COMMENTARY
Do the spatial frequencies of grid cells mold the firing fields of place cells? John L. Kubiea,1 and Steven E. Foxb Departments of aCell Biology and bPhysiology and Pharmacology, State University of New York Downstate Medical Center, Brooklyn, NY 11203
The medial entorhinal cortex, populated by grid cells, projects both directly and indirectly to the CA1 and CA3 cortices, the sites of hippocampal place cells. The study by Ormond and McNaughton (1) investigates a potential mechanism by which grid cells of the medial entorhinal cortex (MEC) exert their influence on the formation of firing fields of place cells. Specifically, the authors propose a “Fourier hypothesis” by which the spatial frequencies from several grid cell modules converge on the place-cell layer, molding the contours of firing fields to a composite of those several frequencies; Ormond and McNaughton tested the theory
Fig. 1. (A) Firing-rate maps for a place cell recorded over 15 min from a rat in a cylinder (Left) and a grid cell recorded over 60 min under similar conditions in a rectangular chamber (Right). Dark pixel colors are high firing rates; yellow pixels are visited with no spikes (from J.L.K. and S.E.F.’s laboratory). (B) Schematic processing layers in the hippocampal formation. Place cells are in CA1 and CA3; grid cells are in the MEC. Although all layers are allocortex, each has unique intrinsic organization. Understanding the information transform across layers is the goal of the hippocampal project.
by inactivating specific input components and obtained fascinating results, largely supporting the model. The work (1) is best understood in the context of the broad “hippocampal cognitive map” project. In 1971, O’Keefe and Dostrovsky, while recording single neurons in the hippocampus of freely moving rats, discovered cells that fired when the animal crossed a restricted region of space; these neurons were called “place cells” (2). A typical place cell will fire in one location, termed the “place field” (Fig. 1A). Soon after, John O’Keefe and Lynn Nadel set the framework of the project with the publication of The Hippocampus as a Cognitive Map (3). The book set out a bold theory that the hippocampus was a map, performing cognitive functions critical for navigation and other cognitive operations including episodic memory. The book also presented an explicit neural model for the creation of place cells. As noted in the book, virtually all of the processing layers were unexplored, and the model highly speculative. The theory set out a goal of understanding how neural dynamics produces cognition (Fig. 1B). In subsequent years many laboratories have attempted to fill the many gaps and improve the model. Notable was the discovery of grid cells in the MEC (4). (The 2014 Nobel Prize in Physiology and Medicine was awarded to John O’Keefe for his discovery of place cells and Edvard and May Britt Moser for their contributions to understanding hippocampal circuitry, most notably the discovery of grid cells in the MEC.) Among other cell types discovered were head-direction cells in the presubiculum (5) and boundary-vector cells in the subiculum (6). Grid cells exhibit remarkable patterns of location-specific firing; an individual cell fires selectively in a set of locations in an environment such that a grid cell’s topographical “map” of the environment has discrete firing rate bumps distributed across accessible space, forming a triangular lattice pattern. Individual grid patterns can be described by
3860–3861 | PNAS | March 31, 2015 | vol. 112 | no. 13
Fig. 2. (A) Lateral view of rat brain illustrating MEC projection to the hippocampus. Grid cells are organized into modules; each has cells of identical scale and orientation. Modules are organized step-wise, dorsal to ventral, in increasing scale. Each region of place cells receives convergent input from several modules. (B) Contour of a 1D place field produced by summing four sinusoids (black line; 1.7 step factor). When the input with the shortest period is eliminated, the place field becomes broader (red). When the input with the longest period is eliminated, there is little change in the shape of the place field (green).
their grid scale (spacing between bumps), orientation (angle of the lattice), and phase (x, y offset of the grid pattern) (Fig. 1A). The pattern is repeated, like a wave, and to an approximation is a 2D cosine wave. The grid scale, or period, is the reciprocal of the spatial frequency of the grid. Within a region of the entorhinal cortex, grid cells share an identical spatial frequency; these groupings are called “modules.” When going from dorsal to ventral the grid scales increase in discrete steps, Author contributions: J.L.K. and S.E.F. wrote the paper. The authors declare no conflict of interest. See companion article on page 4116. 1
To whom correspondence should be addressed. Email: Jkubie@ downstate.edu.
