We first put forward a mathematical framework for quantitatively describing the relationship between the current location of the animal, $x(t)$, and the position represented in the hippocampus at each moment in time, $r(t)$. While this framework considers the population representation, i.e. neuronal sequences during theta, the primary variable to be explained, it is linked to the activity of single place cells as follows. Each cell is considered to posses a ‘true place field’ (Sanders et al., 2015; Lisman and Grace, 2005), which corresponds to the positions represented by the cell, that is, a cell becomes active when the position represented by the population, $r(t)$, falls within the cell’s true place field. Note that due to the prospective and retrospective aspects of the theta phase code, $r(t)$ reaches behind and ahead of $x(t)$, so a place cell will also fire when the animal is located ahead or behind its true place field. This results in a measured place field that is larger than the underlying true place field. To stay consistent with the literature on place cells, in the following, when we refer to ‘place field’, we mean the measured place field.

First, we consider the spatial sweep scheme where the theta phase of a place cell’s spike reflects the distance traveled through the place field (Huxter et al., 2003; Geisler et al., 2007; Cei et al., 2014) or equivalently, some measure of the distance to the field’s preferred position (Jeewajee et al., 2014; Huxter et al., 2008; Drieu and Zugaro, 2019). In a uniform population of such phase coding neurons, the represented position, $r(t)$, sweeps forward within each theta cycle, starting at some distance behind the current location of the animal, $x(t)$, and ending at some distance ahead. We formalize these spatial sweeps by the equation:

where $d_{\theta }$, which we call the theta distance, is a parameter that determines the extent of the spatial sweep, $\theta (t)$ is the instantaneous theta phase in degrees, and ${\theta }_0$ is the theta phase at which the population represents the actual position of the animal. The value of ${\theta }_0$ effectively determines the proportion of the theta cycle that lies behind vs ahead of the animal. We note that none of our results or analyses depends on the particular value of this variable.

At the single-cell level, the spatial sweep model predicts that the phases of a cell’s spikes will advance over a spatial range of $d_{\theta }$. Assuming that the phase advances over the whole theta cycle, the slope of spatial phase precession will be given by ${\displaystyle m=-{360}^{\circ }/d_{\theta }}$ (Equation 13 in Materials and methods). Place field sizes will be given by $s=s_0+d_{\theta }$, where s₀ is the size of the true place field (Equation 14 in Materials and methods). Both phase precession slopes and place field sizes are independent of running speed (Figure 1B), consistent with experimental observations (Huxter et al., 2003; Schmidt et al., 2009; Geisler et al., 2007; Sup. Fig. 1 in Maurer et al., 2012). At the population level, we can define the theta trajectory length as the difference between the maximum and minimum positions that $r(t)$ represents within a theta cycle. To a first-order approximation, that is, assuming that effects from acceleration and higher order derivatives are negligible, the theta trajectory length is given by $L=vT+d_{\theta }$, where $v$ is the running speed, and $T$ is the duration of a theta cycle (Equation 11 in Materials and methods). While the spatial sweep predicts that theta trajectory length increases with running speed, the increase is small since the term $vT$, which represents the change in $x(t)$ due to the animal’s motion during the theta cycle (Figure 1A, see also Figure 1 in Maurer et al., 2012) is smaller than $d_{\theta }$ for usual running speeds. Contrary to this prediction, experimental results show that the theta trajectory length is roughly proportional to running speed (Figure 2 in Maurer et al., 2012; Figure 1C in Gupta et al., 2012 for speeds above 5 cm/s).

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An alternative to the above is the temporal sweep model, where different phases of theta represent the positions that the animals have reached or will reach at some time interval in the past and future, respectively (see Itskov et al., 2008 for a similar model). More formally,

where ${\tau }_{\theta }$ stands for the extent of the temporal sweep. To a first-order approximation, we derived for the theta trajectory length $L=({\tau }_{\theta }+T)v$, which is proportional to running speed (Equation 18 in Materials and methods) (Figure 1C). This is almost exactly the result reported by Maurer et al., 2012; Gupta et al., 2012. However, at the single-cell level, the temporal sweep model predicts that place field sizes increase and phase precession slopes become shallower with running speed (Figure 1D) – in clear conflict with the experimental results mentioned above. In particular, to a first-order approximation, theta phase precesses over the spatial range of $v{\tau }_{\theta }$. Thus, the slope of phase precession is given by ${\displaystyle m=-{360}^{\circ }/(v{\tau }_{\theta })}$, which increases hyperbolically with running speed (Equation 19 in Materials and methods). Place field sizes would in turn increase linearly with speed, determined by $s=s_0+v{\tau }_{\theta }$ (Equation 20 in Materials and methods).

These conflicting results could potentially arise from differences in analytical methods (e.g. definition of key parameters, comparisons within versus across cells) or experimental procedures (e.g. task difficulty due to the use of linear vs. circular or complex mazes). If this was the case, spatial or temporal sweeps could adequately describe the theta phase code, perhaps switching dynamically depending on the circumstances. Therefore, in the subsequent section, we attempt to reproduce and reexamine these different observations in the same dataset.

