TH-I2-4
SCIS&ISIS2006 @ Tokyo, Japan (September 20-24, 2006)
An Autonomous Mobile Robot Controlled by a Spike Neuron Network with one Hidden-Layer Neuron having Spike Timing-Dependent Plasticity Fady Alnajjar and Kazuyuki Murase Department of Human and Artificial Intelligence Systems University of Fukui, Japan E-mail:
[email protected] and
[email protected] Abstract- In this paper we describe an autonomous mobile robot whose sensory-motor connection were made by a three-layered spiking neural network (SNN) with only one hidden-layer neuron that makes synaptic contacts on motor neurons with synapses having spike timing-dependent plasticity (STDP) and presynaptic modulation, and we analyzed the roles of STDP for the autonomous behavior in environment. STDP is known as a kind of Hebbian rules with which the synaptic weights are increased or decreased in accordance with the relative timing of pre- and postsynaptic action potentials. An excitatory synapse may become an inhibitory one, or vise versa, in the STDP. We used one of the SNN models, called the spike response model (SRM), in which the neurons generate spikes, and a spike at presynaptic site generates a delayed, prolonged post synaptic potential (PSP). The postsynaptic neuron fires when the sum of PSPs becomes over a threshold. Once a neuron fires, it goes into a refractory period during which a larger input is necessary to generate the following spike. We considered a mobile robot with left and right front proximity sensors, and left and right wheels driven by independent motors. Each sensor value was converted to the probability of spike generation by the sensory neuron. The outputs from left and right sensory neurons were connected to one hidden-layer neuron with fixed synapses. The output of the hidden-layer neuron makes synaptic contacts on left and right motor neurons, and these synapses have the properties of STDP in regard to the presynaptic modulation signals from sensory neurons. The spiking rates were converted to analog values to drive the corresponding motors. We implemented this SNN in a miniature mobile robot Khepera. In a given environment, it gradually organized the weights and acquired the navigation and obstacle-avoidance behavior. Modeling and experimental data analyses showed that one hidden-layer neuron is sufficient for the task, and that such SNN is suitable for the navigation of mobile robots. Key words: Spiking Neural Network, Spike Response Model, Spike Timing Dependent Plasticity, Time latency, Hebbian learning.
I.
INTRODUCTION
Artificial Neural Network (ANNs) is described as network consisting of neurons interconnected by synapses. There are three generations of ANNs [1]. The ANNs in the first generation are known as threshold gate. It is characterized by treating a neuron as a binary device. This model distinguished only between the occurrence and absence of a spike. The second generations are known as sigmoidal gate in which the
output of a sigmoidal gate is a number that is thought to represent the firing rate of the neuron. In the first two generations of ANNs, synapses are modeled by the magnitudes of weights, and many forms of Hebbian learning have been proposed for the self-organization [2]. The weights are modified with local unsupervised learning rules, and the network activity becomes stabilized during the process. The network can perform tasks such as pattern recognition, clustering and auto association. The third generation of the ANNs is so called the spiking neuron network (SNN), whose properties are now actively investigated and which is applied in tasks that are not well performed by the first and second generation of ANNs [3,4]. The largest advantage of the SNNs is its ability to discriminate events in time domain. The timing of events can be directly dealt with the spike timing [1]. It is therefore thought suitable for the systems that require the analysis of a flow of events, such as moving scenery and optical flow. Synaptic plasticity that depends on the correlations between pre- and postsynaptic firings is called the Hebbian learning (or synapse), as Hebb first recognized this type of plasticity [5]. He postulated that the synaptic efficacy between two neurons is potentiated when the presynaptic neuron contributes to the firing of the postsynaptic neuron. A correlated activity of preand postsynaptic neurons thus results in the long-term potentiation (LTP) of the synaptic efficacy, whereas the uncorrelated activation in long-term depression (LTD). Although such classical Hebbian learning rules based on the synchrony of firings have been proven useful, researches in the last ten years have indicated the precise timing of the firing is often very important, and thus the term spike timingdependent plasticity (STDP) has come into existence. STDP is a form of synaptic modification rule found recently in natural synapses [6,7], and a number of experimental and modeling studies have revealed that the STDP could be responsible to the conventional Hebbian learning, short-term prediction [8], gain adaptation [9] and boosting of temporally correlated inputs [10], etc. Synapses with the properties of STDP can be strengthened or weakened depending on time latency. Inputs that fire the postsynaptic neuron with short latency or that act in coherent manner develop strong synaptic connections. Inputs of longer latency or less effective inputs, in contrast, lead to weaken the synaptic strength [10]. In other words, synapses modifiable with STDP compete for control of
