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Basic concepts Chart concept. A chart is defined here as an imaginary arrangement of a population of place cells on an abstract plane, such that when this plane is appropriately mapped onto an environment, each cell appears to be located at the image of the absolute maximum of its firing rate distribution. Therefore, the total activity distribution on the chart appears to be localized around the image of the animal's head. Moreover, whenever there exists a planar arrangement of cells such that the activity distribution appears to be focused at a particular location within it, it is called an active chart.
Volume 17, Number 15, Issue of August 1, 1997 pp. 5900-5920
Copyright ©1997 Society for NeurosciencePath Integration and Cognitive Mapping in a Continuous Attractor Neural Network Model
Alexei Samsonovich and Bruce L. McNaughton Arizona Research Laboratories Division of Neural Systems, Memory and Aging, The University of Arizona, Tucson, Arizona 85749
ABSTRACT
A minimal synaptic architecture is proposed for how the brain might perform path integration by computing the next internal representation of self-location from the current representation and from the perceived velocity of motion. In the model, a place-cell assembly called a "chart" contains a two-dimensional attractor set called an "attractor map" that can be used to represent coordinates in any arbitrary environment, once associative binding has occurred between chart locations and sensory inputs. In hippocampus, there are different spatial relations among place fields in different environments and behavioral contexts. Thus, the same units may participate in many charts, and it is shown that the number of uncorrelated charts that can be encoded in the same recurrent network is potentially quite large. According to this theory, the firing of a given place cell is primarily a cooperative effect of the activity of its neighbors on the currently active chart. Therefore, it is not particularly useful to think of place cells as encoding any particular external object or event. Because of its recurrent connections, hippocampal field CA3 is proposed as a possible location for this "multichart" architecture; however, other implementations in anatomy would not invalidate the main concepts. The model is implemented numerically both as a network of integrate-and-fire units and as a "macroscopic" (with respect to the space of states) description of the system, based on a continuous approximation defined by a system of stochastic differential equations. It provides an explanation for a number of hitherto perplexing observations on hippocampal place fields, including doubling, vanishing, reshaping in distorted environments, acquiring directionality in a two-goal shuttling task, rapid formation in a novel environment, and slow rotation after disorientation. The model makes several new predictions about the expected properties of hippocampal place cells and other cells of the proposed network.
Key words: hippocampus; CA3; place cells; head direction; dead reckoning; path integration; idiothetic; allocentric; spatial learning; cognitive map; attractor; integrate-and-fire
INTRODUCTION
It is known from individual and multiple parallel recordings of single-neuron activity in freely moving rodents that the dynamics of the rodent hippocampus during active locomotion in a planar maze is essentially two-dimensional in its space of states; furthermore, it is a two-dimensional model of the animal's motion on the maze (O'Keefe and Dostrovsky, 1971
; O'Keefe and Nadel, 1978
Fig. 1. Activity packet on a chart constructed from the experimental data of Wilson and McNaughton (1993)2. The variance of the distribution shown on the figure is ~0.15 M, which is consistent with the observation that ~10
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As experimental data show, the activity packet has the following dynamical properties. (1) It persists and retains its shape during active locomotion regardless of motion parameters and regardless of the stability and immediate availability of sensory cues, e.g., in complete darkness; (2) under certain conditions, the whole representation, rather than a fraction of it, can be spontaneously remapped, without distortion of the intrinsic structure of the chart, and this new mapping may subsequently persist; (3) on entering a novel environment, a new chart (i.e., a new spatial code) appears immediately and normally does not undergo subsequent topographical modifications after exploration or changes in environmental stimuli; and (4) under different behavioral conditions, different charts for the same environment are expressed in the hippocampus, showing uncorrelated arrangements of common place cells.
In the present paper, the term "spatial" is used in a restricted sense to refer to location in a plane and does not include yaw. Several proposals have been made regarding possible explanations of the spatially selective firing of hippocampal pyramidal cells (Zipser, 1985
MATERIALS AND METHODS
To define the MPI model of the hippocampus, the necessary concepts must be introduced, and then the components of the MPI scheme proposed by McNaughton et al. (1996)
In the models proposed by Muller et al. (1991
In different environments, and even in the same environment under different behavioral paradigms or other conditions, alternative charts may be active in which the spatial relations among place fields of the same place cells may be different. Typically, there are no significant correlations between the alternative charts for the whole population of recorded place cells (O'Keefe and Conway, 1978
Attractor map concept. According to the cognitive map concept (O'Keefe and Nadel, 1978
An attractor map concept (cf. Ranck, 1992
An attractor map can be defined as a two-dimensional, quasicontinuous set of attractors (associated with a particular environment or not), with the following dynamical property: the mobility threshold for transitions between neighboring attractors is negligibly small (tends to zero, with the number of units tending to infinity) as compared with the finite threshold for jumps between distant points or outside of the attractor map. Basically this is a generalization of the one-dimensional continuous attractor concept (Amari, 1977
Map-based path integration concept. It is clear from behavioral studies that mammals and many other species possess highly developed path integration capabilities (Mittelstädt and Mittelstädt, 1980
The planar path integration concept involves (1) selecting a physical reference frame, implying the reference location, the reference direction, the metrics, and the clock, and (2) performing integration of the velocity vector over time in this reference frame to update the currently represented, or "perceived," coordinates. This implies two necessary building blocks: an internal representation of the planar coordinates, maintained independently of immediate exteroceptive stimuli, and a mechanism of its updating based on idiothetic information.