However, we also propose an additional scheme that, as it turns out, can potentially reconcile these findings. This third option combines the spatial sweep’s constant place field sizes and phase precession slopes with the temporal sweep’s increase in theta trajectory length with running speed. The former property requires that sweeps are independent of running speed whereas the latter requires that they depend on it. While at first this appears to be a plain contradiction, a resolution is offered if sweeps are made independent of the animal’s instantaneous running speed $v(t)$, but dependent on their ‘characteristic running speed’ at each position $\overline{v}(x(t))$. This characteristic speed could be calculated as as a moving average (e.g. cumulative, exponentially weighted, etc.) as long as it evolves sufficiently slowly, so that it can filter out variation in $v(t)$ from run to run. We refer to this model as the behavior-dependent sweep:

According to this model, at different phases of theta, the place cell population represents the positions that would have been or would be reached at certain time intervals in the past or future, like in the temporal sweep, but under the assumption that the animal runs at the characteristic speed through each location, rather than at the instantaneous running speed (Figure 2A,B). To avoid confusion in the following, when we use the terms ‘speed’ or ‘running speed’ without further qualifiers, we always refer to the instantaneous running speed. In the behavioral-dependent scheme, to a first-order approximation the length of a theta trajectory is given by $L=vT+\overline{v}(x(t)){\tau }_{\theta }$ (Equation 22 in Materials and methods). Thus, the more stereotyped the behavior of the animal is (i.e. $v(t)\approx \overline{v}(x(t))$), the more the behavior-dependent sweep model would share the temporal sweep model’s feature that theta trajectory length increases proportionally with running speed. At the same time, the behavior-dependent sweep model is formally equivalent to the spatial sweep model (Equation 1) with a theta distance that depends on the characteristic running speed at each position: 

Figure 2

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Therefore, place fields, being fixed in space, remain unaltered by the instantaneous running speed, similar to in the spatial sweep model (Figure 2B). However, in regions of higher characteristic running speed, place fields are characterized by larger sizes and shallower phase precession slopes. Specifically, the slope of phase precession is given by ${\displaystyle m=-{360}^{\circ }/(\overline{v}(x(t)){\tau }_{\theta })}$ (Equation 23 in Materials and methods); and the place field size would be $s=s_0+\overline{v}(x(t)){\tau }_{\theta }$ (Equation 24 in Materials and methods).

To better evaluate the conflicting observations across the afore-mentioned studies, we next sought to reproduce their separate observations on the effect of running speed on population-level theta trajectories and single-neuron place field parameters within the same dataset and compare these against predictions from the three theta phase coding models.

To this end, we re-analyzed publicly available simultaneous single unit recordings from CA1 of 6 rats running on linear tracks for water rewards at both ends. Three of the sessions (Mizuseki et al., 2013) were from tracks that were familiar to the animal while the remaining three Grosmark et al., 2016 were from novel tracks. To ensure that spatial representations and theta phase precession were stable for the entirety of the periods under examination, we discarded the first 5 min of the recording on the novel track, allowing sufficient time for hippocampal place fields and phase precession to have stabilized (Frank et al., 2004; Cheng and Frank, 2008).

First, we examined the effect of running speed on theta trajectory length. We performed a Bayesian decoding of positions from the ensemble neuronal firing (Zhang et al., 1998; Davidson et al., 2009) within each theta cycle in overlapping 90° bins with a step size of 30°. Figure 3A shows examples of individual decoded theta trajectories belonging to periods of slow (below 20 cm/s) and fast (above 40 cm/s) running in a sample experimental session. The lengths of the trajectories were measured from best-fit lines. Despite the high level of variability, it was clear in the individual rats’ averages and in the grand average across animals that the trajectory lengths increased roughly proportionately to running speed (Figure 3B). We saw the same effect when we averaged the decoded position probabilities over all cycles belonging to the same speed bin before fitting a line and calculating the theta trajectory length (Figure 3C,D). For this analysis, we defined seven 20 cm/s wide overlapping speed bins starting at 2 cm/s in increments of 10 cm/s. Thus, our results on the dependence of theta trajectory lengths on running speed were consistent with those of Maurer et al., 2012 and Gupta et al., 2012, and appear to be supportive of the temporal sweep model. Alternatively, the results could be accounted for by the behavior-dependent sweep model, if running behavior was sufficiently stereotyped across the track, a question to which we return below.

