Emergence of structured interactions: from a theoretical model to pragmatic robotics Arnaud Revel, Pierre Andry
To cite this version:
Arnaud Revel, Pierre Andry. Emergence of structured interactions: from a theoretical model to pragmatic robotics. Neural Networks, Elsevier, 2009, 22 (2), pp.116-125. hal-00417847
HAL Id: hal-00417847 https://hal.archives-ouvertes.fr/hal-00417847 Submitted on 17 Sep 2010
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Emergence of structured interactions : from a theoretical model to pragmatic robotics A.Revel 1, P.Andry 2 ETIS, CNRS UMR 8051, ENSEA, Univ Cergy-Pontoise, F-95000 Cergy-Pontoise, France
Abstract
In this article, we present two neural architectures for the control of socially interacting robots. Beginning with a theoretical model of interaction inspired by developmental psychology, biology and physics, we present two sub- cases of the model that can be interpreted as “turn-taking” and “synchrony” at the behavioral level. These neural architectures are both detailed and tested in simulation. A robotic experiment is even presented for the “turn-taking” case. We then discuss the interest of such behaviors for the development of further social abilities in robots.
Key words: Robotics, Social Interaction, Neural models, Development
1. Introduction nate the exchange”. To support this approach, we take inspiration from different scientific fields that With the democratisation of robotics, an impor- have to deal with interaction: dynamical systems, tant effort is made to make those “technological” developmental psychology, and neurobiology. machines more attractive for human beings. For in- In this paper, we will show how “synchrony” and “al- stance, a major focus of interest is put on the ex- ternation” can emerge from a simple neural model pressiveness and the aspect of the robots [11,27,34]: of interaction. Next, the underlying dynamical pa- the question is then how to design “cute” robots? rameters of the model will be applied to 2 robotic Nevertheless, if the aim of these approaches is to fa- controllers in order to increase their interactional cilitate human-machine communication, they often abilities. neglect to consider the dynamics of the interaction between two agents. In our opinion, intuitive com- munication (verbal or not) refers to the ability “to 2. Neural model of agents interaction take turn” in the interaction or to be “synchronized” with the other: in summary, to adapt its own dy- 2.1. Inspiration namics to the other’s behavior via the integration of the global dynamical exchange. In consequence, our 2.1.1. Developmental psychology aim is to design robotic controllers that can adapt Developmental psychology aims at understanding to the interaction, embed fundamental dynamical how the sensory, motor and cognitive capabilities properties, and make emerge communicative behav- of the individual evolve from birth. In the frame of iors such as the “ability to synchronize” and “alter- longitudinal studies applied to populations with se- lected developmental ages, researches detect and ob- 1 E-mail: [email protected] serve the emergence, or the disappearance of behav- 2 E-mail: [email protected] iors, allowing to formulate hypothesis and models of
Preprint submitted to Neural Networks 28 October 2008 the underlying sensori-motor and cognitive mecha- tively produce actions, while provided (or not) with nisms. In this discipline, a lot of issues are concerned the vision of similar actions being performed by with the progressive rise of communication among someone else. Results reveal that spontaneous phase babies and young children. synchrony (i.e. in-phase co-ordinated behavior) be- It has been described that human communica- tween two people emerges as soon as they exchange tion can adopt several modalities [47]: language, par- visual information, even if they are not explicitly in- alanguage and kinesic. Paralanguage are the non- structed to co-ordinate with each other. verbal voice and sounds which can be emitted. Ki- In a similar experiment, [26] tries to identify nesic stands for the “body-language” (facial expres- whether an interpersonal motor co-ordination sions, gaze, gestures, postures, head and body move- emerges between two participants when they in- ments, haptics and proxemics). Obviously, nonver- tentionally tried to not co-ordinate their move- bal communication is anterior to language in devel- ments between each other. The goal of the first opment. two situations was for participants to intentionally As our goal is to tackle intuitive communication, co-ordinate or not co-ordinate their movements and as