1YZPPOCAMPUS, VOL. I , NO. 3, PAGES 243-246, JULY 1991
The Hippocampus as a Cognitive Graph (Abridged Version)
slices, the probability that one is presynaptic to the other is about .02. In longitudinal slices (running along the septaltemporal extent of CA3), a given cell is found to contact cells rather far away, although the probability of contact decreases with distance. These “lateral meshwork” synapses are postulated to be the site of storage of the representation. Robert U. Muller,” John L. Kubie,t and In addition to the known features, the model assumes that Russ Saypofr‘ the lateral mesh synapses are modifiable. The modification mechanism is taken to be Hebbian, such that synaptic strength increases when there is the proper temporal conDepartments of *Physiology and l A n a t o m y and Cell junction of pre- and postsynaptic action potentials. More preBiology, Health Sciences Center at Brooklyncisely, it is assumed that lateral mesh synapses show the same SUNY, 450 Clarkson Avenue, Brooklyn, N Y 11203 form of long-term potentiation exhibited by the synapses U.S.A. made by CA3 pyramids on CAI pyramids. The issue of longterm depression is not considered, but we believe that longterm depression could be incorporated into the model without In this brief essay, we bypass many of the interesting issues changing its fundamental properties. that concern the role played by the hippocampus in guiding The basic idea of the model is illustrated in Figure IA. The spatial behavior (O’Keefe and Nadel, 1978) and instead focus large circle shows the boundaries of a cylindrical recording on the nature of the environmental representation that is preapparatus. The small circles show the limits of the firing fields sumably reflected by the ensemble firing of place cells of three CA3 place cells. The fields of cells A and B overIap, (O’Keefe and Dostrovsky, 1971). In particular, we outline a whereas the field of cell C is distant from the others. Note model in which the strengths of synaptic connections among that the locations of the cells in the CA3 surface have nothing CA3 hippocampal pyramidal cells come to encode distance to do with the spatial relationships among the fields; in our within a rat’s surroundings. In its present form, the model is view, there is no topological mapping of behavioral space extremely limited in scope. For example, it ignores most anonto either CA3 or CA1 (Muller and Kubie, 1987; Bostock atomical features of the hippocampus and related structures. et al., 1991). It also neglects the activity of CAI place cells, postsubicular We now imagine that cell A happens to be monosynaptihead direction cells (Taube et al., 1990), and several other cally connected to both cell B and cell C. Under these conclasses of cells certain to be important in selecting trajectories ditions, it is expected that cells A and B will sometimes fire through the environment. Nevertheless, the model shows within a brief enough interval that the synapse from A onto that aspects of space can be represented by a simple neural B will undergo strengthening; the overlap of the firing fields network and is potentially a basis for attacking the question is enough to ensure that pre- and postsynaptic action potenof how rats solve difficult spatial problems (Morris, 1981). tials will occur within the permissive interval for LTP. In The model rests on two known properties of CA3 pyramidal contrast, because the rat cannot move rapidly enough from cells. First, such cells act as “place cells” in a wide variety the region associated with the firing field of cell A to the of behavioral circumstances (O’Keefe, 1979). Place cells are region associated with the field of cell C, the synapse from characterized by “location-specific” firing. Each cell is in- A onto C will remain at its initial strength; the minimal intensely active only when the rat’s head is in a restricted part terval between pre- and postsynaptic action potentials exof the environment, the cell’s “firing field.” To a good first ceeds the LTP permissive interval. Thus, given the existence approximation, the discharge of place cells is independent of of place cells and modifiable lateral mesh synapses in CA3, the rat’s activity (Bostock et al., 1991). In the cylindrical ap- it is only necessary for the rat to move around the environparatus we have used in most of our work, the firing of place ment for the strength of the synapses to encode distance becells is also independent of the rat’s head direction (Bostock tween firing field centers for cell pairs that happen to be conet al., 1988; Muller et al., 1991). In fixed surroundings, it is nected. It is essential to note that in this very reduced first difficult to tell if location-specific firing is a reflection of the model the lateral mesh synapses only store information; they triggering of place cells by certain combinations of the sen- do not codetermine the firing of postsynaptic cells. sory stimulus constellation that occur at only certain places, The plausibility of the basic model has been tested with but this simple sensory view is belied by a variety of manip- computer simulations. The number of cells in the network ulations (Muller and Kubie, 1987; O’Keefe and Speakman, was varied, as was divergence of each cell. The number of 1987; Quirk et al., 1990; Sharp et al., 1990). The model is in synapses is just the product of the number of cells and the no way aimed at explaining how location-specific firing is divergence. In turn, the average convergence of synapses possible; the existence of place cells is taken as a given. onto postsynaptic cells is equal to the average divergence. In The second property of CA3 pyramidal cells essential for the current embodiment of the model, the divergence from the model is the mesh of connections that such cells make each cell was a constant; the convergence onto each cell was with one another. Anatomical (Amaral and Witter, 1989) and also a constant and, therefore, was numerically equal to the physiological (Miles and Wong, 1986) evidence indicates that divergence. Motions of the rat were taken from the sequence of posieach CA3 pyramidal cell makes direct, excitatory synapses with many other CA3 pyramidal cells. When simultaneous tions observed with an automatic TVkomputer rat tracker as recordings are made from pairs of CA3 pyramidal cells in a real rat chased small food pellets scattered into a 76-cm
