6.3. SIMPLE MEMORY 165

terminology is borrowed from classical mechanics, where dynamical systemsmay be described in terms of their potential energy). When visualized in theform of an energy surface, network dynamics are easy to understand: justthink of a ball that is released at a point corresponding to the input and thatstarts rolling downhill until it reaches a local minimum — an attractor.6

In a properly trained network, the energy surface is smooth and eachequilibrium state has a wide basin of attraction; every input that falls withinit will cause the network to end up in the same steady state. Functionally, thisbehavior is precisely what categorization is about. Furthermore, if a networktrained on a few inputs generates wide basins of attraction, it will thereby be-come capable of generalization: the response (equilibrium state) to a stimulus

each model are closely related to knowledge of what isrepresented in different brain regions, as follows.One pos-sible brain region forModels 1 and 2 is the CA3 region ofthe primate hippocampus, which contains spatial viewcells (Georges-Francois et al., 1999; Robertson et al.,1998; Rolls, 1999; Rolls et al., 1997), which show at leastsome continuing firing in the dark (Robertson et al.,1998). The CA3 cells have a highly developed recurrentcollateral system of connections (Ishizuka, Weber, &Amaral, 1990), which show at least associative synapticmodification (Debanne, Gahwiler, & Thompson, 1998).Moreover, whole body rotation cells, which provide thefiring represented in Models 1 and 2 by the head rotationcells, are present in the primate hippocampus (O�Maraet al., 1994). Although eye velocity-related neuronal firinghas not been reported for the hippocampus, eye velocity isthe main form in which signals in the brain related to eyemovements are represented (Berthoz, 2000; Berthoz,Grantyn, & Droulez, 1987). Although the CA1 cells ofprimates do fire better than CA3 cells during update ofspatial view cells by eyemovements in the dark, this couldbe because of associative improvement of the representa-tion at the Schaffer collateral (CA3 to CAl) synapses(Robertson et al., 1998). The CAl region is a less likely re-gion for implementation ofModels 1 and 2, in that it has aless well developed recurrent collateral system than doesthe CA3 system (Rolls & Treves, 1998). It is also possiblethat Models 1 and 2 are implemented at a stage of neuralprocessing prior to the hippocampus, and that the hippo-campal pyramidal cells reflect the inputs received from

this preceding network. It is known that there are spatialview cells in the parahippocampal gyrus (Rolls, 1999;Rolls et al., 1997). With respect to Model 3, if the idio-thetic update of the correct spatial view cell firing in thedark is performed in the hippocampus in the way de-scribed in thatModel, it is relevant that one hippocampalregion with feedforward rather than recurrent collateralconnectivity is the CA1 cell system. For the cells in CA1to be updated directly by idiothetic inputs as describedfor Model 3, the CA1 cells would need to receive headdirection cell inputs (which are represented by the firingof neurons in the primate presubiculum which are them-selves updated correctly by head rotation in the dark(Robertson et al., 1999)), and eye position inputs.

2. Model 1: The recurrent network model with

independent idiothetic inputs

2.1. Architecture and learning in the network

The network architecture is shown in Fig. 1. When vi-sual input is available, each spatial view cell respondswith a Gaussian profile to a view of part of the environ-ment, and that part of the environment can be looked atwith a given gaze angle which is provided by a combina-tion of head direction and eye position signals. The spa-tial view cells are connected by recurrent collateralsynapses that reflect the distance in the state space (inthis case the spatial view) of any two connected cells.

Fig. 1. General network architecture for Model 1, the recurrent continuous attractor network model with independent idiothetic inputs. There is arecurrent layer of spatial view cells in which the cells are mapped onto a regular grid of spatial views. The spatial views are defined by the angle ofgaze of the agent. In the light, individual spatial view cells are stimulated maximally when the agent has an angle of gaze corresponding to theposition of the cell in the grid. The layer of spatial view cells receives external inputs from three sources: (i) the visual system; (ii) a population ofclockwise and anticlockwise rotation cells; and (iii) a population of eye velocity cells. The clockwise and anti-clockwise rotation cells fire according towhether the agent is rotating in the appropriate direction, and with a firing rate that increases monotonically with respect to the speed of rotation.The eye velocity cells fire maximally when an animal moves its eyes in a particular direction, and have firing rates that increase with the speed ofmovement of the eyes. The velocity of the animal�s eyes is denoted by (v, k) where v is the angular speed of movement of the eyes, and k is thedirection of movement of the eyes. The eye velocity cells are mapped onto a circular grid of directions of eye movement, where each eye velocity cellhas a unique direction of eye movement for which the cell fires maximally.

S.M. Stringer et al. / Neurobiology of Learning and Memory 83 (2005) 79–92 81

Figure 6.4 — Left: the neural circuitry of the rodent HIPPOCAMPUS, hand-drawn by Santiago Ramón y Cajal (fromHistologie du Systeme Nerveux de l’Homme et des Vertebretes, A. Maloine, Paris, 1911). Middle: a simple, two-neuron autoassociative ATTRACTOR NETWORK modeled on the CA1-CA3 hippocampal circuit (reproduced fromStringer, Rolls, and Trappenberg, 2005). Note how the projections of the two cells (denoted by the black dots) turnback and connect to the inputs of those same cells, forming the kind of feedback circuit found in the real hippocampusshown on the left (cf. also Figure 6.3 on page 162, bottom). Right: a cartoon illustration of the ENERGY LANDSCAPEof a two-neuron attractor network. The horizontal plane represents the activities of the two neurons; the height ofthe surface stands for the potential energy of the circuit. The shape of the energy surface indicates that the systemhas four stable equilibrium states or attractors (corresponding to the four potential wells), as well as a metastableequilibrium point at the center of the plane. See text for further explanations.