A new model to imitate the foraging behavior of Physarum polycephalum on a nutrient-poor substrate

A new model to imitate the foraging behavior of Physarum polycephalum on a nutrient-poor substrate Yuheng Wu a, Zili Zhang a,b,n, Yong Deng a, Huan Zhou a, Tao Qian a a Faculty of Computer and Information Science, Southwest University, Chongqing 400715, China b School of Information Technology, Deakin University, VIC 3217, Australia a r t i c l e i n f o Article history: Received 6 April 2012 Received in revised form 30 September 2012 Accepted 8 October 2012 Available online 30 July 2014 Keywords: Physarum polycephalum Foraging behavior Spanning tree Euclidean plane a b s t r a c t Researches on Physarum polycephalum show that methods inspired by the primitive unicellular organism can construct an efficient network and solve some complex problems in graph theory. Current models simulating the intelligent behavior of Physarum are mainly based on Hagen–Poiseuille Law and Kirchhoff Law, reaction–diffusion, Cellular Automaton and multi-agent approach. In this paper, based on an assumption that the plasmodium of Physarum forages for food along the gradient of chemo-attractants on a nutrient-poor substrate, a new model is proposed to imitate its intelligent foraging behavior. The key point of the model is that the growth of Physarum is determined by the simple particle concentration field relating the distance to food source and the shape of food source on a nutrient-poor substrate. To verify this model, numerical experiments are conducted according to Adamatzky's experiment. Results in spanning tree construction by this model are almost the same as those of Physarum and Oregonator model. The proposed model can also imitate Physarum to avoid repellents. Furthermore, the Euclidean Spanning tree built by this model is similar to its corresponding Minimal Euclidean Spanning tree. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Physarum polycephalum, a multinucleated single-cell slime mold, inhabits darksome and moist areas. The plasmodium of Physarum is the vegetative stage in its complex life cycle. When positioned in a humidity and shady environment with suitable temperature, the plasmodium of Physarum propagates pseudopo- dia to probe the space around for food and establish network of protoplasmic tubes to connect them all. Since Nakagaki found that the plasmodium of Physarum can solve maze by means of shortest path [1,2], the intelligent adaptive behavior the plasmodium of Physarum has been the focus of many researchers' attention over the past decade. Investigations show that Physarum can construct efficient networks with low cost and high fault tolerance [3,4], and some are even comparable with artificial complex transportation networks [5,6]. Inspired by the organism's intelligent behavior, some new ideas are introduced to engineering applications, such as protocol design in Wireless Sensor Network (WSN) [7] and dynamic reconfiguration in service-oriented internet-ware systems [8]. According to the difference of methodology used in analyzing, explaining and recurring the behavior the plasmodium of Phy- sarum, studies on Physarum can mainly be classified into four aspects: � Nakagaki, Tero et al. constructed mathematical model based on Hagen–Poiseuille Law and Kirchhoff Law to describe the adaptive feature of path finding and the feedback between flux and conductivity of the protoplasm tubes [9–11]. The model could be applied to dynamic navigation [12] and imitate Physarum to design the railway network around Tokyo [6]. � Adamatzky researched Physarum in the perspective of reac- tion–diffusion, he treated it as a bio-realized unconventional computer called Physarum Machine to solve maze, graph problems and design logical gates [13]. Biological experiment indicated that Physarum could achieve the basic operations of Kolmogorov–Uspensky machine [14,15] and be programmed dynamically with light [16]. Through the Oregonator model of Belousov–Zhabotinsky reaction, Adamatzky analyzed the com- mon characteristics between Physarum's foraging behavior and reaction–diffusion [17–19]. � Gunji, Niizatoa, et al. constructed Cellular Automatons to mimic Physarum solving maze, Stainer minimum tree and spanning tree problems [20], simulate Physarum's motion and morphol- ogy behavior [21], and reproduce the adaptive and robust transport network [22]. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing http://dx.doi.org/10.1016/j.neucom.2012.10.044 0925-2312/& 2014 Elsevier B.V. All rights reserved. n Corresponding author at: Faculty of Computer and Information Science, Southwest University, Chongqing 400715, China. E-mail address: [email protected] (Z. Zhang). Neurocomputing 148 (2015) 63–69 www.sciencedirect.com/science/journal/09252312 www.elsevier.com/locate/neucom http://dx.doi.org/10.1016/j.neucom.2012.10.044 http://dx.doi.org/10.1016/j.neucom.2012.10.044 http://dx.doi.org/10.1016/j.neucom.2012.10.044 http://crossmark.crossref.org/dialog/?doi=10.1016/j.neucom.2012.10.044&domain=pdf http://crossmark.crossref.org/dialog/?doi=10.1016/j.neucom.2012.10.044&domain=pdf http://crossmark.crossref.org/dialog/?doi=10.1016/j.neucom.2012.10.044&domain=pdf mailto:[email protected] http://dx.doi.org/10.1016/j.neucom.2012.10.044 � Jones presented a particle-based model to approximate the network formation of Physarum [23] and plan roads [5]. In that model, a population of particle-like agents formed complex emergent pattern formation by a simple rule [24]. In [25], the relationship between various patterns and the model para- meters was discussed. This paper studies and imitates the foraging behavior of the plasmodium of Physarum. The rest of this paper is organized as follows. In Section 2, an assumption of how Physarum foraging on a nutrient-poor substrate is given first. Based on the assumption, a new model imitating the intelligent foraging behavior of Physarum is presented. Experiments conducted referring to [19], and the comparison of the result of our model with that in [19] are described in Section 3. The proposed model is then used to imitate the behavior of Physarum when mixed with repellents on the substrate and construct Euclidean spanning tree in extended experiments. Concluding remarks and further researches are provided in Section 4. 2. Model 2.1. Model supposition When inoculated on a nutrient-rich substrate, Physarum pro- pagates many protoplasmic tubes. While on a nutrient-poor substrate, it propagates its pseudopodia cautiously to connect all food sources with a few protoplasmic tubes [13,19]. One possible explanation is that its growth influenced by the degree of concentration of nutrient-particles. On a nutrient-rich substrate, the nutrient-particles concentration around Physarum is not so distinct but high enough to stimulate it to flourish almost in all directions, while on a nutrient-poor substrate, the concentration is distinguishable in different directions and these several dispersed food sources provide enough space for Physarum to select one direction with more particles. So we assume that Physarum searches for food mostly and simply along the gradient of chemical nutrient-particles field on a nutrient-poor substrate. Thus the foraging behavior of Physarum on a nutrient-poor substrate could be described as follows. Each food source diffuses particles from its surface to the surroundings nearby. The further a point in the space leaves from a food source, the lower its particles concentration. The particles form a superposed and stable con- centration field after the food sources are allocated. When placed on this substrate, Physarum sprouts pseudopodia to forage for food. The front of pseudopodia grows along the gradient of the concentration field. Once a food source covered by pseudopodia, it becomes a part of the Phyasrum. And it cannot emanate particles to the space any more. This leads to the change of concentration distribution, then the position of its protoplasm with maximal concentration is stimulated to forages for the rest food sources along the gradient of the current field. This process stops until all food sources are obtained. In fact, Phyasrum will reorganize its network of protoplasmic tubes after the tree-like tubes stage, but that is beyond this paper's investigation. As discussed above, Physarum's foraging behavior can be abstracted by the simple model below. In the model, we do not concentrate on the details how the particles of chemo-attractants diffuse in the surrounding area of Physarum, it is not easy to calculate the exact value of particles concentration at each point chemically, but adopting relative distance values to express the particle field formed by food sources is an available substitution. 