www.pnas.org/cgi/doi/10.1073/pnas.1503155112
Kubie and Fox
neighbors. For example, a place cell with broad spatial tuning might have track segments with positive correlations up to 5 cm away, whereas a place cell with narrow tuning may only maintain positive correlations for 2 cm; thus, this is a spatial correlation measure. Results were largely, but not completely, consistent with the hypothesis. The most dramatic effect was that, as predicted, inactivating the dorsal part of the MEC caused a broadening of the spatial tuning of place cells. This was a strong, reliable effect, evident at two concentrations of muscimol inactivation. Contrary to prediction, however, inactivating the ventral MEC (the low spatial frequency inputs) did not result in narrower spatial tuning in hippocampal place cells. Rather, there was a weak tendency for place cells to broaden their spatial firing patterns. This result puts the authors (1) in a quandary: If one directly compares responses after dorsal and ventral MEC inactivation, the pattern follows prediction: dorsal inactivation leads to a much larger widening of place cell firing fields and reduced spatial frequency. If, however, one looks at the absolute direction of response, ventral inactivations went the wrong way. Ventral inactivations were predicted to make firing fields smaller (increase frequency) but they made fields slightly larger. Our interpretation is strongly in favor of Ormond and McNaughton’s (1) Fourier hypothesis. This requires a reasonable explanation of the ventral inactivation results; specifically, why does inactivation of grid cell inputs with low spatial frequencies not result in a decrease in the width of place cell firing fields, but rather a modest increase? Ormond and McNaughton suggest that inactivation of part of the MEC, in addition to removing some of the spatial frequency inputs, also causes a decrease in the self-motion signal that drives all grid cells. In principal, a decrease the self-motion signal will cause grid scale expansion throughout the MEC; thus, according to the Fourier model, inactivating any part of the entorhinal cortex will result in expansion of place cell firing fields. This is an attractive idea that can, and should, be tested.
We found a possible alternative explanation, demonstrated with the simple modeling program described above. Recall that because of the modular organization of the MEC, grid cell inputs have spatial frequencies that are separated in discrete ∼1.7-fold steps. In this simple spatial frequency model we found that if the step size is fairly large, the highest frequency input will invariably determine field size. That is, eliminating low-frequency inputs will not cause firing fields to get smaller and sharper. This result is seen in Fig. 2B. Summing four spatial frequencies differing by a ratio of 1.7 yields a “firing field” when the four frequencies align. Eliminating the highest frequency causes the firing field to broaden, as expected. Eliminating the lowest frequency, however, has little effect on the shape of the firing field. In testing a variety of inputs, we find a sharpening effect when eliminating the lowest frequency only when the step-size between frequencies is small, ∼1.2 or lower. Simply put, for a high-frequency peak to create a firing field, it must ride on a broad low-frequency peak. The phase of the low-frequency components determine which high-frequency cycle creates the firing field, but have little effect on field shape. A prediction is that removing low-frequency inputs ought to increase the number of firing fields (perhaps eliciting spatial firing in enclosures where the place cell is quiet) but have little effect on current firing fields or field shape. In summary, Ormond and McNaughton (1) have carefully performed an insightful experiment. Their results support the Fourier hypothesis: the notion that place cells are, in part, formed by a summation of spatial frequency inputs coming from grid cells. Perhaps equally important, their result suggests future work that should clarify grid cell influence on place cells and, perhaps, interactions among the grid cell modules. More broadly, the work is part of an ambitious enterprise aimed at deciphering the complex processing that leads from simple signals to the hippocampal map and cognition.
1 Ormond J, McNaughton BL (2015) Place field expansion after focal MEC inactivations is consistent with loss of Fourier components and path integrator gain reduction. Proc Natl Acad Sci USA 112:4116–4121. 2 O’Keefe J, Dostrovsky J (1971) The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res 34(1):171–175. 3 O’Keefe J, Nadel L (2014) The Hippocampus as a Cognitive Map (Oxford Univ Press, Oxford). 4 Hafting T, Fyhn M, Molden S, Moser M-B, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex. Nature 436(7052):801–806.
5 Taube JS, Muller RU, Ranck JB, Jr (1990) Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J Neurosci 10(2):420–435. 6 Lever C, Burton S, Jeewajee A, O’Keefe J, Burgess N (2009) Boundary vector cells in the subiculum of the hippocampal formation. J Neurosci 29(31):9771–9777. 7 Stensola H, et al. (2012) The entorhinal grid map is discretized. Nature 492(7427):72–78. 8 Solstad T, Moser EI, Einevoll GT (2006) From grid cells to place cells: A mathematical model. Hippocampus 16(12): 1026–1031.
