We evaluated the contribution of sensorimotor coupling to all informational measures by comparing two experimental conditions, one in which sensorimotor coupling was undisturbed and one in which sensorimotor coupling was disrupted—we refer to these two conditions as “fov” and “rnd,” respectively. In condition fov, all sensory, neural, and motor dynamics unfolded without intervention in real time. In condition rnd, a previously recorded motor signal was substituted, resulting in motor activity that was not driven by actual real-time sensory inputs. This enforced dissociation between sensory and motor data was designed to disrupt sensorimotor coupling, while leaving intact the statistical patterns within both sensory and motor domains. Differences in informational measures between these two conditions can be attributed to the presence or absence of coupling between sensory and motor streams. Thus, condition rnd represents an interventional or perturbational approach designed to discern patterns of information flow caused by sensorimotor coupling.

Figure 2 shows maps of entropy, mutual information, integration, and complexity for data collected from array I_R in Roboto. If the sensorimotor interaction was undisturbed (condition fov, Figure 2A), we observed increased cumulative intensity of the color red near the center of the visual field (unpublished data), as well as decreased entropy and increased mutual information, integration, and complexity. All measures exhibited significantly weaker effects if sensorimotor interaction was disrupted (condition rnd, Figure 2B). Very similar informational patterns were observed for neural activation states of Sal, as well as for I_R, I_G, I_B, and Sal in Strider (unpublished data). These results were entirely consistent with those reported earlier for a different robotic pan-tilt platform tracking salient stimuli in color movies [13]. While both platforms shared similar active vision control architectures, their body morphologies were significantly different, as was the nature of their visual stimulation.

Figure 3 summarizes results obtained from an analysis of directed information flow using transfer entropy, performed on the same datasets used for the noncausal analyses shown in Figure 2. The introduction of variable time offsets between sensory (S) and motor (M) time-series data allowed us to plot causal relations between these variables across all time delays (Figure 3A1 and 3B1). When examining the relation between visual inputs (S, array I_R, Figure 3A1) and the amplitude of pan-tilt head movements (M) for condition fov, we found positive transfer entropy in the direction S → M at offsets of +1 and (with decreasing magnitude) for offsets bigger than +1. No transfer entropy was found for offsets less than or equal to zero. In the reverse direction, we found transfer entropy in the direction M → S when M preceded S by at least one time step (time offset = −1) with a falloff towards more negative offsets. For condition rnd, transfer entropy was diminished if not eliminated, in accordance with the experimentally introduced disruption of causal interactions between sensory and motor time series. Residual transfer entropy in the direction M → S persisted in condition rnd, as pan-tilt head movements continued to cause displacements of the visual scene, albeit decoupled from those of the red object. The analysis also revealed the presence (for all time offsets) of elevated transfer entropy near the edges of the stimulus object (“ring-like” structures in Figure 3A and 3B), indicating that discontinuities in the visual image (e.g., at object boundaries) produce state transitions that are more effective in driving changes in motor variables. Figure 3A2 shows plots of transfer entropy across all offsets for the center of array I_R. By comparing the transfer entropy at or near zero time offset to baseline values (z-score maps for transfer entropy; Figure 3A3), we also provided a statistical estimate of image regions that either exerted significant causal effects on motor states (S → M) or were causally affected by motor states (M → S). We found that only the surface representation of the red object caused head displacement, which in turn caused displacement of the entire visual scene (including background). Figure 3B utilized the activity pattern of the saliency map, a neural variable, as the sensory (S) time series. Similar patterns of causality were revealed, with peak transfer entropies that were equal to, if not greater in magnitude than, those obtained analyzing data from I_R.

