The Involvement of Recurrent Connections in Area CA3 in Establishing the Properties of Place Fields: a Model

CA3 neural architecture and dynamics

The main aspect of hippocampal circuitry we actually implement is the CA3 recurrent network (Fig. 1). The model CA3 contains a collection of 1200 pyramidal cells, each connected to all the others through modifiable weights. This high degree of connectivity mimics the extensive recurrent collateral connections of CA3 pyramidal neurons (Ishizuka et al., 1990; Li et al., 1994). Owing to the relatively small number of neurons in the model, the number of connections per cell is still much lower than in reality, although the degree of connectivity is higher. This does not pose a problem, however, as long as the cells a particular neuron connects to can be considered from a functional point of view as a random sample, the number of connections per neuron is high enough, and any one connection is weak enough. In this case, neural responses are determined by averaged population effects, and the actual number of connections only enters the calculations as a constant scaling factor for the individual weights.

Local feedback and feedforward inhibition are thought to play an important and complex role in neural dynamics in CA3. Inhibitory interneurons are spatially much less selective than pyramidal neurons, but their activity during locomotion changes periodically at the theta frequency. We ignore this temporal variation, as well as the diversity of interneurons and patterns of connectivity, and include in the model a single global inhibitory neuron, which fosters competition between stored patterns and keeps global activity levels approximately constant. This cell receives input from all the excitatory neurons and provides inhibitory feedback to each that is proportional to the product of the firing rate of the inhibitory neuron and the depolarization of its postsynaptic target. This nonlinear form of inhibition was chosen because our simulations indicated that, compared with more conventional subtractive inhibition, it leads to improved robustness in the network with respect to variations in weight magnitude (for details on networks with shunting inhibition, seeGrossberg, 1988). It is also consistent with the observed effect of GABA_A receptor activation. We adapt the equations introduced by Wilson and Cowan (1972) to model the dynamics of the CA3 neural population. The following set of equations describes how the membrane potential of CA3 cells in our model changes over time: Equation 1where u_i is the membrane potential of the ith pyramidal cell, v is the membrane potential of the global inhibitory cell (all relative to their resting potentials), τ and τ′ are the membrane time constants for pyramidal neurons and the inhibitory cell, respectively,J_ij is the strength of the connection from neuron j to neuron i, h is the efficacy of inhibition, w represents the strength of the excitatory connection from any one pyramidal cell onto the inhibitory cell, andI_i^PPandI_i^MFare the inputs to cell i through the perforant path and the mossy fibers (MF), respectively.g_u (u) = β [u − μ]₊ is the threshold linear activation function for the pyramidal cells, where […]₊ makes all negative arguments zero but leaves positive numbers unaffected, μ stands for the threshold, and β is the slope of the activation function above the threshold. Similarly, g_v (v) = γ[v − ν]₊ for the inhibitory neuron. As will be described in detail later, some of the terms in these equations are assumed to be influenced by neuromodulatory control and therefore may be absent in certain phases of processing.

The value of the inhibitory time constant τ′ has no effect on the location of the fixed points of the network, although it can change their stability. In the simulations that are described later, we set τ′ = 0, so that v is always equal towΣ_jg_u (u_j ). This simplifies the theoretical treatment of the model and makes the simulations numerically more stable. We conducted simulations to verify that, within a wide range of the parameters, this manipulation does not affect the qualitative dynamical behavior of the model and indeed leads to the same stable patterns of activity. It is worth noting that, in this general class of models (although in a different parameter regimen), setting τ′ > 0 can give rise to oscillations (which, of course, are consistently observed in the hippocampus during active behavior). Even in an oscillatory regimen, however, the mean activities of the units can closely resemble the activities of the units at the fixed points found when τ′ = 0 (Li and Dayan, 1999).

Input representation

Instead of building a detailed model of rodent sensory processing, we consider as inputs to our model the firing rates of pyramidal neurons in superficial layers of entorhinal cortex, which provide most cortical input to the hippocampal formation. Unfortunately, there is relatively little direct experimental evidence about the nature of spatial representations in EC and especially about how these depend on details of the environment. However, there is something of a consensus among modelers (Burgess et al., 1997), which we generally follow. Although entorhinal neurons are found to be spatially selective (Barnes et al., 1990; Quirk et al., 1992), they appear to be much noisier and more broadly tuned than place cells in the hippocampus. Quirk et al. (1992) also found them to be more “sensory bound” than hippocampal cells in that their firing fields transform in a smooth manner after substantial changes in the shape of the environment. This is very unlike the complete remapping seen in place cells under similar circumstances (Muller and Kubie, 1987). The anatomy of the inputs to EC is rather better understood (Burwell and Amaral, 1998). Many of the inputs to EC come from higher order association areas, which contain complex representations of the sensory information available to the animal. In particular, cells may convey information about both the identity of a perceived object and its location with respect to the animal, or, to put it differently, about the location of the rat with respect to particular objects in the world. Such information about multiple objects may be combined in EC to form a more reliable view-based representation of the animal's location in space. Spatial information derived from path integration may also be available and may be combined with visual information to determine EC activities.

