Self-organizing mechanism for development of space-filling neuronal dendrites

Abstract

Neurons develop distinctive dendritic morphologies to receive and process information. Previous experiments showed that competitive dendro-dendritic interactions play critical roles in shaping dendrites of the space-filling type, which uniformly cover their receptive field. We incorporated this finding in constructing a new mathematical model, in which reaction dynamics of two chemicals (activator and suppressor) are coupled to neuronal dendrite growth. Our numerical analysis determined the conditions for dendritic branching and suggested that the self-organizing property of the proposed system can underlie dendritogenesis. Furthermore, we found a clear correlation between dendrite shape and the distribution of the activator, thus providing a morphological criterion to predict the in vivo distribution of the hypothetical molecular complexes responsible for dendrite elongation and branching.

Figure 1. Competitive Interactions between Dendrites Mediate Isoneuronal Avoidance and Tiling

(A) An image of Drosophila larva of NP7028 UAS-mCD8::GFP [11]. Class IV da neurons ddaC (arrows) were visualized with GFP. Dendrites of class IV da neurons almost completely cover the body wall. (B) A high-power image of class IV da neurons at single dendrite resolution (left). Dendrites from the left segment are colored purple; and those from the right segment, green (right). Dendrites of the same class IV da neuron come very close, but hardly overlap each other (isoneuronal avoidance); in addition, minimal overlap was seen between dendrites of neighboring neurons (heteroneuronal avoidance or tiling). (C) Schematic drawing of a filling-in response (adapted from 12). Left: Branches enclosed by the dotted line were severed by laser irradiation (arrow). Right: Neighboring dendrites filled in the open space that had been covered by the detached branches and space-filling pattern was regenerated. Black: dendrites of the operated cell. Gray: dendrites of the neighboring cell. This experiment clarifies an essential role of inhibitory dendro-dendritic interactions in isoneuronal avoidance and tiling. Bar, 50 μm for “A” and 20 μm for “B.”

Figure 2. Schematic Representation of the Cell Compartment Model

(A,B) An activator–suppressor system. Intracellular activator promotes growth of dendrite and produces the suppressor or accelerates secretion of the suppressor from intracellular organelles (“1” in (A)). On the other hand, the suppressor is secreted from the cell and diffuses in extracellular compartments. Binding of its receptor on the plasma membrane triggers signaling to inhibit synthesis of the activator (“2” in (A)). These reactions underlie inhibitory dendro-dendritic interactions (B). (C) Black: core of the cell; dark gray: cell boundary. The cell compartment is represented by collective circular domains around the core with radius R (gray circles). (D) Dynamics of core of the cell (c). The activator promotes cell growth when its concentration is higher than threshold (Tr). a(u) = 0.49 (u ≤ Tr) or a(u) = 0.49 − 2.5(u – Tr) (u > Tr). Upper graph: Both c = 0 and c = 1 are stable equilibrium points. Lower graph: When a(u) < 0, c = a(u) and c = 1 are stable equilibrium points and c = 0 becomes an unstable equilibrium point. Very small positive noise was added to c, so c → 1 quickly. These settings make it possible to store the history of growth of c, because c = 1 is a stable equilibrium point all the time.

Figure 3. Distinctive Dendritic Patterns Obtained from the Computer Simulation of the Cell Compartment Model

(A–C) and (D–F) Two distinct patterns obtained by using different parameter values in the activator–suppressor model. Whole images of dendritic trees (A,D), magnified images of the activator (B,E) and those of the suppressor (C,F). Examples of branch-poor arbors and branch-rich arbors were indicated in yellow and blue, respectively, (A) and (D). Density of the activator is relatively high at the terminal of each branch (arrows in “B”). Alternatively, dendrites elongate as new spots are generated (arrows in “E”) and bifurcate as spots undergo fission (arrowheads in “E”). Parameter values were p_a = 0.9 and p_e = 6.5 (A–C) and p_a = 0.5 and p_e = 2.6 (D–F). Other parameters were p_b = 0.8, d = 30.0, p_h = 1.0, T_r = 1.0, A_max = 30.0, R = 0.004, and γ = 625. The grid size is 800 × 800, dx = 0.02, and dt = 1 × 10⁻⁶.

