Self-organizing Mechanism for Development of Space-filling Neuronal Dendrites
Figure 5
Parameter-Dependency of the Pattern Formation
(A–D) Searches for parameter values for dendritic branch formation on a (pe − pa)-plane. The fixed parameters were pb = 0.8, d = 30.0, ph = 1.0, Tr = 1.0, Amax = 30.0, R = 0.004, and γ = 625. Total calculation time was 4 × 105 steps.
(A) Closed circle: dendritic pattern; square: wide branches; triangle: no second-order branching; and star: no growth. Examples of non-dendritic patterns of square, triangle, and star are shown in (B) (pa = 2.1 and pe = 8.0), (C) pa = 0.5 and pe = 4.0), and (D) (D (pa = 0.7 and pe = 6.5), respectively. Region I satisfies conditions of Turing diffusion-induced instability described by Equations 6a–6d, whereas region II satisfies Equations 6a–6c, but not Equation 6d and region 0 satisfies Equations 6b–6d, but not Equation 6a. In region I, spatially periodic patterns appear in a conventional RD model, whereas homogenous patterns are stable in region II.
(E1–E4) Distributions of the activator that were obtained at different coordinates in the phase diagram (A). pa = 0.7 for all panels; and pe = 3.0 (E1), 3.5 (E2), 4.0 (E3), and 4.5 (E4). The distribution of the activator changes from a punctate pattern (E1) to a more continuous pattern (E4).
(E2) A punctate distribution of the activator in a branch-rich region (enclosed area at right) and a more continuous one in a branch-less region (enclosed area at left).