Bulcsú Sándor

The Sensorimotor Loop as a Dynamical System: How Regular Motion Primitives May Emerge from Self-Organized Limit Cycles

We investigate the sensorimotor loop of simple robots simulated within the LPZRobots environment from the point of view of dynamical systems theory. For a robot with a cylindrical shaped body and an actuator controlled by a single proprioceptual neuron we find various types of periodic motions in terms of stable limit cycles. These are self-organized in the sense, that the dynamics of the actuator kicks in only, for a certain range of parameters, when the barrel is already rolling, stopping otherwise. The stability of the resulting rolling motions terminates generally, as a function of the control parameters, at points where fold bifurcations of limit cycles occur. We find that several branches of motion types exist for the same parameters, in terms of the relative frequencies of the barrel and of the actuator, having each their respective basins of attractions in terms of initial conditions. For low drivings stable limit cycles describing periodic and drifting back-and-forth motions are found additionally. These modes allow to generate symmetry breaking explorative behavior purely by the timing of an otherwise neutral signal with respect to the cyclic back-and-forth motion of the robot.

Spring-block model for a single-lane highway traffic

A simple one-dimensional spring-block chain with asymmetric interactions is considered to model an idealized single-lane highway traffic. The main elements of the system are blocks (modeling cars), springs with unidirectional interactions (modeling distance-keeping interactions between neighbors), static and kinetic friction (modeling inertia of drivers and cars) and spatiotemporal disorder in the values of these friction forces (modeling differences in the driving attitudes). The traveling chain of cars correspond to the dragged spring-block system. Contrary to most of the studies in the field of highway traffic here we focus on a measure characteristic for one car in the row. Our statistical analysis for the spring-block chain predicts a non-trivial and rich complex behavior. As a function of the disorder level in the system a dynamic phase-transition is observed. For low disorder levels uncorrelated slidings of blocks are revealed while for high disorder levels correlated avalanches dominates.

How to test for partially predictable chaos

For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. This process of decorrelation can split into an initial exponential decrease and a subsequent diffusive process on the chaotic attractor causing the final loss of predictability. Both processes can be either of the same or of very different time scales. In the latter case the two trajectories linger within a finite but small distance (with respect to the overall extent of the attractor) for exceedingly long times and remain partially predictable. Standard tests for chaos widely use inter-orbital correlations as an indicator. However, testing partially predictable chaos yields mostly ambiguous results, as this type of chaos is characterized by attractors of fractally broadened braids. For a resolution we introduce a novel 0–1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories. This test robustly discriminates chaos, including partially predictable chaos, from laminar flow. Additionally using the finite time cross-correlation of pairs of initially close trajectories, we are able to identify laminar flow as well as strong and partially predictable chaos in a 0–1 manner solely from the properties of pairs of trajectories.

Closed-loop Robots Driven by Short-Term Synaptic Plasticity: Emergent Explorative vs. Limit-Cycle Locomotion

We examine the hypothesis, that short-term synaptic plasticity (STSP) may generate self-organized motor patterns. We simulated sphere-shaped autonomous robots, within the LPZRobots simulation package, containing three weights moving along orthogonal internal rods. The position of a weight is controlled by a single neuron receiving excitatory input from the sensor, measuring its actual position, and inhibitory inputs from the other two neurons. The inhibitory connections are transiently plastic, following physiologically inspired STSP-rules. We find that a wide palette of motion patterns are generated through the interaction of STSP, robot, and environment (closed-loop configuration), including various forward meandering and circular motions, together with chaotic trajectories. The observed locomotion is robust with respect to additional interactions with obstacles. In the chaotic phase the robot is seemingly engaged in actively exploring its environment. We believe that our results constitute a concept of proof that transient synaptic plasticity, as described by STSP, may potentially be important for the generation of motor commands and for the emergence of complex locomotion patterns, adapting seamlessly also to unexpected environmental feedback. We observe spontaneous and collision induced mode switchings, finding in addition, that locomotion may follow transiently limit cycles which are otherwise unstable. Regular locomotion corresponds to stable limit cycles in the sensorimotor loop, which may be characterized in turn by arbitrary angles of propagation. This degeneracy is, in our analysis, one of the drivings for the chaotic wandering observed for selected parameter settings, which is induced by the smooth diffusion of the angle of propagation.

A versatile class of prototype dynamical systems for complex bifurcation cascades of limit cycles

A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf- and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance.

Chaos on the conveyor belt.