our robots are not verbal, we are essentially between each other. The results revealed in the interested in that latter kind of communication. “not co-ordinate” condition the emergence of an At a behavioral point of view, it has been shown unintended co-ordination in the frequency domain. that in our daily activities the social context is for- What is however interesting to notice is that the re- matted by physical interactions. One example is, for sults also reveal the presence of individual intrinsic instance, the emergence of rhythmic applause in a motor properties (motor signature) in the 2 condi- crowd [38]. Another interesting example is the fact tions. These results indicate that, when there was that we can adopt a similar posture [9] when inter- information sharing, participants could not avoid acting with another. More precociously, adults and (unintentionally) co-ordinating with someone. neonate participate in a social interaction via head This is consistent with the results found by movements [43] what seems to suggest that very [55] in EEG. In a specially designed dual electro- early in development, humans are equipped to deal encephalogram system, pairs of participants execut- with social interaction. ing self-paced rhythmic finger movements with and In particular, Neo-natal imitation (from [32,33,29]), without vision of each other’s actions were continu- has been widely studied for its presumed impli- ously monitored and recorded. After analysis, a pair cation in communication. In neo-natal imitation of oscillatory components (phi1 and phi2) located the observation by the baby of facial expressions above the right centro-parietal cortex distinguished performed by the experimenter gives rise to basic effective from ineffective co-ordination: increase of imitation (tongue protrusion, eye blinking, vocal- phi1 favored independent behavior and increase of izations). As experiments has been done with babies phi2 favored co-ordinated behavior. The author’s aged of a few minutes old, this kind of imitation is hypothesis is that the phi complex rejects the influ- not supposed to be learned. This phenomenon asks ence of the other on a person’s ongoing behavior, an important question concerning the function of with phi1 expressing the inhibition of the human such a low-level imitative behavior. Answering this mirror neuron system and phi2 its enhancement. question, a few developmental psychologists have In this section, we have shown that synchroniza- emphasized that imitation has not only a learn- tion behaviors exist very early in human and per- ing function but also a communication one [36,12]. sists in time. This can suggest that they could have They have suggested that neo-natal imitation is a early and perennial effect on the structure of inter- a pre-linguistic mode of communication that does personal non-verbal communication. not require any interaction protocol but can be the basis of subsequent higher level communication and social abilities. More precisely, Nadel has empha- 2.1.2. Synchrony in biological systems sized the fact that synchrony and “turn-taking” are We have seen that behavioral synchronisation ex- fundamental for communication [37]. ists in social contexts. The question is then: are they Several recent experiments seem to validate this individual mechanisms that can help synchrony be- point of view. Oullier, for instance [41], has proposed tween humans ? a simple experimental paradigm in which pairs of In fact, many biological activities in human are participants facing each other are required to ac- naturally synchronized. For instance, it has been
2 shown that heartbeat, respiration and locomotion in lators embedded in each robot models an internal humans walking or running on a treadmill are syn- propensity to “interact” with the other robot. chronized [40]. The oscillator is supposed to be connected both The role of circadian rhythm has also been proved to sensations and actions: perception modulates this to have effect on the oscillation of hormone secre- oscillator while this latter modifies the actions (in- tion, core body temperature and lymphocyte num- hibition, modulation, activation...). ber [3,35,48]. It has also been shown that we are well equipped 2.2. Formal model of the oscillator to satisfy auto-synchrony (for instance with fingers [22]). What is interesting to note is that the result The oscillator we use is made of 2 neurons (u and is different in function of speed of motion: we are in v) inhibiting each other proportionally to the pa- phase for slow motion, and anti-phase for increasing rameter β. The equations are given in 