-Locomotion is most of the time considered to be the result of top-down control commands produced by the nervous system in response to inputs received via sensory organs from the environment. Locomotion may arise alternatively when attracting states are stabilized in the combined dynamical space made up by the brain, the body and the environment. Cognition is embodied in this case within the sensorimotor loop, viz self-organized. Using a physics simulation environment we show that self-organized locomotion may result in complex phase spaces which include limit cycle corresponding to regular movements and both strong and partially predictable chaos describing explorative behavior.
We used the LPZRobots physics simulation environment [
i R denotes here the (rescaled) radius of the spherical robot and y(xi) = 1=(1 + exp( xi)) the firing rate of the controlling neuron. The membrane potential xi is determined via x_ i = xi + w0 2R x(a) + R i z0 X uj 'j y(xj ) j6=i (1) (2)
STSP does not induce any long-lasting traces (modifications of the synaptic strength), being hence a fully transient form of plasticity which tends to destablize fixpoint attractors.
The here considered robot moves only, as an entity
comprised of body and controlling neurons, when embedded
within the environment. Locomotion corresponds then to
selfstabilizing attractors in the combined phase space of the
controlling neural network, of the body and of the environmental
degrees of freedom it couples to [
Our robot may engage in a rich palette of regular motion patterns, as illustrated in Fig. 2, which are stable either for distinct sets of internal parameters, such as the bare synaptic weights w0 and z0, or simultaneously. Autonomous mode switching corresponding to a rollover from one to another basin of attraction occurs regularly in the latter case upon collision with either an external object, or with another robot. We note, importantly, that limit-cycles corresponding to regular motion, as shown in Fig. 2, are continuously degenerate with respect to the direction and/or to the center of propagation.
Explorative behavior arises when the synaptic weights w0
and z0 are set such that chaotic attractors are formed within the
by the relaxation constant , by the coupling w0 > 0 to
the proprio-sensory reading of xi(a), and with ( z0) < 0 by
the inhibition it receives from the other two neurons. The
interneural inhibition is dynamically modulated
presynaptically by a mechanism known as short-term synaptic plasticity
(STSP) [
U(y) u (uT;uy) '
T'
U (y) (u; y) = = 1 1 + (Umax
1)y Uumyax : Both the effective Ca2+ concentration u and the fraction of available vesicles ' of neurotransmitters relax to unity in the absence of a presynaptic input y, which, when present, tends to increase/decrease u ! Umax and ' ! 0 respectively.
We note that STSP is well known to change synaptic
efficiencies transiently by up-to fifty percent on time scales of
a few hundred milliseconds, as defined by Tu and T'. These
are also the time scales which are relevant for locomotion.
Fig. 1. The simulated robot contains three weights (red, green and blue)
moving along perpendicular rods within a movable sphere. The position of
the three weights is controlled respectively by a single neuron (see Eqs. (1)
and (2)). The small balls at the end of the respective rods are guides to the
eye. [video]
sensorimotor loop. We note that noise is absent for the
simulations shown in Fig. 3, with the seemingly random wandering
of the robot resulting exclusively from the chaotic nature of
the underlying attractor. Two types of chaotic attractors may
be stabilized in addition, denoted respectively as strong and
as partially predictable chaos [
Morphological computation [
The sphere robot does neither perform any form of knowledge acquisition with its brain consisting of only three neurons, nor does its ‘cognitive system’ dispose of higher-level internal drives or motivations. The explorative behavior observed in Figs. 3 and 4 can be explained on the contrary fully in terms of dynamical systems theory. Taking a philosophical perspective our simulated robots hence demonstrate that it is in general impossible for an external observer to deduce reliably the internal settings and motivations of an acting cognitive system.