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the timing of postsynaptic action potentials to be more sensitive to the presynaptic action potentials. Application of the SNN to robotics has drawn much attention recently. A mobile robot, whose sensory-motor connections are made with a SNN and evolved by genetic algorithm, successfully navigates and avoids obstacles [11,12]. Robotic application of SNNs with the STDP, however, has not yet fully investigated though it has drawn much attention [6,7]. In this study, we try to formulate a SNN with a STDP for controller of mobile robots, and test it with a real mobile robot Khepera to examine the robot movement. We here exhibit that only one hidden-layer neuron between the sensory and motor layers is necessary for the self-organization of navigation and obstacle-avoidance behavior. This paper is organized as follows: In section 2, the spike response model (SRM), a kind of SNNs used in this study, is explained. In section 3, we formulate an algorithm based on the STDP. In section 4, we explain the network features. In section 5, experimental setups such as the features of the mobile robot used, tasks given to the robot and the arena, the environment where the robot performed the task, are described. In section 6, the implementation of STDP in the robot is described. Section 7 explains the experimental results and the analyses. We conclude the paper in section 8. II.
SPIKING RESPONSE MODEL (SRM)
Artificial SNNs are classified into two types, pulse-coded neural networks and rate-coded neural networks [3]. And also, there are several models of spiking neurons with various degrees of details [13]. In this study, we focus on the Spiking Response Model (SRM) as a spiking neuron model, which is easiest to understand and implement especially with STDP. SRM is defined as a single variable υi that describes the state of a neuron. As shown in Fig. 1, in the absence of a spike, the variable υi is in its resting value. Here we assume it is zero. Each incoming spike generates a postsynaptic potential that takes time before returning to zero. Mathematically speaking, υi can be represented by equation (1); equations analysis can be found in [11,12]:
∑
∑
∑
Spike
Incoming spikes
Refractory period
Fig. 1. The membrane potential of a spiking neuron in a SNN.
III.
SPIKE TIMING DEPENDENT PLASTICITY (STDP)
Spike timing-Dependent Plasticity (STDP) was originally found in a natural synapses [14]. Synaptic efficacy is modified persistently by a coherent activation of pre- and postsynaptic elements. The degree of potentiation is found to be a function of the time internal between pre- and postsynaptic excitations. When the postsynaptic site is excited just after the presynaptic excitation, for example, the synaptic efficacy is increased. Conversely, it is decreased, or even becomes negative (inhibitory) when the postsynaptic site is excited before the presynaptic excitation. There are two types of computational models of STDP [9]. One is that the change in synaptic strength is a function of the time difference between presynaptic element and postsynaptic neuron as shown in Fig. 2A. That is, when the presynaptic site and postsynaptic neuron are excited at tpresynaptic and tpostsynaptic, respectively, the synaptic weight is modified in proportion to the difference tpre-post, where,
∆ t pre − post = t pre − syn . − t post _ syn .
Another type of the STDP model, which we used in this series of study, is based on the firing rate as shown in Fig. 2B. If presynaptic site and postsynaptic neuron are generating action potentials with time intervals of ∆t presynaptic and ∆t postsynaptic , respectively, the synaptic weight is modified in
(1)
accordance with the difference ∆t pre− post , where,
ε ( s ) = exp[−( s − ∆) / τ m ](1 − exp[−( s − ∆) / τ s ])
(2)
∆t pre− post = ∆t pre−syn. − ∆t post−syn.
η ( s ) = − exp[ − s / τ m ]
(3)
υ i (t ) =
j
ω
t j
f
ε j(s j) +
η i (si )
f
The function є(s) describes the time course of the postsynaptic response generated by an incoming spike. If the summation of the effects of several incoming spikes reaches a threshold (θ), an output spike is triggered. Then the refractory period η(s) will start from a negative value. The neuron can hardly emit a new spike until υi returns gradually to its resting potential. In practice, each synaptic potential has to be terminated in a finite time [11]. The values of exponents were truncated for 20 sampling periods in the experiments below [12].