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HIPPOCAMPUS VOL. 1, NO. 3, JULY 1991
DISTANCE BETWEEN FIELD CENTERS
D
C
VARY LTP TIME
DISTANCE BETWEEN FIELD CENTERS
DISTANCE BETWEEN FIELD CENTERS
Fig. 1 . (A) Idealization of the spatial arrangement of the firing fields of three CA3 place cells in the cylinder. On the assumptions made in the text, the synapse from A to B will strengthen whenever the rat visits the region shared by the fields of cells A and B. In contrast, the synapse from A to C will not strengthen even if the rat moves directly from the field of cell A to the field of cell C, because the distance is so great that the rat cannot get from one place to the other within the longest time compatible with LTP. (B) Synaptic strength as a function of distance between field centers with networks containing different numbers of cells. The near identity of the functions is expected, given that the synaptic strength for each pair of connected cells is independent of all other strengths. (C) Synaptic strength as a function of distance between field centers for networks in which the LTP permissive time was varied from 0.06 to 2.0 seconds. Longer LTP times are associated with slower decays of synaptic strength with distance, but the effect is weak. (D) Synaptic strength as a function of distance between field centers for networks in which the width of the firing fields was varied from 2 to 15 pixels. Larger field diameters are associated with much slower decays of synaptic strength with distance.
THE HIPPOCAMPUS AS A COGNITIVE GRAPH I Muller et al.
diameter cylinder (Muller and Kubie, 1987). Position is measured in pixels; a pixel is a square about 3 cm on a side. The rat’s position was updated from the position sequence at 60 Hz. Synapses were strengthened according to the Hebb-like rule:
where AS,i is the increment of the strength of the synapse made by the i-th cell onto the j-th cell, and fi and fj are the firing frequencies of the two cells averaged over the L T P permissive interval; no limits are imposed on maximum synaptic strength. The initial estimate of 300 ms for the L T P permissive interval was taken from Brown et al. (1988); no attention was paid to the asymmetric relationship between pre- and postsynaptic activity, such that strengthening appears to be more efficient if the presynaptic activity leads the postsynaptic activity. The firing field center for each cell was randomly chosen from a list of possible positions in the apparatus. The firing field for each cell was simulated as a twodimensional Gaussian, such that the rate fell off monotonically in all directions from the field center. Each field was characterized by a peak rate at its center and a width: the rate was set to zero for all pixels in which the rate was lower than 1.0 action potential per second. The action potential sequence for each cell was determined from the firing rate associated with the rat’s current position; a cell “fired” if a properly scaled random number was lower than the cell’s expected rate for a &second interval. Figure IB shows the relationship between synaptic strength and distance between field centers for networks with different numbers of cells (range, 100-3,600). In each case, the divergence was 8, so that the number of synapses is proportional to the number of cells. The synaptic strength is the average for all connected cell pairs, such that the distance (D) between their field centers is in the range d 5 D < d + 1, where d is an integer. The lowest possible value of d is zero: the highest is one pixel less than the diameter of the cylinder. As expected from the model, there is a strong, monotonic decrease of synaptic strength with distance. The fact that the strengthidistance function is the same for all network sizes is also expected, since each synapse is an independent storage unit. For the same reason, the strength/distance function is unchanged if the divergence is varied while holding the number of cells constant (not shown). The effects of varying the duration of the LTP permissive time over the range 60-2,000 ms are shown in Figure IC. The curves are normalized to the maximum synaptic strength for each LTP time. F o r small distances, there is no clear trend, but at distances greater than about 5 pixels long L T P times are associated with somewhat greater synaptic strength. The rather weak effects of changing LTP time stands in contrast to the stronger effect of varying field width over the range 2-15 pixels, as shown in Figure ID; again the curves are normalized to the maximum synaptic strength for each field width. It is clear that synaptic strength increases rather rapidly with field width at constant distance. The demonstrated strength/distance functions using LTP time and field width as parameters are expected from the model, but the relative ef-
245
