Mimicry mechanism model of octopus epidermis pattern by inverse operation of Turing reaction model
11 Mimicry mechanism model of octopus epidermis pattern by inverse operation of Turing reaction model
Takeshi Ishida Department of Ocean Mechanical Engineering, National Fisheries University, Shimonoseki, Yamaguchi, Japan 2-7-1, Nagata-honmachi, Shimonoseki, Yamaguchi, 759-6595, Japan *Corresponding author Takeshi Ishida E-mail: [email protected]
Abstract
Many cephalopods such as octopus and squid change their skin color purposefully within a very short time. Furthermore, it is widely known that some octopuses have the ability to change the color and unevenness of the skin and to mimic the surroundings in short time. However, much research has not been done on the entire mimicry mechanism in which the octopus recognizes the surrounding landscape and changes the skin pattern. It seems that there is no hypothetical model to explain the whole mimicry mechanism yet. In this study, the mechanism of octopus skin pattern formation was assumed to be based on the Turing model. Here, the pattern formation by the Turing model was realized by the equivalent filter calculation model using the cellular automaton, instead of directly solving the differential equations. It was shown that this model can create various patterns with two feature parameters. Furthermore, for the eyes recognition part where two features are extracted from the Turing pattern image, our study proposed a method that can be calculated back with small amount of calculation using the characteristics of the cellular Turing pattern model. These two calculations can be expressed in the same mathematical frame based on the cellular automaton model using the convolution filter. As a result, it can be created a model which is capable of extracting features from patterns and reconstructing patterns in a short time, the model is considered to be a basic model for considering the mimicry mechanism of octopus. Also, in terms of application to machine learning, it is considered that it shows the possibility of leading to a model with a small amount of learning calculation.
Introduction
Many cephalopods such as octopus and squid change their skin color purposefully within a very short time. The blue-ringed octopus (Hapalochlaena fasciata) is famous for threatening its surroundings with emerging a blue ring-shaped pattern on its body when it feels dangerous. David Scheel et al. [1] report that an octopus called Octopus tetricus uses body color patterns as signals in hostile behavior of the same species. These are examples of using the epidermis pattern for communication such as intimidation. Furthermore, it is widely known that some octopuses have the ability to purposefully change the col or and unevenness of the skin and to mimic the surroundings in short time (cf. [2], [3]). The mechanism of these cephalopods to change the skin color is realized by controlling the size of the chromatophore in the epidermis connecting to nerve cells. It has been elucidated that the chromatophore contains pigment granules and regulates color effect with changing the size of the chromatophore. By adjusting the chromatophore with muscles, it is possible to change the color effect in a short time. In addition, below the layer of chromatophore, there are iridescent chromatophores, white chromatophores, and reflex cells, which realize color diversity [4]. However, although the skin color adjustment part has been elucidated, much research has not been done on the entire mimicry mechanism in which the octopus recognizes the surrounding landscape and changes the skin pattern. It seems that there is no hypothetical model to explain the whole mimicry mechanism yet. There have been many papers regarding high learning ability using octopusβ vision (cf. [5]). Considering these facts, it is appropriate to think that mimicry realized with chromatophore controls of the skin by processing the information around it visually within the brain and instructing the nerve cells neighboring skin chromatophore. Regarding the unevenness of the skin, it can be inferred that the information from the sensory organs of the arm is processed by the brain to move the skin muscles. However, the entire mimicry mechanism has not yet been elucidated. Furthermore, it is said that the eyes of octopus can see only blue wavelength color [6], but since the retina is well developed, it is thought that the light and shade pattern can be recognized. Considering the report that there are cells that sense light in the skin [7], and there is the possibility that the light and dark information from these cells and the information from the eyes are combined to sense the color. As mentioned above, there are still many unclear points about the visual recognition of octopus. The purpose of this study is to construct a mathematical model of the octopus mimicry process, which is to characterize the pattern of the surrounding landscape visually and reproduce the pattern on the skin. This model is a hypothetical model. However, just as the Turing model [8] contributed to the elucidation of the morphogenesis of living organisms, it is considered meaningful to construct the hypothetical model in order to elucidate the mechanism of cephalopod mimicry in the future. It is generally said that the lifespan of octopus is about one year (cf. [9]), and it is difficult for octopus with a short lifespan to be able to mimic with a lot of learning and experience. The octopus is likely to have the ability to mimic instantly by nature. It is considered that model constructing of the mimicry mechanism can also be applied to machine learning which has a feature to response based on the aggregate information from external information such as images. Artificial intelligence, which can be processed with little learning, is effective for expanding the range of use of artificial intelligence, and various studies are underway (cf. [10]). Furthermore, concerning engineering applications include artificial camouflage, although research trends in artificial camouflage are shown in Reference [11], it seems that an effective mathematical model for cephalopod mimicry has not yet been reported. From this background, the octopus mimicry model must be built from scratch. The following two models are considered to be the basis for constructing an octopus mimicry model. 