Self-organising continuous attractor networks with multiple activity packets, and the representation of space S.M. Stringer a , E.T. Rolls a, * , T.P. Trappenberg b a Department of Experimental Psychology, Centre for Computational Neuroscience, Oxford University, South Parks Road, Oxford OX1 3UD, UK b Faculty of Computer Science, Dalhousie University, 6050 University Avenue, Halifax, NS, Canada B3H 1W5 Received 21 February 2003; accepted 13 June 2003 Abstract ‘Continuous attractor’ neural networks can maintain a localised packet of neuronal activity representing the current state of an agent in a continuous space without external sensory input. In applications such as the representation of head direction or location in the environment, only one packet of activity is needed. For some spatial computations a number of different locations, each with its own features, must be held in memory. We extend previous approaches to continuous attractor networks (in which one packet of activity is maintained active) by showing that a single continuous attractor network can maintain multiple packets of activity simultaneously, if each packet is in a different state space or map. We also show how such a network could by learning self-organise to enable the packets in each space to be moved continuously in that space by idiothetic (motion) inputs. We show how such multi-packet continuous attractor networks could be used to maintain different types of feature (such as form vs colour) simultaneously active in the correct location in a spatial representation. We also show how high-order synapses can improve the performance of these networks, and how the location of a packet could be read by motor networks. The multiple packet continuous attractor networks described here may be used for spatial representations in brain areas such as the parietal cortex and hippocampus. q 2003 Elsevier Ltd. All rights reserved. Keywords: Continuous attractor neural networks; Multiple activity packets; Spatial representation; Idiothetic inputs; Path integration 1. Introduction ‘Continuous attractor’ neural networks are neural net- works which are able to maintain a localised packet of neuronal activity representing the current state of an agent in a continuous space, for example head direction or location in the environment, without external sensory input (Amari, 1977; Taylor, 1999). They are useful in helping to understand the representation of head direction (Redish, Elga, & Touretzky, 1996; Skaggs, Knierim, Kudrimoti, & McNaugh- ton, 1995; Stringer, Trappenberg, Rolls, & de Araujo, 2002b; Zhang, 1996), place (Redish & Touretzky, 1998; Samsono- vich & McNaughton, 1997; Stringer, Rolls, Trappenberg, & de Araujo, 2002a), and in the primate hippocampus, spatial view (Stringer, Rolls, & Trappenberg, 2003a). Continuous attractor networks use excitatory recurrent collateral connections between the neurons to reflect the distance between the neurons in the state space (e.g. head direction space) of the agent. Global inhibition is used to keep the number of neurons in a bubble of activity relatively constant. In the applications of continuous attractor networks dis- cussed above, where a network is required to represent only a single state of the agent (i.e. head direction, place or spatial view), it is appropriate for the continuous attractor networks to support only one activity packet at a time. In this paper we propose that continuous attractor networks may be used in the brain in an alternative way, in which they support multiple activity packets at the same time. The stability of multiple activity packets in a single network has been discussed previously by, for example, Amari (1977) and Ermentrout and Cowan (1979). Ermen- trout and Cowan (1979) analysed neural activity in a two- dimensional (2D) network, demonstrating the existence of a variety of doubly periodic patterns as solutions to the field equations for the net activity. Amari (1977) considered a continuous attractor neural network in which the neurons are mapped onto a one-dimensional (1D) space x; where there are 0893-6080/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0893-6080(03)00210-7 Neural Networks 17 (2004) 5–27 www.elsevier.com/locate/neunet * Corresponding author. Tel.: þ 44-1865-271348; fax: þ 44-1865- 310447. E-mail address: edmund.rolls@psy.ox.ac.uk (E.T. Rolls). Web pages: www.cns.ox.ac.uk