fects of the two parameters were surprising and are not currently fully understood. Preliminary simulations in which running speed is varied by modifying the position sequence are also in agreement with the model; if the running speed is higher, the strength/distance function is broader. Thus, the model predicts the nature of the relationship between synaptic strength and the distance between the field centers of connected cells. It also predicts how the relationship is affected by several important variables. It therefore appears that the connectivity of CA3 place cells can represent the connectivity of the environment, although it remains to be formally demonstrated that the information contained in the lateral mesh synapses is sufficient to reconstruct the layout of the environment. Several comments may be made about the general nature of the model. First, it is extremely easy to understand. Indeed, something along the proposed lines must happen if the lateral mesh synapses are modifiable. Second, the model is parsimonious: it allows for a representation of the environment using only a small fraction of the neural machinery that is involved in guiding spatial behavior. Third, the model is distinct from other proposals about how place cells represent space; it bears little resemblance t o either the stimulus-response association model of McNaughton (1989) o r the Euclidian model of O’Keefe (1990). The representation has the form of a “directed graph” (see, for example, Harary, 1972), rather than a list of local views and paths (McNaughton, 1989) or a map (O’Keefe, 1990). In this analogy, the cell bodies plus dendrites are nodes of the graph, and the axons are edges. The graph is directed because the path of information flow is one-way along the axons and across synaptic clefts. The notion that the network is a directed graph is potentially valuable because directed graphs are routinely used as the algorithmic basis for solving critical path problems (Ammeraal, 1987). If the strengthidistance representation stores adequate information to reconstruct the rat’s surroundings, the question of retrieving the information arises. In other words, we can ask how to use the representation to find optimal (critical) paths from the rat’s current position to a goal. A possible answer lies in isomorphisms between connectivity in the network and the structure of the environment. For example, short chains of cells are probably preferentially associated with short trajectories. If this is true, short trajectories might correspond to neural chains in which the ability of cells with fields in the starting position to activate cells with fields at the goal is maximized. It is likely, however, that additional machinery is necessary to extract paths from the proposed representation, given that the model attends only to a small part of the network. We conclude this brief note by asking how multiple environments can be represented in the lateral network with minimal interference among the supposedly independent strengthidistance functions. Clearly, if every CA3 place cell were active in each environment, synaptic strength would tend to become homogeneous as the number of environments increases. It is a remarkable property of the place cell population, however, that only a small fraction of the units (the “active subset”) have firing fields in any given environment
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(Kubie and Ranck, 1984; Muller and Kubie, 1987; Thompson and Best, 1989); the preponderance of cells are virtually silent everywhere in the apparatus. In addition, the active subset and its complement are stable in time; each active cell has the same firing field whenever the rat returns to a familiar environment, and each inactive cell is reliably silent. Finally, the cells in the active subset and their field locations appear to be independent if environments are “sufficiently different” (Bostock et al., 1991). It is our contention that each active subset is associated with its own set of weights for the lateral mesh synapses and that the sets of synaptic weights are independent because the active subsets are independent. In this view, the active subset is a real unit of hippocampal organization. The effective size of the hippocampal network is reduced proportionally to 1 - P a , where Pa is the probability that a cell is in the active subset, and the number of potentially modifiable synapses is reduced proportionally to 1 - (Pa)’.In other words, if Pa = . I , a reasonable value from the work of Thompson and Best (1989), the number of cells is only 10% of the population and the number of synapses that can possibly be modified is only 1% of all the lateral synapses. We conclude that it is possible to keep representations independent because each representation utilizes only a small, independent fraction of the storage capacity of the CA3 lateral synapses. Under the extreme constraint that a synapse is never used in more than one representation, the entire lateral network can store at most 100 representations. We end by saying that despite the small fraction of synapses involved in each representation, the number is quite large. If the number of CA3 cells in one hemisphere is 250,000, the connectivity of the environment is stored in 2,500 synapses. Thus, the representations may be detailed as well as independent. References Amaral, D. G . , and M. P. Witter (1989) The three dimensional organization of the hippocampal formation: A review of anatomical data. Neuroscience 31:571-591. Ammeraal, L. (1987) Programs and Daru Structures, Wiley, Chichester, U.K. Bostock, E., J. Taube, and R. U. Muller (1988) The effects of head orientation on the firing of hippocampal place cells. Neurosci. Abstr. 18:127. Bostock, E. J., R. U. Muller, and J . L . Kubie (1991) Experiencedependent modifications of place cell firing. Hippocampus 1: 193206. Brown, T. H . , A. H. Ganong, E. W. Kairass, C . L. Keenan, and S.
