Loxley P N, Robinson P A
School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Oct;76(4 Pt 2):046224. doi: 10.1103/PhysRevE.76.046224. Epub 2007 Oct 31.
Hopfield's Lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation. For two neural populations with reciprocal inhibition and slow adaptation, the dynamics of neural activity is found to include a pair of limit cycles: one for oscillations between states where one population has high activity and the other has low activity, as in rivalry, and one for oscillations between states where both populations have the same activity. Hopfield's Lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle. For a spatially continuous population with lateral inhibition, stable equilibria are found for local regions of high activity (solitons) and for bound states of two or more solitons. Bound states become stable when moving two solitons together minimizes the Lyapunov function, a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function. Lowering the barrier to soliton formation leads to a pattern-forming instability, and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice, which is completely characterized by the soliton width and the spacing between neighboring solitons. Fluctuations due to noise create lattice vacancies analogous to point defects in crystals, leading to activity which is spatially inhomogeneous.
霍普菲尔德的李雅普诺夫函数用于研究视觉竞争和模式形成中神经网络平衡态的稳定性和拓扑结构。对于具有相互抑制和缓慢适应的两个神经群体，发现神经活动的动力学包括一对极限环：一个用于在一个群体活动高而另一个群体活动低的状态之间振荡，如同竞争状态；另一个用于两个群体活动相同的状态之间振荡。霍普菲尔德的李雅普诺夫函数用于找出振荡的动力学机制以及每个极限环的吸引域。对于具有侧向抑制的空间连续群体，在高活动的局部区域（孤子）以及两个或更多孤子的束缚态中发现了稳定平衡。当将两个孤子移到一起使李雅普诺夫函数最小时，束缚态变得稳定，这是当发放率由一个S形函数描述时，高活动峰值之间区域活动降低的结果。降低孤子形成的势垒会导致模式形成不稳定性，并且发现动力学方程的一个非线性解由一个孤子晶格给出，它完全由孤子宽度和相邻孤子之间的间距来表征。由噪声引起的涨落会产生类似于晶体中点缺陷的晶格空位，导致空间上不均匀的活动。
