to other interneurons (Freund

Phase Precession of Hippocampal Place Cells 7

and Buzs´aki, 1996). Skaggs and coworkers (1996)measured the phase of interneuron firing in the CA1 re-gion and found that on average the interneurons do notphase shift. On the other hand, Csicsvari and cowork-ers (1998) found a large, significant cross-correlationbetween pyramidal cell and interneuron pairs, wherethe pyramidal cell firing precedes interneuron firing bytens of milliseconds. This implied high synaptic con-ductance from pyramidal cells to interneurons suggeststhat interneurons could phase precess. This sugges-tion may not be inconsistent with the experimentallyobserved average properties of interneurons (Skaggset al., 1996) if only a subset is shown to transientlyphase precess.

The medial septum is also involved in modulating thetemporal firing patterns of hippocampal place cells andinterneurons. The projection from the medial septum tothe hippocampus is both GABAergic and Cholinergic(Freund and Antal, 1988), and it provides a pacemakerto drive the theta rhythm (Green and Arduini, 1954).We have shown that the interburst frequency of some ofthe neurons in the medial septum is a linear function of arat’s running speed on a linear track (King et al., 1998).

In summary, the observed properties of the phaseprecession phenomenon include the following: (1)place cells fire in only one direction of motion dur-ing running on a linear track; (2) all place cells startfiring at the same initial phase; (3) the initial phase isthe same on each entry of the rat into the place field of aplace cell; (4) the total amount of phase precession is al-ways less than 360 degrees; (5) the cells in the dentategyrus that project to CA3 undergo a smaller numberof cycles of phase precession; (6) in one-dimensionalenvironments, the phase plus firing rate provide moreinformation of the rat’s location than the firing rate ofa place cell; (7) phase is more correlated with the ani-mal’s location in a place field than with the time sincethe animal entered the place field; and (8) backgroundfiring of place cells outside of the place field occursat a fixed phase that is closest to the initial phase thatoccurs when the animal enters the place field.

Several models have been proposed to explain theneural basis of the phase precession phenomenon. Themodel proposed by Tsodyks and coworkers (1996) pro-vides an environment-driven phase precession, whichis certain to be more correlated with position than time.Other models of environment-driven phase preces-sion have been proposed by Wallenstein and Hasselmo(1997) and Jensen and Lisman (1996). In these mod-els, the phase correlation in CA1 pyramidal cells resultsfrom CA3 input that is phase precessing. Kamondi and

coworkers (1998) have proposed a model for phase pre-cession in the CA1 region that does not depend on pre-cisely timed inputs, but instead the phase is a resultof the total amount of depolarization of the place cells.In this model, the total amount of phase shift is muchless than 360 degrees.

In this article, we describe a minimal biophysicaltemporal model for phase precession in the CA3 regionof the hippocampus. It differs from prior models inthat (1) it generates the spatial correlation of the phaseprecession from a single spatial input and from infor-mation about the animal’s running speed; (2) it doesnot require phase precession in the upstream projec-tion cells; (3) it does not require a spatially dependentdepolarization of place cells; (4) it provides a mecha-nism for determining the end of the place field; (5) itprovides a mechanism for up to 360 degrees of phaseshift; and (6) it is consistent with associative memorymodels for the CA3 region.

Alternatively, our model does not include a mecha-nism to account for the firing rate of place cells. Theactivity of place cells includes both rate and timing in-formation. In the present model, we are only concernedwith the timing properties.

This minimal model, which is composed of two neu-rons (one pyramidal cell and one interneuron) and apacemaker input, can explain the onset, occurrence,and end-of-phase precession. The network has two im-portant dynamic patterns. The first is a stable attractingstate, which mimics the behavior of CA3 outside of theplace field. The second dynamic pattern is a transientstate that encodes the behavior of CA3 within the placefield. The main difference between the two states is theinput that controls the firing pattern of the interneuron.In the stable state, the pacemaker input controls in-terneuron firing, while in the transient state, excitationfrom the pyramidal cell does. The seed from the den-tate gyrus switches control from the pacemaker to theplace cell to initiate phase precession, and the durationof phase precession is determined by the duration of thetransient dynamics as the network returns to the stablestate.

The transient place cell firing in the place field is atemporal process in which the phase of firing of the cellin each cycle of the theta rhythm strictly depends onthe phase in the prior cycle of the theta rhythm. Thisis in contrast to all other models of phase precessionin which the phase of firing of place cells depends onthe external inputs arriving at that cycle and not onthe phase in other cycles of the theta rhythm. For thisreason, to account for the spatial correlation of 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).

Phase Precession of Hippocampal Place Cells 9

T projects only toI through fast, GABA-mediatedinhibition.

Our analysis does not depend on a specific model forthe individual neurons; hence, we use a general form forthe model equations for each cell. The few restrictionswe require on the model properties (outlined below) aresatisfied by a large class of neural oscillators, includingthe Morris-Lecar (Morris and Lecar, 1981) equations,which have been successfully used in numerous mod-eling studies (Rinzel and Ermentrout, 1997; Somersand Kopell, 1993; Terman et al., 1998; Terman andLee, 1997), and which we use for our simulations (seeAppendix B).

TheP cell is a general neural oscillator, modeled byequations of the following form:

v′p= f p(vp, wp)

w′p= εgp(vp, wp),(1)

where′ denotes the derivative with respect to timet .The variablevp represents the membrane voltage ofthe place cell and the functionf p is composed of thesum of the ionic and leakage currents found in the cellas well as an applied current. We considerP to havebursting behavior and the variablevp to model the burstenvelope and not to account for individual, fast spikesoccurring during the active phase of the burst. In thisgeneral class of neural models, the variablewp mod-els the activation gating variable of an outward, ioniccurrent, usually potassium-mediated, and the functiongp governs the activation gate dynamics of this con-ductance. The parameterε controls the time scale ofthewp dynamics. Our analysis assumes thatε is small(ε ¿ 1), so that this general neural oscillator behavesas a relaxation oscillator.

The model for the interneuronI also has a generalform, but we considerI to be an excitable cell, ratherthan oscillatory. The model equations are

v′i= fi (vi , wi )

w′i= εgi (vi , wi ),(2)

wherevi represents the membrane voltage ofI andwi

represents the activation gating variable of an outward,ionic current. As described below in terms of its phaseplane, we require that it is able to fire in response toeither excitation received fromP or inhibition receivedfrom T .

The equations for the pacemakerT are completelygeneral. In fact, we simply assume thatT oscillates

between 0 and 1, with a periodTT in the theta range.When the value ofT exceeds a prescribed threshold,then we assume thatT fires, and we measure the dura-tion of its “spike” as the time spent above the threshold.

For the network model, appropriate synaptic currentsare added to the individual cell models. In the synap-tic current variables and parameters, the first subscriptdenotes the presynaptic cell and the second subscriptdenotes the postsynaptic cell. The network equationsfor P are

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

w′p = εgp(vp, wp).(3)

For I , the network equations are

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 synaptic currents has been used inother modeling studies (Ermentrout and Kopell, 1998;Wang and Rinzel, 1992). The variablessip, spi , andsti

govern the dynamics of the synaptic currents, and eachsatisfies an equation of the form

s′ = αH∞(v − vθ )(1− s)− βs, (5)

whereα and β are the rise and decay rates of thesynapse, andvθ is the synaptic activation threshold.The Heaviside step functionH∞ acts as an activationswitch for the synapse;H∞(v− vθ ) is 0 if v <vθ andis 1 if v≥ vθ . The parametersgpi , gti , andgip are themaximal conductances of these synaptic currents, andvpi , vt i , andvi p are the reversal potentials. In particu-lar, vi p andvt i are set to make the associated currentsoutward, whilevpi is set to make the synaptic currentinward. Finally, the parametersσi p andσpi representa synaptic delay fromI to P andP to I , respectively.Without loss of generality, we assume there is no delaybetweenT and I .

2.2. Phase Plane Methods

In this section, we describe the phase plane methodswe use to analyze the model. We first explain how thebehavior of each cell when isolated from the networkcan be studied in the phase plane and then describethe effect of network interactions. An important aspect

10 Bose et al.

of our analysis that is worth mentioning at this initialstage is that we often compare behavior of uncoupledand coupled versions ofP. By the former, we meanthe intrinsic P cell oscillation in isolation from thenetwork. By the latter, we mean the oscillation of thesameP cell when it must now interact withT and Iin the network. One of the important conclusions ofour analysis is that the network dynamics can evolvein dramatically different ways, depending on the firingsequence of the cells.

