Predictability

2

(d) and|κ˜|²(e) share most preva-lent scales (C=0.77^{) (f).}

Before further pursuing this approach, it is interesting to consider the spatial scales on which the softnessκvaries. Using Eqs. (7.15) and (7.17) we rewrite (7.18) into of spatial scales generated through multiple combinations of wavevectorsk_j−k_i. Depending on the attractor, already these constituent wavevectors vary in their modulus within 0 ≤k_j−k_i

≤2kccos(π/2n), i.e. on scales betweenΛ/2 and∞ in the largenlimit, but even larger frequencies are generated from these wavevec-tors by Eq. (7.19). As demonstrated in Fig. 7.11f the range of scales in the power-spectrum of the modification,

∆˜(k)

2(Fig. 7.11d), is shared by the spectrum of the softness, |κ˜(k)|² (Fig. 7.11e). Thus, the phase condition (7.16) in the form of the softnessκcan indeed account for the observed spatial dependency of the induced modification∆.

7.7.2 Predictability

The softness κ provides a means of predicting the effect of stimulation to some extent without knowing the dynamics of the system. The extent to which this is

Figure 7.12: Parameter regime of successful prediction of reorganization. a, Diagram of pre-dictabilityP of modification from softnessκ: Dependence on ICMS durationt_S and strengthδ (in red scales). b, Effect diagram from Fig. 7.8. The strength of induced reorganization can be predicted in the parameter regime of moderate reorganization.

possible, denoted by the predictabilityP, is quantified as the cross-correlation be-tween modification∆(x_S)and softnessκ(x_S). A value ofP close to 1 indicates a high similarity between both quantities. A value close to 0 indicates no predictive power of the softnessκ. The predictabilityP may depend on the durationt_S and strengthδ of ICMS. Comprehensively, this can be studied only by the amplitude equations which are much more tractable both analytically and numerically. There-fore, we first focus on this approximation before investigating the full dynamics.

Fig. 7.12a shows the diagram of predictabilityP for the attractor from Fig. 7.8 to-gether with the diagram of mean modification for the same range of ICMS strength δ and durationt_S as in Fig. 7.8a. Across the parameter regime of moderate modi-fication we observed a high predictabilityP. Values ofP >0.6 were found in the regimeδt_S ≈ 1, especially for small strengths ofδ ≈ 10⁻² and large durations of t_S & 10² with predictabilities up to P = 0.75. The predictability was low only in the regime of no persistent modification and in the regime of maximal modification where all amplitudes are equally activated by a strong external stimulation.

What gives rise to this large predictability in the regimeδt_S ≈ 1? An improved understanding of the effect of ICMS can be obtained by linear response analysis.

Starting point are the amplitude equations in the form Eq. (7.13). Linearizing around a planform solution

Ai = Ae^iφ0i , i=0, . . . ,n−1

Bi = 0 , i =i, . . . ,n−1 (7.21) and splitting the ICMS term into the part acting on the amplitudes and that acting

on the phases leads to equations in the first equation we used r = A²_∑g_{i j}. Thus, their linearized dynamics are decoupled from those of the inactive modesBj. The phases are given by

θ_i(t) =φ_∞−φ⁰_i +sign showing that they are driven towards the ICMS phaseφ_∞. The dynamics of the amplitudesa_i is according to Eq. (7.22) governed by an inhomogeneous system of linear ODEs. The solution is a superposition of the general solution of the homoge-neous system and a particular solution of the full system. Sincegi j =gandgii =1, the homogeneous system has two different growth rates, both stable. The fast rate is given by λ_f = A²(1+ (n−1)g), the slow one byλ_s = A²(1−g) = A² with form asGwithcon the diagonal anddon the off-diagonals leads to

c+ (n−₁)dg = ₁ for the entries of the inverse matrix ofG. The components of the particular solution read

a_i = δr|z_∞|(1+g(n−2))cos(φ_∞ −φ_i)−g∑j6=icos φ_∞−φ_j 2A²(1−g)(1+g(n−1))

= δ|z_∞|(1+g(n−2))cos(φ_∞−φ_i)− |z_∞|/A

2 (7.28)

where we used A² = r/_∑gi j and Eq. (7.15). Thus, the dynamics of amplitudes is decoupled in linear approximation. The particular solution (7.28) represents the amount the original amplitude is shifted under ICMS.

Consider an original amplitude Ai with phaseφ_i = φ_∞ +π, opposite to the ICMS phase. The ICMS strengthδ^∗ suppressing the amplitude completely can be estimated from the conditiona_i =−Aas

δ^∗ = √ ² n 1+√

n+g(n−2) ^(7.29)

where we substituted|z_∞|by its average value h|z_∞|i_x=

r nr

1+ (n−1)g. (7.30)

The temporal scale on which the dynamics relaxes towards the new state is ts = ¹+g(n−1) In particular, for the parameter values used in Figs. 7.8 and 7.12,g = 0.98,n =8, Eq. (7.32) predictsδtS ≈0.6 which agrees well with the estimated value ofδtS ≈1.

This analysis also predicts the minimal magnitude necessary to induce a persis-tent change. The presence of a minimal magnitude explains the deviation from δtS ≈ 1 in the small δ regime. Fig. 7.12 shows that this deviations occurs at δ ≈₁₀⁻³. Following Eq. (7.29) the minimal strength to achieve the necessary am-plitude suppression is on averageδ_min = 1.5·10⁻³. At very susceptible locations this value can be reduced by a factor of≈2 depending on|z_∞(x_S)|.

Following Eq. (7.31) the temporal scale on which the dynamics including ICMS relaxes towards the attractor ists =3.9·10²/rfor the set of parameters used in Fig.

7.12. This accounts for the finding that at least fortS >10³the modification∆and to some extent also the predictabilityP remain constant with increasing duration t_S.

To what extend does the modification∆and the predictability P depend on the initial solution z∞? The linear response analysis is independent of the configura-tionl = (l₀,l₁, . . . ,l_n−1)of active modes suggesting that both quantities are similar for an anisotropic attractor. Indeed, fig. 7.13a shows that both the predictability P and the mean modification ∆ are very similar for a map with an anisotropic spectrum.

Figure 7.13: PredictabilityP (left, in red scales) and modification∆(right, in blue scales) for a map with anisotropic spectrum containingn = 8 active modes (a), a map withn = 9active modes (b),n=₁₇modes (c), and for an attractor of the full dynamics integrated numerically to t = 10⁶ (from Fig. 7.4) (d). Predictions of the linear response analysis areδ_min = 1.5·10^−3, t_S= 3.9·10²(a),δ_min= 4.0·10⁻³,t_S= 2.0·10²(b), andδ_min =4.9·10⁻⁴,t_S=8.3·10²(c).

Note that both the predictability and the modification depend only weakly on the type of map.

Both the minimal strengthδ_min necessary to induce any persistent effect and the relevant time scalets depend on the number of active modes n. The diagrams for a solution withn = 4 modes (Fig. 7.13b) indicate a reduction ofts and an increase inδ_min as predicted from Eq. (7.31) and Eq. (7.29), respectively. Likewise, for an attractor withn =17 modes (Fig. 7.13c)ts was larger andδ_min reduced compared to a solution withn= 8 modes. Generally, the maximal mean modification∆was half the number of active modes and a prediction from the softnessκ was possible in the parameter regime of moderate modification.