Peak transfer entropies depended on a variety of factors, including some key parameters in the behavioral pattern. We varied the “jump frequency” i.e., the frequency with which the object in Roboto's hand was translated to a new randomly chosen position. This “jump” could not be predicted by the head system and acted like a large environmentally driven perturbation that elicited corrective action by the pan-tilt unit to maintain foveation. Peak transfer entropies declined as these jumps became less frequent; peak T(S → M) was 0.29, 0.28, 0.23, 0.12, 0.05 bits, for jump frequencies of 5, 10, 20, 50, and 100 time steps (n = 5 runs). No significant transfer entropy was measured if the object was not moved at all and always remained foveated. Transfer entropy was zero if no changes occurred. This result indicates that transfer entropy increases with the amount of environmental changes causing behavioral responses—even if the perception–action loop is unperturbed (as for condition fov).

Estimates of information structure (entropy, mutual information, integration, and complexity) as well as transfer entropy maps for sensory and motor variables were obtained from a second morphologically and behaviorally different robotic platform, the quadruped Strider. Distributions of information across the visual field were comparable to those obtained with Roboto (unpublished data), indicating increased levels of information structure near the visual fovea and for behaviorally salient feature maps. Summaries of spatial profiles (z-score maps) for transfer entropy obtained from Strider are displayed in Figure 4. While considerably more noisy than maps obtained from Roboto, profiles for T(S → M) and T(M → S) reveal similar causal relations between I_R, Sal, and pan-tilt amplitude, with peaks for transfer entropy near the center of the visual field. Plots of T(S → M) and T(M → S) between sensory maps and leg movement amplitudes display peaks that are laterally displaced, reflecting the efficacy of steering input relayed from laterally located visual targets via the head system to the legs. For example, visual targets on the right side resulted in decreased movement amplitudes of the legs on the right side of the body axis, a causal connection that is retrieved via estimation of transfer entropy between the appropriate sensorimotor variables.

To illustrate the effects of learning on patterns of information flow, we included a value system as part of the neural architecture of Strider (Figure 1C). Based on changes in reward and aversiveness, the value system was capable of modulating saliency factors that, in turn, were used to compute the activation profile of the saliency map controlling head movements. Close encounters with objects resulted in the deployment of a virtual taste sensor. Positive changes in this sensor's activation triggered reward (appetitive taste) and aversive (aversive taste) signals (Figure 1C). Reward and aversive signals, in turn, modulated saliency factors such that encounters of rewarding objects tended to increase the saliency factor for that object's color, while encounters of aversive objects had the opposite effect. If the saliency distribution of objects in the environment changed over time (e.g., previously rewarding objects became aversive, and vice versa), the system adapted through changes in the saliency factors. As saliency factors changed, different objects became capable of “capturing attention” and causing approach or foveation behavior. These changes in saliency and attention could be monitored by recording behavioral and neural data. Figure 5A shows sample traces of average activities of the color maps and the value system collected in the course of a representative experiment lasting 8,000 time steps (approximately 13 minutes of real time). Activations of color maps increased as objects were approached and peaked around the time of object encounter. Sensing of taste triggered a value signal (either rewarding or aversive) coupled with visual inputs relaying the object's color. The comparison of two time segments, before and after a switch in color-taste contingency was made, documents a switch in behavior from approach of red objects to approach of blue objects, as well as changed reward/aversiveness profiles. Figure 5B displays the time course of saliency factors during the same experiment as shown in Figure 5A. The reversal of reward/aversiveness at time step t = 3,000 is accompanied by a reversal of saliency factors for red and blue, due to the modulatory actions of the value system.

Changes also occurred in sensorimotor information flow, i.e., transfer entropy. Transfer entropy maps obtained from a representative experiment (matching Figure 5A and 5B) are shown in Figure 5C. Initially (from t = 1 to t = 3,000), red objects were rewarding, leading to an increase in the corresponding saliency factor η_R, accompanied by a strong flow of information from red visual sensors near the fovea to the pan-tilt head system. Peak transfer entropy from sensory to motor variables was T(S → M) = 0.449 bits, while the peak value in the reverse direction was T(M → S) = 0.447 bits. As reward was switched from red to blue objects (at t = 3,000), the saliency factors adjusted, with η_B becoming dominant around t = 4,000. This adaptive change was reflected in a decrease of information flow emanating from red sensors and an increase of information flow from blue sensors to pan-tilt motors with a peak value of T(S → M) = 0.412 bits. Peak values for transfer entropy from the neural saliency map to motors remained high throughout the experiment, as the map continually drove foveation behavior, even while its afferent connections adapted to changes in the environment.