In the model, each EC cell is assumed to respond to a subset of the available cues. Based on the suggestion that EC is involved in conjunctive coding (Myers et al., 1995), each EC cell in our model combines in a conjunctive manner the sources of spatial information to which it is sensitive. Because the animal's sensory experience depends on both its position and the direction it faces, we assume (in the absence of data either way) that the activity of entorhinal neurons is head direction-, as well as location-, dependent. A model EC cell fires maximally when all the cues it is sensitive to are in the positions corresponding to the preferred location and orientation of the cell, and activity diminishes as some or all of the sources of information signal a different location or orientation. We achieve this by multiplying together gaussian tuning curves, each of which is tied to the location of a different cue and peaks at the preferred location of the cell. We assume that these individual tuning curves can have different variances.

In cases in which the environment has walls, these were found to be important sources of spatial information (O'Keefe and Burgess, 1996). For simplicity, we assume that the activities of EC neurons are completely determined by the rat's position and heading relative to the walls. We also restrict ourselves to rectangular environments and assume that all cells are sensitive to the position of all four walls [whose allocentric bearings will be referred to as “north” (N), “west” (W), “south” (S), and “east” (E)]. The only difference in the cue selectivity of EC cells in our model is that they are assumed to be sensitive to spatial information derived from path integration to different degrees. However, because this last property is only expected to be manifested under special circumstances, we actually ignore this variation in most of what follows and only consider it when we describe the results of our modeling of the experiment of Skaggs and McNaughton (1998). We assume that the tuning curve components tied to the walls of a rectangular apparatus are ridge-like functions with gaussian dependence on the distance from the wall. The variances of these tuning functions may also depend on the location and heading of the animal; in particular, we assume that the variance is lower if the animal is closer to, or facing away from, the wall. The latter dependence is based on the influence of a path integration input whose precision is greater when the animal is coming from somewhere nearer the wall and should have been able to maintain its location accurately using path integration.

The total activation of a model EC neuron as a function of the rat's location and heading is described by the following expression: Equation 2where k indexes the neuron, b is a constant to set the scale, andz_k^ais the component of the tuning function of the neuron tied to walla. The last term describes the dependence on head direction (which is assumed to be independent from the spatial components) as a circular gaussian function (with sharpness parameter ρ_EC) of the difference between the current head direction φ and the preferred heading of the cell φ_k^EC. Equation 2 bears some resemblance to the spatial tuning function used by Touretzky and Redish (1996), in that it also takes the form of a product of terms corresponding to different sources of information. However, they use this tuning function to directly model the spatial response properties of hippocampal place cells, and the parameters change with experience, whereas our EC representation is always the same for a given location and head direction in any particular environment.

The components of the tuning function tied to particular walls have the following functional form: Equation 3where d_a is the actual distance from wall a (a can be N, W, S, or E),d_k^EC,ais the distance from wall a of the preferred location of the neuron, and ς_EC,a is the width of this component, which depends on the current position and heading of the animal according to: Equation 4where φ_a is the direction of walla (0, π/2, π, and −π/2 for N, E, S, and W, respectively), and ς_EC is a constant. Equation3 and the positional-dependence in Equation 4 are similar to the expressions describing the spatial tuning of “sensory” cells in the model of Burgess et al. (1997), and “boundary vector cells” ofHartley et al. (2000). The numerical values of the parameters in the above equations have been chosen suitably for environments of approximately the size used in most relevant experiments.

Figure 2a shows two examples of the spatial and directional dependence of input components in EC, whereas Figure 2, b and c, displays the resulting net spatial and directional tuning for a sample EC neuron.

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Fig. 2.

Input components and net spatial and directional tuning. a, The dependence of a single (essentially one-dimensional) spatial component of the tuning function of cells in EC on the distance of the rat from the wall to which that component is tied. Two examples are shown, with preferred distances of 0.5 and 1.5, respectively; for each preferred distance, thesolid curve is for the case when the rat is facing the wall, and the dashed curve is for the opposite head direction. Note how the width of the curve changes with preferred distance and actual head direction. b, The net two-dimensional tuning of a sample EC neuron in a rectangular box of dimensions 2 × 1; the preferred location of the cell is (0.5, 0.4); the current heading of the rat at each location is the same as the preferred head direction of the cell. c, This polar plot shows the activity of an EC neuron as a function of the difference between its preferred direction and the actual heading of the rat.d–f, Plots similar to a–c, for the MF inputs to CA3; note that both the spatial and the directional tuning is much sharper here because of the orthogonalization property of the dentate gyrus. For all contour plots in this article, darker shading indicates higher activity, and the contour lines are at 20, 40, 60, and 80% of the maximum activity of the given cell or set of cells. Activities are normalized and the absolute values are omitted in most figures because these could be set arbitrarily in the model by changing parameters essentially unconstrained by experimental data.