1 (see also speed. fig.1). When regarding underlying neural circuits, there is also evidence that synchrony exists within brain − · u(n +1) = f (u(n) β v(n)) areas and between them, and is even crucial for (1) · information processing [56,49]. For instance, syn- v(n +1) = f (v(n)+ β u(n)) chronization in the visual cortex seems responsible With f(.) being the activating function of the neu- of binding of related visual features [20,56]. Syn- rons. chronization of oscillatory activity in the sensori- In a first approximation, we consider the function motor cortex serve also for the integration and co- f(.) as the identity function. Equation 1 becomes: ordination of information underlying motor control [31]. Simultaneous spiking in neural population has − · u(n +1) = u(n) β v(n) also been observed in olfaction [51] and touch [50]. (2) In certain cases, synchronization of neural popu- v(n +1) = v(n)+ β · u(n) lations leads to pathological disorders. For instance, Parkinson tremors seem to be due to synchro- By approximating the derivative of a function by: f ′ x ≃ f x − f x nization between the cortex and the basal ganglia ( + 1) ( + 1) ( ). Equation 2 turns into: [15,54]. ′ u (t) = −β · v(t) (3) ′ v (t) = β · u(t) 2.1.3. Dynamical systems In 1665, C.Huygens described how two clocks By deriving the first line of equation and simpli- anti-synchronize after a while by mutually inter- fying equation 3, we get: acting together [23]. Since this discovery, the self- ′′ 2 · organization of synchronization phenomenon has u (t)+ β u(t)=0 been widely studied by nonlinear dynamics [21,39]. Solutions of such an equation have the following In physics, relaxation oscillators have been modeled form: by Van Der Pol [46]. More recently, the synchroniza- u(t)= A · eiβt + cnt tion mechanisms has been deeply studied between 2 or several oscillators, even in the case of chaotic That is an oscillatory function with frequency ν = β systems [45,42,53]. This has lead to a better under- 2π standing and modeling of the mechanisms involved during mutual interaction. 2.2.1. Linking 2 oscillators together As far as we are concerned, this discipline pro- When considering the interaction between the 2 vides a theoretical framework for the study of inter- agents, it can simply be obtained by linking the out- acting/coupled social systems. put of neuron u1 of agent 1 (considered as the “ac- Taking inspiration from non-linear dynamics we tion” performed by agent 1) to the input of neuron propose that basic dynamical social properties may u2 of agent 2 (considered as the “sensation” of agent emerge from the dynamical perception/action inter- 2) and reversely. action of two identical oscillatory systems in inter- In that case, we want to study the form of the action. The main idea is that the value of the oscil- interaction between the 2 systems.
3 2 2 +1 ′ (ω − β ) A (t) = −i · ( 1 ) · A (t)+ ε · A 1 ω 1 2 . N 2 2 (9) 1 ′ (ω − β ) A (t) = −i · ( 2 ) · A (t)+ ε · A 2 ω 2 1 −β +β 2 2 ≃ ≃ (ω −β ) (ω−β)(ω+β) ≃ · − . N As ω β1 β2, = 2 (ω 2 ω ω β1+β2 β1−β2 β). If we set ω = 2 and χ = 2 , equation 9 +1 becomes: ′ · Fig. 1. The oscillator is made of 2 neurons, u and v, with A1(t)= 2iχA1 + ε A2 β (10) a self-connection weighted to 1. A link whose weight is + ′ − · connects v to u, and a link whose weight is −β connects u A2(t) = 2iχA2 + ε A1 to v. As A is a complex amplitude, we can turn it into iΦ Formally, this interaction can be modeled by A = R · e . transforming equation 3 into: After substitution in 10 and simplifications, we get: u′ t −β · v t I · u t 1/2( ) = 1/2 1/2( )+ 21/12 2/1( ) ′ ′ · i(Φ2−Φ1) (4) R (t)+ R1iΦ = 2iχR1 + ε R2e ′ 1 1 (11) v (t) = β · u (t) ′ ′ − 1/2 1/2 1/2 R (t)+ R iΦ = 2iχR + ε · R ei(Φ1 Φ2) 2 2 2 2 1 With I21 the value of the inhibition between agent Grouping real and imaginary terms and consider- I 2 and 1 and 12 the value of the inhibition between ing Ψ = Φ2 − Φ1 leads to: agent 1 and 2. R By deriving the first line of those equations and ′ ′ · 2 R1(t) = εR2 cosΨ Φ1 = 2χ + ε sinΨ by reducing with the second line, we get: R1 (12) ′ ′ R R (t) = εR cosΨ Φ = 2χ − ε · 1 sinΨ 2 1 2 R u′′ t −β2 · u t I · u′ t 2 1 ( ) = 1 1( )+ 21 2( ) ′ ′ ′ (5) As Ψ = Φ − Φ , we finally reach the following ′′ − 2 · · ′ 2 1 u2 (t) = β2 u2(t)+ I12 u1(t) equation: ′ Let us consider to simplify that: I12 = I21 = ε a R (t) = εR cosΨ 1 2 weak value of the interaction. The previous equation ′ R (t) = εR cosΨ becomes: 2 1 (13) ′ R R Ψ = −ε · ( 2 − 1 )sinΨ ′′ − 2 · · ′ u (t) = β u1(t)+ ε u (t) R1 R2 1 1 2 (6) ′′ − 2 · · ′ The equation in Ψ has different behavior accord- u2 (t) = β2 u2(t)+ ε u1(t) ing to the sign of ε.
