Universality in visual cortical pattern formation - Nonlinear Dynamics

Journal of Physiology - Paris 97 (2003) 253–264 www.elsevier.com/locate/jphysparis

Universality in visual cortical pattern formation F. Wolf *, T. Geisel Department of Nonlinear Dynamics, Max-Planck-Institut f€ur Str€omungsforschung and Institute for Nonlinear Dynamics, Fakult€at f€ur Physik, Universit€at G€ottingen, D-37073 G€ottingen, Germany

Abstract During ontogenetic development, the visual cortical circuitry is remodeled by activity-dependent mechanisms of synaptic plasticity. From a dynamical systems perspective this is a process of dynamic pattern formation. The emerging cortical network supports functional activity patterns that are used to guide the further improvement of the network’s structure. In this picture, spontaneous symmetry breaking in the developmental dynamics of the cortical network underlies the emergence of cortical selectivities such as orientation preference. Here universal properties of this process depending only on basic biological symmetries of the cortical network are analyzed. In particular, we discuss the description of the development of orientation preference columns in terms of a dynamics of abstract order parameter fields, connect this description to the theory of Gaussian random fields, and show how the theory of Gaussian random fields can be used to obtain quantitative information on the generation and motion of pinwheels, in the two dimensional pattern of visual cortical orientation columns.
2003 Elsevier Ltd. All rights reserved. Keywords: Area 17; Development; Experience-dependence; Cortical maps; Self-organization

1. Introduction Universality, the phenomenon that collective properties of very different systems exhibit identical quantitative laws, is of great importance for the mathematical modeling of complex systems. Originally, the phenomenon of universality gained widespread recognition when it was realized that the quantitative laws of phase transitions in physically widely different equilibrium thermodynamic systems were determined only by their dimensionalities and symmetries and were otherwise insensitive to the precise nature of physical interactions (for an introduction see [26]). Subsequent research in nonlinear dynamics and statistical physics has uncovered that universal behavior extends far beyond equilibrium thermodynamics and is found for instance in pattern forming systems far from equilibrium (see e.g. [16]), in chaotic dynamics (see e.g. [40]), and in turbulence (see e.g. [20]). It is for two reasons that universal behavior is particularly important for the mathematical modeling of complex systems such as the brain. First, in order to understand the universal properties of a system

*

Corresponding author.

0928-4257/$ - see front matter
2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jphysparis.2003.09.018

it is sufficient to study fairly simplified models as long as they are in the right universality class. Second, predictions for experiments that are derived from universal model properties are critical: because universal properties are insensitive to changing microscopic interactions and numerical parameters or refining the level of detail in a model, verification or falsification of universal predictions can determine whether a certain modeling approach is appropriate or not. This is particularly important in theoretical neuroscience because for the neuronal networks of the brain a complete microscopic characterization of all interactions cannot be achieved experimentally and even if available would preclude comprehensive mathematical analysis. In this chapter we will discuss in detail the universal properties of a paradigmatic process in brain development: the formation of so called orientation pinwheels and the development of orientation columns in the visual cortex. In the visual cortex as in most areas of the cerebral cortex information is processed in a 2-dimensional (2D) array of functional modules, called cortical columns [15,29]. Individual columns are groups of neurons extending vertically throughout the entire cortical thickness that share many functional properties. Orientation columns in the visual cortex are composed

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of neurons preferentially responding to visual contours of a particular stimulus orientation [24]. In a plane parallel to the cortical surface, neuronal selectivities vary systematically, so that columns of similar functional properties form highly organized 2D patterns, known as functional cortical maps. In the case of orientation columns, this 2D organization is characterized by so called pinwheels, regions in which columns preferring all possible orientations are organized around a common center in a radial fashion [8,43] (see Fig. 1). It is a very attractive but still controversial hypothesis that in the ontogenetic development of the brain the emerging cortical organization is constructed by learning mechanisms which are similar to those that enable us to acquire skills and knowledge in later life [28,41,42]. Several lines of evidence strongly suggest that the brain in a very fundamental sense learns to see. First, visual experience is very important for the normal development of sight. If the use of the visual sense is prevented early in life vision becomes irreversibly impaired [17]. Since this is not due to a malformation of the eye or of peripheral stages of the visual pathway, it suggests that in development visual input it used to improve the processing capabilities of the visual cortical networks [17]. In addition, the performance of the developing visual system responds very sensitively to visual experience. In human babies, for instance, already a few hours of visual experience lead to a marked improvement of visual acuity [32]. Second, the synaptic organization of the visual cortex is highly plastic and responds with profound and fast functional and structural reorganization to appropriate experimental manipulations of visual experience [3,49]. These and similar observations suggest that the main origin of perceptual improvement in early development is due to an activity-dependent and thus use-dependent refinement of the cortical network, in which neuronal activity patterns that arise in the processing of visual information in turn guide the refinement of the cortical network. Whereas, theoretically, this hypothesis is very attractive, it is, experimentally, still controversial, whether neural activity actually plays such an instructive role (for discussion see [14,27,34]). In 1998, we discovered that experimentally accessible signatures of such a activity refinement of the cortical network are predicted by universal properties of a very general class of models for the development of visual cortical orientation preference maps [52]. We could demonstrate that if the pattern of orientation preferences is set up by learning mechanisms, then the number of pinwheels generated early in development exhibits a universal minimal value that depends only on general symmetry properties of the cortical network. This implies that in species exhibiting a lower number of pinwheels in the adult pinwheels must move and annihilate in pairs during the refinement of the cortical circuitry. Verification of this intriguing prediction would

