**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

#### **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 . 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.*

#### **Figure 8.**

*The driving functions ξ*ð Þ*t (upper) and double-logarithmic plots of* τ *and f*ð Þτ *(lower) for (a) neurite A and (b) neurite B. These figures are reproduced from Ref. [14] with permission.*

using Neuron J software (a plugin for Image J software). The obtained *x*–*y* coordinates were transferred into the upper half complex plane 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.

We calculated the driving functions *ξ*ð Þ*t* from the coordinates of the neurite traces on using the algorithm mentioned above (**Figure 8a** and **b** (upper figures)). Here, the driving functions were resampled at even time intervals Δ*t* ¼ 10 using the MATLAB signal processing tool box, resample function, because the calculated f g Δ*tn*

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

is generally inhomogeneous as mentioned above. The driving forces are defined here as *δξ*ðÞ¼ *t ξ*ð Þ� *t* þ Δ*t ξ*ð Þ*t* .

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

$$f(\mathbf{r}) = \sqrt{<\left(\Delta\xi(\mathbf{r}) - <\Delta\xi(\mathbf{r})>\right)^2} = \sqrt{\left<\Delta\xi(\mathbf{r})^2\right> - \left<\Delta\xi(\mathbf{r})\right>^2},\tag{9}$$

where Δ*ξ*ð Þ¼ τ *ξ*ð Þ� *t*<sup>0</sup> þ τ *ξ*ð Þ *t*<sup>0</sup> . 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:

$$f(\mathfrak{r}) \sim \mathfrak{r}^a. \tag{10}$$

Here, the linearity of the double-logarithmic plot of τ and *f*ð Þτ indicates that the curves and driving functions are scale-invariant.

The scale exponent *α* is estimated by the slope of the double-logarithmic plot of τ and *f*ð Þτ . It is known that we can classify a given time series into different autocorrelation types by the scaling exponent [21, 22]. When *α* ¼ 0*:*5, *δξ*ð Þ*t* is an uncorrelated time series such as Gaussian noise or Markov processes. The uncorrelated curve is therefore the SLE-produced curve. *α* >0*:*5 indicates the presence of positive correlation in the time series of *δξ*ð Þ*t* , and in particular, *α* ¼ 1*:*0 implies 1/f noise. Conversely, *α*< 0*:*5 indicates anticorrelation.

**Figure 8a** and **8b** (lower figures) show the double-logarithmic plots of τ and *f*ð Þτ for neurite A and B, respectively. Both plots showed almost perfect linearity for 10<sup>1</sup> ≤ τ≤ 102 , where the slopes were *α*<sup>1</sup> ¼ 0*:*94 for neurite A and *α*<sup>1</sup> ¼ 0*:*93 for neurite B. On the other hand, for 10<sup>2</sup> ≤τ≤10<sup>3</sup> , both plots somewhat lost linearity, and the slopes were decreased to *α*<sup>2</sup> ¼ 0*:*51 for neurite A and *α*<sup>2</sup> ¼ 0*:*47 for neurite B. For 10<sup>3</sup> <sup>≤</sup>τ≤104, *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>0</sup>*:*37 for neurites A and *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>0</sup>*:*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 *α*<sup>1</sup> ¼ 0*:*91 � 0*:*02, *α*<sup>2</sup> ¼ 0*:*50 � 0*:*07, and *α*<sup>3</sup> ¼ 0*:*49 � 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 *δξ*ð Þ*t* by the three-dimensional time-delay embedding [23]. **Figure 9a** shows the time series of *δξ*ð Þ*t* for neurite A and **Figure 9b** shows the corresponding reconstructed attractor where the delayed time was set to the resampled time interval Δ*t* ¼ 10. The reconstructed attractors for the other neurites had similar

#### **Figure 9.**

*(a) The time series of δξ*ð Þ*t 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* Δ*t* ¼ 10*. These figures are reproduced from Ref. [14] with permission.*

forms, indicating there exists some deterministic dynamics in the time series of *δξ*ð Þ*t* . 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 ð Þ þ, 0, � , 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.

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

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

#### **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 (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*ð Þτ *for (c) the healthy neurite and (d) the AD neurite. The short-range and long-range exponents, α*<sup>1</sup> *and α*2*, are indicated in each figure. The corresponding values are α*<sup>1</sup> ¼ 0*:*37 *and α*<sup>2</sup> ¼ 0*:*64 *for the healthy neurite and α*<sup>1</sup> ¼ 0*:*55 *and α*<sup>2</sup> ¼ 0*:*66 *for the AD neurite. These figures are reproduced from Ref. [15] with permission.*

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 (**Figure 10b**), similarly to the case of Neuro2A. We then calculated the time-normalized driving forces {*xn*} by Eq. (8) (**Figure 11a** and **b**).

To examine the scaling properties of the obtained time series of {*xn*}, we performed the fluctuation analysis. The scaling exponents *α* were estimated by the slopes of the double-logarithmic plots of τ and *f*ð Þτ (**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, *α*1, and that in the long-time range, *α*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 *α*<sup>1</sup> <0*:*5, that is, anticorrelation for both healthy and AD neurites at most DIV points. *α*<sup>1</sup> 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 *α*<sup>2</sup> >0*:*5, that is, positive correlation for both healthy and AD neurites at all DIV points. *α*<sup>2</sup> 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.

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

#### **Figure 12.**

*DIV-dependent behaviors of the scaling exponents. (a) Plots of DIV vs the short-range scaling exponent α*<sup>1</sup> *for healthy (blue) and AD (red) neurites. (b) Plots of DIV vs the long-range scaling exponent α*<sup>2</sup> *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.*

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.

### **Acknowledgements**

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