PNAS | March 31, 2015 | vol. 112 | no. 13 | 3861
COMMENTARY
with each module having a grid scale (period) ∼1.7-times the module above (Fig. 2A) (7). Shortly after the initial description of grid cells, models emerged illustrating how convergence of grid cell output could produce place cells (8). Inherent in these models is that in a given region of the hippocampus, place cells receive convergent input from grid cells with different spatial frequencies. The meticulous study by Ormond and McNaughton (1) is an attempt to provide physiological evidence for this notion and decipher how it is that place fields actually do emerge from upstream neural signals. The study describes a mechanism for producing place cells from grid cells in what they refer to as the Fourier hypothesis. Specifically, because the grid scale of a module describes a spatial frequency, then a given place cell may arise from summation of the spatial frequencies of several modules. Recall that the entorhinal cortex is organized in modules that are ordered in spatial frequencies from dorsal to ventral. In simplest terms, if each place cell gets input from a single grid cell from four adjacent modules, the place cell results from the summation of the four spatial frequencies, each at a specific phase. The location in the environment where the four grid cells have common peaks will become the firing field of the place cell. Fig. 2B illustrates this simply as a summation of four 1D sinusoids, each with a specific spatial frequency and phase. The 1D place field is produced when the inputs align, producing a sum above threshold. Most importantly, the size and shape of the place cell field will be a function of the spatial frequencies of the input grid modules. Ormond and McNaughton (1) reason that if this model is correct, removing specific sets of grid cell inputs ought to have predictable effects on the size of the fields of place cells. Specifically, they predict that inactivating input from grid cells with high spatial frequency (dorsal MEC) ought to make place fields larger (lower spatial frequency), whereas inactivating input from grid cells with low spatial frequency (ventral MEC) ought to make the place fields smaller (higher spatial frequency). In a clever design, Ormond and McNaughton tested this hypothesis by recording place cells while inactivating either the dorsal or ventral MEC. Rats were tested in a linear track making both grid cell and place cell firing patterns 1D; for place cells, a firing field is a continuous length of the track with high firing rates. Although several analyses were used, the principal analysis is the distance that the mean rate on the track correlated with its
Do the spatial frequencies of grid cells mold the firing fields of place cells? John L. Kubiea,1 and Steven E. Foxb Departments of aCell Biology and bPhysiology and Pharmacology, State University of New York Downstate Medical Center, Brooklyn, NY 11203
The medial entorhinal cortex, populated by grid cells, projects both directly and indirectly to the CA1 and CA3 cortices, the sites of hippocampal place cells. The study by Ormond and McNaughton (1) investigates a potential mechanism by which grid cells of the medial entorhinal cortex (MEC) exert their influence on the formation of firing fields of place cells. Specifically, the authors propose a “Fourier hypothesis” by which the spatial frequencies from several grid cell modules converge on the place-cell layer, molding the contours of firing fields to a composite of those several frequencies; Ormond and McNaughton tested the theory
Fig. 1. (A) Firing-rate maps for a place cell recorded over 15 min from a rat in a cylinder (Left) and a grid cell recorded over 60 min under similar conditions in a rectangular chamber (Right). Dark pixel colors are high firing rates; yellow pixels are visited with no spikes (from J.L.K. and S.E.F.’s laboratory). (B) Schematic processing layers in the hippocampal formation. Place cells are in CA1 and CA3; grid cells are in the MEC. Although all layers are allocortex, each has unique intrinsic organization. Understanding the information transform across layers is the goal of the hippocampal project.