The oscillatory behavior of the isolatedP cell can beanalyzed in thevp−wp phase plane where the trajec-tory of the cell is determined by the nullclines of eachequation (see Fig. 3). Thevp-nullcline, denotedCP,is a cubic-shaped or N-shaped curve found by solvingf p(vp, wp)= 0 forwp. Thewp-nullcline, denotedDP,is a sigmoidal-shaped, nondecreasing curve found bysolving gp(vp, wp)= 0. To ensure that the isolatedPcell is oscillatory, parameters in the functionsf p andgp

are chosen so thatCP andDP intersect uniquely alongthe middle branch ofCP. As in prior models (Somersand Kopell, 1993; Terman and Lee, 1997; Terman et al.,1998), we require that the following condition ongp

holds near the left branch ofCP:

∂gp

∂vp> 0. (6)

Following standard analysis of relaxation oscillators,in the limit ε= 0, the burst trajectory of theP cell canbe traced in the phase plane. As displayed in Fig. 3,the silent phase of the burst occurs as the trajectorytravels down the left-hand branch of the cubic-shapednullclineCP, while during the active phase of the burst,the trajectory travels up the right-hand branch ofCP.The transitions between the active and silent phases ofthe oscillation occur when the trajectory reaches theknees of the nullcline. One cycle of the burst oscil-lation is obtained by tracing along the four different“pieces” of the trajectory. Specifically, settingε= 0 in(1), we obtain the following, fast equations governingthe transitions of theP cell between its active and silentstates:

v′p = f p(vp, wp)

w′p = 0.(7)

In this case,wp acts as a parameter in thevp equa-tion, which is simply a scalar equation and thus easyto analyze. The equations governing the slow variationof vp during its active and silent states are obtained

by rescaling time usingτ = εt in (1) and then settingε= 0:

0 = f p(vp, wp)

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

where˙ denotes the derivative with respect to the slowtime variableτ . To mathematically construct the os-cillation cycle, we denote the left and right knees ofthe nullclineCP by (vLK, wLK) and(vRK, wRK) respec-tively. Results by Mishchenko and Rozov (1980) implythat forε sufficiently small, an actual periodic orbit liesO(ε) close to a so-called singular periodic orbit. Thesingular periodic orbit consists of four pieces, all ofwhich are constructed withε= 0. The first is a solutionof (7) that connects(vLK, wLK) to some point(vR, wLK)

on the right branch ofCP (rising edge of burst). Thenext piece is a solution of (8) that connects(vR, wLK)

to (vRK, wRK) (active phase of burst). The third piece isa solution of (7) that connects(vRK, wRK) to (vL , wRK)

(falling edge of burst). The final piece is a solution of(8) that connects(vL , wRK) back to(vLK, wLK) (silentphase). Note that with respect to the slow time scaleτ , the jumps between the active and silent states of theburst are instantaneous. We denote the period of thesingular periodic orbit byTP.

The nullclines for the model interneuron with ex-citable behavior have the same qualitative shape asthose for theP cell, except model parameters are ad-justed so that the cubic-shapedvi -nullcline, denotedCI , and the sigmoidalwi -nullcline, DI , intersect alongthe left branch ofCI . The intersection point of the null-clines is a stable fixed point along the left-hand restingbranch ofCI , that ensures that the isolatedI cell isunable to oscillate on its own. This change in the null-clines can be obtained, for example, by shifting thehalf-activation voltage of the outward current to lowervoltage levels. A property we assume for the intersec-tion point of the nullclines is that this point be closeto the left knee of the nullclineCI , thus allowing theinterneuron to fire either in response to excitation fromtheP cell or via rebound following inhibition from thepacemakerT . As with P, we assume that near the leftbranch ofCI ,

∂gi

∂vi> 0. (9)

In our geometric analysis in the phase plane, thegeneral effect of introducing a synaptic current is toeither raise or lower the cubic-shaped nullcline of

Phase Precession of Hippocampal Place Cells 11

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12 Bose et al.

the postsynaptic cell without qualitatively changingits shape, as long as the maximum synaptic conduc-tance is not too large relative to the maximum ionicconductances. Since (5) contains no factor ofε, thesynapses turn on and off on the fast time scale gov-erned byt ; thus they are instantaneous with respectto the slow flow of the trajectories governed byτ .To examine the effects of the synaptic currents on theP cell in our network model, we denote the solutioncurve of f p(vp, wp)− sip(t − σi p)gip[vp− vi p] = 0by CPI . Since P receives inhibition,CPI lies belowCP (see Fig. 3). We assume that the nullclineDP andCPI intersect somewhere along the left branch ofCPI .This intersection point will be a stable fixed point forthe inhibitedP cell, with the result that the inhibitedPcell will not spontaneously oscillate. Further note thatthis intersection point will lie below the left knee ofCP

because of our restriction (6)∂gp

∂vp> 0 (see Somers and

Kopell, 1993).The effect onI of the excitatory synaptic current

from P and the inhibitory synaptic current fromT isthat its cubic-shaped nullcline can be raised or lowereddepending on the activation ofspi andsti . For any pos-sible case, we assume that thewi -nullcline DI contin-ues to intersect the synaptically perturbedvi -nullclinesalong their left branches, thus ensuring thatI remainsexcitable. As noted previously, we require thatI beable to fire due to either excitation received fromPor via rebound from inhibition received fromT . Forexample, assume thatI is sufficiently close to the inter-section point of the nullclines when it receives synap-tic input. If the input is excitatory fromP, the cubic-shaped nullcline of theI cell is instantaneously raisedreleasing the trajectory from its left branch. The tra-jectory is then attracted to the right branch of the raisednullcline, thus initiating an action potential. If the inputis inhibitory fromT , the cubic-shaped nullcline is low-ered and the trajectory moves left toward the new stablefixed point at the intersection of the perturbed nullclineandDI . For sufficiently strong inhibition, because ofour restriction (9), the new fixed point will lie belowthe left knee ofCI . When the synaptic inhibition shutsoff, the nullcline rises back again towardCI , and thetrajectory is attracted to the right branch, initiating anaction potential via postinhibitory rebound. The abilityof the I cell to fire due to these two different stimuli iscritical to our analysis.

The mathematical analysis for this article, as wellas the specific equations used in the simulations, canbe found in Appendices A and B, respectively. We

work throughout in the appropriateε= 0 limit and thenuse results of Mischenko and Rozov (1980) to obtaintheε small result. Our proofs depend on the use of atime metric that allows us to define times between cellsand also times over which relevant behavior occurs. Itturns out that we can relate many of the times associ-ated with the network model back to times associatedwith the isolated cells.

3. Results

3.1. System Properties

We first present an overview of our results that providesa qualitative description of how and why the modelgenerates phase precession. In the second part of thissection, we provide the analysis of our model and spe-cific analytical results.

There exist two different, important firing patternsfor the simple network. The first firing pattern repre-sents network behavior when the rat is outside of theplace field. The second represents network behaviorwhen the rat is inside the place field. The differencebetween these two behaviors is directly correlated towhetherT or P controls the firing ofI . An importantaspect of our analysis is to identify mechanisms that al-low the control of firing to be shifted betweenT andP.As described below, the first change in control fromTto P requires a brief external input to the network oc-curring when the rat first enters the place field. Thesecond change in control fromP back toT is deter-mined internally by the network and signals that theplace field has ended.

The firing pattern representing out-of-place field be-havior is a stable periodic state that we call theTIPorbit. In this stable pattern, the pacemakerT controlsthe firing of I , which fires via postinhibitory rebound.In turn, I modulates the firing ofP, so that it firesphase-locked to the underlying theta rhythm. The ef-fect of the inhibition fromI is to either delay or ad-vance the firing ofP, depending on where inP’s cyclethe inhibition occurs. Thus,P is phase-locked to thetheta rhythm and does not phase precess whether itsintrinsic frequency is higher or lower than theta (seeSection 3.2.1). Moreover,P can only fire when re-leased from inhibition fromI .

