7.5 Soft spots in the orientation map
7.5.6 Modification diagram
with width 1/σS. Since this width is large compared to the typical wavenumber kc = 2π/Λwe may set ˜µ(kc) = 1 for simplicity. With Eq. (7.11) the ICMS term (7.10) exhibits a uniform complex phase
φ∞(xS) = arg{z∞(xS)} (7.12) which is determined by the orientation preference at the ICMS site byφ∞(xS) = 2ϑ(xS).
Hence, the amplitude equations accurate at orderr3/2read
∂tAi =rAi−
∑
governing the development of amplitudes Aj = Aj
eiφj under the presence of ICMS betweent = 0 and t = tS. Eq. (7.13) is for 0 ≤ t ≤ tS a set of inhomoge-neous equations. The ICMS term in (7.13) acts as an additional force with constant strength and constant phase on the modes on the active amplitudes Ai. For very small strengthδthis force will mainly shift the phasesφjof the active amplitudes towards the ICMS phaseφ∞(xS). For sufficiently largeδadditional amplitudes on the critical circle may grow or previous active amplitudes are suppressed. In the following, the dependency of the ICMS effect on the two stimulation parameters strengthδand durationtS shall be evaluated using amplitude Eq. (7.13).
7.5.6 Modification diagram
The modification diagram displayed in Fig. 7.8a captures the dependence of the mean modification ∆ = h∆(x)ix induced by ICMS on its two parameters ICMS
Figure 7.8: Modification diagram: Dependence of modification ∆ on ICMS duration tS and strengthδ. a, Mean modification ∆ = h∆(xS)ix
S calculated by the amplitude equations 7.13.
ICMS was applied to a planform solution withn=8active modes (inset). b, Modification∆and its power spectrum
∆˜
2 (illustration as in Fig. 7.4b) for different parameter regimes (marked by stars in a). In the smallδ, tS regime the configuration of active modes remains unchanged under ICMS. For sufficiently largeδ, a single mode is flipped for some ICMS sitesxS(I). Under very strong stimulation,4modes switch on average (IV). Soft spots are found in the parameter regime of moderate modification given by10−1.δtS. 102fortS< 102and10−3 .δ .100 fortS>102. Note that∆is similar for I IandV(c=0.66) and for I I IandV I (c=0.55).
strengthδand durationtS. Calculations were carried out using a planform ofn=8 active modes. For short and weak stimuli the modification was close to zero – the stimulation failed to switch any of the active modes. For large durations tS >
103, the modification depended on δ only. In fact, there was a minimal strength δmin ≈ 10−3 necessary for inducing any modification at all in a map. A nonzero modification was obtained for strengths as small asδ>10−3for durationtS >102 or for tS < 102 for strengths fulfilling δtS > 10−1. For larger δ, the mean effect
∆ was found to increase gradually over more than three orders of magnitude of ICMS strengthsδ. The maximal effect was a switching of more than half of the originally active modes. In this regime ICMS induced an activation of all n = 16 amplitudes to an almost equal amplitude. The selection of the finally active modes after termination of ICMS becomes essentially a random process in which on averagen =4 modes switch.
Thus, soft spots are only found in the parameter regime in which usually a frac-tion of the possible modes are switched. This regime of moderate modificafrac-tion, is therefore of particular interest in the following. As will become apparent later in this chapter, in this regime, the modification can be predicted to some degree from the ICMS site in the map. The strength of the modification∆ increases gradually
Figure 7.9: The modification ∆ largely depends on the product δtS of ICMS strength and duration. a, Traces of cross-correlations C parallel (solid) and perpendicular (dashed) to the level lines of equal mean modification ∆. The four reference modifications at tS = 100 and δ = 10−1, 10−1/2, 100, 101/2 are marked by different symbols. b, Correlations parallel to level lines vs. ICMS durationtS remain large over a broad width. c, Correlations perpendicular to level lines decay rapidly.
with the strengthδ and duration tS of ICMS in this regime. Fig. 7.8b shows the representative maps of modification∆from 6 different regions of the modification diagram. Whereas the spatial dependence of stimulation is essentially random for very large and strong stimulation, in the regime of moderate modification, it ex-hibits an intriguing spatial dependence resembling the results form the numerical calculations of the full dynamics (compare Fig. 7.4 and Fig. 7.7). As for the full dynamics, in this regime the modification ∆ varies on spatial scales smaller and larger than the pattern itself as will be discussed in further detail in Section 7.7.1.
Fig. 7.8 shows a further property of the modification∆that is worth noting. The spatial structure changes only little when moving along the level lines in the mod-ification diagram. The modmod-ification maps II and V were significantly correlated (c =0.66) despite their relatively large distance in parameter space. A similar cor-relation was found forIIIandVI (c = 0.66). To evaluate this more systematically we calculated cross-correlations of the maps of modification parallel and perpen-dicular to the equal effect level lines starting from four maps within the sensitive regime at equal timet=100, but with various strengthδ(Fig. 7.9). Along the lines of equal mean modification their spatial structure was largely reproducible, espe-cially along the line given byδtS = 1 with correlations close to 1 (Fig. 7.9b, solid lines in a). The spatial structure of the modification∆was much less similar along the perpendicular direction and correlations decayed rapidly (Fig. 7.9c, dashed lines in a). Thus, the combination of stimulation strength and duration mainly de-termines the induced modification. For a durationtS <103the result depends to a good approximation only on the productδtS.
Figure 7.10: Layout of orienta-tion map implies the occurrence of soft spots. a, Fourier rep-resentation of map before ICMS (schematic). Active amplitudes with moduli
Aj
(black dots) and phasesφj (blue arrows, shown for two amplitudes). b, c, Phase condi-tion for ICMS sites with small mod-ification ∆. Under ICMS, phases φj (diamonds) are shifted towards the ICMS phaseφ∞ = π (red ar-row) strengthening all active ampli-tudes. d, Phase condition at ICMS site with large ∆. Here, two am-plitudes are suppressed and can therefore be exchanged by different amplitudes.