According to McNaughton et al. (1991)
The map-based path integration concept is represented by a possible MPI scheme of the hippocampal spatial representation system (Fig. 2) (McNaughton et al., 1996 (an array representing the angular velocity of the head), M (an array representing the speed of motion), R [an array sensitive to both horizontal head direction (yaw) and the angular velocity], and I (an array that receives inputs from P, H, and M). The core of the scheme is the P-I path integrator fragment. The foregoing components and the model are described below. For a simpler treatment of the path integration principles in the present scheme, the reader is referred to McNaughton et al. (1991)
Fig. 2. Hippocampal path integration system (according to McNaughton et al., 1996
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Identification of components V. The component referred to as the V network (Fig. 2) can be identified with sensory association cortex, which provides high-level representations of the local sensory information and sends its output to the hippocampus mainly via the entorhinal cortex and the perforant path. Under normal conditions, a neocortical representation of multimodal sensory stimuli can be thought of as a function of the animal's current location x and its head direction in the horizontal plane (yaw) given by the angle
. This function is presumably smooth. For the sake of parsimony, a mnemonic mechanism in the sensory system (e.g., imagery) involved in maintaining the activity of local sensory representations is not assumed. Therefore, in this oversimplified picture the space of states of the V network is three-dimensional (two spatial coordinates plus the yaw angle). The term local view is used here as a shorthand for the entire sensory representation that is typically specific for a particular combination of location and orientation in the environment. Each local view is associated with a corresponding pattern in the P network, because of associative learning in the afferent synapses rather than in the internal P-to-P connections. Different parts of the environment will typically have separate representations in V, thus allowing differential binding of the attractor map coordinates to local cues.
M. The path integration concept implies that the integrator receives information about self-motion (M). Participation of the motor system in the dynamics of hippocampal spatial representations is suggested by the finding that under conditions of movement restraint, both hippocampal place cells (McNaughton et al., 1983
I. To perform spatial path integration, it is necessary to know the speed and direction of motion. In general, hippocampal place cells are known to be relatively nondirectional in a two-dimensional environment, if the behavior does not involve the following of specific routes between discrete reinforcement sites (O'Keefe, 1979
H, H
Actually, the path integration mechanisms proposed in the present paper require information about direction of motion rather than head orientation. Head direction will suffice so long as it remains highly correlated with movement direction. The possibility remains, however, that true "direction-of-motion cells" exist but have not been documented. Such cells could arise from a simple coordinate transformation such as the one that has been suggested by Andersen and his colleagues (Andersen et al., 1985
Knierim et al. (1995) 
Fig. 3. Head direction path integration system (according to McNaughton et al., 1991R, whereas an H cell sends equal outputs to its column in R via WHR. The nonzero WRH connections, also all of equal strength, are established with different angular displacement with respect to their counterparts WHR, depending on the slab of their origin at the R array; namely, this displacement is proportional to the angular velocity represented by the associated H
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Slow rotation of the place-field pattern, which was observed by Knierim et al. (1995)
P. According to the above considerations, the main candidate for the attractor map as a component of the path integrating system is the P network, presumably based on the areas CA3, CA1, and dentate gyrus, with a primary role for CA3 in the origination of the attractor dynamics. Indeed, neuroanatomical data show that CA3 has multiple, long-range excitatory internal connections (Amaral and Witter, 1995
It seems natural to extend the principles underlying the head direction path integration model to two dimensions, taking the locally interconnected two-dimensional array of P cells as the central element; however, experimental data indicate that this extension cannot be made in a straightforward manner (e.g., as was suggested recently by Zhang, 1996
The first problem with a parallel between the spatial and the head direction system is that preferred directions of H cells typically have unique relations to each other, even though the absolute direction selected by an H cell may vary between environments, parts of an environment, and recording sessions (Ranck, 1984
A possible solution to this problem is that the P network may implement many alternative two-dimensional attractor maps at the same time, and a typical P cell participates in a number of these implementations, one of which is selected by the current activity state (McNaughton et al., 1996 Elements and dynamic rules of the MPI model Elements of the above networks were implemented as model integrate-and-fire units interconnected by synapses. In such a network, dynamic variables are "spikes" (S), "EPSPs," or "voltages" (V), and some of the synaptic weights (W) that are slow variables. The latter were mostly assumed fixed in the simulations. The network to which a given variable belongs is marked by a superscript.
Would individual attractor maps not be destroyed by interference? What kind of architecture does this point of view imply? These questions are examined below. Another problem is that of formation of the attractor map. In particular, a new chart can become active in the dark (Quirk et al., 1990
The discrete time approximation is used here, with a time bin = 6 msec, which is bigger than the refractory period, close to axonal plus synaptic delays, and yet smaller than the neuronal integration time (for review, see Shepherd, 1990
There are two ways to achieve this: (1) by adjusting the global inhibition h at each step to match the total number of firing units, and (2) by taking h as a function of the total number of firing units computed at the previous iteration. The latter is consistent with the idea of hidden inhibitory interneurons and may permit the model to exhibit theta oscillations naturally; however, the time bin is too large for the model to be realistic, and it is not the present goal to study the origin of the theta rhythm. For this reason the first approach is adopted.