2.2. Field intensity Food sources, such as oat flake and sugar, can attract Physarum to grow towards them. In bio-experiment, each food source occupies a small area in a Petri dish. In the paper, plane P represents Petri dish with nutrient-poor substrate. The variform sources can be represented by varying size of ellipse area in P. The center of the ellipse has the strongest field intensity, and the points in the same ellipse have the same field intensity. Therefore, we give the mathematic description of the model below. Let Γ ¼ fF1; F2;…; Fng denote the set of field sources (a new name of food source). Let Θ¼ fρ1;ρ2;…;ρng denote the size set of field sources. For ρiAΘ, ρi ¼ ðai; biÞ. ai and bi are the semimajor axis and semiminor axis of Fi respectively. Let O¼ fo1; o2;…; ong denote the center location set of the ellipse standing for field sources. For oiAO, oi ¼ ðxi; yiÞ. Let Φ¼ fϕ1;ϕ2…;ϕng denote the central field intensity set of field sources, where 8ϕiAΦ, ϕi40. Since different kinds of food sources have a different attraction for Physarum, the symbol ϕ reflects the distinction. Thus in a plane with a single field source F, the field intensity at each point pðx; yÞ can be calculated as follows: f ðpÞ ¼ϕe�dpo ð1Þ In Eq. (1), ϕ denotes the central intensity of F , o¼ ðxo; yoÞ is the central coordinate of F , e is the natural logarithm. The power dpo represents the field distance, which relates to the shape of F, can be calculated by Eq. (2), where a and b denote the semimajor axis 2 4 6 8 2 4 6 8 0 0.2 0.4 0.6 0.8 1 x y z Fig. 1. This figure shows surfaces of the field intensity formed by single and multi-source(s). The higher the z value is, the larger the field intensity of point (x, y) is. The curves in the X–Y plane are the contour of field intensity. (a) An elliptic field source with semimajor axis twice than semiminor axis and (b) synthetical field of six circular field sources. Y. Wu et al. / Neurocomputing 148 (2015) 63–6964 and semiminor axis of F . dpo ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x�xo a � �2 þ y�yo b � �2r ð2Þ In general, there exist many field sources in a plane, so the concentration field is the superposition of them. Then we define the field intensity at point p as Eq. (3), in which n is the number of elements in set Γ. Through Eqs. (1)–(3), we can get the surfaces of field formed by different sources as shown in Fig. 1. f sðpÞ ¼ ∑ n i ¼ 1 ϕie �dpoi ð3Þ 2.3. Growth rules The last subsection has introduced the field intensity formed by a single source and multi-source, this part will illustrate the dynamic process of the model. Firstly, an initial point is necessary to active the model, which is called initial growth point corre- sponding to the place where Physarum is inoculated. Then the next growth point is determined by the previous one until a field source is accessed. For any field source Fi and the current growth point pðx; yÞ, if the condition jx�xijrai4 jy�yijrbi is satisfied, we say that the growth point steps into Fi or Fi is accessed. After all sources assigned, the gradient at the growth point pðx; yÞ can be defined as Eq. (4), in which mr jΓj denotes the number of non- accessed field sources in set Γ. The normalized vector Nf s ðpÞ of Gf s ðpÞ can be calculated as the following Eq. (5). Gf s ðpÞ ¼ ∑ m i ¼ 1 ϕie �dpoi ðxi�xÞ dpoi a 2 i ; ϕie �dpoi ðyi�yÞ dpoib 2 i ! ð4Þ Nf s ðpÞ ¼ GfsðpÞ JGfsðpÞJ ð5Þ The growth point moves along the gradient of the field at a speed of δ per time step. In each time step, the point, δ away from the previous growth point along the gradient of the current field, becomes the current growth point. For example, if the current growth point is pjðxj; yjÞ, the next growth point pjþ1ðxjþ1; yjþ1Þ will be calculated by the following equation: pjþ1 ¼ pjþδNf s ðpjÞ ð6Þ After the initial growth point set, the model starts and works as is shown in Fig. 2. At any time, there exists only one growth point and it will decide the next one until another field source is found. When the growth point steps into the area of a field source, the field source will not contribute field intensity to the plane any more. This will lead to the update of field intensity in the plane. Then we choose a new growth point from the accessed source (s) and the initial growth point whose current field intensity is the maximal. Finally, the process will stop after all field sources in set Γ are accessed. 3. Numerical experiment Both contrast and extended experiments were conducted to verify the proposed model. Details of the experiments are pre- sented in the following two subsections. 