provide striking evidence for the activity-dependent generation of the basic visual cortical processing architecture. In the following, we will present a self-contained treatment of the mathematical origin of this kind of universal behavior. The presentation is organized as follows. In Section 2, we introduce the mathematical language used to describe the spatial layout of orientation preference columns in the visual cortex and briefly describe their main features as experimentally observed. In Section 3, the description of the development of the orientation map by an abstract dynamics of order parameter fields is introduced and elementary consequences of its basic symmetry properties are discussed. In Section 4, we discuss how such a dynamics is derived from models of cortical learning processes that explicitly describe how activity patterns restructure the cortical architecture. In Section 5, we present qualitative arguments demonstrating that spontaneous breaking of symmetry in such models will lead to the formation of pinwheels and discuss how the further motion of pinwheels is constrained by topological principles. By the very nature of spontaneous symmetry breaking, the initial pattern of orientation preferences is, mathematically, a random variable. The pattern that will be generated by the breaking of symmetry is undetermined, owing to a lack of knowledge of the microscopic parameters and initial conditions. In spite of this, it may be assumed that the pattern will be one instance from a well-defined ensemble of possible patterns. The ensemble can be characterized by symmetry assumptions. This is the starting point for the examination of the statistically expected initial density of the pinwheels in Section 6. We demonstrate that for every Gaussian ensemble this density has a lower bound that depends only on symmetry properties. In Section 7, we will then show that the assumption of Gaussian statistics for the random pattern is fulfilled for a very general class of dynamical models. In Section 8, we present additional symmetry arguments indicating that the range of validity of our theory also includes models in which the pattern of orientation preferences interacts with the pattern of ocular dominance and the pattern of phases of receptive fields. Because the initial pinwheel density has a lower bound, quantitative limits are placed on the dynamics of the pinwheels in the investigated classes of models. Pinwheel densities that are less than the minimum initial density can occur only as a result of pinwheel annihilation during a phase of development subsequent to the breaking of symmetry. In the discussion (Section 9), we therefore compare the experimentally observed density with the theoretically calculated bounds which identifies species for which annihilation of pinwheels is predicted. We conclude with an outlook on possible extensions of the symmetry based analysis of universal properties in cortical pattern formation.

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264

2. The pattern of orientation preference columns In the following, we will briefly introduce the mathematical description of the spatial layout of orientation columns in the visual cortex in terms of complex valued order parameter fields. Experimentally, the pattern of orientation preferences can be visualized using the optical imaging method [6,8]. In such an experiment, the activity patterns Ek ðxÞ produced by stimulation with a grating of orientations hk are recorded. Here x represents the location of a column in the cortex. Using the activity patterns Ek ðxÞ, a field of complex numbers zðxÞ can be constructed that completely describes the pattern of orientation columns: X zðxÞ ¼ ei2hk Ek ðxÞ: ð1Þ k

The pattern of orientation preferences #ðxÞ is then obtained from zðxÞ as follows: #ðxÞ ¼

1 argðzÞ: 2

ð2Þ

Typical examples of such activity patterns Ek ðxÞ and the patterns of orientation preferences derived from them are shown in Fig. 1. Numerous studies confirmed that the orientation preference of columns is a almost everywhere continuous function of their position in the cortex. Columns with similar orientation preferences occur next to each other in ‘‘iso-orientation domains’’ [46]. Neighboring iso-orientation domains preferring the same stimulus orientation exhibit a typical lateral spacing K in the range of 1 mm, rendering the pattern of preferred orientations roughly repetitive. Furthermore, it was found experimentally that the iso-orientation domains are often arranged radially around a common center. Such an arrangement had been previously hypothesized on the basis of electrophysiological experiments [2,43]. The regions exhibiting this kind of radial arrangement were termed ‘‘pinwheels’’ (see Fig. 1). The centers of pinwheels are point discontinuities of the field #ðxÞ where the mean orientation preference of nearby columns changes by 90. They can be characterized by a topological charge which indicates in particular whether the orientation preference increases clockwise around the center of the pinwheel or counterclockwise: I 1 qi ¼ r#ðxÞ ds; ð3Þ 2p Cj where Cj is a closed curve around a single pinwheel center at xi . Since # is a cyclic variable within the interval ½0; p and up to isolated points is a continuous function of x, qi can in principle only have the values n ð4Þ qi ¼ ; 2

255

where n is an integer number [33]. If its absolute value jqi j is 1/2, each orientation is represented exactly once in the vicinity of a pinwheel center. Pinwheel centers with a topological charge of ±1/2 are simple zeros of zðxÞ. In experiments only pinwheels that had the lowest possible topological charge qi ¼ 1=2 are observed. This means there are only two types of pinwheels: those whose orientation preference increases clockwise and those whose orientation preference increases counterclockwise. This organization has been confirmed in a large number of species and is therefore believed to be a general feature of visual cortical orientation maps [4,5,7,9,10,38,50].