Figure 3

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We next analyzed the effect of running speed on place field size. First, we calculated firing rates separately for periods with different running speeds using the same speed bins as before. As the number of trials in each spatial bin at a given running speeds were limited, we calculated place field sizes only when a field was sufficiently well sampled at a given speed (see Materials and methods). Visual inspection revealed that individual fields tended to maintain similar sizes regardless of running speed (Figure 4A). To quantify the relationship more systematically, we performed linear regression on place field size vs. speed for each field and examined the slopes of the best fitting lines (Figure 4A, bottom row). Figure 4B shows that for each rat, slopes variably took on both negative and positive values, with no clear bias towards either ($p>0.3$, Wilcoxon signed-rank test). Thus, we found no evidence for a systematic increase in place field size with running speed.

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Intriguingly, however, when we combined all place field sizes corresponding to each speed bin, we uncovered a linear relationship between place-field size and speed (Figure 4C; $p<0.02$ for the association between place field sizes combined across fields and speed bin for five of the six animals, Kendall’s $\tau$ test). This increase in place field sizes is consistent with the temporal sweep model, whereas the lack of within-field changes mentioned above supports spatial sweeps. Both findings, however, can be consistent with the behavior-dependent sweep model provided that place fields are larger in track areas characterized by faster running (Figure 2). We examine this question below.

Similarly conflicting results were obtained when analyzing phase precession. We calculated phase precession slopes at each running speed bin for each field separately by pooling spikes across different field traversals according to the instantaneous running speed of the animal when the spikes were emitted. Figure 5A shows the results of this analysis for three example place fields. To ensure that we capture and adequately describe steep phase precession in small place fields (e.g. Figure 1D), we devised a method using orthogonal regression (see Materials and methods). An analysis of all place fields (Figure 5B) revealed that phase precession slopes of individual fields did not change systematically with running speed for any of the six animals ($p>0.05$, Wilcoxon signed-rank test). Remarkably, however, when we combined phase precession slopes across different fields for each speed bin, the combined slopes at slower speeds tended to be steeper than at faster speeds (Figure 5C; $p<0.006$ for the association between phase precession slopes combined across fields and speed for five of the six animals, Kendall’s $\tau$ test).

Figure 5

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In this analysis, when rats accelerate or decelerate through a field, different parts of the field’s phase precession cloud are sampled at different running speeds. This can be seen in the first field (no. 4) in Figure 5A. Since the rat was accelerating through the field, spikes emitted at lower and higher speeds appeared more frequently at the beginning and end of the field, respectively. Phase precession clouds have some curvature (Souza and Tort, 2017; Yamaguchi et al., 2002), so sampling different parts of the field under acceleration or deceleration could lead to a spurious relationship between speed and phase precession slope. To avoid this confound, we further analyzed phase precession slopes in single passes through a field (Schmidt et al., 2009). We only included passes that occurred at approximately constant speed, that is, passes with a coefficient of variation in speed lower than 0.3. Figure 5D shows examples of individual passes through two fields, sorted by running speed. Note how, notwithstanding variability in the slopes, there is no systematic change with running speed, even though the speeds nearly double. There was no significant relationship between phase precession slope and speed for individual fields in any of the three animals with sufficient data for this analysis, that is, more than 10 fields (Figure 5E). Yet again, when we combined phase precession slopes across place fields, we saw that they become shallower with speed ($p<0.04$ for five of the six animals, Kendall’s $\tau$ test; Figure 5F). Hence, the results from single pass phase precession analysis confirm those obtained from the session-wide analysis.

We have shown that theta trajectory lengths and place field sizes increase with running speed, and phase precession slopes become shallower, when combining data-points from different positions along the track or place fields (Figure 3B,D; Figure 4C and Figure 5C,F), but not when fields are analyzed individually (Figure 4B and Figure 5B,E). While these combined findings appear counter-intuitive, the behavior-dependent sweep model offers a simple explanation: short theta trajectories and small place field sizes could occur at locations where animals tend to run slowly, contributing to the low-speed averages, while long theta trajectories and large place field sizes could occur where animals tend to run fast, contributing to the fast-speed averages (Figure 6A). For this explanation to be valid, the following prerequisites must hold: (1) running speeds should differ systematically along the track such that (2) place fields in different positions contribute differently to the low- and high-speed analyses; and (3) theta trajectory lengths, place field sizes and the inverse of phase precession slopes depend linearly on the characteristic speed across different locations (Equation 22-24).

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To test the first prerequisite, we calculated the characteristic speed for each spatial bin along the track. This was defined as the mean speed after discarding trials where the animal stopped running on the track (speeds < 10 cm/s), to prevent atypical pauses from distorting the mean values. The upper panel in Figure 6A shows the characteristic speed for rightward runs in one sample experimental session and Figure 6B shows averages across running directions and sessions. All animals showed similar asymmetric inverted U-shape relationships between characteristic speed and position along the track. Next, as required by the second prerequisite, we confirmed that running speeds at each position were clustered around the characteristic running speed (Figure 6C). We also observed inverted U-shape relationships between position along the track, and theta trajectory lengths and place field sizes (Figure 6D,E). Because theta trajectory lengths and place field sizes should be linearly related to the inverse of phase precession slopes (Equation 22-24), we plotted this variable along the track and found the corresponding U-shape distribution (Figure 6F). Finally, in support of the third prerequisite, we found that theta trajectory lengths and place field sizes increased, and inverse phase precession slopes decreased, roughly in proportion to characteristic speed for up to at least 60 cm/s, which encompassed most of the data (Figure 6G–I). Some flattening of the curves appeared after 60 cm/s, but this was less apparent when considering each animal individually (Figure 6—figure supplement 1 and Figure 6—figure supplement 2), suggesting that the flattening is at least partly caused by combining points from clouds with different slopes and speed ranges.

