Correspondence to: Robert U. Muller, State University of New York- Health Science Center at Brooklyn, 450 Clarkson Ave., Brooklyn, NY 11203. Fax:718-270-3103 E-mail:bob{at}fasthp.hippo.hscbklyn.edu.
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Abstract |
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Changing the angular separation between two visual stimuli attached to the wall of a recording cylinder causes the firing fields of place cells to move relative to each other, as though the representation of the floor undergoes a topological distortion. The displacement of the firing field center of each cell is a vector whose length is equal to the linear displacement and whose angle indicates the direction that the field center moves in the environment. Based on the observation that neighboring fields move in similar ways, whereas widely separated fields tend to move relative to each other, we develop an empirical vector-field model that accounts for the stated effects of changing the card separation. We then go on to show that the same vector-field equation predicts additional aspects of the experimental results. In one example, we demonstrate that place cell firing fields undergo distortions of shape after the card separation is changed, as though different parts of the same field are affected by the stimulus constellation in the same fashion as fields at different locations. We conclude that the vector-field formalism reflects the organization of the place-cell representation of the environment for the current case, and through suitable modification may be very useful for describing motions of firing patterns induced by a wide variety of stimulus manipulations.
Key Words: cognitive maps, stimulus control, place fields
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INTRODUCTION |
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The purpose of this paper is to present a quantitative description of how the firing fields of hippocampal place cells are affected by two kinds of stimulus manipulations (
A full account of these results would be based on a neural network model and would address a variety of questions, including the origins of the firing fields in the standard conditions, the lack of effect of card removals on both the relative positions of fields and firing rates, and the ability of card reconfigurations to induce both relative field position movements and position-independent decreases of firing rate. The geometric theory we present here is less ambitious; it is concerned only with field movements and explains neither why fields exist in the first place nor why reconfigurations cause changes in firing rates.
This theory consists of an empirical vector-field equation that relates the movement of all field centroids to weighted functions of the movements of both cards. Field movement is therefore a smooth function of the initial position of the centroid in the cylinder. Thus, neighboring fields are constrained to move in concert and no allowance is made for individual fields to be coupled to arbitrarily selected combinations of the available stimuli. The model, like our data, is therefore in contrast to the virtually complete independence of fields from each other that was used to describe the effects of counter-rotating distal and proximal cues (
Beyond providing a concise summary of field movements, the vector-field theory is valuable in several ways. First, it serves as a benchmark against which the field-movement predictions of any network theory can be checked. Second, we will show that the vector-field equation predicts additional features of our experimental data that were not included in the equation. In our view, the success of these predictions indicates that the stimulus cards act in a smooth, continuous way on the positional activity distributions of hippocampal neurons located everywhere in the environment and that the undistorted fields in the standard configuration therefore serve to indicate location in space rather than the conjunction of stimuli important for each cell.
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METHODS |
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Our goal is to write an empirical vector-field equation that describes how the center of a firing field anywhere in a cylindrical apparatus moves when a white stimulus card and a black stimulus card on the cylinder wall are both moved or one is deleted and the other is moved. Thus, we want to calculate V, the displacement vector for the field center (Equation 1):
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(1) |
where X is the initial location of the field and VB and VW are vectors that point from the initial to the altered locations of the black and white cards (see Fig 1 for notation).
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We first treat rotations of field centers induced by moving a card. During rotations of a single card (
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(2) |
where R is the radius of the cylinder and r is the distance of the field from the cylinder center; the direction of the vector is such that the angular movement of the field is equal to the angular movement of the black card. Similarly, the rotational field movement that would be induced by the white card is V 'W.
If the two cards always contributed equally to rotations of field centers, the net rotational movement of the field center would be the average of the vectors V 'B and V 'W. We saw from reconfiguration experiments, however, that the contribution of the card near a field was greater than the contribution of the other card. We imagine, therefore, that the contribution of a card is inversely proportional to the distance of the field from that card. The net rotational movement, VR, is the weighted average of the rotational field movements that would be separately induced by each card and is given by:
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(3) |
where dw is the distance from the field center to the white card and db is the distance from the field center to the black card as show in Fig 1. Equation 3 correctly predicts that equal rotations of the two cards causes equal rotations of all field centers. Under the interpretation that deleting a card makes the distance from the card to any field arbitrarily large, Equation 3 also correctly predicts that after removing one card all fields rotate equally with the remaining card. Moreover, it works well for fields near either card. Interestingly, Equation 3 describes how field centroids would move in the case that each card controls an independent component of the field and is considered briefly in the RESULTS.
