2011/02/01 by Mathieu Galtier, Mathieu N. Galtier, Olivier D. Faugeras +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · Neuroscience · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #FOS: Biological sciences #FOS: Physical sciences #Neural dynamics and brain function #Neurobiology and Insect Physiology Research #Neurons and Cognition (q-bio.NC) #Visual perception and processing mechanisms #nlin.AO #q-bio.NC
paper · pdf · doi:10.48550/arxiv.1102.0166
arxiv created 2011/02/01 · openalex publication_date 2011/02/01 · arxiv updated 2011/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show how a Hopfield network with modifiable recurrent connections undergoing slow Hebbian learning can extract the underlying geometry of an input space. First, we use a slow/fast analysis to derive an averaged system whose dynamics derives from an energy function and therefore always converges to equilibrium points. The equilibria reflect the correlation structure of the inputs, a global object extracted through local recurrent interactions only. Second, we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs. Indeed, multidimensional scaling methods make it possible to project the final connectivity matrix on to a distance matrix in a high-dimensional space, with the neurons labelled by spatial position within this space. The resulting network structure turns out to be roughly convolutional. The residual of the projection defines the non-convolutional part of the connectivity which is minimized in the process. Finally, we show how restricting the dimension of the space where the neurons live gives rise to patterns similar to cortical maps. We motivate this using an energy efficiency argument based on wire length minimization. Finally, we show how this approach leads to the emergence of ocular dominance or orientation columns in primary visual cortex. In addition, we establish that the non-convolutional (or long-range) connectivity is patchy, and is co-aligned in the case of orientation learning.