Ординатура / Офтальмология / Английские материалы / Computational Maps in the Visual Cortex_Miikkulainen_2005

Table A.2. Defaults for constant parameters. This table specifies how the parameter values for the different simulations can be constructed, based on the reference values from Table A.1. These parameters have constant values in each simulation. Those with the subscript “i” represent the initial values for parameters that are changed over the simulation, as shown in Table A.3. The table is organized into sections including user-defined network size, connection radius, calculated network size, and input pattern parameters on the previous page, and connection strength, activation, and learning parameters on this page. The numerical values in formulas marked with a dagger (†) were determined empirically in earlier work (Bednar and Miikkulainen 2000b; Sirosh 1995). Parameters that are listed as being used in Table A.2 are temporary variables, introduced to make the notation easier to follow. Those listed as being used in various equations and sections are actual parameters of the LISSOM model itself.

A.3 Choosing Parameters for New Simulations

Despite the seemingly large number of parameters, few of them need to be adjusted when running a new simulation. The most commonly changed parameters are the cortical density Nd and area scale sg, because these parameters directly determine the time and memory requirements of the simulations. The default Nd of 142 representing a 5 mm × 5 mm area is a good match to the density of columns in V1 orientation maps (Bednar, Kelkar, and Miikkulainen 2004), but in practice much smaller

422 A LISSOM Simulation Specifications

A.4 Retinotopic Maps

The retinotopic map simulations in Chapter 4 were based on the default values, except the training Gaussians were unoriented (σa = σb = σu = 3), the sigmoid threshold was lower (θli = 0.035) so that the smaller patterns would produce as much cortical activity as oriented Gaussians do, the RF centers were randomly scattered by 5% in both the x and y directions, and the lateral connections were uniformly random instead of initialized to a Gaussian profile.

The default parameter values result in two Gaussian inputs per iteration. While in principle it would be possible to self-organize the network with patterns that consist of more than two, given that the Gaussians are relatively large compared with the retina, it would be difficult to distribute them uniformly on the retina while enforcing a minimum distance dr between them. After two Gaussians have been placed randomly, the only remaining possible locations are often near the corners. The corners would therefore be trained more often than the other areas of the map, resulting in a distorted organization. More Gaussians could be used if a larger retinal and cortical area was simulated (i.e. with a larger sg), or the Gaussians were narrower.

A.5 Orientation Maps

The LISSOM simulations with oriented Gaussians in Sections 5.3 and 6.2 (Figures 5.5–5.12, “Gaussians” in 5.13, “ON/OFF” in 6.4 and 6.5) were based on the default parameter values, except the inhibitory connection death threshold wd was increased to 6wdorI2o/rI2 for historical reasons. These values were also used in the “Plus/Minus” simulation in Figure 5.13, except the sign of the input scale sb was chosen randomly for each pattern. The other types of orientation map simulations presented in that figure are described in the following subsections.

A.5.1 Disks, Noisy Disks, and Noise

The simulations with circular disks (“Disks” in Figure 5.13) were based on the default LISSOM parameters, except only one disk was drawn per iteration (np = 1), the lateral interactions were stronger to allow long contours to be separated into distinct activity bubbles (γI = 2.0, γE = 1.2), and the LGN afferent scale was slightly stronger (γL = 3/sb) because each activity bubble was slightly weaker than in the oriented Gaussian simulation.

The disk input patterns were fully activated in the circular region (rd = 25sw) around their centers, and the activation then fell off according to a Gaussian pattern (σd = 3). Each input center was separated far enough so that input patterns would never overlap (dr = 1.5rd). Because the disk stimuli are large compared with Ld, the area in which disk centers are chosen was increased so that even the neurons at the edges of the map are equally likely to receive input from all parts of the disks. Thus, the number of inputs per eye was corrected to reflect this larger area (np =

The simulations with noisy circular disks (“Noisy disks” in Figure 5.13) were identical to the simulations with noiseless disks, except uniform random noise in the range [−0.5..0.5) was added to the input pattern. The simulations with uniform random noise alone (“Noise” in Figure 5.13) were based on the noisy disk simulations, except no disks were drawn, the LGN parameters were adjusted to produce stronger LGN activations (σc = 0.75, σs = 3σc, and γL = 5/sb), and the V1 parameters were changed to produced stronger V1 activations (γE = 0.9, γI = 0.9, γA = 3).