1) It is considered that a deep learning convolutional neural network (CNN) can be applied to the part where features are visually extracted. Since CNN (cf. [12]) is a development of a visual model such as the Neocognitron [13] that applies the mechanism of the biological eye, it is considered to be appropriate as a model for the octopusβs eye. 2) Since the appearance of the skin pattern is expected to be the Turing pattern [8], it is considered that the mechanism of pattern emergence by the reaction diffusion model (Turing model) can be applied to the part of the skin pattern creation. There are no reports of studies identifying octopus or squid patterns as Turing patterns, but some striped or speckled patterns appear to be typical Turing patterns. This study is based on these two models, which appear to be completely different algorithms. However, as described in detail in Method, by considering a cellular automaton model equivalent to the reaction-diffusion model and modifying it to make it a filter calculation model, it is possible to reproduce the pattern in the same mathematical frame as convolutional operation of CNN. In the part of the CNN model that extracts the feature parameters from the pattern, the back propagation method is generally used to calculate the weight parameters. However, in this study, it was clarified that it is possible to calculate the feature amount (specify the filter) with a smaller amount of calculation by inverse calculating of the cellular automaton model that emerges the pattern. In this report, it was shown that both the feature extraction model from the image and the pattern reproduction model can be handled by the same mathematical structure. This model makes it possible to extract feature parameters from the Turing pattern in a short time, and to reconstruct the Turing pattern from the feature parameters. Further biological studies are needed to determine if this model is consistent with the information processing mechanisms in the cephalopod body. However, it is expected to be an essential model in that the information processing process from the eyes and the reaction process on the skin can be processed by a model with basically the same structure. This model is also considered to be a hypothetical model for elucidating the mechanism of mimicry. Furthermore, in applications of machine learning, there is a possibility that the amount of calculation to determine the convolution filter can be reduced, which will lead to reduction of the amount of learning of the CNN model.
Methods
Outline of the model In this study, as shown in Figure. 1, the mechanism of octopus skin pattern formation was assumed to be based on the Turing model. Here, the pattern formation by the Turing model was realized by the equivalent filter calculation model using the cellular automaton, instead of directly solving the differential equations. It was shown that this model can create various patterns with two feature parameters. Furthermore, for the part where two features are extracted from the Turing pattern image, our study proposed a method that can be calculated back with small amount of calculation using the characteristics of the cellular Turing pattern model. Figure 1. Outline of the mimicry mechanism model of octopus epidermis pattern. The mechanism of octopus skin pattern formation was assumed to be based on the Turing model based on cellular automaton. The part where two features are extracted from the Turing pattern image, inverse calculation model was proposed using the characteristics of the cellular Turing pattern model. Skin pattern emergent model (Turing pattern model) The Turing pattern model is one type of reaction-diffusion model, which was introduced by Turing in 1952 [8], where he treated morphogenesis as the interaction between activating and inhibiting factors. Typically, this model produces self-organizational patterns through the different diffusion coefficients of two morphogens, equivalent to an activating and an inhibiting factor. The general reaction-diffusion equations can be written as follows: ππ’ππ‘ = π β π’ + π(π’, π£) ππ£ππ‘ = π β π£ + π(π’, π£) (1) where u and v are the morphogen concentrations, functions f and g are the reaction kinetics, and d and d are the diffusion coefficients. Previous studies have considered a range of functions f and g, and models such as the linear model, the Gierer β Meinhardt model [ ], and the Gray β Scott model [ ] have been used to produce typical Turing patterns. Moreover, it is easy to solve such reaction-diffusion equations numerically and create Turing patterns in space by using the difference method. However, deriving the parameters by inversely calculating the differential equation from the pattern generally requires many iterative operations and requires a large amount of calculation. Therefore, in this study, we used a method of estimating parameters on Cellular Two shape parametersScanning the pattern Pattern reproduction
