Center for Complexity Science, Imperial College London, London, SW7 2AZ, UK.
Theoretical Physics of Biology Laboratory, The Francis Crick Institute, London, NW1 1AT, UK.
Bull Math Biol. 2021 Feb 17;83(4):28. doi: 10.1007/s11538-020-00848-2.
Locomotion characteristics are often recorded within bounded spaces, a constraint which introduces geometry-specific biases and potentially complicates the inference of behavioural features from empirical observations. We describe how statistical properties of an uncorrelated random walk, namely the steady-state stopping location probability density and the empirical step probability density, are affected by enclosure in a bounded space. The random walk here is considered as a null model for an organism moving intermittently in such a space, that is, the points represent stopping locations and the step is the displacement between them. Closed-form expressions are derived for motion in one dimension and simple two-dimensional geometries, in addition to an implicit expression for arbitrary (convex) geometries. For the particular choice of no-go boundary conditions, we demonstrate that the empirical step distribution is related to the intrinsic step distribution, i.e. the one we would observe in unbounded space, via a multiplicative transformation dependent solely on the boundary geometry. This conclusion allows in practice for the compensation of boundary effects and the reconstruction of the intrinsic step distribution from empirical observations.
运动特征通常在有界空间内记录，这种约束会引入特定于几何形状的偏差，并可能使从经验观察推断行为特征变得复杂。我们描述了无关联随机游走的统计特性（即稳态停止位置概率密度和经验步概率密度）如何受到有界空间限制的影响。这里的随机游走被视为在这种空间中间歇性运动的生物体的零模型，也就是说，点代表停止位置，步长是它们之间的位移。除了任意（凸）几何形状的隐式表达式外，我们还推导出了一维和简单二维几何形状的闭式表达式。对于无限制边界条件的特殊选择，我们证明经验步分布与内在步分布有关，即我们在无界空间中观察到的分布，通过仅依赖于边界几何形状的乘法变换。这一结论允许在实践中补偿边界效应，并从经验观察重建内在步分布。
