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Divergence alone cannot guarantee stable sparse activity patterns if connections are dense


It is quite well known, and we have recently confirmed it for the olfactory system in insects [1], that sparse activity is necessary for efficient pattern recognition and memory formation. In many brain centers, including the olfactory system, strongly divergent connections with high-threshold post-synaptic neurons are believed to generate these sparse patterns. We have shown in a generic model that this interpretation rests on the assumptions that the connections are not only divergent but also sparse. However, in locusts the connections from the antennal lobe to the mushroom bodies have been found to be not sparse. To the contrary, experimental observations seem to imply connectivities with 50% of all-to-all connections [2]. Our generic model elucidates how such dense connections lead to instabilities with respect to noise and fluctuations in the incoming signals. We then suggest a hypothesis how the original coding idea can be rescued by appropriate feed-forward gain control mechanisms.


There are three minimal requirements for a successful sparse coding strategy: (1) The response patterns should be sparse for the whole range of expected input patterns, (2) different input patterns should not elicit identical responses (confusion), and (3) the baseline activity of input neurons should not trigger any responses. To investigate these requirements we formalized the idea of a divergent layered neuronal system with a set of minimal assumptions. We then investigated for which combinations of two crucial parameters, the density of connections p c and the firing threshold θ KC , these conditions can be fulfilled. Without additional mechanisms of gain control all three conditions cannot be fulfilled simultaneously for dense connections (Figure 1A) while an appropriate gain control mechanism allows it (Figure 1C).

Figure 1
figure 1

Regions where our criteria 1–3 are fulfilled. Dark blue: none are fulfilled, blue: (3) is true, light blue: (2) is true, cyan: (2) and (3) are true, yellow: (1) and (3) are true, orange: (1) and (2) are true, and red: all three are true. Note that for no gain control (A), the red region is located between p c = 0.1 and p c = 0.3 and is very thin. It then moves to higher p c with increasing gain control (B), (C) and (D), where it disappears. The strings of white dots in each of the panels mark the locus of the minimum of the probability of confusion with respect to the threshold θ KC for each given value of p c Depending on the density of connections, a different degree of gain control is necessary to be able to fulfill all three criteria for a suitable sparse activity.


  1. Huerta R, Nowotny T, Garcia-Sanchez M, Abarbanel HDI, Rabinovich MI: Learning classification in the olfactory system of insects. Neural Comput. 2004, 16: 1601-1640. 10.1162/089976604774201613.

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TN acknowledges helpful discussions with Falk Hildebrand. This work was partially funded by the Biotechnology and Biological Sciences Research Council (grant number BB/F005113/1).

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Correspondence to Thomas Nowotny.

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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Nowotny, T. Divergence alone cannot guarantee stable sparse activity patterns if connections are dense. BMC Neurosci 10 (Suppl 1), P188 (2009).

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