================================================================================================== Muller R. U., M. Stead, J. Pach (1996) "The hippocampus as a cognitive graph."J. Gen Physiol 1996 107 pp. 663-694 ================================================================================================== [abstract] A theory of cognitive mapping is developed that depends only on accepted properties of hippocampal function, namely, long-term potentiation, the place cell phenomenon, and the associative or recurrent connections made among CA3 pyramidal cells. 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. Our 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. [Introduction] As an aside, we note that not everyone accepts this reasoning, and that even advocates must maintain a healthy skepticism about maps (see Terrace, 1984). Nevertheless, the behavioral evidence in favor of maps is quite convincing, and the reader is referred to compendious reviews by O'Keefe and Nadel (1978) and Gallistel (1990). - O'Keefe, J, and L. Nadel. 1978. The Hippocampus as a Cognitive Map. Clarendon Press, Oxford, UK. 570 pp. Accepting the existence of maps, it is natural to ask how they are implemented. Rather, in the course of evolution, a variety of mapping 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] This putative map was revealed by recordings from hippocampal neurons in freely moving rats by O'Keefe and Dostrovsky (1971). The seminal discovery of O'Keefe and Dostrovsky was that the discharge of many hippocampal neurons is location specific; they fire rapidly only when the rat's head is in a restricted part of the recording apparatus. Such units, now called "place cells" (O'Keefe, 1976) are pyramidal cells of the CA3 and CA1 regions of the hippocampus. - 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. -------------------------------------------------------------------------------------------------- The seminal discovery of O'Keefe and Dostrovsky was that the discharge of many hippocampal neurons is location specific; they fire rapidly only when the rat's head is in a restricted part of the recording apparatus. Such units, now called "place cells" (O'Keefe, 1976), are pyramidal cells of the CA3 and CA1 regions of the hippocampus. -------------------------------------------------------------------------------------------------- In the view of O'Keefe (see, for example, O'Keefe, 1991), this map is a Euclidean representation of the environment; it allows the computation of distances and angles in the environment, thereby permitting solutions to spatial problems. The position taken in this paper is in fundamental agreement with O'Keefe: We think that place cells reveal a hippocampal map. Our primary purpose is to show that there is a realistic way in which synaptic connections in the hippocampus can store a map-like representation of the environment. By "map-like" we mean that the representation can be used to solve specific, difficult spatial problems. In this scheme, the mapping information is stored in the strengths of CA3 --> CA3 synapses that connect pairs of pyramidal/place cells. [Storing Mapping Information] The central idea in this paper is that key information in the hippocampal map, namely, distance in the environment, is represented as the strength of Hebbian synapses (embodied as N-methyl-D-aspartate [NMDA] based, long-term potentiation [LTP]-modifiable synapses) that connect place cell pairs. Specifically, we propose that the strength of a synapse made by a pair of place cells is a decreasing function of the distance in two-dimensional (2-D) space between the firing fields of the cells (Muller et al., 1991). If there 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 one field to the other, and not the Euclidian distance. -------------------------------------------------------------------------------------------------- If the cells are connected by a Hebbian synapse, the short intervals between pre- and postsynaptic spikes are expected to cause increased synaptic strength. -------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------- Note that synaptic strengths are modified in the desired way during exploration and no explicit teaching mechanism is required. -------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------- If distance in the environment can be encoded as synaptic strength, it is reasonable to ask if such information is sufficient to implement a cognitive map. -------------------------------------------------------------------------------------------------- to explain spatial or nonspatial operations of the hippocampus, we focus on the recurrent or lateral synapses that are made between pairs of CA3 place cells Other possibilities are - the contacts from entorhinal cortex (EC) cells onto dentate granule (DG) cells - the Schaffer collateral projection from CA3 to CA1. Synaptic classes EC -> DG and CA3 -> CA1 are both considered to be NMDA-based, LTP-modifiable synapses with Hebbian logic. There is also growing evidence that CA3 -> CA3 synapses also show NMDA-based LTP There are still other candidate synaptic classes. The mossy fiber projection from dentate granule cells to CA3 pyramidal cells also shows LTP, but the biophysics