Stability of Amygdala Learning System Using Cell-To-Cell Mapping Algorithm

Transcription

1 Stability of Amygdala Learning System Using Cell-To-Cell Mapping Algorithm Danial Shahmirzadi, Reza Langari Department of Mechanical Engineering Texas A&M University College Station, TX USA ABSTRACT In this paper, we study the stability of a biologically motivated system, termed BEL-Brain Emotional Learning, using the Cell-To-Cell mapping method. The BEL system is a learning algorithm, which is mainly applied in control problems, and hence, its stability properties are of most important to investigate. Because of the nonlinearity and complicacy of the governing equations, investigating the stability of the system using analytical methods, e.g. Lyapunov theory, turns out to be very cumbersome. On the other hand, the Cell-To-Cell mapping algorithm is a numerical method to examine the behavior of the systems where the nonlinearities can be easily incorporated. The Cell-To-Cell mapping method is used to investigate the stability of the BEL system. The results enabled us to infer the behavior of the system with respect to changing different parameters of the system and gave a general idea of how to choose the control parameters to ensure stability. KEY WORDS Amygdala Learning, Stability, Cell-To-Cell Mapping 1. Introduction Following the work of Moren and Balkenius [1], the Brain Emotional Learning (BEL) system is developed and its applications are investigated in control and prediction problems [2, 3, 4]. The algorithm is designed to imitate the minimum characteristics of brain emotional processing and consists of four components of the socalled Limbic system of the brain: Amygdala, Orbitofrontal Cortex, Sensory Cortex and Thalamus. Of them, the first two play key roles in processing of emotions [1]. Since the BEL model is mainly aimed for control applications, stability is one of the most important issues to be addressed [5]. This paper reports the preliminary results obtained by applying the Cell-To-Cell mapping on the BEL system including only the Amygdala. A more comprehensive study is progressing to investigate the stability of the system including both Amygdala and Orbitofrontal Cortex subsystems, using Cell-To-Cell mapping. 2. Amygdala System The BEL system has two components of Amygdala and Orbitofrontal Cortex where the main learning of the system happens in these components. The learning of these components is updating rules by which the gains of these components are adapted. In this study, we are focusing on the Amygadala system where the Orbitofrontal Cortex is excluded. The BEL system is mainly designed for being used as an adaptive controller and the Fig. 1 shows the control block diagram of the system. The fundamental idea of the model, following [1, 6], is to generate the output, y, which minimizes an emotional signal, ES, while the system is receiving different sets of sensory inputs, SI. The sensory inputs received by the system represent the situation the system is experiencing and the emotional signals reflect the degree of satisfaction with the performance of the system. Following these ideas, the emotional signal and the sensory input are defined as: SI K. y (1) 1 ES K3.( r y) K4. u (2) where r, u, and K 1 through K 4 are the control reference, plant input, and design parameters. The complete descriptions of the BEL model and its control structure can be found in [5]. After incorporating the above relations in the Amygdala learning rule, the final update rule of the Amygdala is obtained as follow:. V K. y.max 0, K. y K. K. yv. K. yv. (3) where and V are the learning rate and adaptive gain of the Amygdala.