3. Symmetries in the development of orientation columns Owing to the large number of degrees of freedom of a microscopic model of visual cortical development, the description of the development of the pattern of columns by equations for the synaptic connections between the LGN and cortex is very complicated. On the order of 106 synaptic strengths would be required to realistically describe, for example, the pattern of orientation preference in a 4 · 4 mm2 piece of the visual cortex. This complexity and the presently very incomplete knowledge about the nature of realistic equations for the dynamics of visual cortical development demand that theoretical analyzes concentrate on aspects that are relatively independent of the exact form of the equations and are representative for a large class of models. Because on a phenomenological level the pattern of orientation columns can be represented by a simple order parameter field zðxÞ, models with the following form provide a suitable framework: o zðxÞ ¼ F ½zð Þ þ gðx; tÞ: ot

ð5Þ

Here F ½zð Þ is a nonlinear operator and the random term gðx; tÞ describes intrinsic, e.g., activity-dependent fluctuations. In Eq. (5), it is assumed that, except for random effects, changes in the pattern of orientation columns during development can be predicted on the basis of a knowledge of the current pattern. Swindale was the first to study models of this type, with the intent to show that roughly periodical patterns of columns can develop from a homogeneous initial state [44,45,47]. It is biologically plausible to assume that Eq. (5) exhibits various symmetries. Considered anatomically, the cortical tissue appears rather homogeneous. If we look at the arrangement of the cortical neurons and their patterns of connections, there is no region of the cortical layers and no direction parallel to the layers that is distinguishable from other regions or directions [11]. If the development of the pattern of orientation preferences can be described by an equation in the form of Eq.

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(5), it is thus very plausible to require that it is symmetric with respect to translations,

4. From learning to dynamics

F ½ Tby z ¼ Tby F ½z

It is not difficult to construct models with the form of Eq. (5) that represent the features of activity-dependent plasticity in an idealized fashion [52]. One instructive possibility is to start from an equation that describes how the pattern of orientation preferences zðxÞ changes under the influence of a sequence of patterns of afferent activity Ai :

with Tby zðxÞ ¼ zðx þ yÞ

ð6Þ

and rotations, b b z ¼ R b b F ½z with F ½R    cosðbÞ sinðbÞ b R b zðxÞ ¼ z x  sinðbÞ cosðbÞ

ð7Þ

of the cortical layers. This means that patterns that can be converted to one another by translation or rotation of the cortical layers are equivalent solutions of Eq. (5). If the orientation preference of a column is determined by the afferent connections from the lateral geniculate nucleus (LGN), it is also plausible to require that the arrangement of iso-orientation domains contains no information about the orientation preferences of the columns. This is guaranteed by a further symmetry. If Eq. (5) is symmetric with respect to shifts in orientation, F ½ei/ z ¼ ei/ F ½z

ð8Þ

then patterns whose arrangement of iso-orientation domains is the same but whose orientation preference values differ by a given amount, are equivalent solutions of Eq. (5). The three symmetries Eqs. (6)–(8) imply a number of basic properties of Eq. (5). Owing to the symmetry under orientation shifts, zðxÞ ¼ 0 is a stationary solution of Eq. (5) because F ½0 ¼ ei/ F ½0 ) F ½0 ¼ 0:

ð9Þ

Near this state, zðxÞ develops approximately according to a linear equation: o zðxÞ ¼ b LzðxÞ þ gðx; tÞ; ð10Þ ot b is a linear operator. Like F ½ , the operator b where L L must also commute with rotation and translation of the cortical layer and with global shifts of orientation. Thus, the Fourier representation of b L is diagonal and its eigenvalues kðkÞ are only a function of the absolute value of the wave vector k ¼ jkj. A qualitative requirement placed in Eq. (5) is that it be able to describe the spontaneous generation of a roughly repetitive pattern of orientation preferences from an initially homogeneous state zðxÞ 0. This requirement further constrains the class of dynamic equations that are to be considered. Eq. (10), and thus also Eq. (5), describes the generation of a repetitive pattern of orientation preferences when the spectrum kðkÞ has positive eigenvalues, and exhibits them only in one interval of wave numbers ðkl ; kh Þ with 0 < kl < kh . Mathematically such a system is said to exhibit a Turing-type instability of the homogenous state zðxÞ 0 (see e.g. [16]).

Ai

zi ðxÞ!ziþ1 ðxÞ:

ð11Þ

In a minimal model, the changes dzðxÞ ¼ ziþ1 ðxÞ  zi ðxÞ

ð12Þ

in the pattern must be dependent on both the current pattern zi ðxÞ and the patterns of activity Ai : dzðxÞ ¼ f ðx; zi ð Þ; Ai Þ:

ð13Þ

If the maximum absolute value maxA ðjdzðxÞjÞ of a modification induced by a single activity pattern is much smaller than the amplitude of the pattern and if the patterns of afferent activity Ai are random variables with a stationary probability distribution, then the changes in zðxÞ on a long time scale are described by the following equation: o zðxÞ ¼ ½f ðx; zð Þ; AÞA  F ½zð Þ; ot

ð14Þ

where ½ A denotes averaging over the ensemble of activity patterns [21]. This equation has the form of Eq. (5). A requirement that results directly from the activitydependent nature of synaptic plasticity is that only the selectivity of the columns that are activated by A is changed. For cortical activity patterns eðxÞ ¼ eðx; zð Þ; AÞ, this requirement is fulfilled by making the modification proportional to the cortical activity: f ðx; zð Þ; AÞ / eðx; zð Þ; AÞ:

ð15Þ

For simplicity, let us assume that a single activity pattern A forces the activated cortical neuron to take on a certain orientation preference h and a certain orientation selectivity jsj, described by the complex number sðAÞ ¼ jsðAÞj e2ihðAÞ ;

ð16Þ

then the simplest modification rule has the form f ðx; zð Þ; AÞ / ðsðAÞ  zðxÞÞeðx; zð Þ; AÞ:

ð17Þ

Several models have been proposed that can be interpreted in the just described way as an order parameter dynamics [18,19,22,23,37]. These models are defined by modification formulas with the form of Eq. (17). They differ mainly in how the pattern eðxÞ of the activity of the cortex is modeled.