2.2.1.1. Oscillatory solutions We consider here 2.2.1.2. If ε > 0 (positive coupling) The stable that oscillatory solutions can exists. For that pur- solution stands for Ψ = 0=”synchronous case”. R2 − R1 1 pose, we consider the variables (A1 and A2 being Indeed, in that case, supposing that R1 R2 = τ : complex): ′ Ψ ≃−εΨ (14) − ε iωt iωt Whose solution is Ψ = e τ t. x1 = u1 = A1e x2 = u2 = A2e (7) ε> ′ iωt ′ iωt The only stable solution stands for 0. y1 = u1 = iωA1e y2 = u2 = iωA2e 2.2.1.3. If ε < 0 (negative coupling) The stable Considering that 7 can be rewritten as: solution stands for Ψ = π=”phase opposition”.
′′ 2 2 2 ′ Indeed, in that case, supposing that Ψ = π + φ: u (t) − ω = (ω − β ) · u1(t)+ ε · u (t) 1 1 2 (8) ′ ′ ′′ ′ Ψ = φ = −ε sin(π + φ)= ε sin φ ≃ εφ (15) u t − ω2 ω2 − β2 · u t ε · u t 2 ( ) = ( 2 ) 2( )+ 1( ) ε Whose solution is φ = e τ t with a stable solution We can deduce that: only if ε< 0.
4 2.2.2. Interpretation of the model for social Agent 1 Agent 2 interaction Action 1 Environment Action 2 Given this theoretical framework, we can give an . . interpretation of this model at the level of social in- teraction. The oscillator could be an abstraction of Oscillator 1 Oscillator 2 the agent’s motivation for action: if the oscillator is “up”, the system tends to produce actions; if the os- α β cillator is “down”, it does not produce actions. The Noise Noise inhibition coupling is then a model of the interaction inhib inhib between the agents which is supposed to be “per- . . fect” (without loose of energy). Sensation 1 Sensation 2
2.2.2.1. Phase opposition (ε< 0 ) In this case, we Fig. 2. Architecture of the two agents influencing each other. are in the situation where the two oscillators recipro- Each agent is driven by an internal oscillator and produces actions depending on this oscillator. A noise due to the cally alternate in motor production: when an agent environment and the hardware devices appears on the signal acts the other stops any action until this situation between the two oscillators. alternates. It is interesting to notice that in such a case, the alternation itself is an emergent property 3.1.1. Simulation results of coupling. Interpreting this situation as a social in- To test the conditions and the limits of the anti- teraction, the model suggests that a “turn-taking” synchronization between the two oscillators in inter- behaviour, as demonstrated by young children, may action we varied the different parameters in simula- naturally emerge from the dynamical interaction be- tion: the oscillator 1 was kept unchanged (β1 =0.2) tween 2 agents (cf. section 2.1.1). while the oscillator 2 was varied between 0.02 and 0.4 with a step of 0.01. The inhibition between the two systems was also varied between 0. and 0.1 with 2.2.2.2. Phase synchronisation (ε > 0 ) In this a step of 0.02. We observe in figure 3 Left. which − case, the interaction allows to reciprocally exchange plots the difference of frequencies f1 f2 that they energy which results in the synchronization of oscil- get completely synchronized (∆f = 0) in the vicin- lators: both agents performs the same action at the ity of the mean frequency (this is called “frequency same time. It typically corresponds to a minimal in- locking”). In addition, as the inhibition increases, teraction, where each agent is able to produce its the locking region gets wider and wider. The regions motor repertoire, and at the same time to modulate of synchronization observed in simulation are anal- its production in order to adapt to the other’s one. In ogous to the “phase locking domain” described as other words, solving the trade-off between producing “tongues” by Arnold: when two periodic oscillators actions of self and observing those of others, as ob- are coupled together there are parameter regions served among young children naturally performing called “Arnold tongues” where they mode lock and immediate and spontaneous imitations in the frame their motion is periodic with a common frequency of pre-verbal communication (cf. section 2.1.1). [2]. In the following section, we will consider those two Concurrently with “frequency locking”, it can sub-cases in the context of robotics control architec- be observed (an example is given in figure 3-right, tures. which plots u1(n) in function of u2(n) with f1 = f2 and I =0.5) that both oscillators converge to a con- stant phase shift of π. That precisely corresponds to 3. Designing an interactive robotic controller anti-synchronization. This phenomenon is described as “phase locking of periodic oscillators” [52]. 3.1. The case of negative coupling: turn taking 3.1.2. Robotic implementation In this section, we explore the idea that the formal To transpose the conceptual model to a robotics model developed above can make a kind of “turn- platform, the way the oscillators and their interac- taking” behaviour emerge from the dynamical inter- tion could be embodied in a concrete robot should action between 2 agents (see figure 2). be considered. In particular, the reciprocal influ-