2 Fig. 1 Control Block Diagram with BEL Controller In addition to the state equation of the Amygdala system, there is also the state equation for the plant. In this analysis, a linear plant is assumed with the state equation of given in Eq. (4):. y a y b. u 2. y 3. u 2. y 3. K. V. y (4). 1 the former state evolution leads to discovering a new periodic motion, while the latter joins into a previously found periodic trajectory. Therefore, the control system has two states of Amygdala gain, V, and plant output, y. The objective of this study is to investigate the stability of this system using Cell-To- Cell mapping method. 3. Cell-To-Cell Mapping Algorithm The Cell-to-Cell Mapping method is initially developed as an efficient numerical technique for global analysis of nonlinear systems [7, 8] and its applications in different nonlinear analyses are studied [9, 10, 11]. The method is based on discretization of a portion of the state space of the system that is of interest to the problem. This discrete space defines a partition of the state space into a number of small areas, called cells. Then, a cell-to-cell mapping can be evolved based on the dynamic equations of the system. The mapping is generated in the way that one cell is selected as the initial state and then based on the dynamic equations of the system, the next state is determined and this process is continued up until one of the predefined scenarios happen. These scenarios are as follow: The mapping is resulted in a sink cell (the sink cell is a unique cell whose size is exceptionally different from all other regular cells and contains all the area outside of the state space of interest.) The mapping is found to generate a new periodic motion. The mapping falls in the domain of attraction of another periodic cell or reaches the cell itself, which is a previously determined periodic motion. Figure 2 shows typical state trajectories in each of these three scenarios for a 2-dimensional state space. The state trajectory starts from initial state #1, falls into the sink cell after three steps that show an unstable trajectory. On the other hand, the trajectories starting from initial states #2 and #3 are stable trajectories, since they will remain within the boundary of state space of interest. However, Fig. 2 Different scenarios in evolution of a state trajectory After investigating the trajectories starting from all available cells within the state space domain, what we have is different periodic motions with different domain of attractions, where one of them belongs to the domain of attraction of the sink cell, i.e. the motion ends in a point out of the domain of interest. Therefore, as the result of this analysis, the selected domain of state space is divided into regions where if the initial state is anywhere within them, we can infer its trajectory route and so the final state, in particular. In the other word, we can simply recognize the domains where the started motion will be bounded or unbounded. For the purpose of this study, we assume any unbounded motion as unstable and otherwise, stable. Therefore, the dynamics of the system can be efficiently characterized and its behavior is globally analyzed via such mapping [7]. However, the Cell-to-Cell mapping is a universal method which can be principally applied to any nonlinear system, but its practical utilization is usually limited by the huge amount of memory and time required for processing all the cells within the state space of

3 interest [7]. However, methods are proposed to generate cells in a more effective manner to reduce the number of cells and correspondingly the required amount of memory [12]. 4. Stability Analysis of Amygdala System In order to apply the Cell-To-Cell mapping method to investigate the stability of the control system with Amygdala, we assume the state space system possessing the state equations of Eqs. (3) and (4). The next step is to determine a state domain of interest, where we assume a region around the origin where the states are varying from -1 to 1 with the discretization step of Figure 3 shows the results obtained for a set of parameter values given in the figure. The points indicated by a or sign, characterizes stable points where the points depicted by a sign are the points whose trajectories step out of the domain of interest (unstable trajectory). It is realized from the figure that an upward portion of the space around the origin is stable and the rest of the space is unstable. Fig. 4 Time response of the system for two stable and two unstable initial states as depicted in the Fig. 3 The first parameter we are considering is the learning rate of the Amygdala,. Figure 5 shows the stability regions for different values of. As it is realized, by increasing from 0.1 to 1, the stability region is shrinking mostly from the below, i.e. in the domains where the plant output is negative. Fig. 3 Stability analysis of the Amygdala system for the parameters of 0.8, K 1 2, K 3 3, K 4 5 To verify the results obtained from the Cell-To-Cell mapping method, the time responses of the system are determined for some of the stable and unstable initial states. The time responses of Fig. 4 are in agreement with the state space results of Fig. 3, where the responses of the system blow up for two unstable points where for the other stable points, the system show converging responses. The next issue that is of interest is how the parameters of the system affect the stability. In fact, we want to investigate how the stability regions in the state space vary when the parameters of the system change. To this purpose, we keep all the parameters of the system fixed and vary one parameter at each time to realize the effects of that on the stability of the system. Fig. 5 Stability regions of the system for different values of From Eq. (3), it is realized that is in fact the rate of updating the state V which also indirectly affects the evolution of state y, because the state equations are interrelated. Therefore it is expected that by increasing the update step, the convergence of the equations becomes worse and so the trajectories are more likely to jump out of the region of interest. Figures 6 through 8 show the stability regions of the system for different values of K 1, K 3 and K 4, respectively.

