7.7 Predicting soft spots
7.7.3 Predicting the full dynamics
Next we asked to what extent the reorganization ∆of the full dynamics Eq. (7.1) is predicted from the phase condition (7.16). Fig. 7.14a shows the map of mod-ification ∆(xS) (reproduced from Fig. 7.4b) and the map of softnessκ(xS) (Fig.
7.14b) calculated at corresponding cortical locations. The softnessκ reproduced many characteristics of the modification∆including variation on small scales with nearby locations often having very different values, and variation on scales large compared to the typical scale of the column pattern. Statistically, both quantities were related (Fig. 7.14c,C = 0.40). A large ICMS effect was obtained only at sites with large softnessκ. At locations with small softnessκ ICMS did not cause any persistent changes in the layout of the map. Thus, the softness accurately predicts
Figure 7.14: Predicting potential soft spots. a, Modification∆from Fig. 7.4b. b, Softnessκ coded in gray scale. Bright (dark) regions mark low (high) values ofκ. Note that the softnessκ shares structural features of the modification∆. c, Scatter plotκvs.∆for all analyzed locations x0. The modification∆is large only where the softnessκ is large, i.e. a large value ofκ is a necessary condition for significant reorganization.
the ’hard’ spots, where the induction of persistent reorganization is impossible. A large value of softness, instead, is a necessary, but not a sufficient condition for inducing significant reorganization.
Why does the softness fail to provide a sufficient condition? At least partially, this is due to a considerable variation in the amplitudesAjof active modes shown in Fig. 7.15a (orange crosses). Generally, modes with smaller amplitudes were sup-pressed more severely. However, this was not simply due to their smaller size, but rather due to a reduction in stability as indicated by the stronger reduction they experience under ICMS. Fig. 7.15b shows that amplitudes with small modulus are much more often subject to switching than amplitudes with initially larger modu-lus. In fact, two of the larger amplitudes experienced no switching at all. One of the larger amplitudes (no. 5) often encountered successful suppression, despite its relatively small change immediately after ICMS. Fig. 7.15b furthermore shows that the active amplitudes often altered their magnitude through ICMS. Apparently, the range of possible changes was discrete. An explanation for this phenomenon is unclear at present, and could include the loss of degeneration of amplitudes for largeror a shifting of grain boundaries induced by ICMS. A thorough analysis of these phenomenom would be promising but requires also 5th order contributions to the amplitude equations or to allow for spatial variation of amplitudes (envelope equations) which is beyond the scope of this work.
The softnessκcan be refined by incorporating the dependency of the modifica-tion on amplitude differences into its definimodifica-tion. Crucial is a different treatment of amplitudes with phases opposed to the ICMS phase. A simple way to do this is to
Figure 7.15: Modes with small amplitudes switch more likely. a, Absolute values A of the initially active amplitudes in the map from Fig. 7.4a (orange crosses) and their values immedi-ately after ICMS applied at the 50×50 sitesxSshown in Fig. 7.4b (black dots). b, Amplitudes from a (orange crosses) after relaxation from ICMS (black dots). Amplitudes can either decay and switch with new amplitudes or alter their magnitude discretely. Note that modes with small amplitudes are stronger suppressed by ICMS and experience with higher probability complete suppression after relaxation.
Figure 7.16: Contribution of ampli-tude differences to modification. a, Constellation of amplitudes Aj (nor-malized to
Aj
j = 1) at sitexS with large softnessκbut small modification
∆(compare Fig. 7.14c). Illustration as in Fig. 7.10. b, Configuration at site with largeκand∆.
define the softness by the SD
κ0 =SD
A0jcos φj−φ∞
(7.33) where
A0j = A−j q, cos φj−φ∞
>0 A0j = Aj, cos φj−φ∞
≤0 (7.34)
ensuring forq > 0 and a prevalent contribution of amplitudes with smaller abso-lute valuesAjif their phasesφjpoint in opposite direction to the ICMS phaseφ∞. The softnessκ0is small either if no amplitude exhibits a phase opposite to the ICMS phase (as in the case ofκ), or if such amplitudes exhibit large moduli (Fig. 7.16a).
Only if both conditions are satisfied (as in Fig. 7.16b) the softnessκ0becomes large.
