Application of the Loewner Equation for Neurite Outgrowth Mechanism

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.

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, 8, 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 self-organization 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 $\mathbb{H}$ is expressed as a form of time evolution of the conformal map. We consider a simple curve ${\gamma }_{\left[0,t\right]}$ starting from the origin O. Here, ${\gamma }_{\left[0,t\right]}$ is parameterized by time t. The tip of the curve at time t is denoted as ${\gamma }_t$. The following equation, which is called the Loewner equation, yields a family of conformal maps $g_t$ from $\mathbb{H}\backslash {\gamma }_{\left[0,t\right]}$ to $\mathbb{H}$ [4, 5, 16]:

$$\frac{\partial g_t\left(z\right)}{\partial t}=\frac{2}{g_t\left(z\right)-{\xi }_t},g_0=z,z\in \mathbb{H}.$$E1

Here, ${\xi }_t$ is a real-valued time function called the driving function. In the SLE, ${\xi }_t$ is usually chosen as a standard Brownian motion $B_t$ with the diffusivity parameter $\mathrm{\kappa }$, that is, $\sqrt{\mathrm{\kappa }}{B}_t$. The curve is transformed to the driving function by the relationship $g_t\left({\gamma }_t\right)={\xi }_t$ and the inverse transformation ${g_t}^{-1}$ is also available so that they have a one-to-one correspondence (Figure 1a). The transformations between ${\gamma }_t$ and ${\xi }_t$ are often referred to as encoding and decording (Figure 2).

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.

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, 18, 19]. Among them, we here introduce the frequently used zipper algorithm based on the vertical slit map $g_t\left(z\right)={\xi }_t+\sqrt{{\left(z-{\xi }_t\right)}^2+4t}$ [18, 19], which is a solution of the Loewner equation in Eq. (1). This map transforms the line segment from ${\xi }_t$ to ${\xi }_t+2i\sqrt{t}$ , which is called the vertical slit, to a line segment on the real axis, where ${\xi }_t+2i\sqrt{t}$ is transformed to ${\xi }_t$ (Figure 1b). This map has an important role to the calculation algorithm mentioned below.

We define discretized points on an arbitrary simple curve on $\mathbb{H}$ as ${\gamma }_{\left[0,N\right]}=\left\{z_0\left(=0\right),z_1,z_2,\dots ,z_n,\dots ,z_N\right\}$, and we define the time sequence corresponding to each point of ${\gamma }_{\left[0,N\right]}$ as $t_{\left[0,N\right]}=\left\{t_0\left(=0\right),t_1,t_2,\dots ,t_n,\dots ,t_N\right\}\left(t_n\in \mathbb{R}\right)$. The discretized vertical slit map can be expressed as:

$$g_n\left(z\right)=∆{\xi }_{t_n}+\sqrt{{\left(z-∆{\xi }_{t_n}\right)}^2+4∆t_n}.$$E2

Here, $∆{\xi }_{t_n}={\xi }_{t_n}-{\xi }_{t_{n-1}}$ and $∆t_n=t_n-t_{n-1}$. This map corresponds to the conformal map determined by the Loewner equation in Eq. (1) for each step $n$. We consider the complex variable $w_n\left(\equiv ∆{\xi }_{t_n}+2i\sqrt{∆t_n}\right)$, which represents the tip of a short vertical slit. To calculate the sequence of $\left\{∆{\xi }_{t_n}\right\}$ and $\left\{∆t_n\right\}$, we consider the following map shifted from Eq. (2):

$$h_n\left(z\right)\equiv g_n\left(z\right)-∆{\xi }_{t_n}=\sqrt{{\left(z-∆{\xi }_{t_n}\right)}^2+4∆t_n}.$$E3

The iterations of $h_n$ give the increments of the driving function in terms of $w_n$ [18, 19]. In the followings, we describe the details of the algorithm in a step-by-step manner. First, $w_1$ equals to the coordinate $z_1$ of the first point of the curve, i.e.

