The Hippocampus as a Cognitive Graph

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The Hippocampus as a Cognitive Graph

R O B E R T U . M U L L E R , * M A T T S T E A D , *

andJANOS

P A C H ~;

From the *Department of Physiology, State University of New York, Brooklyn, Brooklyn, New York 11203; and

*Department of Computer Science, City College of New York, and Courant Institute of Mathematical Sciences, NewYork University, New York 10012

ABSTRACT A theory of cognitive m a p p i n g is developed that depends only on accepted properties of hippocampal function, namely, long-term potentiation, the place cell p h e n o m e - non, and the associative or recurrent connections made a m o n g CA3 pyramidal ceils. It is proposed that the distance between the firing fields of connected pairs of CA3 place cells is en- coded as synaptic resistance (reciprocal synaptic strength). The encoding occurs because pairs of ceils with coincident or overlapping fields will tend to fire together in time, thereby causing a decrease in synaptic resistance via long-term potentiation; in contrast, ceils with widely separated fields will tend never to fire together, causing no change or perhaps (via long-term depression) an increase in synaptic resistance. A network whose connection pat- tern mimics that of CA3 and whose connection weights are proportional to synaptic resistance can be formally treated as a weighted, directed graph. In such a graph, a "node" is assigned to each CA3 cell and two nodes are connected by a "directed edge" if and only if the two corresponding cells are connected by a synapse. Weighted, directed graphs can be searched for an optimal path between any pair of nodes with standard algorithms. Here, we are interested in finding the path along which the sum of the synaptic resistances from one cell to another is minimal. Since each cell is a place cell, such a path also corresponds to a path in two-dimensional space. O u r basic finding is that minimizing the sum of the synaptic resistances along a path in neural space yields the shortest (optimal) path in unobstructed two-dimensional space, so long as the connectivity of the network is great enough. In addition to being able to find geodesics in unobstructed space, the same network enables solutions to the "detour" and "shortcut" problems, in which it is necessary to find an optimal path around a newly introduced barrier and to take a shorter path through a hole o p e n e d up in a preexist- ing barrier, respectively. We argue that the ability to solve such problems qualifies the proposed hippocampal object as a cognitive map. Graph theory thus provides a sort of existence p r o o f demonstrating that the hippocampus contains the necessary information to function as a map, in the sense postulated by others (O'Keefe,J., and L. Nadel. 1978. T h e Hippocampus as a Cognitive Map. Clarendon Press, Oxford, UK). It is also possible that the cognitive mapping functions of the hippocampus are carried out by parallel graph searching algorithms implemented as neural processes. This possibility has the great attraction that the hippocampus could then operate in m u c h the same way to find paths in general problem space; it would only be necessary for pyramidal cells to exhibit a strong nonpositional firing correlate. Key words: place ceils Q cognitive map Q hippocampal long-term potentiation * graph theory, neural applications 9 rat navigation, c o m p u t e r model

I N T R O D U C T I O N

Cognitive Maps

Dating back at least to the work o f T o l m a n (1932, 1948), the n o t i o n has b e e n e n t e r t a i n e d that rats are en- dowed with m a p q i k e r e p r e s e n t a t i o n s o f their environ- ments. T h e existence o f these "cognitive maps" is in- f e r r e d f r o m the ways in which rats solve certain spatial p r o b l e m s . Because the p r o b l e m s s e e m difficult a n d the Address correspondence to Dr. Robert U. Muller, Department of Physiology, SUNY-Brooklyn, 450 Clarkson Ave., Box 31, Brooklyn, NY 11203. Fax: (718) 270-3103; [email protected]

solutions s e e m efficient a n d intelligent, o n e imagines that rats m u s t use i n f o r m a t i o n a b o u t the overall structure or g e o m e t r y o f their s u r r o u n d i n g s while solving these p r o b l e m s . In short, m a p s are postulated because it is believed that n o simpler problem-solving m e c h a - nism will do.

As an aside, we n o t e that n o t everyone accepts this reasoning, a n d that even advocates m u s t m a i n t a i n a healthy skepticism a b o u t m a p s (see Terrace, 1984).

Nevertheless, the behavioral evidence in favor o f m a p s is quite convincing, a n d the r e a d e r is r e f e r r e d to com- p e n d i o u s reviews by O ' K e e f e a n d Nadel (1978) a n d Gallistel (1990). In this paper, we take for g r a n t e d that 663 J. GEN. PHYSlOL.9 The Rockefeller University Press" 0022-1295/96/06/663/32 $2.00

Volume 107 June 1996 663-694

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various animal species, and particularly rats, have cognitive maps.

Accepting the existence o f maps, it is natural to ask how they are implemented. From comparative (cross- species) behavioral studies, it is clear that map-like rep- resentations can be s u p p o r t e d by nervous systems o f widely varying anatomy; they are f o u n d in insects, am- phibia, reptiles, birds, and mammals (Gallistel, 1990).

T h e strong implication is that there is not a unique way o f representing global information a b o u t the environment. Rather, in the course of evolution, a variety o f m a p p i n g systems seem to have developed, presumably because it is useful to know where you are and how to get to where you want to go.

The Neural Basis of a Cognitive Map

Although comparative work may suggest that maps can be i m p l e m e n t e d in many ways, it is a m o r e difficult task to u n d e r s t a n d how a particular m a p might function. In the first place, it might be difficult to locate a map, even if a particular part o f a nervous system is preferen- tially associated with mapping. For example, a decrease in the ability to solve spatial problems after destruction o f a part o f the brain might result from disconnecting the putative map f r o m its inputs or outputs, as well as from damaging the map itself. Similarly, the existence o f a map would n o t necessarily be revealed by single- cell recordings because the e n c o d i n g o f spatial information might be distributed across cells in a very com- plicated fashion.

Despite these possibilities, there seems to be o n e map that is at least partly localized a n d whose operations are sufficiently simple to be detectable in the discharge o f individual neurons. This putative map was revealed by recordings from h i p p o c a m p a l n e u r o n s in freely moving rats by O'Keefe and Dostrovsky (1971).

T h e seminal discovery o f O'Keefe and Dostrovsky was that the discharge o f many h i p p o c a m p a l n e u r o n s is location specific; they fire rapidly only when the rat's head is in a restricted part o f the r e c o r d i n g apparatus.

Such units, now called "place cells" (O'Keefe, 1976), are pyramidal cells of the CA3 and CA1 regions o f the hippocampus. T h e existence o f place cells has b e e n c o r r o b o r a t e d by many workers (Olton et al., 1978;

Muller et al., 1987), and although their firing is not ide- ally location specific (Muller et al., 1991b), there are circumstances in which discharge is i n d e p e n d e n t o f the direction that the head points in the e n v i r o n m e n t

(Muller et al., 1994).

