Viewed from a dynamical systems perspective, the activity-dependent remodeling of the cortical network described above is a process of dynamical pattern forma-tion. In this picture, spontaneous symmetry breaking in the developmental dy-namics of the cortical network underlies the emergence of cortical selectivities such as orientation preference [61]. The subsequent convergence of the cortical circuitry towards a mature pattern of selectivities can be viewed as the development to-wards an attractor of the developmental dynamics [107]. This is consistent with the interpretation of cortical development as an optimization process. In the fol-lowing, we will briefly describe a model [105] that is based on this view. This model is the starting point of a number of analyses carried out in this thesis.
2.2.1 A pattern formation model
Self-organization has been observed to robustly produce large scale structures in various complex systems. Often, the class of patterns emerging depends on fun-damental system properties such as symmetries rather than on the system specific details. Its formation can therefore be described by an effective model incorporat-ing only these properties. For patterns with a typical scale such a model is
∂tz = F[z]
= LSHz+N2[z] +N3[z] +· · · (2.5) where the linear part is the Swift-Hohenberg operator [23, 89]
LSH =r−k2c +∇22 (2.6)
and z(x,t) is a scalar field. If the bifurcation parameter r < 0, the homogeneous state z(x) = 0 is stable. For r > 0, a pattern with average spacing close to Λ = 2π/kc emerges. The lowest order nonlinearities N2 and N3 are quadratic and cu-bic in z, respectively. The form of these nonlinearities determines the class of the
emerging pattern, whether e.g. hexagons, rotating stripes, spiral waves, or of other type.
In analogy to the theory of complex systems Wolf [105] adopted a model of the form Eq. (2.5) with nonlinearities derived from key features of the visual cortex.
As in experimental recordings, orientation columns are represented by a complex field [90, 107]
z(x) =|z(x)|ei2ϑ(x) (2.7) whereϑis the orientation preference and|z|a measure of the selectivity at location xin the map. The factor 2 in the exponent accounts for theπ-periodicity of stimulus parameter orientation. The model includes the effects of long-range intracortical connections between columns with similar orientation preference (Fig. 2.2). Based on the repetition of columnar circuits across cortex it is assumed that the dynamics is symmetric with respect to translations,
F[Tˆyz] = TˆyF[z] with ˆTyz(x) = z(x+y), (2.8) of the cortical sheet. This means that patterns that can be converted to one another by translation or rotation of the cortical layers are equivalent solutions of Equa-tion. It is further assumed that the dynamics is symmetric with respect to shifts in orientation,
F[eiφz] = eiφF[z], (2.10) such that patterns whose arrangement of iso-orientation domains is the same but whose orientation preference values differ by a given amount, are equivalent so-lutions of Equation. Soso-lutions contain equal representation of all stimulus orien-tations. Moreover, we neglect all possible couplings to other visual cortical repre-sentations such as ocular dominance or retinotopy. Considering only leading order terms up to cubic nonlinearities the nonlinear part reads [105]
N3[z(x)] = (g−1)|z(x)|2z(x) + and N2 = 0. Long-range interactions are mediated through convolutions of a Gaussian controls local and nonlocal influences. The model minimizes an energy functional during development. It is consistent with synaptic models based on Hebb-type plasticity, e.g. [32, 62, 92].
2.2.2 Weakly nonlinear analysis
Close to the finite wavelength instability (atr1), stationary solutions to Eq. (2.5) with (2.6) and (2.11) can be calculated analytically using a perturbation method called weakly nonlinear stability analysis. When the dynamics is close to a finite wavelength instability, the essential Fourier components of the emerging pattern are located on the critical circle. Thus, solutions may be sought in terms of Plan-form patterns
z(x) =
∑
j
Ajeikjx (2.13)
composed of a finite number of Fourier components. By symmetry, the dynamics of amplitudesAiof a planform are governed by amplitude equations
A˙i =rAi−
∑
where j− denotes the index of the mode antiparallel to mode j. The form of Eq.(2.13) is universal for models of a complex fieldzsatisfying symmetry assumptions (2.8-2.10). All model dependences are included in the coupling coefficientsgi j and fi j and may be obtained fromF[z] by multiscale expansion [23, 57]. Denoting the angle between the wave vectorski andkjbyα, the coefficients read
gi j =
Stationary solutions of Eq. (2.14) are given by planforms z(x) =
Figure 2.4: Essentially complex plan-forms with different numbers n = 1, 2, 3, 5, 15of active modes: The pat-terns of orientation preferences θ(x) are shown. The diagrams to the left of each pattern display the position of the wavevectors of active modes on the critical circle. Forn= 3, there are two patterns; forn =5, there are four;
and forn=15, there are612different patterns.