Body shape, limb articulation, as well as the position and density of sensors on the body surface have a large impact on sensory and motor capabilities of an organism. For instance, theoretical and experimental studies demonstrate that the distribution of retinal cells (e.g., cones, rods, ganglion cells) impacts the coding and transmission of the retinal image to higher levels in the visual pathway [16–18]. To take a specific case, can the morphology of visual sensors affect visuo–motor information flow? We recall that the retina of most biological eyes is a variable resolution (space–variant) sensor: the mosaic formed by the photoreceptors (cones and rods) across the retinal surface is inhomogeneous, yielding a spatial resolution that varies across the visual field [19,20]. In primates, the density of cones (used for high acuity vision) is typically greatest in the center (fovea) and falls off with retinal eccentricity (angular distance from the center of gaze). This morphological arrangement simultaneously enables high acuity of some parts of the visual field (achieved by appropriate head/eye movements) and a wide field of view, without requiring an enormous number of sensing elements and processing resources.

Body morphology is shaped in the course of evolution and development and is hard to manipulate systematically in either animals or physical robots. Our approach was to bypass this experimental difficulty by using a simulated mobile robot (Madame). Of all possible implementations of visual sensors, in Madame we implemented variants of retinal morphologies with a “log-polar” distribution of photoreceptors (here, only cones) [21–23]. The log-polar geometry has been shown to model accurately the topographical (retino–cortical) mapping of retinal cells (cones or ganglion cells) to the geniculate body and the striate cortex (area V1) [16,24]. We define the mapping from the “Cartesian retina” (x,y) onto the “cortical” plane (u,v) as the following coordinate change: u(r,θ) = klog(r/a + 1), v(r, θ) = θ where k is a normalization constant, the parameter a determines the density distribution of the retinal cells, and polar coordinates (r,θ) are used to replace Cartesian ones (x,y) in the retina: r = θ = tan⁻¹(y/x). A possible implementation of this arrangement is shown in Figure 6A, where a constant number of photoreceptors (represented by crosses) is arranged so as to give rise to an increase of the cells' spacings with respect to the distance from the central point of the structure. The mapping “template” is composed of nonoverlapping ring sectors (receptive fields) formed by the intersection of rays originating at the center of the retina. The photoreceptors' activities are calculated as the average of the intensities of the photoreceptors within their receptive fields. The larger number of receptors in the foveal part of the retina leads to a visual magnification in the cortical plane (Figure 6B). Magnified areas correspond to receptors that are proportionally more important than others requiring more accurate information processing [25]. The visual magnification factor is defined as the derivative of the mapping k/(r + a), and has a magnitude inversely proportional to distance from the center of the retina. Note that the derivative is radially symmetric, and roughly inversely proportional to the retinal eccentricity, approximating the retinotopic structure of the cat, owl monkey, rhesus monkey, and human visual cortex [16].