Although Equation 3 performs acceptably near the cards, it incorrectly predicts the movements of fields between the two cards. Thus, regardless of whether the cards are moved apart or together, Equation 3 predicts that the net movement of such fields is toward the cylinder center rather than nearly parallel to the motion of the line that connects the card centers. To compensate for this error, we define a translational vector for fields, VT, given by:
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(4) |
The direction of VT is always parallel to the direction of motion of the line that connects the two card centers. The two terms in the denominator of Equation 4 serve different purposes. The first makes VT zero if dw or db becomes arbitrarily large when the corresponding card is removed. The constant c1 is very small so that this term has no effect except during card removal. The second term in the denominator reduces the effect of the translational term as the distance to either card decreases, thereby preserving the effects of the rotational term. The constant c2 allows the amount of translation caused by card movements to be adjusted independent of the amount of rotation caused by card movements. The total motion of the fields, V, is the sum of VR and VT:
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(5) |
The patterns of field movements generated by Equation 5 for apart and together card manipulations are shown in Fig 2 in the normalized coordinate system (
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RESULTS |
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Predictions of a Simple Component Model for Field Centroid Movements
Changes in the size or aspect ratio of a recording box can induce changes in the size and shape of firing fields (
Might card reconfigurations also split fields into two components? We saw no such effect in firing rate maps, but the simplicity of the stimulus arrangement makes it is easy to test the component hypothesis directly. We assume that both putative components are in register when the cards are in their standard positions. We also assume, from card removals, that each component undergoes a pure rotation caused by rotation of the relevant card. By symmetry, the centroid of the composite field must be halfway between the centroids of the two components, so the composite centroid can experience only radial, but not angular movements. In addition, this radial movement can only be inward, regardless of the angular position of the field, an effect clearly at odds with our data.
In this simple component model, the weakening of control with distance between the card and the field is not included. We have, however, already implicitly considered a component model that includes the distance effect, expressed by Equation 3. Solutions to Equation 3 are shown as gray lines in Fig 3, where it is seen that fits to the angular centroid movements are not very bad, but fits to the radial movements are unacceptable. It was the failure of Equation 3 that led us to add the translational term.
From this analysis, we conclude reconfigurations do not split fields into components that are controlled by the two cue cards. It is important to note that this analysis militates against any model in which the two cards separately trigger activity in the place cells. Thus, we argue that schemes in which control over firing resides at some times with one card and at other times with the second card are not accurate. Schemes of this sort include those in which the animal resets the coordinate system relative to one card, and then uses only self-motion information until another reset relative to the other card.
Fit of the Vector Field Equation to Field Centroid Movements
The ability of Equation 5 to reproduce field centroid movements after the cue cards are moved apart is shown by the black lines in Fig 3 A, 1 and 2. The observed angular (Fig 3 A, 1) and radial (2) displacement vector components are the same as in Figure 6 B, 1 and 2, of
A correlational analysis indicates a satisfactory fit of Equation 5 to the angular and radial displacements caused by apart card movements. We first calculated the correlation between two measures of the angular movement of field centers in apart sessions; namely, the observed movement compared with standard sessions and the movement expected from the vector-field equation given field center positions in the previous standard session. According to a t test, the probability that the correlation of 0.774 with 45 df occurred by chance was 1.78 x 10-10. The correlation of 0.721 with 45 df between observed and predicted radial movements associated has a probability of 1.08 x 10-8. Thus, the theory accounts for 60% of the variance in angular field movement and 52% of the variance in radial field movement caused by moving the cards apart without any correction for random movements of field centers between pairs of standard sessions (
The ability of Equation 5 to account for field centroid movements when the cards are moved together is shown by the black lines in Fig 3 B, 1 and 2. Once again, the solutions are for c2 = 83.2 and for the circle that divides the cylinder into a central disk and an annulus of equal area. A correlational analysis confirms that the fit is very good for the angular component of centroid displacement (r = 0.936, df = 62; P = 1.00 x 10-29), but not so good for radial movements (r = 0.296; df = 62; P = 0.017). The theory therefore accounts for 88% of the variance in angular field movement, but only 9% of the variance in radial field movement. We think that the overall performance of the theory is very good, but the relatively poor ability to predict the radial movement appears to be a real discrepancy that is seen again when we attempt to predict the relative amount of field movement parallel to and perpendicular to the line that connects the centers of the cue cards.