Feedforward connections

There are two separate neural pathways from EC to area CA3 (Fig.1), which have quite different characteristics and likely serve different computational purposes (McNaughton and Morris, 1987; Treves and Rolls, 1992). One of these pathways is via the perforant path projection to the DG, which in turn provides a set of feedforward inputs to CA3 through the mossy fibers. Dentate granule cells are spatially selective, and, at least in linearly restricted environments, they have also been found to be sensitive to direction (Jung and McNaughton, 1993). Unlike EC neurons, dentate granule cells have sharper spatial tuning than CA3 place cells, and we assume that they are also sharply tuned for head direction. Episodic memory theories of hippocampal function suggest that an important function of the DG is that of orthogonalization, i.e., reducing the similarity between input patterns to facilitate their discrimination (O'Reilly and McClelland, 1994; Treves and Rolls, 1994), and, in keeping with the theme of linking memorial and spatial processing, we assume it plays a similar role for spatially-based inputs. One way the DG is thought to decrease pattern overlap is to implement a sparser representation (perhaps through direct competitive interactions), and indeed, the proportion of active cells in the DG at any given time is reported to be only ∼0.5% (Jung and McNaughton, 1993; B. L. McNaughton, cited byO'Reilly and McClelland, 1994).

A typical CA3 pyramidal cell receives on the order of 50 MF inputs, which are thought to be relatively powerful (Yamamoto, 1982; McNaughton and Morris, 1987). Combined with the sparseness of the DG representation, this means that a CA3 neuron is very unlikely to have more than one active mossy fiber input at any given time. In circumstances under which CA3 cells are driven primarily by these inputs, place cells essentially inherit the tuning characteristics of their afferent granule cells. We assume, for simplicity, that each CA3 cell has at most one active MF input in any given environment. This defines the base preferred location and direction for that neuron, which, of course, may then be altered by the recurrent connections in CA3. Multiple active MF inputs may explain why some place cells have multiple place fields even in simple environments (Muller et al., 1987); however, we ignore this complexity for the purpose of this paper. In addition, to make better use of the limited number of cells we can implement in our numerical simulations, all our model CA3 pyramidal cells are activated by MF inputs somewhere in any given environment, rather than the 30% or so found in practice (Wilson and McNaughton, 1993).

In its current form, the model considers both the mossy fiber connections and the perforant path connections from EC to DG as being fixed. Because our goal is to model activity in CA3, and that is completely determined by its inputs and internal dynamics, we can therefore skip modeling the dentate gyrus explicitly and proceed by characterizing how the MF input to CA3 (which results from processing in DG) depends on the characteristics of the environment. We assume that, for any single environment, the MF input to CA3 place cells has a similar functional form to the tuning function of EC cells described in the previous section, but both the spatial and the directional tuning is assumed to be sharper as a result of sparsification and orthogonalization in DG (Fig. 2d–f). This can be achieved by replacing the spatial spread parameter ς_EC with a smaller value, ς_MF, and by replacing ρ_EC, characterizing the sharpness of directional tuning, with a larger ρ_MF in Equations 2 and 4. The proposed orthogonalization property of the dentate gyrus becomes more pronounced when we look at multiple environments. We assume that, except when two environments are quite similar, the MF inputs to CA3 in two different environments are completely unrelated. We will return to the case of exceptionally similar environments in a later section.

The perforant pathway (PP) also provides a direct connection between EC and CA3 and has a large degree of divergence and convergence. Thus, CA3 cells can sample the EC representation very effectively. In the model, we implement this property using all-to-all connections between EC and CA3 neurons, although this is obviously a simplification. This pathway is also known to be capable of long-term synaptic plasticity (Breindl et al., 1994). In the model, the strengths of these connections (denoted by W_ik for the connection from entorhinal cell k to CA3 cell i) are initially set to zero, and they are assumed to be modifiable by associative Hebbian learning.

Network dynamics

Although we will shortly be interested in the spatial representation that results from on-line learning during exploration, we first test our model using an idealized set of connection strengths to gain some insight into its dynamical behavior. For this, we just assume that the weights result from an idealized form of Hebbian associative learning, and thus reflect the correlations between connected neurons. It has been noted (Muller et al., 1991; Shen and McNaughton, 1996) that such an associative learning process for spatially selective neurons can lead to connections whose strength is a function of the distance between the preferred locations of the presynaptic and postsynaptic neurons, exactly the sort of connections that can support a place field-like attractor structure in CA3 (Samsonovich and McNaughton, 1997). Here we assume that the CA3 recurrent weights are determined by the correlations between the mossy fiber inputs to the cells, and the perforant path weights between EC and CA3 are given by the correlations between EC activities and MF inputs to CA3. These correlations are calculated as spatial averages (which assumes spatially homogeneous exploration) over all locations and head directions in the part of the environment in which the postsynaptic cell is active, resulting in the following expressions for the recurrent weights J_ij and perforant path weights W_ik : Equation 5whereI_i^MFis the mossy fiber input to neuron i in CA3,z_k is the activity of neuron kin entorhinal cortex, and κ sets the learning rate.