Here, we not only show that morphology shapes information flow but we also provide a quantitative measure for the amount of flow (the behavior of the robot was qualitatively the same throughout all experiments). Figure 7 displays transfer entropy for values of the parameter a = 2^k(k ∈ [−5,8]) evaluated as the average over both a region of the visual field and multiple experimental runs. The transfer entropy was calculated by averaging the transfer entropy between every pixel of a central (and noncentral) 6 × 6 pixel patch (variable S) and the difference between angular speed of left and right wheel (variable M). The error bars in Figure 7 denote standard deviations calculated for five runs of 4,096 samples each. Invariably, T(M → S) was larger than T(S → M) (for both central and noncentral visual patches), i.e., motor variables (e.g., difference between left and right angular speed) were more effective in driving sensor variables (e.g., red color intensity map) than vice versa. A striking result is that the information flow T(M → S) and T(S → M) for a < 0.25 is larger than for a > 2, with a transition between the two regions characterized by a clear inflection in the profile. This inflection is most likely a function of the size of the object on the robot's retina. Such “apparent” (or relative) size is used to regulate the distance of the robot from the object: the closer the artificial creature is to the object, the larger the object looms in front of it, and the more the creature slows down. The “causal” effect is more evident for lower values of a because the visual magnification factor is larger. Clear numerical differences can be also seen in the standard deviation. As a result of the visual magnification effect and in accord with our intuitions, the standard deviation for larger values of a is lower. Notably, the standard deviation for T(M → S) for a = {0.125,0.25,0.5,1.0} is significantly larger than for T(S → M). By contrast, for a > 2 and a < 0.125, the standard deviations for the two conditions are of comparable magnitude. By increasing the visual area over which the transfer entropy was calculated (up to patches of 12 × 12 central pixels), we observed no significant change in the resulting plots (unpublished data). In the case of noncentral visual patches, the graph flattens and the inflection is less pronounced; also the standard deviations are smaller. Given that the visual magnification is inversely proportional to the distance from the center, peripheral areas of the retina are less effective in driving the motor variables, and, vice versa, the effect of movements is less pronounced in the periphery.

These results indicate that eye morphology can affect information structure (here, mutual information) and information flow (here, transfer entropy), and how such effects of morphology can be quantified. Our findings are consistent with the hypothesis that morphology has an effect on information measures capturing statistical interactions and dynamic dependencies between variables.

We used three morphologically and behaviorally different robotic platforms, a fixed miniature humanoid named Roboto (Figure 1A1), a mobile quadruped named Strider (Figure 1A2), and a simulated mobile robot with wheels named Madame (Figure 1A3).

Roboto. For the present experiments we used five of Roboto's 14 kinematic degrees of freedom (DOF), three in the left arm (shoulder, elbow, and wrist), and two in the head system (pan and tilt), which was equipped with a centrally mounted CCD camera. A red object (visual target) was connected to the tip of the arm's most distal link and the arm was moved in a preprogrammed pattern. Initially the arm and object were positioned directly in front of Roboto, in view of the CCD camera. Every ten time steps, the arm was abruptly moved to a randomly chosen new position (a “jump”), selected within a range of the workspace of the individual joints, resulting in a displacement of the object relative to the head. Motor actions of the head were under visual control (see below), and displacement of the visual target resulted in foveation, with the robot tracking the position of the object as the arm was moved (Figure 1B1).

Strider. While Roboto was mounted on a pedestal, Strider (Figure 1A2) was fully mobile and situated within a tub-like environmental enclosure (~1 m diameter) containing a number of stationary colored cubes. Two front-mounted infrared sensors were used for wall avoidance. Locomotion was generated by rhythmic movement of the four legs (12 DOF), using ipsilateral and contralateral phase coupling between the legs [53]. The head system contained two DOF (pan, tilt) and was similar in construction and identical in terms of neural control to that of Roboto (see below). Motor actions of the head system were under visual control, resulting in foveation of colored objects. The position of the head system was relayed to the locomotion controller to steer Strider by modulating the movement amplitudes of two of the legs. This amplitude modulation resulted in gradual orientation of the body axis towards the object, while fixation was maintained by the independent action of the head system (Figure 1B2). The resulting behavior was a series of approaches to colored objects, each lasting for about 20 time steps, with intermittent periods of searching for new targets while navigating through the environment. All experiments were carried out with 12 red and 12 blue objects, initially positioned at random throughout the environment.

