show that using this methodology is a feasible way towards autonomic systems. KEYWORDS. Excitable Media, Autonomic Computing, Self-Configuration. 1.
IADIS International Conference Informatics 2008
EXCITABLE MEDIA: A WAY TOWARDS AUTONOMIC COMPUTING Luis M. Fernández-Carrasco, Hugo Terashima-Marín, Manuel Valenzuela-Rendón Center for Intelligent Systems Instituto Tecnológico y de Estudios Superiores de Monterrey 2501 Eugenio Garza Sada Ave., Monterrey, NL, México
ABSTRACT Nature possesses the incredible ability to generate a number of autonomous processes which amaze scientists and researchers. This paper makes also use of a phenomenon seen in nature which is called excitable media. It is believed that, using this mechanism, self-star properties can be achieved. In this document a novel approach using such occurrence is proposed. An environment simulating an excitation where autonomous agents live was created, and the way this excitation makes these agents behave is presented. The results show that using this methodology is a feasible way towards autonomic systems. KEYWORDS Excitable Media, Autonomic Computing, Self-Configuration.
1. INTRODUCTION According to Richard Murch’s book (Murch, 2004) autonomic computing is, quoting, “the ability to manage one's computing enterprise through hardware and software that automatically responds to the requirements of your business. This means self-healing, self-configuring, self-optimizing, and self-protecting hardware and software that behaves in accordance to defined service levels and policies. Just like the autonomic nervous system responds to the needs of the body, the autonomic computing system responds to the needs of the business.” The quest towards autonomic computing has taken important steps in recent years; yet, such accomplishment is still out of reach. The present document aims at providing another alternative to such goal using excitable media as a way to achieve self-star properties in complex systems. Based on life itself, as complex and diverse as it is, excitable media is an attempt to provide a mathematical model to excitation waves. This is not an easy task to do but many successful models show the common property that they treat any cell as an automaton (Mikhailov, et al., 2006). An automaton is basically a black box which receives information from the surrounding environment and generates output accordingly. In mathematics, the automata are often considered to be discrete (Wuensche, 2005). However, automata with gradually varying input and output signals and a continuous temporal evolution may also be covered. Moreover, automata with stochastic dynamics are possible (Mikhailov, et al., 2006). When an automaton which mimics the individual behavior of a biological organism is constructed, a population of such automata can be considered whose members communicate by generating, sending and receiving signals (Mikhailov, et al., 2006). As such, excitable media is a communicating automaton. This document proposes a new approach to autonomic computing by modeling an excitable media environment based on cellular automata where agents that might represent autonomous entities live. The idea of using this media comes natural as it resembles an electric field. This electric field is the one that starts the process and excites the appropriate entities so that they do what they are supposed to do. As it was said previously, this is a novel proposed approach to autonomic computing and, consequently, there are not any related works. Most publications in the excitable media field are in the biological and medical areas (this is so since it is associated to modeling electric pulses found in nature.) Some examples
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can be found in the work done by Romanes (Romanes, 1876), Mayer (Mayer, 1908) in the biology field. In the medical area, the publications of Mines (Mines, 1913), Lewis (Lewis, 1925), Garrey (Garrey, 1914), and Wiener and Rosenblueth (Wiener, et al., 1946) are some examples. The next sections of this paper present a thorough description of excitable media, the mathematical model used in this implementation. Then, the implementation itself is explained along with the tools used to achieve that. Finally, the experiments, the results and the conclusions of this research are presented.
2. EXCITABLE MEDIA: AN OVERVIEW Excitable media belongs to the field of models of complex coherent action, being mainly spatially distributed systems which have the ability to propagate signals without damping. Let us look at a forest fire, for instance, which travels as a wave from its initiation point, and regenerates with every tree it ignites (Center for Nonlinear Dynamics - Mc.Gill University, 2007). This is in contrast to passive wave propagation, which is characterized by a gradual damping of signal amplitude due to friction. An example of passive wave propagation is sound waves passing through air. An impulse over a certain threshold initiates a wave of activity moving across the excitable media. As each element undergoes an excursion from steady state, it causes its neighbors to move over threshold at a rate determined by the diffusion coefficient (a passive property of the media), and the rate of rise of the diffused species of the excited element (an active property of the media). Figure 1 shows this idea graphically where the arrival of a perturbation initiates a transition from the rest state (white) into the estate of excitation (dark grid). After that, the element goes into the refractory state (light gray) and then returns to the rest state (Mikhailov, et al., 2006).