Simulation results from the simple network whereeach cell is modeled with Morris-Lecar equations areshown in Fig. 4. During the first five bursts of the P cell(top trace), the network is in the stableTIP orbit. The

Phase Precession of Hippocampal Place Cells 13

Figure 4. TIP andPIT orbits for the two cell and pacemaker net-work with each cell modeled by Morris-Lecar equations. TheT spikeis not explicitly shown and is denoted by dashed vertical lines. Thenetwork oscillates in theTIP orbit for the first 5 cycles. ThePITorbit is initiated by the arrival of the memory seed (heavy arrow at535 msec, which lasts for 3 msec) and phase precession occurs overthe next 7 cycles (under heavy bar). The network returns to the stableTIP orbit for the remainder of the simulation. The model equationsand parameter values are given in Appendix B.

sequence of cell firing withT firing first (not explic-itly shown; time of peak indicated by dotted verticallines), followed byI (lower trace) and thenP is evi-dent. During theT spike, I is inhibited as seen by thehyperpolarization in theI voltage trace immediatelyfollowing the peak ofT . During theI spike (after itsrelease from inhibition fromT), P is inhibited as seenby the hyperpolarization in theP voltage trace imme-diately preceding the spike.

Clearly, the out-of-place field firing rate in theTIPorbit is higher than experimentally observed. This highout-of-place field firing leads to a loss of spatial speci-ficity to a downstream detector of firing rate but hasminimal effect on a downstream detector that uses thephase of firing to determine the spatial location of theanimal. The simplest way to reduce or remove the out-of-place field firing is to have a spatial inhibitory signalthat provides inhibition at all points outside of the placefield. This approach is not taken here because it relieson spatial information upstream from the hippocampusabout the size and extent of the place field. The goal of

this model is to generate the spatial phase correlationfrom less spatially specific upstream information. Fol-lowing the description of the properties of the simplemodel, we show in Section 3.2.7 how to completelysuppress the out-of-place field firing, without resortingto additional upstream spatial information (see Fig. 9).

The second firing pattern, representing behaviorwithin the place field, is a transient state that we callthePIT orbit. ThePIT orbit is initiated by a brief, soli-tary dose of excitation toP, representing a memoryseed externally provided from the dentate gyrus. Theseed causesP to fire earlier than it would have in theTIP orbit and thereby allowsP to seize control ofI ’sfiring. TheP cell is able to fire via its intrinsic oscilla-tory mechanisms for as long as it controlsI . Thus, theseed has the effect of switching control ofI from T toP. During thePIT orbit, the period ofP firing, TP, isless than the period ofT , TT , regardless of the intrinsicperiod of P, provided thatTP andTT are not too dis-parate. The reason phase precession occurs in thePITorbit is somewhat subtle and depends both on the intrin-sic periods ofP andT and also on the time duration ofinhibition from I to P. If this inhibition is short-lasting,then duringPIT, P fires at its intrinsic frequency. Thusif TP < TT , it is clear that precession will occur. Al-ternatively, if the inhibition is long-lasting, then duringPIT, P fires at a substantially faster rate than its in-trinsic frequency. Counterintuitively, this increase infiring rate is most strongly dependent on the strengthof the inhibitory synaptic conductance fromI to P. InSection 3.2.2, we clarify how inhibition may speed upthe firing rate ofP. For this case, precession occurswhether the intrinsic periodTP is less than, greater thanor equal toTT . In either case, sinceP controls the fir-ing of I in thePIT orbit, forcing I to fire with everyPburst,I also phase precesses.

Figure 4 illustrates thePIT orbit and phase preces-sion for the Morris-Lecar model network. Arrival ofthe memory seed toP is modeled by a brief, excita-tory, applied current pulse (heavy arrow) that causesthe early firing ofP and the interruption of theTIPorbit. Due to the excitation fromP, I overcomes theinhibition fromT and fires withP, as seen by the slighthyperpolarization in the peak of theP burst. Over thenext six cycles (under the heavy bar),P and I phaseprecess relative toT . The firing of P due to intrinsicoscillatory mechanisms is evidenced by the smooth riseto threshold displayed by its voltage trace.

The phase precession ofI provides an internal mech-anism that returns the network to theTIPorbit. Namely,

14 Bose et al.

in PIT, I continues to receive inhibition with eachTspike, but the timing or phasing of the inhibition is suchthat postinhibitory rebound does not occur at each cy-cle. For example, during the first two spikes of phaseprecession, the inhibition fromT arrives during theIspike, as seen by the lower spike heights, and rebounddoes not occur. Over the next three cycles,T inhibitsI during early portions of its afterhyperpolarizationwhen rebound is not possible. But by the last spikeof phase precession,I is inhibited sufficiently late inits afterhyperpolarization, resulting in a rebound spikethat interrupts thePIT orbit, returning the network tothe stableTIP orbit. The recapture ofI by T signalsthe end of the place field and occurs whenP andI haveprecessed through up to 360 degrees of phase, which isconsistent with experimental data (O’Keefe and Recce,1993). Thus, the precession ofI is necessary fordetermining the end of the place field.

The brief dose of excitation in the form of the mem-ory seed reorganizes the functional roles of the cells intheTIPandPIT orbits. Namely, the excitation switchesthe control ofI from T to P. T is able to regain controlof I only whenP and I have precessed through up to360 degrees. Note that this functional reorganizationoccurs without any changes in the coupling or intrin-sic properties of the cells. We emphasize that, in themodel, phase precession is a transient phenomena thatceases with no further input to the network. Thus, thelength of time over which the pyramidal cell precesses,or alternatively, the length in space over which preces-sion occurs, completely determines the spatial extentof the place field. In this way, the network generatesthe temporal code embodied in phase precession.

In the following analytic results section, we providemathematical justification for the qualitative observa-tions described here (see Table 1 for a list of relevantsymbols). In particular, in Section 3.2.1, we discuss theexistence of theTIP orbit. In Sections 3.2.2 to 3.2.5,we focus on thePIT orbit and determine parameters

Table 1. List of relevant symbols.

TT = intrinsic period of the pacemaker input CP = intrinsic P cell cubic

TP = intrinsic period of theP cell CI = intrinsic I cell cubic

Ä= TT − TP CPI = P cell cubic withsip = 1

TPC= period ofP cell in TIP network CI P = I cell cubic withspi = 1

TP = period ofP cell in PIT network CIT = I cell cubic withsti = 1.

τTon = duration of the inhibition fromT to I τIon = duration of inhibition fromI to P

τP A= duration of the active state of the intrinsicP cell τP R= duration of the silent phase of the intrinsicP cell

σi p = synaptic delay fromI to P σpi = synaptic delay fromP to I

Note: All times are measured atε= 0 in slow time scaleτ .

that most strongly affect this network behavior. InSection 3.2.6, we show how experimentally supporteddata relating running speed to interburst frequency ofthe pyramidal cells can be used to achieve a spatial cor-relation in this temporal model. Finally in Section 3.2.7,we introduce a modification to the simple network thatsuppresses out-of-place field firing.

3.2. Analysis

3.2.1. The Network Oscillates at Theta FrequencyOutside of the Place Field. The trajectories ofP andI in theTIP orbit can be traced along nullclines in theirphase planes. We may assume that atτ = 0, T fires,the trajectory ofP lies on the left branch ofCP withwp(0) = w? and the trajectory ofI lies somewherealong the left branch ofCI . In response to the inhibi-tion received fromT , I falls back toCIT and is releasedfrom inhibition atτ = τTon. We assume thatwi (0) issuch thatwi (τTon) lies below the left knee ofCI . Thusat τ = τTon, I fires by postinhibitory rebound. Allow-ing for synaptic delay fromI to P, atτ = τTon+σi p, Pfeels inhibition fromI and as a result falls back toCPI .At τ = τTon + σi p + τIon, P is released from inhibition.We show in Appendix A thatw? can be chosen suchthatwp(τTon + σi p + τIon) < wLK . With this condition,when P is released from inhibition, it will fire. De-pending on the position of the trajectory ofP on theleft branch ofCP when the inhibition fromI arrives, atτ = τTon+ σi p, the firing of P has either been delayedor advanced relative to its intrinsic frequency. We alsoshow in Appendix A that whenT fires again atτ = TT ,wp(TT ) = w?, thus ensuring that this cycle of firingrepeats, and we show that theTIP orbit is stable. Notethat for P to fire, it must be released from inhibition;it is unable to take advantage of the fact that it is anoscillatory cell.