Madame. The third robot was implemented in simulation. It consisted of a mobile two-wheeled platform (of length L) equipped with seven proximity sensors for obstacle avoidance and a pan-tilt camera unit. The pan and the tilt angles were constrained to vary in an angular interval of 60° relative to the robot's midline. The environment was a square arena bounded by blue walls (of length 40L) containing 20 red-colored floating spheres (of diameter 0.3L) placed in random locations. The elevation of the spheres was L and was affected by a small amount of Gaussian noise. The spheres were relocated to a new position (not too far from the previous position) every 300 time steps. Neural control of the head system was identical to that used in Roboto and Strider (see below). Similar to Strider, Madame's behavior consisted of a series of approaches to colored objects and foveations with intermittent periods of search while moving through the environment. Fixation to the objects was maintained by independent action of head and body (Figure 1B3 illustrates the sensorimotor links).

All three robots used an active vision system (Figure 1C) designed to direct attention—and thus processing resources—to particular locations in space according to their behavioral relevance or saliency, here encoded exclusively by color feature maps [13]. The design of the color and saliency system closely followed the model of Itti et al. [54]. The sampled raw visual images were first luminance-scaled according to standard formulae, and then used to compute color-opponent maps for red, green, blue, and yellow (R, G, B, and Y). Subsequently, an opponent threshold was applied, followed by a “winner-take-all” mechanism resulting in four color-intensity maps Col_RGBY(R), Col_RGBY(G), Col_RGBY(B), and Col_RGBY(Y), which recorded the pixel-wise thresholded intensity of the dominant colors R, G, B, and Y. A color-saliency map normalized to [0, 1] was created by linear summation of the individual intensity maps as Sal = η_RGBY · Col_RGBY, where η_RGBY = [η_R,η_G,η_B,η_Y] and Col_RGBY = [Col_RGBY(R),Col_RGBY(G),Col_RGBY(B),Col_RGBY(Y)], and by scaling to the global maximum. The saliency factors η_R, η_G, η_B, and η_Y encoded the relative saliency of each of the four color components. In the experiments with Roboto and Madame, η_R was set to one, and all other saliency factors were set to zero, which resulted in a strong preference of the active vision system for the color red. In the experiments with Strider, saliency factors were modified dependent upon experience (see below). Once derived, the color saliency map was “block-averaged” to yield a map Sal with lower spatial resolution, whose global maximum determined the spatial location to which eye/camera movements were directed. The spatial coordinates of the maximum were then transformed into servo motor commands relayed to the pan-tilt motors moving the camera and to the servo motors driving the legs (Strider) and wheels (Madame). Camera motion resulted in lateral image shifts and a repositioning of the saliency profile. For stationary objects, camera motion stabilized quickly to direct gaze toward the maximum of the saliency map (foveation).

In the experiments with Strider, the saliency factors were subject to experience-dependent plasticity. Plastic changes were under the control of a value system [55,56] capable of influencing the coupling between Col_RGBY and Sal in the presence of changes in innately salient sensory stimulation, analogous to biological neuromodulatory systems (e.g., [57]). Innately salient sensory inputs were modelled as “virtual taste,” sampled through a small virtual tastepad attached below the camera. Whenever the camera pointed downward, indicative of close approach and foveation of a target object, the virtual tastepad was activated. Taste inputs were either appetitive (T_AP) or aversive (T_AV), depending on the color of the object. Color-taste associations were under experimental control and varied between red-appetitive/blue-aversive and red-aversive/blue-appetitive. The level of appetitive and aversive taste input was transformed into a rewarding [rew(t)] or aversive neural signal [ave(t)], respectively, according to: with Φ(·) denoting a standard sigmoidal function used to scale T_AP and T_AV to the interval [0, 1]. The saliency factors η_RGBY were adjusted by means of the following equation:

P_RGBY corresponded to a binary representation of the activation of the color intensity map Col_RGBY in the center of the visual field (e.g., a red object generated P_RGBY = [1 0 0 0]). The incremental learning rate α was set to 0.2, and the decay rate δ = 0.0005, with η₀ = [0.1, 0.1, 0.1, 0.1] and |η_RGBY (t)| ≥ 0 at all times. At the beginning of every run, the saliency factors were initialized as η_RGBY(0) = [0.25,0.25,0.25,0.25]. During experience, positive changes in the taste input (i.e., the onset of rewarding or aversive sensation) generated phasic and graded reward/aversive signals that were used to increase (in the case of reward) or decrease (in the case of aversiveness) the saliency factors.