T h e place cell p h e n o m e n o n is so striking that it immediately convinced O'Keefe and Dostrovsky that the h i p p o c a m p u s is the locus o f a map. In the view of O'Keefe (see, for example, O'Keefe, 1991), this map is a Euclidean representation o f the environment; it allows the c o m p u t a t i o n o f distances and angles in the en-

vironment, thereby permitting solutions to spatial problems. An alternative view is that the spatial functions o f the h i p p o c a m p u s are special cases of m o r e general computations (see C o h e n and Eichenbaum, 1993). A r e c e n t set o f brief papers (Nadel, 1991) lays out the thoughts o f many workers in this area.

T h e position taken in this p a p e r is in f u n d a m e n t a l a g r e e m e n t with O'Keefe: We think that place cells re- veal a h i p p o c a m p a l map. O u r primary purpose is to show that there is a realistic way in which synaptic connections in the h i p p o c a m p u s can store a map-like representation o f the environment. By "map-like" we m e a n that the representation can be used to solve specific, difficult spatial problems.

T h e map-like representation is built from place cells, long-term potentiation, and the circuitry o f the CA3 portion o f the hippocampus. In this scheme, the mapping information is stored in the strengths o f CA3 --) CA3 synapses that c o n n e c t pairs o f p y r a m i d a l / p l a c e cells. T h e scheme is both parsimonious and precise, but is not comprehensive. T h a t is, we attempt to prove formally that the r e q u i r e d information could be stored in the stated way but do not a t t e m p t to explain either how place cells come to exist n o r how the stored information could be extracted by accepted neural operations. Retrieval as it might go on in the nervous system is considered only in the Discussion.

It is useful to c o m m e n t also o n the supposition by O'Keefe that m a p p i n g is the sole function of the hippocampus. A contrary supposition has b e e n expressed by E i c h e n b a u m that m a p p i n g is a special case of a m o r e general computational process. We take it as a major strength o f the ideas p r o p o s e d h e r e that they fit either of these views. O'Keefe's position is strength- e n e d if it is true that pyramidal cells act strictly as place cells. If pyramidal cells can also represent nonspatial aspects of the situation, the same circuitry permits solutions o f nonspatial problems, and E i c h e n b a u m ' s position is strengthened. In this paper, we deal only with place cells and location-specific firing but regard as o p e n the question of w h e t h e r the h i p p o c a m p u s may serve m o r e general functions.

Storing Mapping Information

T h e central idea in this p a p e r is that key information in the h i p p o c a m p a l map, namely, distance in the environment, is r e p r e s e n t e d as the strength o f H e b b i a n synapses ( e m b o d i e d as N-methyl-D-aspartate [NMDA] 1- based, long-term potentiation [LTP]-modifiable synapses) that c o n n e c t place cell pairs. Specifically, we propose that the strength o f a synapse made by a pair

IAbbreviations used in this paper: EEG, electroencephalogram; LTD.

long-term depression; LTP, long-term potentiation; NMDA, N-methyl- o-aspartate.

664 The Hippocampus as a Cognitive Graph

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o f place cells is a decreasing function o f the distance in two-dimensional (2-D) space between the firing fields o f the cells (Muller et al., 1991). If t h e r e is a barrier between the two fields when the synaptic resistance is set, the distance is how far the rat must go to get from o n e field to the other, and n o t the Euclidian distance.

T h e a r g u m e n t for supposing that synaptic strength should decrease with distance between field pairs is as follows:

(a) Consider a pair o f place cells with c o i n c i d e n t firing fields, as shown in Fig. 1, A1 and A2. Because the fields are coincident, the two cells will often fire in close temporal order. T h e timing o f the firing o f the cell in A 2 relative to that in A1 is seen in the point-process cross-correlation in Fig. 1 B, which shows that there are many short ( < 5 0 0 ms) intervals. If the cells are c o n n e c t e d by a H e b b i a n synapse, the short intervals between pre- and postsynaptic spikes are e x p e c t e d to cause increased synaptic strength. As is stated m o r e fully below, when it is possible for the strength o f the synapse to be modified, the strength is assumed never to get so great that discharge o f the presynaptic cell is an i m p o r t a n t d e t e r m i n a n t o f discharge o f the postsynaptic cell. Thus, we imagine that the short intervals indicate only that the fields are n e a r each o t h e r (similar to c o m m o n stimulus driving) and do not indicate a causal relationship between presynaptic a n d postsynaptic action potentials.

(b) Now consider a pair o f cells whose fields are far apart, as in Fig. 2, A1 and A2. In this case, the rat can- n o t move from o n e field to the o t h e r in a time short e n o u g h to p e r m i t the two cells to fire in close temporal order, as is visible in Fig. 2 B. Since the cells rarely if ever fire together, the H e b b i a n synapse should remain weak. For intermediate cases, synapses should have intermediate strengths.

In previous work, it was shown that synaptic strength changes (ASij) m a d e according to a simple H e b b i a n rule lead to a relationship o f the e x p e c t e d form between distance and synaptic strength (Muller et al., 1991a):

A~.j = f . ~ , (1)

where fl is the firing f r e q u e n c y o f the presynaptic place cell and f is the firing f r e q u e n c y o f the postsynaptic place cell. This rule is unrealistic since it permits synaptic strength to increase without limit. In the present work, however, the f o r m o f the strengthening rule is n o t critical for two reasons. First, having established that even a minimal rule allows distance to be e n c o d e d as synaptic strength, we now use explicit functions to set synaptic strength from the distance. T h e issue o f how to d o the e n c o d i n g m o r e realistically is left in abey- ance. Second, we show in Results that the m e t h o d o f storing m a p p i n g information works as long as synaptic

strength decreases with distance, regardless o f the ex- act strength--distance function. T h e implication is that there are few constraints on the strengthening rule.

In the example using Eq. 1, the probability that each cell discharges in a time interval is strictly d e t e r m i n e d by the position o f the rat's head. T h e probability function is maximal at the c e n t e r of the firing field and decreases in Gaussian fashion in all directions away from the center. T h e f r e q u e n c y o f each cell is averaged over a time span called the "LTP permissive interval," which was taken as 300 ms from the work o f Brown et al.

(1989). T h e sequences o f h e a d positions is from paths rats actually took as they retrieved r a n d o m l y scattered f o o d pellets in a cylindrical apparatus.