(4)
(5)
We used this second model, assuming that a sensory input generates a barrage of presynaptic action potentials, and that the firing lasts longer than that required for the generation of postsynaptic spiking. After the synaptic modification, the postsynaptic neuron would tend to respond more quickly to the presynaptic spiking. In some natural synapses, the amount of a neurotransmitter released from the presynaptic site can be modified by another substance that acts on the presynaptic site, the phenomenon called the presynaptic modulation. Especially in synapses with plasticity, a diffusible substance released by other neuron(s), such as nitric oxide and carbon oxide, acts presynaptic site, and regulates the magnitude of potentiation (or depression) of the synaptic efficacy in cooperation with the intracellular conditions of the presynaptic site [15].
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In plastic synapses receiving such presynaptic modulation, therefore, we can consider that such “pre-presynaptic” activity cooperate with the presynaptic activity, and determines the synaptic efficacy and the following postsynaptic activity. Such cooperative mechanism could be modeled with the scheme of the STDP. That is, the signal received by postsynaptic neuron is a function of the difference between the “pre-presynaptic”-spike interval and the presynaptic-spike interval. In this study, we implemented this presynaptic modulation at the synapses on motor neurons.
∆tpost. Post-syn.
Post-syn. Pre-syn.
Pre-syn.
eight infrared (IR) proximity sensors and two motors. Six sensors are located on the front side of the robot, while the remaining two are on the rear. We summed up the values of sensors 1 and 2 to present the left sensor value (L) of the robot, while of sensors 3 and 4 to present the right sensor value (R). These two sensors should be sufficient for Khepera to avoid front, right and left obstacles. Khepera communicates via a serial port with a host computer. The high level processes, such as SNN activation and genetic operations, run on the host computer, while the low level processes such as sensor reading and motor control run on the Khepera processor. The task was to navigate in an arena by avoiding obstacles. The size of arena was 60×60 cm, and obstacles are distributed in the environment as shown in Fig. 4B. The walls and obstacles were made of white plastic in order to increase the performance of the IR sensors. L.sensor
R.sensor
∆tpre. (A)
(B)
Fig. 2. (A) One type of STDP based on the spike time interval between pre- and postsynaptic sites. (B) Another type of STDP based on the difference in the firing rates in pre- and postsynaptic sites.
IV.
L.motor
R.motor (A)
Fig.4. (A) Schematic drawing illustrates the position of the sensors and motors on Khepera. (B) The arena and the connection with an external host computer.
THREE-LAYERED SNN FOR MOBILE ROBOT WITH ONE HIDDEN-LAYER NEURON
Assume a three-layered SNN that controls mobile robot. In this study, we explore the possibility that only one neuron is necessary in the hidden layer for the navigation and obstacleavoidance of a mobile robot. The network structure is illustrated in Fig. 3. Consider a mobile robot with minimum structure, consisting of left and right proximity sensors and left and right motors. Each of the left and right proximity sensor values is feed to the corresponding input-layer neuron. One hidden-layer neuron receives synaptic inputs from both inputs-layer neurons, and sends spikes to the two output-layer neurons, one of which drives left motor and another the right motor. L.sensor
R.sensor
Fixed weight synapses
Input Layer
Hidden Layer Presynaptic modulation
STDP synapses L.motor R.motor
Output Layer
Fig 3. SNN with one hidden-layer neuron for mobile robot
(B)
VI.
SYNAPTIC PROPERTIES
A. Synaptic on hidden-layer neuron Each sensory neuron in the input layer makes a synaptic contact with the hidden-layer neuron. These synaptic weights were set initially at fixed values, and not modified. B. Synaptic on output-layer neurons The hidden layer neuron makes a synaptic contact on each of the output-layer neurons. Both synapses had the properties of STDP. As shown in Fig. 3, the presynaptic sites on the left and right output-layer neurons received “pre-presynaptic” contacts from the left and right input layer neurons, respectively. The synaptic strengths of both synapses were modified in accordance with the difference in averaged spike-time intervals at the presynaptic sites and “pre-presynaptic” modulator input from the input-layer neurons. That is, the STDP took place in accordance with Eq(5) where ∆tpre-synapstic referred to the firing of input-layer (sensor) neurons and ∆tpresynapstic to that of the hidden-layer neuron. For example, if the left input-layer neuron spikes at time = 2,6,10 and 14, and the hidden-layer neuron spikes at time = 3, 7, 11 and 15, then, the average of both ∆tpre-synapstic and ∆tpost-synapstic =4. In this case, the time latency ∆tpre-post of this pair is 0. That is, we calculated the time latency ∆tpre-post with the following equation. N
N
V.
