Neural Maps and Topographic Vector Quantization Corresponding Author Running Title

Acknowledgements The authors are grateful to R. Der and T. Martinetz for useful discussions and to the reviewers for valuable comments. Abstract Neural maps combine the representation of data by codebook vectors, like a vector quantizer, with the property of topography, like a continuous function. While the quantization error is simple to compute and to compare between diierent maps, topography of a map is diicult to deene and to quantify. Yet, topography of a neural map is an advantageous property, e.g. in the presence of noise in a transmission channel, in data visualization, and in numerous other applications. In this paper we review some conceptual aspects of deenitions of topography, and some recently proposed measures to quantify topography. We apply the measures rst to neural maps trained on synthetic data sets, and check the measures for properties like reproducability, scalability, systematic dependence of the value of the measure on the topology of the map etc. We then test the measures on maps generated for four real-world data sets, a chaotic time series, speech data, and two sets of image data. The measures are found to do not a perfect, but an adequate job in selecting a topographically optimal output space dimension, while they consistently single out particular maps as non-topographic.