and possibly the logic of the modifiability are different (Jaffe and Johnston, 1990). Moreover, pathways from entorhinal cortex directly to CA3 and CA1 exist and show LTP (Buzsaki, 1988), although the nature of the modifiability is not well characterized. Given this embarrassment of riches, there are several reasonable ways in which mapping information might be distributed across synapses. Nevertheless, we believe that there is a major advantage to focusing on the CA3 -> CA3 network. ======================================================== mapping information might be distributed across synapses ======================================================== [The Connectivity of Networks and the Connectivity of Space] In the feedforward network ... there is no way to make paths in the environment correspond to paths in the network. Thus, smooth paths in neural space need not correspond to smooth or even possible paths in the surroundings. In our theory of synaptic strengthening, strength remains near zero if the firing fields of two cells are sufficiently far apart. This does not mean that the representation of map can generate direct, efficient paths,... [What Spatial Problems Must Be Solvable to Call a Representation a Map?] [Searching for Paths in a Model of the CA3 Recurrent Network] In Methods, an algorithm is described that allows optimal paths to be found according to sequences of synaptic weights. -------------------------------------------------------------------------------------------------- The critical question is then whether optimal paths in neural space are also optimal paths in 2-D space. -------------------------------------------------------------------------------------------------- A directed graph (A --> B does not imply B --> A) is said to be "strongly" connected if it is possible to walk from any node to any other node using a sequence of properly directed edges. We mention two other related notions of connectedness. A directed graph is called "unilaterally" connected if, for any two nodes, it is possible to walk from at least one of them to the other using a sequence of properly directed edges. Finally, we say that a directed graph is "weakly" connected if one can walk from any node to any other node not necessarily respecting the direction of edges; [METHODS] [Experimental Foundations] 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. * 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). - Muller, R.U., and J.L. Kubie (1987) "The effects of changes in the environment on the spatial tiring of hippocampal complex-spike cells. J. Neurosci. 7:1951-1968. => not available In brief, place cell recordings were made as rats ran around in walled apparatuses of simple geometric shape. * Place cell properties. * Building the network (1) - create N cells every cell should have same number of n outgoing edge (only one connection to one other neuron, no me to me) - test for strong connectivity it should be possible to walk from any node to any other node (A surprisingly low divergence is necessary to virtually ensure that the network is strongly connected) each cell is assingned a location in 2-D space at rondom real field is represented by pixcels then synapse is given a strength according to the distance between the field centers * Finding optimal paths in synaptic resistance space and 2-D space. [Results] [IS the CA3 Network Strongly Connected?] The stated properties are compatible with the idea that the strength of a Hebbian synapse that connects a pair of place cells should decrease with the distance between the firing fields of the cells (Muller et al., 1991a). Because the real strength-distance function is unknown (if indeed one exists), synaptic strengths in networks are calculated from one of several explicitly stated functions of the distance between firing fields. Several effects of varying the strength-distance function, or, more correctly, the reciprocal "resistance-distance" function, are shown in Results. The reason for using resistance-distance functions is stated below. There is no question that this assumption is wrong in detail since it is agreed that the density of recurrent CA3 -> CA3 connections varies with the position of the presynaptic cell in the pyramidal cell layer (Miles and Wong, 1986; Ishizuka et al., 1990; Li et al., 1993; Bernhard and Wheal, 1994). The same workers agree, however, that recurrent connections are widespread and massive. In the absence of a specific role for the partial specificity of connections, our main interest is in whether networks of the size of CA3 and connection density of CA3 are likely to be strongly connected, i.e., that it is possible to "walk" along a chain of cell ~ synapse ~ cell ~ synapse, etc., and reach any cell from any starting cell. As stated in the Introduction, strongly connected networks share with unobstructed 2-D space the property that it is possible to get from any place to any other place. This property underlies our analysis, and, accordingly, it is the first topic dealt with in Results. Since each cell is a place cell, any path in the graph corresponds to a path in 2-D space. [Fig. 4 Caption] 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 theme 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 