3.1. Contrastive experiment To check the validity of this model, the input data of numerical experiment is set the same as [19], and the result of the experi- ment is compared with that of [19]. The field sources are arranged as the configuration of food sources in [19]. The southern field source is chosen as the initial growth point because the Physarum is inoculated there in [19]. All the field sources are limited in the scenario with the size of 10�10. Because [19] uses oat flake as the only kind of food source, the field sources in our experiment are set the equal central field intensity ϕ¼ 1. Meanwhile we set each field source the same size a¼ b¼ 0:1, and the growth speed δ¼ 0:01. As shown in Fig. 3(a), the solid diamond, at the south of the domain, is the initial growth point. The other six solid dots are field sources, which form a superposed field in the domain. After the process starts, the first edge grows gradually along the gradient of current field from the initial growth point. While the first field source is found at t¼164 during the process, the current field distribution changes (Fig. 3(b)). After the second field source is accessed at t¼413, the location of accessed field source with the maximal field intensity is chosen as the new growth point and grows a new edge from t¼413 to t¼708. The process stops at t¼1683, while the initial growth point and six field sources are connected together with six edges (Fig. 3(g)). To compare the result, we refer to Fig. 3(h) and (i) from [19]. Spanning tree in Fig. 3 (h) is approximated by the two-variable Oregonator equation in [19]. Fig. 3(i) is the result of the real biologic experiment. Although the model builds a spanning tree the same as Oregonator Model does in [19], there is a little difference that Physarum can branch off at tube. As a result, we modify the growth rule like this: when the growth point step into the area of any field source, the point belonging to the current edges with the maximal field intensity is selected as the new growth point. Applying the modified rule, we execute the experiment again. We record the approximate time when branches are generated as shown in Fig. 4. In the same scenario, the modified rule consumes less time (Fig. 4(d)) to construct spanning tree and mimic Physarum's forming branches in protoplasmic tubes. Fig. 2. The process of the model. Y. Wu et al. / Neurocomputing 148 (2015) 63–69 65 Fig. 3(a)–(g) is the snapshots of the experiments totally following the model described in Section 2. The only difference between Figs. 3 and 4 is the selection of the next growth point after a field source accessed. In Fig. 3, when a field source is accessed, the accessed source with the maximum field intensity is chosen as the next growth point. However, in Fig. 4 the point belonging to the growth traces and with the maximum field intensity is chosen as the next growth point once a field source is accessed. Thus all the brunches appear only at nodes in Fig. 3, while brunches can appear at edges in Fig. 4. 3.2. Extended experiment In Section 2.2, the central field intensity of the source is restric- ted to a positive value to represent food sources. However there exists the other kind of material called repellent in Physarum's world, such as salt and oil. Repellent restrains Physarum from vegetating, but Physarum can propagate away from it foraging for food sources. If the central field intensity ϕ is set to be negative, the field source represents repellent, named negative field, will appear on the substrate. The particles emitted from repellent and nutrient-particles from food sources, blending together and satis- fies the principle of superposition in Eq. (3), co-affect Physarum's behavior on the substrate. Because repellent with too strong effect could inhibit Physarum's growth thoroughly, we assign a negative value near zero to repellent's central field intensity. Consequently by altering the central field intensity of one and two field sources to �0.2 in Section 2.2, we restart the model in both cases. The results of them shown in Fig. 5 indicate that the edges of the trees avoid repellent(s) and connect all other sources successfully. On the other hand, it demonstrates that this model can also simulate Physarum's foraging behavior in some degree with repellents existing. It is well-known that a minimal spanning tree (MST) which uses the minimal total weight of edges to connect all nodes in a graph is a very important structure in graph theory, as well as in real life. The model exhibits the ability to construct spanning tree in Euclidean plane. To explore the relation between the Euclidean spanning trees and its corresponding Euclidean minimal spanning tree (EMST), we designed three groups of experiments with 10, 50 and 200 nodes respectively. The locations of nodes are produced randomly, and the node first generated is chosen as the initial growth point. We set the size of each group's domain equal to the square of its node number, other parameters are the same as the first experiment in Section 3.1. Then we utilize the proposed model to construct Euclidean spanning tree (EST) of them. To obtain the corresponding EMST of each group, we