In Fig. 5, we superimpose the phase portrait of thecoupledP cell onto that of the uncoupledP cell to

Phase Precession of Hippocampal Place Cells 15

Fig

ure

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16 Bose et al.

clarify the differences in their trajectories. In the fig-ure, we have used lowercase letters to denote timesthat these cells spend in parts of their orbit where theirtrajectories differ. The letters with bar superscripts cor-respond to the coupledP cell. Note thatTPC − TP =c+ d − (b− b). SinceTPC = TT , a necessary con-dition for theTIP orbit to exist is that the difference inintrinsic periods ofP andT ,Ä = TT−TP, must satisfy

Ä = c+ d − (b− b). (10)

The differenceb− b> 0 is related to the magnitude of∂gp

∂vp. Since∂gp

∂vp> 0, cells evolve through the same Eu-

clidean distance onCP more slowly than onCPI . Thusb− b measures the additional time that the uncoupledP cell must spend to cover the same Euclidean dis-tance along the left branch ofCP that the coupledPcell covers onCPI .

In the case whenÄ>0, a stableTIPorbit is obtainedif c+ d> b− b. This can be achieved by choosingwp(τTon+ σi p), and hencew?, such that the trajectoryof P enters a neighborhood of the stable fixed pointat the intersection ofCPI and DP within the durationof τIon. The trajectory ofP will remain near the fixedpoint until P is released from inhibition. In this way,the firing of P is delayed andP can be phase-lockedto the theta rhythm.

When Ä<0, a stableTIP orbit is achieved ifb− b > c+ d. In this case,wp(τTon+ σi p), and hencew?, are chosen so that the trajectory ofP remains suf-ficiently far from the fixed point at the intersection ofCPI and DP during τIon. The trajectory displayed inFig. 5 is representative of this case. Since cells evolvefaster onCPI than onCP, the timeb+ c+ d can beshorter than the timeb. This shows that inhibition canbe used to speed upP. Thus,P can be phase-lockedto the theta rhythm ifTP is greater thanTT , as long asthe periods are not too disparate.

Observe that (10) can also be rewritten as

Ä = τIon + d − b. (11)

For fixedÄ, bothbandd may increase asτIon increases.An increase inb means thatP must feel inhibition athigher values ofwP along the left branch ofCP. In turn,this means that inTIP the phase difference betweenTandP will be greater. Hence, the durationτIon controlsthe phase difference betweenT andP in theTIP orbit.

In summary, theTIP orbit can be obtained whetherthe differenceÄ= TT − TP is positive, negative, orzero. The effect of the inhibition fromI to P depends

not on its durationτIon but on its timing during the cycleof P. We note that while theTIP orbit can be achievedwith either positive or negativeÄ, independent of thedurationτIon, other results of our model—namely, ob-taining thePIT orbit—require that for a short-lastingτIon,Ä must be positive.

3.2.2. Within the Place Field, Both P and I PhasePrecess. Similar to above, we analyze thePIT orbitby describing the trajectories ofP andI in their phaseplanes. In the following analysis, we assume that nearthe right branch ofCP, ∂gp

∂vp= 0. Thus the speed at which

P evolves along the right-hand branches ofCP andCPI

will be the same. Our analysis continues to hold for∂gp

∂vp> 0 but sufficiently small. We also assume that near

the right branch ofCI ,∂gi

∂vp= 0.

For simplicity, we assume that the seed arrives atτ = 0, so that bothT and P fire simultaneously. Thetrajectory ofP jumps to the right branch ofCP. The Icell receives inhibition fromT during the time interval(0, τTon), but, atτ = σpi , I fires due to excitation fromP. At τ = σpi + σi p, P receives inhibition fromI andfalls to the right branch ofCPI . What happens nextis most strongly determined byτIon, the time lengthover whichP feels inhibition fromI . The timeτIon,however, is hard to define precisely as it changes duringeach cycle of the phase precession (see Appendix A forclarifications of this definition). We describe how thePIT orbit is obtained in the cases whenτIon is short andwhen it is long relative to the duration of the active stateof P.

First consider the case whenτIon is short, in partic-ular, τIon <τP A, the duration of the active state of theuncoupledP cell. At τ = σpi + σi p, let the distancealongCPI from the position of the trajectory ofP tothe right knee be greater thanτIon. Then the inhibi-tion to P shuts off beforeP jumps down from the rightknee ofCPI . So atτ = σpi + σi p + τIon, P returns to theright branch ofCP and leaves the active state throughthe right knee of its intrinsic cubicCP. In Fig. 6A, wehave drawn the trajectory thatP follows one cycle af-ter the seed is given. It follows this trajectory until thetime at which the network returns to theTIP orbit. Wedenote byTP the period ofP during phase precession.In this case,TP = TP. Thus, a necessary condition forphase precession in this scenario isTP < TT orÄ>0.Here, the sole factor that drives the precession is thedifferenceÄ in the intrinsic periods ofP andT .

It is instructive to contrast the phase portrait ofI withthat of P duringPIT. As opposed to the clean and

Phase Precession of Hippocampal Place Cells 17

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18 Bose et al.

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Phase Precession of Hippocampal Place Cells 19

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20 Bose et al.

unchanging trajectory ofP, the trajectory ofI changeswith each cycle of the precession. As we noticed inFig. 4, the inhibition fromT to I occurs at progressivelyearlier phases duringPIT. In terms of the phase portrait(see Fig. 6B), this means that the inhibition fromT to Ioccurs at different places in phase space. In particular,during the first cycle,T inhibits I when I is on theright branch ofCI P and still far away from the rightknee. At the next cycle,T inhibits I while I is stillon the right branch but now closer to the right kneeat a higherwi value. Progressing in this manner, theinhibition fromT eventually occurs whenI is back onthe left branch but still refractory. Ultimately,I receivesinhibition when it is close enough to the left knee ofCI to be captured intoTIP, thus ending precession andsignaling the end of the place field.

Phase precession occurs for a more complicated rea-son whenτIon >τP A. In this case, we assume thatPreaches the right knee ofCPI before the inhibition fromI shuts off. Note that the right knee ofCPI lies belowthe right knee ofCP. So the length of timeP stays in itsactive state during thePIT orbit is less than the length oftime it stays in its active state during theTIPorbit. Thisdecrease in time is the primary reason why, in this case,phase precession occurs. This can be seen by consid-ering the phase plane ofP. There are several subcasesdepending on the time length ofτIon (see Appendix Afor details). In Fig. 6C, we have drawn the trajectorythat P follows one cycle after the seed is given forone of these subcases. Letτ = τ1 be the timeP hits theright knee ofCPI . ThenP jumps back to the left branchof CPI and evolves downCPI until τ = τ1+ σpi + σi p.We have denoted by lowercase letters various times ofevolution for the uncoupledP cell and the coupled pre-cessingP cell. Note thatTP − TP = kr + kl +m− m.There are two sources of time reduction. The larger ofthe two is the timekr + kl . This time is related to themaximum conductance for the synaptic current fromIto P, gip. If gip is small, then the cubicCPI will notbe too far belowCP, and thus the timekr + kl is small.As gip increases, so doeskr + kl . The second source oftime reduction, and thus phase precession, arises dueto the differencem− m. This difference is controlledby the magnitude of∂gP

∂vpnear the left branch ofCP. For

this case, phase precession in thePIT orbit is achievedwhetherÄ is positive, negative, or zero.

3.2.3. The Memory Seed Initiates Phase Precession.Granule cells in the dentate gyrus make excitatorysynapses on CA3 pyramidal cells, and granule cells

have a spatially specific (Jung and McNaughton, 1993)and theta rhythm phase specific (Skaggs et al., 1996)firing pattern. In our model, the arrival time of theexcitatory input, representing the memory seed fromthese granule cells, need not be so precise. In fact,there exists an open interval of potential seed timings(or phases) that result in phase precession. The seedmust be timed to arrive whenP is sufficiently closeto the left knee ofCP. To determine an expressionfor the appropriate seed arrival timeτseed, let T fireat τ = 0 and assume that the duration of the seed issmall. The seed excitation momentarily adds a cur-rent of magnitudegdg[vp− vdg] to the current balanceequation for theP cell, wheregdg is the maximal con-ductance. Phase precession occurs if the seed arrives atτseed∈ (−δ, τTon+ σi p) whereδ >0. The lower bound−δ is related to the parametergdg. For larger values ofgdg,−δ can be larger (in magnitude). The upper boundon τseed is a sufficient condition for phase precessionsince theP cell will certainly precess if it is excitedbefore it receives inhibition fromI . As this intervalfor τseedgives only a sufficient condition, seeds that ar-rive slightly afterτTon+ σi p may actually produce phaseprecession.

3.2.4. The Total Amount of Phase Precession Dependson the Phase of Memory Seed Arrival.In additionto determining whether phase precession is initiated,the timing or phase of the memory seed also affectsthe total amount of phase change in theP cell dur-ing its precession. Since bothP and I phase precessand since, in theTIP orbit, both cells fire at differentfixed phases afterT , we need to consider the amountof precession for each of these cells separately.