Since theta trajectory lengths, place field sizes and inverse phase precession slopes all displayed matching distributions, we tested their mutual relationships. We found that the three of them displayed strikingly linear relationships to one another (Figure 6—figure supplement 3), underscoring the intimate relationships that hold between these systematically interrelated measures of place cell activity. It is also worth noting that theta trajectory length correlates slightly better with characteristic running speed than with either mean place field size or mean inverse phase precession slope at the locations where the trajectories occurred (R = 0.35 vs 0.29 for the latter two; Figure 6—figure supplement 3B,C).

Since speed and location on the track are correlated, we also tested whether place field parameters could be best explained by distance to the nearest end of the track, rather than by characteristic running speed. We performed a likelihood ratio test that indicated that both variables contributed significantly to the prediction of place field sizes and inverse phase precession slopes ($p<{10}^{-6}$). However, the fraction of the total variance accounted for differed substantially between the two variables. A model combining both variables could account for 29% and 28% of the variance in place field sizes and inverse phase precession slopes, respectively. Out of this, characteristic running speed uniquely accounted for 13% and 11% of the variance, whereas distance to the border uniquely accounted for 3% and 7%, respectively. Therefore we conclude that although both variables are related and contribute to determining place field parameters, characteristic running speed plays a more prominent role.

Taken together, these findings account for and reconcile the speed-dependence of average theta trajectory lengths, place field sizes and phase precession slopes with the speed-independence of individual fields, and satisfy the prerequisites of the behavior-dependent sweep model.

Our analyses so far suggest that neither the spatial nor the temporal sweep models can account for all of the experimental observations, whereas the behavior-dependent sweep appears consistent with all of them. However, these analyses might involve too many simplifying assumptions about how running speeds, theta coupling and place field parameters are co-distributed. To address this issue, we developed generative models based on the experimental LFP and animal tracking data to generate spikes according to each of the different theta phase coding schemes. The parameters in each model were adjusted to account for as many features in the experimental data as possible. We analyzed the effect of instantaneous running speed on theta trajectory lengths, place field sizes and phase precession slopes in each of the simulated datasets alongside the experimental data (Figure 7). This approach further allowed us to examine whether any of the models could provide a simple unified account of the data, using a single set of parameters.

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The temporal sweep model roughly captures the observed increase in theta trajectory lengths, place field sizes and phase precession slopes combined across different positions and fields based on instantaneous running speed. However, it did so for the wrong reasons, since it incorrectly predicts changes in individual fields with speed (Figure 7B). By contrast, the spatial sweep model accounts for the speed-independence of individual fields, but severely undershoots the increase in theta trajectory length with running speed and fails to capture the increases in place field sizes and phase precession slopes (Figure 7C). Only the behavior-dependent sweep model is able to replicate the correct combination of qualitative effects (Figure 7D). Furthermore, the behavior-dependent sweep produced values with the lowest mean squared deviation to the experimental data for four of the five variables considered (Figure 7; numbers in corners).

Further proof for the validity of the behavior-dependent sweep model comes from the fact that we fitted one of its crucial parameters based on a different subset of the data as the one used to evaluate the models. In particular, we estimated ${\tau }_{\theta }$ based on Equation 23 from the experimentally observed relationship between phase precession slopes and characteristic running speed (Figure 6I), yielding τ_θ = 0.57 s. With this value, the model provided a good fit to the relationship between instantaneous running speed and theta trajectory lengths, place field sizes and phase precession slopes (Figure 7D). This highlights the ability of the model to account for the relationships and internal consistency of multiple measurements of place cell activity.

Finally, a subtle prediction of the behavior-dependent sweep model is that fields will skew differently depending on whether the animal is typically accelerating or decelerating through the field–that is when the characteristic running speed changes within the field. We therefore analyzed the relationship between mean acceleration and place field skew in the experimental data and found a good qualitative fit with the predictions from the model (Figure 7—figure supplement 1). Negatively skewed fields Mehta et al., 2000 were observed at all acceleration values, but positively skewed fields occurred mainly at positive accelerations, resulting overall in a significant correlation between mean acceleration and skew ($p=4\times {10}^{-8}$, Kendall’s $\tau$ test). A similar observation was made by Diba and Buzsáki, 2008, who reported positive skews at the beginning of the track, where rats are usually accelerating, and negative skews towards the end, where rats decelerate.

