Figure 1.
Diagram of the control system and the experimental set up.
The system is ‘virtual’ and is controlled through a joystick interface. The participant receives visual feed-back information about the system position through a dot presented on a real oscilloscope. The joystick position defined the system's: position (0 order system), its velocity (1st order system), or its acceleration (2nd order system). While controlling the system, participants were asked to track the position of a second dot displayed on the oscilloscope. The four possible step sequences (uni- and reversed directional step to the left or to the right) of the pursuit target are illustrated by the red line. First and second stimuli are separated by an inter-step interval (ISI), double stimuli are separated by an approximate recovery period (ARP). When applied to a model (as shown in Fig. 7), this sequence is applied as a set-point disturbance.
Figure 2.
Representative responses; reconstruction of the set-point (stage 1).
Panels show representative examples of positional joystick responses over time (blue solid lines) in: Zero Order (top row), First Order (middle row), and Second Order (bottom row) conditions. Left panels show independent responses without interference, right panels show trials with interference between responses to the second and first stimulus. The dashed line (red) shows the time-invariant optimized ARMA fit corresponding to the original/actual double step stimulus (dark blue dashed line). The dotted line (green) shows the best fitting ARMA model corresponding to the non-time-invariant optimised step sequence (dark green dotted line). Estimates of first (RT1 in blue horizontal bar) and second (RT2 in green horizontal bar) delays hover above, and span the interval between the actual and optimised step sequence. System position is displayed by the solid gray line; this is the same as joystick position in the Zero Order case.
Table 1.
Selected ISIs.
Figure 3.
The system condition is Zero Order. Panels A and B show the distribution of RT1 (panel A) and second RT2 (panel B) for all step-pairs (light) and reversed step-pairs only (dark). Panels C and D show individual values of RT1 as a function of the Recovery Period (panel C), and the inter-step interval (ISI, panel D). Panel E shows individual values of RT2 against ISI.
Figure 4.
The inter-participant (13 in each system condition) ranges (5–95%) in RT1 (blue) and RT2 (green). Each box shows, the median range (central mark), the 25th and 75th percentile range (the edges of the box are), and the most extreme data points not considered outliers (the whiskers) of these ranges. The maximum whisker length is 1.5. Data points are drawn as outliers ‘+’ if they are larger than q3 + w(q3 - q1) or smaller than q1 - w(q3 -q1), where w is the whisker length and q1 and q3 are the 25th and 75th percentiles, respectively.
Figure 5.
Group results: statistical analysis of Mean delays (stage 2).
The four panels: Zero Order System (A), First Order System (B), Second Order marginally stable System (C), Second Order unstable System (D), show the inter participant mean RT1 (blue) and RT2 (green) against ISI for the reversed step-pairs stimuli only (for details regarding the box plot’s constituents see caption Fig. 4). The P-values of the ANOVA’s post hoc test are display above each ISI level (black if significant, gray if not). The blue dotted line shows the mean RT1, the dashed green line shows the unconstrained regression linear fit between (interfered) RT2 and ISIs.
Figure 6.
Estimates of the refractory duration.
The four panels: Zero Order System (A), First Order System (B), Second Order marginally stable System (C), Second Order unstable System (D), show -in the right axes- how each metric is calculated. Plotted as a function of ISI are mean RT1 (blue circles) and mean RT2 (green circles). The gray area describes the lower and upper limits of the ANOVAs ‘general linear model’ significance, the cyan solid line shows the least mean squares fit between RT2 dependent upon ISI with slope constrained to −1, the magenta dashed line shows the unconstrained regression linear fit between RT2 dependent upon ISI, crosses displayed on the y-axes show the intercepts of these function, and the dotted blue line shows the mean RT1 which served as a baseline for the left axes that summarizes the four estimates of the refractory duration: the intercept of the unconstrained regression fit (magenta), the intercept of the −1 regression fit (cyan), the ANOVA metric (brown), and the Range in RT2 (yellow).
Figure 7.
General model of intermittent control.
The intermittent predictive controller is based on continuous control as a special case [12], [36], [43], but generally the predicted system state is only used intermittently to update the time varying control signal sent from the generalized “hold” to the actuator. “Trig.” detects when the control trajectory is to be updated and this event trigger requires three conditions: (i) a single event must be detected (i.e. all events within the sampling delay (Δs) are considered as one), (ii) a minimum open-loop interval (ΔOL) must have elapsed since the previous event and (iii) an error signal must exceed a threshold [12], [36], [43]. Scalar signals are represented by solid lines, vector signals are represented by dashed lines. The participant's neuro-muscular dynamics are modelled (linear) in the “NMS” block with input u(t). The linear external controlled system with output y(t) (represented by the “System” block) is driven by signals ue(t) and d(t) representing the externally observed control signal and the disturbance signal. The state of the composite “NMS” and “System” blocks is estimated xo(t) by the “observer” block. Sampling is preceded by an anti-aliasing low-pass filter “LP” of the subtracted set point disturbance w(t) and subject to an event delay “ΔS” between event and sampling. The trigger for the sampling times ti is provided by the event detector block labelled “trig”. Sampling xw(t) takes place at discrete times ti. Sampled signals (represented by the dotted lines) are defined only at the sample instants ti. The future state error xp(ti) is provided by the “predictor” block. The various delays in the human controller are accounted for by a pure time delay of td represented by the “delay” block. The block labelled “hold” is a system-matched hold, that provides the delayed version of the continuous-time signal that is multiplied by the feedback gain vector k (block “State FB”) to give the feedback control signal u(t). This figure and its caption are based on [12].
Figure 8.
Model based interpretation (stage 3).
Discriminating serial ballistic (intermittent) from continuous control and identifying several possible relationships between RT2 and inter-step interval (ISI). The simulated system is zero order. The open-loop interval (delta OL) is 0.35 s and feedback time delay (td) is 0.14 s. For four models: A) continuous LTI, B) externally-triggered intermittent control, C) internally-triggered intermittent control (triggered to saturation), and D) externally-trigger intermittent control supplemented with a sampling delay. The following is shown as a function of ISI: (A–D), mean RT2 (joined green circles), black dotted horizontal shows mean RT2 without interference, black dotted vertical shows the set open-loop interval, gray dotted shows unconstrained regression linear fit between RT2 and ISI using ISIs smaller than delta OL.