On rhythmic and discrete movements: reflections, definitions and implications for motor control

Abstract

At present, rhythmic and discrete movements are investigated by largely distinct research communities using different experimental paradigms and theoretical constructs. As these two classes of movements are tightly interlinked in everyday behavior, a common theoretical foundation spanning across these two types of movements would be valuable. Furthermore, it has been argued that these two movement types may constitute primitives for more complex behavior. The goal of this paper is to develop a rigorous taxonomic foundation that not only permits better communication between different research communities, but also helps in defining movement types in experimental design and thereby clarifies fundamental questions about primitives in motor control. We propose formal definitions for discrete and rhythmic movements, analyze some of their variants, and discuss the application of a smoothness measure to both types that enables quantification of discreteness and rhythmicity. Central to the definition of discrete movement is their separation by postures. Based on this intuitive definition, certain variants of rhythmic movement are indistinguishable from a sequence of discrete movements, reflecting an ongoing debate in the motor neuroscience literature. Conversely, there exist rhythmic movements that cannot be composed of a sequence of discrete movements. As such, this taxonomy may provide a language for studying more complex behaviors in a principled fashion.

Appendices

Appendix 1: Definition of recurrent

To precisely define “approximately equal” and “significantly different”, consider a particular value in the range of the function and denote the corresponding argument by t _i . Let t _i+ denote the closest larger value of the argument at which the function differs by a small constant ε from that value,

$$t_{i+}=t_{i}+\delta_{i+}$$

where δ _i+ is the maximum positive constant such that

$$\Big|y(t)-y(t_{i})\Big| < \varepsilon \text{ for all }t\text{ in the interval }t_{i}\leq t\leq t_{i}+\delta_{i+}$$

and let t _i− denote the closest smaller value,

$$t_{i-}=t_{i}-\delta_{i-}$$

where δ _i− is the maximum positive constant such that

$$\Big|y(t)-y(t_{i})\Big| < \varepsilon \text{ for all }t\text{ in the interval }t_{i}-\delta_{i-}\leq t\leq t_{i}.$$

The function is recurrent if there exists a set of arguments {t _j } for which

$$\Big|y(t_{i})-y(t_{j})\Big| < \varepsilon \text{ and }t_{j} < t_{i-}\text{ or }t_{j} > t_{i+}.$$

This might be termed minimally recurrent as it may be satisfied by as few as two values of the argument. To precisely define “a large number” of recurrences, we require the number to grow without bound if the observation interval grows without bound. For any candidate t _j identified above, find t _j+ and t _j− as above to identify the interval in which the function remains approximately equal and assign all t in the interval t _j− ≤ t ≤ t _j+ as one occurrence. Let N denote the number of members of the set {t _j } and D denote the interval of observation, 0 ≤ t ≤ D. Then the function is indefinitely recurrent if ${{\mathop {\lim}\limits_{D \to \infty}}N = \infty.}$ White noise is a theoretical extreme case which might be termed infinitely recurrent; in a finite interval of observation, all values in the amplitude distribution recur an infinite number of times with probability approaching unity.

Appendix 2: Smoothest discrete and cyclic movements

Using mean-squared-jerk as a measure, the problem of identifying the smoothest movement may be formulated using optimization theory as that of finding the function y(t) that minimizes the scalar

$${msj = \frac{1}{{t_{2} - t_{1}}}{\int\limits_{t_{1}}^{t_{2}} {\frac{1}{2}u^{2} {\rm d}t}}}$$

subject to the dynamic constraints

$${\dot{y} = v, \dot{v} = a, \dot{a} = u}$$

where y, v, a and u are position, velocity, acceleration and jerk, respectively. Using the method of Lagrange multipliers, the constraints may be added to the scalar to be minimized as

$${msj = \frac{1}{{t_{2} - t_{1}}}{\int\limits_{t_{1}}^{t_{2}} {{\left\{{\frac{1}{2}u^{2} + \lambda_{y} {\left({v - \dot{y}} \right)} + \lambda_{v} {\left({a - \dot{v}} \right)} + \lambda_{a} {\left({u - \dot{a}} \right)}} \right\}}{\rm d}t}}}$$

where λ _y , λ _v and λ _a are functions to be determined. For convenience, form the Hamiltonian

$${H = \frac{1}{2}u^{2} + \lambda_{y} v + \lambda_{v} a + \lambda_{a} u}$$

so that

$${msj = \frac{1}{{t_{2} - t_{1}}}{\int\limits_{t_{1}}^{t_{2}} {{\left({H - \lambda_{y} \dot{y} - \lambda_{v} \dot{v} - \lambda_{a} \dot{a}} \right)}{\rm d}t}}}.$$