Differences of Field Movements Induced by Moving the Cards Apart and Together
As seen in Fig 2, the vector-field equation makes two specific predictions concerning the movements of fields after apart and together card manipulations. First, movements parallel to the horizontal diameter (Fig 4) should be in opposite directions and, second, there should be no average movement in the vertical direction. In addition, numerical solutions of Equation 5 indicate that the magnitude of the movement in the horizontal direction should be slightly greater for apart sessions than together sessions. The source of this difference is in the translational term of Equation 5. The mean horizontal displacement from the translational term for the initial field positions of the theoretical vectors in Fig 2 is -4.16 cm for apart sessions and +3.80 cm for together sessions. Since the mean contribution of the rotational term of Equation 5 is equal in magnitude (1.10 cm) for apart and together sessions and of the same sign as for the translational term, the total expected horizontal displacement is -5.95 cm for apart sessions and 5.59 cm for together sessions.
The observed horizontal and vertical centroid displacements for apart and together sessions are summarized in Fig 4. In agreement with theory, the average vertical displacements of field centroids for apart (0.147 cm) and together (0.387 cm) sessions were not reliably different from each other [t = 0.38; df = 109; P(t 0.38) = 0.71], nor was either reliably different from zero. Also, as expected, the mean horizontal centroid displacement was negative for apart sessions (-4.84 cm) and positive for together sessions (+1.80 cm), and the magnitude of the horizontal displacement for apart sessions was greater than the magnitude of the horizontal displacement for together sessions [t = 4.82; df = 109; P(t
4.82) = 4.6 x 10-6]. Thus, the difference in horizontal displacement is in the expected direction, but is much larger than predicted by theory. The origin of this discrepancy is the small average horizontal displacement caused by together card movements. We showed this by determining, for each observed horizontal displacement, the corresponding predicted displacement. We then did paired t tests between observed and expected horizontal displacements for apart and together sessions. For apart sessions, the average difference between observation and expectation was 0.55, so that the observations were somewhat smaller than expected. This discrepancy was not, however, statistically reliable [paired t = 1.14; df = 46; P(t
1.14) = 0.26]. In contrast, the average difference between the observed and expected horizontal displacements for together sessions was -3.77 cm, so that the observed displacements were considerably smaller than expected [paired t = 5.67; df = 63; P(t
5.67) = 3.6 x 10-9]. Similar paired t calculations for the vertical displacements show that the average difference between observation and expectation is not reliably different for either apart or together card movements.
In summary, the directions and magnitudes of the horizontal and vertical field centroid displacements caused by apart and together card manipulations are in good agreement with theory. The only discrepancy is the smaller than expected average horizontal movement seen after together manipulations, an effect we are currently unable to explain. One way of handling this discrepancy would be to separately analyze apart and together sessions, allowing us to choose a larger value of the constant c2 in Equation 5 for together sessions. In the absence of a reason to expect different horizontal movements and for parsimony, however, we used only a single form of Equation 5. In this regard, we note that the accuracy of additional predictions from the vector-field equation (see below) might have been improved by treating apart and together manipulations separately. It turns out, however, that we can make accurate predictions of changes in firing field shapes and positional firing patterns of hippocampal theta cells (interneurons) without complicating the theory. We believe that these predictions are not sensitive to the relatively small field displacements caused by together manipulations. This insensitivity arises because the additional analyses are made after field centroid displacements are subtracted by superimposing the centroid in a manipulated session onto the centroid for a standard session.
Changes of Field Shapes Caused by Card Reconfigurations
Up to now, we have considered only movements of firing field centroids caused by changing the angular distance between the cards. A key finding is that the direction and magnitude of these centroid movements depends on the initial centroid location, so that the representation of the environment seems to be topologically distorted. Firing fields are not points, but occupy significant fractions of the apparatus surface. Moreover, the linear dimensions of firing fields are substantial fractions of the diameter of the cylinder. Imagine, for example, a circular field whose diameter is, say, one third the diameter of the cylinder. Imagine also two other fields whose centroids happened to lie on the opposite ends of the diameter of the circular field. In general, the distance between the centroids of the two other fields would change after the cards were reconfigured.
Does this "tidal" effect apply only to field centroids, or does it operate on an entire field to distort its shape? In other words, do card reconfigurations cause the field to move as a rigid object or does the vector field operate in a smooth, continuous way to stretch the firing field in a predictable fashion? Observing this sort of deformation would be a powerful indication that the vector-field description is valid.
To test whether card reconfigurations deform the positional firing in a way predicted by the vector-field equation, we calculated in two ways the similarity (defined below) of fields recorded in a standard session and in a card reconfiguration session. In the first method, we calculated the similarity when the field in the standard session was moved relative to the reconfigured field as a rigid object, so that neither its shape nor its firing rate contours were altered. In the second method, the field in the standard session was moved according to the vector-field equation, distorting its positional firing pattern. Based on the ability of the vector-field equation to account for the movements of field centroids, we expected that applying the equation to each point in the field would yield a higher similarity.
