**1. Introduction**

Various growth phenomena in physics have been discussed for decades mainly in the field of fluid dynamics [1, 2]. The previous studies, including those on Laplacian growth and diffusion-limited aggregation [1, 2], mathematically involve conformal dynamics derived from Riemann mapping theorem. The contribution of the Loewner differential equation [3] to this field is remarkable as notably shown by Stochastic Loewner evolution (SLE), which was discovered by Schramm [4, 5]. The SLE is a kind of growth processes described by the Loewner equation having a stochastic driving function, which is used as a model for random curves in statistical mechanics. A

typical example is the phase interface of the Ising model at a critical temperature [6]. On the other hand, there exist many types of self-organized curves in biological systems, e.g. veins of leaves, skin patterns of animals, axons of neurons, and so on. However, the exact theory for their morphogenesis is still unclear, and the SLE has never been used to explain it. Above all, neurite morphogenesis is one of the most informative processes when considering production processes of complex curves as well as neural development processes.

The neurite curves are very diverse and inherently constitute ambiguous messages in their forms, being closely related to their functions and development. Specifically, morphological neurite disorders are hallmarks of the pathologies of various neurodegenerative diseases. Alzheimer's disease (AD) is a typical example, where neurite disorders are considered a key factor in its pathology. The main characteristics representing the abnormality of AD neurons are dystrophic neurites (DNs) and neurofibrillary tangles (NFTs). These morphological disorders are associated with the accumulation of specific proteins in AD neurons and also in those of other neurodegenerative diseases.

The quantification methods of neurite morphology, including morphological disorders, have not been valid, and disorders such as DNs and NFTs are mainly evaluated by visual observations [7–9]. Thus, ambiguity remains in the morphological definition of neurite disorders, and diagnosing neurodegenerative diseases based solely on morphological characteristics is still a difficult work for biological research. Therefore, several mathematical and physical methods have been suggested to quantify neurite morphology (e.g. fractal dimensions [10], stochastic methods [11], or differential equations [12, 13]). Quantifying pathological states of neurite morphology, however, requires further improvements or alternatives of these models. Therefore, a systematic and theoretically plausible method for examining the degree of morphological abnormalities is needed to discuss morphological neurite disorders.

Recently, we have proposed a statistical-physical approach to analyze neurite morphology based on the Loewner equation mentioned above, which leads to not only a physical interpretation neurite outgrowth mechanism but also a new description of selforganization mechanism of complex curves. In this chapter, we introduce such a recent approach of us. First, we briefly review the concept of the Loewner equation and its calculation algorithm with some calculation examples. We next describe analyses of neurite morphology of neuroblastoma cells (Neuro2A) using the Loewner equation to show the efficacy of our approach. Here, we show that the neurite outgrowth mechanism can be described by the Loewner equation having a deterministic (chaotic) driving function, which differs from the SLE [14]. Finally, we describe similar analyses of neurite morphology of human-induced pluripotent stem cell (iPSC)-derived neurons and discuss the possibility of a medical application of our approach [15].

### **2. Loewner equation and its calculation algorithm**

A growth process of a simple curve, which does not intersect, on the upper half complex plane is expressed as a form of time evolution of the conformal map. We consider a simple curve *γ*½ � 0, *<sup>t</sup>* starting from the origin O. Here, *γ*½ � 0, *<sup>t</sup>* is parameterized by time *t*. The tip of the curve at time *t* is denoted as *γt*. The following equation, which is called the Loewner equation, yields a family of conformal maps *gt* from n*γ*½ � 0, *<sup>t</sup>* to [4, 5, 16]:

*Application of the Loewner Equation for Neurite Outgrowth Mechanism DOI: http://dx.doi.org/10.5772/intechopen.108377*

$$\frac{\partial \mathbf{g}\_t(\mathbf{z})}{\partial t} = \frac{2}{\mathbf{g}\_t(\mathbf{z}) - \boldsymbol{\xi}\_t}, \text{ g}\_0 = \mathbf{z}, \; \mathbf{z} \in \mathbb{H}. \tag{1}$$

Here, *ξ<sup>t</sup>* is a real-valued time function called the driving function. In the SLE, *ξ<sup>t</sup>* is usually chosen as a standard Brownian motion *Bt* with the diffusivity parameter κ, that is, ffiffiffi <sup>κ</sup> <sup>p</sup> *Bt*. The curve is transformed to the driving function by the relationship *gt γ<sup>t</sup>* ð Þ¼ *ξ<sup>t</sup>* and the inverse transformation *gt* �<sup>1</sup> is also available so that they have a oneto-one correspondence (**Figure 1a**). The transformations between *γ<sup>t</sup>* and *ξ<sup>t</sup>* are often referred to as *encoding* and *decording* (**Figure 2**).