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264

257

latter assumption implies that the expectation of zðxÞ is equal to zero:

5. Pinwheel generation and motion It is easily shown that within this class of models pinwheels will typically form during the initial symmetry breaking phase of development. If the eigenvalues kðkÞ are real, which is expected when zðxÞ develops to a stationary state, then beginning with a homogeneous state zðxÞ 0 the real and imaginary parts of zðxÞ initially develop independently of each other. In particular, the zero lines of the real and imaginary parts will develop independently of each other and thus typically intersect at points xi . These points are simple zeros of zðxÞ and therefore are the centers of pinwheels with a topological charge qi ¼ 1=2. It is important to note that the possible forms of a change in the pinwheel configuration over time fxi ; qi g are already constrained by assuming an equation for the developmental dynamics with the form of Eq. (5). Since the field zðxÞ can only be a continuous function of time, the entire topological charge of a given area A with a boundary ðAÞ, I X 1 QA  r#ðxÞ ds ¼ qi ð18Þ 2p ðAÞ xi 2A is invariant as long as no pinwheel transgresses the boundary of the area [33]. If the pattern contains only pinwheels with qi ¼ 1=2, then only three qualitatively different modifications of the pinwheel configuration are possible. First, movement of the pinwheel within the area; second, generation of a pair of pinwheels with opposite topological charges; third, the annihilation of two pinwheels with opposite topological charge when they collide. Only these transformations conserve the value of QA and are therefore permitted.

6. Random orientation maps In order to estimate the pinwheel density which results from the initial breaking of symmetry, we will, in this section, start by examining an ensemble of random fields zðxÞ. Such an ensemble can be characterized by its spatial correlation functions: Cðx; yÞ ¼ hzðxÞzðyÞi;

ð19Þ

C  ðx; yÞ ¼ hzðxÞzðyÞi:

ð20Þ

Here angular brackets, h i, represent the expectation value for the ensemble. The form of these correlation functions can be constrained by symmetry assumptions. Because of the symmetries equations (6)–(8), we assume that the ensemble is statistically invariant with respect to translations and rotations and that the patterns that can be transformed into each other by a global orientation shift zðxÞ ! ei/ zðxÞ occur with the same probability. The

hzðxÞi ¼ 0:

ð21Þ

This means that any orientation preference can occur at any location x in the cortex. Moreover, invariance under orientation shifts implies that the correlation function (20) is also equal to zero, since only in this case can the following relation be fulfilled for any /: hzðxÞzðyÞi ¼ hei/ zðxÞ ei/ zðyÞi:

ð22Þ

Re ðzðxÞÞ describes the patterns of columns that prefer horizontal and vertical stimuli. Im ðzðxÞÞ describes the patterns of columns that prefer oblique stimuli. These two patterns are not correlated and both have the same correlation function because the correlation function (20) is zero: C  ðx; yÞ ¼ hzðxÞzðyÞi ¼ hRe ðzðxÞÞRe ðzðyÞÞi  hIm ðzðxÞÞIm ðzðyÞÞi þ iðhRe ðzðxÞÞIm ðzðyÞÞi þ hIm ðzðxÞÞRe ðzðyÞÞiÞ ¼ 0:

ð23Þ

Invariance with respect to translations and rotations implies that the correlation function CðrÞ is a function only of the distance r ¼ jx  yj of the respective pair of locations in the cortex: Cðx; yÞ ¼ Cðjx  yjÞ ¼ CðrÞ:

ð24Þ

The correlation function CðrÞ mainly provides information about the characteristic wavelength and the correlation length of the pattern. The characteristic wavelength K of the pattern can be defined using the Fourier transform of CðrÞ: Z 1 P ðjkjÞ ¼ d2 xCðxÞ eikx ð25Þ 2p which is called the power spectral density. It is of advantage to use the mean wavenumber k to define the characteristic wavelength: 2p 2p K ¼  ¼ R1 : dkkP ðkÞ k 0

ð26Þ

Without loss of generality, the power spectral density is R1 assumed to be normalized as follows: 0 dkP ðkÞ ¼ 1. There is an infinite number of ensembles of maps that have the same two-point correlation function. In the following considerations, we will assume that the ensembles considered consists of Gaussian random fields. In the next section, we will show that this assumption is fulfilled for a large class of dynamic models, owing to the central limit theorem. The centers of the pinwheels are the zeros of the field zðxÞ. The pinwheel density is therefore obtained from the number of these zeros in an area A,

258

N¼

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264



oðRe zðxÞ; Im zðxÞÞ

; d2 xdðzðxÞÞ



oðx1 ; x2 Þ A

Z

! rzrz

d ðrzÞ exp  2

ðrz þ rzÞ cg !

Z zz

2 ; ð38Þ  ðrz  rzÞ d zdðzÞ exp  ca

1 q¼ 3 2 p cg ca

ð28Þ

where jðrz þ rzÞ  ðrz  rzÞj is the Jacobian of zðxÞ. This integral is easily evaluated by converting to the spherical coordinates g 2 ½0; 1Þ, h 2 ½0; pÞ and /1 /2 2 ½0; 2pÞ in gradient space with the volume element

where oðRe zðxÞ; Im zðxÞÞ oRe zðxÞ oIm zðxÞ ¼ oðx1 ; x2 Þ ox1 ox2 oRe zðxÞ oIm zðxÞ  ox2 ox1

is the Jacobian of zðxÞ. The expectation value of the number of zeros in an ensemble is

 Z

oðRe zðxÞ; Im zðxÞÞ

d2 x dðzðxÞÞ

ð29Þ hN i ¼

oðx1 ; x2 Þ A from which follows that



oðRe zðxÞ; Im zðxÞÞ



q ¼ dðzðxÞÞ

oðx1 ; x2 Þ

Z

ð27Þ

ð30Þ

4

d4 ðrzÞ ¼ g3 dgj cosðhÞ sinðhÞj dh d/1 d/2 :