5 0.20 1.0 movements are. The vision system allows to detect
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-60 Fig. 3. a) Difference of frequencies f1 − f2 for oscillators 1 -80 and 2 in function of f2 (f2 ∈ [0.01, 0.4])and I the mutual 0 1 2 3 4 5 6 7 inhibition (I ∈ [0, 0.1]) between oscillators (f1 is fixed to 0.2). b) Lissajou plotting of u1(n) in function of u2(n) for the first 500 steps of the simulation. Fig. 5. Left : Difference of frequencies in function of the reciprocal inhibition. Right : Time plot of agent1 and agent2 ence between the two oscillators is then mediated activities. Both internal oscillators rapidly anti-synchronize by the environment. In the robotic platform ADRI- thanks to the perception/action interaction. ANA (ADaptable Robotics for Interaction ANAly- sis – see [28] for details), dedicated to the study of We have first tested this architecture and setup interaction and communication features, the detec- with two robots having the same oscillator’s pa- tion of movements and a moving arm are used to rameter (β1 = β2). After a transition period, the interact (see fig.4). two behaviors of the two robots stabilise in anti- synchronization: when one robot moves the other stops and conversely, the two robots take turns. This is shown by figure 5 Left. where both internal os- cillators rapidly anti-synchronize thanks to the per- ception/action interaction (this plot is equivalent to the Lissajou representation mention above, along the time axis). As in simulation, we have tested the interaction between the 2 agents for several values of f2 and I. 32 experiments were conducted with the robots in order to test the following set of parameters : β2 the frequency ratio β1 took the values 1, 1.2, 1.4, 1.6, 1.8, 2, 4 and 6, (only β2 varies) the reciprocal inhibition I took the values 0, 0.1, 0.3 and 0.5. The difference of frequencies is plotted in figure 5-right. Fig. 4. In the present experiment ADRIANA is customized As in simulation, we can see wide region of frequency with a single arm for each robot and a webcam which enables the robot to see the other robot but not to see its own locking which varies with I. arm. The two robots move their arms according to their own dynamic. They influence each other by seeing the movement of the other robot. 3.1.3. Summary This emerging dynamics have been shown to tol- Each robot has one arm and the camera is put erate a wide range of parameters. This robustness in front of the arm so as each robot faces the other allows the 2 systems to remain in interaction even robot in order to perceive its arm movements. The if the conditions are not ideal: the agent’s dynam- architecture controlling each robot is made of two ics and environment may be modified along time. It parts: the oscillator described in the previous section suggests that the emergence of turn taking between which controls the arm movements ; the image pro- two systems is a very robust dynamical attractor cessing system which computes the inhibitory signal. that can be trusted in order to develop further social The frequency of the arm movements directly de- and communicative functionalities. pends on the activation of the oscillator: the higher This view fits physics results on chaotic oscillators the activation of u1 is, the more frequent the arm which support the idea that the more chaotic oscil-
6 lators are synchronized, the more information they sequence. Of course, a reliable demonstration of the exchange ([7]). learned sequence requires that no input is given con- currently to the system (the experimenter, or the learner stops stimulating the robot), otherwise the 3.2. The case of positive coupling: synchrony system will be confused between selecting an action corresponding to its own prediction (the response to In the previous section, we have seen the effects its own predictions) or to its inputs (the response to of adding an oscillator, as a component of the ar- the other’s stimulations). chitecture, that modulates the perceptions and the Agent 1 Agent 2 action of the agents. In the previous robotic exper- Action 1 Environment Action 2 iment, the perceptions and the actions of the ar- . . chitecture are limited to a coding of the perception (amount of movement perceived) and of the action (speed of the movement of the arm) expressed in one dimension. Nevertheless, if we take into account Noise Noise more elaborated control architectures such as the Sequence Sequence one that apply to complex robotic systems, it can Learning and Learning and reproduction reproduction appear that adding explicitly one internal oscilla- . . tor is not so simple, because the information flow is Sensation 1 Sensation 2 of higher dimension, and because these systems are often able to switch from a particular dynamics to Fig. 6. Architecture of the two agents influencing each other. another (for example systems that have a particular Each agent is now able to learn