$$w_1=z_1.$$E4

From the real and imaginary parts of $w_1$, we obtain $∆{\xi }_{t_1}$ and $∆t_1$, while they determine the map $h_1$ expressed by Eq. (3). Subsequently, $w_2$ is determined by $z_2$ as:

$$w_2=h_1\left(z_2\right).$$E5

Similarly, the real and imaginary parts of $w_2$ provide the increments of the driving function $∆{\xi }_{t_2}$ and $∆t_2$, and they determine the next map $h_2$. Repeating this procedure successively, $w_{n+1}$ is obtained as the following:

$$w_{n+1}=h_n\circ h_{n-1}\circ \cdots \circ h_1\left(z_{n+1}\right).$$E6

By applying this recursive relation to the coordinates of ${\gamma }_{\left[0,N\right]}$ up to $n=N-1$, the increments of the driving function $\left\{∆{\xi }_{t_n}\right\}$ , called the driving forces, and $\left\{∆t_n\right\}$ can be calculated. By summing up these, the driving function $\left\{{\xi }_{t_n}\right\}$ and $\left\{t_n\right\}$ 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 $\left\{{\xi }_{t_n}\right\}$ using the inverse transformation ${g_n}^{-1}$. Figure 3 shows an obtained curve. Here, we employed the discretized driving function $\sqrt{\mathrm{\kappa }}{B}_t=\sqrt{\mathrm{\kappa \tau }{\sum }_{i=1}^n{W}_i}$ like the SLE, where $W_i$ denotes the Gaussian noise with mean 0 and variance 1 ($B_0=0)$ and τ is a sufficiently small time step interval satisfying $=n\mathrm{\tau }$.

Figure 3.

Numerically calculated curve on $\mathbb{H}$ from the Loewner equation having the driving function $\sqrt{\mathrm{\kappa }}{B}_t$.κ$=$0.6.

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:

$$H=-\sum _{i,j}{\mathrm{\sigma }}_i{\mathrm{\sigma }}_j.$$E7

Here, ${\mathrm{\sigma }}_i$ and ${\mathrm{\sigma }}_j$ are the nearest neighboring spins, which take the value of ${\mathrm{\sigma }}_{i,j}\in \left\{-1,1\right\}$ 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 $\mathbb{H}$ (Figure 4), and then the corresponding driving force $\left\{∆{\xi }_{t_n}\right\}$ was calculated by the above algorithm. Figure 5 shows the calculated driving force at each temperature. We should note here that the driving force is often normalized or resampled at even time intervals because the calculated $\left\{∆t_n\right\}$ is generally inhomogeneous. In Figure 5, the driving forces are time-normalized as:

Figure 4.

Simulation result of the 2D Ising model. (a) Example of the spin configurations of the Ising model at $T=T_{\mathrm{C}}$. $T_{\mathrm{C}}$shows a critical temperature. The yellow sites represent the spins of ${\mathrm{\sigma }}_i=1$, and the blue sites represent those of ${\mathrm{\sigma }}_i=-1$. 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 $\mathbb{H}$ 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 {$x_n$} corresponding to the interfaces of the 2D Ising model. (a) Example of the time series of {$x_n$} at $T=0.2T_{\mathrm{C}},0.6T_{\mathrm{C}}\text{and}1.0T_{\mathrm{C}}$ from the top to bottom. $T_{\mathrm{C}}$shows a critical temperature. The data length and $x_n$-range increase as $T\to T_{\mathrm{C}}$. (b) Time series of {$x_n$} at $T=0.2T_{\mathrm{C}}$ 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.

$$x_n=\frac{∆{\xi }_{t_n}}{\sqrt{∆t_n}},x_0=0.$$E8

In this study, we found that the normalized driving forces {$x_n$} are based on deterministic (chaotic) dynamics, not stochastic dynamics. Figure 6 shows the Poincaré map for {$x_n$} at each temperature, which shows the existence of attractors for the dynamics of {$x_n$}. Therefore, the dynamics of {$x_n$} 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 {$x_n$} 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.

Figure 6.

Poincaré maps of the driving forces {$x_n$} corresponding to the interfaces of the 2D Ising model. (a) $T=0.2T_{\mathrm{C}}$, (b) $T=0.6T_{\mathrm{C}}$, and (C) $T=1.0T_{\mathrm{C}}$. These figures are reproduced from Ref. [20] with permission.

3. Analyses of neurite morphology of Neuro2A using the Loewner equation

Recently, we have proposed a statistical-physical approach to analyze neurite morphology based on the Loewner equation. To show the efficacy of our approach, we first analyzed neurite morphology of neuroblastoma cells (Neuro2A) using the Loewner equation [14].

The prepared Neuro2A was derived from mice (regions of spinal cord). The cells were cultured in Eagle’s minimum essential medium (MEM) with 10% fetal bovine serum (FBS). The cultured medium was replaced to MEM with 2% FBS and retinoic acid (10 μM) was added on the day in vitro 2 (DIV2).