In summary, we found that the behavior-dependent sweep model provided the best overall fit to the experimental data with a single set of model parameters.

Why were the relationships between characteristic speed, hippocampal sweeps and place field parameters not evident in previous studies? Two previous studies (Maurer et al., 2012; Gupta et al., 2012) pooled theta trajectories across different positions when showing the effect of speed on theta trajectory length, therefore masking the role of position in mediating the relationship between the two variables. In other studies, the dependence of place field parameters on the characteristic running speed through the field did not emerge because analyses focused on relative measures between pairs of cells (Geisler et al., 2007; Diba and Buzsáki, 2008) or individual fields (Schmidt et al., 2009; Huxter et al., 2003; Maurer et al., 2012) that remain unaffected.

We note that one other study (Chadwick et al., 2015) previously proposed a model to account for the increase in theta trajectory length with running speed despite stable place field sizes and phase-precession. The model achieved this by increasing the precision of the theta phase versus position relationships of cells with running speed. This feature, however, results in unusually noisy phase-precession at low speeds (see Neuronal sequences during theta rely on behavior-dependent spatial maps for results on a model that operates based on the same principle and comparisons to experimental data). Furthermore, this kind of model cannot account for the dependence of average place field size on characteristic running speed.

On the other hand, two models which are consistent with the increase in place field sizes with characteristic speed did not address theta phase coding. Slow feature analysis attempts to extract slowly varying signals from the sensory input, and has been found to account for different properties of place, head direction and view cells (Franzius et al., 2007). Extracting place-specific features that vary on the order of several seconds, for instance, would lead to larger place fields at locations of higher speeds, as we have observed. This observation is also consistent with the successor representation model of place-fields (Stachenfeld et al., 2017). In this model, the activity of a cell is determined by the expected discounted future occupancy of the cell’s preferred location, which proves useful for planning. Thus, a cell with a preferred location where average running speed is higher would begin firing relatively further away from that place. The behavior-dependent theta sweep model shares this prediction with these two models, and further accounts for theta-phase and trajectory relationships.

Given the purported role of theta phase coding in prediction and planning, we can compare our three coding schemes in terms of their benefit for these functions. Representing future positions based on the distance to them, as in the spatial sweep, could allow an animal to calculate the speed necessary to reach those positions in a given amount of time. This representation might also be convenient because our bodies and sensory organs limit the spatial range at which we can interact with the world. In contrast, looking forward based on an interval of time rather than space, as in the temporal sweep model, could increase the code’s efficiency by matching the extent of the look-ahead to the speed of progression through the environment. For instance, while driving a car on a highway, we might have to look ahead by several hundred meters, since those positions are just seconds away, while such a distance is not useful when strolling in a park. Plans made too far in advance are very likely to be discarded upon unexpected changes or as new information comes to light, which is conceivably related to the use of a discount factor in reinforcement learning models (Sutton and Barto, 1998). Temporal sweeps also seem more plausible for phase-coding in non-spatial domains (Pastalkova et al., 2008; Lenck-Santini et al., 2008; Robinson et al., 2017; Terada et al., 2017; Aronov et al., 2017). However, temporal sweeps require a fine control by the instantaneous running speed, and there are indications that the hippocampal network lacks the the flexibility required to modify the time lags between pairs of cells at the theta timescale (Diba and Buzsáki, 2008; Shimbo et al., 2021).

The behavior-dependent sweep might thus represent the best solution compatible with network constraints by taking advantage of the relative simplicity of spatial sweeps while still enjoying the benefits of temporal sweeps in situations in which behavior is stereotyped. Thus, the look ahead on a highway remains a few hundred meters even if on occasion we are stuck in a traffic jam. Finally, behavior-dependent sweeps could make predictions of future positions more robust through the averaging of past experience.

Despite the potential benefits of behavior-dependent sweeps mentioned here, these sweeps are probably best construed as a default or baseline mode of operation, which could be further modulated by ongoing cognitive demands and task contingencies (Wikenheiser and Redish, 2015; Shimbo et al., 2021; Zheng et al., 2021).

All theta sweep models and concepts of prospective and retrospective coding, and extra-field spikes in general, imply that place cells also fire outside their true place fields. In other words, an experimentally measured place field encompasses the extent of the true place field, as well as look-ahead and look-behind activity. This link between place fields and prospective/restrospective representations at the theta timescale provides a lower bound on place field sizes for a given task based on its characteristic running speed. For sweeps that are centered around the current position, a ${\tau }_{\theta }$ of $\sim 0.6$ s means that theta sweeps on average end in positions arrived at in $\sim 300$ ms. Representing those future positions would be of use only if there was enough time for the animal to modify its behavior in response to them. Reaction times for rats performing simple and choice tasks range between 150 ms and 500 ms (Baunez et al., 2001; Brown and Robbins, 1991), and stop signal reaction times stand at around 360 ms (Eagle et al., 2009). Therefore, place field sizes associated with a look-ahead time of 300 ms probably stand towards the lower end of what can be used to control behavior through rapid decision-making. However, place fields (Kjelstrup et al., 2008) increase in size along the dorsoventral axis of the hippocampus. The largest place fields recorded in rats stand at about 10 m, which would correspond to representing a time window of > 10 s. Thus, we can conceive of a hierarchy of subregions corresponding to predictions at different temporal scales.

