**Abstract**

Stochastic Loewner evolution (SLE), which was discovered by Schramm, is a kind of growth processes described by the Loewner equation having a stochastic driving function. The SLE is used as a model for random curves in statistical mechanics. On the other hand, there exist many types of self-organized curves in biological systems. Among them, the neurite curves are very diverse and inherently constitute ambiguous messages in their forms, being closely related to their functions and development. Recently, we have proposed a statistical-physical approach to analyze neurite morphology based on the Loewner equation, which leads to not only a physical interpretation of neurite outgrowth mechanism but also a new description of self-organization mechanism of complex curves. In this chapter, we first review the concept of the Loewner equation and its calculation algorithm. We next show that neurite outgrowth process can be described by the Loewner equation having a deterministic (chaotic) driving function, which differs from the SLE. Based on this point of view, we finally analyze induced-pluripotent stem cell (iPSC)-derived neurons from a healthy person and an Alzheimer's disease (AD) patient and discuss pathological neurite states and the possibility of a medical application of our approach.

**Keywords:** Stochastic Loewner evolution, Loewner equation, driving function, neurite morphology, neurite outgrowth mechanism, fluctuation analysis, scaling exponent, induced-pluripotent stem cell