Figure 7 shows two examples of analyzed neurites, which are denoted here as neurite A and neurite B. The microscope images were captured on DIV8. From the obtained images, we semi-automatically extracted the x-y coordinates of the neurites using Neuron J software (a plugin for Image J software). The obtained x–y coordinates were transferred into the upper half complex plane $\mathbb{H}$ so that the starting points of neurite growth corresponded to the origin O and the real-axis coordinates of the tips of the neurites were 0.

Figure 7.

Two examples of analyzed neurites of cultured Neuro2A, which are denoted here as (a) neurite A and (b) neurite B. The left photographs are the microscope images captured on DIV8. The starting points of neurite growth and their tips are marked with “O” and “Tip,” respectively. The right figures are the neurite traces on the upper half complex plane $\mathbb{H}$. The starting points of neurite growth correspond to the origin O, and the real axis-coordinates of the tips of the neurites are 0. These figures are reproduced from Ref. [14] with permission.

We calculated the driving functions $\xi \left(t\right)$ from the coordinates of the neurite traces on $\mathbb{H}$ using the algorithm mentioned above (Figure 8a and b (upper figures)). Here, the driving functions were resampled at even time intervals $∆t=10$ using the MATLAB signal processing tool box, resample function, because the calculated $\left\{∆t_n\right\}$ is generally inhomogeneous as mentioned above. The driving forces are defined here as $\mathit{\delta \xi }\left(t\right)=\xi \left(t+∆t\right)-\xi \left(t\right)$.

Figure 8.

The driving functions $\xi \left(t\right)$ (upper) and double-logarithmic plots of τ and $f\left(\mathrm{\tau }\right)$ (lower) for (a) neurite A and (b) neurite B. These figures are reproduced from Ref. [14] with permission.

Since the Loewner equation has an encoding property, we can consider that the topological properties of the curves are encoded into the corresponding driving functions [16]. Therefore, we can directly determine the morphological features of neurites by examining the properties of the driving functions. To investigate the statistical features of the driving functions, we used the root mean square (r.m.s.) fluctuation analysis [21]. (Note here that we actually used detrended fluctuation analysis (DFA) [22], a modified r.m.s. fluctuation analysis, which is often used for a “trend”-included time series.) For the r.m.s. fluctuation analysis, we first calculate

$$f\left(\mathrm{\tau }\right)=\sqrt{<{\left(∆\xi \left(\mathrm{\tau }\right)-<∆\xi \left(\mathrm{\tau }\right)>\right)}^2}=\sqrt{⟨∆\xi {\left(\mathrm{\tau }\right)}^2⟩-{⟨∆\xi \left(\mathrm{\tau }\right)⟩}^2},$$E9

where $∆\xi \left(\mathrm{\tau }\right)=\xi \left(t_0+\mathrm{\tau }\right)-\xi \left(t_0\right)$. The brackets denote the average values over all reference time $t_0$. We can examine the time autocorrelations of the driving functions by considering the following relationship:

$$\begin{array}{c}f\left(\mathrm{\tau }\right)\sim {\mathrm{\tau }}^{\alpha }.\end{array}$$E10

Here, the linearity of the double-logarithmic plot of $\mathrm{\tau }$ and $f\left(\mathrm{\tau }\right)$ indicates that the curves and driving functions are scale-invariant.

The scale exponent $\alpha$ is estimated by the slope of the double-logarithmic plot of $\mathrm{\tau }$ and $f\left(\mathrm{\tau }\right)$. It is known that we can classify a given time series into different autocorrelation types by the scaling exponent [21, 22]. When $\alpha =0.5$, $\mathit{\delta \xi }\left(t\right)$ is an uncorrelated time series such as Gaussian noise or Markov processes. The uncorrelated curve is therefore the SLE-produced curve. $\alpha >0.5$ indicates the presence of positive correlation in the time series of $\mathit{\delta \xi }\left(t\right)$, and in particular, $\alpha =1.0$ implies 1/f noise. Conversely, $\alpha <0.5$ indicates anticorrelation.

Figure 8a and 8b (lower figures) show the double-logarithmic plots of $\mathrm{\tau }$ and $f\left(\mathrm{\tau }\right)$ for neurite A and B, respectively. Both plots showed almost perfect linearity for ${10}^1\le \mathrm{\tau }\le {10}^2$, where the slopes were ${\alpha }_1=0.94$ for neurite A and ${\alpha }_1=0.93$ for neurite B. On the other hand, for ${10}^2\le \mathrm{\tau }\le {10}^3$, both plots somewhat lost linearity, and the slopes were decreased to ${\alpha }_2=0.51$ for neurite A and ${\alpha }_2=0.47$ for neurite B. For ${10}^3\le \mathrm{\tau }\le {10}^4$, ${\alpha }_3=0.37$ for neurites A and ${\alpha }_3=0.59$ for neurite B, which were slightly different each other. We analyzed 13 neurites sampled in the same culture condition and found that the slopes were ${\alpha }_1=0.91\pm 0.02$, ${\alpha }_2=0.50\pm 0.07$, and ${\alpha }_3=0.49\pm 0.14$ (mean ± SD). Thus, the obtained scaling exponents differed between the short- and long-time ranges, which is sometimes referred to as the crossover phenomenon [22]. Because the slope should be 0.5 in the whole time range for the SLE curve, these results show the neurite outgrowth process differs from the SLE, similar to the growth processes of the Ising interface mentioned above.

