Journal of Computational Neuroscience 9, 5–30, 2000 °c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.
A Temporal Mechanism for Generating the Phase Precession of Hippocampal Place Cells
AMITABHA BOSE, VICTORIA BOOTH AND MICHAEL RECCE Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ, 07102-1982 [email protected]
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 theta rhythm systematically precesses as the animal runs through the region in an environment called the place field of the cell. We present a minimal biophysical model of the phase precession of place cells in region CA3 of the hippocampus. The model describes the dynamics of two coupled point neurons—namely, a pyramidal cell and an interneuron, the latter of which is driven by a pacemaker input. Outside of the place field, the network displays a stable, 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, simulating entrance into the place field, reorganizes the functional roles of the cells in the network for a number of cycles of the theta rhythm. In the reorganized network, the pyramidal cell drives the interneuron at a higher frequency than the theta frequency, thus causing a systematic precession relative to the theta input. The frequency of the pyramidal cell can vary to account for changes in the animal’s running speed. The transient dynamics end after up to 360 degrees of phase precession when the pacemaker input to the interneuron occurs at a phase to return the network to the stable background firing pattern, thus signaling the end of the place field. Our model, in contrast to others, reports that 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 has crossed the place field, and it explains the observed nearly 360 degrees of phase precession. We also describe how this 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 group of neurons in region CA3 of the hippocampus of freely moving rats can generate a spatially encoded There is considerable current interest in the ways in output, called phase precession, using only limited and which the temporal firing pattern of neurons may pro- spatially nonspecific inputs. vide additional information that is not conveyed by the It has been proposed that hippocampal place cells averaged firing rate alone. This interest has led to a provide information for downstream neurons through search for ways in which temporal firing properties the phase relationship between neuronal activity and of neurons are generated and detected in the central the hippocampal EEG (O’Keefe and Recce, 1993). nervous system (for a review, see Rieke et al., 1997; Place cells were first described in the CA1 region of Recce, 1999). The main goal of this article is to use freely moving rats (O’Keefe and Dostrovsky, 1971), minimal biophysical modeling to demonstrate how a 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 spike relative to the hippocampal theta rhythm. C: Hippocampal theta activity recorded at the same time as the hippocampal unit. D: Half sine wave fit 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- firing rate of the cell alone. A downstream system that cation in an environment (O’Keefe, 1976; Muller et al., measures the phase of place cell activity would then 1987). The preferred firing location of a place cell is have more information about the location of the an- called its place field. imal in the environment and may be able to ignore During locomotor activity, the hippocampal EEG out of place field firing that occurs preferentially at has a characteristic 6 to 12 Hertz sinusoidal form called a phase that is different from the range found in the the theta rhythm (Green and Arduini, 1954), and the place field. Place cells in both the CA1 region and phase and frequency of this rhythm is highly correlated the upstream CA3 region are found to undergo phase across the CA1 region of the hippocampus (Bullock precession (O’Keefe and Recce, 1993). et al., 1990). As the rat runs through a place field, on a Skaggs and coworkers (1996) confirmed this find- linear runway, the phase of the theta rhythm at which ing and additionally found that the initial phase was a place cell fires systematically precesses. Each time consistent among a large number of place cells in the the animal enters the place field, the firing begins at CA1 region. They also found that dentate granule cells the same phase, and over the next five to ten cycles that project to CA3 undergo a small number of cy- of the theta rhythm it undergoes up to 360 degrees of cles of phase precession and therefore provide a syn- phase precession (O’Keefe and Recce, 1993; Skaggs chronized, timed excitation to the CA3 pyramidal cells. et al., 1996). The maximum observed phase preces- Marr (1971) proposed that this input from the dentate sion was 355 degrees (O’Keefe and Recce, 1993). So, granule cells provides a seed input for a memory re- by the phrase “up to 360 degrees of phase precession” trieval and pattern completion process that is driven we mean strictly less than but in a neighborhood of by the excitatory feedback among pyramidal cells in 360 degrees. An example of the phase precession of a the CA3 region (Treves and Rolls, 1994; Gibson and place cell is shown in Fig. 1. During the run along the Robinson, 1992; Hirase and Recce, 1996). The phase track, the phase of hippocampal place cell firing is more precession of place cells may be an essential part of correlated with the animal’s location within the place this pattern completion process. field than with the time that has passed since it entered The hippocampal regions also contain a variety the place field (O’Keefe and Recce, 1993). This sug- of interneurons that contribute to the generation of gests that the phase of place cell activity provides more hippocampal oscillations. These interneurons project information on the location than is available from the to place cells as well as to other interneurons (Freund Phase Precession of Hippocampal Place Cells 7 and Buzsaki,´ 1996). Skaggs and coworkers (1996) coworkers (1998) have proposed a model for phase pre- measured the phase of interneuron firing in the CA1 re- cession in the CA1 region that does not depend on pre- gion and found that on average the interneurons do not cisely timed inputs, but instead the phase is a result phase shift. On the other hand, Csicsvari and cowork- of the total amount of depolarization of the place cells. ers (1998) found a large, significant cross-correlation In this model, the total amount of phase shift is much between pyramidal cell and interneuron pairs, where less than 360 degrees. the pyramidal cell firing precedes interneuron firing by In this article, we describe a minimal biophysical tens of milliseconds. This implied high synaptic con- temporal model for phase precession in the CA3 region ductance from pyramidal cells to interneurons suggests of the hippocampus. It differs from prior models in that interneurons could phase precess. This sugges- that (1) it generates the spatial correlation of the phase tion may not be inconsistent with the experimentally precession from a single spatial input and from infor- observed average properties of interneurons (Skaggs mation about the animal’s running speed; (2) it does et al., 1996) if only a subset is shown to transiently not require phase precession in the upstream projec- phase precess. tion cells; (3) it does not require a spatially dependent The medial septum is also involved in modulating the depolarization of place cells; (4) it provides a mecha- temporal firing patterns of hippocampal place cells and nism for determining the end of the place field; (5) it interneurons. The projection from the medial septum to provides a mechanism for up to 360 degrees of phase the hippocampus is both GABAergic and Cholinergic shift; and (6) it is consistent with associative memory (Freund and Antal, 1988), and it provides a pacemaker models for the CA3 region. to drive the theta rhythm (Green and Arduini, 1954). Alternatively, our model does not include a mecha- We have shown that the interburst frequency of some of nism to account for the firing rate of place cells. The the neurons in the medial septum is a linear function of a activity of place cells includes both rate and timing in- rat’s running speed on a linear track (King et al., 1998). formation. In the present model, we are only concerned In summary, the observed properties of the phase with the timing properties. precession phenomenon include the following: (1) This minimal model, which is composed of two neu- place cells fire in only one direction of motion dur- rons (one pyramidal cell and one interneuron) and a ing running on a linear track; (2) all place cells start pacemaker input, can explain the onset, occurrence, firing at the same initial phase; (3) the initial phase is and end-of-phase precession. The network has two im- the same on each entry of the rat into the place field of a portant dynamic patterns. The first is a stable attracting place cell; (4) the total amount of phase precession is al- state, which mimics the behavior of CA3 outside of the ways less than 360 degrees; (5) the cells in the dentate place field. The second dynamic pattern is a transient gyrus that project to CA3 undergo a smaller number state that encodes the behavior of CA3 within the place of cycles of phase precession; (6) in one-dimensional field. The main difference between the two states is the environments, the phase plus firing rate provide more input that controls the firing pattern of the interneuron. information of the rat’s location than the firing rate of In the stable state, the pacemaker