ð39Þ

Integration then yields  Z p Z 1 1 g2 5 dgg exp  2 dhj cosðhÞ sinðhÞj2 q¼ 3 2 p cg ca 0 cg 0 Z 2p d/1 d/2 j cosð/1 Þ sinð/2 Þ  cosð/2 Þ sinð/1 Þj 0

is the density of the pinwheels. This expectation value will now be evaluated. Superficially, q as defined in Eq. (30) is the expectation value for an ensemble of fields zðxÞ and thus is a functional integral. It is, however, important to note that this expectation value depends on locally defined parameters only, namely the values of the field zðxÞ and its derivatives rzðxÞ at a given location x. To integrate Eq. (30) it is, therefore, sufficient to know the joint probability density pðzðxÞ; rzðxÞÞ. Since this is also Gaussian, it is given by the cross-correlations and autocorrelations of zðxÞ and rzðxÞ. Most of these correlations are equal to zero owing to the symmetry requirements: hzðxÞrzðxÞi ¼ 0;

ð31Þ

hzðxÞzðxÞi ¼ 0;

ð32Þ

hrzðxÞrzðxÞi ¼ 0;

ð33Þ

hzðxÞrzðxÞi ¼ 0;

ð34Þ

hzðxÞzðxÞi ¼ ca :

ð35Þ

hrzðxÞrzðxÞi ¼ cg :

ð36Þ

Eqs. (31)–(33) result from the statistical invariance of the ensemble under an orientation shift. Eq. (34) results from the rotation invariance of the ensemble. Eq. (35) defines the scale on which the order parameter is measured. Thus, ca and cg are the only nontrivial correlations of the distribution pðzðxÞ; rzðxÞÞ. Because there is no correlation between zðxÞ and rzðxÞ, the distribution is composed of two factors: ! ! 1 rzrz zz pðz; rzÞ ¼ 3 2 exp  2 exp  : ð37Þ p cg ca cg ca Because this is true for any location x, the argument of the field zðxÞ is omitted here and in the following equations. Because Eq. (37) is factored, the expectation in Eq. (30) is also factored. Thus,

ð40Þ 1 cg ¼ : 4p ca

ð41Þ

Since in the following discussion the dependence of this expression on the power spectral density P ðkÞ is of interest, we express ca and cg as functionals of P ðkÞ: Z ð42Þ ca ¼ d2 kP ðkÞ; Z 2 cg ¼ d2 kjkj P ðkÞ: ð43Þ The pinwheel density is then given by R 2 1 d2 kjkj P ðkÞ R 2 : q¼ 4p d kP ðkÞ

ð44Þ

The exact form of the correlation function CðrÞ and the structure function P ðkÞ at the beginning of development is not known. In particular, it is to be expected that these functions vary from species to species and from individual to individual. In spite of this, the following argument shows that Eq. (44) implies a quantitative estimate of the initial pinwheel density. Since it may be assumed that the pinwheel density is inversely proportional to the square of the characteristic wavelength q / K2 , we shall rewrite the expression for the density: R1 R1 k 2 dkk 3 P ðkÞ dkk 3 P ðkÞ p 0 0 ð45Þ q¼ R1 3 ¼ 2  R 1 3 : 4p K dkkP ðkÞ dkkP ðkÞ 0 0 Owing to Jensen’s inequality [39], Z 1 3 Z 1 3 dkk P ðkÞ P dkkP ðkÞ 0

ð46Þ

0

it follows that q has a lower bound: q¼

p ð1 þ aÞ; K2

ð47Þ

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264

259

where a > 0. Thus, the exact form of the power spectral density influences the expected pinwheel density only via the positive-definite functional Z 1 Z 1 2 3 ðk  kÞ ðk  kÞ a¼3 dk 2 P ðkÞ þ dk 3 P ðkÞ; ð48Þ k k 0 0

Eq. (51) with the initial condition z0 ðxÞ 0 at time t ¼ 0 is solved to yield Z Z t   zL ðx; tÞ ¼ d2 y dt0 Gðy  x; t  t0 Þ nðy; t0 Þ þ z0 ðxÞdðt0 Þ :

where a is zero only when the power spectral density is the Dirac delta distribution P ðkÞ ¼ dðk  kÞ, i.e., when the correlation length of the pattern diverges. Thus, it can be seen that a Gaussian random pattern of orientation preferences has a minimum pinwheel density, p qm ¼ 2 ; ð49Þ K independently of the exact form of its spatial correlations. Because two-dimensional Gaussian random fields are ergodic [1], this lower bound is also valid for the pinwheel density in an individual realization of such a field.