Using these expressions, we can calculate the weights resulting from even exposure to a rectangular box (with one side twice as long as the other). Then, lettingI_i^PP= Σ_kW_ikz_k andI_i^MF= 0, where z_k is the EC activity pattern corresponding to a particular location and heading in the environment, we simulate the neural dynamics described by the full Equations 1 for a fixed number of iterations (using Euler's method). We find that, within a broad range of model parameters, the network always settles into a stable state by the end of the iterations. Furthermore, for most initial CA3 activity patterns, the same final state is reached for given feedforward inputs. This shows that these states are actually attractors of the neural dynamics and that they have suitably large basins of attraction. The final state of the network was determined for different input patterns in EC, representing different positions and head directions of the animal over a grid that covered the whole environment. The firing rate map for a given cell is defined as the final activity of that cell as a function of the actual location and head direction of the animal.

Throughout the paper, two different kinds of plots are used to display the activities of neurons (and their inputs). Quantities characterizing single cells as a function of actual position and heading (such as firing rate maps) are shown in a “single-cell plot,” which is the kind of plot traditionally used to describe the spatial activity patterns of place cells. A single-cell plot may contain multiple subplots to represent different headings at any given location. The second kind of plot we use is the “population plot,” which describes the behavior of all the cells with the actual position and heading of the animal kept fixed. In the population plot, we arrange cells with a given preferred direction on an imaginary plane according to their preferred locations (for CA3 place cells, this is defined as the preferred location of their active mossy fiber input). A complete population plot would include eight subplots, one for each population of cells with a different preferred direction, but we typically show only one, two, or four of these, depending on the degree of variation with preferred direction in that particular case. Population plots in this paper are marked with P in the bottom right corner for easy identification.

The results of the simulations with the “ideal” weights are summarized in the population plots of Figure3, which display activities in EC, net perforant path inputs, and final activities in CA3 for all cells with two particular (opposite) preferred head directions, when the model rat is at a given location, facing in a particular direction. Figure 3shows that the final states of the model CA3 network resemble thresholded two-dimensional gaussian bumps of activity in the population plot. This type of solution can emerge spontaneously from the network dynamics even in the absence of external inputs, in which case the location of the bump is random, i.e., determined by the initial neural activities, as well as various other factors, including the distribution of preferred locations and directions of the neurons. Although the network only has a finite number of point attractors (possible stable activity patterns) in the absence of input, when there is even a small perforant path input to CA3, the location of the bump is determined by this input so that the activity profile provides the best possible fit to the input. The position of the peak varies continuously, and the shape of the activity profile is essentially constant. This holds in our model if the net feedforward input to the most active CA3 neurons is between ∼1 and 30% of the summed input they receive from other CA3 cells; in most simulations, we set the relative efficacies of perforant path and recurrent synapses so that this ratio is ∼5%.

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Fig. 3.

The formation of nondirectional place fields.a, b, The bottom plot in each case shows the actual position (indicated by thecross) and head direction (indicated by thearrow) of the rat in the environment. The other plots are population plots (as defined under Results, Network dynamics, and marked with P), and they show, at the location and direction in the bottom plot, the activities of cells in EC, the net PP inputs to CA3 neurons (I_i^PP), and the final activities of the same place cells (marked CA3), as a function of the preferred location of the neuron; the twocolumns in both a and b are for cells with preferred head direction indicated by the arrow aboveeach column.

Figure 3a illustrates how inputs are used by the network to effectively select one of the possible final states. First of all, the EC activity pattern (which is determined by sensory features in the environment as already described) gives rise to a pattern of perforant path inputs to CA3, which is centered on neurons with preferred locations close to the actual position of the rat, although the profile is even broader than the activity profile in EC. This is the consequence of plasticity of the perforant path in the learning phase, which establishes an association between EC cells and CA3 neurons with similar preferred locations and head directions. Based on the learned weights, the PP projection also reduces directionality substantially, so that inputs to CA3 already depend less on the preferred head direction of the cell than neuronal activities in EC. The shape of the final activity profile across place cells is, however, essentially determined by the CA3 internal dynamics, resulting in a spatial activity profile that is much more sharply peaked than the feedforward inputs. Furthermore, the final activities of the cells are essentially independent of their preferred head direction. The resulting model place fields (i.e., single-cell activity maps) possess many of the characteristics of real place cell firing patterns recorded in open environments. As we will see later (see Fig. 6a), they are unimodal, approximately gaussian with circular symmetry, and essentially nondirectional.

Figure 3 reveals how nondirectional place fields result despite the directional input representation in EC. The two parts of the figure compare the activities of EC neurons, the PP inputs to CA3 place cells, and the final activities of place cells as the model rat faces in two opposite directions at the same location. Because of the properties of the PP projection discussed above, place cells receive relatively similar inputs in the two cases. More importantly, however, this leads to the emergence of the same, nondirectional, attractor in CA3, making the place fields independent of head direction. It should be emphasized that, given the dominance of internal connections in determining the final state of the system, even PP inputs as similar as those in Figure3, a and b, could easily lead to fundamentally different patterns of final activity if the two input patterns biased the system toward different attractors. Indeed, these same two EC input patterns do actually give rise to two very dissimilar final patterns if the weights are set up during a directed search task like the one described later (instead of the omnidirectional random exploration assumed here).