We note that the meaning of the term information differs depending on the particular context. Here, information is used in the Shannon sense, that is, to quantify statistical patterns in observed variables. We applied a set of five informational measures, all of them fundamentally based on Shannon entropy [58,59]. Four of these measures (entropy, mutual information, integration, and complexity) capture statistical regularities between random variables without taking into account temporal precedence. These measures as well as some of the details of their computational derivation are discussed in more detail in [13]. To estimate directed information flow, we used a measure developed for time-series analysis, transfer entropy [15].

Shannon entropy. Given a time series x_t that can assume N states, entropy provides a measure of the average uncertainty, or information, calculated from the state probability distribution according to: where P_X(i) is the probability of x_t being in the i^th state. Entropy is maximal if all states occur with equal probability (maximal disorder or uncertainty), while deviations from equal probability result in lowered entropy (increased order and decreased uncertainty).

Mutual information. Mutual information is a general measure of association between two or more random variables, naturally encompassing both linear and nonlinear dependencies. The formal definition of mutual information in terms of single and joint state probability distributions is If X and Y are two statistically independent random variables, P_XY(i, j) = P_x(i)P_Y(j) and MI(X,Y) = 0. In a sense, mutual information quantifies the error we make in assuming X and Y as independent variables, i.e., any statistical dependence between X and Y yields MI(X,Y) > 0. In general, statistical dependency as measured by mutual information is insufficient to disclose directed interactions (e.g., causal relationships) between X and Y, or between Y and X, thus requiring the use of special techniques (see below).

Integration. Integration (or multi-information; [60]) is the multivariate generalization of mutual information and captures the total amount of statistical dependency among a set of random variables X_i forming elements of a system X = {X_i }. Integration [61] is defined as the difference between the individual entropies of the elements and their joint entropy: As for mutual information, if all elements X_i are statistically independent, I(X) = 0. Any amount of statistical dependence leads to I(X) > 0.

Complexity. If a system X has positive integration, i.e., some amount of statistical dependence, we may ask how such statistical dependence is distributed within the system. If the system consists of locally segregated and globally integrated components, we would expect to find statistical dependence among units at specific spatial scales. A system combining local and global structure has high complexity: where H(X_i|X – X_i) is the conditional entropy of one element X_i given the complement X – X_i composing the rest of the system. Previous studies (e.g., [62]) have shown that complexity is high for systems that effectively combine local and global order, e.g., systems that are both functionally segregated and functionally integrated. On the other hand, complexity is low for systems that are entirely random or entirely uniform.

Transfer entropy. In addition to these noncausal informational measures, we used a measure that aims at extracting directed flow or transfer of information (also referred to as “causal dependency”) between time series, called transfer entropy [15]. Given two time series x_t and y_t, transfer entropy essentially quantifies the deviation from the generalized Markov property: p(x_t+1|x_t) = p(x_t+1|x_t,y_t), where p denotes the transition probability. If the deviation from a generalized Markov process is small, then the state of Y can be assumed to have no (or little) relevance on the transition probabilities of system X. If the deviation is large, however, then the assumption of a Markov process is not valid. The incorrectness of the assumption can be expressed by the transfer entropy, formulated as a specific version of the Kullback-Leibler entropy [15]: where the sums are over all amplitude states, and the index T(Y → X) indicates the influence of Y on X. The transfer entropy is explicitly nonsymmetric under the exchange of X and Y—a similar expression exists for T(X → Y)—and can thus be used to detect the directed exchange of information (e.g., information flow, or causal influence) between two systems. As a special case of the conditional Kullback-Leibler entropy, transfer entropy is non-negative, any information flow between the two systems resulting in T > 0. In the absence of information flow, i.e., if the state of system Y has no influence on the transition probabilities of system X, or if X and Y are completely synchronized, T(Y → X) = 0 bit.

