Stationary solutions and their stability

The simplified interaction kernels in equations2.3.1and2.3.2have the advantage of being described with only three bounded parameters. Therefore, for all possible combinations ofS,Wandgequation 2.2.2can be used to find the patterns that are stationary and equations2.2.6 and2.2.8to calculate their stability comprehensively screening all exact solutions for all parameters combinations.

Unselective state

As stated before,A=0 is always a stationary solution independent of the combination of parameters.

Using the definition of ˜r as in equation2.2.3, λmax(k) = r > 0, so the unselective state is not stable and the selectivityAwill increase with time.

Perfectly periodic and uniformly selective state

The condition forz= A_ke^ikx to be stationary isA_k >0 in equation2.2.2. Since there is no orientation selective inhibition, this condition is fulfilled when λ(k) > 0 in equation2.2.6. With the reduced interaction kernels the expression forλis independent ofg

λ(k) =r˜+Se^−12k2W2−(1−^S)e^{−12k2(1−W)2} (2.3.3) The range of patternskthat makeλ(k)>0 depends on the choice ofrin equation2.2.3. The focus is on the maximum ofλsince it is always positive even with very smallr. λ(k)has two extreme points

kI =0 and kI I = vu utlog₍

1−^S)(1−^W)² SW²

1

2−^{W (2.3.4)}

They exchange characteristics depending on the choice ofSandW:

Figure 2.3.2:Stability of stationary patterns. a) - c) The growth rate of the stationary patternλ(k)is plotted for a combination of interactions parameters belonging to a) type I, b) type II and c) type III. On top three examples of the growth rate of a perturbation over the stationary patternµ(k,Q)is plotted, withQ→ 0 (dotted line), with the maximum of µfor a givenk: Qmax (short dashed line), and with Q → ^∞(long dashed line). The patterns with wavenumberkwhere the growth rate of the instability is positive for anyQare unstable and are marked with gray.

• Type I interaction: For ^(1−S_SW⁾⁽¹⁻2 ^W)2 ≤ ^{1 and S} ≥ ¹₂^,kI is the maximum and ˜r = r−(2S−¹). With these parameters the uniformly selective statek=0 is stationary.

• Type II interaction: For ^(1−S_SW⁾⁽¹⁻2 ^W)2 >_{1 andW} < ¹

2,kI I is maximum and

˜

r =r− ^S(1−S_SW^)(1−2W)2

W2

−¹+2W

−(1−^S)^(1−S_SW⁾⁽¹⁻2^W)2

⁽₋¹⁻₁₊^W_2W⁾²!

. ChoosingSandW in this range makes the perfectly periodic state stationary.

• Type III interaction: ForS = 0 or for S ≤ ¹₂ andW ≥ ¹₂, the single maximum at finite wave number is lost and an infinite number of modes satisfyλ(k)>0. In this case, even with smallr, there is a continuum of periodic patterns with arbitrary high wave numbers that are stationary.

In every interaction type the stability of the stationary patterns will be analyzed. An example of this procedure for a limited number of perturbations is shown in figure2.3.2a) to c). In every figure the growth rate of an ordered solutionλ(k)in equation2.3.3is plotted for different interaction pa-rameters belonging to interaction type I, II and II respectively. The bifurcation parameter isr = 0.1.

Plotted on top of each curve are three examples of the growth rate of the perturbation µ(k) from equation2.2.8, with Q→ 0 (dotted line), with the maximum ofµfor a givenk: Qmax(short dashed line), and withQ → ^∞(long dashed line). In all examples, wheneverµ(k)is positive for any given Qthe patternz = Ae^ikx is unstable and is marked in the figures with gray. Using only this limited number of instabilitiesQin the example, with type I interaction in a) the stable pattern with fastest growth is k = 0, with type II interaction in b) the stable pattern has a finite wavelengthk > _{0, and} with interaction type III in c) all ordered patterns are unstable. Having this in mind, next the different interaction types will be treated separately showing that the results of this example are general for the whole interaction parameters in each type. As stated, the focus is on the maximum of λ(k) in equation2.3.4, since it is always positive even with very smallr.