In line with earlier work, we define similarity as the z transform of the pixel-by-pixel correlation coefficient (r) for a pair of positional firing rate patterns (
The statistical analysis of similarity values was done by combining apart and together sessions in the following way. Cells recorded in only an apart (13) or a together (32) session were included. For cells recorded in both apart and together sessions (27), a random choice was made to select either type. In the end, the sample consisted of 29 apart and 43 together sessions. Apart and together sessions were combined because similarity averages were nearly equal for both session types.
When the firing field in the standard was rigidly superimposed on the field in the reconfigured session, the average similarity was 1.14 (r = 0.781; r 2 = 0.61). In contrast, when the vector transformation was first applied to the field in the standard session, the average similarity increased to 1.30 (r = 0.844; r 2 = 0.71); this analysis is summarized in the bar graphs of Fig 5. A paired t value for the rigid and vector transform similarities showed that the vector values were reliably higher [t = 5.79; df = 71; P(t > = 5.79) = 2.1 x 10-7].
The significantly higher similarity after vector transformation indicates that Equation 5 mimics the observed effects of card reconfigurations, but does not indicate the accuracy of the prediction. To address this question, we calculated for each cell the similarity between a pair of standard sessions. Since the similarity of firing fields in identical conditions is limited by accuracy of tracking, discrimination of action potentials and nonideal positional firing ( 0.11) = 0.91] (see Fig 5). Thus, applying the vector-field transform to a standard session produces a distorted pattern that resembles the reconfigured pattern as closely as two standard sessions resemble each other. Thus, within experimental error, the vector-field equation does a virtually perfect job in accounting for the effects of moving the cards closer together or further apart.
Card Reconfigurations Induce Predictable Changes in the Firing Patterns of Hippocampal Interneurons
In addition to pyramidal cells, recordings are often made from interneurons encountered in the CA1 region of the hippocampus. In our experience, these cells are usually found in stratum oriens or superficial stratum pyramidale. On the assumption that we detect cell bodies but not dendrites, these cells are likely to be basket cells (
In addition to differences in electrophysiological and temporal firing properties, theta cells show different positional firing properties than pyramidal cells. Most strikingly, since theta cell discharge never shows the long silent intervals characteristic of place cells, theta cells discharge everywhere in the available space. Nevertheless, there are clear, cell-specific variations in positional firing patterns, even for simultaneously recorded theta cells; the firing in higher rate regions is 22.5x higher than in low-rate regions. On this basis, and because the sizes and shapes of high rate regions resemble place cell firing fields,
With this background, we asked whether the effects of card reconfigurations could be detected in the positional firing patterns of theta cells. By inspection, these firing patterns were distorted in the fashions expected from the movements of firing field centroids; two examples are given in Fig 6. We therefore calculated the similarity of a standard session and a reconfigured session for each cell in two ways, by rigidly shifting the pattern in the standard session and by applying the vector-field transformation before shifting. In either case, the standard session pattern was moved relative to the reconfigured pattern and the correlation between the patterns calculated at each step. As before, the similarity was the z transform of the maximum correlation. We calculated both similarities for 13 theta cells, six recorded during apart sessions and seven during together sessions.
The average similarity after rigid movement of the firing pattern in the standard session onto the reconfigured pattern was 0.583. This is reliably lower than the average similarity of 0.643 seen after applying the vector-field transform to the pattern in the standard session [paired t = 2.39; df = 12; P(t 2.39) = 0.034]. Thus, the vector-field transform improved the similarity. In the case of theta cells, however, the improvement was less than optimal. To show this, we calculated similarities for pair of standard sessions. The mean of 0.719 was significantly higher than for the rigid movement of one standard session onto the reconfigured session [paired t = 3.13; df = 12; P(t
3.13) = 0.0087], but not significantly different from the similarities based on the vector transformation [paired t = 1.32; df = 12; P(t
1.32) = 0.21]. Despite the lack of statistical significance, however, we think that the accuracy of the vector-field transform is lower for theta cells than place cells. The somewhat poorer predictions may be due to the relatively strong dependence of theta cell activity on behavior as well as on position (
In addition to combining the apart and together sessions for theta cells, they can be treated separately to see whether the average direction and magnitude of pattern movements are similar to the movements of firing field centroids. To this end, we used the shift of position necessary to maximize the similarity to estimate firing pattern movements. For apart sessions, the mean horizontal shift (Fig 1 A) was -4.62 cm, in excellent agreement with the average horizontal movement of -4.84 cm for centroids of place-cell fields during apart sessions. During together sessions, the average horizontal movement of theta cell firing patterns moved +1.49 cm, again in excellent agreement with the corresponding value for place cells of 1.84 cm. Thus, the effects of apart and together sessions on theta cells show strong parallels to the effects on place cells. An interesting exception is the lack of change of theta cell firing rate after reconfigurations despite the decrease of place-cell activity.