For simplicity, assume a time scale such that t ₁= 0 and t ₂ = 1. Application of variational calculus shows that within the interval 0 ≤ t ≤ 1 the partial derivative of H with respect to each of its arguments must be zero. The dynamic constraint equations (or state equations) are recovered from

$${\left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial \lambda_{y}}}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{y}}} \right|_{o} = v = \dot{y}, \left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial \lambda_{v}}}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{v}}} \right|_{o} = a = \dot{v}}\text{ and }{\left. {{\partial H} \mathord{\left/{\vphantom {{\partial H} {\partial \lambda_{a}}}} \right. \kern-\nulldelimiterspace} {\partial \lambda_{a}}} \right|_{o} = u = \dot{a}.}$$

The Lagrange multipliers are determined by the co-state equations

$${\left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial y}}} \right. \kern-\nulldelimiterspace} {\partial y}} \right|_{o} = 0 = - \dot{\lambda}_{y}, \left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial v}}} \right. \kern-\nulldelimiterspace} {\partial v}} \right|_{o} = \lambda_{y} = - \dot{\lambda}_{v} }\text{ and }{\left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial a}}} \right. \kern-\nulldelimiterspace} {\partial a}} \right|_{o} = \lambda_{v} = - \dot{\lambda}_{a}.}$$

The optimal solution is defined by

$${\left. {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial u}}} \right. \kern-\nulldelimiterspace} {\partial u}} \right|_{o} = 0 = u^{o} + \lambda_{a}.}$$

Combining equations, −λ _a is jerk, λ _v is snap (the fourth time derivative of position), −λ _y is crackle (the fifth time derivative of position) and the minimal mean-squared jerk movement is defined by

$${{{\rm d}^{6} y} \mathord{\left/ {\vphantom {{{\rm d}^{6} y} {{\rm d}t^{6}=0}}} \right. \kern-\nulldelimiterspace} {{\rm d}t^{6} = 0}.}$$

Integrating yields a fifth-order polynomial

$$y^{0}(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}+a_{5}t^{5}$$

with six coefficients to be determined by the boundary conditions at the ends of the time interval. A discrete movement starting from rest at y(0) = 0 and ending at rest at y(1) = 1 yields

$$a_{0}=a_{1}=a_{2}=0,\;a_{3}=10,\;a_{4}=-15\;\text{and }a_{5}=6$$

so that the general solution is

$${y^{0} {\left(t \right)} = y{\left(0 \right)} + A{\left[ {10{\left({\frac{t}{d}} \right)}^{3} - 15{\left({\frac{t}{d}} \right)}^{4} + 6{\left({\frac{t}{d}} \right)}^{5}} \right]}}$$

for 0 ≤ t ≤ 1 where y(0) is initial position, A is movement amplitude and d is movement duration. The minimal value of the msj measure for this movement is

$${msj^{0} = 360{A^{2}} \mathord{\left/ {\vphantom {{A^{2}} {d^{6}}}} \right. \kern-\nulldelimiterspace} {d^{6}}.}$$

To identify the smoothest cyclic movement between two positions, consider two adjacent intervals, i.e., a “forth” movement from the first to the second position in the interval 0 ≤ t ≤ m, and a “back” movement from the second to the first position in the interval m ≤ t ≤ p, where p is the period of the cycle and 0 < m < p is the passage time (to be determined) at which the second position, y ₂, is passed. The details of movement between these positions may be found by solving an optimization problem with an “interior-point constraint” N on position at the passage time m such that

$$N=y(m)-y_{2}=0.$$

Again using the method of Lagrange multipliers, this constraint may be appended to the measure to be minimized to form

$${msj_{c} = \pi N + \frac{1}{p}{\int\limits_0^p {\frac{1}{2}u^{2} {\rm d}t}}.}$$