construct the complete graph of it, calculate Euclidean distance of any two nodes to build its weight matrix and compute the EMST by Prim algorithm. The results of the three groups are shown in Fig. 6. Fig. 3. Process of constructing the spanning tree and its contrastive result. (a)–(g) Exhibit the access sequence of field sources and how a spanning tree is built. (h) and (i) quote from [19] by courtesy of Adamatzky, (h) is the result of two-variable Oregonator Model after erosion operation on diffusion wave and (i) is the enhanced picture of Physarum constructing the spanning tree. (a) t¼0, (b) t¼164, (c) t¼413, (d) t¼708, (e) t¼1006, (f) t¼1341, and (g) t¼1683. Y. Wu et al. / Neurocomputing 148 (2015) 63–6966 The first group produces an EST the same as EMST, the latter two groups reveal just a little difference between its EST and EMST on topology at southwest and top-middle region of them respectively. 4. Conclusion Oregonator model is used to describe the Belousov–Zhabo- tinsky (BZ) reaction. In [19], the excitation waves of Oregonator model are utilized to imitate the pseudopodia of Physarum. In that approach, a site of the simulated medium is affected only by the nearest active source. The wave-fronts move on the substrate induced by the active sources. After all sources accessed by the waves-fronts, the image overlaid by wave-fronts of every 100 time steps is obtained. Finally, by repeated erosion post-processing operations on the image, the spanning tree appears as shown in Fig. 3(h). In the partial differential equations of Oregonator Model, the parameter related to source-dependent gradient should be regulated cautiously to the size of medium and the configuration of the sources of nutrients. If the parameter does not fit for a Fig. 4. Use modified rule to construct spanning tree. (a)–(c) Show the approximate time step when branches form and (d) is the final spanning tree. (a) t¼430, (b) t¼770, (c) t¼1360, and (d) t¼1629. Fig. 5. The trees constructed by the model when mixed with negative field source(s). The red stars represent this field sources with negative field intensity. (a) Spanning tree when mixed one negative field source and (b) a circular shape when mixed two negative field sources in the center. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) Y. Wu et al. / Neurocomputing 148 (2015) 63–69 67 certain scenario, the wave-fragments might disappear before traveling all sources of nutrients. Oregonator Model has the features of paralleling computing, while the proposed model uses a serial computing method. Both models are used to imitate the Physarum, but in different views. The former is based on reaction- diffusion; however, the later adopts the way to search along the gradient of the concentration of nutrient particles. The proposed model uses simpler equations to express the synthesized impacts of all active sources in the medium and the trajectory obtained by the model is the exactly spanning tree without any other post-operation. It not only simulates both attractants and repellents in Physarum's world, but also imitates Physarum to forage for food and avoid repellents successfully. Through little modification, the model can imitate more features of Physarum's foraging behavior, such as forming branches on tubes. From the contrastive experiment, we discover that the model can imitate Physarum to construct spanning tree, which is very similar to the one did by Physarum and almost the same as the one did by the Oregonator model. In the extended experiment, the model shows a very interesting feature that it can build a spanning tree in Euclidean plane, which is very close to EMST on topology and on morphology. Moreover, if extending Eq. (2) to 3 dimensions, this model can be transformed to construct spanning tree in 3D space. Until now, the model can imitate Physarum to construct spanning tree and avoid repellents on a nutrient-poor substrate. However, the model employs a serial method to simulate the main protoplasmic tubes of Physarum based on the supposition of the gradient of particles concentration. Further researches are required to sustain the supposition on its foraging behavior. In addition, the Physarum never stops after the spanning tree built. It is found that the plasmodium of Physarum constructs graphs partially along the Toussaint hierarchy presented in Chapter 6 of [13]. Thus improving the model to imitate plasmodium to construct complex graphs and exploring its relationship to Toussaint hierarchy can be the future work. Besides, we expect to modify this model to a parallel one and solve some practical problems. Fig. 6. The comparison