4 K3 parameters, respectively, where the initial state values are assumed as Fig. 6 Stability regions of the system for different values of K 1 As it is realized from the Fig. 6, increasing K 1 impairs the stability of the system where for the values of K 1 greater than 20, the system is completely unstable. The reason for such paramount diverging effects of K 1 on the state equations is that this coefficient appears in both state equations and even with the squared terms. Therefore, it is strongly affecting the convergence of the equations. Fig. 8 Stability regions of the system for different values of K 4 These figures are corresponding to the Figs. 6 and 7, respectively. It is realized that the point of (0.25, 0.25) is located within the stable regions for K 1 values of 0.2 and 2, while it falls into unstable region for K 1 value of 20. This is in agreement with the stable and unstable outputs observed in Fig. 9. On the other hand, for K3 values of 0.2, 2 and 20, the point (0.25, 0.25) remains in the stable region that are verified by the stable outputs of the system given in Fig. 10. Another issue to be considered in the BEL model is the effects of the max function in the learning of the Amygdala on the stability of the system. The max function comes from the biological system where the Amygdala learning capability is monotonic [13]. Fig. 7 Stability regions of the system for different values of K 3 By the similar reasoning, the behaviors of the system with respect to changes in coefficients K 3 and K 4, which are shown in Figs. 7 and 8, can be described. By increasing these coefficients, the update step for the state V will be increased. Therefore, for the larger values of K 3 and K 4, the stability regions become smaller. In particular, for large values of K 3 and K 4, the stability regions lie on upper and lower domains of the state space, respectively. This is due to the negative sign of K 3 coefficient that causes the update direction opposite. To verify the results achieved from Cell-To-Cell mapping analysis, we examine the outputs of the system in time domain when the parameters are changed. Figures 9 and 10 show the two outputs of the system for varying K 1 and Fig. 9 Time simulations for the initial state of (0.25,0.25) for different values of K 1 To investigate the effects of max function, we analyze the system both with and without max function whose results are shown in Fig. 11. As it is observed from the figure, the max function expands the domain of stability of the system.

5 Fig. 11 Stability regions of the same system with and without max function This is directly related to the expression K 3. y K1. K4. yv. K1. yv. in the max function of Eq. (3). For the parameters of the system in this case, this expression is simplified to 3 8. V. y. It is easily verified that this expression is negative in the upper region of interest. So when the max function is assumed in the formula, these negative values become zero and so, as the results show, prevent the trajectory from going unstable. Fig. 10 Time simulations for the initial state of (0.25,0.25) for different values of K 3 5. Conclusion In this paper, we studied the stability of an adapting system using Cell-To-Cell mapping algorithm. The adapting system was the learning mechanism of Amygdala which is considered in conjunction with a control system. Due to the nonlinearity and complicacy of the equations, analyzing the stability of the system using analytical methods, e.g. Lyapunov function, turns out to be cumbersome. Therefore, we examined the behavior of the system using the numerical method of Cell-To-Cell mapping which enabled us to recognize the domains of stability for different parameters of the system. This gives an insight on how the parameters of the system affect its convergence and can be a rough baseline for designing parameters of the system. As an example for the specific system under study, by looking at different stability patterns, it can be inferred that the origin will be a stable point for the design parameters of.9, K 2, K 3, K 5, whereas the parameter , K1 2, K3 300, K4 values of 5 lead to an unstable design. As a general conclusion, increasing the parameters in the state equations limits the stability regions, with different levels of impact, though. To verify the results achieved from Cell-To-Cell mapping analysis, we obtained the time responses of the system for some stable and unstable points where the results of time domain and state space were in agreement. Finally, we examined the effects of the max function in the Amygdala learning rule by applying the Cell-To-Cell mapping algorithm on the system with and without max function. It was realized that the max function expands the stability regions of the system, i.e. some previously unstable points become stable when the max function is included in the learning rule. References: [1] J. Moren & C. Balkenius, A Computational Model of Emotional Learning in the Amygdala, Proc. 6 th International Conference on the Simulation of Adaptive Behavior, Cambridge, MA, The MIT Press, [2] C. Lucas, D. Shahmirzadi, & N. Sheikholeslami, Introducing BELBIC: Brain Emotional Learning Based Intelligent Controller, International Journal of Intelligent Automation and Soft Computing, 10(1), [3] D. Shahmirzadi, R. Langari, L. J. Ricalde & E. N. Sanchez, Intelligent versus Sliding Mode Control in Rollover Prevention of Tractor-Semitrailers, International Journal of Vehicle Autonomous Systems, Special Issue on Intelligent Mobile Machines, To Appear.

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