Analysis of the elastic net model applied to the formation of ocular dominance and orientation columns

Abstract

The development and structure of orientation (OR) and ocular dominance (OD) maps in the primary visual cortex of cats and monkeys can be modelled using the elastic net algorithm, which attempts to find an 'optimal' cortical representation of the input features. Here we analyse this behaviour in terms of parameters of the feature space. We derive expressions for the OR periodicity, and the first bifurcation point as a function of the annealing parameter using the methods of Durbin et al (Durbin R, Szeliski R and Yuille A 1989 Neural Computation 1 348-58). We also investigate the effect of the relative order of OR and OD development on overall map structure. This analysis suggests that developmental order can be predicted from the final OR and OD periodicities. In conjunction with experimentally measured values for these periodicities, the model predicts that (i) in normal macaques OD develops first, (ii) in normal cats OR develops first and (iii) in strabismic cats OD develops first.

Coverage, continuity, and visual cortical architecture

“…However, for all stationary solutions the regions of high gradient in the orientation map coincide with low gradient regions (the folds of the grid) in the retinotopic map. This is precisely what is generally expected from a dimension-reducing mapping [21,62,63,92]. In the following section, we will investigate which of these solutions become optimal depending on the two parameters σ/Λ and η r that parametrize the model.…”

“…From prior work, the coverage continuity compromise appears to be a promising candidate for a principle to explain visual cortical functional architecture. Firstly, many studies have reported good qualitative agreement between the layout of numerically obtained dimension reducing maps and experimental observations [19,21,42,[55][56][57][59][60][61][62][63][64][65][66][67][68][69][70][71]. Secondly, geometric relationships between the representations of different visual features such as orientation, spatial frequency and ocular dominance have been reproduced by dimension reduction models [25, 56, 63-65, 68, 72, 73].…”

“…Since the seminal work of Durbin & Mitchison [21], the EN model has been widely used to study visual cortical representations [25, 42, 62-65, 69-72, 80]. The EN model possesses an explicit energy functional which trades off a matching constraint which matches cortical cells to particular stimulus features via Hebbian learning, with a continuity constraint that minimizes euclidean differences in feature space between neighboring points in the cortex [63]. In two ways, the EN model's explicit variational structure is very appealing.…”

Coordinated Optimization of Visual Cortical Maps (II) Numerical Studies

“…Many optimization models of the form presented in Eq. (4) have been studied [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. The concrete dynamics in Eq.…”

“…In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference . Many attempts have been made to explain and understand the spatial organization of these maps as optima of specific energy functionals the brain minimizes either during development or on evolutionary timescales [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In part (I) of this study we presented an analytical approach to study the coordinated optimization of interacting pairs of visual cortical maps where maps are described by real and complex valued order parameter fields [39] .…”