of the three groups' results. (a), (c) and (e) are the results of EST constructed by our method, (b), (d) and (f) are their corresponding EMST. Y. Wu et al. / Neurocomputing 148 (2015) 63–6968 Acknowledgments We thank Prof. Adamatzky for discussions on Oregonator Model. This work was supported by Chongqing Natural Science Foundation, Grant no. CSTC, 2010BA2003. References [1] T. Nakagaki, H. Yamada, Á. Tóth, Maze-solving by an amoeboid organism, Nature 407 (2000) 470. [2] T. Nakagaki, H. Yamada, Á. Tóth, Path finding by tube morphogenesis in an amoeboid organism, Biophys. Chem. 92 (2001) 47–52. [3] T. Nakagaki, H. Yamada, M. Hara, Smart network solutions in an amoeboid organism, Biophys. Chem. 107 (2004) 1–5. [4] T. Nakagaki, R. Kobayashi, Y. Nishiura, T. Ueda, Obtaining multiple separate food sources: behavioural intelligence in the Physarum plasmodium, Proc. R. Soc. B: Biol. Sci. 271 (2004) 2305–2310. [5] A. Adamatzky, J. 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Adamatzky, Physarum machines: encapsulating reaction-diffusion to com- pute spanning tree, Naturwissenschaften 94 (2007) 975–980. [19] A. Adamatzky, If BZ medium did spanning trees these would be the same trees as Physarum built, Phys. Lett. A 373 (2009) 952–956. [20] Y.P. Gunji, T. Shirakawa, T. Niizato, T. Haruna, Minimal model of a cell connecting amoebic motion and adaptive transport networks, J. Theor. Biol. 253 (2008) 659–667. [21] T. Niizato, T. Shirakawa, Y.-P. Gunji, A model of network formation by Physarum plasmodium: interplay between cell mobility and morphogenesis, BioSystems 100 (2010) 108–112. [22] Y.-P. Gunji, T. Shirakawa, T. Niizato, M. Yamachiyo, I. Tani, An adaptive and robust biological network based on the vacant-particle transportation model, J. Theor. Biol. 272 (2011) 187–200. [23] J. Jones, Influences on the formation and evolution of Physarum polycephalum inspired emergent transport networks, Nat. Comput. 10 (2011) 1345–1369. [24] J. Jones, The emergence and dynamical evolution of complex transport networks from simple low-level behaviours, Int. J. Unconv. Comput. 6 (2010) 125–144. [25] J. Jones, Characteristics of pattern formation and evolution in approximations of Physarum transport networks, Artif. Life 16 (2010) 127–153. Yuheng Wu received the Bachelor degree in Computer Science from Southwest University (Chongqing, China) in 2010. He is currently a postgraduate student in the same university, majoring in Network and Distributed Computing. His research interests include complex network, bio-inspired algorithm, machine learning, and data mining. Zili Zhang is a Professor at Southwest University, Chongqing, China, and a Senior Lecturer at Deakin University, Australia. He received his B.Sc. from Sichuan University, M.Eng. from Harbin Institute of Technology, and Ph.D. from Deakin University, all in Computing. He authored or co-authored more than 90 refereed papers in International Journals or Conference Proceedings, one monograph, and four textbooks. His research interests include agent-based computing, hybrid intel- ligent systems, and agent-data mining interaction and integration. Yong Deng, male, born in 1975. He earned his doctor degree from Shanghai Jiao Tong University in 2003. He is a Full Professor in School of Computer and Informa- tion Science of Southwest University, Chongqing, China and now a visiting professor in School of Engineering of Vanderbilt University, TN, USA. He supervised several Ph.D. students in Shanghai Jiao Tong University and in Southwest University. He won the Chenxing Scholar- ship Youth Found of Shanghai Jiao Tong University (2008), Program for New Century Excellent Talents in University (2008), Shanghai Rising-Star Program (2009), Chongqing Science Found for Distinguished Young Scholars (2010). His research field includes uncertain information modeling, risk and reliability analysis, information fusion and optimization under uncertain environment. He published about 50 peer reviewed papers in IEEE Transaction on System, Man and Cybernetics B, Decision Support Systems and Applied Soft Computing. Huan Zhou is a Master candidate in Southwest Uni- versity, Chongqing, China. Her research interests include bionic approach and dynamic reconfiguration and adaptive network design. Contact her at iamz- [email protected]. Tao Qian received the Engineer degree in Software Engineering in 2010 from Southwest University, Chongqing, China. He is currently a postgraduate stu- dent, majoring in the technology of computer applica- tion. His research focuses on the bionic algorithm in computer science. Y. 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