Turing pattern model
Inverse Turing pattern modelBrain Skin
Eyes
Automata (CA) model that reproduces the Turing pattern, instead of directly solving the differential equations. CA models are discrete in both space and time. The state of the focal cell is determined by the states of the adjacent cells and the transition rules. The advantage of CA models is that they can describe systems that cannot be modeled using differential equations. Historically, various interesting CA patterns have been discovered. Markus [16] demonstrated that a CA model could produce the same output as reaction-diffusion equations. The Young model [17] is one of the two-dimensional totalistic models that bridges the reaction-diffusion equations and CA model, which can produce Turing patterns. Some other example models to produce Turing patterns are below. Adamatzky [18] studied a binary-cell-state eight-cell neighborhood two-dimensional cellular automaton model with semitotalistic transitions rules. Dormann [19] also used two-dimensional outer-totalistic model with threes states to produce a Turing-like pattern. Tsai [20] analyzed a self-replicating mechanism in Turing patterns with a minimal autocatalytic monomer β dimer system. The proposed model was based on the Youngβs CA model. This
Youngβs model uses a real number function (Figure 2 (b)) to derive the distance effects, with two constant values ; u (positive) and u (negative). The calculation begins by randomly distributing black cells on a rectangular grid (Figure 2 (a)). Then, for each cell at position R in two-dimensional fields, the next cell state (black or white) of R dues to the sum of the functions value of black cells at positions of R i . R i is assumed to be a black cell within radius r from R cell, and i is the number of black cells within radius r from R . The function v(r) is a continuous step function, as shown in Figure 2 (b). The activation area, indicated by u , has a radius of r and the inhibition area, indicated by u , has a radius of r (r > r ) (Figure 2 (a)). At position R , when R i is assumed to be a grid within r , function v(|R - R i |) value is added up. Function |R β R i | indicates the distance between R and R i . If β i v (|R β R i |) > 0, the grid cell at point R is marked as a black cell. If β i v (|R β R i |) < 0, the grid cell becomes a white cell. If β i v (|R β R i |) = 0, the grid cell does not change state [17]. Using these functions, Young described that a Turing pattern can be generated. Spot patterns or striped patterns can be created with relative changes between u and u . Figure 2.
Outline of Youngβs model. (a) The activation area has a radius r and the inhibition area has an outer radius r . (b) Function v(r) is a continuous step function representing the activation area and inhibition area. In this Young model, let u = 1, u = w (here 0 < w < 1), and further, if the state of the cell is set to 0 (white) and 1 (black), this model can be arranged as follows. The state of cell i is expressed as c i (t) (c i (t) = [0, 1]) at time t. The following state c i (t + 1) at time t + 1 is determined by the states of the neighboring cells. Here, N is the sum of the states of the domain within s meshes of the focal cell. Similarly, N is the sum of the states of the domain within s meshes of the focal cell, assuming that s < s . π = β π π (π‘) π π=1 π = β π π (π‘) π π=1 (2) Here, S and S is the number of cells within s and s meshes from focal cell. In addition, s = 2s was assumed in this paper. Figure 3 shows the schematic of the total sum of states N and N . The next time state of the focal cell is determined by the following expression (3). Here, there are two parameters, w and s, that determine the pattern of the Turing pattern. Cell state at the next time step = { 1: if π > π Γ wUnchange: if π = π Γ w 0: if π < π Γ w (3) Figure 3. Schematic diagram of the summing of states N and N . Each grid cell has state 0 (white) or 1 (black). The inner area has a domain within s grids of the focal cell and the outer area has a domain within 2s grids. Furthermore, the operation of counting state 1 cells as N and N is equivalent to the convolution operation, which uses the filter with range of s as 1 and with range of 2s as w (shown in Figure 4) to sum of the values obtained by AND calculation of the filter and the mesh space. This is a process equivalent to CNN of performing convolution operations on the mesh space. N =Number of black cell with inthe blue circle N =Number of black cell with in the red circle2ss Figure 4. Overview of convolution operation by the filter. The operation of counting state 1 cells in the area N and N is equivalent to the convolution operation, which uses the filter with range of s as 1 and with range of 2s as w to sum of the values obtained by AND calculation of the filter and the mesh space. The essence of the Turing pattern model is that the short- and long-range spatial scales are each affected by separate factors, and pattern formation emerges from nonlinear interactions between the two factors. Turing used two chemicals with different diffusion coefficients to create these short- and long-range spatial effects. However, as long as there exists a difference between long- and short-range effects, other implementation methods can be applied. This model used two ranges of s mesh and 2s mesh to create a difference effect. It is therefore thought that this model is essentially the same as an reaction-diffusion model.