Figure 4. Typical Changes of u and v at the Tips of Branches

(A,B) Representative data of the values of u and v were indicated on the phase plane, where direction of dynamics and null-cline of u and v are also shown (“A” for the cell of Figure 3A and “B” for that of Figure 3D). The data is sampled in every grid from the branch terminals. v increases linearly with u from the cell boundary to the interior of the cell. Solid lines and dashed lines represent f(u,v) = 0 and g(u,v) = 0, respectively. We have found that an elongation speed is about twice slower in rugged dendrites than in well-aligned ones (compare Videos S1 and S3). The difference in the positioning of the u − v values relative to the isoclines may potentially explain the difference in a velocity of pattern formation (our unpublished data).

Figure 5. Parameter-Dependency of the Pattern Formation

(A–D) Searches for parameter values for dendritic branch formation on a (p_e − p_a)-plane. The fixed parameters were p_b = 0.8, d = 30.0, p_h = 1.0, Tr = 1.0, A _max = 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) (p_a = 2.1 and p_e = 8.0), (C) p_a = 0.5 and p_e = 4.0), and (D) (D (p_a = 0.7 and p_e = 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). p_a = 0.7 for all panels; and p_e = 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).

Figure 6. Comparative Analysis of Conditions for Dendritic Pattern Formation and Those for Dot, Stripe, and Reverse-Dot Pattern Generation

(A) Condition of parameters for neuronal branching and Turing condition on a (p_e − A _max)-plane. Simulations of the no-compartment model showed that dot, stripe, and reverse-dot domains are mapped to different parameter regions (D, S, and R, respectively); and dotted lines roughly represent boundaries between the domains. Closed circles: dendritic patterns obtained by our cell-compartment model. The fixed parameters were p_a = 0.7, p_b = 0.8, d = 30.0, p_h = 1.0, R = 0.004, and γ = 625. We used Tr = 0.95 (when A _max = 1.0), Tr = 0.85 (when A _max = 0.9), and Tr = 1.0 (otherwise). Total calculation time was 4 × 10⁵ steps. (B–E) Typical examples in the stripe domain (B,C): p_e = 4.0, A _max = 1.1, and Tr = 1.0, and in the reverse-dot domain (D,E): p_e = 3.6 and Tr = 0.95. Dendrites (B,D) and the distribution of the activator (C,E) are shown.

Figure 7. Patterns Generated by the Linear Dynamics

(A) Region I indicates a parameter region on a (p_a − p_b)-plane that satisfies the classical Turing condition. p_c does not appear in the Turing condition. Region II is as described in the legend of Figure 5. (B,C) and (D,E) Examples of patterns generated in region II and region I, respectively. Magnified images of dendrites (B,D) and those of activator distributions (C,E). Parameter values were p_a = 0.5, p_b = 2.5, and p_c = 0.16 (B,C) and p_a = 0.6, p_b = 2.9, and p_c = 0.2 (D,E). Other parameters were the same as in Figure 3A, except for A _max = 10.0 and Tr = 0.75. Conditions of simulation were as described in the legend of Figure 3.

Figure 8. Two Distinctive Branching Patterns of Real Neurons

(A,B) Smooth and well-aligned type. A neuron in thalamic nuclei in monkey (A) and Purkinje cell at postnatal day 25 (B). (C,D) Rugged and less-aligned type. A neuron of inferior olivary complex of monkey (C) and a remodeled class I da neuron at a pupal stage, which was visualized with ppk-GAL4 UAS-mCD8::GFP [49] (D). Images in (A and C) were taken from [50]. (E) Quantification of DOB in the generated patterns in our computer simulation. Data are presented as the means ± SD. A single asterisk indicates p < 0.01 (t-test), and double asterisks indicate p < 0.001 (t-test).

Figure 9. Procedures for Quantification of DOB

Step 1: Skeletonize images with ImageJ. Step 2: Crop four pairs of squares. Step 3: Approximate each dendritic segment between the two branching points by a line segment connecting those points. Measure the angle of a branch segment with respect to the horizontal direction. Repeat measurement for all segments in each local area and calculate the coefficient of variation, DOB.