First considerP. If T fires atτ = 0, then duringthe TIP orbit, P fires atτ = τTon + σi p + τIon. Wecan translate these times of firing to phases of firingby multiplying the times by 360◦/TT . The earlier theseed arrives, with respect to the time ofP firing, thesmaller the amount thatP precesses. Letτadv= τTon+σi p + τIon − τseedbe the amount of time (or phase) thefiring of P is advanced by the memory seed. The totalamount of phase precession is a decreasing functionof τadv. In particular, if the seed arrives in the interval(−δ, 0) (that is, if τadv is large),P precesses throughless than 360 degrees. Whereas if the seed arrives closetoτ = τTon+ σi p (τadvsmall),P precesses through closeto 360 degrees, provided that the duration of inhibitionfrom I to P, τIon, is sufficiently short. Theoretically, theP cell can only phase precess through a full 360 degrees

Phase Precession of Hippocampal Place Cells 21

Figure 7. Amount of phase traversed by theP cell during phaseprecession in the Morris-Lecar-based network model. When zerophase is defined by the peak of theT cell spike,P cell fires (bursttimes indicated by circles, filled circles indicate Fig. 4 results) at152 degrees duringTIP (first five bursts). Seeds arriving earlier (firstseed at 500 msec) result in less phase precession than seeds arrivinglater (last seed at 550 msec). Note also that the number of cycles ofphase precession increases with later seed arrival.

in the limit τseed→ τTon+ σi p + τIon from below. ForτIon large, such a seed would fall well outside of theinterval described in Section 3.2.3 and thus would beunlikely to produce phase precession. Thus, a lowerbound on the maximal amount of phase precession is360◦(1− τIon

TT).

An example of the effect of the seed timing on theamount of phase precession is shown in Fig. 7 for theMorris-Lecar-based network model. In the figure, thefilled circles indicate the phase of eachP cell burstduring theTIP andPIT orbits displayed in Fig. 4. Dur-ing TIP, P fires at approximately 152 degrees (dashedhorizontal lines) when 0 degrees is defined by theTspike. The memory seed advancesP cell firing to ap-proximately 77 degrees and thenP precesses through285 degrees until returning to theTIP orbit and re-suming firing at 152 degrees. The open circles indicatethe phase ofP firing when the seed arrives at differ-ent times. For the earliest seed arrival,P precessesthrough approximately 170 degrees, and for the latestseed,P precesses through roughly 340 degrees.

In contrast to the restriction on the amount ofphase precession of theP cell, the I cell can pre-cess through the full 360 degrees. This is possible ifτseed∈ (τTon− σpi , τTon+ σi p). The lower bound occursif the seed arrives at such a time so that the excitationfrom P to I arrives exactly at the momentT releasesI from inhibition.

3.2.5. The Number of Cycles of Phase Precession De-pends on the Phasing of the Memory Seed and on theDuration of the Inhibition from I to P. There aretwo possible formulae for the number of cycles of pre-cession. The duration of inhibition fromI to P, τIon,determines which of the two formulae applies to a givensituation. In the caseÄ>0 (TP < TT ), either formulamay apply withτIon actually determining which one touse. IfÄ<0 (TP > TT ), then only the second formulacan be used.

If τIon <τP A, recall from Section 3.2.2, that thePITorbit exists only forÄ>0. The number of cycles isgiven simply by

n = TT − τadv

Ä, (12)

wheren is the nearest integer to the ratio value. Here,it is the difference between the intrinsic periods ofTand P, together withτadv, that determines the num-ber of cycles of precession. The network model thatdisplayed the results in Figs. 4 and 7 generates aPITorbit by satisfying these conditions onÄ andτIon andthe number of cycles of phase precession is accuratelypredicted by (12), as shown in Table 2. The valuesTT = 100.3 ms andTP = 87.4 ms were used.

If Ä is negative withτIon <τP A, then after the seed,(12) predicts and simulations corroborate (not shown)that theP and I cells phaserecessback to theTIPperiodic orbit.

If τIon > τP A, as described in Section 3.2.2, there isno requirement on the sign ofÄ= TT − TP to obtainthe PIT orbit. In this case, the number of cycles ofphase precession is given by the following expressionwhere the times are defined in Fig. 6C:

n = TT − τadv

Ä+ kr + kl +m− m. (13)

Recall that the quantitykr + kl +m− m is positive,which compensates for a potentially negativeÄ. The

Table 2. Number of cycles of phase precession predictedby Eq. (12) and observed in simulations of Morris-Lecar-based network (parameters as in Figs. 4 and 7) for differentarrival times of the memory seed.

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 Bose et al.

time m− m is related to the magnitude of∂gp

∂vpnear

the left branch ofCP. The magnitude of the timekr + kl is of the order ofgip and arises because the burstwidth and interburst interval are longer for the isolatedP cell than in thePIT orbit. We ran several simula-tions (not shown) of the Morris-Lecar-based networkmodel withTP > TT (specifically,TT = 100.3 ms andTP = 102 ms). Withgip = 1.0, we foundn= 18, andfor gip = 1.5, n= 9, thus corroborating the strong de-pendence of the number of cycles of phase precessionon gip.

Reiterating, ifÄ > 0, either (12) or (13) may holddepending on the duration of the inhibition fromI to P.By continuous dependence on parameters, there ex-ists an intermediate value forτIon at which the networkswitches from (12) to (13). Note that near this switch-ing point, small changes in the active duration ofIhave large ramifications with respect to the occurrenceof precession and to the number of cycles of precession.

3.2.6. The Spatial Correlation of Phase PrecessionIs a Speed-Corrected Temporal Phenomena.Othermodels of phase precession rely explicitly on the as-sumption that phase precession is a spatial phenomena.We show here that the apparent spatial dependence ofphase precession can be accounted for by our tempo-ral model. More precisely, if the interburst frequencyof the place cell and the frequency of the underlyingtheta rhythm are linear functions of the animal’s speed,then phase precession can be a temporal phenomenonand also be more correlated with the animal’s locationthan with the time that has passed since it entered theplace field.

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

are positive constants. The velocityv is set tov0+1v,wherev0 is the minimum velocity needed for the thetarhythm to exist, and1v is the positive deviation fromthis baseline velocity. Without loss of generality, thefrequency fB is a common baseline for1v= 0. Letγ = γ2− γ1, then f P = fT + γ v, whereγ >0. Thetime of thenth theta cycle peak isnTT , and the time ofthenth P cell burst,τn, is nTP = nTT

1+ TTγ v. The amount

of phase shift after thenth burstφn is the differencenTT − nTP divided by the period of the theta rhythm;thusφn= 360◦ nTTγ v

1+ TTγ v. The position of the rat at the

time of thenth burst isxn= vτn= vnTT1+ TTγ v

. By in-spection,φn= 360◦γ xn, or equivalently, the phase isspatially correlated.

Figure 8. Phase shift ofP firing during each cycle of precessionchanges linearly with changes in animal’s running speed, modeledby changing intrinsic frequency ofP. In Morris-Lecar-based net-work model, phase difference ofP bursts during precession (circles,filled circle corresponds to Fig. 4 results) increased linearly as in-trinsic period ofP was decreased, corresponding to an increase inÄ

(applied current toP, I p= 95, 98, 100, 103 and 105µA/cm2). TheperiodTT = 100 ms was fixed throughout.

The phase shift at each cycle is given by1φ =360◦ ÄTT

. Thus the phase shift is a linear function ofthe difference in period, and for fixedTT it is a linearfunction of TP. This linear relationship holds in ourMorris-Lecar-based network model as shown in Fig. 8.The filled circle showing the largest phase shift dur-ing each cycle of phase precession corresponds to theparameter values in Figs. 4 and 7. To model a de-crease in running speed, the intrinsic frequency of theP cell was decreased. In response to this increase inTP, corresponding to a decrease inÄ, the phase shift1φ decreases linearly.

3.2.7. Suppression of Out-of-Place Field Firing.The primary objective of our model is to illustrate amechanism for generating the phase precession of placecells. In the model, phase precession only occurs in theplace field, but the out-of-place field firing is too high.This implies that a downstream phase-based detectorcould precisely determine the animal’s location at allphases except for the phase-locked one.