R . Kelso (1990) Long-term potentiation in two synaptic systems of the hippocampal brain slice. In Neural Models of Plasricir~,J. H. Byrne and W. 0. Berry, eds., pp. 266-306, Academic Press. Sari Diego, CA. Harary, F. (1972) Graph Theory. Addison-Wesley. Reading. MA. Kubie, J. L., and J. B . Kanck. Jr. (1984) Hippocampal neuronal firing, c!f M ~ m o r y .L. R. context, and learning, in Ne~irop.sycko/o~,v Squire and N. Butters, eds.. Guilford Press, New York, NY. McNaughton, B. L. (1989) Neuronal mechanisms for spatial computation and information storage. In Neural Connrcrions, Mentcrl Computution, L. Nadel. L. A. Cooper, P. Culicover and R. M. Harnish, eds., pp. 285-350, MIT Press. Cambridge, MA. Miles, R.. and R . K . S. Wong (1986) Excitatory synaptic connections between CA3 neurones in the guinea pig hippocampus. J. Physiol. (Lond). 373:397-418. Morris, R. G. M. (1981) Spatial localization does not require the presence of local cues. Learn. and Motiv. 12:239-260. Muller, R. U . , and J . L. Kubie (1987) The effects of changes in the environment on the spatial firing of hippocampal complex-spike cells. J . Neurosci. 7:1951-1968. Muller, R. U., J . L . Kubie, E . M. Bostock, J. S . Taube, and G . J . Quirk (199 I ) Spatial firing correlates of neurons in the hippocampal formation of freely moving rats. In Brain and Spuce. J. Paillard, ed., Oxford University Press, Oxford. In press. O’Keefe, J . (1979) A review of the hippocampal place cells. Prog. Neurobiol. 13:419-439. O’Keefe, J. (1990) A computational model of the hippocampal cog~ Bruin n ~ j iThroirgh ? g the Hippocrimpcis: nitive nmp. In ~ ~ ~ i ~ r s t ~the The Hippocampus Region us u Model,for Sfitdving Brain S/r/i(,ii4re rind Fii,ic,rion. J. Storm-Mathisen, J . Zimmer. and 0. P. Ottersen. eds., Prog. Brain Res. 83:301-312, Elsevier, Amsterdam. Holland. O’Keefe, J., and J . Dostrovsky (1971) The hippocampus as spatial map: Preliminary evidence from unit activity in the freely moving rat. Brain Res. 34:171-175. O’Keefe, J.. and L . Nadel (1978) The Hippoc,tirnpiis u s o CoKnitiw Map, Clarendon Press, London. O’Keefe. J . , and A. Speakman (1987) Single unit activity in the rat hippocampus during a spatial memory task. Exp. Brain Res. 68: I 27. Quirk, G. J., R. U. Muller, and J. L. Kubie (1990) The Firing of hippocampdl place cells in the dark depends on the rat’s recent experience. J . Neurosci. 10:2008-2017. Sharp, P. E., J . L. Kubie. and R. U . Muller (1990) Properties of hippocampal neurons in a visually symmetric environment: Contributions of multiple sensory cues and mnemonic processes. J . Neurosci. 10:3093-3105. Taube, J . S., K. U. Muller, and J . B. Kanck, Jr. (1990) Head-direction cells recorded from the post-subiculum in freely moving rats. I . Description and quantitative analysis. J . Neurosci. 10:420-435. Thompson, L. T., and P. J . Best (1989) Place cells and silent cells in the hippocampus of freely-behaving rats. J. Neurosci. 9:2382-2390.
The Hippocampus as a Cognitive Graph (Abridged Version)
slices, the probability that one is presynaptic to the other is about .02. In longitudinal slices (running along the septaltemporal extent of CA3), a given cell is found to contact cells rather far away, although the probability of contact decreases with distance. These “lateral meshwork” synapses are postulated to be the site of storage of the representation. Robert U. Muller,” John L. Kubie,t and In addition to the known features, the model assumes that Russ Saypofr‘ the lateral mesh synapses are modifiable. The modification mechanism is taken to be Hebbian, such that synaptic strength increases when there is the proper temporal conDepartments of *Physiology and l A n a t o m y and Cell junction of pre- and postsynaptic action potentials. More preBiology, Health Sciences Center at Brooklyncisely, it is assumed that lateral mesh synapses show the same SUNY, 450 Clarkson Avenue, Brooklyn, N Y 11203 form of long-term potentiation exhibited by the synapses U.S.A. made by CA3 pyramids on CAI pyramids. The issue of longterm depression is not considered, but we believe that longterm depression could be incorporated into the model without In this brief essay, we bypass many of the interesting issues changing its fundamental properties. that concern the role played by the hippocampus in guiding The basic idea of the model is illustrated in Figure IA. The spatial behavior (O’Keefe and Nadel, 1978) and instead focus large circle shows the boundaries of a cylindrical recording on the nature of the environmental representation that is preapparatus. The small circles show the limits of the firing fields sumably reflected by the ensemble firing of place cells of three CA3 place cells. The fields of cells A and B overIap, (O’Keefe and Dostrovsky, 1971). In particular, we outline a whereas the field of cell C is distant from the others. Note model in which the strengths of synaptic connections among that the locations of the cells in the CA3 surface have nothing CA3 hippocampal pyramidal cells come to encode distance to do with the spatial relationships among the fields; in our within a rat’s surroundings. In its present form, the model is view, there is no topological mapping of behavioral space extremely limited in scope. For example, it ignores most anonto either CA3 or CA1 (Muller and Kubie, 1987; Bostock atomical features of the hippocampus and related structures. et al., 1991). It also neglects the activity of CAI place cells, postsubicular We now imagine that cell A happens to be monosynaptihead direction cells (Taube et al., 1990), and several other cally connected to both cell B and cell C. Under these conclasses of cells certain to be important in selecting trajectories ditions, it is expected that cells A and B will sometimes fire through the environment. Nevertheless, the model shows within a brief enough interval that the synapse from A onto that aspects of space can be represented by a simple neural B will undergo strengthening; the overlap of the firing fields network and is potentially a basis for attacking the question is enough to ensure that pre- and postsynaptic action potenof how rats solve difficult spatial problems (Morris, 1981). tials will occur within the permissive interval for LTP. In The model rests on two known properties of CA3 pyramidal contrast, because the rat cannot move rapidly enough from cells. First, such cells act as “place cells” in a wide variety the region associated with the firing field of cell A to the of behavioral circumstances (O’Keefe, 1979). Place cells are region associated with the field of cell C, the synapse from characterized by “location-specific” firing. Each cell is in- A onto C will remain at its initial strength; the minimal intensely active only when the rat’s head is in a restricted part terval between pre- and postsynaptic action potentials exof the environment, the cell’s “firing field.” To a good first ceeds the LTP permissive interval. Thus, given the existence approximation, the discharge of place cells is independent of of place cells and modifiable lateral mesh synapses in CA3, the rat’s activity (Bostock et al., 1991). In the cylindrical ap- it is only necessary for the rat to move around the environparatus we have used in most of our work, the firing of place ment for the strength of the synapses to encode distance becells is also independent of the rat’s head direction (Bostock tween firing field centers for cell pairs that happen to be conet al., 1988; Muller et al., 1991). In fixed surroundings, it is nected. It is essential to note that in this very reduced first difficult to tell if location-specific firing is a reflection of the model the lateral mesh synapses only store information; they triggering of place cells by certain combinations of the sen- do not codetermine the firing of postsynaptic cells. sory stimulus constellation that occur at only certain places, The plausibility of the basic model has been tested with but this simple sensory view is belied by a variety of manip- computer simulations. The number of cells in the network ulations (Muller and Kubie, 1987; O’Keefe and Speakman, was varied, as was divergence of each cell. The number of 1987; Quirk et al., 1990; Sharp et al., 1990). The model is in synapses is just the product of the number of cells and the no way aimed at explaining how location-specific firing is divergence. In turn, the average convergence of synapses possible; the existence of place cells is taken as a given. onto postsynaptic cells is equal to the average divergence. In The second property of CA3 pyramidal cells essential for the current embodiment of the model, the divergence from the model is the mesh of connections that such cells make each cell was a constant; the convergence onto each cell was with one another. Anatomical (Amaral and Witter, 1989) and also a constant and, therefore, was numerically equal to the physiological (Miles and Wong, 1986) evidence indicates that divergence. Motions of the rat were taken from the sequence of posieach CA3 pyramidal cell makes direct, excitatory synapses with many other CA3 pyramidal cells. When simultaneous tions observed with an automatic TVkomputer rat tracker as recordings are made from pairs of CA3 pyramidal cells in a real rat chased small food pellets scattered into a 76-cm