Figure 1. The schematic representation of a cycle of an excitable element
Excitable media can be found in natural events such as the heart palpitation. Moreover, the propagation of electrical activity in cardiac muscle involves the interaction of different ion species across a combination of active and passive ion channels and diffusion of charge through a heterogeneous substrate with dynamically changing conductance (Center for Nonlinear Dynamics - Mc.Gill University, 2007). Despite complexity inherent in conduction at microscopic scales, the heart can be approximated as a continuous excitable media. A variety of cardiac tachycardia have been attributed to formation of large scale patterns of excitation such as the formation and break up of spiral waves (Center for Nonlinear Dynamics - Mc.Gill University, 2007). Excitable media can be modeled using both partial differential equations and cellular automata, and some examples are the works of Lewis (Lewis, 1925), Wiener and Rosenblueth (Wiener, et al., 1946), Zykov and Mikhailov (Mikhailov, et al., 1994). This last one is explained in the next section.
3. EXCITABLE MEDIA MATHEMATICAL MODEL The model proposed by Zykov-Mikhailov (Mikhailov, et al., 1994) is the one to be used in the implementation phase. Its simplicity and few numbers of variables make it ideal to test the design. Moreover, it allows agent-based implementation simulating cellular automata. Let us consider a two-dimensional square lattice occupied by excitable entities. Each lattice site is ), where is the size of the square. The state of an specified by a pair of coordinates and ( individual element at a discrete moment of time is described by two variables and . The first of ) corresponds to the rest these variables, , represents the integer phase of an entity. The zero phase ( state. The phases in the interval correspond to excited states ( is the duration of the excitable
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period). The refractory states correspond to the interval , where is usually the duration of the refractory period. The second variable specifies the accumulated energy received by the element from its neighbors. If this variable happens to exceed a certain threshold , the entity goes from the state of rest into the first excitable state and its cycle is thus initiated. The accumulated signal of the element ( ) is obtained by summation of signals coming from its neighbors. Also, to consider temporal accumulation, it is assumed that of the signal. Hence, there is (Mikhailov, et al., the signal at the next time step includes a fraction 2006):
Equation 1
where the coefficients determine the range of spatial summation. The variables if the respective element ( ) is currently in the state of excitation, i.e., otherwise. The updating laws for the phase variable are (Mikhailov, et al., 2006):
are defined here as , and
Equation 2
Both Equation 1 and Equation 2 mean that the element operates like a clock: the cycle is started if the threshold is exceeded and then lasts for the time , returning the element to its original state (Mikhailov, et al., 2006). Although the model of excitable media above is the one that is to be used, this does not mean that other possible models would not be considered. The next section of this paper presents an analogy that is the support for proposing the use of excitable media as a platform for autonomic computing.
4. ANALOGY AND JUSTIFICATION How can all that has been mentioned in the lines above be associated to autonomic computing and its components? The answer is simple if each program is considered as an excitable element. Let us develop this idea further. In excitable media one element reaches a certain level of excitement and propagates such energy to its neighbors and the process continues until it dissipates (maybe it runs out of energy or the model is built to do so). In the computing field, when one launches a program is like exciting it, and, at the same time, it excites other programs or components that are needed for its proper running, and the excitement continues up to the point where the program stops running and its task is done. With all this two things are achieved, a pattern to remember (this could be a graphical representation or a mathematical function) representing the path followed by the program, and way to achieve self-organization by changing the agent to excite. Moreover, when modeling the media, one can include obstacles which in the computing world might be associated to hardware limitations. Thus, it is believed that these properties can help achieve autonomic computing not only in the software area, but also in the hardware one since the model proposed is based on electric intensities. The next section of this paper describes the implementation of the model and the simulation setup to test the idea.
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5. IMPLEMENTATION The implementation of the simulated environment was done using TurtleKit (Michel, 2002), a MadKit plugin for multiagent simulation (MadKit, 2005). This plugin offers a world where agents live in a built square world of dimensions 300 300. Consequently, there are two aspects to consider here: the agents involved and the world.
5.1 Simulated Agents Two kinds of agents were developed in order to make use of the excitable media model. One kind was responsible of exciting the world. From the computer system’s point of view, these agents could be considered as the initiators of a process, either because a user just launched an application or because an autonomous process took place. These agents were placed horizontally centered in the simulated world. These agents are fixed and do not move. The other kinds of agents, randomly placed in the simulated world, are simple agents with the only assigned task of finding the agent that excited the environment. They achieve this by reading two variables embedded in the world, phase and energy which were presented in Equation 1 and Equation 2. The greater the phase and energy variables the closer the agent is to finding the origin of the excitation. In a real system, these agents could represent the other components associated to the launching of a program, such as dll files. In the simulated world, one thousand and five hundred of these agents were instantiated. This number could have been less or more, as it does not influence the way the excited waves propagate in the media. Figure 2 presents the initial state of the agents in the world.
