Thomas Peter J, Cowan Jack D
Departments of Mathematics, Biology and Cognitive Science, Case Western Reserve University, Cleveland, OH 44106, USA.
Math Med Biol. 2006 Jun;23(2):119-38. doi: 10.1093/imammb/dql006. Epub 2006 Apr 20.
The most prominent feature of mammalian striate cortex (V1) is the spatial organization of response preferences for the position and orientation of elementary visual stimuli. Models for the formation of cortical maps of orientation and 'retinotopic' position typically rely on a combination of Hebbian or correlation-based synaptic plasticity, and constraints on the distribution of synaptic weights. We consider a simplified model of orientation and retinotopic specificity based on the geometry of the feed-forward synaptic weight distribution from an 'unoriented' layer of cells to a first weakly oriented layer. We model the feed-forward weight distribution as a system of planar Gaussian receptive fields each elongated in the direction matching the preferred orientation of the postsynaptic cell. Under the constraint of presynaptic weight normalization (each cell in the oriented layer receives the same net synaptic weight) and a uniform retinotopic map (displacement of centres of mass of receptive fields in the unoriented layer is strictly proportional to the displacement of the corresponding cells in the oriented layer), we find that imposing a pattern of orientation preference forces the system to violate postsynaptic weight normalization (each cell in the unoriented layer no longer sends forth the same net synaptic weight). We study this deviation from uniformity of the postsynaptic weight, and find that the deviation has a distinct form in the vicinity of the 'pinwheel' singularities of the orientation map. We show that uniform synaptic coverage of the unoriented layer can be restored by introducing a distortion in the retinotopic locations of the receptive fields. We calculate, to first order in the relative elongation of the receptive fields, the retinotopic distortion vector field. Both the pattern of postsynaptic weight non-uniformity and the corrective retinotopic distortion vector field fail to possess the reflection symmetry commonly assumed to relate orientation singularities with topological index +/- pi. Hence, we show that 'right-handed' and 'left-handed' orientation singularities are funda-mentally distinct anatomical structures when full 2D synaptic architecture is taken into account. Finally, we predict specific patterns of retinotopic distortion that should obtain in the vicinity of +/- pi-fold orientation singularities, if uniform pre- and post-synaptic weight constraints are strongly enforced.
哺乳动物纹状皮层（V1）最显著的特征是对基本视觉刺激的位置和方向的反应偏好的空间组织。用于形成方向和“视网膜拓扑”位置的皮层图谱的模型通常依赖于基于赫布或相关性的突触可塑性组合，以及对突触权重分布的约束。我们考虑一个简化的方向和视网膜拓扑特异性模型，该模型基于从“无方向”细胞层到第一个弱方向层的前馈突触权重分布的几何结构。我们将前馈权重分布建模为一个平面高斯感受野系统，每个感受野在与突触后细胞的偏好方向匹配的方向上拉长。在突触前权重归一化（方向层中的每个细胞接收相同的净突触权重）和均匀视网膜拓扑图谱（无方向层中感受野质心的位移与方向层中相应细胞的位移严格成比例）的约束下，我们发现施加方向偏好模式会迫使系统违反突触后权重归一化（无方向层中的每个细胞不再发出相同的净突触权重）。我们研究了这种突触后权重均匀性的偏差，发现该偏差在方向图谱的“风车”奇点附近具有独特的形式。我们表明，通过在感受野的视网膜拓扑位置引入畸变，可以恢复无方向层的均匀突触覆盖。我们在感受野的相对伸长的一阶近似下计算视网膜拓扑畸变矢量场。突触后权重不均匀模式和校正性视网膜拓扑畸变矢量场都不具有通常假设的将方向奇点与拓扑指数±π相关联的反射对称性。因此，我们表明，当考虑完整的二维突触结构时，“右手性”和“左手性”方向奇点是根本不同的解剖结构。最后，我们预测，如果严格执行均匀的突触前和突触后权重约束，在±π倍方向奇点附近应该会出现特定的视网膜拓扑畸变模式。