A.5.2 Natural Images

The orientation map simulations trained on natural images (“Nature” in Figure 5.13) were based on the default LISSOM parameters, except the retinal density was doubled to provide more image resolution (Rd = 48), only one image was drawn per iteration (np = 1), the input density scale was higher (sd = 4) because each input resulted in about four activity bubbles on average, the number of iterations was fixed to the default value (st = 0.5) instead of being adjusted automatically for the input density scale, the LGN afferent scale was increased to produce activity for lowcontrast inputs (γL = 4.7/sb), and the sigmoid threshold was lower (θli = 0.076) to allow responses to low-contrast stimuli.

The lateral interaction strengths and learning rates were also adjusted during training rather than being fixed. In the default simulation, the lateral inhibitory weights self-organize into only a few small regions, because Gaussian patterns have no long-range correlations. In contrast, natural images have significant long-range correlations, and inhibitory weights spread over a larger area. To keep the balance between excitatory and inhibitory lateral weights approximately constant, the lateral inhibitory strength was set to γI = 1.75 at first, increased to 2.2 at iteration 1000st and 2.6 at iteration 2000st. The lateral inhibitory learning rate αI was

these modifications, the orientation map self-organizes robustly with natural image input.

A.6 Ocular Dominance Maps

The ocular dominance simulations in Section 5.4 (Figures 5.15–5.17, “Dimming” in Figure 5.19) were based on the default LISSOM parameters, except four LGN regions (nA = 4) and two eyes were included, the sigmoid threshold was lower (θli = 0.035) so that the smaller patterns would produce as much cortical activity as oriented Gaussians do, the training Gaussians were unoriented (σu = 4.25), and the input brightness scale sb was randomly chosen from the range [0..1] for the left eye and calculated as 1 − sb for the right eye for each input.

The strabismic ocular dominance simulations (Figure 5.18, “Strabismic” in Figure 5.19) were otherwise similar, except the inputs were scattered randomly between

424 A LISSOM Simulation Specifications

the eyes (ss = 1.0). The “Mild” ocular dominance simulations in Figure 5.19 were identical to “Strabismic”, except the input scale was constant (sb = 1.0), and the inputs were scattered slightly between the eyes (ss = 0.2). The “Moderate” ocular dominance simulations were identical to “Mild”, except the inputs were scattered more between the eyes (ss = 0.4).

A.7 Direction Maps

The direction map simulations in Section 5.5 (Figures 5.21–5.24, “Speed 1” in Figure 5.25) were based on the default LISSOM parameters, except eight LGN regions were included (nA = 8), only a single pattern was presented per iteration (sd = 1) to avoid overlapping patterns that are moving in different directions, and the LGN afferent scale was slightly stronger (γL = 2.38/sb) because the response is weaker when inputs are not perfectly aligned in each eye. Because sd = 1, st = 0.5, which means that these simulations were run for 20,000 iterations with one pattern apiece rather than the default 10,000 with two patterns apiece. As long as input patterns can be placed randomly on the retina without overlap, and the responses are localized in V1 (as they usually are), using more patterns in fewer iterations is equivalent to using fewer patterns in more iterations.

The simulations with different speeds (Figure 5.25) were otherwise identical except the LGN afferent scale was adjusted for each speed to keep the average response comparable: γL = 2.33/sb for speed 0, γL = 2.38/sb for speed 1, γL = 2.53/sb for speed 2, and γL = 2.80/sb for speed 3.

A.8 Combined Orientation / Ocular Dominance Maps

The combined orientation and ocular dominance simulations in Section 5.6.2 (Figures 5.27 and 5.28) were based on the “Gaussians” orientation-only simulation, except four LGN regions (nA = 4) and two eyes were included. For each input, the input scale sb was chosen randomly for the left eye and calculated as 1 − sb for the right eye.

A.9 Combined Orientation / Ocular Dominance / Direction Maps

In Section 5.6, the Gaussian-trained combined orientation, ocular dominance, and direction simulations (Figure 5.29, “Gaussians” in Figure 5.32) were based on the “Gaussians” orientation-only simulation, except 16 LGN regions (nA = 16) and two eyes were included, only a single pattern was presented per iteration (sd = 1) to avoid overlapping patterns that are moving in different directions, and the LGN afferent scale was adjusted to match the value computed for the direction-only map (γL = 2.38/sb). For each input, the input scale sb was chosen randomly for the left

eye and calculated as 1 − sb for the right eye. The speed of the moving patterns was 1, i.e. subsequent frames differed by 1.0 retinal units.