input controls in- a place cell; (7) phase is more correlated with the ani- terneuron firing, while in the transient state, excitation mal’s location in a place field than with the time since from the pyramidal cell does. The seed from the den- the animal entered the place field; and (8) background tate gyrus switches control from the pacemaker to the firing of place cells outside of the place field occurs place cell to initiate phase precession, and the duration at a fixed phase that is closest to the initial phase that of phase precession is determined by the duration of the occurs when the animal enters the place field. transient dynamics as the network returns to the stable Several models have been proposed to explain the state. neural basis of the phase precession phenomenon. The The transient place cell firing in the place field is a model proposed by Tsodyks and coworkers (1996) pro- temporal process in which the phase of firing of the cell vides an environment-driven phase precession, which in each cycle of the theta rhythm strictly depends on is certain to be more correlated with position than time. the phase in the prior cycle of the theta rhythm. This Other models of environment-driven phase preces- is in contrast to all other models of phase precession sion have been proposed by Wallenstein and Hasselmo in which the phase of firing of place cells depends on (1997) and Jensen and Lisman (1996). In these mod- the external inputs arriving at that cycle and not on els, the phase correlation in CA1 pyramidal cells results the phase in other cycles of the theta rhythm. For this from CA3 input that is phase precessing. Kamondi and reason, to account for the spatial correlation of phase 8 Bose et al. precession, the present model requires that the amount is as spatially precise as the phase precessing output. of phase change during a theta cycle depends linearly Another prediction, which is easier to address experi- on changes in the animal’s running speed. Neurons in mentally, is that a subset of the interneurons transiently the medial septum have been found to have an inter- phase precesses. The model also strongly depends on, burst frequency that is a linear function of the running and thus predicts, continued evidence for a linear fre- speed of a rat (King et al., 1998). Also, the theta fre- quency dependence of pyramidal cells in CA3 and pro- quency in the hippocampus of rats running on a linear jection cells in the medial septum on running speed of track has been found to be highly correlated with the the animal. rat’s speed (Recce, 1994). Since the phase-precession data includes a range of running speeds (O’Keefe and Recce, 1993), this implies the interburst frequency of 2. Model and Methods place cells is also a function of the animal’s running speed. These data provide a mechanism to maintain a 2.1. Model spatially correlated phase precession in the proposed temporal model. Our minimal biophysical model for phase precession The model generates two different types of spatial in CA3 consists of a pyramidal or place cell, an in- information. First, it determines the length of the place terneuron, and a pacemaker input, denoted P, I , and T , field. The place field ends when phase precession ends, respectively. The pacemaker input T may be thought which occurs when the pacemaker regains control of of as an individual cell or perhaps a conglomeration the interneuron from the place cell. Second, the model of inputs that produce the theta rhythm. The synap- determines the location of the animal within the place tic connections among P, I , and T reflect the net- field. Note that these two types of spatial information work architecture of CA3 (see Fig. 2). Specifically, are generated from the seed, a single location specific the pyramidal cell P receives fast, GABA-mediated input, and the velocity of the animal, a nonspatially inhibition from the interneuron I and projects to I specific input. via fast, glutamatergic excitation. The pacemaker input Our model supports two themes that have arisen in other neural modeling studies. The first is that the con- trol of the dynamics in a network may switch among different neurons over time. These switches can occur in the absence of any synaptic changes. Instead, dy- namics are governed by the timing of events relative to one another (Ermentrout and Kopell, 1998; Nadim et al., 1998). The second theme concerns the role of inhibition in networks of neurons. In recent studies (Terman et al., 1998; Rubin and Terman, 1999; van Vreeswijk et al., 1994), it is shown that inhibition may have counterintuitive effects on the network, such as synchronizing mutually coupled inhibitory cells. In the present study, we use inhibition to speed up the oscil- lations of the pyramidal cell. The primary prediction of this model is that phase precession is a spatially specific output that results from an integration of a spatial signal from the dentate gyrus at a single point and a nonspatial velocity coded signal originating from the medial septum. We predict that no upstream physiological signal exists that contains as much information about the animal’s spatial location Figure 2. Two cell and pacemaker network model consisting of a pyramidal or place cell (P), an interneuron (I), and a pacemaker input as the phase of hippocampal place cell firing in CA3. (T). Synaptic connections reflect CA3 network architecture (arrow This prediction differentiates our model from all other indicates excitatory and filled circle indicates inhibitory synaptic models of phase precession that require an input that connections). Phase Precession of Hippocampal Place Cells 9
T projects only to I through fast, GABA-mediated between 0 and 1, with a period TT in the theta range. inhibition. When the value of T exceeds a prescribed threshold, Our analysis does not depend on a specific model for then we assume that T fires, and we measure the dura- the individual neurons; hence, we use a general form for tion of its “spike” as the time spent above the threshold. the model equations for each cell. The few restrictions For the network model, appropriate synaptic currents we require on the model properties (outlined below) are are added to the individual cell models. In the synap- satisfied by a large class of neural oscillators, including tic current variables and parameters, the first subscript the Morris-Lecar (Morris and Lecar, 1981) equations, denotes the presynaptic cell and the second subscript which have been successfully used in numerous mod- denotes the postsynaptic cell. The network equations eling studies (Rinzel and Ermentrout, 1997; Somers for P are and Kopell, 1993; Terman et al., 1998; Terman and v0 = (v ,w ) − ( − σ ) v − v , Lee, 1997), and which we use for our simulations (see p f p p p gipsip t ip [ p ip] (3) Appendix B). w0 = ²g (v ,w ). The P cell is a general neural oscillator, modeled by p p p p equations of the following form: For I , the network equations are v0 = (v ,w ) p f p p p v0 = (v ,w ) − ( − σ ) v − v (1) i fi i i gpi spi t pi [ i pi ] w0 = ²g (v ,w ), p p p p − gtisti(vi − vti), (4) 0 0 w = ² (v ,w ). where denotes the derivative with respect to time t. i gi i i The variable vp represents the membrane voltage of This form of the synaptic currents has been used in the place cell and the function f p is composed of the sum of the ionic and leakage currents found in the cell other modeling studies (Ermentrout and Kopell, 1998; as well as an applied current. We consider P to have Wang and Rinzel, 1992). The variables sip, spi , and sti govern the dynamics of the synaptic currents, and each bursting behavior and the variable vp to model the burst envelope and not to account for individual, fast spikes satisfies an equation of the form occurring during the active phase of the burst. In this 0 s = αH∞(v − vθ )(1 − s) − βs, (5) general class of neural models, the variable wp mod- els the activation gating variable of an outward, ionic current, usually potassium-mediated, and the function where α and β are the rise and decay rates of the synapse, and vθ is the synaptic activation threshold. gp governs the activation gate dynamics of this con- ductance. The parameter ² controls the time scale of The Heaviside step function H∞ acts as an activation the w dynamics. Our analysis assumes that ² is small switch for the synapse; H∞(v − vθ ) is0if v