A c o m p u t e d example of the relationship between the synaptic strength a n d distance between firing field centers is shown in Fig. 3. As expected, the synaptic strength decreases with distance and falls to zero if the distance is great e n o u g h . Note that synaptic strengths are modified in the desired way d u r i n g exploration and n o explicit teaching mechanism is required. Muller et al. (1991a) showed that the broadness o f the s t r e n g t h - distance function is very sensitive to changes in field size. In contrast, changing the LTP permissive interval over a fairly wide range had little effect on the shape o f the strength-distance function. This is likely because the time-average firing rate is so strongly d e t e r m i n e d by w h e t h e r or not the rat is in the field so long as the averaging time is short c o m p a r e d with the time spent in the field. T h e insensitivity o f the strength-distance function to the LTP interval is e n c o u r a g i n g because it suggests that the theory will be robust as u n d e r s t a n d i n g o f the temporal properties o f LTP advances. It is also e n c o u r a g i n g that the various time and distance scales are mutually compatible; there is n o n e e d to use un- physiologic values for LTP interval, firing rate, speed of m o v e m e n t by the rat, or field size.

If distance in the e n v i r o n m e n t can be e n c o d e d as synaptic strength, it is reasonable to ask if such information is sufficient to i m p l e m e n t a cognitive map. T o sharpen up the question, however, we really ask whether such an e n c o d i n g can be used to calculate efficient paths t h r o u g h the environment. O u r main c o n t e n t i o n is that the answer is yes. We will use the m e t h o d s o f graph theory to show that the information is in fact available. T h e a r g u m e n t is straightforward: We d e m o n - strate that the requisite information is present by solving certain difficult spatial problems with graph-searching algorithms applied to a network of place cells. If the information is n o t there, n o algorithm can c o m p u t e the required paths.

T h r e e limitations on inferences that can be drawn from the p r o o f must be noted. First, the p r o o f assumes that the e n c o d i n g actually takes place. T h e arguments that point towards the e n c o d i n g are attractive b u t in n o

6 6 5 MULLER ET AL.

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sense g u a r a n t e e that the postulated i n f o r m a t i o n is stored. Second, even if the i n f o r m a t i o n is there, it is n o t necessarily available to the rat. Finally, even if the i n f o r m a t i o n is used by the rat, the p r o o f does n o t necessarily m e a n that the n e u r a l process o f extracting m a p p i n g i n f o r m a t i o n bears a close similarity to the g r a p h algorithm.

An additional r e q u i r e m e n t o f the m o d e l is that synaptic strength m u s t n e v e r get so great that presynaptic action potentials can cause 1:1 driving o f postsynapfic action potentials. T h e r e are two u n f o r t u n a t e conse- q u e n c e s if such driving is possible. First, b e c a u s e H e b - bian c o n j u n c t i o n o f pre- a n d postsynaptic activity always occurs, synapfic strength will go to the m a x i m u m value a n d stay there.

T h e s e c o n d difficulty with 1:1 driving is that the firing field o f the presynaptic cell would b e c o m e p a r t o f the field o f the postsynaptic cell; specificity would be lost (Hasselmo a n d Bower, 1993; Hasselmo a n d Schnell, 1994). By dealing only with writing b u t n o t r e a d i n g o f distance i n f o r m a t i o n , we do n o t have to c o n f r o n t any o f these issues immediately. It is clear, however, that if the i n f o r m a t i o n is stored in synapses b e t w e e n place cells, the i n f o r m a t i o n can be r e a d only if the presynaptic cells play a m a j o r role in discharging the postsynaptic cell. For the theory to be c o m p l e t e , it is t h e r e f o r e necessary that synaptic modifiability be t u r n e d o f f w h e n distance i n f o r m a t i o n is read. In the discussion, we speculate o n h o w r e a d i n g a n d writing are s e p a r a t e d

in time.

T h e r e are indications that the contacts between presynaptic pyramidal cells a n d their postsynapfic p a r t n e r s are i n d e e d quite weak. T h e r e is evidence that a given pyramidal cell m a k e s no c o n t a c t or j u s t o n e contact with i n t e r n e u r o n s . In addition, the statistics o f quantal release between pairs o f pyramidal cells again suggest

at m o s t o n e contact, and, f u r t h e r m o r e , that the c o n t a c t releases at o n e t r a n s m i t t e r q u a n t u m at m o s t f o r each action potential (Balshakov a n d Siegelbaum, 1995).

Which Synapses Store the Map Information ?

A second m a i n idea in this p a p e r c o n c e r n s the identity o f the synapses in which the m a p p i n g i n f o r m a t i o n is p r o p o s e d to be stored. As is true of m a n y o t h e r schemes to explain spatial or nonspatial o p e r a t i o n s o f the hip- p o c a m p u s , we focus o n the r e c u r r e n t or lateral synapses that are m a d e b e t w e e n pairs o f CA3 place cells (see T r a u b a n d Miles, 1991). This synaptic class is by n o m e a n s the only candidate in which to store m a p p i n g information. O t h e r possibilities are the contacts f r o m en- torhinal cortex (EC) cells o n t o d e n t a t e g r a n u l e (DG) cells a n d the Schaffer collateral projection f r o m CA3 to CA1. Synaptic classes EC --~ D G a n d CA3 ~ CA1 are b o t h c o n s i d e r e d to be NMDA-based, LTP-modifiable synapses with H e b b i a n logic (see, for e x a m p l e , Brown et al., 1989). T h e r e is also growing evidence that CA3 --~ CA3 synapses also show NMDA-based LTP (Miles a n d Wong, 1987; Jaffe a n d J o h n s t o n , 1990; Jeffreys a n d T r a u b , 1993).

T h e r e are still o t h e r c a n d i d a t e synaptic classes. T h e mossy fiber projection f r o m d e n t a t e g r a n u l e cells to CA3 pyramidal cells also shows LTP, b u t the biophysics a n d possibly the logic o f the modifiability are different (Jaffe a n d J o h n s t o n , 1990). Moreover, pathways f r o m e n t o r h i n a l cortex directly to CA3 a n d CA1 exist a n d show L T P (Buzsaki, 1988), a l t h o u g h the n a t u r e o f the modifiability is n o t well characterized.

Given this e m b a r r a s s m e n t o f riches, t h e r e are several reasonable ways in which m a p p i n g i n f o r m a t i o n m i g h t be distributed across synapses. Nevertheless, we believe that there is a m a j o r advantage to focusing on the CA3 ---) CA3 network. We c o n t e n d that a n e t w o r k o f connec-

FIGURE 1. Description of the discharge properties of a pair of simultaneously r e c o r d e d hippocampal place cells with overlapping firing fields. (A1) The first cell had its field against the apparatus wall at ,'-q 1:30. The discharge rate of this cell is relatively low, as shown by the color code to the left, which indicates the median rate (in spikes/s) for each color category. (A2) The field of the second cell is somewhat larger than that of the first, and the first rate is considerably greater (see color scale) but is in almost the same part of the apparatus. There are two very strong indications that the two cells are independent. First, they were recorded from different microwires. Second, the timing of action potentials was very different on the short scale (ms). (B) Cross-channel spike histogram. Each count in the histogram denotes the existence of an interval between a spike fired by the first cell and a spike fired by the second cell. The key features of the histogram are the great excess of counts at short intervals (< 1 s) and the strong peak near zero. If one of the two cells directly contacted the other via an LTP- modifiable synapse, the existence of many short intervals would tend to cause synaptic strength to increase (synaptic resistance to decrease).

