A representative example of a developing orientation map is presented in Fig. 6.1.
After t = 102, the state already resembled an orientation map observed in the vi-sual cortex. The initially broad spectrum was narrowed due to the strong damping of the linear part Eq. (5.2) further away from the critical circle |k| = kc. On the critical circle, modes grew initially exponential with rate r until their amplitudes were of order∼r1/2. For the choice ofrand the amplitude of the initial conditions used, this phase typically ended at t ≈ 101 (Fig. 5.1). The subsequent phase of nonlinear competition among modes lead to the development of isolated ampli-tudes distributed around the critical circle att=102, and to a stable configuration of amplitudes att=104. The solution consisted of a finite number of active modes as predicted by the amplitude formalism (see Chapter 5 and 2). Different from the planform solutions, however, phasesφi of the amplitudes Aj = |Aj|eiφj were not degenerate anymore. In fact, phasesφi still changed after t = 104 and complete stationarity was not always reached even att =106.
The amplitude formalism provides an approximate description for the second phase of mode competition. Within this framework modes develop on a time scale
Figure 6.1: A developing orientation map. A band-pass filtered Gaussian random map as initial condition att = 0was advanced untiltf = 106. Integration of Eq. (5.1) and parameters as in Fig. 5.1 (r = 0.1, g = 0.98,σ = 1.7Λ, Γ = 17). The upper row shows a subregions of the angle map at initial, final and intermediate times (A= (4.25Λ)2, smoothed for illustration) with beneath the power spectrum of the entire map (origin of Fourier-space in center of annulus).
Note that the map is considerably stable already aftert=102.
ofO(r). Thus, during the first two phases orientation maps might approximately develop on time scaleO(r). Stable solutions of the amplitude equations of third order are degenerate in phasesφj. Since the next correction is of 5th order the adjustment of phases will happen on a time scale of at leastO(r2). This suggests that in maps the selection of phases φj takes place on a much larger time scale consistent with the long term dynamics depicted in Fig. 6.1 and in the following figures. To compare results obtained for different values ofr during the phase of mode competition, they were presented using the rescaled timet(Eq. (5.19)), as in previous sections.
A more quantitative view on the map development governed by Eq. (5.1) is revealed by tracking the similarity across different stages of development. For this purpose consider the time dependent cross-correlation
C(t,t0) =
s Re(R d2x z(x,t)z¯(x,t0)) pR
d2x|z(x,t)|2R d2x|z(x,t0)|2 (6.4) between the map at timetand the map at a fixed reference timet0. Fig. 6.2 shows the correlationCfor three different reference timest0and for three values ofr. Each cross-correlation was averaged overN =10 different solutions.
Forr =0.1 (red curves), maps developed rapidly during the early phase and the
correlation between the initial and the developing map dropped considerably be-foret =103(dashed curve). Interestingly, however, even at very late stages, maps did not become uncorrelated from the initial maps, but maintained a correlation of C ≈ 0.2. Moreover, very late maps att = 106 were withC ≈ 0.5 still moderately correlated with their early states att=102(dashed-dotted curve), a time still at the beginning of the phase of mode competition. When the modes reached their final configuration att =104 maps were already very similar to their very late states at t = 106. During phase relaxation, correlations did not change substantially indi-cating that on average, phases experience only small changes, at least untilt=106 (solid curve). Thus, largest changes occurred beforet ≈ 103. Aftert ≈ 104, maps were basically stable. The map layout strongly resembles the final layout even be-fore that time, after t = 102 as shown by the representative example in Fig. 6.1 (cross-correlation between maps at t = 102 and t = 106, C = 0.45). Consistent with these results, several experimental studies observed a persistence of the basic layout of the orientation map during development. Experimental paradigms for testing the predicted dynamical changes during development will be proposed at the end of this chapter.
Figure 6.2: Quantified changes during map development. Cross-correlation C(t,t0)(Eq. (6.4)) between the map and the map at fixed reference times t0 = 100 (dashed lines), t0 = 102 (dashed-dotted),t0 =106 (solid) for different val-ues of r. Each curve represent an av-erage over N = 10 cases. Parame-ters as in Fig. 5.1. Note that already after t ≈ 102, maps are considerably correlated with their very late states at t =106.
A dependence of map stability on r was observed especially during the inter-mediate and late stages of development. Solutions obtained for different values of r showed identical time courses of C(t) during the initial phase of development (dashed curves). Approximately betweent = 103 and t = 104, a dependence on rbecame apparent as highlighted by the dashed-dotes curves. Whereas for small r, solutions remained relatively similar to the early state att = 1·102 during this stage of development (C ≈ 0.6), they diverged stronger from the early state for largeras indicated from the decrease ofCbelow 0.4. At late stagest>1·104, cor-relations stabilized. Consistent with a faster phase dynamics for large r, changes during this phase were stronger for solutions with a larger value ofr(solid curves).
Following these results, the development of orientation maps may be subdivided into three stages. First, a phase of linear growth taking place for 0 ≤t≤101when
amplitudes near the critical circle grow until reaching their maximum. Second, the phase of nonlinear competition among modes between 101 < t < 104 when the finally active amplitudes are selected. Third, the relaxation of phases of the selected amplitudes fort >104. This defines two time scales central for the study in this section: The end of the linear phase denoted by to ≈ 101 and the end of mode competition called the cross-over phaset∗ ≈104.