Figure 2. A view of the simulated agents and world.
5.2 The Simulated World As mentioned previously, the world is a square lattice of 300 300 which behaves as a cellular automaton. The TurtleKit Platform is the simulation engine used to process the rules presented in Equation 2. This engine offers tools for modeling, using and exploiting multiagent simulations based on agents that evolve in a discretized world and act on it. Moreover, one characteristic that helped a lot in the simulation process was the fact that, thanks to the plugin, each cell in the world could hold variables (i.e., phase and energy). These variables change at each time step. In this simulated world one has the ability to excite any of the already mentioned exciting agents and see how such excitation affects the other agents. In order to better visualize the wave propagation, a coloring based on the phase variable is imposed on each cell of the world. The darker the color, the higher the phase is. Figure 2 shows the initial state of the world. As it can be seen, the state in all cells is zero; consequently, no excitation has taken place yet.
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The next section of this document describes the experiments that were conducted on this simulated world.
6. EXPERIMENTS In order to proceed with the experimentation, some initial values had to be set (see Equation 1 and Equation 2). The values for these variables were taken following Zykov and Mikhailov's model (Mikhailov, et al., 1994), and the examples presented by Mikhailov and Calenbur's book (Mikhailov, et al., 2006). Thus:
The addition of and gives as a result eight; this last number provides the number of phases the world has. This condition holds true for all cases meaning that the world may have as many phases as needed. The variable represents the threshold that has to be passed in order to be ready for excitation. The lower this number is, the easier to excite the media. At the beginning the world is set up with the phase and state variables set to zero. As expected, once one of the exciting agents is disturbed, wave propagation begins as shown in Figure 3, and the surrounding agents are attracted to the origin of the disturbance. It is clearer that the agents are attracted to the center of the disturbance, in other words, they are configuring according to the exciting agent.
Figure 3. Initial steps of wave propagation.
As it was mentioned, the agents move towards the origin of the disturbance and organize accordingly. A better view of such instance is presented by Figure 4 as they make their way towards the first exciting agent; and almost all agents have found the origin of the excitation.
Figure 4. Agents starting to find and organize according to the disturbance
For the present experiment, many runs were conducted disturbing consecutively each exciting agent; meaning that, one exciting agent would be disturbed for a fixed number of times, after, the world would be reset (i.e., phase and energy variables to zero), and the next disturbing agent would be excited to see the
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response of the agents. This process was repeated until all exciting agents have been disturbed by a fixed number of times. In general, the result was an almost perfect cluster of all 1500 agents as they find the origin of the disturbance and configuring according to it. Figure 5 shows one of such many ending instances. All agents had been attracted to the top exciting agent forming an almost perfect cluster.
Figure 5. All agents found the origin of the disturbance.
The next section presents the results of the measurements taken as these simulations were conducted and, finally, the conclusions of the present work.
7. RESULTS & ANALYSIS There were two things that were measured in the already presented experiments, the existing distance among the simple agents, and the information entropy. These two characteristics were chosen as they are considered to be task independent ways to measure organization. Regarding distance, the obtained results are the ones that were expected. The distance, as time goes by, decreases as the agents are configured to find the origin of the disturbance. Thus, the distance among them reduces as they get closer to the exciting agent. Out of all the many experiments conducted, the figures that are presented next are the most representative ones, as they did not change that much from one experiment to the other. Figure 6 presents the maximum and minimum distance among all agents at each time step. This was done by measuring the Euclidean distance between one agent and the others. These results were stored in a vector . Of each vector the maximum and the minimum distance were calculated; and then, the biggest and smallest value out of all these was assigned to the current step. In other words,
Figure 6. Maximum and minimum distance among agents as they find the origin of the disturbance.
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As ones goes over Figure 6, one notices peaks in the plot. Each peak of course means an variation in the distance. This was so because of the process explained in the previous section. After a fixed time, the world is reset and the next disturbing agent is excited. While the agents try to organize to find the origin of the new disturbance, they tend to separate more. However, as time goes by, this distance is reduced. Following the same idea, Figure 7 shows the averaged Euclidean distance at each step. This expressed formally is,
Figure 7. Averaged distance among agents as they find the origin of the disturbances.