243
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HIPPOCAMPUS VOL. 1, NO. 3, JULY 1991
DISTANCE BETWEEN FIELD CENTERS
D
C
VARY LTP TIME
DISTANCE BETWEEN FIELD CENTERS
DISTANCE BETWEEN FIELD CENTERS
Fig. 1 . (A) Idealization of the spatial arrangement of the firing fields of three CA3 place cells in the cylinder. On the assumptions made in the text, the synapse from A to B will strengthen whenever the rat visits the region shared by the fields of cells A and B. In contrast, the synapse from A to C will not strengthen even if the rat moves directly from the field of cell A to the field of cell C, because the distance is so great that the rat cannot get from one place to the other within the longest time compatible with LTP. (B) Synaptic strength as a function of distance between field centers with networks containing different numbers of cells. The near identity of the functions is expected, given that the synaptic strength for each pair of connected cells is independent of all other strengths. (C) Synaptic strength as a function of distance between field centers for networks in which the LTP permissive time was varied from 0.06 to 2.0 seconds. Longer LTP times are associated with slower decays of synaptic strength with distance, but the effect is weak. (D) Synaptic strength as a function of distance between field centers for networks in which the width of the firing fields was varied from 2 to 15 pixels. Larger field diameters are associated with much slower decays of synaptic strength with distance.
THE HIPPOCAMPUS AS A COGNITIVE GRAPH I Muller et al.
diameter cylinder (Muller and Kubie, 1987). Position is measured in pixels; a pixel is a square about 3 cm on a side. The rat’s position was updated from the position sequence at 60 Hz. Synapses were strengthened according to the Hebb-like rule:
where AS,i is the increment of the strength of the synapse made by the i-th cell onto the j-th cell, and fi and fj are the firing frequencies of the two cells averaged over the L T P permissive interval; no limits are imposed on maximum synaptic strength. The initial estimate of 300 ms for the L T P permissive interval was taken from Brown et al. (1988); no attention was paid to the asymmetric relationship between pre- and postsynaptic activity, such that strengthening appears to be more efficient if the presynaptic activity leads the postsynaptic activity. The firing field center for each cell was randomly chosen from a list of possible positions in the apparatus. The firing field for each cell was simulated as a twodimensional Gaussian, such that the rate fell off monotonically in all directions from the field center. Each field was characterized by a peak rate at its center and a width: the rate was set to zero for all pixels in which the rate was lower than 1.0 action potential per second. The action potential sequence for each cell was determined from the firing rate associated with the rat’s current position; a cell “fired” if a properly scaled random number was lower than the cell’s expected rate for a &second interval. Figure IB shows the relationship between synaptic strength and distance between field centers for networks with different numbers of cells (range, 100-3,600). In each case, the divergence was 8, so that the number of synapses is proportional to the number of cells. The synaptic strength is the average for all connected cell pairs, such that the distance (D) between their field centers is in the range d 5 D < d + 1, where d is an integer. The lowest possible value of d is zero: the highest is one pixel less than the diameter of the cylinder. As expected from the model, there is a strong, monotonic decrease of synaptic strength with distance. The fact that the strengthidistance function is the same for all network sizes is also expected, since each synapse is an independent storage unit. For the same reason, the strength/distance function is unchanged if the divergence is varied while holding the number of cells constant (not shown). The effects of varying the duration of the LTP permissive time over the range 60-2,000 ms are shown in Figure IC. The curves are normalized to the maximum synaptic strength for each LTP time. F o r small distances, there is no clear trend, but at distances greater than about 5 pixels long L T P times are associated with somewhat greater synaptic strength. The rather weak effects of changing LTP time stands in contrast to the stronger effect of varying field width over the range 2-15 pixels, as shown in Figure ID; again the curves are normalized to the maximum synaptic strength for each field width. It is clear that synaptic strength increases rather rapidly with field width at constant distance. The demonstrated strength/distance functions using LTP time and field width as parameters are expected from the model, but the relative ef-
245