Therefore, the dynamic equations describing an isolated network of leaky integrate-and-fire units with reciprocal interconnections (e.g., P or H component) can be written as:
(1) Here i is the unit number, N is the total number of units in this subnetwork, t is the discrete time,
(2) is the neuronal integration time, W is the synaptic matrix, and
is the Heaviside step function. The global threshold ht is an implicit function of the set of variables {V} defined by the right Equation 2, where M is the given total number of active units.
Multichart architecture of the P network. Now the multichart architecture of the P component is introduced. Consider all P cells distributed on an abstract plane according to the relative locations of their place-field centers. This arrangement is called a chart; however, there may be multiple such arrangements of the same P cells that are uncorrelated and can be used to represent different environments. Therefore, consider n possible arrangements of the same P cells. In this model they are random permutations of each other. In general, a P cell may not be found on some charts. In the present model, however, it is assumed that each chart is composed of all model units. To obtain the matrix of internal connections WPP of the P network, local interconnections were created on each chart (local in the sense that weights decay rapidly with distance between units on the chart, rather than with the anatomical distance), and then the sum over all charts was taken. The result is a multichart architecture. Although charts, being defined in terms of firing rate distributions, make sense without relation to any connectionist model, the structure of the synaptic matrix is an important feature of the MPI model. It is given by the formula: where rijk is the distance between units i and j on the chart k, or infinity, if at least one of the two units is missing on this chart (also see Muller et al., 1991
(3) is a fixed parameter. The resultant multichart architecture is illustrated by Figure 4.
Fig. 4. Multichart architecture. A, The set of n charts, composed of the same P units. Activity that is well localized on one of the charts (1) looks scattered on other charts (2, 3, n). B, C, Two arbitrarily selected charts with some interconnections between units on them. Local interconnections on chart B (solid lines) appear to be random (nonlocal) on chart C; local interconnections on chart C (dashed lines) are random (nonlocal) on chart B. Therefore, on a given chart, contributions from other charts to the synaptic matrix can be treated, approximately, as random noise.
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Although this model has all-to-all, symmetric, excitatory connections according to Equation 3, the matrix W can be called sparse because most of its elements are very close to zero, if L, where L is the chart dimension. Quantities rijk,
Summary of assumptions of the MPI model. In summary, the proposed MPI model (Fig. 2) is based on the following basic assumptions about the hippocampal formation. (1) The architecture of the P network is preconfigured as a sum of uncorrelated, quasi-two-dimensional architectures. This is most likely to be the architecture of internal CA3 connections; however, other implementations in anatomy would not invalidate the concept. (2) The primary driving mechanism for the activity packet on a chart is based on internal dynamics and is attributable to asymmetry in the connections from I cells to P cells. (3) Activation of I cells is controlled jointly by representations of the speed of motion (M) and head direction (H) and by return projections from the P network. (4) Learning results in selective strengthening of V to P connections. This enables stimulation of the P layer by the V array to determine on which chart and at which location the activity packet emerges on entry into a familiar environment. (5) Connections between H and I cells are also preconfigured and fixed. In other words, each chart has a built-in compass. Because there is only one chart in the H network, the layered structure of the I network must be the same for all charts. Numerical implementation of the MPI model The MPI model described above has been implemented numerically on SUN Sparc-20 and Ultra-Spark work stations as a system of networks of integrate-and-fire units. In the simulations of Figure 9 (see Results) each layer of the P-I system was composed of n = 256 × 192
It follows from the last assumption that although spatial relations between place fields of I cells may be different for different environments, relations between their preferred directions must be the same in all environments and for all representations. This is an untested prediction of the theory. 45,000 model neuronal units distributed in a square lattice on a torus (i.e., a rectangle with periodic boundary conditions).
Fig. 9. Simulation results for the integrate-and-fire modelI. Snapshots in all rows except B and C are taken at a constant phase of the theta rhythm. A, Self-focusing, formation, and propagation of the activity packet in a six-chart network. The network consists of P and I layers. Each pixel on the figure represents a P unit. Each plate consists of 256 × 192 pixels. Boundary conditions are periodic for all charts. The four plates A1-A4 show the four sequential theta cycles that correspond to different stages of spontaneous self-focusing of activity on the chart 1. Spikes arranged according to this chart are represented by red; the background is blue. When the same units are arranged according to chart 2 (not shown), their spikes appear almost uniformly scattered and some are grouped into small patches. B, Simulated phase precession. The self-focused activity packet propagates to the right; the simulated rat location (same on all 4 plates) is shown by the white arrow. Only one chart is represented. B1 through B4 correspond to the four phases of the same theta cycle (0, 90, 180, and 270°). The center of the distribution clearly oscillates in the direction of motion, which resembles the phenomenon shown in C. C, Real phase precession of the activity packet in CA1 reconstructed from experimental data. Color on each plate represents an average firing rate distribution on a chart, where the momentary rat location and head direction is shown by the arrow in the center. High activity is coded by red. The two ends of the arrow are images of the two infrared light-emitting diodes attached to the rat's head, spaced 0.15 m from each other (for details, see Wilson and McNaughton, 1993
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The role of sensory input: accelerated motion and jumps of activity packet The next numerical study (Fig. 9E-G) examined the question of correction of the activity packet position by the visual input. Now the simulated rat moves to the right with a constant velocity. The rat's position (projected onto the chart) is marked by the cross. It differs from the activity packet position from the beginning of motion, because of an initially introduced error. The input to the P layer from V representations is absent in Figure 9E, relatively weak in F, and relatively strong in G. Therefore, the V system has no effect on the activity packet motion in E. (See Materials and Methods.)
