New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Temporal delays among place cells determine the frequency of population theta oscillations in the hippocampus

Edited* by Nancy J. Kopell, Boston University, Boston, MA, and approved March 5, 2010 (received for review October 29, 2009)
Abstract
Driven either by external landmarks or by internal dynamics, hippocampal neurons form sequences of cell assemblies. The coordinated firing of these active cells is organized by the prominent “theta” oscillations in the local field potential (LFP): place cells discharge at progressively earlier theta phases as the rat crosses the respective place field (“phase precession”). The faster oscillation frequency of active neurons and the slower theta LFP, underlying phase precession, creates a paradox. How can faster oscillating neurons comprise a slower population oscillation, as reflected by the LFP? We built a mathematical model that allowed us to calculate the population activity analytically from experimentally derived parameters of the single neuron oscillation frequency, firing field size (duration), and the relationship between withintheta delays of place cell pairs and their distance representations (“compression”). The appropriate combination of these parameters generated a constant frequency population rhythm along the septo–temporal axis of the hippocampus, while allowing individual neurons to vary their oscillation frequency and field size. Our results suggest that the fasterthantheta oscillations of pyramidal cells are inherent and that phase precession is a result of the coordinated activity of temporally shifted cell assemblies, relative to the population activity, reflected by the LFP.
Neural oscillations detected in the local field potential (LFP) provide important information about the cooperative activity of neuronal populations (1 –3). In the simplest case, the firing rate of a subset of neurons oscillates with a particular mean frequency, and this seed population functions as the pacemaker and biases the discharge phase of the remaining majority. Examples include the various rhythms of the thalamocortical system, where individual neurons fire strictly at a specific phase of the LFP (4). A similar scenario has been hypothesized to be realized in the hippocampal–entorhinal system, with the medial septum serving as the pacemaker of the prominent theta rhythm (5–10 Hz) (5, 6). However, hippocampal neurons in the exploring rat are active in short bouts, typically representing a particular place (7). During such highfiring epochs, place cells oscillate faster than the frequency of theta LFP (8 –10), the result of which is a progressive phase precession of place cell spikes (8, 11). The frequency of the LFP theta is constant across the whole hippocampus, even though the size of place fields increases whereas the oscillation frequency of place cells decreases along the septo–temporal (dorsal–ventral) axis of the hippocampus (12 –14). Furthermore, the LFP theta is highly coherent at different stages of the hippocampal–entorhinal loop (15 –19).
The discrepancy between the oscillation frequencies of spiking place cells and the global rhythm reflected in the LFP led us to pose several questions. Given that the output of hippocampal pyramidal cells is in tune with their targets, how do phase precessing place cells generate a rhythmic activity at theta frequency? Furthermore, how does the neuronal coordination allow for large differences of place cell oscillation frequencies across the hippocampal regions? To address these questions, we designed an analytical model and confronted the model predictions with experimentally derived parameters.
Results
All data were collected while the rats were running on various tracks (14, 19 –21). Fig. 1A illustrates the spatial distribution and temporal course of spiking of example neurons recorded in a single session. Spikes of active place cells showed robust theta phase precession as the rat moved across the place field (Fig. 1 B and C ), and the oscillation frequency of individual place cells was faster than that of the simultaneously recorded LFP theta (Fig. 1D ). As reported previously (9, 11, 20, 22, 23), the thetascale time lag (τ) and the distance between the preferred locations of cell pairs were correlated (Fig. 1 E and F ). The linear part of the curve was confined to the temporal window of approximately a half theta period (ca. 60 ms), representing ≈50 cm distance (12, 20, 22). The distances between the place fields correlate with the time it takes for the rat to run between the place fields (“real” time; T in Fig. 1E ). This relationship reflects an ≈20fold temporal compression, assuming a constant running speed of 50 cm/s.
Despite the systematic, positiondependent forward phase shift of spikes, the number of spikes fired by CA1 place cells per theta cycle increased then decreased as the rat traveled through the field (Fig. 1B ), such that the highest firing rate occurs at the center of the place field, coinciding with the maximal firing at the trough of the LFP theta (Fig. 1C and Fig. S1; see also refs. 22, 24).