connection is determined by distance in 2-D space. [Results] Our argument that the CA3 network is strongly connected rests on calculations done on random graphs 000 with 250,000 nodes, which is about the number of cells in the CA3 region of rats (Amaral et al., 1990). For each of 20,000 random graphs, the divergence was initially set to 1 and the graph was tested for strong connectedness. If the graph was not connected, the divergence of each node was increased by 1 and connectedness was again tested. This task is possible with relatively limited computing resources because graphs with 250,000 nodes are virtually certain to be strongly connected with divergences as small as 20 or so. An astronomically large majority of all connection patterns are strongly connected with 250,000 cells and a divergence of 6,000. Put another way, if only ~0.5% of the connections were random, strong connectedness is virtually ensured. [Are There Enough Synapses in CA3 for the Proposed Representation ?] [Problem 1: Finding a Straight Path between Any Pair of Points in Free Space] The fundamental mapping problem is to calculate the shortest path between any pair of points in the environment, using information stored in the map. Each node (cell equivalent) of the graph is associated with a position in 2-D space and is therefore an analog of a place cell. 2-D space is divided into equal area square pixels, and equal numbers of cells are assigned to each pixel. This mimics the even distribution of firing fields over the surface of the apparatus (Muller et al., 1987). At least one node is associated with each pixel. Given 250,000 CA3 cells, ~330 cells would be assigned to each pixel if we attempted to preserve the size of the neural system. The divergence determines not only whether the graph is strongly connected, but also how closely shortest paths correspond to straight lines. The best available path in 2-D space is found as follows. - First, a starting point and a goal point are selected. - Next, the graph is searched for a node whose associated 2-D position is at the start; there must be at least one because every pixel has at least one associated node. - If several nodes are found, one is randomly selected. - In the same way, a node whose 2-D position is at the goal is selected. - Next, the shortest path between the nodes is found by minimizing the sum of the synaptic resistances along the path. - Finally, the path in the network is converted to a path in 2-D space by listing the positions of the nodes. It is then possible to draw the 2-D path or to calculate the total distance traversed by summing the distances from node position to node position. my-paper ================================================================================================== The topic of animal's navigation by hippocampus in brain and it's neurophsiological simulation are beyond of this article. So let's take a breif look at it. To simply put, animal creates a cognitive map in its hippocampus in its brain. Cognitive map is a map like representations of animal's surrondings. Here our consideration is principally based on Muller et al (1996) among others in which long-term potentiation, place cell phenomenon, and recurrent connections among CA3 pyramidal cells play an important role for animals path-finding in its environment. O'keefe and Dostrovsky (1971) found that pyramidal cells of the CA3 and CA1 regions of the hippocampus, now called place cells fire rapidly only when rat is in a specific area. It is proposed that the distance between the firing fields of connected pairs of CA3 place cells is encoded as synaptic resistance (reciprocal synaptic strength). 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. Our basic finding is that minimizing the sum of the synaptic resistances along a path in neural space yields the shortest path in un-obstructed two-dimensional space, so long as the connectivity of the network is great enough. Such nervous system seems to be developed in the course of Darwinian evolution since it is useful to have an ability to know where you are and how to get to where you wnat to go. In this system, the mapping information is stored in the strength of synapses from CA3 to CA3 -------------------------------------------------------------------------------------------------- As the rat runs from its starting position to the goal during learning, place cells along the route fire in sequence so that synapses connecting place cells with adjacent fields are strengthened. When the rat is again at the starting point of such a sequence, the whole series of place cells can be replayed, therby allowing the rat to follow the path from cell to cell so that it eventually gets to the goal. In contrast, connections between place cells with widely separated field would be weak because such cells cannot fire together. This encoding of distnace is a consequence of connectiong place cells by Hebbian-like syanpse and could occur in the recurrent pyramidal cell to pryramidal cell connection in CA3. This model has the advantage that it does not rely on previously taken paths: it permits the animal to find an optimal path from any starting point to any goal location in a familiar environment.