In the simulations of Figure 9A,B,E,F,G, the external inputs to I from H and M were assumed fixed, meaning that the model rat velocity was constant. In a more general case (see Fig. 9D) the I array is modulated by the internal representations of velocity in the M and H systems. The P-I system evolves according to the following system of equations constructed on the basis of Equations 1-3: Here t is the discrete time with a 6 msec time bin
(4) = 200 pixels, rik is the fixed coordinate of unit i on a chart k, and µ is the efficacy of V-to-P connections, which was zero in the simulations of Figure 9A,B,D,E, 0.01 in F, and 0.1 in G. The last term in the second equation of (4) describes modulation of the I array by the H system (v is a unit vector pointing in the direction of motion).
Fig. 5. The continuous model. The dynamic state of the whole path integrator system is described by the following variables: the currently selected chart number k, the activity packet coordinates on this chart (y1, y2), and the "perceived" direction of motion i, dynamically stored in the H array (the symbol i
.
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The set of constant, random vectors {bi} is the same as the set {bj} in Equation 8. The whole I component can be viewed as containing cells with a continuous range of gaussian-like directional tuning functions at each chart coordinate. An interval on this range is selected by the "perceived" direction of motion v. The M, H, and V systems represented in Equation 4 by MI, v, and x, respectively, were assumed to be consistent with the model rat motion and were not simulated explicitly. The total activities MP and MI are periodic functions of time: with A = 1000, B = 500, C = 250, D = 200: the relative values A/B and C/D were adjusted to fit the experimental data on the population theta rhythm in CA1 and in DG, respectively (Skaggs, 1995
(5)
All synaptic connections W in Equation 4 are excitatory; the role of interneurons implicitly present in the model consists of adjustment of the global thresholds h at each iteration, according to Equation 4. Thus, in the terminology of Amit (1989)
(6)
(7) Here
(8) is the static synaptic noise (centered at 0 and <1 in absolute value) that results from the numerical algorithm (see below) and simulates the anatomical irregularities in the connectivity matrix; n is the number of charts; the vector rijk connects the nodes i and j on chart k; the constant, random gaussian vector bj, which represents the asymmetry in the I-P connections, is centered at zero and has a variance of
Fig. 10. Simulation results for the integrate-and-fire model
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In the simulations of Figure 10 the same model was implemented, but with the following differences in detail. Instead of placing exactly one unit at each node of the square lattice on each chart, as in Figure 9, each unit i was assigned random coordinates rik on each chart k; all vectors rik were generated independently of each other, with a uniform probability distribution. In addition, an explicit division of the I array into L = six head direction layers (Fig. 2) was used, so that the vector b in Equations 4 and 8 was the same for all I units in each layer and was rotated by 60° from layer to layer. The fixed absolute value of this vector b = six pixels was the same for all I units. The number N of P units was the same as the number of I units in each head direction layer; therefore, the total number of I units was six times larger than the number of P units in this implementation. Only one I layer was active in the simulation of Figure 10C. The static synaptic noise Reduced continuous model Because the simulation of complex experimental results using the integrate-and-fire scheme was severely limited by computational constraints, an alternative strategy was to characterize the dynamical state of each component by a set of "macroscopic" variables, instead of a detailed description in terms of variables S and V, and to formulate dynamic laws in terms of these "macroscopic variables." The rationale for this approach is described below.
To implement the synaptic matrix numerically, an additional, intermediate array and a two-dimensional filtering procedure were used. At each iteration, for each chart, the output of P and I units was added to the intermediate layer, according to the unit arrangement on this chart. After that, a noisy gaussian filtering procedure was applied, which consisted of redistribution of spike densities among lattice cells in the intermediate layer, and the result was taken as input to P and I units, together with external inputs from V, H, and M according to Equation 4. Two parallel filtering methods were combined: random gaussian filtering that created additional static synaptic noise and the standard gaussian filtering with the kernel (0.09, 0.24, 0.34, 0.24, 0.09). This algorithm substantially reduced the necessary computer memory and enabled the performance of O(N · n) operations per iteration instead of O(N2), which would be the case for an explicit implementation of synaptic connections. On the other hand, the number of charts that could be simulated was limited by the computation time.
For the present purpose, the critical results of the integrate-and-fire model of one- and multichart networks (see Results) can be summarized as follows. (1) Under the given conditions, the distribution of activity in a model network possessing an attractor map (P or H network) is described by an activity packet, which retains its shape and moves smoothly, remaining on the same chart. (2) In the two-layer model (I-P fragment), it is possible to control the velocity and direction of motion of the activity packet with reasonable precision via modulation of activity of the I array by the outputs of the M and H arrays. The activity packet in the R-H system can be similarly controlled. (3) The activity packet position on the chart can be smoothly corrected by an additional stimulation of the substrate array (P or H) by the output of the V array. (4) With the introduction of a strong local stimulation of the substrate array, distant from the current activity packet location, the activity packet may jump, with some probability, to the stimulated area on the same or different chart.