Since zL ðx; tÞ is the sum of linear transforms of the random fields z0 ðxÞ and nðx; tÞ, it will always have Gaussian statistics when z0 ðxÞ and nðx; tÞ are also Gaussian. This is independent of the form of their correlation functions:

7. Random orientation maps from a Turing instability The assumption was made in the previous section that the ensemble of possible initial patterns show Gaussian statistical properties. We will now show that this is actually the case for a large class of models. This establishes that the lower bound for the pinwheel density calculated in the previous section places a quantitative constraint on the dynamics of pinwheels in this class of models. For this purpose, we will use the class of models defined in Section 3 that can be represented by an equation for the dynamics of the order parameter field zðxÞ: o zðxÞ ¼ F ½zð Þ þ nðx; tÞ: ð50Þ ot The assumed symmetries of this equation imply that zðxÞ ¼ 0 is a stationary solution. In the vicinity of this point, the dynamics of zðxÞ is approximately linear: o zðxÞ ¼ b LzðxÞ þ nðx; tÞ; ð51Þ ot where b L is a linear operator. This equation generates an orientation map from an initially homogeneous state. As discussed in Section 3, the class of operators b L is further constrained by the requirement that Eq. (50) describe the spontaneous generation of a pattern of orientation preferences starting with an initially homogeneous state. Equation (51), and thus also Eq. (50), describes the generation of a repetitive pattern of orientation preferences if the spectrum kðkÞ has positive eigenvalues only within a single interval ðkl ; kh Þ of kvalues. Using the Green’s function Z 1 d2 k eikxþkðjkjÞt ; Gðx; tÞ ¼ ð52Þ 2p

0

ð53Þ

Cz0 ðrÞ ¼ hz0 ðxÞz0 ðx þ rÞi;

ð54Þ

Cn ðr; tÞ ¼ hnðx; t0 Þnðx þ r; t0 þ tÞi:

ð55Þ

In general, the statistical properties of zL ðx; tÞ are also Gaussian for a much larger class of random processes. The field zL ðx; tÞ is an integral over a number of random variables. When this integral consists of a large number of independent terms then the central limit theorem [39] gives the conditions under which the statistical properties of zL ðx; tÞ are Gaussian, even if z0 ðxÞ and nðx; tÞ are not Gaussian. We will now briefly present these conditions and discuss their biological significance. For simplicity we will limit ourselves to sufficiently smooth functions z0 ðxÞ and nðx; tÞ such that the right side of Eq. (53) can be expressed approximately by a sum: X  zL ðx; tÞ Dyi2 Dtj Gðyi  x; t  tj Þ nðyi ; tj Þ i;j

 þ z0 ðxi Þdj;0 :

ð56Þ

The central limit theorem applies if the distributions of z0 ðxÞ and nðx; tÞ fulfill the Lindeberg criterion: Z lim dww2 P ðwÞ ¼ 0; ð57Þ

b!1

jwj>b

where the variable w represents either z0 ðxÞ or nðx; tÞ at an arbitrary location x at any time t and P ðwÞ is the probability density. Eq. (57) is fulfilled when jz0 ðxÞj and jnðx; tÞj are bounded or have a finite variance. Both appear to be plausible for the observed biological process. There is very little orientation selectivity, which is observed in the visual cortex considerably before the eyes open [12]. Thus, it can be taken as certain that the jz0 ðxÞj field, which describes this early selectivity, is bounded. A similar argument can be made for the fluctuations nðx; tÞ. All available evidence indicates that the process in which the initial orientation map is generated is continuous [12]. This speaks for the assumption that fluctuations in this process are bounded or at least have a finite variance. In order to determine the conditions under which zL ðx; tÞ consists of a large number of independent terms, it is necessary to compare the correlation times and lengths of z0 ðxÞ and nðx; tÞ with the temporal and spatial

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scales of the Green’s function Gðx; tÞ. The characteristic time scale of Gðx; tÞ is s ¼ 1=kðkmax Þ;

ð58Þ

L. The ampliwhere kðkmax Þ is the largest eigenvalue of b tude jGð0; DtÞj of GðDx; DtÞ increases with time with time constant s. The Green’s function also has a characteristic spatial scale L on which GðDx; DtÞ decays as a function of the distance jDxj for a given time difference Dt. In order to estimate L, let us assume that kðkÞ has a quadratic maximum between kl and kh . The spatial e Fourier transform Gðk; DtÞ of the Green’s function GðDx; DtÞ is then approximated in the vicinity of the maximum kmax by a Gaussian function: e Gðk; DtÞ ¼ ekðjkjÞt

2 ! ! t :



ð59Þ

jkj  kmax ð60Þ ðkh  kl Þ=2 pffiffiffiffiffiffiffiffiffiffi The bandwidth Dk ¼ jkh  kl j=2 Dt=s of this function is a function of time. Its inverse provides an estimate of L: pffiffiffiffiffiffiffiffiffiffi 2 Dt=s : ð61Þ L jkh  kl j

exp kmax 1 

Thus, the field zL ðx; tÞ Eq. (53) integrates fluctuations nðx; t  DtÞ that occurred in the past at time t  Dt within a distance L. The size of this p volume ffiffiffiffiffiffiffiffiffiffi as given by Eq. (61) increases diffusively: L / Dt=s. If it is assumed that the correlation function Cz0 ðrÞ decays on a spatial scale Lz0 and the same is assumed for Cz0 ðrÞ on the spatial and temporal scales Ln and sn , so that Cz0 ðrÞ 6 Az0 ejxj=Lz0 ; Cn ðr; tÞ 6 An e

ð62Þ

jxj=Ln jtj=s

ð63Þ

are fulfilled with positive constants Az0 and An , then the number of independent terms in zL ðx; tÞ is large if sn  s

ð64Þ

and maxfLz0 ; Ln g 

the statistical properties of the initial pattern can be assumed to be Gaussian.

