![[Image: 1n_Ns1b10.gif]](EN/1n_Ns1b10.gif){kind=small}
GIF image (6K)MPG animation (182K)
AVI animation (892K)
This page describes the application of generalized elastic nets, a probabilistic model, to cortical maps in primary visual cortex. We have extended the elastic net model of Durbin and Willshaw to include arbitrary quadratic regularization terms. These are derived from discretized differential operators of any order, such as (for first order) the forward-difference s = [0 -1 1] or the central-difference s = [-1/2 0 1/2]. We are working on the theoretical analysis of this model and its simulation, mainly applied to modeling maps of retinotopy, ocular dominance and orientation in primary visual cortex.
References:
Carreira-Perpiñán, M. Á. and Goodhill, G. J.: "Generalised elastic nets". Unpublished manuscript, Aug. 14, 2003, arXiv:1108.2840v1 [q-bio.NC].
[external link] [paper preprint] [Matlab implementation]
Carreira-Perpiñán, M. Á., Dayan, P. and Goodhill, G. J. (2005): "Differential priors for elastic nets". 6th Int. Conf. Intelligent Data Engineering and Automated Learning (IDEAL'05), pp. 335-342, Lecture Notes in Computer Science vol. 3578, Springer-Verlag.
[external link] [paper preprint] [Matlab implementation] [© Springer-Verlag]
Carreira-Perpiñán, M. Á., Lister, R. J., and Goodhill, G. J. (2005): "A computational model for the development of multiple maps in primary visual cortex". Cerebral Cortex 15(8):1222-1233.
[external link] [paper preprint] [Matlab implementation]
We model the combined development of 5 maps of primary visual cortex (retinotopy, ocular dominance, orientation, direction and spatial frequency) using the elastic net model, as well as the effects of monocular deprivation and single-orientation rearing. We also predict that the stripe width of all maps (OR, DIR, SF) increases slightly under monocular deprivation.
Carreira-Perpiñán, M. Á. and Goodhill, G. J. (2004): "Influence of lateral connections on the structure of cortical maps". J. Neurophysiology 92(5):2947-2959.
[external link] [paper preprint] [Matlab implementation]
Using a generalised elastic net model of cortical maps, we show that the number of excitatory and inhibitory oscillations of a Mexican-hat cortical interaction function has a remarkable effect on the geometric relations between the maps of ocular dominance and orientation. We predict that, in biological maps, this function oscillates only once (central excitation, surround inhibition).
The following pictures and animations were obtained with a Matlab simulator I have written, the Generalized Elastic Nets Matlab Toolbox.
1D net with nonperiodic boundary conditions in a 2D space, beta = 10. Note the bunching bifurcations along the horizontal direction before the bifurcation along the vertical direction, as well as the different effect of the stencils used.
1st-order forward-difference stencil [0 -1 1]
GIF image (6K)
MPG animation (182K)
AVI animation (892K)
2nd-order forward-difference stencil [1 -2 1]
![[Image: 1n_Ns2b10.gif]](EN/1n_Ns2b10.gif){kind=small}
GIF image (6K)
MPG animation (222K)
AVI animation (967K)
3rd-order forward-central-difference stencil [-1 2 0 -2 1]; note the sawteeth
![[Image: 1n_Ns3b10.gif]](EN/1n_Ns3b10.gif){kind=small}
GIF image (10K)
MPG animation (333K)
AVI animation (1161K)
4th-order forward-difference stencil [1 -4 6 -4 1]
![[Image: 1n_Ns4b10.gif]](EN/1n_Ns4b10.gif){kind=small}
GIF image (6K)
MPG animation (254K)
AVI animation (1083K)
1D net with periodic boundary conditions in a 2D space, beta = 100.
1st-order forward-difference stencil [0 -1 1]
![[Image: 1n_Ps1b100.gif]](EN/1n_Ps1b100.gif)
GIF image (5K)
MPG animation (168K)
AVI animation (1361K)
2nd-order forward-difference stencil [1 -2 1]
![[Image: 1n_Ps2b100.gif]](EN/1n_Ps2b100.gif)
GIF image (6K)
MPG animation (209K)
AVI animation (1440K)
3rd-order forward-central-difference stencil [0 -1 3 -3 1]
![[Image: 1n_Ps3b100.gif]](EN/1n_Ps3b100.gif)
GIF image (6K)
MPG animation (274K)
AVI animation (1520K)
4th-order forward-difference stencil [1 -4 6 -4 1]
![[Image: 1n_Ps4b100.gif]](EN/1n_Ps4b100.gif)
GIF image (6K)
MPG animation (302K)
AVI animation (1602K)
1D net with periodic boundary conditions in a 3D space, beta = 100.