These results allow the introduction of a simple, continuous description of the above components. In this approach the state of the H network is described by one angular variable
(9) where µ and
(10) are constants that represent activity packet mobility; i
is acting on y;
U is the effective attractive potential for the activity packet, and
and
In the unchanged familiar environment
It is assumed here that in the unchanged environment, U is a gaussian, with some variance
(11)
(12)
Next, the differential binding assumption is made. Behavioral studies indicate independent binding of the cognitive map to individual local parts of the environment. For example, Collett et al. (1986)
Therefore, it is assumed that in a geometrically changed environment, the gaussian (Eq. 11) is split into several components, each of which is bound to some rigid part of the environment, and their relative strengths depend on the rat's position x: Here ai is the vector of displacement of the ith part of the environment, and the coefficients Ci are smooth functions of x. They reach their maxima at those environmental locations to which the corresponding terms Ui are bound.
(13)
A similar assumption about the stimulation function was made by O'Keefe and Burgess (1996)
To complete the definition of the model, the initial conditions must be specified. It is assumed that when the rat finds itself in a familiar environment (modified or not), the activity packet jumps under the influence of a strong external stimulus to the location on the corresponding chart that is mostly stimulated by V, that is, to the absolute maximum of U taken over all charts. The same assumption is made for the H system: the first perceived allocentric head direction in the entered environment is determined by the strongest visual cue:
(14) This "initial condition rule" applies not only at the moment of entry or waking up. It is assumed that under certain circumstances the state of the path integration system can be reset during active running. This means a jump of the activity packet to another location on the same or a different chart. Although there is only one chart in the head direction system, chart switching may happen in the P-I system and appears to do so during shuttling tasks on linear tracks in which the rats follow routes back and forth between goals. In such cases, the charts representing journeys in opposite directions appear to be different, leading to an appearance of directional dependence of place fields (McNaughton et al., 1983
where y is the current activity packet location on chart 1; U1 is taken at y on chart 1, and U2 is taken on chart 2 at its absolute maximum (the expected new activity packet location on chart 2 is at the absolute maximum of U2); p and Ut are constants. The same formula describes possible jumps of the activity packet between two distant points on the same chart, if one takes U1 = U2 = U.
(15) Understanding place fields in stretched and shrunken environments To develop an intuitive understanding, it is useful to consider a further simplification of the model (represented by Eqs. 9-16). The dynamics of the activity packet in this simplified model is described by Equation 9, if the last two terms in it are neglected: the path integration term and the noise
The shape of the activity packet is not a dynamical property in this reduced model; however, it affects place-field shape and dimension and therefore needs to be specified. Thus, a fixed gaussian shape of the activity packet is assumed, with the variance , and if the activity packet is centered at y, then the relative firing rate for a cell located at a coordinate z on the currently active chart is:
Equations 9-16 define the continuous model that was implemented numerically to reproduce the basic experimental results, including the results in unstable environments described below. Learning within this model will be considered afterward.
(16) where l is the length of the stretched/shrunken environment and 2a is the amount of stretching, which becomes negative in the case of shrinking: if l0 is the original length, then l = (1 + 2a)l0. This situation is depicted in Figure 6.
(17) 
Fig. 6. Splitting of the stimulated region on the chart in linearly shrunken and stretched environments. a, b, Original environment; c, d, shrunken environment; e, f, stretched environment. a, c, e, The rat is in the first half of the journey; b, d, f, the rat is in the second half. The activity packet location is marked by the black dot. The amount of stretching a is defined as shown in e. On each figure, the top bar represents the environment, and the bottom bar represents the chart. Given two reference points L and R (walls, landmarks, reward sites, etc.), the current rat location x is associated with two locations on the chart, yL and yR, according to the original map anchored at L or UR, respectively: yL = x + a and yR = x
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Then the activity packet coordinate y, as a function of x and a, is given by the transcendent equation: This equation for y has one or two stable roots, depending on the values of x and a. Figure 7 shows the activity packet coordinate y as a function of x, when x changes monotonically from
(18) 
Fig. 7. Transformation of the map in distorted environments. These plots show solutions to Equation 18, i.e., stable (with respect to corrections by visual information) activity packet coordinate y as a function of the rat coordinate x in an environment of the variable length l = (1 + 2a)l0 in the attractor map model approximation. a, Shrunken environment; b, stretched environment. The straight diagonal line represents the map in the unchanged environment (a = 0). Values of the parameters are the variance of the "visual gaussian"
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Numerical implementation of the continuous model The model (represented by Eqs. 9-16) was implemented numerically using the simple Euler scheme, which is sufficiently accurate for the qualitative purposes of this model. Typical parameter values were
Suppose a P cell is located in the middle of the chart (y = 0), which is mapped to the middle of the box (x = 0). Then, in the original environment (Fig. 7, diagonal line) this cell will fire near one location only: x = 0. The size of the place field is determined by the size of the activity packet 2>1/2 = 0.002; the value of
Fig. 11. Simulation results for the continuous model. a-f, Predictions for the place-field modifications in two-dimensional environments computed according to Equations 9, 10, 12-14, 16. The two reference points L and R (Fig. 6) are the left and right walls of the box on a-f. In the case of stretching (a-c) studied by O'Keefe and Burgess (1996)
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Fig. 12. Numerical counterpart of the Sharp et al. (1990)
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Finally, the continuous model is extended on the basis of Equations 9 and 10 to incorporate learning mechanisms. To do that, the H circle, the P plane, and the V volume (Fig. 8) were divided into lattice cells to which virtual Boolean synapses are attached. In this sense the model is not continuous anymore but is still referred to here as "continuous" to distinguish it from the original network of integrate-and-fire units. Potentials U and u are computed as follows. Given the coordinates (x, 
Fig. 8. Binding of sensory features to charts. This figure illustrates the learning rules of the model. Each P unit is virtually connected to any V unit, and the list of nonzero connections is stored in each P unit, and similarly for the H units. There is a limit of connections per unit in each array. Given the activity distribution in V centered at the unit 1, the activity packet in P centered at the unit 2, and the activity packet in H centered at the unit 3, the connections 1-2 and 1-3 can be potentiated (i.e., added to the list) with some probability. At the same time, connections 1-5, 4-2, 1-7, and 6-3 can be depressed (i.e., deleted from the list).