Different complicated features could be added to themodel to reduce or remove the out-of-place field fir-ing. For example, there are many types of interneu-rons in the CA3 region that have a diverse patternof connections and an incompletely determined rolein network activity. In particular, some of the in-terneurons make synaptic connections only on otherinterneurons (Freund and Buzs´aki, 1996). By introduc-ing one additional interneuron into our current model,

Phase Precession of Hippocampal Place Cells 23

Figure 9. The voltage traces for the three cell and pacemaker net-work. Phase precession begins att roughly equal to 500 msec withthe arrival of the memory seed. The cellsP andI2 then phase precessfor the next 6 cycles. Outside of the place field, the longer lastinginhibition from I1, which is periodically reinforced, forcesP to staybelow threshold.

the out-of-place field firing can be suppressed as shownin Fig. 9. In this more complex modelT andP are asbefore, but now there exist two interneurons,I1 andI2.Both interneurons receive fast GABA-mediated inhi-bition fromT and are capable of firing rebound spikes.The interneuronI1 now provides a slowly decayingGABA-mediated inhibitory current toP and receivesa similar current fromI2. The interneuronI2 receivesfast glutamatergic excitation fromP. As before,Pmay also have a feedback connection fromI2 that caneither be a fast or slow inhibitory current. With thisnetwork architecture, we require that the slow inhibi-tion decay in roughly one theta cycle—that is, roughly100 msec.

Outside of the place field,I1 and I2 fire by postin-hibitory rebound in response toT . Note that the synap-tic delay fromI2 to I1 allows the latter to rebound before

the inhibition from I2 can take effect. Moreover, thisslow inhibition decays away over one theta cycle thusallowing I1 to again be in a position to fire by rebound.The inhibition fromI1 to P, which lasts the entire thetacycle is strong enough to preventP from firing duringthe cycle. Since this inhibition is renewed at each cy-cle, P never fires. We call this theT I1I2{P} orbit andnote thatT controls the dynamics of the network.

As in the simple model, the externally providedmemory seed firesP and interrupts theT I1I2{P} orbit.Excitation from P causesI2 to fire. The subsequentslowly decaying inhibition fromI2 to I1 now comesbefore the fast inhibition fromT . Thus I1 is stronglyinhibited and initially will be unable to respond toTinput. In the reorganizedP I2{I1}T orbit, phase pre-cession occurs withP controlling I2 directly, causingit to fire with eachP burst and thus phase precess. Inthis orbit,P also controlsI1, indirectly throughI2. Asin the simple model, the timing and effect of the in-put from T to both I1 and I2 changes from cycle tocycle. The effect onI2 is similar to that discussedin Section 3.2.2 and shown in Fig. 6B. The inhibitionfrom T chasesI2 around its phase plane. WhenI2 hasprecessed through up to 360 degrees, it can be recap-tured byT . For precession to end, however,I1 must berecaptured byT .

BecauseP and I2 are phase precessing, the inhibi-tion from I2 to I1 comes at progressively earlier phases.This means that at each cycle of theta, there exists pro-gressively more time for this inhibition to decay beforethe T input to I1 arrives. Similar to the basic model,phase precession ends whenT recaptures control ofI1.This occurs when the input fromI2 to I1 comes earlyenough in theT cycle so that the inhibition decaysaway sufficiently to allowI1 to rebound in response toT input. Once this happens, the slow inhibition fromI1 to P is reinitiated, thus suppressingP. Moreover,since P has been functionally removed from the cir-cuit, I2 must now wait forT input to fire. As discussedabove, this input occurs at a phase that allowsI1 to fireperiodically.

The complex model described here operates in a fun-damentally similar fashion to the basicTIP/PIT model.As before, the role of inhibition differs inside and out-side of the place field. Indeed, control of the networkof interneurons is again crucial for determining thenetwork output. Moreover, our analytical results inSections 3.2.2 through 3.2.6 carry over with little orno modification. In this model, it appears thatI1 hasan “antiplace field” in that it fires everywhere except

24 Bose et al.

within the place field. In a less reduced and more re-alistic network model, this may not be the case. As aresult, we do not intend our results to be interpreted aspredicting the existence of antiplace fields.

4. Discussion

We described a minimal biophysical model of the phaseprecession of hippocampal place cells that is consis-tent with the essential empirically determined proper-ties of the phenomenon. We identified mechanismswhereby temporal control of phase precession can oc-cur. In particular, we proposed a mechanism for chang-ing the firing pattern of place cells as the animal enters,crosses, and then leaves the place field. This mecha-nism of changing control of the interneuron fromT toP switches the network from a stable state to a tran-sient state. The precession of the interneurons pro-vides a second mechanism to switch the control of theinterneuron back toT and thus return the network to astable state.

The initial form of the model accounts for most of theempirical observations that were listed in the introduc-tion. Namely, (1) since we provide the seed only whenthe rat moves in one direction, place cells only fire inone direction of motion; (2) all place cells, only thesingleP cell in our case, start firing at the same initialphase; (3) the initial phase is the same on each entry ofthe rat in the place field since the seed always occursat the same phase; (4) since the change in control inthe network occurs before 360 degrees of precession,it is impossible for there to be more than this amountof phase precession; (5) while the cells in the dentategyrus may undergo a small number of cycles of pre-cession, our model requires only one cycle; additionalcycles would not change the results; (6) the model doesnot account for the increase in firing rate (a less reducednetwork model that accounts for individual spikes maybe able to include firing rate information; we believethat the mechanisms we have uncovered for phase pre-cession in our idealized bursting neurons would carryover to neurons that instead exhibit bursts of spikes);and (7) the linear dependence on frequency of pyra-midal cell firing is used here to maintain a correlationwith location rather than time that has passed since theanimal entered the place field.

The initial model, however, had a high out-of-placefield firing rate. This out-of-place field activity wouldresult in a loss of spatial specificity if a downstream sys-tem uses firing rate to determine the animal’s location

in the environment. However, the phase relationshipbetween place cell activity also provides informationon the location of the animal, and the background fir-ing outside of the place field may not be a substantialproblem for a downstream system that uses phase to de-termine the animal’s location. We presented a method(not necessarily unique), using an additional interneu-ron, to remove the out-of-place field firing without re-quiring upstream information on the size and extent ofthe place field.

From a mathematical point of view, this work showsthe importance of determining functionality of the sub-pieces of the model. We have shown how control withinthe network can be switched from one network memberto another and then back again. We have demonstratedthat geometric analysis is well suited to identify themechanisms that change control. Our work also high-lights the changing and perhaps nonintuitive effects ofinhibition in the network. In particular, we have shownhow a faster oscillator can entrain a slower one usingonly inhibition. This depends critically on the timing ofthe inhibitory input from the faster oscillator, which isconsistent with the theme that timing is of fundamentalimportance in temporal code generation. We have alsoshown how the network can use inhibition to function-ally enhance or remove the effects of a chosen member.

4.1. Related Work

Other models of phase precession differ from ours inimportant ways. In prior work (Tsodyks et al., 1996;Jensen and Lisman, 1996; Wallenstein and Hasselmo,1997; Kamondi et al., 1998), phase precession is mod-eled as a spatial phenomena. Each of these models insome way utilizes external input that already encodesfor the phase precession. Thus none of the models trulyexplains the genesis of phase precession. Namely, themodels do not address how a singleP cell behavesboth outside and inside its place field or what causesthe transition in behavior of the cell between these twoareas.

The above models require either a network of exci-tatory cells or a highly parameterized description of theneuron to achieve precession. In the work of Jensen andLisman (1996), a one-dimensional chain of pyramidalcells is considered, and phase precession occurs dueto the local unidirectional excitatory synapses betweenthese cells. Selective excitation to a specific memberof the chain, occurring at each theta cycle, providesphase-precessed input to the network. In the model of

Phase Precession of Hippocampal Place Cells 25

Tsodyks et al. (1996), strong excitatory connectionsbetween place cells is also required. In this model,precession is driven by asymmetric synaptic weightsbetween these cells. The total amount of excitatory in-put a cell receives is at a maximum in the center of theplace field and decreases monotonically as the distancefrom the center increases. Tsodyks et al. (1996) callthis “directional tuning.” A similar idea is employedby Kamondi et al. (1998). There a ramplike depolariz-ing current is provided to the place cell to mimic pas-sage through the place field. Thus in these two models,the running animal is given external information aboutits position in space that is encoded in the spatiallydependent depolarization. Kamondi et al. (1998) andWallenstein and Hasselmo (1997) both use multicom-partment descriptions of the place cells. Precessionin these models depends critically on a more com-plex, multiparameter description of each neuron. More-over, the model of Wallenstein and Hasselmo (1997)also requires phase-precessed external input at eachtheta cycle as in Jensen and Lisman (1996) to achieveprecession.