2.3 The transition from order to disorder 35

Type I interaction InsertingkI = 0 in equation2.2.8leads toΓ(0,Q) = ^Υ_±(0,Q) = 0 and|^A0|² = λ(0). The eigenvalue of the perturbation can be written as

µ(k =0,Q) =λ(Q)−^λ(0)

Since in interaction type Iλ(0)is maximum,µ ≤ ^{0 for all}Q, making the continuous pattern stable in this regime. In this interaction regime the emerging patterns will always be a uniform orientation preference.

Type II interaction For the interaction regime type II where there is a single maximum atkI I = s one equation as in type I. Here the focus is on three different perturbation types for the unselective interaction case (g=0),Q→0,QmaxandQ→^∞. In all those cases the pattern with wavenumberkI I

will be shown to be stable. In the next section the stability of the map will be investigated numerically for moreQandg6=0, showing that the results also holds for those cases.

Long wavelength perturbationQ→ ^0: Since many derivatives are involved in the next steps, the following notation will be used

ψ(k) = Se^−12k2W2

ξ(k) = (1−^S)e^{−12k2(1−W)2}

Equation2.2.8will be expanded in Taylor series around zero, such thatQ≃0. Only even terms ofQ will be non-zero because of the symmetry of the system. Up to second order the terms are

µQ→⁰(k,Q) = ^k Inserting the maximumkI I leads to

µQ→⁰(kI I,Q) =−^SW2

Finite wavelength perturbation Qmax: For the finite wavelength perturbation the extrema of the eigenvalueµwith respect ofQwill be found and their effect on the stationary pattern will be calcu-lated

To set equation 2.3.5to zero, first notice thatγ = ±1 only when A = 0, which is not a stationary pattern. The derivative consists of two parts, where only the sign of Q is changed. µ is even, so

∂µ(k,Q)

∂Q is odd and the zeros will be symmetric. The zeros of the first term will be modified by the zeros of the second term and the other way around. Since there is not an analytical expression that sets both parts to zero simultaneously, the location of the maximum is approximated using only the term that gives the positive zeros. Those are

Q1= k Q2,3= k±^kI I. with real wave numberkin equation2.3.4. Since in interaction regime II this holds, the pattern z(x) = A(kI I)e^{ikI Ix} is stable under this perturbation. This expression is negative forW < ¹

2, so ^(1−S_SW⁾⁽¹⁻2^W)2 > 1, making the wave numberkI I stable with this interaction type II.

2.3 The transition from order to disorder 37

InsertingkI I this gives the normalization constant, rI I =S is stable under this kind of perturbations.

Type III interaction For the interaction regime where there is no single maximum a short wave-length perturbation is used,Q→^{∞. Forλ}(k)>0, the growth rate of the perturbation can be written as The eigenvalue of the instability decrease exponentially withk. ForS=0 the growth rate is positive for allk. ForS>0 the condition for instabilityµQ→^∞(k)>0 can be rewritten as With g = 0, i.e. with no orientation specific connections, for S < ¹

2 the left side hand is positive (zero for S = ¹₂) and for W > ¹

2 the right hand side negative (zero for W = ¹₂), satisfying the condition of instability for all k in this interaction type. With g > 0, the second part of the right hand side is zero atk = 0 and otherwise positive with maximum log(1.5) for large k and g = ¹₄. This only modifies the stability whenS ≃ ¹₂ andW ≃ ¹₂ for smallk. In general, in this interaction regime all patterns, although being stationary solutions, are not stable under perturbations. Even with orientation selective interactions no ordered arrangement of neurons is possible in this regime, so the continuum limitN → ^∞can not be used and the neurons have to be defined individually as zi = Aie^i2θi.