and reproduce a sequence dynamics during a learning phase, and a second one of actions. This “learn and reproduce” mechanism induces during the reproduction phase). an internal dynamics that will drive the behavior of each agent. If we suppose that an agent learns a cyclic sequence But interestingly, the dynamics of such systems is (for example ABCDA), it will then reproduce the sequence often already rich enough to provide oscillating sig- in loop, and the looping of the prediction of the sensori-mo- nal without the use of any explicit ad-hoc oscillators. tor transitions will act as an endogenous multi-stable oscil- Traditional works on motor control have for exam- lator. Interestingly, the first property that emerges from the ple shown the benefit of using oscillating primitives interaction of 2 systems is the synchronization of the sen- sori-motor behaviors. as the basis of the building of more complex behav- iors [8,25,24,13,10]. In previous works, we have pro- As a result, one could qualify such an architec- posed a NN architecture able to learn, predict and ture as “blind” when reproducing the sequence : in reproduce on-line one simple sequence of sensori- order to obtain a correct reproduction, the system motor events. Such a system has been used in navi- executes its predictions without “listening” to any gation (from [5] to [19]), planning [4,14], imitation other inputs. Therefore, an important question is with an eye-arm system [17,30], with different vari- how to obtain the emergence of interactive capabili- ations of the NN model used to categorize and pre- ties with any architecture having a possible conflict dict the sensori-motor flow. between perceptive and predictive inputs? From the On these systems, the behavior (i.e the sequence of theoretical model described in section 2.1.3, the idea sensori-motor actions) learned was most of the time is to start from the connection of two systems sim- cyclic, in order to obtain a system that provides con- ulating a robot-robot interaction (see Fig 6), and to tinuous demonstration of the learned sequence. For exploit for free the dynamics of the interaction. Our example a system that learned the sequence ABCD working hypothesis is that the predictive layer plays (each letter symbolizing a different sensory-motor the role of an endogenous oscillator, producing in- attractor) associated the transitions A-B, B-C, C-D, ternal energy (the predictions) for the motor out- D-A, in order to reproduce cyclically the sequence put, while the perceptive input plays the role of the ABCD. As a result, after an on-line learning period, modulation link allowing to add or subtract energy and once they are able to predict a sensory-motor to the motor layer and favour a global synchrony or event, our robots start to reproduce it in loop, driven anti-synchrony of both agents. Once again, the solu- by their step by step predictions [18,1]. This pro- tion is inspired by the C. Huyggens “entrainment” vides systems that start to reproduce the behavior phenomenon: our system’s perception is considered they have learned, once they are able to predict a as an energy playing the same role than the phys-
7 ical wave transmitted by the support in Huyggens Proprioceptive Pathway experiment. Motor In the next sections, we will describe more pre- Output cisely the details of the architecture , and the inter- action experiment. + 3.2.1. Energy based architecture Prediction Time Base Cells Cells The Perception-Action architecture studied is a Sensory−Motor PO TB Neural network with the following properties (see Pathway the architecture Fig. 7): A reflex pathway links actions (the Motor Output, MO) with perceptions (the Input group, Input). Perception is processed by the Input Group and sent d/dt TD to MO. If the resulting MO potential overcome a given Motor threshold it triggers a response (an ac- tion) from the MO neuron. Inputs The output of MO neurons is computed as fol- Transition Learning lows: and Prediction
P otMO = (W Input · ActInput) (16) Fig. 7. Details of the Perception-Action architecture. The i X i,j i,j j Input-Motor Output link represents a “reflex” pathway. A second loop pathway allow a Transition Learning and Pre- MO MO diction group to learn the sensori-motor flow under the form Act = fMO P ot (17) i,j i,j of a sequence of sensori-motor transition. As a result, after a learning phase and TLP is able to recall step by step the 1 if x(t) >θMO transition and to replay the learned sequence. fMO (x (t)) = (18) 0 otherwise To each neuron in MO corresponds a detector in TD (a simple derivator to detect new events), and This direct Input-MO connection, ensures simple to each neuron of MO corresponds one battery of reflex behaviors at the basis of sensori-motor explo- TB neurons firing with different time constants de- ration and task learning [16,44]. In order to exploit pending of j: this emerging “spontaneous” imitative behavior, our · architecture is composed of a second pathway con- n Θ · t if t