Task-dependence of directionality

We have already described how random exploration in an open environment can lead to nondirectional place fields (an example of which is shown in Fig. 6a), through the establishment of appropriate attractors in CA3. In agreement with the recent modeling study by Brunel and Trullier (1998), we found that the ability of the recurrent network to suppress the directionality of the inputs depends critically on the set of locations and head directions experienced by the rat during learning. Place cells become direction-independent only in situations in which the animal is exposed to a wide range of directions at a particular location. On the other hand, when the behavioral task or the environment itself constrains the set of directions experienced at a given location, as in a radial maze or when the rat is required to follow a specific route in an open field, place cells retain their intrinsic directionality. Even in these cases, the width of directional tuning can, however, be modified by the recurrent network. These results are in good agreement with experimental findings (Muller et al., 1994; Markus et al., 1995). The dependence of directionality on movement patterns is illustrated in Figure 6b, which shows the place field of the same model CA3 cell that appears in Figure 6a, for a rat that has performed a different behavioral task in the same environment. In this task, which can be thought of as a simplified version of the directed search task described by Markus et al. (1995), the rat is required to run back and forth between the two shorter walls of the environment to obtain reward. For the idealized case, we model this by assuming during exploration that the rat is now exposed only to the two directions parallel to the long walls instead of all directions at each location. Everything else in the simulations is left the same. This change affects the correlations between place cells in the learning phase, resulting in altered weight structure, which, in turn, changes the attractors. In agreement with experimental data, the new attractors do not eliminate the directionality of the inputs to the place cells. In fact, two very distinct sets of attractors are established, one corresponding to each of the two directions sampled during learning. To illustrate this point, in the bottom half of Figure 6, we plotted the maximum activity of the place cell shown at thetop of the figure, as a function of the animal's heading. In the rat trained using random exploration, the activity of the cell is essentially direction-independent; however, if we train the rat in the shuttling task instead, the cell fires at a high rate for all directions with a westward component and is completely silent for all directions with an eastward component. Cells that prefer direction east behave in exactly the opposite way. Once more, the results of simulations with the on-line learning procedure are similar to those obtained using the ideal weights, although the differences in directionality between the two training paradigms are generally somewhat reduced, and activity changes with head direction tend to be more graded. For further discussion on how different behavioral paradigms might lead to spatial representations with different degrees of directionality, see Brunel and Trullier (1998).

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Fig. 6.

The task-dependence of directionality. The contour plots show the place field of a CA3 cell that prefers the left direction, when the rat faces in the direction indicated by thearrows, and the polar plots show the maximum firing rate (indicated by the crosses, and relative to the maximum rate when averaged across directions) of the same neuron as a function of head direction; a, in a model rat which explored the environment randomly during the learning phase; b, in a model animal that always ran in one of the directions parallel to the long walls of the box during learning. The top plot is empty in b because the cell does not fire at all in that direction in this case. The effect of the attractor dynamics is very prominent in the all-or-none nature of activity in the directional plot in b (all the points in the bottom half of the plot collapsed to the origin). The maxima of thetop and bottom contour plots correspond to the crosses at 270 and 90°, respectively, in the polar plots.

Very different environments

Experiments in which the firing rate maps of place cells are recorded in multiple environments that are similar to a controlled degree can provide valuable information about how input representations depend on details of the environment, how they are transformed into the place cell representation, and also about possible interference between representations of different environments realized by the same network of place cells. The general pattern of results is that radically different environments give rise to very different and apparently unrelated place cell representations (O'Keefe and Conway, 1978; Muller and Kubie, 1987; Bostock et al., 1991). On the other hand, when a previously familiar environment is subjected to subtle alterations, the place cell representation often stays basically the same (O'Keefe and Conway, 1978; Bostock et al., 1991) or changes according to the transformation of the environment (Muller and Kubie, 1987; O'Keefe and Burgess, 1996).

To test our model in the first type of situation, we added another model environment to the one described in the previous section and tested whether these two environments can be learned and recalled simultaneously without interference. The two environments are very different in terms of visual appearance; the new environment has a circular shape and is assumed to carry visual features that are dissimilar to the ones in the rectangular box. Therefore, we assume that the spatial characteristics of both EC neuronal activities and mossy fiber inputs to CA3, as well as their relationships, are completely independent in the two environments, i.e., for instance, knowing the relative locations of maximum activity for two EC neurons in one environment carries no information about the relationship of their preferred locations in the other environment. However, as a worst case scenario, we use exactly the same neuronal populations to represent the two environments; if these populations are distinct to any extent, this can only improve the separability of the two environments. Because we are interested in interactions between different environments and not in extending our input model to curved walls and other cues, we derive the inputs in the circular environment assuming that there is a very salient square box (which looks very different from the rectangular box) surrounding the circular arena so that the inputs are determined by distances from the walls of the square box in the same way as before.