CPI intersect somewhere along the left branch of CPI . This intersection point will be a stable fixed point for We first present an overview of our results that provides the inhibited P cell, with the result that the inhibited P a qualitative description of how and why the model cell will not spontaneously oscillate. Further note that generates phase precession. In the second part of this this intersection point will lie below the left knee of CP section, we provide the analysis of our model and spe- ∂g because of our restriction (6) p > 0 (see Somers and cific analytical results. ∂vp Kopell, 1993). There exist two different, important firing patterns The effect on I of the excitatory synaptic current for the simple network. The first firing pattern repre- from P and the inhibitory synaptic current from T is sents network behavior when the rat is outside of the that its cubic-shaped nullcline can be raised or lowered place field. The second represents network behavior depending on the activation of spi and sti. For any pos- when the rat is inside the place field. The difference sible case, we assume that the wi -nullcline DI contin- between these two behaviors is directly correlated to ues to intersect the synaptically perturbed vi -nullclines whether T or P controls the firing of I . An important along their left branches, thus ensuring that I remains aspect of our analysis is to identify mechanisms that al- excitable. As noted previously, we require that I be low the control of firing to be shifted between T and P. able to fire due to either excitation received from P As described below, the first change in control from T or via rebound from inhibition received from T .For to P requires a brief external input to the network oc- example, assume that I is sufficiently close to the inter- curring when the rat first enters the place field. The section point of the nullclines when it receives synap- second change in control from P back to T is deter- tic input. If the input is excitatory from P, the cubic- mined internally by the network and signals that the shaped nullcline of the I cell is instantaneously raised place field has ended. releasing the trajectory from its left branch. The tra- The firing pattern representing out-of-place field be- jectory is then attracted to the right branch of the raised havior is a stable periodic state that we call the TIP nullcline, thus initiating an action potential. If the input orbit. In this stable pattern, the pacemaker T controls is inhibitory from T , the cubic-shaped nullcline is low- the firing of I , which fires via postinhibitory rebound. ered and the trajectory moves left toward the new stable In turn, I modulates the firing of P, so that it fires fixed point at the intersection of the perturbed nullcline phase-locked to the underlying theta rhythm. The ef- and DI . For sufficiently strong inhibition, because of fect of the inhibition from I is to either delay or ad- our restriction (9), the new fixed point will lie below vance the firing of P, depending on where in P’s cycle the left knee of CI . When the synaptic inhibition shuts the inhibition occurs. Thus, P is phase-locked to the off, the nullcline rises back again toward CI , and the theta rhythm and does not phase precess whether its trajectory is attracted to the right branch, initiating an intrinsic frequency is higher or lower than theta (see action potential via postinhibitory rebound. The ability Section 3.2.1). Moreover, P can only fire when re- of the I cell to fire due to these two different stimuli is leased from inhibition from I . critical to our analysis. Simulation results from the simple network where The mathematical analysis for this article, as well each cell is modeled with Morris-Lecar equations are as the specific equations used in the simulations, can shown in Fig. 4. During the first five bursts of the P cell be found in Appendices A and B, respectively. We (top trace), the network is in the stable TIP orbit. The Phase Precession of Hippocampal Place Cells 13
this model is to generate the spatial phase correlation from less spatially specific upstream information. Fol- lowing the description of the properties of the simple model, we show in Section 3.2.7 how to completely suppress the out-of-place field firing, without resorting to additional upstream spatial information (see Fig. 9). The second firing pattern, representing behavior within the place field, is a transient state that we call the PIT orbit. The PIT orbit is initiated by a brief, soli- tary dose of excitation to P, representing a memory seed externally provided from the dentate gyrus. The seed causes P to fire earlier than it would have in the TIP orbit and thereby allows P to seize control of I ’s firing. The P cell is able to fire via its intrinsic oscilla- tory mechanisms for as long as it controls I . Thus, the seed has the effect of switching control of I from T to P. During the PIT orbit, the period of P firing, TP ,is less than the period of T , TT , regardless of the intrinsic period of P, provided that TP and TT are not too dis- Figure 4. TIP and PIT orbits for the two cell and pacemaker net- parate. The reason phase precession occurs in the PIT work with each cell modeled by Morris-Lecar equations. The T spike orbit is somewhat subtle and depends both on the intrin- is not explicitly shown and is denoted by dashed vertical lines. The sic periods of P and T and also on the time duration of network oscillates in the TIP orbit for the first 5 cycles. The PIT inhibition from I to P. If this inhibition is short-lasting, orbit is initiated by the arrival of the memory seed (heavy arrow at then during PIT, P fires at its intrinsic frequency. Thus 535 msec, which lasts for 3 msec) and phase precession occurs over < the next 7 cycles (under heavy bar). The network returns to the stable if TP TT , it is clear that precession will occur. Al- TIP orbit for the remainder of the simulation. The model equations ternatively, if the inhibition is long-lasting, then during and parameter values are given in Appendix B. PIT, P fires at a substantially faster rate than its in- trinsic frequency. Counterintuitively, this increase in firing rate is most strongly dependent on the strength sequence of cell firing with T firing first (not explic- of the inhibitory synaptic conductance from I to P.In itly shown; time of peak indicated by dotted vertical Section 3.2.2, we clarify how inhibition may speed up lines), followed by I (lower trace) and then P is evi- the firing rate of P. For this case, precession occurs dent. During the T spike, I is inhibited as seen by the whether the intrinsic period TP is less than, greater than hyperpolarization in the I voltage trace immediately or equal to TT . In either case, since P controls the fir- following the peak of T . During the I spike (after its ing of I in the PIT orbit, forcing I to fire with every P release from inhibition from T ), P is inhibited as seen burst, I also phase precesses. by the hyperpolarization in the P voltage trace imme- Figure 4 illustrates the PIT orbit and phase preces- diately preceding the spike. sion for the Morris-Lecar model network. Arrival of Clearly, the out-of-place field firing rate in the TIP the memory seed to P is modeled by a brief, excita- orbit is higher than experimentally observed. This high tory, applied current pulse (heavy arrow) that causes out-of-place field firing leads to a loss of spatial speci- the early firing of P and the interruption of the TIP ficity to a downstream detector of firing rate but has orbit. Due to the excitation from P, I overcomes the minimal effect on a downstream detector that uses the inhibition from T and fires with P, as seen by the slight phase of firing to determine the spatial location of the hyperpolarization in the peak of the P burst. Over the animal. The simplest way to reduce or remove the out- next six cycles (under the heavy bar), P and I phase of-place field firing is to have a spatial inhibitory signal precess relative to T . The firing of P due to intrinsic that provides inhibition at all points outside of the place oscillatory mechanisms is evidenced by the smooth rise field. This approach is not taken here because it relies to threshold displayed by its voltage trace. on spatial information upstream from the hippocampus The phase precession of I provides an internal mech- about the size and extent of the place field. The goal of anism that returns the network to the TIP orbit. Namely, 14 Bose et al. in PIT, I continues to receive inhibition with each T that most strongly affect this network behavior. In spike, but the timing or phasing of the inhibition is such Section 3.2.6, we show how experimentally supported that postinhibitory rebound does not occur at each cy- data relating running speed to interburst frequency of cle. For example, during the first two spikes of phase the pyramidal cells can be used to achieve a spatial cor- precession, the inhibition from T arrives during the I relation in this temporal model. Finally in Section 3.2.7, spike, as seen by the lower spike heights, and rebound we introduce a modification to the simple network that does not occur. Over the next three cycles, T inhibits suppresses out-of-place field firing. I during early portions of its afterhyperpolarization when rebound is not possible. But by the last spike 3.2. Analysis of phase precession, I is inhibited sufficiently late in its afterhyperpolarization, resulting in a rebound spike 3.2.1. The Network Oscillates at Theta Frequency that interrupts the PIT orbit, returning the network to Outside of the Place Field. The trajectories of P and the stable TIP orbit. The recapture of I by T signals I in the TIP orbit can be traced along nullclines in their the end of the place field and occurs when P and I have phase planes. We may assume that at τ = 0, T fires, precessed through up to 360 degrees of phase, which is the trajectory of P lies on the left branch of CP with ? consistent with experimental data (O’Keefe and Recce, wp(0) = w and the trajectory of I lies somewhere 1993). Thus, the precession of I is necessary for along the left branch of CI . In response to the inhibi- determining the end of the place field. tion received from T , I falls back to CIT and is released τ = τ w ( ) The brief dose of excitation in the form of the mem- from inhibition at Ton . We assume that i 0 is w (τ ) ory seed reorganizes the functional roles of the cells in such that i Ton lies below the left knee of CI . Thus τ = τ the TIP and PIT orbits. Namely, the excitation switches at Ton , I fires by postinhibitory rebound. Allow- τ = τ +σ the control of I from T to P. T is able to regain control ing for synaptic delay from I to P,at Ton ip, P of I only when P and I have precessed through up to feels inhibition from I and as a result falls back to CPI . τ = τ + σ + τ 360 degrees. Note that this functional reorganization At Ton ip Ion , P is released from inhibition. occurs without any changes in the coupling or intrin- We show in Appendix A that w? can be chosen such w (τ + σ + τ ) Table 1. List of relevant symbols. TT = intrinsic period of the pacemaker input CP = intrinsic P cell cubic TP = intrinsic period of the P cell CI = intrinsic I cell cubic Ä = − = = TT TP CPI P cell cubic with sip 1 = = = TPC period of P cell in TIP network CIP I cell cubic with spi 1 = = = TP¯ period of P cell in PIT network CIT I cell cubic with sti 1. τ = τ = Ton duration of the inhibition from T to I Ion duration of inhibition from I to P τPA = duration of the active state of the intrinsic P cell τPR = duration of the silent phase of the intrinsic P cell σip = synaptic delay from I to P σpi = synaptic delay from P to I Note: All times are measured at ² = 0 in slow time scale τ. Phase Precession of Hippocampal Place Cells 15 P line The . P C coupled dashed of the The knee that left time the coincide. reach to remaining . cell trajectories the P P o C tw of equals ¯ c these intrinsic branch time the where for right The P . I the C P space of C on of part part phase in branch dotted dashed left places the the r r the e e v v o o of l e v represent part time tra es the to curv es dashed tak equals the bold it b n o t time u solid time b the The The wn. equals cell. distance, ¯ d P sho are time ertical v cell intrinsic The P . same I the P the C to the r of of e v line co to branch dotted cell left trajectories the P the and of cell, coupled part uncoupled P the and es dashed tak coupled the it coupled the time to along The es the . 5 is olv e v ¯ b e igur F corresponds cell time 16 Bose et al. τ clarify the differences in their trajectories. In the fig- not on its duration Ion but on its timing during the cycle ure, we have used lowercase letters to denote times of P. We note that while the TIP orbit can be achieved that these cells spend in parts of their orbit where their with either positive or negative Ä, independent of the τ trajectories differ. The letters with bar superscripts cor- duration Ion , other results of our model—namely, ob- respond to the coupled P cell. Note that TPC − TP = taining the PIT orbit—require that for a short-lasting ¯ + ¯ − ( − ¯) = τ Ä c d b b . Since TPC TT , a necessary con- Ion , must be positive. dition for the TIP orbit to exist is that the difference in Ä = − intrinsic periods of P and T , TT TP , must satisfy 3.2.2. Within the Place Field, Both P and I Phase Precess. Similar to above, we analyze the PIT orbit Ä =¯c + d¯ − (b − b¯). (10) by describing the trajectories of P and I in their phase ¯ planes. In the following analysis, we assume that near The difference b − b > 0 is related to the magnitude of ∂g ∂ ∂ p = gp gp the right branch of CP , ∂v 0. Thus the speed at which . Since > 0, cells evolve through the same Eu- p ∂vp ∂vp P evolves along the right-hand branches of CP and CPI clidean distance on CP more slowly than on CP . Thus ¯ I will be the same. Our analysis continues to hold for b − b measures the additional time that the uncoupled ∂ gp > ∂v 0 but sufficiently small. We also assume that near P cell must spend to cover the same Euclidean dis- p ∂gi the right branch of CI , = 0. tance along the left branch of CP that the coupled P ∂vp For simplicity, we assume that the seed arrives at cell covers on CPI . In the case when Ä>0, a stable TIP orbit is obtained τ = 0, so that both T and P fire simultaneously. The ¯ ¯ if c¯ + d > b − b. This can be achieved by choosing trajectory of P jumps to the right branch of CP . The I ? wp(τT + σip), and hence w , such that the trajectory cell receives inhibition from T during the time interval on ( ,τ ) τ = σ of P enters a neighborhood of the stable fixed point 0 Ton , but, at pi , I fires due to excitation from P.At τ = σpi + σip, P receives inhibition from I and at the intersection of CPI and DP within the duration of τI . The trajectory of P will remain near the fixed falls to the right branch of CPI . What happens next on τ point until P is released from inhibition. In this way, is most strongly determined by Ion , the time length τ the firing of P is delayed and P can be phase-locked over which P feels inhibition from I . The time Ion , to the theta rhythm. however, is hard to define precisely as it changes during When Ä<0, a stable TIP orbit is achieved if each cycle of the phase precession (see Appendix A for − ¯ > ¯ + ¯ w (τ + σ ) clarifications of this definition). We describe how the b b c d. In this case, p Ton ip , and hence ? τ w , are chosen so that the trajectory of P remains suf- PIT orbit is obtained in the cases when Ion is short and ficiently far from the fixed point at the intersection of when it is long relative to the duration of the active state CP and DP during τI . The trajectory displayed in of P. I on τ Fig. 5 is representative of this case. Since cells evolve First consider the case when Ion is short, in partic- ¯ ¯ τ <τ faster on CP than on CP , the time b +¯c + d can be ular, Ion PA, the duration of the active state of the I τ = σ + σ shorter than the time b. This shows that inhibition can uncoupled P cell. At pi ip, let the distance be used to speed up P. Thus, P can be phase-locked along CPI from the position of the trajectory of P to τ to the theta rhythm if TP is greater than TT , as long as the right knee be greater than Ion . Then the inhibi- the periods are not too disparate. tion to P shuts off before P jumps down from the right τ = σ + σ + τ Observe that (10) can also be rewritten as knee of CPI .So at pi ip Ion , P returns to the right branch of CP and leaves the active state through Ä = τ + ¯ − . Ion d b (11) the right knee of its intrinsic cubic CP . In Fig. 6A, we have drawn the trajectory that P follows one cycle af- Ä ¯ τ For fixed , both b and d may increase as Ion increases. ter the seed is given. It follows this trajectory until the An increase in b means that P must feel inhibition at time at which the network returns to the TIP orbit. We w higher values of P along the left branch of CP . In turn, denote by TP¯ the period of P during phase precession. = this means that in TIP the phase difference between T In this case, TP¯ TP . Thus, a necessary condition for τ < Ä> and P will be greater. Hence, the duration Ion controls phase precession in this scenario is TP TT or 0. the phase difference between T and P in the TIP orbit. Here, the sole factor that drives the precession is the In summary, the TIP orbit can be obtained whether difference Ä in the intrinsic periods of P and T . the difference Ä = TT − TP is positive, negative, or It is instructive to contrast the phase portrait of I with zero. The effect of the inhibition from I to P depends that of P during PIT. As opposed to the clean and Phase Precession of Hippocampal Place Cells 17 ). values along he precessing w is the time to cover this l Continued on next page k ( cell is shown in the case that P sing . The time P C around its phase plane. The shows I "chasing" figure T is the time to cover this same difference in to the right knee of m I P C , and I P C from the right knee of P C is short. B: A depiction of the inhibition from on I τ is the time along the right branch of r k is the time to cover the dashed portion on the left branch of ¯ m cell is shown in the case that P . The time cell. The time P . P C P C A cell trajectory is also superimposed. The solid bold curves represent places where these two trajectories coincide. The dashed line corresponds to t P . A: The trajectory of the precessing is long. The intrinsic cell and the dotted line to the intrinsic on I P Figure 6 how this inhibition changes at each cycle beginning on the right branch and culminating near the knee of the left branch. C: The trajectory of the preces τ same distance, now on the left branch of the dotted portion of the left branch of 18 Bose et al. ). Continued on next page B .( Figure 6 Phase Precession of Hippocampal Place Cells 19 C .) Continued .( Figure 6 20 Bose et al. unchanging trajectory of P, the trajectory of I changes have a spatially specific (Jung and McNaughton, 1993) with each cycle of the precession. As we noticed in and theta rhythm phase specific (Skaggs et al., 1996) Fig. 4, the inhibition from T to I occurs at progressively firing pattern. In our model, the arrival time of the earlier phases during PIT. In terms of the phase portrait excitatory input, representing the memory seed from (see Fig. 6B), this means that the inhibition from T to I these granule cells, need not be so precise. In fact, occurs at different places in phase space. In particular, there exists an open interval of potential seed timings during the first cycle, T inhibits I when I is on the (or phases) that result in phase precession. The seed right branch of CIP and still far away from the right must be timed to arrive when P is sufficiently close knee. At the next cycle, T inhibits I while I is still to the left knee of CP . To determine an expression on the right branch but now closer to the right knee for the appropriate seed arrival time τseed, let T fire at a higher wi value. Progressing in this manner, the at τ = 0 and assume that the duration of the seed is inhibition from T eventually occurs when I is back on small. The seed excitation momentarily adds a cur- the left branch but still refractory. Ultimately, I receives rent of magnitude gdg[vp − vdg] to the current balance inhibition when it is close enough to the left knee of equation for the P cell, where gdg is the maximal con- CI to be captured into TIP, thus