FIGURE 2. Description o f the discharge properties of a pair of simultaneously recorded h i p p o c a m p a l place cells with separated firing fields. (A1) The first cell h a d its field n e a r the apparatus wall at "-~7:30. This cell fired quite briskly; the m e d i a n rate in the highest rate category (purple) was 17.5 spikes/s. (A2) The second cell is the same as in Fig. 1 A 2 its field is well away from that of the first unit. (B) Cross- c h a n n e l spike histogram. In contrast to the clear peak n e a r t = 0 for the overlapping firing fields in Fig. 1 B, there is a m i n i m u m near t = 0 a n d two m a x i m a at ~ - 3 s a n d +3.5 s. If o n e o f the two cells directly contacted the o t h e r via an LTP-modifiable synapse, the large n u m b e r of long intervals a n d the n e a r absence of short intervals would tend to leave synaptic strength u n c h a n g e d or to produce a reduction via LTD. Note that the asymmetry of the histogram is caused by the behavior of the rat; there are more counts at positive t h a n negative intervals because the rat t e n d e d to walk m o r e often from the field of the first cell to the field of the second cell.

667 MULLER ET AL.

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1,0

0.8

9 ~.. 0.6 r~ ~ 0.4 N

~ 0.2

0.0

0 5 ^.^.^.^. ^~ ^.^.^.^. ^k ^.^.^.^. ⁱ ^. ^. ^, ^,310

10 15 20 25

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A

CA3 -> CA1 Schaffer Collateral Projection

FIGURE 3. An example of a calculated strength-distance function. The strength of a given synapse is calculated from the simple H e b b i a n rule of Eq. 1. Firing rate was strictly d e t e r m i n e d from the animal's position in the following way. (a) The c u r r e n t position was taken from a time series of positions recorded as a real rat ran inside of a cylindrical apparatus. (b) T h e time-averaged firing rate at the c u r r e n t position was a Gaussian function of the distance from a field center for the cell. In the present case, the standard deviation of the Gaussian was three pixel-edge lengths a n d the peak rate was 30 spikes/s. (c) The time-average rate was used in conjunction with a r a n d o m n u m b e r generator to d e t e r m i n e if the cell did or did n o t fire in the c u r r e n t 1 / 6 0 t h s. (d) If the firing rate of b o t h the pre- a n d postsynapfic cells was greater than zero averaged over 300 ms, the strength of the synapse was increased according to Eq. 1. In the actual simulation, a total of 600 cells were scattered across the surface of the cylinder, a n d each cell was connected to eight o t h e r cells for a total of 4,800 synapses. All synaptic strength calculations were d o n e at once, using a single time series of positions. In the graph, strength at a given distance is the m e a n strength of many synapses, such that the distance between the field centers of the pre- a n d postsynaptic cells was in the range n < d <

n + 1. Normalization was d o n e after taking the averages.

t i o n s a m o n g c e l l s o f a s i n g l e k i n d h a s s t r o n g e r i s o m o r - p h i s m s t o 2-D s p a c e t h a n a n e t w o r k i n w h i c h c o n n e c - t i o n s a r e m a d e i n o n e d i r e c t i o n f r o m c e l l s o f o n e c l a s s o n t o c e l l s o f a d i f f e r e n t class. W e n o w a t t e m p t t o j u s t i f y t h i s c l a i m .

The Connectivity of Networks and the Connectivity of Space C o n s i d e r t h e s u i t a b i l i t y o f t w o d i f f e r e n t k i n d s o f n e u r a l c o n n e c t i o n s f o r r e p r e s e n t i n g t h e p r o p e r t i e s o f s p a c e . T h e n e t w o r k i n Fig. 4 A h a s t w o l a y e r s i n a f e e d f o r w a r d a r r a n g e m e n t t h a t r e s e m b l e s t h e C A 3 ---> CA1 p r o j e c t i o n ( o r g e n e r a l l y , t h e p r o j e c t i o n f r o m o n e c l a s s o f c e l l s t o a s e c o n d c l a s s o f c e l l s ) . T h e r e a r e n o i n t e r a c t i o n s b e - t w e e n c e l l s i n a l a y e r . A l t h o u g h t h e y a r e n o t d r a w n , c e l l s i n t h e f i r s t l a y e r r e c e i v e i n p u t s f r o m s o m e o t h e r s o u r c e a n d c e l l s i n t h e s e c o n d l a y e r s e n d o u t p u t s t o s o m e o t h e r r e g i o n . N e t w o r k s c o n n e c t e d i n t h i s w a y h a v e b e e n s h o w n t o b e u s e f u l as p a t t e r n s o r t i n g a n d

CA3 -> CA3 Recurrent Connections

FIGURE 4. Drawings of the c o n n e c t i o n schemes for a two-layer system (A) a n d a r e c u r r e n t or associational system (B). (A) T h e two-layer system captures the form of the projection from CA3 to CA1 pyramidal cells. If each cell is imagined to be a place cell, the two-layered system allows associations between arbitrarily selected pairs of points in the environment. O n the o t h e r hand, there is n o natural equivalent of a path in the environment, since only one step can be taken without leaving the system. (B) The r e c u r r e n t system captures the form of the CA3 to CA3-associational connections. T h e connections once again allow associations between arbitrarily selected pairs of points in the environment. Note, however, that if there are e n o u g h connections (as is true here), the recur- r e n t system is isomorphic to real space in the sense that it is possible to get from any cell to any o t h e r cell,just as it is possible to get from any place to any o t h e r place in unobstructed 2-D space. In addition, if each cell is a place cell, then a walk along a sequence of cells corresponds to a path in 2-D space, although the 2-D path in general will not be smooth if the walk along the cell sequence is chosen only according to connectivity. A central t h e m e in this paper is that optimal paths in 2-D space can be found from optimal paths in connectivity space if the strength of the c o n n e c t i o n is de- t e r m i n e d by distance in 2-D space.

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recognition devices (Kohonen, 1984), especially if there are three cell layers c o n n e c t e d by two sets o f projec- tions, as in basic back p r o p a g a t i o n schemes (Rumel- hart et al., 1986).