distributed equidistantly on the upper half of the critical circle and binary values lj = ±1 determining whether the mode with wave vector kj or with wavevector
−kj is active. These planforms cannot realize a real valued function and are called essentially complex planforms (Fig. 2.4). For such planforms the third term in Eq.
(2.14) vanishes and the effective amplitude equations for the active modes reduce to a system of Landau equations
A˙i =rAi−
∑
j
gi j
Aj
2Ai (2.19)
with stationary solutions (2.17) with amplitudes of equal modulus
|Ai| = s r
∑jgi j (2.20)
and an arbitrary phase φi independent of the mode configuration lj. If the dy-namics is stabilized by long-range nonlocal interactions (g < 1, σ > Λ), large n planforms are the only stable solutions. In this long-range regime, the order n grows linearly with the interaction range 2πσ/Λ. For a given order n, different planforms are degenerated in energy. This is a consequence of a fourth symmetry of the nonlinear part (2.11) namely the permutation symmetry
N3(u,v,w) = N3(w,u,v). (2.21) This symmetry implies that the relevant stable solutions are essentially complex planforms which in turn guarantees that all stimulus orientations are represented in equal parts. The property of multistability is characteristic for this model class and will play an important role in Chapter 7.
2.2.3 Pinwheel density
Figure 2.5: Pinwheel densities of the constitu-tively different essentially complex planforms for n=3, 4, ..., 17. In the case ofn=3, a periodic pattern is formed in which the pinwheel densi-ties are exactly equal to 2 cos(π/6) ' 1.73 and 6 cos(π/6) ' 5.2. The points mark the numerically determined pinwheel densitiesρiof the constitutively different planforms for different numbers of active modes n. With increasing numbers of active modes the pinwheel densi-ties occur within an interval of allowed pinwheel densities1.4<ρ<3.5.
Whereas degenerate in energy, essentially complex planforms of order n vary substantially in their pinwheel densities. Fig. 2.5 shows the pinwheel densitiesρi, i.e. the number of pinwheels per unit areaΛ2, of the essentially complex planforms from Fig. 2.4 of various ordern. Each value represents an average over the ensem-ble of phasesφj. Pinwheel densities fill a band of values between 1.5 and 3.5 with the majority of values between 2 and 3.5. Pinwheel density selection is analyzed in experimental data in Chapter 4, and in the described model in Chapters 5 and 6.
in the visual cortex
3.1 Motivation
It is a necessary prerequisite for calculating pinwheel densities to determine the column spacingΛwith high precision. Owing to its definition,ρ=ρΛˆ 2where ˆρis the pinwheel density in units 1/mm2, the column spacingΛcontributes quadrat-ically to the pinwheel densityρ. A relative error in Λ, therefore, results in a rel-ative error twice as large forρ. Furthermore, since there is no apriory reason to assume that the column spacing is constant within a map, it is important to es-timate column spacings locally. Visual inspection of orientation maps suggests a considerable variation of local column spacing at least in some cases.
In this chapter, we investigate the column spacing locally within a set of complete visual areas V1 and V2 in cat visual cortex, both containing a complete represen-tation of the binocular contralateral visual field [97, 98]. We adopt methods that we used previously for an analysis of genetic influence on columns spacing [44, 45]
and calculate a map of local column spacingΛ(x)for each orientation map express-ing the spatial variation of local column spacexpress-ing in this map. Decomposexpress-ing Λ(x) into the three components average, systematic variation, and individual variation reveals that in V1 and V2 and in the left and right brain hemisphere, column spac-ings covary in regions representing the same part of the visual field. Surprisingly, this matching of column spacing even applies to individual brains and consolidates with age. The analysis demonstrates the precision of the estimation of local column spacings.