Assuming a constant locomotion velocity within and across trials, travel distances can be expressed in the time domain as travel durations. Only trials in which the rat ran in a continuous forward motion (minimal speed 10 cm/s) and the LFP was sufficiently theta modulated (see Methods ) were included in the analysis. We characterized the activity of pyramidal cells using their oscillation frequencies f _{0}, their relative frequencies with respect to the LFP theta f _{0}/f_{θ} , and their placefield sizes in seconds L. Further, the travel time between place fields T and the theta time scale time lag between place cell pairs τ are correlated, and their relationship defines the “compression factor” c (Fig. 2 and Fig. S2). These experimentally derived measures were subsequently used in the modeling section of this report (below). In the example session shown in Fig. 2, CA1 pyramidal cells during running in the maze and in the runningwheel were recorded in the same session (Fig. 2 A and B ). Note that both the LFP and pyramidal cells oscillate slower during wheel running than in the maze and that the firing fields L are longer in the wheel.
As a surrogate of direct measurement of phase precession of single cells, we calculated the relative oscillation frequencies of single neurons with respect to corresponding LFP segments (9, 19, 25). Overall, 84% of CA1 and 78% of CA3 pyramidal cells oscillated faster than the simultaneously recorded LFP theta (Fig. S3). Interneurons in general sustained a high and relatively stable firing rate during the run and oscillated at the same frequency as the LFP theta (but see refs. 9, 26).
Because nearly all active CA1 and CA3 pyramidal cells oscillated faster than the LFP, it is important to clarify how their population behavior relates to the LFP theta. Therefore, we analyzed not only the single unit activity but also the population output of place cells (POP) consisting of the combined spikes of the simultaneously recorded pyramidal cells. When sufficiently large numbers of pyramidal cells were active simultaneously in a given epoch (see also Fig. S4), POP showed oscillatory fluctuations at the same frequency as the simultaneously recorded theta LFP (Fig. 3 A–E and H ). At the same time, individual pyramidal cells of the same population oscillated at a higher frequency than the LFP and the POP (Fig. 3 F and G ), indicating that the frequency of the global output of place cells is slower than that of the constituent neurons.
To understand better why the summed activity of neurons, reflected by the LFP, oscillates slower than the frequency of the individual cells, we built a firing rate–based model of a population of place cells that allows an analytical calculation of the model POP (mPOP). In the model shown in Fig. 4, each single place cell oscillates at 8.6 Hz for ≈1.5 s duration with increasing and decreasing rate/cycle to mimic the firing patterns of representative place cells in the dorsal hippocampus (see Methods and ref. 27). All model neurons are identical, and they cover together the entire “trial” interval (2 s) evenly. However, their oscillation cycles are shifted with respect to each other according to the compression factor c = 0.075, meaning that the time offset between the oscillatory cycles τ of the respective neurons depends on the time difference between the centers of their place fields T as τ = cT (as in Fig. 1E and Fig. 2). The model neurons form a chain of overlapping firing fields (Fig. 4 A and B ), and their summed activity produces an oscillation of constant amplitude A whose frequency, f_{θ} = 8 Hz, is slower than the frequency of the individual model neurons (Fig. 4B ). As a result of the phase interference between the faster single cells and the slower population oscillation, the phase between these two oscillators shows a systematic forward shift (phase precession), with the highest discharge probability at 180° (Fig. 4C and Eq. S2 ), corresponding to the trough of the theta cycle, as observed experimentally (Fig. S1).
We assume that the activity of place cells can be approximated by a sine wave of frequency f _{0} multiplied by a Gaussian place field of width and centered at time T (see Eq. 2 ). The population frequency f_{θ} and the oscillation amplitude A are calculated from the compression factor c, the singlecell oscillation frequency f _{0}, and the place field size L (Eqs. 6 and 7 and Fig. S4). The oscillation frequency of the mPOP f_{θ} = f _{0}(1 − c) (Eq. 7 ) is directly linked to the delays between the individually oscillating place cells, represented by the compression factor c. The oscillation amplitude is (Eq. 6 and Eq. S1 ) and goes to zero for infinitely large place fields (corresponding to large L), with fixed single cell oscillation frequency f _{0} and compression factor c, because the phaseshifted oscillators average each other out. Furthermore, the phase shift between the oscillation of the single place cell and mPOP at the center of the place field (T_{n} ) is always zero (Eq. S2 ), corresponding to the trough of LFP theta. The population frequency can be written as f_{θ} = f _{0} – 1/L (Eq. 8 and Fig. S5), and the amplitude is a constant (Eq. S7 ), if we constrain the size of the place field to 360° phase precession (refs. 8, 28; but see Fig. S6).
