Fig 1.
Red and blue represent ON and OFF sub-regions of the receptive field of a simple cell, respectively. (A) The classic energy model of a complex cell. The linear response of the input convolved with the filter is passed to a two-sided nonlinear function (a power function in this case). The outputs of two nonlinear functions are then summed to generate the response for the complex cell. (B) The equivalent hierarchical structure of the energy model of a complex cell. The response of the complex cell sums over the responses of simple cells.
Fig 2.
Graphical representation of the model.
I is the identity matrix that represents self-excitation. Red and green arrows represent excitatory and inhibitory connections, respectively. Upward and downward arrows represent feedforward and feedback pathways. Parameters are defined in Table 1.
Table 1.
Model symbols and parameters.
Fig 3.
The structure of Nonlinear Input Model (NIM).
The filter and input nonlinearities determine how it responds to the visual input. The sum of responses of all filters are then passed to a spiking nonlinear function to generate the response of the model.
Fig 4.
Complex cell C18 trained using the modified BCM rule with N = 15.
(A) Each block is a 16 × 16 synaptic field (defined in Eq 12). Values in each block are normalized to the range [−1 1] when plotting the figure. (B) Orientation tuning curves. (C) Spatial phase tuning curves. Solid lines are for simple cells in the subspace. The dotted line is for complex cell C18. S represents simple cell and the following number is the index of the simple cell.
Fig 5.
Scatter plots that investigate the diversity of learned complex cells.
Left: scatter plot of simple-complex cell connections for the model. The dots in each row represent the indices of simple cells that have substantial weights (>0.4) with the complex cell indicated by an index on y-axis. Right: scatter plot of F1/F0 vs. preferred orientation for all model complex cells. (A) Modified BCM rule. (B)-(D) Modified NBCM rule with β = 11 (B), β = 12 (C), and β = 13 (D).
Fig 6.
Histograms of F1/F0 for models based on the modified NBCM rule.
(A) Experimental complex cells [57]. Model complex cells learned with (B) β = 11, (C) β = 12, and (D) β = 13.
Fig 7.
Histograms of half-bandwidth for models based on the modified NBCM rule.
(A) Experimental complex cells [57]. Model complex cells learned with (B) β = 11, (C) β = 12, and (D) β = 13.
Fig 8.
Examples of model complex cells based on the modified NBCM rule.
Left: each block is a 16 × 16 synaptic field (defined in Eq 12) for simple cells in the subspace and values in each block are normalized to the range [−1 1] when plotting the figure. Middle: orientation tuning curve. Right: spatial phase tuning curves. Solid lines are for simple cells in the subspace. Dotted line is for complex cell. S represents simple cell and the following number is the index of the simple cell. (A) Complex cell that is invariant to all spatial phases. (B) Complex cell that shows invariance to perturbations in orientation. (C) Complex cell that is invariant to orientation but not spatial phase.
Fig 9.
Comparison between experimental data [11] (left) and model data trained using the modified NBCM rule (middle).
Right: histograms, where filled points indicate differences that are significant (p-value<0.05; Welch’s t-test). (A) Orientation breadth (°). (B) Spatial frequency breadth (cycles per degree). (C) Spatial phase breadth (°).