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In terms of the original model, the matrices of synapses WVP and WVH are implemented as sparse matrices of Boolean values. According to the selected rules, the number of nonzero synapses per unit is limited for all units. In the learning regime, the probabilities of a synapse switching at a given iteration from zero to one (P+) and from one to zero (P
(19) where
(20) is the rate of learning; mV and mP are the total numbers of synapses per unit (i,j) in V and in P; mVmax and mPmax are their maximal allowed values;
is the Kronecker
y] and [(x,
RESULTS
Simulated dynamics of the integrate-and-fire MPI model
Observation of a multichart attractor map First, the simulation exhibited self-focusing of activity on a chart in the P component (Fig. 9A, 10A,B). In this architecture, within a certain parameter range, the only dynamically stable configurations of cell activity are gaussian-like activity packets on one of the charts. Such an activity packet was found to be stable in the sense that it maintains its uniqueness and shape, and given constant external input to the P-I system (no input from V) it moves on the chart with a persistent velocity. An activity packet forms spontaneously when the network is started with random activity. The process of self-focusing in the network of 45,000 units in each layer and with six charts is represented on Figure 9A; the variance of the activity distribution as a function of time for a network of 300,000 units in each layer with 20 stored charts is represented on Figure 10A; and the same for a network of 30,000 units with 100 and 200 charts is represented on Figure 10B. Figure 10A shows relatively fast relaxation that lasts less than one theta cycle: ~40 msec. It takes 0.8 sec for 100 charts (Fig. 10B, solid line), and it does not happen at all in a network of 30,000 units with 200 charts (Fig. 10B, dashed line). The activity packet remains stable on the same chart while exploring the chart for at least 6 sec (Fig. 10D).Simulated propagation of the activity packet Next, the details of the activity packet dynamics under the condition of constant external input to the I component, corresponding to a fixed "perceived" direction and speed of motion, were studied. The result is shown in Figures 9B and 10C; the latter represents propagation of the center of the distribution in a network of 300,000 units with 20 charts; the variance of the distribution oscillates with theta rhythm at the same level as in Figure 10A after self-focusing. The related Figure 9C was constructed from the actual experimental data of Wilson and McNaughton (1993)
These observations indicate the existence of an attractor map in the P component: the two-dimensional set of fixed-point attractors at zero input to the I component corresponds to the two-dimensional set of possible activity packet locations on a chart. The estimated storage capacity in terms of the maximal number of stable charts stored in the same network is thus ~0.004 × N. A similar value for a simpler model was obtained analytically by Samsonovich (1997)
Comparison of the two figures (Fig. 9B,C; see also Fig. 10C) shows that the model reproduces qualitatively the phase precession phenomenon, which can be defined as follows: the firing phase of hippocampal neurons relative to the local theta rhythm advances systematically through almost 360° as the rat passes through the place field of each cell (O'Keefe and Recce, 1993
Thus, the phase precession phenomenon occurs as a byproduct of path integration in this model. This mechanism is quite different from that suggested by Tsodyks et al. (1995) Simulated path integration Next, the activity packet was allowed to perform path integration (Figs. 9D, 10D) in a circular motion, in the absence of input from V. The simulated rat moves counterclockwise around a circle. The activity packet repeats this motion with a certain degree of error, which accumulates with time (here, by assumption, the starting position of the activity packet on the chart coincides with the image of the model rat's starting position in the environment). This result shows that the velocity and the direction of motion of the activity packet can be controlled effectively by appropriate stimulation of the I array, and therefore the activity packet position can be accurately updated by idiothetic information; however, after a few seconds, a small correction of the activity packet position becomes necessary.
As can be seen in Figure 10C, the average group velocity of the activity packet remains constant given a constant input to the I layer. In addition, over multiple trials, the average speed of motion appeared to be a monotonic function of the I layer activity.
