Takeuchi A, Amari S
Biol Cybern. 1979 Nov 2;35(2):63-72. doi: 10.1007/BF00337432.
Topographic connections are found in many parts of the vertebrate nervous systems, known for example as retinotopy. The self-organizing ability of Hebb type modifiable synapses plays an important role in forming, at least in refining, the topographic connections. We present a mathematical analysis of a revised version of the Willshaw-Malsburg model of topographic formation, solving the equations of synaptic self-organization coupled with the field equation of neural excitations. The equilibrium solutions are obtained and their stability is studied. It is proved that two cases exist depending on parameters. In one case, the smooth topographic organization is obtained as a stable equilibrium of the equations. In the other case, this solution becomes unstable, and instead the topographic organization with columnar microstructures appears. This might explain the columnar structures in the cerebrum. The theory is confirmed by computer simulated experiments.
在脊椎动物神经系统的许多部位都发现了拓扑连接，例如视网膜拓扑。赫布型可修改突触的自组织能力在形成（至少是细化）拓扑连接中起着重要作用。我们对拓扑形成的威尔肖 - 马尔堡模型的修订版进行了数学分析，求解了突触自组织方程与神经兴奋场方程的耦合。获得了平衡解并研究了它们的稳定性。证明了根据参数存在两种情况。在一种情况下，平滑的拓扑组织作为方程的稳定平衡而获得。在另一种情况下，该解变得不稳定，取而代之的是出现具有柱状微结构的拓扑组织。这可能解释了大脑中的柱状结构。该理论通过计算机模拟实验得到了证实。