Information processing model of octopus eyes and brain ; Model to extract the shape parameters by inverse Turing operation
A general method for extracting features parameters from patterns is to build a predictive model from training data consisting of images and features parameters by machine learning such as CNN model. However, this method requires a large amount of training data and a large amount of calculation due to the back-propagation method. It is unlikely that the octopus is using the same method, as it naturally possesses the ability to mimic without learning. There may be a mechanism for acquiring features parameters in a short time without learning and without many optimization calculations. In this study, it was considered to calculate the features parameters by back-calculating of the CA model that creates the Turing pattern described in previous w Filter Γ Grid
Convolutional operation
1 section. However, the parameter values cannot be calculated by simply performing the reverse operation. This is because the reverse process of the part that counts the state 1 in a certain range cannot be determined in a single way. In this study, we devised the method described below. In the CA model algorithm that can emerge the Turing pattern described in previous section, the edge lines of the patterns are characterized in that N = N Γ w holds. Therefore, it is possible to calculate w from the ratio of N and N in the cell on the pattern edge lines. ο½οΌ N οΌ N (4) However, since the values of N and N change with value s, and value w on the patterns edges also changes with s. When s is properly determined, N / N (=w) match on all pattern edges and w is also specified, but when s is different from the true value, the values of w are not fixed on the pattern edges. On the contrary, if s is adjusted to true value, the deviation of the values of w on the edges disappears and becomes constant, so that it is possible to determine s and w at the same time. As shown in Figured 5, as the space is discrete in the edges of patterns, where the white cell and the black cell are in contact, there is actually no cells in which the value w is equal to N / N , the w values are w
2 Figure 5. Example of the situation of the boundary between black cell (state 1) and white cell (state 0); Black cell (1) in contact with white cell (0) has N -N w> 0, and white cell (0) in contact with black cell (1) has N -N w <0. As a specific calculation method, s and w are specified by calculation of the index value, which starts from a large initial value s, and find s at which the index becomes maximum while gradually decreasing s. This is a much smaller amount of computation than the backpropagation method using a neural network. Model implementation and calculation conditions The calculation program was implemented in Python. The program is divided into three parts; the first part that sets the parameters w and s and calculates the Turing pattern, the second part that estimates the parameters w and s from the obtained Turing pattern image, and the third part that calculates the Turing pattern again from the estimated w and s. The grid was a two-dimensional space of 40 Γ 40 cells, and the boundary of calculation space was periodic boundary condition. The calculation conditions were carried out in the cases of w = 0.175, 0.20, 0.25, 0.30, 0.325, and s = 3, 4, 5, 6, 7, 8. In this paper, the studies only focus for Turing patterns, whether non-Turing patterns or random texture images can be reconstructed is not dealt with in this study and it is future subject. Results
Results of Turing pattern emerging model The emergent pattern was calculated by the parameters w and s using the skin pattern formation model based on the Young model. Figure 6 shows the results for the expression (1) model using a rectangular grid. As the initial conditions, state 1 cells
Border; N -N Γ w=0 Field of the condition N -N Γ w>0 Field of the condition N -N Γ w<0
3 were set randomly in the calculation field 40 Γ 40. Turing-like patterns emerged as parameter w and s changed. When w was relatively large, spot patterns were observed, whereas at middle range of w, stripe patterns emerged. When w was 0.15 or lower, all cells in the field had a value of 1 (shown in black). Changing the parameters s merely changed the scale of the patterns. The calculation starts from a random state and settles in a steady pattern in about 10-time steps. Since 0 and 1 are set randomly in the initial arrangement, the pattern will not be exactly the same even with the same parameters, but the pattern characteristics (width of the striped pattern and size of the speckled pattern) will be the same. 4
Figure . Results for the model of expression (1) in a square grid. As the initial condition, state 1 cells were set randomly in the calculation field. Turing-like patterns emerged as parameter w changed. When w was relatively large, spot patterns were observed, whereas at middle range of w, stripe patterns emerged. When w was 0.15 or lower, all cells in the field had a value of 1 (shown in black).