T h e single-layer, r e c u r r e n t network in Fig. 4 B is pat- t e r n e d after the CA3 ---) CA3 circuitry b u t could equally well stand for any neural system in which cells o f a single class are mutually i n t e r c o n n e c t e d . No interactions with cells in o t h e r layers are drawn, although again o n e expects t h e r e to be inputs a n d outputs. R e c u r r e n t or

"peer-to-peer" networks function as autocorrelators that can do p a t t e r n completion, where a f r a g m e n t o f a stimulus configuration allows recall o f the entire configuration (Kohonen, 1984).

In the feedforward network of Fig. 4 A, only sequences o f two cells are possible, regardless o f the n u m b e r o f cells in each layer a n d o f the density o f connections. Since place cell firing fields occur with a b o u t equal frequency everywhere in the e n v i r o n m e n t (Muller et al., 1987), if the c o n n e c t i o n density between cell pairs is high e n o u g h , the network can store distances between every pair of points in the environment. Nev- ertheless, there is no way to use a two-layered structure to calculate paths in 2-D space, since only pairwise b u t n o t h i g h e r o r d e r sequences of cells o c c u r in network space. 2 In o t h e r words, t h e r e is n o way to make paths in the e n v i r o n m e n t c o r r e s p o n d to paths in the network.

T h e r e c u r r e n t network in Fig. 4 B also permits the storage of distances between pairs o f points in the envi- r o n m e n t . In addition, however, the r e c u r r e n t network provides a direct analogy between environmental and network paths. If the network is c o n n e c t e d richly e n o u g h , it is possible to find a path from any cell to any o t h e r cell. Since each cell is a place cell, a path in neural space immediately corresponds to a path in the sur- roundings. Thus, the r e c u r r e n t network allows for chains o f arbitrary length, in the same way that arbitrarily long paths are g e n e r a t e d by locomotion. More- over, because firing fields o c c u r everywhere in the envi- r o n m e n t , and because there are so many CA3 place cells (~250,000 p e r side; Amaral et al., 1990), if it is possible to find a neural path from any cell to any o t h e r cell, there must be c o r r e s p o n d i n g paths in the environ- m e n t from any place to any o t h e r place; the network shares with 2-D space the p r o p e r t y that any place is accessible from any o t h e r place (you can get from there f r o m h e r e ) .

T h e existence o f paths t h r o u g h network space and the existence o f c o r r e s p o n d i n g paths t h r o u g h 2-D

2yeckel and Berger (1990) d e m o n s t r a t e d that a single shock to the p e r f o r a n t path can excite the same hippocampal elements two or m o r e times because of loops that involve the hippocampus. Pathways of this kind, or a r e c u r r e n t network in CA1 (Christian and Dudek, 1988; T h o m s o n and Radpour, 1991), are possible alternatives to the r e c u r r e n t CA3 network considered here.

669 MULLER ET AL.

space is a key p r o p e r t y o f r e c u r r e n t place cell networks.

T h e possibility o f paths t h r o u g h 2-D space does not, however, necessarily m e a n that the paths are physically reasonable. Imagine that the probability o f c o n n e c t i o n between a pair o f cells is i n d e p e n d e n t o f where their firing fields are in the environment. U n d e r these circumstances, going from a presynaptic cell to a postsynaptic cell might be associated with a large j u m p in 2-D space. Thus, smooth paths in neural space n e e d n o t c o r r e s p o n d to smooth or even possible paths in the sur- roundings.

This difficulty is resolved by taking into a c c o u n t n o t just w h e t h e r two cells are c o n n e c t e d , b u t also the strength of the connection. In particular, if the synaptic strength approaches zero, two cells can be c o n s i d e r e d to be u n c o n n e c t e d even if the anatomical j u n c t i o n exists. In o u r theory o f synaptic strengthening, strength remains near zero if the firing fields o f two cells are sufficiently far apart. At once, this means that cell sequences in neural space such that the synaptic weights are all strong c o r r e s p o n d fairly well to real paths in the environment; j u m p s o f arbitrarily great distance no longer occur in the representation. This does n o t m e a n that the representation or m a p can generate direct, efficient paths, a matter that remains to be d e m o n - strated. What it does m e a n is that a r e c u r r e n t network o f place cells can be strongly isomorphic to 2-D space if the strength o f the r e c u r r e n t connections decreases with distance, as may h a p p e n if the place cells are con- n e c t e d by LTP-modifiable synapses.

What Spatial Problems Must Be Solvable to Call a Representation a M a p ?

We now turn to a key question c o n c e r n i n g the proposed e m b o d i m e n t of a map: Does the map contain e n o u g h information to p e r m i t solutions o f spatial problems? T o answer this question affirmatively, it is necessary only to show that there is some m e t h o d , no m a t t e r how unrealistic, that can generate the r e q u i r e d solutions using the stored information; the solutions can- n o t be g e n e r a t e d if the information is n o t there.

A m u c h m o r e difficult p r o b l e m is to find a plausible neural mechanism that is capable o f finding solutions.

It is yet m o r e difficult to show that any p r o p o s e d mechanism is actually used to generate the paths rats are ob- served to take. In this paper, we deal mainly with the easiest issue: w h e t h e r the r e c u r r e n t network stores e n o u g h information a b o u t the structure o f the environ- m e n t to solve three spatial problems. These problems are selected because the ability of rats to solve t h e m suggests the existence o f maps in the first place.

T h e first p r o b l e m concerns the ability to find the straight-line path between any pair o f points in the environment, so that any p o i n t can serve as a starting location and any o t h e r p o i n t can serve as a goal. This ability

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is closely associated with the h i d d e n goal p r o b l e m ex- emplified by the Morris swimming task (Morris, 1981) a n d especially the variant designed by Whishaw (1985).

T h e s e c o n d p r o b l e m c o n c e r n s the ability to find an o p t i m a l d e t o u r w h e n a m o r e efficient r o u t e is suddenly b l o c k e d (Poucet et al., 1983). T h e selection o f the best d e t o u r should be d o n e by operations on the existing m a p a n d m u s t n o t require the g e n e r a t i o n of a new map. If the p r o p o s e d r e p r e s e n t a t i o n is n o t flexible e n o u g h to find detours, in o u r j u d g m e n t it would n o t be a p p r o p r i a t e to call it a m a p .

T h e final p r o b l e m involves the capacity to find short- cuts w h e n a p a t h is suddenly o p e n e d that is m o r e efficient t h a n the c u r r e n t best p a t h (Poucet, 1993). Again, it is critical w h e t h e r the system can p r o d u c e the shortcut without g e n e r a t i n g a new m a p . I f this is possible, it may be c o n c l u d e d that the substrate for the new p a t h already exists in the m a p , as would be true if the m a p r e p r e s e n t e d the overall structure of the e n v i r o n m e n t .

