Generalized elastic nets

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:

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]
[Image: 1n_Ns1b10.gif] GIF image (6K)
MPG animation (182K)
AVI animation (892K)

2nd-order forward-difference stencil [1 -2 1]
[Image: 1n_Ns2b10.gif] 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] GIF image (10K)
MPG animation (333K)
AVI animation (1161K)

4th-order forward-difference stencil [1 -4 6 -4 1]
[Image: 1n_Ns4b10.gif] 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] GIF image (5K)
MPG animation (168K)
AVI animation (1361K)

2nd-order forward-difference stencil [1 -2 1]
[Image: 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] GIF image (6K)
MPG animation (274K)
AVI animation (1520K)

4th-order forward-difference stencil [1 -4 6 -4 1]
[Image: 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] GIF image (10K)
MPG animation (263K)
AVI animation (1123K)

2nd-order forward-difference stencil [1 -2 1]
[Image: 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] GIF image (11K)
MPG animation (333K)
AVI animation (1183K)

4th-order forward-difference stencil [1 -4 6 -4 1]
[Image: 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] GIF image (6K)
MPG animation (254K)
AVI animation (1083K)


Miguel A. Carreira-Perpinan
Last modified: Sun Aug 28 22:32:19 PDT 2011

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