The present model takes an entirely different ap-proach. First, we believe phase precession is a tem-poral mechanism that, as demonstrated, can accountfor changes in the rat’s running speed. Second, we donot require any external precessed input to the network.Instead, we simply need a one time dose of excitationthat mimics a memory seed and changes the networkfrom the TIP to the PIT orbit. We do not require anetwork of excitatory cells or multicompartment mod-els for the cells. Finally, our analysis gives a differentinterpretation of the significance of phase precessionthan all of the previous studies. Namely, in our modelthe end of phase precession signals the end of the placefield. In the other studies, the end of the place fieldsignals the end of the precession. Thus our model pro-vides an internal mechanism for determining the end ofthe place field (via the end of precession), whereas theother models require another external input to notifythe pyramidal cell that its place field has ended.

4.2. Experimental Support

Several studies have demonstrated that the mean phaseof activity of hippocampal interneurons is just prior tothe mean phase of pyramidal cell activity (Fox et al.,1986; Buzs´aki and Eidelberg, 1982). This implies thatduring the phase precession, the activity of a place cellmust pass through the phase of theta rhythm at which

interneurons are most strongly inhibiting pyramidalcells. In the present model the pyramidal cell con-trols the firing of the interneuron during the phaseprecession process, and its activity is not adverselyinhibited by the interneuron.

In recent recordings of a larger number of pyra-midal cells and interneurons (Csicsvari et al., 1998),pyramidal cells were found that had large, significantcross-correlation peaks at times on the scale of tenmilliseconds that preceded the activity of simultane-ously recorded interneurons. These correlations findthat there is a tight coupling in the time at which pyra-midal cells and interneurons are active and suggeststhat the activity of a subset of the interneurons mightprecess along with place cells. The experimental ev-idence of the high level of correlation between placecells and interneurons demonstrates that there existsa sufficiently strong conductance between subsets ofthese two types of neurons. These findings are consis-tent with the predictions in the present model in whichthe phase precession of interneurons may only occur ina subset of interneurons and only when they are beingdriven by a phase precessing place cell.

4.3. Consistency of Biophysicaland Functional Models

In a model of phase precession, there are several im-portant larger goals and anatomical considerations thatshould be taken into account. There is considerabledata that demonstrates that the hippocampus has arole in episodic memory. We have proposed a func-tional model that describes how the spatial and mem-ory roles of the hippocampus can be combined (Recceand Harris, 1996; Recce, 1999). In this view, whena rat returns to a previously experienced environment,the set of coactive place cells corresponds to the recallof an egocentric map of the environment. The memoryrecall is performed by a pattern completion process thatis controlled through the excitatory feedback pathwayin the CA3 region of the hippocampus. The seed forthis recall enters the CA3 region through the entorhi-nal cortex to the dentate gyrus and then in the mossyfiber projection from the dentate granule cells to theCA3 pyramidal cells and interneurons. The temporalcontrol of the recall process, in this view, is providedby the pacemaker input from the medial septum.

The feedback connections between CA3 pyramidalcells are relatively sparse, which can limit the num-ber of egocentric maps of space that can be stored if

26 Bose et al.

the region is modeled as an autoassociative memory.However, Gardner-Medwin (1976) has demonstratedthat this limitation can be overcome if the recall pro-cess is performed over a sequence of steps. Hiraseand Recce (1996) found that the optimal performancein a multistep autoassociative memory model of theCA3 region can be achieved if an interneuron networkcontrols the recall process by providing an inhibitoryinput to the pyramidal cells that is a linear function ofthe number of simultaneously active pyramidal cells.Further, it has been proposed that the phase precessioncorresponds to the multiple steps in this recall processand that the phase precession suggests a method forthe concurrent recall of several egocentric memories(Recce, 1999).

In the present work, we have begun the processof systematically addressing these issues. The presentmodel for phase precession, while conceptually quitesimple, is not necessarily the one that most easily pro-duces output that phase precesses. Instead, our mod-eling is based on the larger considerations of identify-ing mechanisms that can allow multiple memory recallprocesses to be simultaneously performed. The basicframework described above, which accounts for pre-cession, is compatible with a functional model of thespatial and episodic memory roles of CA3 place cells.

Appendix A. Analysis: Existence and Stabilityof TIP Periodic Orbit

Outside the place field, the network is in theTIPconfig-uration. We show that there is a unique, stable periodicorbit to which the cells are attracted. Assume thatTfires atτ = 0. We will locate an initial condition forP and show that whenT fires again atτ = TT , P hasreturned exactly to this initial condition. The analysiswill be in terms of a time metric that will measure thetimes of evolution ofP on various parts of its orbit inphase space. These times depend on various biophys-ical parameters that are related to both intrinsic andsynaptic properties of the cells and the network.

Theorem 1. There exists a locally unique asymptoti-cally stable TIP periodic orbit with a period O(ε) closeto TT .

Proof The analysis below occurs atε= 0, but usingthe work of Mischenko and Rozov (1980), the resultsactually hold forε >0 and sufficiently small. Considerthe left branch ofCP betweenwRK andwLK . Associ-ated to each point on this curve, denotedwp, is the time

1(wp) it takesP to reachwLK starting fromwp. Inparticular,1(wRK)= τP R and1(wLK)= 0. It is clearthat1 is a monotone decreasing function ofwp. Thedescription of theTIP network given earlier shows thatthere is a one-dimensional map5 that takes the initialposition ofP at τ = 0 along the left branch ofCP andreturns the position ofP at τ = TT .

Proposition 1. The map5 defines a uniform con-traction on a subinterval(wlo,whi ) of (wLK,wRK). Ifw? is the resulting locally unique asymptotically stablefixed point, then the network possesses a locally uniqueasymptotically stable singular TIP periodic orbit withperiod TT , wherewp(0)=w?.Proof We prove the existence of a fixed point by com-paring the behavior of the uncoupledP cell to its cou-pled counterpart. SinceTT = TP +Ä, we can comparethe period of the coupledP cell to TT . First we locatea suitablewhi . Associated with it is a timeThi fromwhi towLK on CP. At τ = τTon+ σi p, P receives inhi-bition and jumps back toCPI . At τ = τTon+ σi p + τIon,the inhibition toP shuts off. We choosewhi such thatwp(τTon+ σi p + τIon)=wLK . Next let us consider theevolution of theP cell, solely onCP in the absence ofany synaptic input. Since∂gp

∂vp> 0 near the left branch

of CP andCPI , the P cell moves downCP at a slowerrate than onCPI . In other words,P takes a longertime to cover the same Euclidean distance onCP thanonCPI . Thus atτ = τTon+ σi p + τIon, the uncoupledPcell will be abovewLK . It is seen that inhibition ac-tually shortens the time distance to the knee for thisinitial condition. This is very similar to, but timewisethe opposite of, the idea of virtual delay of Kopell andSomers (1995). There they showed that excitation hadthe opposite effect of increasing the time to the knee.The coupled and uncoupled versions ofP follow thesame trajectory in the active state.

Assume momentarily thatÄ= 0. For the uncoupledP cell,1(5(whi ))= Thi . But for the coupledP cell,1(5(whi ))< Thi . Thus the time it takes for the coupledP cell to return to its initial positionwhi is Tpc< TP. IfÄ is sufficiently small, thenTpc< TP +Ä = TT . Notethat Thi > τTon + σi p + τIon. How big the differencebetween these times needs to be depends on the size of∂gp

∂vp, which determines the difference in the rates along

CP andCPI . Thus we have located an initial conditionwhose time distance to the knee compresses over oneoscillation.

We next locate an initial condition whose time dis-tance to the knee expands. Now letwlo = wp(0) such

Phase Precession of Hippocampal Place Cells 27

that for the uncoupledP cell wP(τTon+ σi p)=wLK .Assume that the initial conditions forT andI are iden-tical to the above case. Thus at the momentP wereto fire, it receives inhibition fromI . It will thus fallback toCPI and move down this cubic toward the fixedpoint onCPI for a timeτIon. At τ = τTon+ σi p + τIon,P is released from inhibition and jumps to the rightbranch. Thew value of the point to where it jumps liesbelowwLK . On the the right branch, let the time be-tween this point andwLK be denoted byτD. OncePpasses throughwp=wLK on the right branch ofCP, itwill follow the same orbit as the uncoupledP cell. ThecoupledP cell has been delayed by a timeτIon+ τD.So it will return to its initial position at a timeTpc> TP,if Ä= 0. Thus ifÄ is sufficiently small,Tpc> TT . Forthis initial condition, the time distance to the knee hasbeen expanded—that is,1(5(wlo))>1(wlo).