∆ t pre − post =
IMPLEMENTATION OF THE SNN INTO A REAL MOBILE ROBOT KHEPERA.
We have implemented the above-mentioned SNN into a real mobile robot Khepera [16] Fig. 4A. Khepera is provided with
∑ ∆t i =0
i ( pre − synaptic )
N −1
−
∑ ∆t i=0
i ( post − synaptic )
(6)
N −1
In order to take the advantage of synaptic modification using STDP, we supposed that ∆t should be in a range of [-0.5,+0.5]. We called this range a short latency region. During the
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lifetime of the synapse, strengthening or weakening of a synapse should lead ∆t to be in the above range. C. Motor control signals Each output-layer neuron (motor neuron) should generate signal for the corresponding motor in proportional to its membrane potential value. As the measure of the membrane potential value, we instead used the spike interval at the synapse. Note that, although the same hidden-layer neuron sends out spikes to both of the output-layer neurons, the presynaptic modification signals from the input-layer neurons are different each other. And thus, the spike rates received by two output-layer neurons become different each other. The signal to each motor was thus calculated with the following equation, where ∆t referred the spike interval. M _ value = ( 0 .5 − ∆ t ) x10
presynaptic weights. These correlations between both synapses weights in the same pair cause the postsynaptic neuron to fire sensitively to the presynaptic neuron firing time as shown in figure 6II. From the figure the postsynaptic weight was adapted and ∆t increased from –2.3 to –0.2 to be in the short latency region. In this case the robot performed the obstacle avoidance behavior successfully as will be explained in the next experiment. Weight
Pre-syn. weight
Post-syn. weight
(7) Number of Experiment
VII. EXPERIMENTS AND RESULTS To verify the proposed method, we conducted a number of experiments to examine the performance of the network and the robot’s behavior. In all the experiments done in this section; the initial synapse’s weights were selected randomly from a range between [0,10]. The postsynaptic modification rate is (+/- 0.1). The program’s goal is to lead ∆t to be in the short latency region [-0.5,0.5]. A. Experiment 1 1) Experimental Setting: The purpose of this experiment is to show how the proposed method to lead ∆t to be in the short latency region, can adapt the postsynaptic weight to force the postsynaptic neuron into a balanced, i.e. irregularly firing regime in which it is sensitive to the timing of the presynaptic action potentials it receives. In this experiment, first, the network was initialized with random weights as mentioned above. Then, the robot started to move in the arena. The postsynaptic weight was adjusted in every time robot hit the obstacle until ∆t reached the short latency region. This experiment was repeated 10 times, and each time, we started Khepera from different positions and reinitialized the network with different random weights. 2) Experimental Results and Analysis: Figure 5A shows the initial random synapse weights of the left pair of pre- and post-synapses of the network. From the two curves in the figure it is clear that the initial random presynaptic weights have no correlation with the random postsynaptic, i.e. ∆t is out of the region. The lake of a correlation between the preand post-synaptic weight causes the postsynaptic neuron to fire irregularly and insensitive to the presynaptic neuron firing time as shown in Fig. 6I. Figure 6I represents the firing time of pre and postsynaptic neurons when the pre- and postsynapses’ weights were selected randomly 4.3 and 1.7, respectively. In this case ∆t = -2.3, which is out of the short latency region that we specified. In this case, the robot motors’ outputs were not sufficient to perform the obstacle avoidance task. Figure 5B shows the modified synapses’ weights with STDP. From the figure, it is clear that the postsynaptic weights were adapted and built correlations with the
(A)
Weight
Number of Experiment (B) Fig. 5. (A). Initial random left pre-and post-synapses weights of the network. (B) Pre- and post-synapses weights after it modified with STDP. The two curves in Fig. B show clearly the new correlation between the pre- and postsynapses weights.
