Branching pattern formation that reflects the history of signal propagation

In living organisms, branching structures are often observed in open systems. During the process of structure formation/deformation, signal propagation can be observed. Branching paths often deform depending on the history of signal propagation. To gain a better understanding of the process of pattern formation that results in characteristic geometrical paths, we adopt a system in which the dynamics of path formation are correlated with signal propagation. This model involves both branch-generation dynamics and signal-propagation dynamics, and we introduced positive feedback between these two dynamic processes. We studied the geometrical properties of path deformation and the pattern of signal propagation using a discretized reaction-diffusion model. The proposed model can qualitatively reproduce different branching patterns and means of signal propagation. One remarkable result is that the mutual interaction of these two dynamic processes leads to autonomous wave generation, similar to a pacemaker or the generation of spiral waves. Because the autonomous wave generation in the signal is spontaneous, the shapes of the branching paths become distorted. We discuss the correlation between path deformation and signal propagation as a first step in understanding signal processing for such complex deformable paths.

Figure 1

Morphological map of the patterns observed for the algorithm without interactions. Black cells correspond to $w=1$ [17].

Figure 2

(a) Typical phase-plane diagram for a differential equation model of an excitable medium. (b) Schematic of the state evolution of a cellular automaton [18].

Figure 3

(Color) Time-evolution of non dense pattern formation at time steps 1000, 3300, 5600, and 7900. The parameters are $R_a=2$ and $f_0=68$. In the top row, the black, yellow, and red cells correspond to $w=1$, $u=1$, and $a=1$, respectively. The bottom row shows corresponding profiles of the excitation wave $u$ in a quasi-three-dimensional representation. The dark region corresponds to $w=1$.

Figure 4

(Color) Time evolution of dense pattern formation at time steps 70, 290, 510, and 730. The parameters are $R_a=5$ and $f_0=77$.

Figure 5

Morphological map of the patterns observed in the algorithm with interaction. The region of the initial growth-factor value is the same as that in Fig. 1. Black cells correspond to $w=1$.

Figure 6

Morphological map of the patterns observed in the algorithm that included interaction. The region shown for the initial growth-factor value is higher than that in Fig. 1. Black cells correspond to $w=1$.

Figure 7

Radius of gyration method for $R_f=9$, (○): $R_a=2$, $f_0=68$, and (◼): $R_a=5$, $f_0=77$. The fractal dimensions are 1.65 and 1.81, respectively.

Figure 8

Fractal dimensions of patterns observed with changes in the parameters $f_0$ and $R_a$.

Figure 9

Numbers of pattern branches observed with changes in the parameters $f_0$ and $R_a$.

Figure 10

Growth rates of the patterns observed with changes in the parameters $f_0$ and $R_a$.

Figure 11

(Color) Frame format for the process of spiral wave generation at the end of a path. Yellow and black regions correspond to the wave front (excited region) and wave back (unexcitable refractory region), respectively. Gray and red regions correspond to the existing and newly formed neutral branching paths, respectively.

Figure 12

Morphological map of the patterns observed with the algorithm that includes interaction, with the first transient external inputs. Black cells correspond to $w=1$.

Figure 13

Probability of formation of branching patterns observed with changes in the parameters $f_0$ and $R_a$.