The integral has two components, the first with a variable end time, the second with a variable start time,

$${msj_{c} = \pi N + \frac{1}{p}{\left[ {{\int\limits_0^m {\frac{1}{2}u^{2} {\rm d}t}} + {\int\limits_m^p {\frac{1}{2}u^{2} {\rm d}t}}} \right]}.}$$

Applying variational calculus, within each interval the movement is described by a quintic polynomial as above. Thirteen coefficients (six for each interval plus the passage time, m) have to be determined from the boundary conditions at the end of each interval. For simplicity, assume p = 2. The position, velocity and acceleration (indeed, all derivatives) at the beginning of the “forth” interval and end of the “back” interval are identical:

$$y(2)=y(0)=0,\;v(2)=v(0),\;a(2)=a(0)$$

and so on; this is a basic requirement for a cyclic movement. At the passage time, position is known but none of its time derivatives are. In general, the Hamiltonian and each of the Lagrange multipliers λ _y , λ _v and λ _a may be discontinuous at time m. Using self-evident subscripts for the first and second intervals, a general boundary condition at time m is

$$ {\rm d}y{\left({- \lambda_{{y1}} + \pi + \lambda_{{y2}}} \right)} + {\rm d}v{\left({- \lambda_{{v1}} + \lambda_{{v2}}} \right)} + {\rm d}a{\left({- \lambda_{{a1}} + \lambda_{{a2}}} \right)} + {\rm d}m{\left({H_{1} - H_{2}} \right)} = 0. $$

As each of the infinitesimal differentials dy, dv, da and dm are unspecified, this reduces to

$${H_{1}=H_{2},\; \lambda_{a_1}=\lambda_{a_2},\; \lambda_{v_1}=\lambda_{v_2},\;}\text{ and }{\lambda_{y_1}=\pi + \lambda_{y_2}.}$$

Thus crackle may be discontinuous at the passage time but the Hamiltonian, jerk and snap are continuous and (by integration) acceleration, velocity and position are also continuous. If crackle is discontinuous and the Hamiltonian is continuous then velocity must be zero at the passage time.

While this analysis yields sufficient equations to determine the unknown coefficients, evaluating them requires solving a large number of simultaneous, nonlinear, algebraic equations. A simpler approach is to assume that the “forth” and “back” movements have the same (unknown) shape, though perhaps different durations. If so, the jerk measure for each interval is proportional to the square of movement amplitude and inversely proportional to the sixth power of interval duration so that

$${msj_{c} = \frac{1}{p}{\left[ {\frac{{CA^{2}}}{{m^{5}}} + \frac{{CA^{2}}}{{{\left({p - m} \right)}^{5}}}} \right]},}$$

where C is a constant that depends on the details of the shape. Setting p = 2 and minimizing with respect to the passage time, m, yields a sixth-order polynomial with only one real-valued root at m = 1. Thus the smoothest cyclic movement has equal-duration “forth” and “back” segments, each a mirror-image of the other. Boundary conditions for the “forth” movement are

$$y(0)=0,\;y(1)=1,\;v(0)=-v(1),\;a(0)=-a(1),\;u(0)=-u(1)\text{ and }s(0)=-s(1),$$

where s is snap. They yield

$${a_{0}=a_{1}=0,\; a_{2}=5 \mathord{\left/{\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2,\; a_{3}=0,\; a_{4}=-5 \mathord{\left/{\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}\text{ and }a_{5}=1$$

so that the general solution is

$${y^{0} {\left(t \right)} = y{\left(0 \right)} + A{\left[ {\frac{5}{2}{\left({{\left({\frac{t}{d}} \right)}^{2} - {\left({\frac{t}{d}} \right)}^{4}} \right)} + {\left({\frac{t}{d}} \right)}^{5}} \right]}}$$

for 0 ≤ t ≤ d where y(0) is initial position, A is movement amplitude and d is the duration of the outbound movement, so that p = 2d. The minimal value of the msj measure for this movement is

$${msj^{0} = 60{A^{2}} \mathord{\left/ {\vphantom {{A^{2}} {d^{6}}}} \right. \kern-\nulldelimiterspace} {d^{6}}.}$$