In summary, applied to sensory, neural, and motor datasets, entropy quantifies the average uncertainty (or self-information) about the state of individual elements, while mutual information measures the statistical dependency between two elements. Integration (or multi-information) serves as the multivariate extension of mutual information, capturing the degree to which two or more elements share information. The degree to which individual elements are specialized (representing statistical independence) while also sharing information (through global interdependence) is captured by complexity. Transfer entropy is designed to detect “directed” information exchange or coupling between two elements or parts of a system.

In Roboto and Strider, visual sensory data was collected at a frame rate of approximately 10 Hz by head-mounted CCD video cameras and separated into red, green, and blue components. The raw visual images were sampled at a resolution of 240 × 320 pixels, and downsampled to 55 × 77 pixels (arrays I_R, I_G, I_B). Motor data was collected as motor commands were issued to the robot's servos and saved as angular positions for each separate servo motor. Servo positions were issued and recorded at a resolution of 256 steps per approximately 100° rotation. Time series of individual motor positions were transformed into movement amplitudes and directions by temporal differencing. For Roboto and Strider, we collected data from five runs per experimental condition. For Roboto, all runs had a length of 1,000 time steps (approximately 100 s). For Strider, runs involving learning had a length of 8,000 time steps (approximately 800 s). For Madame, the raw visual images were sampled at a resolution of 80 × 80 pixels and downsampled to 40 × 40 pixels. As for both physical robots, sensor and motor data (speed of left and right wheel and angular displacement of pan and tilt motor) were collected as motor commands were issued to the agent's actuators. We collected data from five runs with a length of 4,096 simulation steps each (note that although the sampling frequency cannot be meaningfully expressed as updates/s (in Hz), it can be taken to be equivalent to the sampling frequencies of the two physical robots, 10 Hz).

On a more general note, the simulation time step (sampling period) needs to “match” the behavioral/neural time scale. Here, we chose the minimal possible sampling period, which is one time step. There is no possible sampling period below one time step (as time steps cannot be subdivided). Having designed the systems, we know that this time scale matters, because it is the time scale at which sensors sample the environment and at which motors change position.

All numerical computations for data analysis were carried out in Matlab (Mathworks, http://www.mathworks.com). To calculate entropy and mutual information, datasamples were discretized (16 states, 4 bits) to allow robust estimates of probability distributions. Mutual information, integration, and complexity were calculated from differenced datasets, i.e., the original time series x_t was replaced with its first-order temporal derivative, y_t = x_t – x_t−1. Differenced datasets remove trends while exhibiting improved stationarity. In addition, the use of the first temporal derivative mimics the sensitivity of visual neurons to spatial and temporal changes in visual inputs, resulting in more stable representations of object properties especially in the presence of object motion. To estimate integration and complexity, we used statistical formulae that allow the calculation of entropies from the covariance matrix, under the assumption that these covariances were generated by a stationary Gaussian random process with zero mean and unit variance [58]. All differenced datasamples were examined for Gaussian state distributions (by fitting state histograms) as well as stationarity (by ensuring stable means and standard deviations across time). Nonstationary datasets were excluded from analysis. To calculate transfer entropy, time series were discretized to eight states (3 bits) and joint probabilities and conditional probabilities were approximated by means of kernel density estimation. As in [13], we chose a step kernel. Temporal delays across time series were introduced by shifting one time series relative to the other, thus allowing the evaluation of directed relationships across variable time offsets (or delays). Such delays could potentially be introduced due to the discrete nature of the updating of the control architecture and due to the temporal persistence of sensory and motor states. All results reported in this paper were qualitatively robust with respect to the specific choice of state space over a broad range of discretization (32 to four states).