Note that the three tests o f the m a p p i n g s c h e m e are all variants o f what m a y be called the "geodesic problem," in which o p t i m a l solutions are simply shortest paths. It is n o t clear if the m a p p i n g system deals with motivation or time as well as g e o m e t r y in the process o f p a t h selection, b u t in the c u r r e n t t r e a t m e n t only geom- etry is considered.

Searching for Paths in a Model of the CA3 Recurrent Network Having stated the criteria for d e t e r m i n i n g if a network contains e n o u g h i n f o r m a t i o n to be c o n s i d e r e d a m a p , it is necessary to decide h o w to l o o k for the r e q u i r e d paths. T h e m e t h o d used h e r e is to treat the network as a graph, in which cells are i n t e r p r e t e d as n o d e s (or ver- tices) a n d axons plus synapses are i n t e r p r e t e d as edges.

O n e n o d e is c o n n e c t e d to a n o t h e r by a directed edge if a n d only if an a x o n b r a n c h o f the n o d e c o r r e s p o n d i n g to the first cell makes a synaptic contact with the n o d e c o r r e s p o n d i n g to the second cell. T h e graphs o f interest are weighted because each synapse has a certain strength a n d are directed because i n f o r m a t i o n flows in only o n e direction across a synapse. 3 In Methods, an al- g o r i t h m is described that allows o p t i m a l paths to be f o u n d a c c o r d i n g to sequences of synaptic weights. T h e critical question is t h e n w h e t h e r o p t i m a l paths in neural space are also o p t i m a l paths in 2-D space.

It is worth n o t i n g that the intuitive distinctions drawn above between feedforward a n d r e c u r r e n t networks have f o r m a l parallels in g r a p h theory (Harary, 1969). A directed g r a p h (A --~ B does n o t imply B ~ A) is said to be "strongly" c o n n e c t e d if it is possible to walk f r o m any

3This does n o t preclude the possibility o f a retrograde signal sent during modification o f synaptic strength via LTP, Information flow is m e a n t to include only the effect that the presynaptic cell has o n t h e likelihood o f discharge o f the postsynaptic cell.

n o d e to any o t h e r n o d e using a s e q u e n c e o f p r o p e r l y directed edges. We m e n t i o n two o t h e r related notions o f connectedness. A directed g r a p h is called "unilater- ally" c o n n e c t e d if, for any two nodes, it is possible to walk f r o m at least o n e o f t h e m to the o t h e r using a se- q u e n c e o f p r o p e r l y directed edges. Finally, we say that a directed g r a p h is "weakly" c o n n e c t e d if o n e can walk f r o m any n o d e to any o t h e r n o d e n o t necessarily re- specting the direction of edges; this is likely true o f feedforward networks such as the CA3 ---) CA1 projection. We a r g u e that the strong c o n n e c t e d n e s s of the CA3 --~ CA3 network allows it to mimic the c o n n e c t e d - ness o f space, s o m e t h i n g that the weak c o n n e c t e d n e s s o f the CA3 ~ CA1 projection does n o t permit. Experi- m e n t a l results (Miles a n d Wong, 1983) a n d n u m e r i c a l calculations (Traub a n d Miles, 1991; see also Results) indicate that the anatomical divergence a n d convergence in the r e c u r r e n t CA3 --~ CA3 network is great e n o u g h to m a k e the network strongly connected.

To conclude, we refer to s o m e origins of the ideas p r e s e n t e d here. As far as we know, the first s t a t e m e n t that t e m p o r a l coincidence o f firing m i g h t p r o d u c e functional aggregates of place cells via LTP was by Bliss (1979), in a c o m m e n t a r y on the work o f O ' K e e f e a n d Nadel (1979). T h e aggregates were c o m p o s e d of all the cells that fired in a given place, a n d their significance was t h o u g h t to be increased accuracy o f localization o f the rat. A second source for the p r e s e n t work is the to- pological m a p p i n g theory o f Deutsch (1960) a n d the p r e s e n t a t i o n of the theory by Gallistel (1980). This theory has n o specific neural e m b o d i m e n t , b u t it states that navigation may d e p e n d on associations between n e i g h b o r i n g regions o f space a n d on the a b s e n c e o f associations between distant regions o f space.

M E T H O D S

Experimental Foundations

One requirement for implementing the graph model is a description of place cell discharge. We begin by briefly describing the behavioral situation for recording and then summarize some important aspects of place cell activity.

Behavioral conditions. Detailed methods used for training rats, implanting electrodes, discriminating and recording single cells, and tracking rats are given elsewhere (Muller et al., 1987). In brief, place cell recordings were made as rats ran around in walled apparatuses of simple geometric shape. The most com- monly used apparatus was a cylinder 76 cm in diameter and 50 cm high. The wall of the cylinder was gray except for a white cue card that covered about one-fourth of the circumference. Hun- gry rats were trained to scamper over the whole surface of the cylinder to retrieve 20-mg food pellets, so that place cell firing rate could be measured everywhere. Since the rats ran almost contin- ually, positional firing variations cannot easily be ascribed to ten- dencies of rats to do discharge-related things in certain places.

Under the stated circumstances, place cells have several well- characterized properties, which are considered next.

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Place cellproperties. (a) Place cell discharge is location specific.

This is the d e f i n i n g property o f place cells; t h e i r firing rate is largely d e t e r m i n e d by the position of the rat's h e a d in the envi- r o n m e n t . T h e r e are known deviations from ideal location-specific firing. For example, the positional firing p a t t e r n is m o r e precise w h e n the h i p p o c a m p a l e l e c t r o e n c e p h a l o g r a m (EEG) is in "theta" m o d e t h a n otherwise (Kubie et al., 1985), a n d firing ceases if the rat is immobilized (Foster et al., 1989). Nevertheless, firing is intense only w h e n the rat's h e a d is in a delimited firing field (see below).

(b) In the cylinder, firing is i n d e p e n d e n t of h e a d direction (Muller et al., 1994). In this paper, we consider only omnidirectional firing. We note, however, t h a t place cells are often direc- tionally selective w h e n r e c o r d e d o n a n eight-arm maze or a linear runway ( M c N a u g h t o n et al., 1983; O'Keefe a n d Recce, 1993;

Muller et al., 1994). We argue in the discussion that the g r a p h model handles directional a n d omnidirectional firing equally well.

(c) Positional firing patterns are stationary over time intervals o f weeks o r m o n t h s (Muller et al., 1987; T h o m p s o n a n d Best, 1990). If a given place cell is r e c o r d e d with the rat in a familiar e n v i r o n m e n t , its positional firing p a t t e r n seems to be the same n o m a t t e r how m a n y times the rat is removed a n d replaced into the e n v i r o n m e n t .