Therefore, we have located a set of initial conditionssuch that the lower and upper boundaries of the setare mapped into the interior of the set by the map5.By continuous dependence on initial conditions, thisimplies that there exists at least one element of the setthat remains fixed under the map5. To prove that thispoint is unique and attracting, we need to show that5 is a contraction mapping. We follow two differentversions of theP cell, denotedP1 andP2, and show thatthe dynamics of theTIPnetwork bring these cells closertogether after one iterate of5. Let w1<w2 denoteany two points in(wlo, whi ) such thatwp1(0)=w1 andwp2(0)=w2. The cellsP1 andP2 do not interact withone another but receive common inhibition from thesameI cell.

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

−5(w1)|<λ|w2−w1|.Proof: From the above analysis, it is clear that in theTIP network, P1 and P2 will jump to the active statefrom a point onCPI that lies below the left knee ofCP.This occurs atτ = τTon+ τIon+ σi p. Let δ denote thetime between the cells atτ = 0. As the cells evolvealong the left branch ofCP and thenCPI , the time be-tween them remains invariant. However, the Euclideandistance between the cells decreases. This is a standardfact for two cells that are evolving along the same one-dimensional curve toward a fixed point. The longerthe cells spend near the critical point formed by the in-tersection ofCPI andDP, the more the Euclidean dis-tance is compressed. Atτ = τTon+ τIon+ σi p, the cellsare released from inhibition and jump horizontally tothe right branch ofCP. The same mechanism that fos-

ters synchrony due to fast threshold modulation (seeSomers and Kopell, 1993) implies that there is com-pression across the jump. The rational is the follow-ing. Across the jump, the Euclidean distance does notchange. However, the rate of evolution of the cells onthe left branch is much slower due to the fact that thesecells are evolving near a critical point. On the rightbranch the cells are very far way from thew-nullclineDp, so the rate of evolution is much faster. Thus thecells travel through the same Euclidean distance in amuch shorter time. Thus the new time between cellsis less thanδ. Since the cells always satisfy the samedifferential equation, the time between them remainsinvariant on the right branch ofCP. Moreover, sincethey both jump down from the right knee of theCP,the time is invariant across the down jump. WhenP1

andP2 return to the left branch ofCP, P2 will lie belowP1. So whenP2 returns to the initial position ofP1 atw1, the time between the cells will be less than whenthey started, so the Euclidean distance between themmust have compressed. This proves that5 is a con-traction. The constantλ can roughly be approximatedby the ratio of the speed at the jump on point on theright branch to the speed at the jump off point on theleft branch (Somers and Kopell, 1993).

We have made some implicit assumptions about thebehavior ofI . Namely, we have not proved thatI ac-tually returns to its initial position atτ = TT . We areassuming that the time on the left branch ofCI fromw=wRK of CI to a neighborhood of the fixed pointon CI is bounded from below and above. The upperbound arises from the fact that we wantI to be able torespond toT at τ = TT . The lower bound comes fromthe fact that we do not wantI to fire for the time periodτP A+ σi p. Provided these conditions are met, the ex-istence of an asymptotically stable fixed point impliesthe existence of an asymptotically stable singular peri-odic orbit. It is now fairly standard to show that the map5 perturbs smoothly forε >0, but sufficiently small,thus yielding the actualTIP periodic orbit (Mischenkoand Rozov, 1980). 2

Let us now address more carefully what we mean bythe active state of the interneuron. InPIT, it is not pos-sible to locate the precise place from whichI jumps up.Thus the precise location at which it reaches the rightbranch is not determinable, and therefore we cannotprecisely define its active duration. To give an esti-mate first on the jump up point we need to define twomore cubics. One isCI P , which occurs whensP I = 1

28 Bose et al.

andsT I = 0. The second isCI PT , which occurs whenbothsP I = 1 andsT I = 1. Note thatCI PT lies betweenCI P andCIT . Depending on the parametersgpi , gti , vpi ,andvt i , it may lie above or belowCI . For this argumentassume it lies aboveCI . Then the maximumw valueto which I can jump is thew value of the left knee ofCI P , call it wi M . The minimum value is thew value ofthe intersection ofCI andDI , call itwim. Since∂gi

∂vi= 0

near the right branches, the speeds of evolution alongall the right branches are identical. For this argumentthe ordering of the right knees of the relevant cubics isRIT < RI < RI PT < RI P , where the notation should beclear. So, by a long active phase of the interneuron, wemean the time fromwi M to RIT is longer thenτP AI . Bya short active phase we mean the time fromwim to RI P isless thanτP AI . Note that these times are simply boundsfor the length of the active phase. They do not implythat I necessarily jumps down from a specific knee.Indeed,I ’s jump down point may change each cycledepending on the timing of inhibition fromT . Finally,if the length of the active state ofI is too long—forexample, if the timewi M to RI is much longer thanτP AI + σpi , then the analysis presented above breaksdown. This is becauseI will not be in a position torespond to the next bout of excitation fromP.

Appendix B. Model Equationsand Parameter Values

We have numerically implemented our two cell andpacemaker network using the Morris-Lecar equations(1981) to model the neurons. The current balance equa-tions for the P and I cells are

Cmdvp

dt=−gCam∞(vp)(vp− vCa)− gKwp(vp− vK )

−gL(vp − vL)− sipgIP(vp − vIP)+ I p,

Cmdvi

dt=−gCam∞(vi )(vi − vCa)− gKwi (vi − vK )

− gL(vi − vL)− spi gP I (vi − vPI)

− sti gTI(vi − vTI)+ Ii ,

where vX (in mV) is the membrane voltage in thepyramidal cell (X= p) and the interneuron (X= i ).The maximal conductances for the calcium and potas-sium currents in the cells are the same (gCa= 4.4,gK = 8.0). The reversal potentials for the ionic cur-rents areVCa= 120 mV andVK = −84 mV. The leakconductance density in each cell isgL = 2mS/cm2, and

the leak reversal potential isVL = −60 mV. The mem-brane capacitance isCm= 20 µF/cm2. The appliedcurrent values (inµA/cm2) areI p= 105 andIi = 120.In our implementation of the model, theT cell is mod-eled by the same equations as an isolatedP cell, exceptapplied current is set to 92µA/cm2. In order to ob-tain frequencies in the theta range, time was scaled by4.5. The model equations were numerically integratedusing XPP (information on the program available athttp://www.pitt.edu/∼phase).

The gating kinetics of the potassium conductancesin each cell (X= p and i ) are governed by equationsof the following form:

dwX

dt= φw∞,X(v)− wX

τw,X(v),

whereφ= 0.005 corresponds toε in the general formof the equations. The steady-state activation and inac-tivation functions and the voltage-dependent time con-stant functions for the calcium and potassium currentsin each cell (X= p andi ) are given 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].

The half-activation voltages for the gating functions are(in mV) v1,p= v1,t = − 1.2, v3,p= 2, andv3,i =−25.The activation and inactivation sensitivities for the gat-ing functions are (in mV)v2,p= v2,i = 18, v4,p= 30,andv4,i = 10.

The synaptic currents in each cell are governed bythe variablesXY whereX indicates the presynaptic cellandY indicates the postsynaptic cell. These variablessatisfy equations of the form

s′XY = α(1− sXY)1

2

(1+ tanh

(vX − v5

v6

))− β sXY,

where the Heaviside function in the general form of theequations forsXY is replaced by a sigmoidlike functionthat depends on the presynaptic voltage (half-activationv5= 0 and activation sensitivityv6= 10). For the threesynaptic currents in the model, governed byspi , sip,and sti , the rise and decay rates of the synapse are

Phase Precession of Hippocampal Place Cells 29

the same (α= 2, β = 1). In the current balance equa-tions, the effect of the synapse is determined by thereversal potentials of the synaptic currents (vpi = 80,vi p = vt i =−80), and the maximal conductances ofthe synaptic currents are all equal (gpi = gip = gti = 1).We tookσi p = σpi = 0 in the simulations, thus showingthat the existence of delays are not crucial for the aboveresults.

For the simulation in Fig. 9, the following parame-ters were used: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 Technology under grants 421540 (AB),421590 (VB), and 421920 (MR). V. Booth also ac-knowledges support from the National Science Foun-dation (grant IBN-9722946).

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