Solutions 2 and 3
4.8 Phase diagram
gj,k = g(|π
n(k−j− 1 2|) f˜j,k = f(|π
n(k−j−1 2|)
Assuming intrinsic stability of the solution(B0±, B1±, . . . , B±n−1), the criterion for extrinsic stability becomes
(1±�)−2�B+2+B−2� �
k
˜
gj,k∓2�B+2−B−2� �
k
f˜j,kcos4π
n (k−j− 1 2)<0 or, since this expression does not explicitly depend on the indexj,
(1±�)−2�B+2+B−2� �
k
˜
g0,k∓2�B+2−B−2� �
k
f˜0,kcos4π n (k−1
2)<0. (4.59) In conclusion, stability of a solution under extrinsic perturbations requires that the solution is intrinsically stable and that, in addition, condition (4.59) is fulfilled.
Energies
Consistent with the strong interindividual variability of orientation maps in the brain, the dynamics Eq.(4.13) exhibits a potentially exceedingly high number of multistable solutions. The energy of essentially complex planforms can be calculated from Eq.(4.32). For type 1 solutions
En(I) = − n
2�kg0k(1 +�2/�∗), (4.60) for solutions 2 and 3 the energy is given by
En(II) =− n 2�kg0k
(1 +�)2
1 +�∗ (4.61)
and
En(III) =− n 2�kg0k
(1−�)2
1 +�∗ , (4.62)
respectively and does not depend on the variableslj which identify a particular ECP. Due to the growth of antiparallel modes with increasing |�|patterns for all different realizations lj with phasesφj :=ljΦj+14(1−sign(�))(1−lj)π (Φj arbitrary but fixed) eventually collapse in a single statez(x)∝�nj=0−1zje/o(x,φj) (Fig.4.2a).
4.8 Phase diagram
To answer how SSB affects pattern selection we calculated the phase diagram for the model specified in Eqns.(4.4-4.6) for various values of �. To this end we calculated the energy of all
4.9 Pinwheel Densities solutions found above, that is plane wave solutions, rhombic pinwheel crystals and generalized ECPs up to n= 25. For a dense mesh of parameter values in the(σ, g)-plane and fixed�. From this comprehensive list of fixed point energies we obtained the energetic ground state of the model in different regions of the (σ, g)-plane. Fig. 4.6 shows the configurations of ECPs and rPWCs minimizing the energy. Plane waves are progressively replaced by rPWCs with increasing SSB. Depending on the location in parameter space and on �, a particular angleαminimizes the energy (c.f. Fig. 4.1c). Large n−ECPs are selected when the dynamics is stabilized by long-range interactions (g < 1, σ >Λ). In this parameter regime plane waves and pinwheel crystals are unstable. The degree of SSB q manifested in a given n−ECP attractor depends on � and on the location in the phase diagram. Above a critical line defined by |�∗(n, g,σ)|=|�|antiparallel modes are maximal and |q|= 1(gray area), below that line |q|≤1. Figs. 4.6 and 4.7 show the high sensitivity of the dynamics to even small amounts of SSB, a substantial area in phase is occupied by ECPs with |q|= 1even for �= 0.02.
For the biologically most interesting quasi-periodic ECPs we also determined the regions in the (σ, g)-plane for which a fixedn-ECP is dynamically stable. To this end we used the criteria for extrinsic and intrinsic stability derived in Section 4.7 and calculated the corresponding stability matrices. Our results (shown in Fig.4.7 forn= 20) reveal that the region of stability of an ECP covers a much larger portion of the (σ, g)-plane than the range in which it is the ground state.
The overall shape and position of this stability region for largenwas found to be insensitive to the strength of shift symmetry breaking.
4.9 Pinwheel Densities
The main impact of shift symmetry breaking on aperiodic pattern solutions is the collapse of multistability between different ECPs at the critical point�∗. The bifurcation analysis given above established that this transition is continuous such that the different n-ECPs become gradually more similar with increasing �until they are identical for�=�∗. It is thus an interesting question how statistical properties of the spatially irregular pinwheel layouts change with�. To answer this question we calculate the pinwheel densities of essentially complex planforms for arbitrary degree of shift symmetry breaking. We consider the ensemble of solutionsz(x),
z(x) = �2 n
n�−1 j=0
ei2πnj[�1 +qcos(ljkjx+φj) + i�1−qsin(ljkjx+φj)]
which is identical to the definition (4.54) up to the normalization factor, which we can freely choose for later convenience, since a rescaling of z does not affect the pinwheel configuration.
Here, the phases φj are random variables, the n-tuplelj which identifies the active modes of the planforms is held fixed and q denotes the degree of shift symmetry breaking. As shown in Section 3.11 the pinwheel density can be calculated from the joint probability distribution of the field and its gradient p(z,∇z). We set, without loss of generality, x= 0 and omit the argument
81
Figure 4.6: Phase diagrams of the model, Eqns.(4.4-4.6), near criticality for variable SSB �. The graph shows the regions of the g −σ/Λ plane in which n-ECPs and rPWCs have minimal energy (n= 1−25, n >25dots). Regions of maximally broken shift symmetry [�≥�∗(N, g,σ)] shaded ingray.