Initial learning in the rectangular environment is performed using the on-line procedure described in the previous section, and the resulting place cell firing patterns are determined as before. Then the weights are modified by running a learning phase in the circular environment, and spatial firing distributions during recall are determined in both environments to assess interference caused by exposure to the other environment.

Figure 7 shows the firing rate maps of five model CA3 cells in the rectangular apparatus before any exposure to the circular environment (top row) and in the rectangular and the circular apparatus after learning in both environments (middle and bottom rows). In general, there is no systematic relationship between the location of place fields in the two different environments, which indicates that several different sets of attractors can be stored and recalled independently in the model.

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Fig. 7.

Very different environments. This figure shows the place fields of five selected place cells in a rectangular and a circular apparatus that have very different sensory features. Thetop row (R1) shows the place fields after learning in the rectangular apparatus but before any experience in the circular one, and the middle and bottomrows show the place fields in the circular and rectangular environments (C and R2, respectively) after the rat has become familiar with both. There is no obvious relationship between place fields of the same cell in the two environments. The effect of encoding a second environment on the place cell representation in the first environment can be assessed by comparing the top and bottom rows. Although there are some visible changes, these tend to be small and do not affect the general structure of the spatial representation. One of the few exceptions is shown on the far right in which a place cell that had been silent in the rectangular environment becomes active there after experience in the circular environment.

Comparing the top and bottom rows of Figure 7reveals that most place cells have very similar firing rate maps in the rectangular box before (R1) and after (R2) training in the circular environment. In particular, for the majority of CA3 cells, the location of maximal firing, the size, shape, and directionality (data not shown) of the place field are all virtually unchanged. Consequently, the overall structure of the spatial representation is essentially unaffected by exposure to a different environment. However, for a minority of place cells, experience in the circular environment resulted in a more radical change in the firing rate map in the rectangular box (as in the last example in Fig. 7). The most commonly observed types of change were the appearance of a new place field and the disappearance of one previously present. These probably occurred when the changes in the net input received by the cell (resulting from the weight changes that took place in the other environment) caused the neuron to cross the dynamic threshold for activation. Learning to represent a new, orthogonal environment can be thought of as introducing noise into both the feedforward and the recurrent weights as far as the representation of the original environment is concerned.

To quantify the change caused by exposure to a different environment, we computed the overlap between the overall CA3 spatial representations in the rectangular box before and after learning in the circular environment. To obtain a scale against which we can measure differences in overlap and also to facilitate direct comparison with experimental data, we generated from our firing rate maps a large number of spike count samples, assuming independent Poisson noise for all cells and bins. Maximum firing rates, bin sizes, and session time were similar to those in experiments (Muller et al., 1987). The correlation coefficient between samples from the R1 and R2 spatial representations was found to be 0.754 ± 0.001 (mean and SD), which is significantly lower (t test, p < 0.0001) than the correlation between different samples from R1 (0.911 ± 0.001) but significantly higher (p < 0.0001) than the correlation between samples from R1 and a version of R1 in which place cells have been randomly reshuffled (−0.005 ± 0.0005). These figures confirm our observation that, although there is a certain degree of degradation, the spatial representation after learning in an orthogonal environment remains quite similar to the original one. Furthermore, because the number of neurons and connections is much larger in the real hippocampus than in the model and not all neurons are active in any particular environment, interference between representations of different environments is likely to be less severe, and the number (and perhaps the spatial extent) of environments that can be stored is probably larger.

Finally, our model would also produce orthogonal place cell representations for environments that differ only in shape (Muller and Kubie, 1987), even from nonorthogonal input representations (Quirk et al., 1992), provided that the DG can separate the input patterns effectively, and the two environments are perceived as different so that learning is initiated in both environments.

Geometric manipulations

We also investigated what happens to place fields in our model if the environment undergoes some simple geometric transformation. We chose to model the experiment of O'Keefe and Burgess (1996) because of its relatively complex pattern of results. In this experiment, a rat, which has been thoroughly familiarized with a rectangular box, is transferred into a new box that differs from the original one only in the length of one or both sides. We will concentrate on the case when the second environment is a larger square box that can be obtained by stretching the original box by a factor of two. In this case, stretching the environment had one of the following general effects (O'Keefe and Burgess, 1996): some fields remained fixed with respect to one of the walls of the apparatus; some changed their location and/or shape in correspondence with the transformation of the box; others developed a second peak in the direction of stretching. Many of the cells with two-peaked or stretched fields also developed directional-dependence, i.e., the location of maximum activity depended on the heading of the rat, usually in the way that the subfield closer to a wall was more active when the rat was facing away from that wall.