Avg. ∆tpre-syn.= 3.2
Avg. ∆tpost-syn.= 5.5
Pre-syn. Weight = 4.3
Post-syn. Weight = 1.7
(Initial random pre-and post-synapses weight) ∆tpre-post = 3.2 – 5.5 = -2.3 (I) Avg. ∆tpre-syn.= 3.2
Pre-syn. Weight = 4.3
Avg. ∆tpost-syn.= 3.4
Post-syn. Weight = 3.8
(Modified post-synapses weight) ∆tpre-post = 3.2 – 3.4 = -0.2 (II) Fig. 6. (I) An Example of the firing time of pre and postsynaptic neuron taken from figure 5A. (II) An Example of the firing time of pre and postsynaptic neuron taken from figure 5B.
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B. Experiment (2) 1) Experimental Setting: The purpose of this experiment is to prove that one hidden-layer neuron in the network of spiking neuron having STDP can be remarkably sufficient to generate autonomous behavior in a Khepera robot. In this experiment, weights between the nodes in the network were selected randomly as shown in Fig. 7B. The Khepera started to move in the environment with fixed speed from a position shown in Fig. 7A. Whenever any of Khepera’s sensors were activated, the sensor’s neuron started to emit spikes, and then the program calculated ∆t from the activated pre- and post-synapses and adjusted the postsynaptic weight to drive ∆t to meet the short latency region. L.sensor
L.sensor
W=2 ∆tpre = 5.33
R.M = -18.3
Left side ∆t = -1.34
(A)
(B) L.sensor
R.sensor
W=7 ∆tpost = #
W=2 ∆tpre = 5.33
W=6.7 ∆tpost = 3
W=2.2 ∆tpre = 6.33 L.M = -5
W=6.7
L.motor
W=6.7 ∆tpost = 3
L.M = -8.3
W=7
W=1.7
W=7 ∆tpost = #
W=1.7 ∆tpre = 6.67
R.sensor W=2
R.sensor
R.M = -18.3 Left side ∆t = -1
R.motor
(C)
(D) L.sensor
(A)
R.sensor
(B)
2) Experimental Results and Analysis: Figure 8 shows the number of steps of Khepera trajectories and synapse weight changes from the initial time of the robot movement until the robot performed the task. Figure 8A shows the robot when its left sensor was activated. At that time the value of ∆t and Khepera’s motors were calculated using equations (6) and (7) as shown in Fig. 8B. From the figure, ∆t is not in the region and both of the motors give negative values. Fig. 8C & 8D show the position of the Khepera, and the new synapses and motors values after two time steps of synapse adaptations in the network. It is clear that only the left postsynaptic is changed while the right one is not affected. Figures 8E& 8F show the position of the Khepera and the final left postsynaptic weight in the network. The program drove ∆t to reach the short latency region and the correlation between pre and postsynaptic weights were built. In addition, the left motor value was adjusted to always be positive whenever the left sensor was activated. Figure 9 shows the final synapse weights between the nodes in the network after the post-synapses were modified with STDP. Figure 9A shows the network output and the Khepera trajectory whenever the left sensor was activated, i.e. the left motor output is a positive value while the right motor output is negative, allowing Khepera to avoid the left obstacles by turning to the right. On the other hand, Figure 9B shows the synapse values and the network output whenever the right sensor was activated, which will give a positive value to the right motor and a negative value to the left motor.
W=7 ∆tpost = #
W=2 ∆tpre = 5.33
Fig. 7. (A) The initial position of the Khepera. (B). Schematic drawing of the network clearly shows the initial synapses weights. Note that for simplicity, in next figures, we removed the presynaptic modulation connection.
W=6.7 ∆tpost = 3
W=3.1 ∆tpre = 5.33 L.M = 5
R.M = -18.3 Left side ∆t = 0
(E)
(F)
Fig. 8. Khepera’ trajectories (left) and synapses modification steps (right). L.sensor
R.sensor
W=7 ∆tpost = #
W=2 ∆tpre = 5.33
W=5.8 ∆tpost = 3.67
W=3.1 ∆tpre = 5.33 L.M = 5
R.M = -2
(A)
L.sensor
R.sensor
W=7 ∆tpost = 1.67
W=2 ∆tpre = # W=3.1 ∆tpre = 2.67
W=5.8 ∆tpost = 1.67
L.M = -5
R.M = 5
(B)
Fig. 9. The final synapses’ weights and motors’ values. (A) Whenever left sensor was activated the left motor got a value larger than the right motor. The simulator shows the movement of the Khepera. (B) Whenever the right sensor was activated the Khepera turned to the left by giving the right motor a value larger than the left motor.