Identification of feature parameters by inverse Turing calculation Tables 1 to 3 show the calculation results of the feature parameters s and w by the inverse Turing operation and the pattern reproduction results based on the estimated parameters. The image calculated from the pattern emergence model, the result of
W0.3750.350.3250.300.2750.250.2250.200.1750.15 3 4 5 6 7 8 S
5 estimating the parameters s and w from the image, and the result of reconstructing the image from the estimated parameters are shown under each condition in the tables. From this result, it can be seen that the image reproducibility is good under most calculation conditions. However, under the condition that the value of w and s are low, which forms a small white spot pattern on black background, it seemed that the reproducibility of the image is poor. This is considered to be due to the large error in the value of the "index" when there are only a few small white spots on the black background in the image. The same can be said when there are small number of small black spots on a white background. The pattern emergence model is characterized in that when the w value is 0.25 or less, the pattern changes rapidly from a striped pattern to a solid black color. As shown in Figure. 6, the pattern changes from a speckled pattern to a black background when w is between 0.175 and 0.15. This feature also causes errors of index, and it is considered that a slight difference in the estimated w value causes a large change in the reproduction pattern image. Table 4 shows the results of calculating the index values from the images created by each w value at s = 5.0. For each w value, starting from a large s value ( = 8) and looking at the change in the index value, the index value is the maximum at the part of s = 5.0 except when w = 0.175. It was found that the value of s can be specified by the calculation method from the index value proposed in this study. In addition, the calculation for finding the parameters is determined by iterative calculation of several steps, and it takes about several seconds even on a personal computer that does not use a general GPU. 6 Table 1. Calculation results (1) Table 2. Calculation results (2) ο½ ο½ ο½ ο½ γ γ γ γ γ γ γ γ Set value Image calculatedwith the setvalue Predicted value Image calculatedwith predictedvalue Reproducibilityof image ο½ ο½ ο½ ο½ γ γ γ γ γ γ γ γ Γ Set value Image calculatedwith the setvalue Predicted value Image calculatedwith predictedvalue Reproducibilityof image
7 Table 3. Calculation results (3) ο½ ο½ ο½ ο½ Γ β³ γ γ γ Γ Γ β³ γ γ γ γ Γ Set value Image calculatedwith the setvalue Predicted value Image calculatedwith predictedvalue Reproducibilityof image
8 Table4. The results of calculating the index values from the images created by each w value at s = 5.0.
Discussion
Using the proposed model, it was possible to infer the feature parameters from the Turing pattern and show that the similar Turing pattern can be reconstructed from the feature parameters under the certain conditions. In terms of calculation time, unlike the CNN model, it is not necessary to perform a large amount of calculation, and it is possible to calculate the feature parameters with a small amount of calculation. This model will serve as a hypothetical model for considering the mechanism of cephalopod mimicry, and it will be useful for future biological research. However, the shade pattern on the seabed is not necessarily the Turing patterns. It is necessary to examine whether the non-Turing pattern can be reproduced with this model in the future. In this model, the filter shape was perfect circle (Figure. 3), but if a distorted elliptical filter is used, it is thought that more diverse shapes can be created, therefore it is necessary to examine various filter shapes. What was clarified by this model is that it is possible to reproduce the Turing pattern by using a perfect circle filter processing, and it is possible to create a model that can recognize the Turing pattern without a lot of learning. At present, it is difficult to distinguish various geometric shapes such as circles and squares in this model. Although it is inferior in grasping the detailed shape, it is considered that a model suitable for rapid parameter processing of specific patterns in the surrounding space can be constructed. In terms of contributing to machine learning research, there is a possibility that the discrimination performance will be improved by preparing not only a perfect circle filter ο½
9 but also directional filters such as an elliptical filter, and these are future issues. Even in the CNN model, it is common to use dozens of filters, and it is thought that the cognitive ability is enhanced by combining various anisotropic filter patterns. Furthermore, by integrating CNN model, which learns the weights of the filter from random values, and this model, which uses a specific pattern filter, it may be possible to construct a new image recognition model with a small amount of learning and high recognition ability.
Conclusion
In this study, it was constructed a reaction-diffusion CA model that generates Turing patterns and an arithmetic model that extracts features parameters from Turing pattern images. These two calculations can be expressed in the same mathematical frame based on the cellular automaton model using the convolution filter. As a result, it can be created a model which is capable of extracting features from patterns and reconstructing patterns in a short time, the model is considered to be a basic model for considering the mimicry mechanism of octopus. Also, in terms of application to machine learning, it is considered that it shows the possibility of leading to a model with a small amount of learning calculation.
Acknowledgment
This research was supported by grants from Japan Society for the Promotion of Science, KAKENHI Grant Number 19K04896.
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