It is obviously difficult to gauge the value of the bifurcation parameter r for the development of the orientation map in the visual cortex, and the possibility that the system is in a regime further away from the bifurcation point must not be ne-glected. It is unclear, however, whether the results valid close to criticality estab-lished in the previous Section 5.3 generalize to the regime further away from the bifurcation point. To assess the robustness of these results it is therefore important to study the regimer >0.
In fact, a regime r > 0 is suggested from the following considerations. First, at least for the tree shrew, the layout of the pattern along the boundary suggests a rather short ranging influence within a few column spacingsΛ. (Fig. 2.1). This is,
Figure 5.5: Two examples of orientation maps obtained at intermediate timet=3·102(left in a, b) and at late timet= 3·104(right in a, b) withtrescaled byT =1/r. Integration of 5.1 and parameters as in Fig. 5.1. The upper row shows subregions of the angle map (A= (4.25Λ)2, smoothed for illustration). Below each angle map the power spectrum of the entire map is depicted (origin of Fourier-space in center of annulus). Att =3·102unstable active modes are present. In a an example of a high pinwheel density map is depicted with almost equidistance of active modes on the critical circle. b shows an example of a low pinwheel density map with most modes confined to one half of the critical circle.
however, inconsistent with the stiffness of the solutions implied byr 1. As men-tioned above, only forr>0, the critical circle is sufficiently wide to allow for such sharp transitions in the layout. Second, even in orientation maps of intermediate size with aspect ratios of Γ ≈ 20−30, large-scale inhomogeneities are observed that may be interpreted as different domains (see e.g. Fig. 4.1). Clearly, such domains would only occur in a regime further away from the bifurcation point, since owing to the widthδk ∝ √
rof the critical circle, the size of such a coherent structure isO(1/√
r). Third, assuming thatris not constant in time in a cortex per-manently driven by visual stimuli, in the caserwere close to the bifurcation point, these fluctuations ofrcould cause occasional transitions from maps of orientation selective neurons to unselective neurons. Such large scale fluctuations, however, were never observed in recordings nor do they appear very beneficial for vision.
Two representative solutions of Eq. (5.1) forr = 0.1 are depicted in Fig. 5.5 at intermediate and late timest(rescaled by (5.19)). In transient states att =3·102all active amplitudes were already confined to the critical circle, but not yet in a stable configuration. In the near stationary states at t = 3·104, in contrast, all finally active amplitudes were selected and the further development mainly consisted in the relaxation of the phases of these amplitudes. Whereas the final state depicted in
Fig. 5.5a exhibited a high pinwheel density with almost equidistant active modes on the critical circle (small anisotropyξ, Eq. (5.43)), a map of lower pinwheel den-sity with most modes confined to one side of the spectrum (large anisotropyξ) is shown in Fig. 5.5b.
5.5.1 Transient states
Transient states exhibited average pinwheel densitieshρivery similar to those ob-served in the visual cortex in Chapter 4. As shown in Fig. 5.6a, after an interme-diate integration time t = 3·102 the average pinwheel density was hρi ≈ π for various values ofrand interaction rangesσ. However, pinwheel densities did not mimic the fluctuation withσ found for planforms (Fig. 5.6b), but appeared rather independent ofσ. The deviation from the planforms was even more apparent for the variabilitysof pinwheel densitiesρacross individual maps (SD, Fig. 5.6b).
The origin of this deviation from planform solutions is that this developmental stage still falls into the phase of nonlinear competition. The finally active modes have not been fully selected yet. Especially, the low pinwheel density realizations with active modes only in one half of the critical circle leading to high anisotropies ξ occur only in much later stages. Therefore, pinwheel densities at this stage ex-hibit slightly higher values and less variation compared to planforms. Map devel-opment is further discussed in Chapter 6.
5.5.2 Near stationary states
The pinwheel densities observed for the near stationary states agreed for small values of r well with the experimental results and those obtained for planforms.
Fig. 5.6c shows the average pinwheel densitieshρifor the same initial conditions as in Fig. 5.6a, but after t = 3·104. Above r ≥ 0.03, however, a deviation of hρi relative to the planform densities became evident. Solutions for a givenrdiverged stronger from the asymptotic density near criticality,hρi=π, for larger interaction rangesσ. Also the variation sfollowed the results for planforms only for small r and interaction ranges 1.0 ≤ σ ≤ 2.0 (Fig. 5.2d), but was more pronounced for large values ofσ andr.
The interpretation of these findings is that at this stage the phase of nonlinear competition has come to an end. Planform-like solutions are selected, however, with a bias towards larger anisotropiesξand therefore to smaller average pinwheel densities and a larger variation (see Section 5.3). Different from planforms, for r > 0 solutions do not appear to have degenerated energy anymore. Striving the amplitude expansion to higher than 3rd order may provide insights into an energy splitting or other mechanism controlling the selection of solutions.
Figure 5.6: Pinwheel densities of the full model Eq. (5.1). a, Average pinwheel densities hρifor different values of bifurcation parameterr and interaction rangeσ. Numerical solutions were obtained aftert = 3·102 starting from the identical set (N=50) of initial conditions z0 at t = 0 for each parameter set (see Methods; mesh size, 128×128 using periodic boundary conditions; g = 0.98, Γ = 21forσ = 1.0, 1.4, Γ = 24forσ = 1.7, 2.0, Γ = 25forσ = 2.5).
Pinwheel densitieshρiof planform solutions (Fig. 5.2) shown atσ =n/1.5π(diamonds). Their asymptotic densityhρi=π is indicated by the dashed line. b, SDsof pinwheel densities. c, d, as in a, b, respectively, but at a much larger timet =3·104at which solutions are close to the attractor.
5.5.3 Robust selection
The importance of long-range connections for the selection of pinwheel densities is illustrated in Fig. 5.7. If long-range interactions are absent, low pinwheel den-sity solutions are preferred and pinwheels annhilate after a short period of time [107, 108]. In the presence of long-range interactions, pinwheel densities are ro-bustly selected. Pinwheel densities and pinwheel dynamics during development are further studied in Chapter 6.
Figure 5.7: Robust dynamical pin-wheel density selection. a, Upper map,r = 0.1,g = 0.98andσ = 1.7. Lower map, r = 0.1, g = 2. b, Pinwheel densities ρ in the presence (blue) and absence (green) of long-range interactions (N = 30, param-eter as in a).