It was observed that the addition of a gaussian-shaped stimulation to the P layer from V changes the activity packet velocity, thus correcting the activity packet position. Namely, a smooth acceleration of the activity packet occurred at a relatively weak V input (Fig. 9F), and an abrupt jump to the stimulated region occurred when V input was relatively strong (Fig. 9G). Both of these phenomena have recently been observed experimentally (Gothard et al., 1996 Continuous model: reproduction of experimental results
Place fields in stretched and shrunken environments O'Keefe and Burgess (1996)
In contrast, Gothard and colleagues (1996) performed a set of experiments involving shuttling between two food sites on a linear track. During the experiment, the distance between reward sites was reduced by varying degrees on a random schedule over trials. As a result, place fields became compressed, but not always in the right proportion, and in some cases disappeared.
It is remarkable that when two place fields that originally did not overlap became overlapped in the shrunken environment, the two cells actually did not fire simultaneously, as shown by cross-correlation analysis (Gothard et al., 1996
In both cases (Gothard et al., 1996
First, numerical results for the continuous model (represented by Eqs. 9-16) are presented, with the potential U given by Equation 17. The predicted place-field modifications computed according to these equations for the cases studied experimentally by O'Keefe and Burgess (1996)
The effect of finite noise is broadening of the place field and later appearance of the hysteresis (not shown). Removal of the path integration term changes nothing in the limit of slow motion, which is the only case when the above simplified consideration (Fig. 6) may be valid. In the opposite limit of fast motion, however, path integration becomes necessary to speed up the activity packet, which because of its finite mobility under a given driving force cannot follow the stimulated area without an additional driving force.
The two components of the doubled place field (Fig. 11c) are highly directional: each component fires when the rat is facing the more distant wall. This directionality results from two mechanisms: (1) the hysteresis in activity packet motion on the chart (Fig. 7) and (2) the path integration together with "position fixes" (McNaughton, 1996
In the case of a shrunken environment (Fig. 11f), the place field vanishes instead of doubling; however, doubling could occur with slightly different coefficients in Equation 17, as well as according to a pure path integration mechanism, depending on the location of the field (McNaughton, 1996
If a cell has its place field closer to one end of the box, then, as can be derived from Figure 7, the place field may be doubled with increased stretching or not doubled at all if its location is very near the wall. In the case of shrinking, such a place field may acquire directionality without doubling.
The question of directionality requires special attention in the case of the Gothard et al. (1996)
Place-field behavior on the shrinking rail (Gothard et al., 1996 , depending on the direction of motion); and (3) the moving part of the environment, the box, was assumed to have limited visibility described by an exponentially decaying rather than linear coefficient in Equation 17, with the characteristic length of 0.3l.
The results (Fig. 11g,h) reproduce the principal observations: vanishing and shifting of place fields, including "fractional slope" place fields, i.e., place fields that show smaller displacement than that of the box. The relative influence of the moving box on place field location in this case is different for inbound and outbound journeys, in agreement with the experiment. This difference illustrates the effect of path integration. Dependence of place field location on the entry site By assumption, on entry into a familiar environment, the activity packet appears on the chart that has been associated with this environment, and at the location that is mostly stimulated by the V representation of the local view. Thus, if the position of the stimulated domain is determined by a cue card and the cue card is doubled, then there will be two stimulated domains on the chart. The stimulation originating from the most clearly visible cue card at the moment of entry will be stronger and therefore, according to the assumption of the model, will determine the activity packet location on the chart. Subsequently, place fields will remain bound to the selected cue card. This implies that the positions of the place fields may depend on the entry site and/or the orientation of the rat when first placed into the cylinder. This corresponds to results observed by Sharp et al. (1990)
Path integration has a clear influence on place fields in this case. It becomes necessary for the activity packet to move when the limited range of visibility of the box behind the rat is taken into account by introducing a finite-range rather than exponential coefficient into (17), although in this case the activity packet is accelerated by the other stimulated area on the chart, without path integration. It is unlikely, however, that this mechanism of acceleration by chance would result in the correct activity packet velocity. 
Fig. 13. Conflict between visual cues and path integrator in the case of disorientation in a familiar environment. A, The activity packet and the gaussian (the localized stimulation from the V network) get together at the center of the circle. The activity packet trajectory is y(t); the trajectory of the gaussian (which is the image of the rat trajectory) is x(t). The activity packet would be captured immediately in the absence of path integration. Knierim et al. (1995)
[View Larger Version of this Image (12K GIF file)]
Learning a novel environment Following the scheme described in Materials and Methods, the two synaptic matrices WVP and WVH with sizes of 400,000 × 90,000 and 400,000 × 10 were implemented. The maximal number of nonzero synapses per unit was 1 for the V array, 3 for the P array, and 40,000 for the H array; the value of
Fig. 14. Simulation of learning of a novel environment. a-d, Numerical counterpart of the Wilson and McNaughton (1993)
[View Larger Version of this Image (87K GIF file)]
First, the Wilson and McNaughton (1993)
In agreement with the experimental results, the average error in the new part of the environment was higher than in the familiar part at the beginning of exploration and was reduced to the same level after a certain learning period. At the same time, the average error in the familiar part remained approximately on the same level before, during, and after exploration of the new part.