1 : jkh  kl j

ð65Þ

If zL ðx; tÞ is determined mainly by nðx; tÞ, then either 1 sn  s or Ln  jkh k is a sufficientpcondition. If zL ðx; tÞ is ffiffiffiffi lj 2

8. Hidden parameters The pattern of orientation preferences does not develop completely independently of the patterns of other properties of the neurons in the visual cortex. In the previous analyzes, it was assumed that the initial formation of a orientation preference pattern can be described by an autonomous system. Observations and theoretical arguments, however, indicate that the pattern of orientation preferences interacts with other selectivity patterns. For example, iso-orientation lines tend to cross the boundaries of ocular dominance columns at right angles (Fig. 2). Moreover, oriented receptive fields not only have an orientation preference, but they also have a phase (Fig. 3). Because in microscopic models the forms of the receptive fields govern the development of the afferent connections, these models imply that the orientation preference pattern and the pattern of the phases of receptive fields interact [35,36]. In this section we show based on further symmetry arguments that the conclusions of the previous section are also justified even if it is assumed that the orientation preference pattern interacts with these two patterns. For the discussion of the effects of interaction between the ocular dominance patterns, the receptive field phases, and the developing system of orientation columns, we will represent the two patterns by additional order parameter fields. The pattern of ocular dominance columns can be described by a real field oðxÞ, where oðxÞ > 0 represents the columns that prefer the left eye and oðxÞ < 0 represents the columns that prefer the right eye. The phase W pattern of the receptive fields can also be represent by a field. The phase of a receptive field is a cyclic variable. It can, therefore, be described by a complex scalar. The locations of the cortical neurons are given by xi and the neuronsÔ phases are Wi . The coarse grained pattern of phases is then given by pðxÞ ¼

X

hðxi  xÞ eiWi ;

ð66Þ

i

t=s

determined by z0 ðxÞ, then Lz0  jkh kl j is sufficient, where

t is the time at which the amplitude of zðx; tÞ saturates. Especially the first of the two cases is compatible with the biological situation. Fluctuations caused by afferent activity patterns, like the activity patterns themselves, can be correlated only over time intervals of a few hundred milliseconds. Since the first pattern of orientation preferences is formed over a period of several hours to several days, the number of independent activity-induced fluctuations is surely large. Therefore,

where hðDxÞ is an arbitrary window function. This function is to be chosen so that only neurons in the vicinity of x contribute to the value of pðxÞ. The magnitude of this field then indicates whether the phases of these neurons are similar. It goes to zero when the distribution of these phases is random and has a value of one when the phases are identical. When the orientation preference pattern interacts with the phase and ocular dominance patterns, Eq. (50) must be replaced by

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264

261

Fig. 1. Patterns of orientation columns in the primary visual cortex of a tree shrew visualized using the optical imaging method (figure modified from [10]). The column gratings resulting from horizontally, vertically, and diagonally oriented patterns are shown in (A). White bars depict the orientation of the visual stimulus. Activated columns are labeled dark gray. The used stimuli activate only columns in area 17 (V1 in the lower left part of (A)). The patterns thus end at the boundary between areas 17 and 18 (V2). The pattern of orientation preferences calculated from the activity patterns in (A) is shown in (B). The orientation preferences of the columns are color coded; for example, columns that prefer horizontal stimulation are red and those that prefer vertical stimulation are light blue. Typical elements of orientation preference patterns are shown in (C): left, an area in which iso-orientation domains are in parallel strips; right, two pinwheels with opposite topological charges.

N

o zðxÞ ¼ F ½zð Þ; pð Þ; oð Þ þ n: ð67Þ ot Thus in general, both pðxÞ and oðxÞ must be known in order to predict the development of zðxÞ. The initial formation of pinwheels is now determined by an initially homogeneous state z0 ðxÞ 0; ð68Þ 

Fig. 2. Iso-orientation domains and boundaries between ocular dominance columns in the primary visual cortex of a cat (modified from [30]). The color code for the pattern of orientation preference is given in Fig. 1. The black lines mark the boundaries between the columns activated by the left and right eyes. Iso-orientation domains (areas with the same color) tend to intersect these boundaries perpendicularly.

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invariant with respect to inversion of the receptive fields pðxÞ ! pðxÞ: F ½zðxÞ; pðxÞ; oðxÞ ¼ F ½zðxÞ; pðxÞ; oðxÞ:

Fig. 3. Orientation and phase of receptive fields. The receptive fields of cortical neurons are made up of subfields that receive on- and offcenter input from the LGN. These fields usually have an orientation and a spatial phase. The figure shows two examples with the same orientation but different phases. In (a) and (b) the areas of the field of view from which the cell receives on input are light and those from which it receives off input are dark. The form of typical receptive fields can be described by Gabor functions with the form gðrÞ ¼ r=2 cosðKr þ WÞ erR [25], where the vector r is the location in the field of view relative to the center of the receptive field, the vector K defines the orientation and spatial frequency of the receptive field, R is a positivedefinite, symmetric matrix that governs the size of the field, and W is the phase of the receptive field. (a) shows an arbitrary example; (b) shows the receptive field that results by exchanging the on and off inputs. Exchanging the on and off inputs is equivalent to a phase shift by p.

p0 ðxÞ 0;

ð69Þ

oðxÞ 0:

ð70Þ

With the initial condition p0 ðxÞ 0, it is assumed that the phases are initially in a random state. With the initial condition o0 ðxÞ 0, it is assumed that the projections from the two eyes initially overlap to a large extent. In general, z0 ðxÞ ¼ 0; p0 ðxÞ ¼ 0; oðxÞ ¼ 0 is a stationary state of the development dynamics: F ½0; 0; 0 ¼ 0:

ð71Þ

This is justified by the fact that the homogeneous state should have no tendency to develop a specific pattern of orientation preferences. Such a tendency would be the case if F ½0; 0; 0 6¼ 0. In the vicinity of this stationary state, a linear equation describes the development: o zðxÞ ¼ b LzðxÞ þ b L p pðxÞ þ b L o oðxÞ þ nðx; tÞ: ot