(d) Positional firing patterns are characterized by "firing fields." A place cell discharges rapidly only w h e n the h e a d is in a continuous, restricted p o r t i o n of the apparatus. Outside such a field, the firing rate is virtually zero. Examples of firing fields are shown in Figs. 1, 2, a n d 13. Most cells have only o n e field, b u t a few have two (Muller et al., 1987; Sharp et al., 1990;Jung a n d Mc- N a u g h t o n , 1993). Here, we d o n o t consider cells with m o r e t h a n o n e field since it is clear that they will interfere with the g r a p h searching scheme. As noted, for example, by Shapiro a n d Heth- e r i n g t o n (1993), the existence a n d significance of cells with multiple fields must be settled for a theory to be complete.

(e) Firing fields vary in several ways i n c l u d i n g size (area), in- tensity (peak discharge rate), a n d shape. Nevertheless, for simplicity of c o m p u t a t i o n , Muller et al. (1991a) i m a g i n e d that the iso-rate c o n t o u r s are circular or are circles t r u n c a t e d by the wall of the cylinder. T h e same assumptions are m a d e here.

T h e stated properties are compatible with the idea that the strength of a H e b b i a n synapse t h a t connects a pair of place cells s h o u l d decrease with the distance between the firing fields o f the cells (Muller et al., 1991a). Because the real s t r e n g t h - d i s t a n c e function is u n k n o w n (if i n d e e d o n e exists), synaptic strengths in networks are calculated from o n e of several explicitly stated functions of the distance between firing fields. Several effects of varying the s t r e n g t h - d i s t a n c e function, or, m o r e correctly, the reciprocal "resistance--distance" function, are shown in Results. T h e reason for using resistance-distance functions is stated below.

Building the network. For simplicity, the networks to be ana- lyzed are m o d e l e d as r a n d o m graphs, in which the probability of a c o n n e c t i o n is the same for all pairs of cells. T h e r e is n o question that this assumption is wrong in detail since it is a g r e e d that the density o f r e c u r r e n t CA3 --4 CA3 c o n n e c t i o n s varies with the position of the presynaptic cell in the pyramidal cell layer (Miles a n d Wong, 1986; Ishizuka et al., 1990; Li et al., 1993; B e r n h a r d a n d Wheal, 1994). T h e same workers agree, however, that recur- r e n t c o n n e c t i o n s are widespread a n d massive. In the absence of a specific role for the partial specificity of connections, o u r main interest is in w h e t h e r networks of the size of CA3 a n d c o n n e c t i o n

density of CA3 are likely to b e strongly c o n n e c t e d , i.e., t h a t it is possible to "walk" a l o n g a chain of cell ~ synapse ~ cell ~ synapse, etc., a n d reach any cell from any starting cell. As stated in the I n t r o d u c t i o n , strongly c o n n e c t e d networks share with u n o b - structed 2-D space the property that it is possible to get from any place to any o t h e r place. This property underlies o u r analysis, and, accordingly, it is the first topic dealt with in Results.

A r a n d o m network is characterized by two parameters, namely, the n u m b e r of cells and the n u m b e r of o u t p u t connections made by each cell. In g r a p h theory, the n u m b e r of o u t p u t c o n n e c t i o n s is called outdegree, a n o t i o n t h a t c o r r e s p o n d s precisely to the neu- roanatomical idea of divergence. (Similarly, "indegree" corresponds to convergence.) T h e total n u m b e r o f o u t p u t c o n n e c t i o n s is the p r o d u c t of the n u m b e r of cells a n d the divergence. Because synapses are m a d e between cell pairs, the total n u m b e r of o u t p u t c o n n e c t i o n s is equal to the total n u m b e r of i n p u t c o n n e c t i o n s ( n u m b e r of cells times average convergence) (see Bollobas, 1985).

O n c e the n u m e r i c p a r a m e t e r s for a network are chosen, the units are randomly c o n n e c t e d with the following constraints:

(a) Every cell has the same divergence (is presynaptic to the same n u m b e r of postsynaptic cells). A l t h o u g h the m e a n convergence must equal the divergence, the convergence varies from cell to cell. For small networks, the convergence will have a Pois- son distribution. For networks the size of CA3, the convergence would have a n o r m a l distribution.

(b) A cell c a n n o t contact itself, which m e a n s t h a t "autapses"

are precluded. In g r a p h theory, the c o n c e p t equivalent to a n au- tapse is a loop, in which a n o d e has a n edge with itself. C h a n g i n g this c o n d i t i o n would have little effect o n any of o u r results.

(c) A cell is n o t allowed to contact a n o t h e r cell twice. (In g r a p h theory, this is equivalent to saying that t h e r e are n o parallel edges.) T h e r e is empirical evidence that suggests this is true (Balshakov a n d Siegelbaum, 1995). If multiple contacts d o in fact occur, p e r m i t t i n g only o n e contact can be viewed as l u m p i n g to- g e t h e r all the synapses. This a p p r o x i m a t i o n is n o t necessarily cor- rect, d e p e n d i n g o n how the multiple synapses are distributed o n the dendritic tree.

T h e resulting network is t h e n tested for strong connectivity. If it is n o t strongly c o n n e c t e d , it is discarded a n d a new r a n d o m network is built. A surprisingly low divergence is necessary to virtually ensure that the network is strongly c o n n e c t e d (see Results).

O n c e a network is known to be strongly c o n n e c t e d , a weight is assigned to each synapse in a two-step process. First, each cell is assigned a location in 2-D space for its firing field. This does n o t imply any relationship between the identity of a cell a n d the location of its firing field. To the contrary, the a s s i g n m e n t is d o n e randomly, in accord with o u r belief that the pyramidal cell layer is n o t topographically m a p p e d o n t o the apparatus floor (Muller et al., 1987; Kubie et al., 1992). Accessible space is divided into pixels, a n d at least o n e cell is assigned to each pixel. In the model, each pixel is taken to b e the field c e n t e r for o n e or m o r e place cells. In the p r e s e n t calculations, the accessible space is a circle that contains 756 pixels. T h e radius o f the cylinder is ~ 1 5 . 3 pixel-edge lengths. O u r considerations pertain to a scale such that a pixel is a square ~ 3 . 3 cm o n a side, a n d the d i a m e t e r of the circle is ~ 1 0 0 cm. T h e size of the circle falls within the range of cylinder diameters (76-200 cm) in which we have r e c o r d e d place cells. In this range, place cell properties are nearly constant, al- t h o u g h fields in larger d i a m e t e r cylinders are somewhat larger (Muller a n d Kubie, 1987).

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After field locations are assigned, each synapse is given a strength according to the distance between the field centers of the cells it connects. As stated above, reciprocal strengths (synaptic resistances) are assigned using one of several resistance-distance functions, which share the property that resistance is a monotonically increasing function of the distance between field centers. At this point the network is complete and ready to be tested for whether it contains sufficient information to solve the navigational problems proposed in the Introduction as hallmarks of mapping.