In addition, the results of learning with (Fig. 14e) and without (Fig. 14f) the path integration term in Equation 9 were compared, using a "random start" for WVP and WVH. Without path integration, the activity packet moves on the chart, as the animal moves in the environment, being guided by V representations via preexisting V-to-P connections (which initially were taken as random). This process, which is similar to the Kohonen learning scheme, after a given period of time (which is too short for the Kohonen learning) results in formation of a "map," which is not one-to-one. The results of the simulation show that despite the preexistence of an attractor map in P, appropriate binding of this attractor map to the environment requires a path integration mechanism.
DISCUSSION
Principal findings
The principal findings from the integrate-and-fire model concern the formation, shape, stability, oscillations, two-dimensional controllable motion, and controllable jumps of the activity packet in the proposed multichart MPI model. These results are obtained on the basis of numerical simulations of networks of leaky integrate-and-fire model neurons in discrete time, with predetermined total activity.
The principal finding from the continuous model is the reproduction of several basic experimental facts about pace fields, including doubling, vanishing, reshaping in distorted environments, acquiring directionality in a two-goal shuttling task, rapid formation in a novel environment, and slow rotation after disorientation. Simulation results strongly suggest that a path integration mechanism is involved in the foregoing processes. Significance
The theoretical framework proposed here provides a plausible explanation for how hippocampal place-cell activity may be updated by self-motion cues and how self-motion itself may provide the metrics for representation of spatial relationships. It also provides new insight into how multiple "maps" may be stored within the same synaptic matrix with minimal interference. A multichart architecture implies that the activity of a given place cell has meaning only in the context of the ensemble of other cells that are active with it at a given location on a given chart. In particular, the cooperative interactions that lead to an activity packet suggest that it is misleading and, in general, not useful to think of individual place cells as representing external features at all, in the manner that one thinks of sensory neurons. The multichart architecture easily explains the generally all-or-none "remapping" effects that have been observed in experiments involving cue rearrangements and deletions. Comparison of the simulation results with available experimental data unambiguously supports the underlying general theoretical framework, i.e., the activity packet self-localization property, the multichart property, and the path integration property. The theoretical framework itself provides new insight into how path integration and local view-based mechanisms may interact to update hippocampal spatial representations, and it begins to explain some of the peculiar effects that are observed when path-integration and external sensory inputs are placed in conflict with one another, such as during geometrical distortion of the environment.
The question arises as to how many independent charts can be represented in a globally interconnected network such as CA3 (Amaral and Witter, 1995
The proposed anatomical implementation of the MPI model is not the only one possible. In principle, the P-I path integrator loop of the MPI model could be based on other structures, such as presubiculum and parasubiculum or entorhinal cortex. One difficulty with this interpretation is that these structures appear not to express multiple charts (e.g., Quirk et al., 1992 Alternative theoretical approaches and related works
The trajectory learning concept, which has not yet been discussed here, is in some sense opposite to the concept of a cognitive map. According to this point of view (Blum and Abbott, 1995
The model of path integration recently proposed by Touretzky and Redish (1996)
In another paper (Redish and Touretzky, 1997
O'Keefe and Burgess (1996)
A model called the "cognitive graph," proposed by Muller et al. (1991 Predictions
The proposed MPI model predicts (1) the specific, persistent, microscopic architecture of the CA3 network (although, as stated above, the model may have a different implementation in anatomy); (2) that the directional relations between the I cells will be the same for all environments; (3) a possibility of "translational disorientation" in a nontrivial, translationally symmetric environment; (4) "activity packet jumps" occurring at the scale of one to two theta cycles; and (5) coherent behavior of neighboring cells on the active chart during distortions of the environment provided that chart switching does not occur [this prediction distinguishes the present model from that of O'Keefe and Burgess (1996)
A corollary of prediction 5 is that in a distorted environment two place cells with overlapping place fields cannot be active simultaneously if their original place fields do not overlap and there is no chart switching. This prediction is consistent with recent observations (Gothard et al., 1996
Another critical test would consist of an observation of place-cell firing during head turns in a stretched box (O'Keefe and Burgess, 1996 Generalizations
The above picture of the hippocampal model of space argues for a general concept of an attractor-map-based internal cognitive model. Many cognitive tasks can be represented effectively on the basis of abstract mathematical models such as graphs (e.g., Muller et al., 1996
Therefore, attractor maps are likely to be found in various brain areas in addition to the hippocampus; e.g., they may underlie cortical mental rotations (Georgopoulous et al., 1989 Conclusions
Hippocampal spatial representations can be described efficiently in terms of charts and self-localized activity packets. This representation inspires a new concept of an attractor map, which may have broad applications elsewhere. The attractor map concept together with the path integration concept lead to a plausible connectionist model (MPI) of the hippocampal spatial representation system, which uses previously suggested ideas of multiple wired charts (Muller et al., 1991
FOOTNOTES
Received Feb. 24, 1997; revised May 14, 1997; accepted May 15, 1997.
This work was conducted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (A.S.) and supported by National Institue of Neurological Disorders and Stroke Grant NS20331 and The Office of Naval Research. We are grateful to Drs. C. A. Barnes, K. M. Gothard, J. J. Knierim, W. B. Levy, L. Nadel, J. O'Keefe, W. E. Skaggs, D. S. Touretzky, M. V. Tsodyks, and A. D. Redish for comments on this manuscript and useful discussions.
Correspondence should be addressed to Dr. Bruce L. McNaughton, Room 344, Life Sciences North Building, University of Arizona, Tucson, AZ 85724.
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