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DISCUSSION |
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We showed that a single vector-field equation accounts for the movements of firing field centers caused by three kinds of stimulus manipulations; namely, rigid rotations of both cards, card removals, and card reconfigurations. We further showed that the vector-field equation predicts three additional effects of card reconfigurations. (a) Field movements parallel to the motion of the line that joins the cards should be greater for apart than together reconfigurations. (b) If the vector-field affects entire firing fields and not just the center, firing fields should stretch according to the same rule that describes motions of field centroids. This prediction was confirmed by showing that superimposing a standard field onto a reconfigured field was more accurate when done with the vector-field equation than when done with rigid translation. (c) Assuming that theta cells (hippocampal interneurons in stratum oriens and stratum radiatum) receive convergent location-specific input from place cells (
Why should a formulation as minimal as the vector-field model perform so nicely? In our view, it reflects the nature of the hippocampal representation of our simplified environment. There are really two issues concerning the nature of the representation. First, it may be true that hippocampal pyramidal cells can encode nonspatial aspects of the environment under more complex circumstances, but, in the pellet-chasing task, the hippocampal pyramidal cells act in many ways as nearly ideal place cells, at least in the spatial domain (
The second aspect of the representation in line with the vector-field approach is more controversial: we think that the mathematical language of the vector-field equation, with its assumption of local smoothness and its inclusion of both stimuli as controllers for fields everywhere in the environment, reflects the underlying nature of the representation. In short, we think the data and model imply that the place-cell representation is truly map-like, so that place-cell discharge takes place in a framework that incorporates features of two- (and possibly three-) dimensional space.
This view differs from the combinatorial (or relational) theory proposed by Eichenbaum and colleagues (see, for example,
Another difference between our model and the combinatorial model was alluded to in the preceding paper (
Two other lines of evidence in favor of the vector-field model arise from the distortions of field shape for place cells and overall positional firing pattern distribution for theta cells. Both the existence and precise nature of such "tidal" effects are predicted by the vector-field equation. In contrast, there is no basis from the strictly qualitative combinatorial theory to predict that such effects should exist, and, if so, what form they might take. Incorporating stretching of fields into the combinatorial theory is possible only by specifying the exact nature of the relationships among the triggering stimuli for each place cell. If it turns out necessary to propose that the relationships are the same for all cells or that the relationships vary systematically with position, the combinatorial model will come to resemble the vector-field theory.
In addition to differences from the combinatorial theory, our approach also diverges in several ways from the work of
A more important issue is the decision by
The issue of further testing the vector-field theory raises the additional problem that the form of the theory presented here is extremely specific; it is useful only for two stimuli and for two stimuli inside a cylinder. One direction for future research is to test whether the effects of other stimulus manipulations (e.g., changing the aspect ratio of a rectangle or modifying the cylinder into an elliptically shaped chamber) are also amenable to a vector-field approach. If field centroids move as though the representation of the apparatus floor is again being topologically stretched, it will be important to try to develop more general forms of the vector-field equation. In turn, such a generalization should permit the design of additional experiments to test the model even more rigorously.
Two other extensions of the two-card experiment and its mathematical description are worth considering. First, the two-card experiment serves as a useful (but by no means unique) method of exploring information processing within the hippocampus and its related structures (
The final point we raise concerns the nature of the neural network that is responsible for place-cell activity and that may be used to guide navigational behavior. In our experiments, every place cell was affected by both stimulus cards if both were present, and removing either card left fields unchanged and under the full control of the remaining card. We therefore chose to develop a numerical theory that emphasizes the global control of all cells by each identified stimulus. Our choice of such a model reflects our belief that the place-cell population represents a large scale, unitary construct, the environment, rather than an agglomeration of separate, smaller-scale features of the environment such as pairwise relationships between objects (
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Acknowledgements |
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The authors thank Dr. Emerson Hawley and Mr. Lawrence Eberle for technical support and a great deal of friendly assistance. We also thank Dr. David Touretzky for a very helpful critical review of the manuscript.
This work was supported by National Institutes of Health grants NS20686 and NS37150.
Submitted: 13 October 1999
Revised: 15 June 2000
Accepted: 15 June 2000
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References |
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