The dynamics of a spring-block train placed on a moving conveyor belt is investigated both by simple experiments and computer simulations. The first block is connected by a spring to an external static point and, due to the dragging effect of the belt, the blocks undergo complex stick-slip dynamics. A qualitative agreement with the experimental results can be achieved only by taking into account the spatial inhomogeneity of the friction force on the belts surface, modeled as noise. As a function of the velocity of the conveyor belt and the noise strength, the system exhibits complex, self-organized critical, sometimes chaotic, dynamics and phase transition-like behavior. Noise-induced chaos and intermittency is also observed. Simulations suggest that the maximum complexity of the dynamical states is achieved for a relatively small number of blocks (around five).

A spring-block analogy for the dynamics of stock indexes

A spring-block chain placed on a running conveyor belt is considered for modeling stylized facts observed in the dynamics of stock indexes. Individual stocks are modeled by the blocks, while the stock-stock correlations are introduced via simple elastic forces acting in the springs. The dragging effect of the moving belt corresponds to the expected economic growth. The spring-block system produces collective behavior and avalanche like phenomena, similar to the ones observed in stock markets. An artificial index is defined for the spring-block chain, and its dynamics is compared with the one measured for the Dow Jones Industrial Average. For certain parameter regions the model reproduces qualitatively well the dynamics of the logarithmic index, the logarithmic returns, the distribution of the logarithmic returns, the avalanche-size distribution and the distribution of the investment horizons. A noticeable success of the model is that it is able to account for the gain-loss asymmetry observed in the inverse statistics. Our approach has mainly a pedagogical value, bridging between a complex socio-economic phenomena and a basic (mechanical) model in physics.

Attractor metadynamics in terms of target points in slow-fast systems: adiabatic versus symmetry protected flow in a recurrent neural network

In dynamical systems with distinct time scales the time evolution in phase space may be influenced strongly by the fixed points of the fast subsystem. Orbits then typically follow these points, performing in addition rapid transitions between distinct branches on the time scale of the fast variables. As the branches guide the dynamics of a system along the manifold of former fixed points, they are considered transiently attracting states and the intermittent transitions between branchescorrespond to state switching within transient-state dynamics. A full characterization of the set of former fixed points, the critical manifold, tends to be difficult in high-dimensional dynamical systems such as large neural networks. Here we point out that an easily computable subset of the critical manifold, the set of target points, can be used as a reference for the investigation of high-dimensional slow-fast systems. The set of target points corresponds in this context to the adiabatic projection of a given orbit to the critical manifold. Applying our framework to a simple recurrent neural network, we find that the scaling relation of the Euclidean distance between the trajectory and its target points with the control parameter of the slow time scale allows to distinguish an adiabatic regime from a state that is effectively independent from target points.

Kick control: using the attracting states arising within the sensorimotor loop of self-organized robots as motor primitives

Self-organized robots may develop attracting states within the sensorimotor loop, that is within the phase space of neural activity, body and environmental variables. Fixpoints, limit cycles and chaotic attractors correspond in this setting to a non-moving robot, to directed, and to irregular locomotion respectively. Short higher-order control commands may hence be used to kick the system from one self-organized attractor robustly into the basin of attraction of a different attractor, a concept termed here as kick control. The individual sensorimotor states serve in this context as highly compliant motor primitives. We study different implementations of kick control for the case of simulated and real-world wheeled robots, for which the dynamics of the distinct wheels is generated independently by local feedback loops. The feedback loops are mediated by rate-encoding neurons disposing exclusively of propriosensoric inputs in terms of projections of the actual rotational angle of the wheel. The changes of the neural activity are then transmitted into a rotational motion by a simulated transmission rod akin to the transmission rods used for steam locomotives. We find that the self-organized attractor landscape may be morphed both by higher-level control signals, in the spirit of kick control, and by interacting with the environment. Bumping against a wall destroys the limit cycle corresponding to forward motion, with the consequence that the dynamical variables are then attracted in phase space by the limit cycle corresponding to backward moving. The robot, which does not dispose of any distance or contact sensors, hence reverses direction autonomously.

A self-organized one-neuron controller for artificial life on wheels.

We study simulated animats in terms of wheeled robots with the most simple neural controller possible – a single neuron per actuator. The system is fully self-organized in the sense that the controlling neuron receives uniquely the actual angle of the wheel as an input. Non-trivial locomotion results in structured environments, with the robot determining autonomously the direction of movement (time-reversal symmetry is spontaneously broken). Our controller, which mimics the mechanism used to transmit power in steam locomotives, abstracts from the body plan of the animat, working without problems also in the presence of noise and for chains of individual two-wheeled cars. Being fully compliant our controller may be also used, in the spirit of morphological computation, as a basic unit for higher-level evolutionary algorithms.