Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex
Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex
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DOI：
10.1098/rstb.2000.0769
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发表时间：
2001-03-29
影响因子：
6.3
通讯作者：
Wiener, MC
中科院分区：
文献类型：
--
作者：
Bressloff, PC;Cowan, JD;Wiener, MC
这篇论文关注的是一种惊人的视觉体验：看到几何图形的视觉幻觉。Kluver将幻觉图像分为四组，称为形式常数，包括(I)格子、格子、绒毛、细丝、蜂窝和棋盘，(Ii)蜘蛛网，(Iii)隧道、漏斗、小巷、圆锥体和容器，以及(Iv)螺旋。本文描述了对它们起源的数学研究，其基础是假定视网膜和纹状皮质(以下称为V1)--视网膜皮质图--之间的连接模式和V1中的神经元回路，包括局部和侧向？确定它们的几何形状。在本文的第一部分中，我们表明，当在V1坐标下观察时，形式常数本质上对应于平面波的组合，其波长是人类Hubel-Wiseel超柱宽度的整数倍，约1.33-2 mm。接下来，我们用相互连接的超柱的格子的连续极限来描述V1的大尺度动力学，每个超柱本身由许多相互连接的等向柱组成。然后，我们证明了V1中的互连模式表现出一种非常有趣的对称性，即它们是不变的旋转、反射和平移。新奇的是，V1的横向连通性使得需要一个新的群作用来表示它的性质：由于它的各向异性，它对于平面的某些平移和扭曲是不变的。正是这种平移-扭曲不变性产生了E(2)的新表示。假定横向连接的强度弱于局部连接的强度，然后利用Rayleigh-Schrodinger微扰理论计算了皮质动力学的本征值和本征函数。结果表明，在没有横向连接的情况下，本征函数是简并的，包括Phi中的正弦的偶数和奇数组合、取向偏好的皮质标号和皮质位置坐标r中的平面波。横向相互作用打破了简并性，选择偶数或奇数本征函数。在本文的第二部分中，我们研究了特征函数或平面的各种奇偶组合的性质，它们的对称性使得它们在我们施加的E(2)的特定作用下保持不变。这些对称性对应于E(2)的某些子群，即所谓的轴子群。轴子群很重要，因为等变分支引理表明，当对称动力系统变得不稳定时，会出现新的解，其对称性与基本对称群的轴子群相对应。这正是本文研究的案例。因此，我们研究了当我们的模型V1在假定的迷幻剂或闪烁的光的作用下变得不稳定时出现的各种平面。我们证明了在平移-扭转作用下，平面形式对应于E(2)的轴子群。然后，我们计算这样的平面在视野中会是什么样子，给定视网膜皮质图的扩展，以包括其在局部边缘和轮廓上的作用。最有趣的是，根据我们对V1平面形式和感知模式之间对应关系的解释，平面形式集生成了所有形式常量的代表。同样值得注意的是，从我们的连续统模型得到的平面形式自然地将V1划分为所谓的线性区域，在这些区域中，图案具有几乎恒定的取向，这让人想起通过光学成像构建的等向块。这些区域的边界形成的裂缝，其交点对应于众所周知的‘风车’。为了完成这项研究，我们接着使用包括Liapunov-Schmidt约化和Poincare-Lindstedt摄动理论在内的非线性稳定性分析方法来研究平台的稳定性。我们发现了稳定平面和形状常数之间的剂量对应关系。这些结果对侧向连通性的详细说明很敏感，并暗示了一种有趣的可能性，即产生几何视觉幻觉的皮质机制，如果主要位于V1，与参与边缘和轮廓处理的机制密切相关。
This paper is concerned with a striking visual experience: that of seeing geometric visual hallucinations. Hallucinatory images were classified by Kluver into four groups called form constants comprising (i) gratings, lattices, fretworks, filigrees, honeycombs and chequer-boards, (ii) cobwebs, (iii) tunnels, funnels, alleys, cones and vessels, and (iv) spirals. This paper describes a mathematical investigation of their origin based on the assumption that the patterns of connection between retina and striate cortex (henceforth referred to as V1) -the retinocortical map-and of neuronal circuits in V1, both local and lateral? determine their geometry.In the first part of the paper we show that form constants, when viewed in V1 coordinates, essentially correspond to combinations of plane waves, the wavelengths of which are integral multiples of the width of a human Hubel-Wiesel hypercolumn, ca. 1.33-2 mm. We next introduce a mathematical description of the large-scale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hypercolumns, each of which itself comprises a number of interconnected iso-orientation columns. We then show that the patterns of interconnection in V1 exhibit a very interesting symmetry i.e. they are invariant rotations, reflections and translations. What is novel is that the lateral connectivity of V1 is such that a new group action is needed to represent its properties: by virtue of its anisotropy it is invariant with respect to certain shifts and twists of the plane. It is this shift-twist invariance that generates new representations of E(2). Assuming that the strength of lateral connections is weak compared with that of local connections, we next calculate the eigenvalues and eigenfunctions of the cortical dynamics, using Rayleigh-Schrodinger perturbation theory The result is that in the absence of lateral connections, the eigenfunctions are degenerate, comprising both even and odd combinations of sinusoids in phi, the cortical label for orientation preference, and plane waves in r, the cortical position coordinate. 'Switching-on' the lateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected. These results can be shown to follow directly from the Euclidean symmetry we have imposed.In the second part of the paper we study the nature of various even and odd combinations of eigenfunctions or planforms, the symmetries of which are such that they remain invariant under the particular action of E(2) we have imposed. These symmetries correspond to certain subgroups of E(2), the so-called axial subgroups. Axial subgroups are important in that the equivariant branching lemma indicates that when a symmetrical dynamical system becomes unstable, new solutions emerge which have symmetries corresponding to the axial subgroups of the underlying symmetry group. This is precisely the case studied in this paper. Thus we study the various planforms that emerge when our model V1 dynamics become unstable under the presumed action of hallucinogens or flickering lights. We show that the planforms correspond to the axial subgroups of E(2), under the shift-twist action. We then compute what such planforms would look like in the visual field, given an extension of the retinocortical map to include its action on local edges and contours. What is most interesting is that, given our interpretation of the correspondence between V1 planforms and perceived patterns, the set of planforms generates representatives of all the form constants. It is also noteworthy that the planforms derived from our continuum model naturally divide V1 into what are called linear regions, in which the pattern has a near constant orientation, reminiscent of the iso-orientation patches constructed via optical imaging. The boundaries of such regions form fractures whose points of intersection correspond to the well-known 'pinwheels'.To complete the study we then investigate the stability of the planforms, using methods of nonlinear stability analysis, including Liapunov-Schmidt reduction and Poincare-Lindstedt perturbation theory. We find a dose correspondence between stable planforms and form constants. The results are sensitive to the detailed specification of the lateral connectivity and suggest an interesting possibility, that the cortical mechanisms by which geometric visual hallucinations are generated, if sited mainly in V1, are closely related to those involved in the processing of edges and contours.