ending precession and ductance. Phase precession occurs if the seed arrives at τ ∈ (−δ,τ + σ ) δ> signaling the end of the place field. seed Ton ip where 0. The lower bound Phase precession occurs for a more complicated rea- −δ is related to the parameter gdg. For larger values of τ >τ −δ son when Ion PA. In this case, we assume that P gdg, can be larger (in magnitude). The upper bound τ reaches the right knee of CPI before the inhibition from on seed is a sufficient condition for phase precession I shuts off. Note that the right knee of CPI lies below since the P cell will certainly precess if it is excited the right knee of CP . So the length of time P stays in its before it receives inhibition from I . As this interval active state during the PIT orbit is less than the length of for τseed gives only a sufficient condition, seeds that ar- τ + σ time it stays in its active state during the TIP orbit. This rive slightly after Ton ip may actually produce phase decrease in time is the primary reason why, in this case, precession. phase precession occurs. This can be seen by consid- ering the phase plane of P. There are several subcases τ 3.2.4. The Total Amount of Phase Precession Depends depending on the time length of Ion (see Appendix A for details). In Fig. 6C, we have drawn the trajectory on the Phase of Memory Seed Arrival. In addition that P follows one cycle after the seed is given for to determining whether phase precession is initiated, the timing or phase of the memory seed also affects one of these subcases. Let τ = τ1 be the time P hits the the total amount of phase change in the P cell dur- right knee of CPI . Then P jumps back to the left branch τ = τ + σ + σ ing its precession. Since both P and I phase precess of CPI and evolves down CPI until 1 pi ip. We have denoted by lowercase letters various times of and since, in the TIP orbit, both cells fire at different evolution for the uncoupled P cell and the coupled pre- fixed phases after T , we need to consider the amount cessing P cell. Note that T − T ¯ = k + k + m −¯m. of precession for each of these cells separately. P P r l τ = There are two sources of time reduction. The larger of First consider P.If T fires at 0, then during the TIP orbit, P fires at τ = τ + σ + τ .We the two is the time kr + kl . This time is related to the Ton ip Ion can translate these times of firing to phases of firing maximum conductance for the synaptic current from I ◦ by multiplying the times by 360 /TT . The earlier the to P, gip.If gip is small, then the cubic CPI will not be too far below C , and thus the time k + k is small. seed arrives, with respect to the time of P firing, the P r l τ = τ + As g increases, so does k + k . The second source of smaller the amount that P precesses. Let adv Ton ip r l σ + τ − τ time reduction, and thus phase precession, arises due ip Ion seed be the amount of time (or phase) the to the difference m −¯m. This difference is controlled firing of P is advanced by the memory seed. The total ∂ gP amount of phase precession is a decreasing function by the magnitude of ∂v near the left branch of CP .For p τ this case, phase precession in the PIT orbit is achieved of adv. In particular, if the seed arrives in the interval (−δ, ) τ whether Ä is positive, negative, or zero. 0 (that is, if adv is large), P precesses through less than 360 degrees. Whereas if the seed arrives close τ = τ + σ τ to Ton ip ( adv small), P precesses through close 3.2.3. The Memory Seed Initiates Phase Precession. to 360 degrees, provided that the duration of inhibition τ Granule cells in the dentate gyrus make excitatory from I to P, Ion , is sufficiently short. Theoretically, the synapses on CA3 pyramidal cells, and granule cells P cell can only phase precess through a full 360 degrees Phase Precession of Hippocampal Place Cells 21 3.2.5. The Number of Cycles of Phase Precession De- pends on the Phasing of the Memory Seed and on the Duration of the Inhibition from I to P. There are two possible formulae for the number of cycles of pre- τ cession. The duration of inhibition from I to P, Ion , determines which of the two formulae applies to a given situation. In the case Ä>0(TP < TT ), either formula τ may apply with Ion actually determining which one to use. If Ä<0(TP > TT ), then only the second formula can be used. τ <τ If Ion PA, recall from Section 3.2.2, that the PIT orbit exists only for Ä>0. The number of cycles is given simply by Figure 7. Amount of phase traversed by the P cell during phase precession in the Morris-Lecar-based network model. When zero − τ = TT adv , phase is defined by the peak of the T cell spike, P cell fires (burst n Ä (12) times indicated by circles, filled circles indicate Fig. 4 results) at 152 degrees during TIP (first five bursts). Seeds arriving earlier (first where n is the nearest integer to the ratio value. Here, seed at 500 msec) result in less phase precession than seeds arriving later (last seed at 550 msec). Note also that the number of cycles of it is the difference between the intrinsic periods of T phase precession increases with later seed arrival. and P, together with τadv, that determines the num- ber of cycles of precession. The network model that displayed the results in Figs. 4 and 7 generates a PIT Ä τ τ → τ + σ + τ orbit by satisfying these conditions on and Ion and in the limit seed Ton ip Ion from below. For τ the number of cycles of phase precession is accurately Ion large, such a seed would fall well outside of the interval described in Section 3.2.3 and thus would be predicted by (12), as shown in Table 2. The values = . = . unlikely to produce phase precession. Thus, a lower TT 100 3msand TP 87 4 ms were used. Ä τ <τ bound on the maximal amount of phase precession is If is negative with Ion PA, then after the seed, τ 360◦(1 − Ion ). (12) predicts and simulations corroborate (not shown) TT An example of the effect of the seed timing on the that the P and I cells phase recess back to the TIP amount of phase precession is shown in Fig. 7 for the periodic orbit. τ >τ Morris-Lecar-based network model. In the figure, the If Ion PA, as described in Section 3.2.2, there is Ä = − filled circles indicate the phase of each P cell burst no requirement on the sign of TT TP to obtain during the TIP and PIT orbits displayed in Fig. 4. Dur- the PIT orbit. In this case, the number of cycles of ing TIP, P fires at approximately 152 degrees (dashed phase precession is given by the following expression horizontal lines) when 0 degrees is defined by the T where the times are defined in Fig. 6C: spike. The memory seed advances P cell firing to ap- T − τ proximately 77 degrees and then P precesses through = T adv . n Ä + + + −¯ (13) 285 degrees until returning to the TIP orbit and re- kr kl m m suming firing at 152 degrees. The open circles indicate Recall that the quantity kr + kl + m −¯m is positive, the phase of P firing when the seed arrives at differ- which compensates for a potentially negative Ä. The ent times. For the earliest seed arrival, P precesses through approximately 170 degrees, and for the latest seed, P precesses through roughly 340 degrees. Table 2. Number of cycles of phase precession predicted In contrast to the restriction on the amount of by Eq. (12) and observed in simulations of Morris-Lecar- phase precession of the P cell, the I cell can pre- based network (parameters as in Figs. 4 and 7) for different arrival times of the memory seed. cess through the full 360 degrees. This is possible if τ ∈ (τ − σ ,τ + σ ) τadv (ms) 54 39 29 19 14 9 3 seed Ton pi Ton ip . The lower bound occurs if the seed arrives at such a time so that the excitation npredicted 455677 8 from P to I arrives exactly at the moment T releases n 456778 8 I from inhibition. observed 22 Bose et al. ∂ time m −¯m is related to the magnitude of gp near ∂vp the left branch of CP . The magnitude of the time kr + kl is of the order of gip and arises because the burst width and interburst interval are longer for the isolated P cell than in the PIT orbit. We ran several simula- tions (not shown) of the Morris-Lecar-based network model with TP > TT (specifically, TT = 100.3msand TP = 102 ms). With gip = 1.0, we found n = 18, and for gip = 1.5, n = 9, thus corroborating the strong de- pendence of the number of cycles of phase precession on gip. Reiterating, if Ä>0, either (12) or (13) may hold Figure 8. Phase shift of P firing during each cycle of precession depending on the duration of the inhibition from I to P. changes linearly with changes in animal’s running speed, modeled by changing intrinsic frequency of P. In Morris-Lecar-based net- By continuous dependence on parameters, there ex- work model, phase difference of P bursts during precession (circles, τ ists an intermediate value for Ion at which the network filled circle corresponds to Fig. 4 results) increased linearly as in- switches from (12) to (13). Note that near this switch- trinsic period of P was decreased, corresponding to an increase in Ä = , , , µ / 2 ing point, small changes in the active duration of I (applied current to P, Ip 95 98 100 103 and 105 A cm ). The = have large ramifications with respect to the occurrence period TT 100 ms was fixed throughout. of precession and to the number of cycles of precession. 