The dependence of place field parameters and theta trajectories on characteristic speed raises the question of how and when this speed might be computed in the brain, and how it is used to control hippocampal sweeps. One intriguing possibility is that the animal can estimate the speed at which it will typically traverse some region of an environment based on general knowledge of similar environments (e.g. knowing that speed will tend to be lower around boundaries). Another possibility is that the characteristic speed is taken to be the speed of the firsts traversals through the environment, in which theta sequences are reportedly lacking or reduced (Feng et al., 2015; Tang et al., 2021), and then updated incrementally.

The latter suggestion fits naturally with network connectivity models of phase precession (Jensen and Lisman, 1996; Tsodyks et al., 1996; Drieu and Zugaro, 2019). In these models, theta sequences are produced by the propagation of activity through the population, perhaps owing to the recurrent connectivity in area CA3 (Cheng, 2013; Azizi et al., 2013). The network is cued with the current position and recalls upcoming positions. As the animal moves forward, positions will activate at earlier phases of the theta cycle, producing phase precession. In this family of models, it is easy to imagine sweeps varying based on average speed. Place cells could be connected with a higher synaptic strength the closer in time they become activated, leading to stronger and further reaching connectivity in areas through which the animals run faster on average. If the speed of propagation in the network is then made dependent on synaptic strength, activity would propagate further in the network within each theta cycle where speed had been higher, producing longer theta trajectories in those areas. Alternatively, based on the view that theta sequences arise from independently phase-precessing cells (Chadwick et al., 2015), behavior-dependent sweeps could simply result from a behavior-dependent spatial map that yields larger place fields in areas of faster running.

In the dataset analyzed, changes in running speed arose naturally from animals running on a finite track, but were not controlled by the experimenter. Further confirmation for the behavior-dependent sweep model could come from experiments in more complex environments that can produce variable running speeds at different specific locations (Kropff et al., 2021), such as slowing down in previously fast segments and speeding up in slow ones. This would allow for the comparison between behavior-dependent sweeps, and alternative models for which the look-behind and look-ahead distances vary depending on other variables. In particular, it would be interesting to dissociate the effect of characteristic running speed from that of distance to the environmental boundaries, since the boundary vector cell (Barry et al., 2006) and Laplace transform (Sheehan et al., 2021) models suggest that it could be some non-linear function of distance to the border or trial start, rather than characteristic running speed, that determines place field parameters. Other factors that could play a role are the distance to reward locations (Hollup et al., 2001), the distance to the goal (Wikenheiser and Redish, 2015), the density of sensory cues, the spatial rate of change of visual input (Tanni et al., 2021), the complexity of the maze, or the amount of retrospection and prospection required for the task.

Although we have focused primarily on the effect of speed, it is reasonable to also expect acceleration to have an effect on theta phase coding. We discuss three specific mechanisms. First, acceleration could introduce a second-order term in the equation for $r(t)$ such that theta trajectories curve in areas through which the animal typically accelerates or decelerates. This might cause trajectories in areas of positive acceleration to reach further ahead than behind, and trajectories in areas of negative acceleration to reach further behind than ahead, consistent with the findings of Gupta et al., 2012 and Bieri et al., 2014. This effect would also be modulated by the phase at which the current position is represented (${\theta }_0$ in our equations), since this controls the ratio of look-ahead to look-behind. For instance, the more look-ahead there is, the longer the theta trajectories would become in areas of positive acceleration. Second, acceleration leads to different characteristic running speeds in different parts of a place field. Our model predicts that this would cause place fields to skew differently in areas of positive and negative acceleration, a prediction for which we have found some preliminary experimental support (Figure 7—figure supplement 1). Additionally, the model indicates that changes in characteristic speeds through fields cause corresponding changes in theta phase precession. If characteristic speed increases across a field, the slope of phase precession would become increasingly shallow, resulting in concave phase precession clouds, whereas if characteristic speed decreases across the field, the slope of phase precession would be increasingly steep, producing convex ‘banana shaped’ phase precession clouds (Souza and Tort, 2017; Yamaguchi et al., 2002). Future studies are required to test these two predictions, since the range of accelerations in the current dataset was relatively small, especially for positive values, and strong acceleration and deceleration were concentrated over small regions at the start and end of the track, respectively. Third, theta frequency has recently been shown to increase with positive acceleration, but not deceleration or speed (Kropff et al., 2021). One might wonder whether shorter theta cycles due to positive acceleration, rather than the lower running speed, could explain the shorter theta trajectories we observed at the beginning of the track. However, we also observe shorter theta trajectories at the end of the track, where running speeds are also lower, but acceleration is negative and theta cycles are longest (Figure 6B,C). Furthermore, as shown by Kropff et al., 2021, the increase in theta frequency is accompanied by an increase in the intrinsic firing frequency of cells which would provide a compensatory effect on theta trajectories. Even without such compensation, the reduction in theta trajectory length due to shortened theta cycle is small and insufficient to account for the observed effect.