1st-order forward-difference stencil [0 -1 1]
![[Image: 1p_Ps1b100.gif]](EN/1p_Ps1b100.gif)
GIF image (10K)
MPG animation (263K)
AVI animation (1123K)
2nd-order forward-difference stencil [1 -2 1]
![[Image: 1p_Ps2b100.gif]](EN/1p_Ps2b100.gif)
GIF image (10K)
MPG animation (278K)
AVI animation (1144K)
3rd-order forward-central-difference stencil [-1 2 0 -2 1]; note the sawteeth
![[Image: 1p_Ps3b100.gif]](EN/1p_Ps3b100.gif)
GIF image (11K)
MPG animation (333K)
AVI animation (1183K)
4th-order forward-difference stencil [1 -4 6 -4 1]
![[Image: 1p_Ps4b100.gif]](EN/1p_Ps4b100.gif)
GIF image (11K)
MPG animation (322K)
AVI animation (1181K)
2D net with nonperiodic boundary conditions in a 5D space, beta = 100. In cortical map modeling, the 5D space represents retinotopy (VFx,VFy), ocular dominance (OD) and orientation (ORt,ORr). The images and animations represent sections of that 5D space.
1st-order forward-difference stencil [0 -1 1] (horizontal and vertical)
![[Image: 2_Ns1b100.gif]](EN/2_Ns1b100.gif)
GIF image (122K)
Ocular dominance map: MPG animation (116K) AVI animation (415K)
Contours of ocular dominance and orientation maps: MPG animation (823K) AVI animation (557K)
Orientation polar map: MPG animation (137K) AVI animation (579K)
Net in retinotopic space: MPG animation (837K) AVI animation (535K)
2nd-order forward-difference stencil [1 -2 1] (horizontal and vertical)
![[Image: 2_Ns2b100.gif]](EN/2_Ns2b100.gif)
GIF image (199K)
Ocular dominance map: MPG animation (122K) AVI animation (416K)
Contours of ocular dominance and orientation maps: MPG animation (1092K) AVI animation (620K)
Orientation polar map: MPG animation (155K) AVI animation (640K)
Net in retinotopic space: MPG animation (776K) AVI animation (504K)
3rd-order forward-central-difference stencil [0 -1 3 -3 1] (horizontal and vertical)
![[Image: 2_Ns3b100.gif]](EN/2_Ns3b100.gif)
GIF image (207K)
Ocular dominance map: MPG animation (133K) AVI animation (444K)
Contours of ocular dominance and orientation maps: MPG animation (1225K) AVI animation (636K)
Orientation polar map: MPG animation (170K) AVI animation (657K)
Net in retinotopic space: MPG animation (650K) AVI animation (458K)
4th-order forward-difference stencil [1 -4 6 -4 1] (horizontal and vertical)
![[Image: 2_Ns4b100.gif]](EN/2_Ns4b100.gif)
GIF image (214K)
Ocular dominance map: MPG animation (143K) AVI animation (471K)
Contours of ocular dominance and orientation maps: MPG animation (1313K) AVI animation (641K)
Orientation polar map: MPG animation (182K) AVI animation (665K)
Net in retinotopic space: MPG animation (581K) AVI animation (432K)
1D net with nonperiodic boundary conditions in a 2D space, beta = 5000, for the 2nd-order forward-difference stencil [1 -2 1], but without annealing the scale parameter. Notice the phenomenon of loop elimination.
![[Image: second_K0.016.gif]](EN/nonannealed/second_K0.016.gif){kind=small}
GIF image (5K)
MPG animation (5217K)
AVI animation (5905K)
2D net with nonperiodic boundary conditions in a 5D space, beta = 10, for the 2nd-order forward-difference stencil [1 -2 1] (horizontal and vertical), but without annealing the scale parameter. Notice the annihilation of pinwheels in the orientation map and of stripes in the ocular dominance map (corresponding to the loop elimination in 1D).
![[Image: s2b10.gif]](EN/nonannealed/s2b10.gif)
GIF image (99K)
Ocular dominance map: MPG animation (136K) AVI animation (1542K)
Contours of ocular dominance and orientation maps: MPG animation (1703K) AVI animation (1610K)
Orientation map: MPG animation (199K) AVI animation (2218K)
Orientation polar map: MPG animation (178K) AVI animation (2253K)
1D net in a 2D space, beta = 10, applied to a traveling salesman problem (TSP). Note that the sawteeth central-difference stencil can be used for a 2-TSP.
1st-order forward-difference stencil [0 -1 1] with nonperiodic boundary conditions
![[Image: TSP_o_n_f.gif]](EN/TSP_o_n_f.gif)
GIF image (6K)
MPG animation (182K)
AVI animation (892K)
1st-order central-difference stencil [-1/2 0 1/2] with nonperiodic boundary conditions
![[Image: TSP_o_n_c.gif]](EN/TSP_o_n_c.gif)
GIF image (6K)
MPG animation (222K)
AVI animation (967K)
1st-order forward-difference stencil [0 -1 1] with periodic boundary conditions
![[Image: TSP_o_p_f.gif]](EN/TSP_o_p_f.gif)
GIF image (10K)
MPG animation (333K)
AVI animation (1161K)
1st-order central-difference stencil [-1/2 0 1/2] with periodic boundary conditions
![[Image: TSP_o_p_c.gif]](EN/TSP_o_p_c.gif)
GIF image (6K)
MPG animation (254K)
AVI animation (1083K)