1φ = 3.2.6. The Spatial Correlation of Phase Precession The phase shift at each cycle is given by 360◦ Ä . Thus the phase shift is a linear function of Is a Speed-Corrected Temporal Phenomena. Other TT models of phase precession rely explicitly on the as- the difference in period, and for fixed TT it is a linear sumption that phase precession is a spatial phenomena. function of TP . This linear relationship holds in our We show here that the apparent spatial dependence of Morris-Lecar-based network model as shown in Fig. 8. phase precession can be accounted for by our tempo- The filled circle showing the largest phase shift dur- ral model. More precisely, if the interburst frequency ing each cycle of phase precession corresponds to the of the place cell and the frequency of the underlying parameter values in Figs. 4 and 7. To model a de- theta rhythm are linear functions of the animal’s speed, crease in running speed, the intrinsic frequency of the then phase precession can be a temporal phenomenon P cell was decreased. In response to this increase in Ä and also be more correlated with the animal’s location TP , corresponding to a decrease in , the phase shift 1φ than with the time that has passed since it entered the decreases linearly. place field. Assume the linear relationship fT = f B + γ1v, 3.2.7. Suppression of Out-of-Place Field Firing. = + γ v f P¯ f B 2 , where fT is the theta frequency, f P¯ The primary objective of our model is to illustrate a is the frequency of the precessing P cell and γ2 >γ1 mechanism for generating the phase precession of place are positive constants. The velocity v is set to v0 + 1v, cells. In the model, phase precession only occurs in the where v0 is the minimum velocity needed for the theta place field, but the out-of-place field firing is too high. rhythm to exist, and 1v is the positive deviation from This implies that a downstream phase-based detector this baseline velocity. Without loss of generality, the could precisely determine the animal’s location at all frequency f B is a common baseline for 1v = 0. Let phases except for the phase-locked one. γ = γ − γ = + γv γ> 2 1, then f P¯ fT , where 0. The Different complicated features could be added to the time of the nth theta cycle peak is nTT , and the time of model to reduce or remove the out-of-place field fir- nTT the nth P cell burst, τn,is nT ¯ = . The amount ing. For example, there are many types of interneu- P 1 + TT γv of phase shift after the nth burst φn is the difference rons in the CA3 region that have a diverse pattern nTT − nT ¯ divided by the period of the theta rhythm; of connections and an incompletely determined role P γv φ = ◦ nTT thus n 360 1 + T γv. The position of the rat at the in network activity. In particular, some of the in- T v = vτ = nTT time of the nth burst is xn n + γv. By in- terneurons make synaptic connections only on other ◦ 1 TT spection, φn = 360 γ xn, or equivalently, the phase is interneurons (Freund and Buzsaki,´ 1996). By introduc- spatially correlated. ing one additional interneuron into our current model, Phase Precession of Hippocampal Place Cells 23 the inhibition from I2 can take effect. Moreover, this slow inhibition decays away over one theta cycle thus allowing I1 to again be in a position to fire by rebound. The inhibition from I1 to P, which lasts the entire theta cycle is strong enough to prevent P from firing during the cycle. Since this inhibition is renewed at each cy- cle, P never fires. We call this the TI1 I2{P} orbit and note that T controls the dynamics of the network. As in the simple model, the externally provided memory seed fires P and interrupts the TI1 I2{P} orbit. Excitation from P causes I2 to fire. The subsequent slowly decaying inhibition from I2 to I1 now comes before the fast inhibition from T . Thus I1 is strongly inhibited and initially will be unable to respond to T input. In the reorganized PI2{I1}T orbit, phase pre- cession occurs with P controlling I2 directly, causing it to fire with each P burst and thus phase precess. In this orbit, P also controls I1, indirectly through I2.As in the simple model, the timing and effect of the in- put from T to both I1 and I2 changes from cycle to cycle. The effect on I2 is similar to that discussed in Section 3.2.2 and shown in Fig. 6B. The inhibition from T chases I2 around its phase plane. When I2 has precessed through up to 360 degrees, it can be recap- tured by T . For precession to end, however, I1 must be recaptured by T . Figure 9. The voltage traces for the three cell and pacemaker net- Because P and I2 are phase precessing, the inhibi- work. Phase precession begins at t roughly equal to 500 msec with tion from I2 to I1 comes at progressively earlier phases. the arrival of the memory seed. The cells P and I2 then phase precess This means that at each cycle of theta, there exists pro- for the next 6 cycles. Outside of the place field, the longer lasting gressively more time for this inhibition to decay before inhibition from I1, which is periodically reinforced, forces P to stay below threshold. the T input to I1 arrives. Similar to the basic model, phase precession ends when T recaptures control of I1. This occurs when the input from I2 to I1 comes early the out-of-place field firing can be suppressed as shown enough in the T cycle so that the inhibition decays in Fig. 9. In this more complex model T and P are as away sufficiently to allow I1 to rebound in response to before, but now there exist two interneurons, I1 and I2. T input. Once this happens, the slow inhibition from Both interneurons receive fast GABA-mediated inhi- I1 to P is reinitiated, thus suppressing P. Moreover, bition from T and are capable of firing rebound spikes. since P has been functionally removed from the cir- The interneuron I1 now provides a slowly decaying cuit, I2 must now wait for T input to fire. As discussed GABA-mediated inhibitory current to P and receives above, this input occurs at a phase that allows I1 to fire a similar current from I2. The interneuron I2 receives periodically. fast glutamatergic excitation from P. As before, P The complex model described here operates in a fun- may also have a feedback connection from I2 that can damentally similar fashion to the basic TIP/PIT model. either be a fast or slow inhibitory current. With this As before, the role of inhibition differs inside and out- network architecture, we require that the slow inhibi- side of the place field. Indeed, control of the network tion decay in roughly one theta cycle—that is, roughly of interneurons is again crucial for determining the 100 msec. network output. Moreover, our analytical results in Outside of the place field, I1 and I2 fire by postin- Sections 3.2.2 through 3.2.6 carry over with little or hibitory rebound in response to T . Note that the synap- no modification. In this model, it appears that I1 has tic delay from I2 to I1 allows the latter to rebound before an “antiplace field” in that it fires everywhere except 24 Bose et al. within the place field. In a less reduced and more re- in the environment. However, the phase relationship alistic network model, this may not be the case. As a between place cell activity also provides information result, we do not intend our results to be interpreted as on the location of the animal, and the background fir- predicting the existence of antiplace fields. ing outside of the place field may not be a substantial problem for a downstream system that uses phase to de- termine the animal’s location. We presented a method 4. Discussion (not necessarily unique), using an additional interneu- ron, to remove the out-of-place field firing without re- We described a minimal biophysical model of the phase quiring upstream information on the size and extent of precession of hippocampal place cells that is consis- the place field. tent with the essential empirically determined proper- From a mathematical point of view, this work shows ties of the phenomenon. We identified mechanisms the importance of determining functionality of the sub- whereby temporal control of phase precession can oc- pieces of the model. We have shown how control within cur. In particular, we proposed a mechanism for chang- the network can be switched from one network member ing the firing pattern of place cells as the animal enters, to another and then back again. We have demonstrated crosses, and then leaves the place field. This mecha- that geometric analysis is well suited to identify the nism of changing control of the interneuron from T to mechanisms that change control. Our work also high- P switches the network from a stable state to a tran- lights the changing and perhaps nonintuitive effects of sient state. The precession of the interneurons pro- inhibition in the network. In particular, we have shown vides a second mechanism to switch the control of the how a faster oscillator can entrain a slower one using interneuron back to T and thus return the network to a only inhibition. This depends critically on the timing of stable state. the inhibitory input from the faster oscillator, which is The initial form of the model accounts for most of the consistent with the theme that timing is of fundamental empirical observations that were listed in the introduc- importance in temporal code generation. We have also tion. Namely, (1) since we provide the seed only when shown how the network can use inhibition to function- the rat moves in one direction, place cells only fire in ally enhance or remove the effects of a chosen member. one direction of motion; (2) all place cells, only the single P cell in our case, start firing at the same initial phase; (3) the initial phase is the same on each entry of 4.1. Related Work the rat in the place field since the seed always occurs at the same phase; (4) since the change in control in Other models of phase precession differ from ours in the network occurs before 360 degrees of precession, important ways. In prior work (Tsodyks