Geodesic
Self-Organized Maps on Continuous Bump Attractors
Matematičeskaâ biologiâ i bioinformatika, Tome 8 (2013) no. 1, pp. 234-247.

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In this article, we describe a simple binary neuron system, which implements a self-organized map. The system consists of $R$ input neurons ($R$ receptors), and $N$ output neurons of a recurrent neural network. The neural network has a quasi-continuous set of attractor states (one-dimensional “bump attractor”). Due to the dynamics of the network, each external signal (i.e. activity state of receptors) imposes transition of the recurrent network into one of its stable states (points of its attractor). That makes our system different from the “winner takes all” construction of T. Kohonen. In case, when there is a one-dimensional cyclical manifold of external signals in $R$-dimensional input space, and the recurrent neural network presents a complete ring of neurons with local excitatory connections, there exists a process of learning of connections between the receptors and the neurons of the recurrent network, which enables a topologically correct mapping of input signals into the stable states of the neural network. The convergence rate of learning and the role of noises and other factors affecting the described phenomenon has been evaluated in computational simulations. 
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K. P. Solovyeva. Self-Organized Maps on Continuous Bump Attractors. Matematičeskaâ biologiâ i bioinformatika, Tome 8 (2013) no. 1, pp. 234-247. http://geodesic.mathdoc.fr/item/MBB_2013_8_1_a8/

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