Figure 7.17: Improved prediction of potential soft spots. a, Modification∆from Fig. 7.4b. b, Softnessκ0 coded in gray scales. Bright (dark) regions mark low (high) values ofκ0. Note that the softnessκ0 shares more structural features of the modification∆asκ(Fig. 7.14). c, Scatter plotκ0 vs. ∆for all analyzed locationsxS. As in the case ofκ, the modification∆is large only where the softnessκ0 is large. In addition, a large value ofκ0 implies a large ∆. Thus, a large value ofκis a necessary and sufficient condition for significant reorganization.
For the full dynamics the softnessκ0 improves the predictability P of modifica-tion∆. Fig. 7.17a, b compares both quantities for the map from Fig. 7.4a. Not only appear the dark regions (low values) to overlap, as in the case forκ, but now also the bright regions (high values). Both quantities were considerably correlated as shown by the scatter plot in Fig. 7.17c (C = 0.56). In contrast to Fig. 7.14c, most sites with large softness showed indeed significant modification. Even higher pre-dictabilities could be obtained by e.g. completely eliminating the contribution of amplitudes with large magnitude and phase opposite to ICMS. However, since the generalization to solutions obtained with other parameter and different dynamics is not clear, this shall not pursued further. This shows that by incorporating also the magnitude of amplitudes into the definition of the softness, as in Eqs. (7.33) with (7.34), even larger predictabilities can be attained, at least for the analyzed model.
7.8 Discussion
We explored ICMS in a dynamical model of the reorganization of the orientation map. We found that local stimulation can induce both transient and persistent changes dependent on the stimulation site. In general, the dynamical rearrange-ment continued after termination of local stimulation. ICMS induced pinwheel mo-tion including stimulamo-tion-driven generamo-tion and annihilamo-tion of pinwheel pairs.
The degree of reorganization showed a complex and sensitive dependence on the location of the ICMS site that was relatively robust against variation of model pa-rameters leading to the notion of soft spots in the map at which ICMS induces
spatially widespread and persistent reorganization. Since in a large and relevant parameter regime, the location of soft spots depends only little on the parameters of the model but on the layout of the map, it was possible to predict the reorgani-zation to a substantial degree from the layout of the map.
In the analyzed model, soft spots are a consequence of the aperiodic structure of the orientation map and the property of multistability by mode competition. In-ducing large reorganization by ICMS translates into a switching between different attractors which in turn requires switching among active and inactive modes. This can be achieved by suppressing a fraction of the active modes such that inactive modes can grow and finally prevail over the formerly active ones. Due to the en-forcing nature of ICMS, a suppression of several active modes is realized only at particular locations in the map. The stimulation in Fourier representation acts with a phase given by the preferred orientation at the ICMS site. Since it is the preferred orientation, most Fourier components in the map exhibit a similar phase and are therefore enforced by this stimulation. However, if a fraction of modes exhibits an opposite phase, it will be suppressed by such a stimulation. Thus, following this interpretation, soft spots are at least partially determined by the phase of the stim-ulation relative to the phases of the active modes and therefore by the location of the ICMS site in the map.
This interpretation was formalized by the softness κ which quantifies the effi-ciency of a stimulation in suppressing active modes based on the map before ICMS.
For the model, the softness enables to predict the ICMS sites where no significant reorganization is induced. Thus, it identifies the ICMS sites of potentially high im-pact. This has interesting consequences for the experiment circumventing two diffi-culties a prediction of soft spots during an experiment based on a dynamical model would encounter. First, an analysis of soft spots based on this model requires many hours of computation time, even on a modern computer cluster (for instance, Fig.
7.4 required 50 days of computation on a Pentium 4 processor). Thus, without con-siderable improvements of numerical methods or computer hardware, it appears impossible to predict the soft spots online, during an ICMS experiment. The soft-ness, in contrast, can be calculated from a map within minutes since it is based on the layout of the map, in fact, the only specific quantity being measured before an ICMS experiment. Second, the softness is more general than the model. Its ap-plicability requires multistability by mode competition for solutions with multiple active modes. This property is not necessarily restricted to the class of models with permutation symmetry albeit different classes fulfilling this requirement are not known at present.