et al., 1996; it is impossible for there to be more than this amount Jensen and Lisman, 1996; Wallenstein and Hasselmo, of phase precession; (5) while the cells in the dentate 1997; Kamondi et al., 1998), phase precession is mod- gyrus may undergo a small number of cycles of pre- eled as a spatial phenomena. Each of these models in cession, our model requires only one cycle; additional some way utilizes external input that already encodes cycles would not change the results; (6) the model does for the phase precession. Thus none of the models truly not account for the increase in firing rate (a less reduced explains the genesis of phase precession. Namely, the network model that accounts for individual spikes may models do not address how a single P cell behaves be able to include firing rate information; we believe both outside and inside its place field or what causes that the mechanisms we have uncovered for phase pre- the transition in behavior of the cell between these two cession in our idealized bursting neurons would carry areas. over to neurons that instead exhibit bursts of spikes); The above models require either a network of exci- and (7) the linear dependence on frequency of pyra- tatory cells or a highly parameterized description of the midal cell firing is used here to maintain a correlation neuron to achieve precession. In the work of Jensen and with location rather than time that has passed since the Lisman (1996), a one-dimensional chain of pyramidal animal entered the place field. cells is considered, and phase precession occurs due The initial model, however, had a high out-of-place to the local unidirectional excitatory synapses between field firing rate. This out-of-place field activity would these cells. Selective excitation to a specific member result in a loss of spatial specificity if a downstream sys- of the chain, occurring at each theta cycle, provides tem uses firing rate to determine the animal’s location phase-precessed input to the network. In the model of Phase Precession of Hippocampal Place Cells 25 Tsodyks et al. (1996), strong excitatory connections interneurons are most strongly inhibiting pyramidal between place cells is also required. In this model, cells. In the present model the pyramidal cell con- precession is driven by asymmetric synaptic weights trols the firing of the interneuron during the phase between these cells. The total amount of excitatory in- precession process, and its activity is not adversely put a cell receives is at a maximum in the center of the inhibited by the interneuron. place field and decreases monotonically as the distance In recent recordings of a larger number of pyra- from the center increases. Tsodyks et al. (1996) call midal cells and interneurons (Csicsvari et al., 1998), this “directional tuning.” A similar idea is employed pyramidal cells were found that had large, significant by Kamondi et al. (1998). There a ramplike depolariz- cross-correlation peaks at times on the scale of ten ing current is provided to the place cell to mimic pas- milliseconds that preceded the activity of simultane- sage through the place field. Thus in these two models, ously recorded interneurons. These correlations find the running animal is given external information about that there is a tight coupling in the time at which pyra- its position in space that is encoded in the spatially midal cells and interneurons are active and suggests dependent depolarization. Kamondi et al. (1998) and that the activity of a subset of the interneurons might Wallenstein and Hasselmo (1997) both use multicom- precess along with place cells. The experimental ev- partment descriptions of the place cells. Precession idence of the high level of correlation between place in these models depends critically on a more com- cells and interneurons demonstrates that there exists plex, multiparameter description of each neuron. More- a sufficiently strong conductance between subsets of over, the model of Wallenstein and Hasselmo (1997) these two types of neurons. These findings are consis- also requires phase-precessed external input at each tent with the predictions in the present model in which theta cycle as in Jensen and Lisman (1996) to achieve the phase precession of interneurons may only occur in precession. a subset of interneurons and only when they are being The present model takes an entirely different ap- driven by a phase precessing place cell. proach. First, we believe phase precession is a tem- poral mechanism that, as demonstrated, can account for changes in the rat’s running speed. Second, we do 4.3. Consistency of Biophysical not require any external precessed input to the network. and Functional Models Instead, we simply need a one time dose of excitation that mimics a memory seed and changes the network In a model of phase precession, there are several im- from the TIP to the PIT orbit. We do not require a portant larger goals and anatomical considerations that network of excitatory cells or multicompartment mod- should be taken into account. There is considerable els for the cells. Finally, our analysis gives a different data that demonstrates that the hippocampus has a interpretation of the significance of phase precession role in episodic memory. We have proposed a func- than all of the previous studies. Namely, in our model tional model that describes how the spatial and mem- the end of phase precession signals the end of the place ory roles of the hippocampus can be combined (Recce field. In the other studies, the end of the place field and Harris, 1996; Recce, 1999). In this view, when signals the end of the precession. Thus our model pro- a rat returns to a previously experienced environment, vides an internal mechanism for determining the end of the set of coactive place cells corresponds to the recall the place field (via the end of precession), whereas the of an egocentric map of the environment. The memory other models require another external input to notify recall is performed by a pattern completion process that the pyramidal cell that its place field has ended. is controlled through the excitatory feedback pathway in the CA3 region of the hippocampus. The seed for this recall enters the CA3 region through the entorhi- 4.2. Experimental Support nal cortex to the dentate gyrus and then in the mossy fiber projection from the dentate granule cells to the Several studies have demonstrated that the mean phase CA3 pyramidal cells and interneurons. The temporal of activity of hippocampal interneurons is just prior to control of the recall process, in this view, is provided the mean phase of pyramidal cell activity (Fox et al., by the pacemaker input from the medial septum. 1986; Buzsaki´ and Eidelberg, 1982). This implies that The feedback connections between CA3 pyramidal during the phase precession, the activity of a place cell cells are relatively sparse, which can limit the num- must pass through the phase of theta rhythm at which ber of egocentric maps of space that can be stored if 26 Bose et al. the region is modeled as an autoassociative memory. 1(wp) it takes P to reach wLK starting from wp.In However, Gardner-Medwin (1976) has demonstrated particular, 1(wRK) = τPR and 1(wLK) = 0. It is clear that this limitation can be overcome if the recall pro- that 1 is a monotone decreasing function of wp. The cess is performed over a sequence of steps. Hirase description of the TIP network given earlier shows that and Recce (1996) found that the optimal performance there is a one-dimensional map 5 that takes the initial in a multistep autoassociative memory model of the position of P at τ = 0 along the left branch of CP and CA3 region can be achieved if an interneuron network returns the position of P at τ = TT . controls the recall process by providing an inhibitory input to the pyramidal cells that is a linear function of Proposition 1. The map 5 defines a uniform con- the number of simultaneously active pyramidal cells. traction on a subinterval (wlo,whi) of (wLK,wRK).If Further, it has been proposed that the phase precession w? is the resulting locally unique asymptotically stable corresponds to the multiple steps in this recall process fixed point , then the network possesses a locally unique and that the phase precession suggests a method for asymptotically stable singular TIP periodic orbit with ? the concurrent recall of several egocentric memories period TT , where wp(0) = w . (Recce, 1999). In the present work, we have begun the process Proof We prove the existence of a fixed point by com- of systematically addressing these issues. The present paring the behavior of the uncoupled P cell to its cou- = + Ä model for phase precession, while conceptually quite pled counterpart. Since TT TP , we can compare simple, is not necessarily the one that most easily pro- the period of the coupled P cell to TT . First we locate w duces output that phase precesses. Instead, our mod- a suitable hi. Associated with it is a time Thi from w to w on C .At τ = τ + σ , P receives inhi- eling is based on the larger considerations of identify- hi LK P Ton ip bition and jumps back to C .At τ = τ + σ + τ , ing mechanisms that can allow multiple memory recall PI Ton ip Ion w processes to be simultaneously performed. The basic the inhibition to P shuts off. We choose hi such that w (τ + σ + τ ) = w . Next let us consider the framework described above, which accounts for pre- p Ton ip Ion LK evolution of the P cell, solely on CP in the absence of cession, is compatible with a functional model of the ∂ gp > any synaptic input. Since ∂v 0 near the left branch spatial and