by inactivating specific input components and obtained fascinating results, largely supporting the model. The work (1) is best understood in the context of the broad “hippocampal cognitive map” project. In 1971, O’Keefe and Dostrovsky, while recording single neurons in the hippocampus of freely moving rats, discovered cells that fired when the animal crossed a restricted region of space; these neurons were called “place cells” (2). A typical place cell will fire in one location, termed the “place field” (Fig. 1A). Soon after, John O’Keefe and Lynn Nadel set the framework of the project with the publication of The Hippocampus as a Cognitive Map (3). The book set out a bold theory that the hippocampus was a map, performing cognitive functions critical for navigation and other cognitive operations including episodic memory. The book also presented an explicit neural model for the creation of place cells. As noted in the book, virtually all of the processing layers were unexplored, and the model highly speculative. The theory set out a goal of understanding how neural dynamics produces cognition (Fig. 1B). In subsequent years many laboratories have attempted to fill the many gaps and improve the model. Notable was the discovery of grid cells in the MEC (4). (The 2014 Nobel Prize in Physiology and Medicine was awarded to John O’Keefe for his discovery of place cells and Edvard and May Britt Moser for their contributions to understanding hippocampal circuitry, most notably the discovery of grid cells in the MEC.) Among other cell types discovered were head-direction cells in the presubiculum (5) and boundary-vector cells in the subiculum (6). Grid cells exhibit remarkable patterns of location-specific firing; an individual cell fires selectively in a set of locations in an environment such that a grid cell’s topographical “map” of the environment has discrete firing rate bumps distributed across accessible space, forming a triangular lattice pattern. Individual grid patterns can be described by
3860–3861 | PNAS | March 31, 2015 | vol. 112 | no. 13
Fig. 2. (A) Lateral view of rat brain illustrating MEC projection to the hippocampus. Grid cells are organized into modules; each has cells of identical scale and orientation. Modules are organized step-wise, dorsal to ventral, in increasing scale. Each region of place cells receives convergent input from several modules. (B) Contour of a 1D place field produced by summing four sinusoids (black line; 1.7 step factor). When the input with the shortest period is eliminated, the place field becomes broader (red). When the input with the longest period is eliminated, there is little change in the shape of the place field (green).
their grid scale (spacing between bumps), orientation (angle of the lattice), and phase (x, y offset of the grid pattern) (Fig. 1A). The pattern is repeated, like a wave, and to an approximation is a 2D cosine wave. The grid scale, or period, is the reciprocal of the spatial frequency of the grid. Within a region of the entorhinal cortex, grid cells share an identical spatial frequency; these groupings are called “modules.” When going from dorsal to ventral the grid scales increase in discrete steps, Author contributions: J.L.K. and S.E.F. wrote the paper. The authors declare no conflict of interest. See companion article on page 4116. 1
To whom correspondence should be addressed. Email: Jkubie@ downstate.edu.
www.pnas.org/cgi/doi/10.1073/pnas.1503155112
Kubie and Fox
neighbors. For example, a place cell with broad spatial tuning might have track segments with positive correlations up to 5 cm away, whereas a place cell with narrow tuning may only maintain positive correlations for 2 cm; thus, this is a spatial correlation measure. Results were largely, but not completely, consistent with the hypothesis. The most dramatic effect was that, as predicted, inactivating the dorsal part of the MEC caused a broadening of the spatial tuning of place cells. This was a strong, reliable effect, evident at two concentrations of muscimol inactivation. Contrary to prediction, however, inactivating the ventral MEC (the low spatial frequency inputs) did not result in narrower spatial tuning in hippocampal place cells. Rather, there was a weak tendency for place cells to broaden their spatial firing patterns. This result puts the authors (1) in a quandary: If one directly compares responses after dorsal and ventral MEC inactivation, the pattern follows prediction: dorsal inactivation leads to a much larger widening of place cell firing fields and reduced spatial frequency. If, however, one looks at the absolute direction of response, ventral inactivations went the wrong way. Ventral inactivations were predicted to make firing fields smaller (increase frequency) but they made fields slightly larger. Our interpretation is strongly in favor of Ormond and McNaughton’s (1) Fourier hypothesis. This requires a reasonable explanation of the ventral inactivation results; specifically, why does inactivation of grid cell inputs with low spatial frequencies not result in a decrease in the width of place cell firing fields, but rather a modest increase? Ormond and McNaughton suggest that inactivation of part of the MEC, in addition to removing some of the spatial frequency inputs, also causes a decrease in the self-motion signal that drives all grid cells. In principal, a decrease the self-motion signal will cause grid scale expansion throughout the MEC; thus, according to the Fourier model, inactivating any part of the entorhinal cortex will result in expansion of place cell firing fields. This is an attractive idea that can, and should, be tested.