Finding optimal paths in synaptic resistance space and 2-D space.

Formally, the completed network is a random, directed, weighted graph. It consists of a set of nodes, the cells, which are connected by a set of edges, the axons and the endings made by axons on other cells. The graph is random because the connections between cell pairs are set up randomly. It is directed because information can move in only one direction across synapses. It is weighted because the strength of the synaptic connections can vary.

The central problem is whether the graph can be used to find best paths from a start to a goal in 2-D space. The proposed solu- tion is to use a standard algorithm to find best paths in the graph.

Since each cell is a place cell, any path in the graph corresponds to a path in 2-D space. The question is whether optimal paths in the graph correspond to optimal paths through the environment. If the correspondence exists, it will have been proved that the network stores enough information to act as map.

Optimal paths in the strongly connected weighted graphs were found with Dijkstra's algorithm (Sedgewick, 1987; Even, 1979).

Dijkstra's algorithm finds the path from a start node to an end node that minimizes the sum of the weights. The algorithm works by constructing a simplification of the original graph in the form of a "tree" rooted at the starting node. The tree is simpler than the original graph because it contains no cyclic path such that it is possible to get from a node back to itself. Once built, the tree contains the minimal path from the starting node to every other node and is called a minimum spanning tree.

To build the minimal spanning tree, nodes are divided into three classes: those already known to be part of the tree, those in a fringe that has been visited but that are not yet part of the tree, and, finally, those not yet known to exist. The current state of the fringe is maintained in a list called a priority queue in which the values for nodes determine how the search is made. Initially, the spanning tree consists of only the starting node, and the priority queue is empty. In the first step, all the nodes adjacent (reachable) from the starting node are put onto the priority queue. Next, one of these nodes is attached to the spanning tree (according to the priorities in the queue), and all nodes attached to it are put into the queue. This cycle is repeated until there are no unvisited nodes left. The tree is then finished by attaching the rest of the nodes in the fringe to the tree.

The preceding description of finding a spanning tree makes it clear that the art form is in the assignment of priorities to nodes in the queue. It is not in the scope of this paper to explain how the assignments are made, but it may be clear that the searching process can be varied by changing assignments. For example, if nodes are added to the fringe by always looking at nodes adjacent to the first node on the queue, a "depth-first" search results. If, instead, nodes are added by looking at nodes adjacent to all nodes currently on the queue, a "breadth-first" search is per- formed. For Dijkstra's algorithm, a more complex assignment of

priority is made (Sedgewick, 1987). Dijkstra's algorithm is not optimal for sparse graphs, our main interest, but it is fast enough for small sparse graphs with available computers. For sparse graphs, algorithms exist that run in time proportional to (E + N) log N, where E is the n u m b e r of edges and N is the n u m b e r of nodes.

O n e methodological problem remains to be considered. The path-searching algorithms are designed to find paths along which the sum of the weights is minimized. Clearly, something is wrong if the weights are taken to be synaptic strengths, since a path in the network of minimal synaptic strengths would be a long path in the environment.

It is also clear, however, that the difficulty arises only because synapses are usually characterized by strength and not by its reciprocal, which may be called synaptic resistance (see, for example, Hebb, 1949). If synaptic resistance is used to specify the weight of each connection, then a search algorithm will find paths along which the sum of the synaptic resistances is minimal, and these will be short paths in the environment. It is more con- venient to use synaptic resistance in Dijkstra's algorithm simply because large resistances are associated with long distances in the environment and small resistances with short distances in the environment. It is important to realize that substituting synaptic resistance for synaptic strength in no way compromises the graph model. There is nothing more fundamental about synaptic strength than synaptic resistance; they are related in just the same way as electrical conductance and electrical resistance.

R E S U L T S

IS the CA3 Network Strongly Connected?

As s t a t e d i n M e t h o d s , r a n d o m g r a p h s a r e u s e d to m i m i c t h e C A 3 r e c u r r e n t c o n n e c t i o n n e t w o r k . I n s u c h g r a p h s , t h e p r o b a b i l i t y t h a t a c e l l c o n t a c t s a n y o t h e r c e l l is a c o n s t a n t . It is c l e a r , h o w e v e r , t h a t t h e p r o b a b i l - ity t h a t a C A 3 c e l l c o n t a c t s a n o t h e r v a r i e s w i t h t h e l o c a - t i o n o f b o t h t h e p r e s y n a p t i c c e l l a n d p o s t s y n a p t i c c e l l i n t h e C A 3 layer. F o r e x a m p l e , r e c o r d i n g s f r o m p y r a m i - d a l cells i n l o n g i t u d i n a l slices o f C A 3 r e v e a l e d t h a t m o n o s y n a p t i c C A 3 ~ C A 3 c o n t a c t s a r e m a d e o v e r l o n g s e p t a l to t e m p o r a l d i s t a n c e s a n d o v e r t h e w h o l e w i d t h o f C A 3 f r o m t h e h i l u s to t h e b o r d e r w i t h C A 2 ( M i l e s e t al., 1 9 8 8 ) . N e v e r t h e l e s s , t h e y f o u n d a g r a d i e n t a l o n g t h e l e n g t h o f t h e h i p p o c a m p u s , s u c h t h a t t h e p r o b a b i l - ity o f c o n t a c t was m a r k e d l y l o w e r i f t h e cells w e r e s e p a - r a t e d b y t w o - t h i r d s o f t h e l e n g t h c o m p a r e d w i t h o n e - t h i r d o f t h e l e n g t h . R e c e n t l y , Li e t al. ( 1 9 9 3 ) t r a c e d t h e c o n n e c t i o n s o f i n d i v i d u a l C A 3 p y r a m i d a l cells a n d f o u n d s e v e r a l p a t t e r n s o f c o n t a c t s p e c i f i c i t y . F o r e x a m - p l e , t h e y saw t h a t c e l l s in a b a n d o f C A 3 p a r a l l e l to t h e s e p t a l - t e m p o r a l axis t e n d e d to c o n t a c t o t h e r cells in t h e s a m e b a n d . I n t e r e s t i n g l y , t h e y a l s o s h o w e d t h a t t h e d e - c r e a s e o f c o n t a c t p r o b a b i l i t y a w a y f r o m t h e c e l l b o d y is s o m e t i m e s n o t m o n o t o n i c ; t h e y f o u n d c l e a r o s c i l l a t i o n s o f t h e n u m b e r o f c o n t a c t s f o r s o m e cells a l o n g t h e s e p t o - t e m p o r a l axis.

I n l i g h t o f c o n t a c t s p e c i f i c i t y , a r a n d o m g r a p h c a n n o t