episodic memory roles of CA3 place cells. p of CP and CPI , the P cell moves down CP at a slower rate than on C . In other words, P takes a longer Appendix A. Analysis: Existence and Stability PI of TIP Periodic Orbit time to cover the same Euclidean distance on CP than τ = τ + σ + τ on CPI . Thus at Ton ip Ion , the uncoupled P w Outside the place field, the network is in the TIP config- cell will be above LK. It is seen that inhibition ac- uration. We show that there is a unique, stable periodic tually shortens the time distance to the knee for this orbit to which the cells are attracted. Assume that T initial condition. This is very similar to, but timewise fires at τ = 0. We will locate an initial condition for the opposite of, the idea of virtual delay of Kopell and Somers (1995). There they showed that excitation had P and show that when T fires again at τ = TT , P has returned exactly to this initial condition. The analysis the opposite effect of increasing the time to the knee. will be in terms of a time metric that will measure the The coupled and uncoupled versions of P follow the times of evolution of P on various parts of its orbit in same trajectory in the active state. Ä = phase space. These times depend on various biophys- Assume momentarily that 0. For the uncoupled 1(5(w )) = ical parameters that are related to both intrinsic and P cell, hi Thi. But for the coupled P cell, 1(5(w )) < synaptic properties of the cells and the network. hi Thi. Thus the time it takes for the coupled P cell to return to its initial position whi is Tpc < TP .If Ä < + Ä = Theorem 1. There exists a locally unique asymptoti- is sufficiently small, then Tpc TP TT . Note >τ + σ + τ cally stable TIP periodic orbit with a period O(²) close that Thi Ton ip Ion . How big the difference to TT . between these times needs to be depends on the size of ∂gp ∂v , which determines the difference in the rates along ² = p Proof The analysis below occurs at 0, but using CP and CPI . Thus we have located an initial condition the work of Mischenko and Rozov (1980), the results whose time distance to the knee compresses over one actually hold for ²>0 and sufficiently small. Consider oscillation. the left branch of CP between wRK and wLK. Associ- We next locate an initial condition whose time dis- ated to each point on this curve, denoted wp, is the time tance to the knee expands. Now let wlo = wp(0) such Phase Precession of Hippocampal Place Cells 27 w (τ + σ ) = w that for the uncoupled P cell P Ton ip LK. ters synchrony due to fast threshold modulation (see Assume that the initial conditions for T and I are iden- Somers and Kopell, 1993) implies that there is com- tical to the above case. Thus at the moment P were pression across the jump. The rational is the follow- to fire, it receives inhibition from I . It will thus fall ing. Across the jump, the Euclidean distance does not back to CPI and move down this cubic toward the fixed change. However, the rate of evolution of the cells on τ τ = τ + σ + τ point on CPI for a time Ion .At Ton ip Ion , the left branch is much slower due to the fact that these P is released from inhibition and jumps to the right cells are evolving near a critical point. On the right branch. The w value of the point to where it jumps lies branch the cells are very far way from the w-nullcline below wLK. On the the right branch, let the time be- Dp, so the rate of evolution is much faster. Thus the tween this point and wLK be denoted by τD. Once P cells travel through the same Euclidean distance in a passes through wp = wLK on the right branch of CP ,it much shorter time. Thus the new time between cells will follow the same orbit as the uncoupled P cell. The is less than δ. Since the cells always satisfy the same τ + τ coupled P cell has been delayed by a time Ion D. differential equation, the time between them remains So it will return to its initial position at a time Tpc > TP , invariant on the right branch of CP . Moreover, since if Ä = 0. Thus if Ä is sufficiently small, Tpc > TT .For they both jump down from the right knee of the CP , this initial condition, the time distance to the knee has the time is invariant across the down jump. When P1 been expanded—that is, 1(5(wlo)) > 1(wlo). and P2 return to the left branch of CP , P2 will lie below Therefore, we have located a set of initial conditions P1. So when P2 returns to the initial position of P1 at such that the lower and upper boundaries of the set w1, the time between the cells will be less than when are mapped into the interior of the set by the map 5. they started, so the Euclidean distance between them By continuous dependence on initial conditions, this must have compressed. This proves that 5 is a con- implies that there exists at least one element of the set traction. The constant λ can roughly be approximated that remains fixed under the map 5. To prove that this by the ratio of the speed at the jump on point on the point is unique and attracting, we need to show that right branch to the speed at the jump off point on the 5 is a contraction mapping. We follow two different left branch (Somers and Kopell, 1993). versions of the P cell, denoted P1 and P2, and show that We have made some implicit assumptions about the the dynamics of the TIP network bring these cells closer behavior of I . Namely, we have not proved that I ac- together after one iterate of 5. Let w1 = =− and sTI 0. The second is CIPT , which occurs when the leak reversal potential is VL 60 mV. The mem- = = = µ / 2 both sPI 1 and sTI 1. Note that CIPT lies between brane capacitance is Cm 20 F cm . The applied v µ / 2 = = CIP and CIT . Depending on the parameters gpi , gti, pi , current values (in A cm ) are I p 105 and Ii 120. and vti, it may lie above or below CI . For this argument In our implementation of the model, the T cell is mod- assume it lies above CI . Then the maximum w value eled by the same equations as an isolated P cell, except to which I can jump is the w value of the left knee of applied current is set to 92 µA/cm2. In order to ob- w w CIP , call it iM . The minimum value is the value of tain frequencies in the theta range, time was scaled by ∂gi the intersection of CI and DI , call it wi . Since = 0 4.5. The model equations were numerically integrated m ∂vi near the right branches, the speeds of evolution along using XPP (information on the program available at all the right branches are identical. For this argument http://www.pitt.edu/∼phase). the ordering of the right knees of the relevant cubics is The gating kinetics of the potassium conductances < < < = RIT RI RIPT RIP , where the notation should be in each cell (X p and i) are governed by equations clear. So, by a long active phase of the interneuron, we of the following form: w τ mean the time from iM to RIT is longer then PAI .By a short active phase we mean the time from w to R is dw w∞, (v) − w im IP X = φ X X , τ less than PAI . Note that these times are simply bounds dt τw,X (v) for the length of the active phase. They do not imply that I necessarily jumps down from a specific knee. where φ = 0.005 corresponds to ² in the general form Indeed, I ’s jump down point may change each cycle of the equations. The steady-state activation and inac- depending on the timing of inhibition from T . Finally, tivation functions and the voltage-dependent time con- if the length of the active state of I is too long—for stant functions for the calcium and potassium currents w example, if the time iM to RI is much longer than in each cell (X = p and i) are given by τ + σ , then the analysis presented above breaks PAI pi · µ ¶¸ down. This is because I will not be in a position to 1 v − v1 respond to the next bout of excitation from P. m∞(v) = 1 + tanh , 2 v2 · µ ¶¸ 1 v − v3,X w∞,X (v) = 1 + tanh , Appendix B. Model Equations 2 v4,X and Parameter Values τ (v) = 1 . w,X (v − v )/ v We have numerically implemented our two cell and sech[ 3,X 2 4,X ] pacemaker network using the Morris-Lecar equations (1981) to model the neurons. The current balance equa- The half-activation voltages for the gating functions are v = v =− . v = v =− tions for the P and I cells are (in mV) 1,p 1,t 1 2, 3,p 2, and 3,i 25. The activation and inactivation sensitivities for the gat- ing functions are (in mV) v , = v , = 18, v , = 30, dvp 2 p 2 i 4 p C =−g m∞(v )(v − v ) − g w (v − v ) v = m dt Ca p p Ca K p p K and 4,i 10. The synaptic currents in each cell are governed by −gL (vp − vL ) − sipgIP(vp − vIP) + I p, the variable sXY where X indicates the presynaptic cell dvi Cm =−gCam∞(vi )(vi − vCa) − gK wi (vi − vK ) and Y indicates the postsynaptic cell. These variables dt satisfy equations of the form − gL (vi − vL ) − spi gPI(vi − vPI) µ µ ¶¶ − s g (v − v ) + I , v − v ti TI i TI i 0 = α( − )1 + X 5 − β , sXY 1 sXY 1 tanh sXY 2 v6 where vX (in mV) is the membrane voltage in the pyramidal cell (X = p) and the interneuron (X = i). where the Heaviside function in the general form of the The maximal conductances for the calcium and potas- equations for sXY is replaced by a sigmoidlike function sium currents in the cells are the same (gCa = 4.4, that depends on the presynaptic voltage (half-activation gK = 8.0). The reversal potentials for the ionic cur- v5 = 0 and activation sensitivity v6 = 10). For the three rents are VCa = 120 mV and VK =−84 mV. The leak synaptic currents in the model, governed by spi , sip, 2 conductance density in each cell is gL = 2 mS/cm , and and sti, the rise and decay rates of the synapse are Phase Precession of Hippocampal Place Cells 29 the same (α = 2, β = 1). 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