The neurite outgrowth process had a correlation nearly corresponding to 1/f noise in the short-time range, although the correlation decayed in the long-time range. This implies that the driving forces have deterministic (chaotic) properties. To confirm it, we constructed the attractors of $\mathit{\delta \xi }\left(t\right)$ by the three-dimensional time-delay embedding [23]. Figure 9a shows the time series of $\mathit{\delta \xi }\left(t\right)$ for neurite A and Figure 9b shows the corresponding reconstructed attractor where the delayed time was set to the resampled time interval $∆t=10$. The reconstructed attractors for the other neurites had similar forms, indicating there exists some deterministic dynamics in the time series of $\mathit{\delta \xi }\left(t\right)$. Additionally, we calculated Lyapunov exponents for the reconstructed attractors using an often used method (Figure 9c) [24, 25]. Lyapunov exponents (λ₁, λ₂, λ₃) for each attractor showed a set of $\left(+,0,-\right)$, for example, they were (0.71, 0.02, and 1.36) for neurite A and (0.65, −0.03, and −0.74) for neurite B. These results suggest that the neurite outgrowth process is based on deterministic (chaotic) dynamics.

Figure 9.

(a) The time series of $\mathit{\delta \xi }\left(t\right)$ for neurite A, (b) the corresponding reconstructed attractor and (c) the Lyapunov exponents. The delayed time for the attractor was set to the resampled time interval $∆t=10$. These figures are reproduced from Ref. [14] with permission.

From this study, we found that neurite morphology can be quantified by the scaling properties and chaotic features of the driving functions obtained from the Loewner equation, and neurite outgrowth mechanism can be also analyzed based on them. We next applied similar analyses to neurite morphology of human iPSC-derived neurons and considered their possibility of a medical application [15].

4. Analyses of neurite morphology of human iPSC-derived neurons using the Loewner equation

We purchased neural precursor cells derived from human iPSCs from ReproCELL (Japan), which were obtained from a healthy person (ReproNeuro) and an Alzheimer’s disease (AD) patient (ReproNeuro AD-patient-1). The AD patient has the R62H mutation in PS2 gene. The cells were cultured according to a protocol of ReproCELL. Briefly, the cells were seeded in the culture plates and incubated in a CO₂ incubator (37°C, 5% CO₂). The medium was replaced on DIV3 and DIV7. To promote neural precursor cell differentiation into neurons, the culture medium (ReproNeuro Culture Medium, ReproCELL) used for seeding the cells and the medium replacements were mixed with Additive A (ReproCELL).

Microscopic images of the iPSC-derived neurons were captured on DIV3, DIV5, DIV7, DIV10, and DIV14 (Figure 10a). We calculated the driving functions from the coordinates of the neurite traces on the upper half complex plane $\mathbb{H}$ (Figure 10b), similarly to the case of Neuro2A. We then calculated the time-normalized driving forces {$x_n$} by Eq. (8) (Figure 11a and b).

Figure 10.

(a) Microscope images of a healthy neurite (left) and a AD neurite (right). The starting points of the neurites growth and their tips are marked with “O” and “Tip,” respectively, which are connected with red dashed lines. (b) Neurite traces on the upper half complex plane $\mathbb{H}$(left: healthy neurite, right: AD neurite). The real (horizontal) and imaginary (vertical) axes are shown by “Re” and “Im,” respectively. The starting points of neurite growth correspond to the origin O, and the real axis-coordinates of the tips of the neurites are 0. Each black dashed line shows the imaginary axis, which corresponds to the red dashed line in (a).These figures are reproduced from Ref. [15] with permission.

Figure 11.

