Explaining savings for visuomotor adaptation: linear time-invariant state-space models are not sufficient

Abstract

Adaptation of the motor system to sensorimotor perturbations is a type of learning relevant for tool use and coping with an ever-changing body. Memory for motor adaptation can take the form of savings: an increase in the apparent rate constant of readaptation compared with that of initial adaptation. The assessment of savings is simplified if the sensory errors a subject experiences at the beginning of initial adaptation and the beginning of readaptation are the same. This can be accomplished by introducing either 1) a sufficiently small number of counterperturbation trials (counterperturbation paradigm [CP]) or 2) a sufficiently large number of zero-perturbation trials (washout paradigm [WO]) between initial adaptation and readaptation. A two-rate, linear time-invariant state-space model (SSM(LTI,2)) was recently shown to theoretically produce savings for CP. However, we reasoned from superposition that this model would be unable to explain savings for WO. Using the same task (planar reaching) and type of perturbation (visuomotor rotation), we found comparable savings for both CP and WO paradigms. Although SSM(LTI,2) explained some degree of savings for CP it failed completely for WO. We conclude that for visuomotor rotation, savings in general is not simply a consequence of LTI dynamics. Instead savings for visuomotor rotation involves metalearning, which we show can be modeled as changes in system parameters across the phases of an adaptation experiment.

FIG. 1.

Illustration of superposition for two-rate, linear time-invariant state-space model (SSM_LTI,2) [a_slow = 0.992, b_slow = 0.02, a_fast = 0.59, b_fast = 0.21, taken from the empirical estimates reported by Smith and colleagues (2006)] in the counterperturbation paradigm (CP). See Eq. 2 in the main text for the state-space model. The left column shows the input perturbation functions (abscissa is movement number n), whereas the right column shows both the outputs [equivalently in terms of directional error (red) and net sensorimotor map (black)] and the state variables [slow (blue) and fast (green)]. The rows correspond to a decomposition of the net input [(from top): initial adaptation stimulus, counterperturbation stimulus, readaptation stimulus, summed inputs]. As a consequence of superposition, the shaded plot in the bottom right corner is equal to both 1) the sum of the outputs to the separate inputs (sum down right column) and 2) the output to the summed inputs (transform from left to right in the bottom row). In the CP paradigm, superposition leads to obvious savings (i.e., a faster apparent rate of adaptation during the readaptation phase compared with the initial adaptation phase). Perturbation function for this CP paradigm: 0° for 1 ≤ n ≤ 10, 30° for 11 ≤ n ≤ 100, −30° for 101 ≤ n ≤ 103, 30° for 104 ≤ n ≤ 150.

FIG. 2.

Illustration of superposition for SSM_LTI,2 in the washout paradigm (WO). Formatting and system constants are the same as in Fig. 1. The rows correspond to a decomposition of the net input [(from top) initial adaptation stimulus, washout phase, readaptation stimulus, summed inputs]. Because a sufficient number of washout trials was inserted between the initial adaptation and readaptation stimuli to bring the state vector close to its naive value of (in this case) 0, the apparent rates of adaptation during the readaptation and initial adaptation phases are not nearly as distinguishable as in CP (Fig. 1). However, the directional error output (and thus savings) in response to the summed perturbations can be predicted simply from the superposition of the directional errors caused by the individual perturbations (this superposition being the red curve in the shaded plot), without additional concern for the values of the slow and fast components of the state vector (blue and green curves, respectively, in shaded plot). Perturbation function for this WO paradigm: 0° for 1 ≤ n ≤ 10, 30° for 11 ≤ n ≤ 100, 0° for 101 ≤ n ≤ 200, 30° for 201 ≤ n ≤ 280.

FIG. 3.

The perturbation r[n] (thick gray line), across-subject averaged directional error e[n] (open diamonds) and SSM MLE fits (black line: SSM_LTI,1, green line: SSM_LTI,2, blue line: SSM_VP,1, red line: SSM_VP,2) for the (A) CP and (B) WO paradigms.

FIG. 4.

The across-subject averaged Akaike Information Criterion (AIC, black) and −2 log_e (likelihood) (gray) values (to within an additive constant) of the candidate models for the (A) CP and (B) WO paradigms. Both measures assess model fit, but only the AIC penalizes for the number of parameters. For each measure, it is only the differences between models (within paradigm) that are meaningful. The units of both the AIC and −2 log_e (likelihood) are information, in the information-theoretic sense.

FIG. 5.

The fast (red line), slow (green line), and net = fast + slow (black line) state variables estimated from the fit of SSM_VP,2 to the across-subject averaged directional error e[n] are plotted for the (A) CP and (B) WO paradigms. The perturbation r[n] (thick gray line), i.e., the deterministic input to the system, is also plotted.