We assume that learning is triggered by exposure to the novel situation of the initial, rectangular box and that the transformed environment in this case is similar enough to the original one so that no significant learning occurs subsequently. Therefore, the attractors established in the first environment are the final states of the network dynamics in the new environment as well, and place fields are determined by the way that the inputs (as a function of location and direction) in the new environment select attractors established in the old environment.

Figure 8a shows the place fields of four model CA3 neurons in the rectangular box that was used during initial learning and in the larger square box. The place fields follow the transformation of the box; that is, their centers remain at the same relative distance from opposite walls, and their shapes become elongated along the direction of stretching. As revealed by Figure8b, the fields consist of directional subcomponents with the observed relationship between subfield position and preferred direction.

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Fig. 8.

Place fields in transformed environments.a, The place fields of four selected cells in the original and the stretched environment in our simulation of the experiment by O'Keefe and Burgess (1996); the firing rates shown are averages over all head directions. b, Directionality of the place field shown in the bottom right corner ofa; the place field depends on the heading of the rat (indicated by the arrows). This dependence on head direction is induced by the transformation of the environment; place fields in the original environment are essentially nondirectional (like the one shown in Fig. 6a).

We can understand some of the characteristics of transformed place fields by looking at attractor selection in our model. Attractors have a regular, compact shape if place cells are characterized by their preferred locations in the original environment; on the other hand, we have no a priori knowledge about what they look like as a function of preferred locations in the new environment. Thus, it is much easier to understand the transformations occurring in the system if we look at activities in the new environment (the square box) as a function of the preferred coordinates of the neurons in the old environment (the rectangle). This is illustrated in Figure9, which shows the activities of EC neurons, PP inputs to CA3, and final activities (after recurrent processing) in CA3 at three different locations in the square box, all as functions of preferred locations in the rectangular box. The activities of EC cells are determined by multiplying together (gaussian-tuned) components whose activities depend on the animal's heading and its position with respect to the walls. Because the walls have moved relative to each other, the different components lead to different estimates of position in the old coordinate system. Combining such inputs conjunctively leads to an EC activity profile that peaks somewhere between the positions indicated by individual walls. For instance, when the rat is halfway between the two walls that have been moved apart, listening to one of these walls would indicate that the animal is located at the opposite wall, and the resulting EC activity profile is centered on neurons like the middle of the rectangular box (Fig. 9, bottom left contour plot). Because the PP connections were established in the rectangular box, the PP input pattern to CA3 cells is centered around the same location as the EC activity pattern if both are viewed as a function of preferred coordinates in the rectangle (Fig. 9, compare first andsecond columns). The recurrent connections then sharpen the activity profile considerably but leave the location of the bump (in the old coordinate system) essentially unchanged. The final activities of CA3 cells as a function of location in the square box define the place fields in the new environment. We can see that, as the rat moves around in the new environment, the activity packet also moves smoothly on the plane defined by the preferred locations of place cells in the rectangular box. This results in a smooth transformation of place fields between the two environments. In addition, the activity packet moves more slowly in the stretched direction in the old coordinate system than the actual speed of the rat in the new environment, or, in other words, the rat needs to travel approximately twice as much in the square box than in the rectangular box for the activity profile to shift by the same amount; consequently, place fields become elongated in the direction of stretching.

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Fig. 9.

Place field stretching. The population plots of this figure show the neuronal activities in EC, the PP inputs to place cells, and the CA3 final activities as a function of the preferred location of the neuron in the original, rectangular box, for three different positions of the rat in the square box, indicated by thecrosses in the plot on the left. The plots only show cells with preferred direction north, and the model rat faces west in all cases.

The emergence of directional subcomponents can be understood by looking at how the activities of EC cells and the resulting activities of CA3 neurons depend on the head direction of the rat. This is depicted in Figure 10, which shows that, because of the dependence on head direction of the rat's confidence in the inputs from different walls (as described earlier), conflicting sources of information are weighted differently depending on which way the rat faces. The EC activity profile and, consequently, the CA3 activity profile, shift as the rat turns around in the square box, and the result is that a given place cell fires maximally at different locations depending on head direction. In the example shown in Figure10, the activity profile shifts north when the rat faces north, and shifts south when the rat faces south. In an apparent paradox, from the perspective of a single place field, this actually has the opposite effect (Fig. 8b), that the center of the place field is farther south when the animal faces north, and vice versa. The easiest way to see this is to ask where in the environment the rat has to be when it is facing in a particular direction, to arrange for exactly the population activity across CA3 shown in the middle of Figure10. The answer to this will tell us how the favored location of the most active cell in this population depends on direction. When the rat faces north, the activity profile shifts north, so the rat must be displaced relatively south to compensate for this. Thus, the location for the peak response of the cell is shifted south. The converse is also true; when the rat actually faces south, then the place field moves north.

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Fig. 10.