fects of the two parameters were surprising and are not currently fully understood. Preliminary simulations in which running speed is varied by modifying the position sequence are also in agreement with the model; if the running speed is higher, the strength/distance function is broader. Thus, the model predicts the nature of the relationship between synaptic strength and the distance between the field centers of connected cells. It also predicts how the relationship is affected by several important variables. It therefore appears that the connectivity of CA3 place cells can represent the connectivity of the environment, although it remains to be formally demonstrated that the information contained in the lateral mesh synapses is sufficient to reconstruct the layout of the environment. Several comments may be made about the general nature of the model. First, it is extremely easy to understand. Indeed, something along the proposed lines must happen if the lateral mesh synapses are modifiable. Second, the model is parsimonious: it allows for a representation of the environment using only a small fraction of the neural machinery that is involved in guiding spatial behavior. Third, the model is distinct from other proposals about how place cells represent space; it bears little resemblance t o either the stimulus-response association model of McNaughton (1989) o r the Euclidian model of O’Keefe (1990). The representation has the form of a “directed graph” (see, for example, Harary, 1972), rather than a list of local views and paths (McNaughton, 1989) or a map (O’Keefe, 1990). In this analogy, the cell bodies plus dendrites are nodes of the graph, and the axons are edges. The graph is directed because the path of information flow is one-way along the axons and across synaptic clefts. The notion that the network is a directed graph is potentially valuable because directed graphs are routinely used as the algorithmic basis for solving critical path problems (Ammeraal, 1987). If the strengthidistance representation stores adequate information to reconstruct the rat’s surroundings, the question of retrieving the information arises. In other words, we can ask how to use the representation to find optimal (critical) paths from the rat’s current position to a goal. A possible answer lies in isomorphisms between connectivity in the network and the structure of the environment. For example, short chains of cells are probably preferentially associated with short trajectories. If this is true, short trajectories might correspond to neural chains in which the ability of cells with fields in the starting position to activate cells with fields at the goal is maximized. It is likely, however, that additional machinery is necessary to extract paths from the proposed representation, given that the model attends only to a small part of the network. We conclude this brief note by asking how multiple environments can be represented in the lateral network with minimal interference among the supposedly independent strengthidistance functions. Clearly, if every CA3 place cell were active in each environment, synaptic strength would tend to become homogeneous as the number of environments increases. It is a remarkable property of the place cell population, however, that only a small fraction of the units (the “active subset”) have firing fields in any given environment
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(Kubie and Ranck, 1984; Muller and Kubie, 1987; Thompson and Best, 1989); the preponderance of cells are virtually silent everywhere in the apparatus. In addition, the active subset and its complement are stable in time; each active cell has the same firing field whenever the rat returns to a familiar environment, and each inactive cell is reliably silent. Finally, the cells in the active subset and their field locations appear to be independent if environments are “sufficiently different” (Bostock et al., 1991). It is our contention that each active subset is associated with its own set of weights for the lateral mesh synapses and that the sets of synaptic weights are independent because the active subsets are independent. In this view, the active subset is a real unit of hippocampal organization. The effective size of the hippocampal network is reduced proportionally to 1 - P a , where Pa is the probability that a cell is in the active subset, and the number of potentially modifiable synapses is reduced proportionally to 1 - (Pa)’.In other words, if Pa = . I , a reasonable value from the work of Thompson and Best (1989), the number of cells is only 10% of the population and the number of synapses that can possibly be modified is only 1% of all the lateral synapses. We conclude that it is possible to keep representations independent because each representation utilizes only a small, independent fraction of the storage capacity of the CA3 lateral synapses. Under the extreme constraint that a synapse is never used in more than one representation, the entire lateral network can store at most 100 representations. We end by saying that despite the small fraction of synapses involved in each representation, the number is quite large. If the number of CA3 cells in one hemisphere is 250,000, the connectivity of the environment is stored in 2,500 synapses. Thus, the representations may be detailed as well as independent. References Amaral, D. G . , and M. P. Witter (1989) The three dimensional organization of the hippocampal formation: A review of anatomical data. Neuroscience 31:571-591. Ammeraal, L. (1987) Programs and Daru Structures, Wiley, Chichester, U.K. Bostock, E., J. Taube, and R. U. Muller (1988) The effects of head orientation on the firing of hippocampal place cells. Neurosci. Abstr. 18:127. Bostock, E. J., R. U. Muller, and J . L . Kubie (1991) Experiencedependent modifications of place cell firing. Hippocampus 1: 193206. Brown, T. H . , A. H. Ganong, E. W. Kairass, C . L. Keenan, and S.