We found a possible alternative explanation, demonstrated with the simple modeling program described above. Recall that because of the modular organization of the MEC, grid cell inputs have spatial frequencies that are separated in discrete ∼1.7-fold steps. In this simple spatial frequency model we found that if the step size is fairly large, the highest frequency input will invariably determine field size. That is, eliminating low-frequency inputs will not cause firing fields to get smaller and sharper. This result is seen in Fig. 2B. Summing four spatial frequencies differing by a ratio of 1.7 yields a “firing field” when the four frequencies align. Eliminating the highest frequency causes the firing field to broaden, as expected. Eliminating the lowest frequency, however, has little effect on the shape of the firing field. In testing a variety of inputs, we find a sharpening effect when eliminating the lowest frequency only when the step-size between frequencies is small, ∼1.2 or lower. Simply put, for a high-frequency peak to create a firing field, it must ride on a broad low-frequency peak. The phase of the low-frequency components determine which high-frequency cycle creates the firing field, but have little effect on field shape. A prediction is that removing low-frequency inputs ought to increase the number of firing fields (perhaps eliciting spatial firing in enclosures where the place cell is quiet) but have little effect on current firing fields or field shape. In summary, Ormond and McNaughton (1) have carefully performed an insightful experiment. Their results support the Fourier hypothesis: the notion that place cells are, in part, formed by a summation of spatial frequency inputs coming from grid cells. Perhaps equally important, their result suggests future work that should clarify grid cell influence on place cells and, perhaps, interactions among the grid cell modules. More broadly, the work is part of an ambitious enterprise aimed at deciphering the complex processing that leads from simple signals to the hippocampal map and cognition.
1 Ormond J, McNaughton BL (2015) Place field expansion after focal MEC inactivations is consistent with loss of Fourier components and path integrator gain reduction. Proc Natl Acad Sci USA 112:4116–4121. 2 O’Keefe J, Dostrovsky J (1971) The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res 34(1):171–175. 3 O’Keefe J, Nadel L (2014) The Hippocampus as a Cognitive Map (Oxford Univ Press, Oxford). 4 Hafting T, Fyhn M, Molden S, Moser M-B, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex. Nature 436(7052):801–806.
5 Taube JS, Muller RU, Ranck JB, Jr (1990) Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J Neurosci 10(2):420–435. 6 Lever C, Burton S, Jeewajee A, O’Keefe J, Burgess N (2009) Boundary vector cells in the subiculum of the hippocampal formation. J Neurosci 29(31):9771–9777. 7 Stensola H, et al. (2012) The entorhinal grid map is discretized. Nature 492(7427):72–78. 8 Solstad T, Moser EI, Einevoll GT (2006) From grid cells to place cells: A mathematical model. Hippocampus 16(12): 1026–1031.
PNAS | March 31, 2015 | vol. 112 | no. 13 | 3861
COMMENTARY
with each module having a grid scale (period) ∼1.7-times the module above (Fig. 2A) (7). Shortly after the initial description of grid cells, models emerged illustrating how convergence of grid cell output could produce place cells (8). Inherent in these models is that in a given region of the hippocampus, place cells receive convergent input from grid cells with different spatial frequencies. The meticulous study by Ormond and McNaughton (1) is an attempt to provide physiological evidence for this notion and decipher how it is that place fields actually do emerge from upstream neural signals. The study describes a mechanism for producing place cells from grid cells in what they refer to as the Fourier hypothesis. Specifically, because the grid scale of a module describes a spatial frequency, then a given place cell may arise from summation of the spatial frequencies of several modules. Recall that the entorhinal cortex is organized in modules that are ordered in spatial frequencies from dorsal to ventral. In simplest terms, if each place cell gets input from a single grid cell from four adjacent modules, the place cell results from the summation of the four spatial frequencies, each at a specific phase. The location in the environment where the four grid cells have common peaks will become the firing field of the place cell. Fig. 2B illustrates this simply as a summation of four 1D sinusoids, each with a specific spatial frequency and phase. The 1D place field is produced when the inputs align, producing a sum above threshold. Most importantly, the size and shape of the place cell field will be a function of the spatial frequencies of the input grid modules. Ormond and McNaughton (1) reason that if this model is correct, removing specific sets of grid cell inputs ought to have predictable effects on the size of the fields of place cells. Specifically, they predict that inactivating input from grid cells with high spatial frequency (dorsal MEC) ought to make place fields larger (lower spatial frequency), whereas inactivating input from grid cells with low spatial frequency (ventral MEC) ought to make the place fields smaller (higher spatial frequency). In a clever design, Ormond and McNaughton tested this hypothesis by recording place cells while inactivating either the dorsal or ventral MEC. Rats were tested in a linear track making both grid cell and place cell firing patterns 1D; for place cells, a firing field is a continuous length of the track with high firing rates. Although several analyses were used, the principal analysis is the distance that the mean rate on the track correlated with its