The calculation of the driving function from an arbitrary simple curve requires appropriate discretization of the Loewner equation, and several calculation methods were proposed so far [17–19]. Among them, we here introduce the frequently used zipper algorithm based on the vertical slit map *gt* ð Þ¼ *z ξ<sup>t</sup>* þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>z</sup>* � *<sup>ξ</sup><sup>t</sup>* ð Þ<sup>2</sup> <sup>þ</sup> <sup>4</sup>*<sup>t</sup>* q [18, 19], which is a solution of the Loewner equation in Eq. (1). This map transforms the line segment from *<sup>ξ</sup><sup>t</sup>* to *<sup>ξ</sup><sup>t</sup>* <sup>þ</sup> <sup>2</sup>*<sup>i</sup>* ffiffi *<sup>t</sup>* <sup>p</sup> , which is called the vertical slit, to a line segment on the real axis, where *<sup>ξ</sup><sup>t</sup>* <sup>þ</sup> <sup>2</sup>*<sup>i</sup>* ffiffi *<sup>t</sup>* <sup>p</sup> is transformed to *<sup>ξ</sup><sup>t</sup>* (**Figure 1b**). This map has an important role to the calculation algorithm mentioned below.

We define discretized points on an arbitrary simple curve on as *γ*½ � 0, *<sup>N</sup>* ¼ f g *z*0ð Þ ¼ 0 , *z*1, *z*2, … , *zn*, … , *zN* , and we define the time sequence corresponding to each point of *γ*½ � 0, *<sup>N</sup>* as *t*½ � 0, *<sup>N</sup>* ¼ f g *t*0ð Þ ¼ 0 , *t*1, *t*2, … , *tn*, … , *tN* ð Þ *tn* ∈ . The discretized vertical slit map can be expressed as:

#### **Figure 1.**

*Schematic illustration of the relationship (a) between the curve and the driving function and (b) between the vertical slit and the driving function. See the text in detail.*

**Figure 2.** *Calculation scheme of the Loewner eqation. See the text in detail.*

$$\mathbf{g}\_n(\mathbf{z}) = \Delta \xi\_{t\_n} + \sqrt{\left(\mathbf{z} - \Delta \xi\_{t\_n}\right)^2 + 4\Delta t\_n}.\tag{2}$$

Here, Δ*ξtn* ¼ *ξtn* � *ξtn*�<sup>1</sup> and Δ*tn* ¼ *tn* � *tn*�1. This map corresponds to the conformal map determined by the Loewner equation in Eq. (1) for each step *n*. We consider the complex variable *wn* � <sup>Δ</sup>*ξtn* <sup>þ</sup> <sup>2</sup>*<sup>i</sup>* ffiffiffiffiffiffiffi Δ*tn* � � <sup>p</sup> , which represents the tip of a short vertical slit. To calculate the sequence of Δ*ξtn* � � and f g <sup>Δ</sup>*tn* , we consider the following map shifted from Eq. (2):

$$h\_n(\mathbf{z}) \equiv \mathbf{g}\_n(\mathbf{z}) - \Delta \xi\_{t\_n} = \sqrt{\left(\mathbf{z} - \Delta \xi\_{t\_n}\right)^2 + 4\Delta t\_n}.\tag{3}$$

The iterations of *hn* give the increments of the driving function in terms of *wn* [18, 19]. In the followings, we describe the details of the algorithm in a step-by-step manner. First, *w*<sup>1</sup> equals to the coordinate *z*<sup>1</sup> of the first point of the curve, i.e.

$$w\_1 = z\_1.\tag{4}$$

From the real and imaginary parts of *w*1, we obtain Δ*ξ<sup>t</sup>*<sup>1</sup> and Δ*t*1, while they determine the map *h*<sup>1</sup> expressed by Eq. (3). Subsequently, *w*<sup>2</sup> is determined by *z*<sup>2</sup> as:

$$w\_2 = h\_1(z\_2).\tag{5}$$

Similarly, the real and imaginary parts of *w*<sup>2</sup> provide the increments of the driving function Δ*ξ<sup>t</sup>*<sup>2</sup> and Δ*t*2, and they determine the next map *h*2. Repeating this procedure successively, *wn*þ<sup>1</sup> is obtained as the following:

$$
\omega\_{n+1} = h\_n \circ h\_{n-1} \circ \cdots \circ h\_1(z\_{n+1}).\tag{6}
$$

By applying this recursive relation to the coordinates of *γ*½ � 0, *<sup>N</sup>* up to *n* ¼ *N* � 1, the increments of the driving function Δ*ξtn* � � , called the driving forces, and f g <sup>Δ</sup>*tn* can be calculated. By summing up these, the driving function *ξtn* � � and f g *tn* are also calculated. Source code for this calculation algorithm is available, for example, on LOEW-Schramm Loewner evolutions for Python in GitHub.