R . Kelso (1990) Long-term potentiation in two synaptic systems of the hippocampal brain slice. In Neural Models of Plasricir~,J. H. Byrne and W. 0. Berry, eds., pp. 266-306, Academic Press. Sari Diego, CA. Harary, F. (1972) Graph Theory. Addison-Wesley. Reading. MA. Kubie, J. L., and J. B . Kanck. Jr. (1984) Hippocampal neuronal firing, c!f M ~ m o r y .L. R. context, and learning, in Ne~irop.sycko/o~,v Squire and N. Butters, eds.. Guilford Press, New York, NY. McNaughton, B. L. (1989) Neuronal mechanisms for spatial computation and information storage. In Neural Connrcrions, Mentcrl Computution, L. Nadel. L. A. Cooper, P. Culicover and R. M. Harnish, eds., pp. 285-350, MIT Press. Cambridge, MA. Miles, R.. and R . K . S. Wong (1986) Excitatory synaptic connections between CA3 neurones in the guinea pig hippocampus. J. Physiol. (Lond). 373:397-418. Morris, R. G. M. (1981) Spatial localization does not require the presence of local cues. Learn. and Motiv. 12:239-260. Muller, R. U . , and J . L. Kubie (1987) The effects of changes in the environment on the spatial firing of hippocampal complex-spike cells. J . Neurosci. 7:1951-1968. Muller, R. U., J . L . Kubie, E . M. Bostock, J. S . Taube, and G . J . Quirk (199 I ) Spatial firing correlates of neurons in the hippocampal formation of freely moving rats. In Brain and Spuce. J. Paillard, ed., Oxford University Press, Oxford. In press. O’Keefe, J . (1979) A review of the hippocampal place cells. Prog. Neurobiol. 13:419-439. O’Keefe, J. (1990) A computational model of the hippocampal cog~ Bruin n ~ j iThroirgh ? g the Hippocrimpcis: nitive nmp. In ~ ~ ~ i ~ r s t ~the The Hippocampus Region us u Model,for Sfitdving Brain S/r/i(,ii4re rind Fii,ic,rion. J. Storm-Mathisen, J . Zimmer. and 0. P. Ottersen. eds., Prog. Brain Res. 83:301-312, Elsevier, Amsterdam. Holland. O’Keefe, J., and J . Dostrovsky (1971) The hippocampus as spatial map: Preliminary evidence from unit activity in the freely moving rat. Brain Res. 34:171-175. O’Keefe, J.. and L . Nadel (1978) The Hippoc,tirnpiis u s o CoKnitiw Map, Clarendon Press, London. O’Keefe. J . , and A. Speakman (1987) Single unit activity in the rat hippocampus during a spatial memory task. Exp. Brain Res. 68: I 27. Quirk, G. J., R. U. Muller, and J. L. Kubie (1990) The Firing of hippocampdl place cells in the dark depends on the rat’s recent experience. J . Neurosci. 10:2008-2017. Sharp, P. E., J . L. Kubie. and R. U . Muller (1990) Properties of hippocampal neurons in a visually symmetric environment: Contributions of multiple sensory cues and mnemonic processes. J . Neurosci. 10:3093-3105. Taube, J . S., K. U. Muller, and J . B. Kanck, Jr. (1990) Head-direction cells recorded from the post-subiculum in freely moving rats. I . Description and quantitative analysis. J . Neurosci. 10:420-435. Thompson, L. T., and P. J . Best (1989) Place cells and silent cells in the hippocampus of freely-behaving rats. J. Neurosci. 9:2382-2390.