Lastly, our behavior-dependent sweep model could be applied to describing the activity of place cells in two-dimensional environments, and potentially also that of grid cells, since theta phase coding has been demonstrated in both of these cases (Huxter et al., 2008; Jeewajee et al., 2014; Hafting et al., 2008). For grid cells, our model predicts that sizes of individual firing fields within the grid will vary according to running speeds characteristic of the corresponding maze regions. By contrast, the distance between firing field peaks would remain constant, since the locations of the underlying true fields remain fixed in the model.

In conclusion, our analyses have highlighted the heretofore unresolved tension lurking in the field between experimental results and interpretations explicitly or implicitly supporting spatial and temporal sweeps. Our novel interpretation reconciles apparently contradictory findings by making spatial sweeps dependent on the average behavior at each position, combining features and benefits of both kinds of sweep. In doing so, the model advances our understanding of how the mammalian brain represents past, present and future spatial positions for learning and planning, and provides a quantitative framework that can stimulate further research.

We re-analyze data from two publicly available data sets (Mizuseki et al., 2013; Grosmark et al., 2016) containing recordings of male Long-Evans rats running on linear tracks of various lengths for water rewards at both ends. From the first data set, we limited the analysis to the data from three animals, described in Mizuseki et al., 2009, which contained sufficient parallel recordings for Bayesian decoding of position. The second data set, described in Grosmark and Buzsáki, 2016, contains data from four animals running on a novel track. Data from one animal was excluded due to issues with local field potential recordings (see below), resulting in a total of six animals included in this study. The first 5 min of the recordings in a novel track were discarded since place fields of hippocampal place cells might be unstable in the first 5 min of novel experience (Frank et al., 2004) and we wanted to only include periods with stable spatial representations. Five of the animals had one to three experimental sessions available. The remaining animal had a disproportionately large number of experimental sessions, so a random subset was chosen. The exact number and identity of the sessions analyzed for each animal are reported in Figure 6—figure supplement 2A. In these data, neural activity was recorded using multi-shank silicon probes targeting region CA1 of dorsal hippocampus. After spike sorting, cells were classified as principal neurons or interneurons based on monosynaptic interactions, wave-shapes and burstiness.

We generated surrogate data sets by taking the tracking and LFP data from experimental sessions and substituting the experimental spikes with those generated by our models under different coding assumptions. We discretized the session into 2 ms time bins and calculated for each bin the position that the cell population represents, $r(t)$, according to each coding scheme at that particular phase of theta, $\theta (t)$. For the temporal sweep model, $r(t)=x(t+{\tau }_{\theta }(\theta (t)-180)/360)$, where $x(t)$ is the position obtained from the tracking and ${\tau }_{\theta }$ is the time interval swept over in each theta cycle. We manually tuned ${\tau }_{\theta }$ iteratively to minimize the deviations in theta trajectory lengths, place field sizes and phase precession slopes between experimental and simulated data (Figure 7). A good compromise was obtained for a ${\tau }_{\theta }$ of 0.55 s. For the spatial sweep model, ${\displaystyle r(t)=x(t)+d_{\theta }(\theta (t)-180)/360}$, where $d_{\theta }$ is the spatial distance swept over in each theta cycle. We tuned $d_{\theta }$ as described for ${\tau }_{\theta }$, obtaining a value of 30 cm. For the behavior-dependent sweep model, ${\displaystyle r(t)=x(t)+\overline{v}(x(t)){\tau }_{\theta }(\theta (t)-180)/360}$, where $\overline{v}(x(t))$ was taken to be the experimentally observed characteristic running speed at each position, calculated independently for each experimental session and running direction (e.g. Figure 6A). ${\tau }_{\theta }$ was obtained from fitting Equation 23 to the relationship between inverse phase precession slopes and characteristic running speed (Figure 6I), resulting in a value of 0.57 s.

Then, for each time bin, we generated spikes for each of 20 simulated place cells uniformly distributed along the track. Spikes were generated based on the activation level of each cell $i$ at the represented position, as determined by the cell’s true place field. These true place fields were defined as Gaussian activation functions, f_i, with a peak value of 1, discretized in 1 cm steps. The standard deviation of the Gaussian, $\sigma$, determines the size of the true place fields, which controls how tight the relationship is between firing at a particular phase and the time or distance to the peak’s center. A small true place field size leads to very crisp phase precession clouds, whereas a bigger one results in wider clouds with a noisier relationship between phase and position. For the temporal and spatial sweep models, the standard deviation was set to 7 cm, which corresponds to true place field sizes of 27 cm. With this value, the model produced phase precession clouds approximately matching the experimental ones. For behavior-dependent sweeps, the standard deviation was heterogeneous across the population, set to $\max (4,\mathrm{\hspace{0.25em}0.3}\overline{v}(c_i){\tau }_{\theta })$, where c_i is the center of each place field.