Directionality of stretched place fields. Population plots of EC neuronal activities and CA3 final activities (of the same sets of cells as in Fig. 9), as a function of the preferred locations of the neurons in the rectangular box, for different headings of the rat (indicated by the arrows) at a single location in the square box (marked by the middle crossin Fig. 9). The middle row of plots is for both directions east and west because these lead to the same activities for the neurons displayed here. The position of the input peak changes as the rat faces in different directions (because of the dependence on head direction of the breadths of the input components tied to different walls), and the position of the final activity profile in CA3 changes accordingly. This shift can be compensated for by changes in location (as seen in Fig. 9), resulting in the directional subfields shown in Figure 8b.

Very similar environments

Skaggs and McNaughton (1998) conducted an experiment designed specifically to probe the relationship between spatial representations in environments with a high degree of similarity. In this experiment, animals explored an apparatus that consisted of two visually identical boxes connected by a corridor. Many place cells were found to have similar place fields in the two regions, whereas others had uncorrelated place fields. This finding challenges the idea that there is a predefined set of uncorrelated attractors wired into the recurrent connections in CA3 (Samsonovich and McNaughton, 1997), because such a model would predict either identical or orthogonal firing patterns in different environments or different parts of the same environment. This particular problem may be solved by postulating a hierarchy of fixed attractors with various degrees of overlap (Samsonovich et al., 1998); however, it still remains to be explained why similar representations are selected in very similar environments. On the other hand, the attractors established in our model are input-dependent, which in principle allows attractors with an arbitrary degree of similarity, and directly defines the association between attractors and environments. Therefore, we simulated the experiment by Skaggs and McNaughton (1998)in our model to study the spatial representations in very similar environments.

We still do not model the different sources of spatial information explicitly. We assume that there are some inputs (e.g., signals derived from path integration) that allow the two boxes to be distinguished, whereas other inputs to the system (e.g., local visual cues) are identical at corresponding locations in the two boxes. Because cells in EC are assumed to respond to different inputs to a randomly varying extent and to encode these inputs conjunctively, we applied the following scheme to determine activities in EC at locations inside the two boxes. EC cells are now characterized by a preferred location (and also a preferred head direction) based on visual inputs (this is now actually a set of two locations, one in each box), as well as a polarization index (P), which is defined as the maximum firing rate for the cell in the north box minus the maximum firing rate in the south box, divided by the maximum rate in any of the boxes. P is always between −1 and 1, its magnitude indicates how much that particular cell is influenced by cues that distinguish the two boxes, and its sign shows which box the neuron prefers. We assign P values to EC cells randomly from a uniform distribution. The firing rate of an EC neuron is then given byz_k = (1 +P_k )z_k ′ in the north box and z_k = (1 −P_k )z_k ′ in the south box, where z_k ′ is a function of coordinates within the current box, and it depends on spatial position and head direction the same way as z_k in Equation 2. We assume that the MF inputs to CA3 can be characterized similarly; however, because of the orthogonalizing properties of the dentate gyrus, P values do not vary continuously but only take the values −1, 0, and 1, each with probability of a third. This means that there is a population of cells in CA3 that receives the same input at corresponding locations in the two boxes during learning, whereas another population receives different inputs. Because the first time the rat is introduced into the apparatus it is allowed to explore it entirely, we do not treat the two halves of the environment differently during the learning phase.

Some examples of the place fields that develop in this model are shown in Figure 11. There are cells that have similar firing rate patterns in the two boxes, whereas others are active in only one of the boxes, in accordance with experimental observations. In other words, our model has no difficulty storing and recalling partially overlapping spatial representations. In the model, the degree of overlap is determined by the extent of orthogonalization occurring in DG, i.e., what proportion of granule cells distinguishes between the two boxes; CA3 cells simply inherit the selectivity of their MF inputs as attractors are established during the learning phase. Most EC neurons are active in both boxes, although to a different extent (Fig. 12a). Consequently, all CA3 cells that are active in this environment get a substantial PP input in both boxes (Fig. 12b); however, the activity patterns encoded during learning are restored by the recurrent connections and feedback inhibition, and the PP input only determines which of these patterns emerges. The figures also show that although EC neurons have relatively broad tuning curves, and this results in CA3 cells receiving feedforward input that is even more broadly tuned, the final tuning of CA3 neurons is considerably sharper because of recurrent activity. The attractor network also renders place cells directionally nonselective, just as before.

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Fig. 11.

Place fields in our simulation of Skaggs and McNaughton (1998). The figure shows the place fields of five CA3 place cells in the two identical boxes; activity in the corridor connecting the boxes was not simulated.

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Fig. 12.

Input representation and inputs to CA3 in our simulation of Skaggs and McNaughton (1998). a, The activities, as a function of location in the apparatus, of three entorhinal neurons that have the same preferred (visual) location within the boxes but different degrees of polarization (as defined under Results, Very different experiments; the polarization indices are −0.01, 0.25, and −0.70, respectively). b, This part of the figure, which displays the perforant path inputs to the first three CA3 place cells of Figure 11, shows that, as a result of learning in the perforant pathway, some place cells receive similar inputs at corresponding locations in the two boxes, whereas others receive inputs of different magnitudes, setting the stage for the CA3 recurrent network that makes these differences much more pronounced (as seen in Fig. 11).