The reason why these plots were generated was mainly to see the evolution of the distance as the agents configure according to the disturbance in the media. The distance is considered to be a task independent way to measure organization among agents. It is interesting to see that, at the beginning of the simulation, all agents seemed to be confused; thus, greater distances among them. However, as the simulation continued, the distance, let it be maximum, minimum or average, decreases. This could be an indication of better configuration and organization as they find the origin of the disturbance. Another measurement that was considered was information entropy. Figure 8 shows the relation of information entropy as the simulation evolves throughout time. This picture tells us that the closer the agents get, the more information entropy there is. As the agents find the origin of the disturbance, the information entropy gain is increased. Again, the valleys that are observed in Figure 8 were caused by the reset of the world and the disturbance of the next exciting agent in the world.
Figure 8. Information entropy as simulation takes place.
The next section of this document presents the conclusions that the experimentation and analysis phase provided.
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8. CONCLUSIONS The main conclusion that can be given is that excitable media is an interesting and promising way to achieve autonomic computing systems. The simple excitation of an element can cause the organization of the system according to the needs of the system and/or to the agent that originates the disturbance. This, then, comes to be a way towards self-configuration and self-organization in autonomic systems. More specifically, the conclusions are: • Agents or entities can be organized depending on the source of disturbance in an excitable environment. • Configuration of agents is giving by two variables, phase and energy , embedded in the world, which tell the agents how to behave. • There are ways to optimize the behavior of the system by tuning the variables of the excitable media model. Different values may produce different agent behaviors that could meet the requirements of the system. • Since excitable media models electric pulses, these models can be a way to build hardware that controls the way the system behaves by adapting or changing the energy sent to a device or a peripheral. • Excitable media proposes simple models to complex action. In the current research agents could be organized by just giving them a rule. However, there is yet one aspect that needs to be tested and is how agents would behave to energy and phase values unseen before (i.e., adaptation). One way could be to include fuzzy rules in the agent’s engine, or statistical learning algorithms. This document is the result of a starting research project where, in a general way, models of complex action, and more specifically, excitable media, offer a great opportunity to get to autonomic computing and its features.
ACKNOWLEDGEMENT This research project is funded by Tecnológico de Monterrey, Research Chair CAT 010.
REFERENCES Center for Nonlinear Dynamics - Mc.Gill University. 2007. Excitable Media. [Online] 2007. http://www.cnd.mcgill.ca/bios/bub/excitablemain.html. Garrey, W. 1914. The Nature of Fibrillary Contraction of the Heart. Its Relation to Tissue Mass and Form. American Journal of Physiology. 1914, Vol. 23, 397. Lewis, Th. 1925. The Mechanism and Graphic Registration of the Heart Bea. London : Shaw and Sons, 1925. MadKit. 2005. The MadKit Project. [Online] 2005. http://www.madkit.org/. Mayer, A. G. 1908. Rhythmical pulsation in scyphomeduse. Papers from the Tortugas Laboratory. 1908, Vol. I, pp. 115 - 131. Michel, Fabien. 2002. Introduction to TurtleKit: A Platform for Building Logo Based Multi-Agent Simulations with MadKit. s.l. : Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier, 2002. RR LIRMM 02215. Mikhailov, A. S. and Zykov, V. S. 1994. Foundations of Synergetics I. Distributed Active Systems. Germany : Springer, 1994. Mikhailov, Alexander S. and Calenbuhr, Vera. 2006. From Cells to Societies: Models of Complex Coherent Action. s.l. : Springer, 2006. Mines, C. R. 1913. On Dynamic Equilibrium in the Heart. Physiology. 1913, Vol. XLVI, 349. Murch, Richard. 2004. Autonomic Computing. s.l. : IBM Press, 2004. Romanes, C. J. 1876. Preliminary Observations on the Locomotor System of Meduse. Phil. Trans. Roy. Soc. 1876, Vol. CLXVI, 269. Wiener, N and Rosenblueth, A. 1946. The Mathematical Formulation of the Problem of Conduction of Impulses in a Network of Connected Excitable Elements, Specifically in Cardiac Muscle. Arch. Inst. Cardiol. 1946, Vol. 16, 205. Wuensche, Andrew. 2005. Discrete Dynamic Lab: Tools for Investigating Cellular Automata and Discrete Dynamical Networks. [book auth.] Andrew Adamatzky and Maciej Komosinski. Artificial Life Models in Software. s.l. : Springer, 2005, pp. 263 - 297.
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