Regions where rPWCs prevail is shaded inblue,intensity level codes for the included angleα. (light blue:
π/4≤α≤π/2 :dark blue)
4.9 Pinwheel Densities
Figure 4.7: Stability regions of ECPs withn= 20active modes. Depicted is the region in theg−σ/Λ plane for which these planforms are a stable solution of the dynamics for�= 0,0.02and0.2and coexist with planforms of nearbyn, e.g. n−1,n+ 1. Beyond that region the solution is unstable with respect to intrinsic or extrinsic perturbations, i.e. growth of additional modes or decaying of active modes, respectively. Inside of the region defined by the black lines then= 20 solution minimizes the energy. For
�>0the q values range in[0,1] as displayed by the green colorcode. Dashed yellow line denotes the critical line�=�∗(N, g,σ), above whichq= 1.
in the following, such that
z = �2 n
n�−1 j=0
ei2πnj[�1 +qcosφj+i�1−qsinφj]
∇z = �2 n
n�−1 j=0
ei2πnjljkj[i�1−qcosφj−�1 +qsinφj].
In the following we decomposez into its real and imaginary part, such that
z=R+i I.
For large nthe distributionp(z,∇z)can be approximated by a Gaussian
P(v) = 1
(2π)3√
detW exp�−1
2vTW−1v� (4.63)
where
v=(R, I,∂xR,∂yR,∂xI,∂yI)T and W denotes the corresponding covariance matrix
Wij =�vivj�,
83
the average being performed over the angles φj. First we evaluate the moments involving just which depend on theplanform anisotropy,
�
a measure for how anisotropically the active modesljkj are distributed on the critical circle. The modulus of �χis small for an isotropic distribution of wavevectorsljkj. Without loss of generality we can choose the coordinate system such thatχ� =χ·(1,0), which implies that �I∂yR� and
�R∂yI� vanish. An estimate for the upper bound on χ := |χ�|for a given n can be given by considering the most anisotropic case, lj = 1j = 0. . . n−1, which for q = 0 corresponds to planforms where all active modes are distributed on a semicircle
χmaxn = 1
4.9 Pinwheel Densities For kc =|k|= 1, which is assumed in the following,0≤χn≤1 sinceχmaxn ≤χmax1 =kc. Note, however, that pinwheels only exist for n≥2 modes, such that in our case of interest the values of χn are actually bounded by
0≤χn≤χmax2 =�1/2.
Finally, the nonvanishing moments with two derivatives are
�(∂xR)2�= 1
2 =�(∂yR)2�=�(∂xI)2�=�(∂yI)2�. All together the covariance matrix reads
W =
1 . . . �1−q2χ .
. 1 −�1−q2χ . . .
. −�1−q2χ 1/2 . . .
. . . 1/2 . .
�1−q2χ . . . 1/2 .
. . . . . 1/2
The pinwheel density is obtained as described in 3.11 by calculating the expectation value of ρ=�δ(z)|∂xR∂yI−∂yR∂xI|�
with respect to the probability densityP(v), Eq.(4.63). The average can be performed using polar coordinates,
∂xR=r cosθ,∂xI =r sinθ,
∂yR=scosψ,∂yI =ssinψ and yields the result
ˆ
ρ(χn) :=ρ(χn)Λ2=π�1−2(1−q2)χ2n (4.64) whereΛ= 2π/kc = 2π denotes the typical wavelength of the pattern.
Distribution of Planform Anisotropies
The energies of essentially complex planforms, however, do not depend on the planform anisotropy.
Since they are degenerate with respect to the n-tuplel= (l0, . . . ln−1), each of the2n possible sets occurs with the same likelihood. What can we say about the resulting distribution of �χ? In the large nlimit it can be approximated by a Gaussian distribution with mean zero, since
�χ�n� = 1 n
n�−1 j=0�ljkj�
= 0
85
Figure 4.8: (a)-(c)Pinwheel densities for all realizations of ECPs with3≤n≤17and different degrees of shift symmetry breaking�. (d)Pinwheel densities for n= 17(dots)and forn→ ∞in the Gaussian approximation(gray region).
4.9 Pinwheel Densities
Using Eq.(4.64) it is possible to expressχnin terms of ρn χn= 1 Together with Eq.(4.65) one obtains for the probability density of ρn
Pρ(ρn) = Pχ(χn(ρn))����dχn
whereΦi denotes the imaginary part of the error function. For the second moment we obtain
�ρ2n� =
since
n→∞lim�ρn�=π,
which means that valuesρn are sharply peaked around π for sufficiently large n. Also, since 0≤χn≤ 1
n| 2
1−eiπn| :=χmaxn and
nlim→∞χmaxn = 2 π
one obtains the following estimate for the range of observable pinwheel densities π
� 1− 8
π2(1−q2)≤ lim
n→∞ρn≤π.
This predicted range of pinwheel densities agrees well with densities numerically calculated from ECPs. Figure 4.8 depicts that the range of pinwheel densities found for different planforms continuously shrinks with increasing strength of shift symmetry breaking and collapses to a single unique pinwheel density at �=�∗.