Spikes were generated from a Poisson process with probability equal to an instantaneous firing rate times the width of the time bin:

where $v(t)$ is the instantaneous running speed in cm/s. The first term sets the maximum firing rate and the values were chosen to approximate the increase in peak firing rate with running speed observed in the experimental data (Figure 7—figure supplement 2). The second term introduces an inhibitory modulation by the amplitude of the theta oscillation with $m_{\theta }=0.35$ approximating the experimentally observed value in the experimental data set (not shown).

For the model predictions shown in the the schematics in Figure 1 and Figure 2, we simulated runs at constant speed with a regular 8 Hz theta oscillation, and used the models as described above, except for ${\tau }_{\theta }=0.5$ s and $\sigma =6$ cm.

For completeness, we also considered whether the experimental results could be accounted for by a fourth model similar to the one proposed by Chadwick et al., 2015. Cells participate in spatial sweeps, but the coordination between cells increases with running speed. At lower speeds, the lack of coordination between cells should bias the decoder toward the current position, leading to shorter decoded theta trajectories.

We modeled this by introducing the following modifications to the spatial sweep model. To increase the coordination between pairs of cells at the theta timescale with speed, we introduced into each cell a certain amount of independent theta phase noise that decreased with running speed:

where $N(0,{\sigma }_{\theta })$ are samples obtained from a Gaussian distribution with standard deviation ${\sigma }_{\theta }$; with $k=120$ and $\gamma =0.025$. This is the crucial feature of the model. Additionally, to get a more realistic amount of variance in place field sizes and phase precession slopes, we used a slightly different theta distance for each cell:

with $d=35$, and ${\alpha }_i$ and ${\beta }_i$ derived from Gaussian distributions $N(1,0.3)$ and $N(0,10)$, respectively, and fixed for each cell for the entire simulation. The standard deviation setting the width of each true place fields was $\max (3,\mathrm{\hspace{0.25em}0.25}{d}_{\theta ,i})$.

As expected, the model provides a good fit for the increase in theta trajectory lengths with running speed. However, the model does not account for the increase in average place field sizes (Appendix 1—figure 1A,B). Furthermore, unlike the experimental data, the model produces atypically noisy phase precession clouds at low running speeds (Appendix 1—figure 1C) which become sharper with speed, as measured by decreasing residuals in the phase precession cloud fits (Appendix 1—figure 2A,B).

Appendix 1—figure 1

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Appendix 1—figure 2

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Even if phase precession clouds were noisier at lower speeds, cell coordination, and hence theta trajectory lengths could remain unaffected, if noise affected all cells in the same way. Therefore, we measured the effect of speed on cell coordination directly, by calculating the variance in phase differences between spikes of pairs of cells with overlapping place fields. First, we found cells whose place fields overlapped by more than 20 cm. Then, when rats were within 10 cm of the overlapping region of two cells, we collected, for each theta cycle individually, the phases at which spikes from each cell were emitted, and calculated all pairwise phase differences between them. Finally we calculated their circular variance (Fisher, 1993) as:

where ${\varphi }_i$ is $i$-th pairwise phase difference between the two cells.

If the cells code for positions in a coordinated fashion, the variance in phase differences between their spikes should be low. We found that the level of variance present in the variable noise model at low speeds was unrealistically high as compared to the experimental data (Appendix 1—figure 2C,D). Furthermore, in the experimental data, the variance in phase differences for spikes of a cell pair did not significantly change with running speed for any of the animals ($p>0.15$; Wilcoxon signed-rank test), whereas it did so for five of the six animals in the simulated data ($p<0.05$; Wilcoxon signed-rank test).

Overall, we conclude that these results rule out the possibility that changes in cell coordination at the theta time scale play a major role in explaining the changes observed in theta trajectory lengths.

Supplementary file 1 contains a table with a summary of all statistical tests performed for the analyses of Figure 4, Figure 5 and Figure 6. The table indicates animal names, number of samples (N), test types and p-values.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Loren Frank for helpful discussions and Laia Serratosa Capdevila for feedback on the draft. We thank the anonymous reviewers for their thoughtful and extensive feedback which led to a clearer and more comprehensive article.

1. Preprint posted: April 14, 2021 (view preprint)
2. Received: May 12, 2021
3. Accepted: October 15, 2021
4. Accepted Manuscript published: October 18, 2021 (version 1)
5. Version of Record published: November 3, 2021 (version 2)

© 2021, Parra-Barrero et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.