Using measured parameters from experiments (e.g., f _{0}, c, and L), we predicted other experimental parameters (e.g., f_{θ} ) with the model. There is an excellent agreement between the model (using Eq. 7 ) and experiment in both the examples (compare Figs. 2 and 5) and in 14 additional sessions in which sufficiently large numbers of neurons were available for the estimation of the compression factor (Fig. 6A ). In additional sessions in which c could not be directly measured, it was estimated from the relationship between firing field size L and the oscillation frequency f _{0} (Eq. 9 ). The prediction of the model agreed well with the experimental observations for these 680 neurons (Fig. 6 B–E ).
Up until now, we assumed in the model that the place field size and single neuron oscillation frequency were relatively uniform. However, the place field size can be very variable within the dorsal hippocampus, and the mean place field size increases along the septo–temporal axis, parallel with a decreased oscillation frequency of place cells (12 –15, 18). How can a population of neurons, which differ in oscillation frequency, place field size, and temporal offsets generate a coherent theta frequency population output? To allow for the decrease in single neuron oscillation frequency along with uniform LFP theta, our model predicts that the compression factor c decreases along the long axis of the hippocampus according to c = 1/Lf _{0} (Eq. 9 ). In the illustration shown in Fig. 7, model neurons have randomly variable place field sizes L, single neuron frequencies f _{0}, and temporal offsets τ according to c = τ/T = 1/Lf _{0} (Eqs. 1 and 9 ). Each neuron can be conceived as representing a particular segment along the axis of the hippocampus (see color code in Fig. 7B ). Because of the interactive nature of the parameters (Eqs. 8 and 9 ), the frequency of the mPOP remains constant in the entire hippocampus, despite the large heterogeneity of the properties of the constituent neurons (Fig. 7).
Discussion
We built a ratebased model that allowed us to calculate the population activity analytically, depending on the single neuron oscillation frequency f _{0}, the compression factor c, and the firing field size (duration) L. The appropriate combination of these three parameters, obtained from experiments, generated a constant frequency population rhythm, while allowing individual neurons to vary their oscillation frequency and field size. The key features of the model are the transient Gaussian place field activity and the delays between the oscillation cycles of pairs of neurons, while the rat travels across the respective place fields. Using experimentally derived parameters, the model accurately predicted several fundamental features of hippocampal network activity, including singlecell oscillation frequency, phase precession, and the temporal offsets among neurons at various locations along the septo–temporal axis of the hippocampus.
Cell Assembly Model of Theta Dynamics.
Our model provides a framework for understanding the principles that support the temporal organization of singlecell and population dynamics in the hippocampus. The main assumptions of the model are that (i) the activity of a single cell can be approximated by a sine wave with a Gaussian envelope, (ii) the temporal distance between firing fields (T) and the temporal offset between the spikes of the respective neurons within the theta cycle (τ) are linked through the sequence compression factor (c) as c = τ/T, (iii) the extent of phase precession (ΔΦ) is the same for all neurons (set here as 360°, but see Fig. S6), and (iv) the density of place cells is sufficiently high (Fig. S7). On the basis of these assumptions, we derived a relationship between the single neuron oscillation frequency (f _{0}), population oscillation frequency (f_{θ} ), and the firing field size (L). All these relationships could be correctly predicted from parameters derived from data. If model neurons with identical firing field size and oscillation frequency were connected with random time delays, the amplitude of population fluctuation (i.e., theta oscillation) would converge to zero. However, assuming that the various parameters are controlled by interrelated physiological mechanisms, it is perhaps not surprising that the experimentally measured parameters have a mathematically definable relationship.