Based on this algorithm, we can also calculated the coordinates of the corresponding curve from an arbitrary driving function *ξtn* � � using the inverse transformation *gn* �1. **Figure 3** shows an obtained curve. Here, we employed the discretized *Application of the Loewner Equation for Neurite Outgrowth Mechanism DOI: http://dx.doi.org/10.5772/intechopen.108377*

#### **Figure 3.**

*Numerically calculated curve on from the Loewner equation having the driving function* ffiffiffi <sup>κ</sup> <sup>p</sup> *Bt.*<sup>κ</sup> <sup>¼</sup>*0.6.*

driving function ffiffiffi <sup>κ</sup> <sup>p</sup> *Bt* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κτP*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*Wi* p like the SLE, where *Wi* denotes the Gaussian noise with mean 0 and variance 1 (*B*<sup>0</sup> ¼ 0Þ and *τ* is a sufficiently small time step interval satisfying ¼ *n*τ.

We show a transformation from a curve to the corresponding driving function below. Here, we considered the 2D ferromagnetic Ising model whose Hamiltonian is described as follows:

*Chaos Theory – Recent Advances, New Perspectives and Applications*

$$H = -\sum\_{i,j} \sigma\_i \sigma\_j. \tag{7}$$

Here, σ*<sup>i</sup>* and σ*<sup>j</sup>* are the nearest neighboring spins, which take the value of σ*i*,*<sup>j</sup>* ∈f g �1, 1 on the square lattice. We calculated the driving force corresponding to the calculated phase interface of the Ising model [20]. The interface was set on the upper half complex plane (**Figure 4**), and then the corresponding driving force Δ*ξtn* � � was calculated by the above algorithm. **Figure 5** shows the calculated driving

#### **Figure 4.**

*Simulation result of the 2D Ising model. (a) Example of the spin configurations of the Ising model at T* ¼ *T*C*. T*<sup>C</sup> *shows a critical temperature. The yellow sites represent the spins of* σ*<sup>i</sup>* ¼ 1*, and the blue sites represent those of* σ*<sup>i</sup>* ¼ �1*. The center of the bottom side and that of the upper side are denoted as points A and B, respectively. (b) Extracted interface for the spin configulation in (a). The interface is set on the upper half plane so that point A corresponds to the origin. The inset displays an enlarged view of the interface. These figures are reproduced from Ref. [20] with permission.*

#### **Figure 5.**

*Time series of the driving forces {xn} corresponding to the interfaces of the 2D Ising model. (a) Example of the time series of {xn} at T* ¼ 0*:*2 *T*C,0*:*6 *T*<sup>C</sup> and1*:*0 *T*<sup>C</sup> *from the top to bottom. T*<sup>C</sup> *shows a critical temperature. The data length and xn-range increase as T* ! *T*C*. (b) Time series of {xn} at T* ¼ 0*:*2 *T*<sup>C</sup> *in an enlarged view. The red points and blue dotted line display the data points and trajectories of the time series, respectively. These figures are reproduced from Ref. [20] with permission.*

*Application of the Loewner Equation for Neurite Outgrowth Mechanism DOI: http://dx.doi.org/10.5772/intechopen.108377*

**Figure 6.**

*Poincaré maps of the driving forces {xn} corresponding to the interfaces of the 2D Ising model. (a) T* ¼ 0*:*2 *T*C*, (b) T* ¼ 0*:*6 *T*C*, and (C) T* ¼ 1*:*0 *T*C*. These figures are reproduced from Ref. [20] with permission.*

force at each temperature. We should note here that the driving force is often normalized or resampled at even time intervals because the calculated f g Δ*tn* is generally inhomogeneous. In **Figure 5**, the driving forces are time-normalized as:

$$\mathbf{x}\_{\mathfrak{n}} = \frac{\Delta \mathfrak{E}\_{\mathfrak{t}\_{\mathfrak{n}}}}{\sqrt{\Delta \mathfrak{t}\_{\mathfrak{n}}}}, \ \mathbf{x}\_{0} = \mathbf{0}. \tag{8}$$

In this study, we found that the normalized driving forces {*xn*} are based on deterministic (chaotic) dynamics, not stochastic dynamics. **Figure 6** shows the Poincaré map for {*xn*} at each temperature, which shows the existence of attractors for the dynamics of {*xn*}. Therefore, the dynamics of {*xn*} arises from some deterministic law, although the maps are more complicated than simple unimodal maps. Interestingly, the observed maps have a nested structure: the map at the low temperature is a part of that at the high temperature (see the parts shown by red squares). These results show that the dynamics of {*xn*} have chaotic features, which was also confirmed from the calculated Lyapunov exponents (see ref. [20]). Thus, this study suggested that growth processes of some curves, such as the Ising interface, can be different from the SLE.