ð72Þ

Lp Two plausible symmetries cause the two operators b and b L o to be zero simultaneously. b L o is zero if exchanging the columns of the left and right eyes does not influence the development of the ocular dominance pattern, i.e., when F ½zðxÞ; pðxÞ; oðxÞ ¼ F ½zðxÞ; pðxÞ; oðxÞ:

ð73Þ

All models that have been proposed for the coordinated development of orientation preference columns and ocular dominance columns fulfill this condition [48]. In this case, Eq. (72) must also be invariant with respect to the inversion oðxÞ ! oðxÞ. This is possible only if b L o is zero. Analogously, b L p must also be zero when Eq. (67) is

ð74Þ

Receptive fields of cortical neurons consist of subfields that receive input from on- and off-center neurons. The exchanging of the on- and off-center inputs is mathematical equivalent to a phase shift by p, i.e., a change in sign of the field pðxÞ (Fig. 3). Eq. (67) can, therefore, be invariant when the on- and off-center inputs are exchanged only when the conditions of Eq. (74) and b L p ¼ 0 are fulfilled. Thus, even with the assumption that the orientation preference pattern interacts with the pattern of ocular dominance columns and the phases of the receptive fields, Eq. (51) can describe the formation of the first orientation preference pattern. The reason for this is that the interaction with the other column patterns is determined first by nonlinear terms when the assumed symmetries are present. These interactions, therefore, cannot influence the initial formation of pinwheels.

9. Discussion The analyzes presented above demonstrate that the density of the pinwheels in a Gaussian random pattern of orientation preferences is always larger than qm ¼ p=K2 , where K is the characteristic wavelength of the pattern. This feature is universal, meaning that it applies, no matter what the detailed structure of the fields’ statistical correlations are. In addition, we showed that orientation preference patterns generated by a dynamic instability have Gaussian statistical properties in a large class of models owing to the central limit theorem, and that this property is not affected by interaction with two other patterns of neuronal selectivities. These results imply that the dynamics of pinwheels is quantitatively constrained in a large class of models for the development of orientation preference patterns. Independent of modeling details, pinwheel densities that are less than the lower bound for the initial density qm ¼ p=K2 can develop under the given conditions only by pairwise annihilation of pinwheels. This implies that pinwheels must move during development and must be pairwise annihilated in those species in which low pinwheel densities are observed in adults (Fig. 4). Because of this, it is of interest to compare densities in adult animals with the calculated lower limit. The characteristic wavelength K differs considerably from species to species. Values of the relative pinwheel density ^ ¼ qK2 that are less than p imply pinwheel annihilation q during development. Observed relative densities range from 2.0 to 4.0 (see Fig. 4). Macaques have the highest relative density. The lowest relative densities observed so far are in the visual cortex of adult tree shrews (Tupaia).

F. Wolf, T. Geisel / Journal of Physiology - Paris 97 (2003) 253–264

Fig. 4. Constraints on the development of the pinwheel density by symmetries. The range of permissible trajectories of the scaled pin^ðtÞ is shaded gray. Two permissible trajectories are wheel density q shown. The y-axis on the right shows the scaled pinwheel densities observed in several species. Densities that are less than p can develop only by annihilation of pinwheels (modified from [52]).

The relative densities of both tree shrews and cats are distinctly less than p. Thus, pinwheel movement and annihilation is implied for both species by the theory developed above. Pinwheel movement cannot be excluded in ferrets and squirrel monkeys. This would be especially the case when the initial pinwheel density is substantially larger than the minimum. The high pinwheel density observed in macaques can occur as a result of a dynamic instability without any subsequent rearrangement of the pattern. Thus, tree shrews are the species in which the most extensive rearrangement is expected during development. The ontogenetic development of the cerebral cortex is a process of astonishing complexity. In every mm3 of cortical tissue in the order of 106 neurons must be wired appropriately for their respective functions such as the analysis of sensory inputs, the storage of skills and memory, or motor control [11]. In the brain of an adult animal, each neuron receives input via about 104 synapses from neighboring and remote cortical neurons and from subcortical inputs [11]. At the outset of postnatal development, the network is formed only rudimentarily: In the cat’s visual cortex, most neurons have just finished the migration from their birth zone lining the cerebral ventricle to the cortical plate at the day of birth [31]. The number of synapses in the tissue is then only 10% and at the time of eye-opening, about two weeks later, only 25% of its adult value [13]. In the following 2– 3 months the cortical circuitry is substantially expanded and reworked and the individual neurons acquire their final specificities in the processing of visual information [17]. It is intriguing, that the motion and annihilation of pinwheels, a global aspect of such remodeling, can be predicted without reference to the details of the biological mechanisms involved in this process. It is natural to ask whether the symmetry based approach used above to derive this prediction can be extended to study further aspects of cortical pattern formation. This is indeed possible, however, under more restrictive conditions. Mathematically, we have used primarily the linear equation, Eq. (10), which describes

263

the generation of a pattern of orientation columns from an initially homogeneous state. Eq. (5), which models the entire process of development must be nonlinear in order to describe the saturation of the orientation selectivity jzðxÞj and a possible subsequent reorganization of the pattern. The symmetry with respect to shifts of orientation means that when F ½  is represented by a power series only terms with an odd power can occur. If the parameter values in Equation (5) are assumed to be in the vicinity of an instability of the homogeneous state zðxÞ ¼ 0 and the transition to repetitive patterns is continuous, then also the stationary patterns that bifurcate from zðxÞ ¼ 0 can be studied perturbatively in a largely model independent fashion. Using such an approach general conditions for the final stability of pinwheel patterns can be obtained [51].

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