A fundamental assumption of the model is that singlecell parameters and their interactions, namely oscillation frequency of the single cell f _{0}, the place fields size L, and the time delay between the oscillation cycles of pairs of place cells τ, need to be dynamically adjusted to support a coherently oscillating mPOP in the entire hippocampus (15, 12, 18, 13, 19). The three key parameters (f _{0}, L, and τ) are “builtin” parameters in the model without currently known physiological solutions. We hypothesize that the delays between sequentially active place cells is brought about by feedforward inhibition. Faster spiking/oscillating cells can recruit stronger feedforward inhibition, inducing longer delays and, consequently, a larger difference between the oscillation frequency of the constituent neurons and the population. The mechanisms that determine the duration (“lifetime”) of activity of place cells are not known either. Shortterm synaptic plasticity and spikingdependent adjustment of spike threshold (29, 30) are potential candidates for controlling the lifetime of place cells on theoretical grounds because they operate at the time scale of seconds (31 –34). Our model does not address the origin of single neuron oscillation. Oscillatory activity may be a response to a pacemaker input, such as the medial septum (5, 35, 6, 36, 37), or may represent a locally emerging network phenomenon (38 –42). However, neither pacemaker nor network models explain why active neurons, such as place cells, fire only transiently and at a frequency faster than the pacing or population rhythm. Therefore, we hypothesize that the oscillation of single neurons is a consequence of the same dynamic mechanisms that are also responsible for the phase delays, the finite size of place fields, and the global theta oscillations.
An intuitive explanation for the inverse relationship between the oscillation frequency of place cells and the firing field size is that larger deviations from the population oscillation frequency are less “tolerated” by the network than small differences, and therefore the activity of faster oscillating neurons is terminated sooner by stronger synaptic depression. According to the model, place cells with very large firing fields should oscillate at frequencies close to the POP (as reflected by the LFP theta) and with very small temporal offsets between the neurons (13, 14). Furthermore, pyramidal neurons perfectly phaselocked to the population oscillation should have infinitely large place fields. In support of this model prediction, pyramidal cells sustain prolonged spiking activity at a particular theta phase while the rat is running in a wheel without memory requirement (43, 44, 21). Current phase precession models explicitly assume that phase interference occurs between two independent oscillatory mechanisms, for example between single neurons and a pacemaker input (45 –49, 10, 50). Our model does not require a reference pacemaker. Phase precession of single neurons emerges even if all constituent neurons have the same frequency and firing field size (Fig. 5), and results from the difference between the oscillation frequency of single neurons and their population output. Thus, there is a bidirectional relationship between single cells and their cooperative product: activity of single cells gives rise to a population rhythm, which in turn competes with the faster firing constituents by limiting their duration of activity.
Contribution of Place Cells to LFP Theta.
How do transiently active, faster oscillating place cells contribute to the LFP theta? We have demonstrated in both experiment and model that the summed spikes of the active neurons yield a population output slower than the mean oscillation frequency of the constituent neurons. It follows that, for example, the CA3 population output can generate an 8Hz rhythmic sink in the stratum radiatum of CA1 even if all CA3 place cells individually oscillate faster than 8 Hz. Similarly, the population effect of layer II grid cells (51) under the same behavioral conditions should also be 8 Hz, even though grid cells also show phase precession and thus oscillate faster than the population rhythm (10, 52, 25, 19). We predict that the relationships between the single neuron oscillation frequency (f _{0}), grid size (L), and temporal delays (τ) defined through the compression factor (c) also apply to neurons of of the entorhinal cortex.
Methods
Experimental procedures, data acquisition, and analyses are described in SI Methods , available online.
Compression Factor.
The “sequence compression” has been defined as the ratio between the pairwise time lag of place cells firing within one theta cycle and the distance between the centers of the respective place fields (11, 22, 9, 20). Under the assumption of constant running speed, we define here the “sequence compression factor” as the ratio between the time lag τ of firing within one theta cycle and the time T it takes the animal to run between the respective place fields in the environment (see also Fig. 1E ): where d = distance between place fields and s = running speed. The compression factor c is defined for a population of neurons and is generally nonzero.
Population Rate.
N place fields are evenly distributed along a time interval T_{N} , and each place field is associated with one place cell. The discharge probability of the n_{th} place cell is approximated by the product of a sinewave of frequency f _{0} and a Gaussian of width σ: where τ_{n} is the theta time lag, and T_{n} is the time of the center of the place field, both times are with respect to a chosen origin. The Gaussian is normalized to an integral of unity.
The rate of the population output of place cells (mPOP) is the sum of Eq. 2 over all neurons n: where δ is the cell density with δ = 1/ΔT = c/Δτ, where Δx = x_{k} – x_{k} _{–1}, x = {T, τ}, k = {2, …, N}. In the limit of continuous variable and substituting (see Eq. 1 ) the population rate (Eq. 3 ) can be written as: Eq. 4 can be solved exactly: The population rate (Eq. 5 ) describes an oscillation with unit mean, an oscillation amplitude and an oscillation frequency
Under the assumption that the total phase precession interval ΔΦ = 360° (see also SI Methods ), the oscillation frequency is where L is the place field size, implying that
Acknowledgments
We thank Asohan Amarasingham, Horacio Rotstein, and Anton Sirota for comments and helpful discussions. This work was supported by National Institutes of Health Grants NS34994 and MH54671, National Science Foundation Grant SBE 0542013, and by the James S. McDonnell Foundation.