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VIII. CONCLUSION In this study we describe an autonomous mobile robot whose sensory-motor connection were made by a threelayered spiking neural network (SNN) with only one hiddenlayer neuron that makes synaptic contacts on motor neurons with synapses having spike timing-dependent plasticity (STDP) and presynaptic modulation, and we analyzed the roles of STDP for the autonomous behavior in environment. The experimental results show the effectiveness of our method: the initial random network’s weights were gradually modified using Hebbian learning based on STDP and the robot acquired the desired behavior, i.e., navigates successfully and avoids obstacles in an environment. Building correlation between pre and postsynaptic weights using STDP drives the postsynaptic neuron into a balance “positively” with the presynaptic neuron in the same pair and “negatively” with presynaptic neuron in the opposite pair. Therefore, one hidden-layer neuron in SNN is remarkably sufficient under STDP to control the behavior of the Khepera robot. REFERENCES [1]. W. Maass, “Networks of Spiking Neural Network, the Third Generation of Neural Networks Models”, Neural Networks, vol. 10, pp. 1659-1671, 1997. [2]. S. Becker and M. Plumbley, “Unsupervised Neural Network Learning Procedures for Feature Extraction and Classification”, J. Applied Intelligence, vol. 6, pp.1-21, 1996. [3]. W. Gerstner, Spiking Neurons, In Maass, W. & Bishop, C. M. (eds.), Pulsed Neural Networks, MIT-press, 1999. [4]. J. Vreeken, “Spiking Neural Networks, an Introduction”, Technical report, UU-CS-2003-008, Institute for Information and Computing Sciences, Utrecht University, 2003. [5]. D.O. Hebb, The organization of behavior, Wiley, New York, 1949. [6]. L.F. Abbott and K.L. Blum, “Functional Significance of Long-term Potentiation for Sequence Learning and Prediction”, Cerebral Cortex, vol. 6, pp. 406-416, 1996. [7]. Yao, H. & Dan, Y., “Stimulus Timing-Dependent Plasticity in Cortical Processing of Orientation”, Neuron, vol. 32, pp. 315-323, 2001. [8]. R.P.N. Rao and T.J. Sejnowski, “Predictive Coding, Cortical Feedback, and Spike Timing Dependent Plasticity”, In R.P.N, Rao, B.A. Olshauser and M.S. Lewicki, (eds.), Probabilistic Models of the Brain, pp. 297–315, MIT Press, Cambridge, 2002. [9]. S. Song, K.D. Miller and L.F. Abbott, “Competitive Hebbian Learning through Spike-Timing Dependent Synaptic Plasticity”, Nature Neuroscience, vol. 3, pp. 919-926, 2000. [10]. R. Gutig, R. Aharonov, S. Rotter and H. Sompolinsky, “Learning Input Correlations Through Non-linear Temporally Asymmetric Hebbian Plasticity”, Neuroscience, vol. 23, pp.3697-3714, 2003. [11]. D. Floreano and C. Mattiussi, “Evolution of Spiking Neural Controllers for Autonomous Vision-Based Robots”, in Proc. Lecture Notes in Computer Science (LNCS) 2217, Tokyo, Japan, 2001, pp. 38-61. [12]. F. Alnajjar and K. Murase, “Use-dependent Synaptic Connection Modification in SNN Generating Autonomous Behavior in a Khepera Mobile Robot.”, in Proc. Int. conf. Robotics, Automation and mechatronics(RAM), Bangkok, Thailand, 2006, pp. 34-39. [13]. W. Gerstner and W.M. Kistler, Spiking Neuron Models, Cambridge University Press, 2002. [14]. Abbott, L. F. and Nelson, S. B. “Synaptic Plasticity: taming the beast”, Nature Neuroscience, vol. 3, pp. 1178-1183, 2000. [15]. H. Ikeda and K. Murase, “Glial Nitric Oxide-Mediated Long-Term Presynaptic Facilitation Revealed by Optical Imaging in Rat Spinal Dorsal Horn”, Neuroscience, vol. 24, no. 44, pp. 9888-9896, 2004. [16]. F. Mondada, E. Franzi, and P. lenne, “Mobile Robot Miniaturization: A tool for investigation in control algorithms”, in Proc. 3rd Int. conf. Symposium on Experimental Robotics, Kyoto, Japan, 1993, pp. 501-513.
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