(a) and (b) Time series of the driving forces corresponding to (a) the healthy neurite and (b) the AD neurite shown in Figure 10. (c) and (d) Log–log plots of τ and $f\left(\mathrm{\tau }\right)$ for (c) the healthy neurite and (d) the AD neurite. The short-range and long-range exponents, ${\alpha }_1$ and ${\alpha }_2$, are indicated in each figure. The corresponding values are ${\alpha }_1=0.37$ and ${\alpha }_2=0.64$ for the healthy neurite and ${\alpha }_1=0.55$ and ${\alpha }_2=0.66$ for the AD neurite. These figures are reproduced from Ref. [15] with permission.

To examine the scaling properties of the obtained time series of {$x_n$}, we performed the fluctuation analysis. The scaling exponents $\alpha$ were estimated by the slopes of the double-logarithmic plots of $\mathrm{\tau }$ and $f\left(\mathrm{\tau }\right)$ (Figure 11c and d). The crossover phenomenon was seen in most of the obtained plots, that is, most of the obtained scaling exponents differed between the short- and long-time ranges.

Figure 12a and b show the DIV-dependent changes in the scaling exponent in the short-time range, ${\alpha }_1$, and that in the long-time range, ${\alpha }_2$, respectively. The scaling exponents for the healthy and AD neurites are shown as the blue and red plots, respectively. The number of neurites ranged from 534 to 834 for each plot. The total number of neurites was 3055 for the healthy neurites and 4004 for the AD neurites. As a result, the short-range exponent revealed ${\alpha }_1<0.5$, that is, anticorrelation for both healthy and AD neurites at most DIV points. ${\alpha }_1$ for the AD neurites, however, is closer to 0.5 than that for the healthy neurites. In other words, in the short-time range, the correlation is lower for the AD neurites. The long-range exponent revealed ${\alpha }_2>0.5$, that is, positive correlation for both healthy and AD neurites at all DIV points. ${\alpha }_2$ for the AD neurites, however, is closer to 0.5 than that for the healthy neurites especially at the earlier stages (DIV3-10). In other words, in the long-time range, the correlation is also lower for the AD neurites.

Figure 12.

DIV-dependent behaviors of the scaling exponents. (a) Plots of DIV vs the short-range scaling exponent ${\alpha }_1$ for healthy (blue) and AD (red) neurites. (b) Plots of DIV vs the long-range scaling exponent ${\alpha }_2$ for healthy (blue) and AD (red) neurites. Significant differences for each cell type (healthy or AD) on each DIV are indicated (*p<0.05, **p<0.01, ***p<0.001). The data are expressed as the mean ± SEM. These figures are reproduced from Ref. [15] with permission.

Thus, the scaling exponents enable us to quantify neurite morphology of human iPSC-derived neurons. Because the scaling exponents were different between the healthy and AD neurites, they enable us to identify a neurite as healthy or not [26]. Interestingly, their difference between the healthy and AD neurites were seen in the earlier stages of development, which was earlier than the expressions of aggregations of specific proteins, β-amyloid (Aβ), and phosphorylated tau (p-tau), of the AD neurons (see ref. [15] in detail). This result therefore suggests that a quick identification of neurodegenerative diseases is possible using the scaling exponents as an indicator for neurite morphological disorders.

It is still unclear how the scaling exponents are related with neurite morphological disorders. However, we assume as follows. The scaling exponents for the AD neurites generally were close to 0.5, which means that the outgrowth process of the AD neurites has lower autocorrelations than that of the healthy neurites. This could be due to a lost stability of neurite cytoskeleton, such as microtubule, induced by aggregations of Aβ and/or p-tau. Confirming this assumption is one of our future works.

5. Conclusions

In this chapter, we intoduced our statistical-physical approach to analyze neurite morphology based on the Loewner equation.

We showed that neurite outgrowth process can be described by the Loewner equation having a deterministic (chaotic) driving function, which differs from the SLE. Such a deterministic (chaotic) driving function is also seen in a physical system, the phase interface of 2D Ising model. Therefore, this feature could be ubiquitous in production processes of complex curves in nature.

Based on this point of view, we showed that neurite morphology can be quantified by the scaling properties and chaotic features of the driving functions obtained from the Loewner equation. Such analyses lead to a physical interpretation of neurite outgrowth mechanism and morphological neurite disorders. Our work using human iPSC-derived neurons, for example, showed that the outgrowth process of the AD neurites has lower autocorrelations than that of the healthy neurites, suggesting that the stability of neurite cytoskeleton is lost in the AD neurites. We thus expect that our approach will lead to a medical application, such as identification of neurodegenerative diseases and elucidation of their causes, in the near future.

Acknowledgments

We thank Yusuke Shibasaki, a student of the doctoral course of our laboratory, with whom the original works in this chapter were performed.