Footnotes
 ^{2}To whom correspondence may be addressed. Email: buzsaki{at}andromeda.rutgers.edu or caroline.geisler{at}cin.unituebingen.de.

Author contributions: C.G. and G.B. designed research; C.G., K.D., E.P., K.M., and S.R. performed research; C.G. contributed new reagents/analytic tools; C.G., K.D., E.P., K.M., and S.R. analyzed data; and C.G. and G.B. wrote the paper.

The authors declare no conflict of interest.

↵*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/cgi/content/full/0912478107/DCSupplemental.
References
 ↵
 Friston KJ
 ↵
 Logothetis NK
 ↵
 Buzsáki G,
 Draguhn A
 ↵
 Steriade M,
 McCormick DA,
 Sejnowski TJ
 ↵
 ↵
 ↵
 O'Keefe J,
 Nadel L
 ↵
 ↵
 Geisler C,
 Robbe D,
 Zugaro M,
 Sirota A,
 Buzsáki G
 ↵
 ↵
 ↵
 ↵
 Kjelstrup KB,
 et al.
 ↵
 Royer S,
 Sirota A,
 Patel J,
 Buzsáki G
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Diba K,
 Buzsáki G
 ↵
 Pastalkova E,
 Itskov V,
 Amarasingham A,
 Buzsáki G
 ↵
 ↵
 ↵
 ↵
 ↵
 Maurer AP,
 Cowen SL,
 Burke SN,
 Barnes CA,
 McNaughton BL
 ↵
 Samsonovich A,
 McNaughton BL
 ↵
 ↵
 Spruston N,
 Schiller Y,
 Stuart G,
 Sakmann B
 ↵
 ↵
 ↵
 Mongillo G,
 Barak O,
 Tsodyks M
 ↵
 ↵
 ↵
 ↵
 ↵
 Wang XJ
 ↵
 ↵
 ↵
 Kocsis B,
 Bragin A,
 Buzsáki G
 ↵
 Gillies MJ,
 et al.
 ↵
 Rotstein HG,
 et al.
 ↵
 ↵
 ↵
 ↵
 Magee JC
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
Citation Manager Formats
Article Classifications
 Biological Sciences
 Neuroscience
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Biological Sciences
Neuroscience
Related Content
Cited by...
 Routing information flow by separate neural synchrony frequencies allows for "functionally labeled lines" in higher primate cortex
 Efficient phase coding in hippocampal place cells
 Degeneracy in hippocampal physiology and plasticity
 Spatially dispersed synapses yield sharplytuned place cell responses through dendritic spike initiation
 Independent Theta Phase Coding Accounts for CA1 Population Sequences and Enables Flexible Remapping
 Local Fields of the Hippocampus: More than Meets the Eye
 Nested sequences of hippocampal assemblies during behavior support subsequent sleep replay
 Movement Enhances the Nonlinearity of Hippocampal Theta
 Mechanisms Responsible for Cognitive Impairment in Epilepsy
 Back to the Future: Preserved Hippocampal Network Activity during Reverse Ambulation
 Optogenetic activation of septal cholinergic neurons suppresses sharp wave ripples and enhances theta oscillations in the hippocampus
 Modeling Inheritance of Phase Precession in the Hippocampal Formation
 Spatially Distributed Local Fields in the Hippocampus Encode Rat Position
 Controlling Phase Noise in Oscillatory Interference Models of Grid Cell Firing
 Hippocampal Phase Precession from Dual Input Components
 Spatial Information Outflow from the Hippocampal Circuit: Distributed Spatial Coding and Phase Precession in the Subiculum
 CrossFrequency PhasePhase Coupling between Theta and Gamma Oscillations in the Hippocampus
 Cosine Directional Tuning of Theta Cell Burst Frequencies: Evidence for Spatial Coding by Oscillatory Interference
 A Model of Intracellular Theta Phase Precession Dependent on Intrinsic Subthreshold Membrane Currents