The combined simulation with noisy disks (“Noisy disks” in Figure 5.32), used the inputs from the “Noisy Disks” orientation-only simulation, drawn at speed 2. The parameters were the same as for the “Gaussians” combined OR/OD/DR simulation, except a higher input density scaling was used (sd = 2) because the inputs more regularly activated the cortex, γI and αI were adjusted as for the orientation-only simulation with natural images (Section A.5.2) to keep excitation and inhibition balanced, and the input scaling for the LGN was increased (γL = 3.0/sb) to keep the LGN responses comparable.

The combined model trained with natural images (Figure 5.31, “Nature in Figure 5.32) was identical to the orientation-only natural image simulation in Section A.5.2 except 16 LGN regions (nA = 16) and two eyes were included, the input density scale was increased without changing the number of iterations (sd = 2 and st = 1), and the LGN afferent scale was adjusted to match the value computed for the direction-only map (γL = 5.6/sb). For each input, the input scale sb was chosen randomly for the left eye and calculated as 1 − sb for the right eye. The speed of the moving patterns was 2.0.

B

Reduced LISSOM Simulation Specifications

As described in Section 6.2.3, as long as the inputs to the model consist of abstract patterns instead of natural images, simulations that utilize only the ON channel (or bypass the LGN entirely) lead to the same results as simulations that include both the ON and OFF channels of the LGN explicitly. Such reduced LISSOM networks allow demonstrating many phenomena efficiently and clearly, as was seen in Chapters 6, 7 and 11–15.

The reduced LISSOM simulations were based on the default LISSOM parameter values, except there was only one input sheet (instead of ON and OFF; thus nA = 1). This sheet was mapped directly to the cortex like the LGN sheets (Figure A.1), and sized like the LGN sheets (R = L). The most important adjustment is to set the sigmoid thresholds so that the total cortical response is the same (on average) as with ON and OFF channels, to compensate for the higher average value of retinal activity compared with the LGN activity. In the reduced LISSOM simulations, the input threshold was therefore set to θli = 0.1. These values are taken as the default reduced LISSOM parameters, occasionally overridden in individual simulations as specified in the following sections and in Appendices D and F.

B.1 Plasticity

The retinal lesion experiments in Section 6.3 were run with the default reduced LISSOM parameters, except the inhibitory connection death threshold wd was increased to 6wdorI2o/rI2 for historical reasons, and the afferent learning rate αA was increased to 0.003 after iteration 10,000; all other parameters remained the same as at the end of the self-organization. Faster learning makes the changes more visible, and also models the increased plasticity that might result during recovery from injury (Emery, Royo, Fischer, Saatman, and McIntosh 2003; Kaas 2001a,b). The cortical lesion experiments in Section 6.4 were based on the same parameters as the retinal lesion experiments, except the afferent learning rate remained at its default value even after the end of self-organization.

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B.2 Tilt Aftereffect

The tilt aftereffect experiments in Chapter 7 were based on the default reduced LISSOM parameters, except for historical reasons the cortical density was slightly larger (Nd = 192), the final excitatory radius rEf was reduced to 1.5, the inhibitory connection death threshold wd was decreased to 0.00005, and only a single pattern was presented per iteration (sd = 1). Also for historical reasons, the input patterns were blurred with a uniform 3 × 3 convolution kernel before they were presented to the network.

For the average tilt aftereffect plots in Figures 7.5 and 7.6, each input line was presented at nine different locations chosen from the nodes of a regular 3 × 3 grid centered in the retina, with each grid step four retinal units wide.

B.3 Scaling

The scaling equation and GLISSOM simulations in Chapter 15 were based on the tilt aftereffect parameters (Section B.2), scaled as described in each section of Chapter 15. For instance, the area scaling simulation (Figure 15.1) compares two sim-

ulations with Ld = 24 and Nd = 44, differing by their area scales (sg = 1 and sg = 4).

The scaling simulations were run on a 600 MHz Intel Pentium III Linux machine with 1024 MB of RAM. The timing results are user CPU times reported by the GNU “time” command; the CPU time is essentially the same as the elapsed wallclock time because the CPU utilization was always over 99%.