Stabilization of cat paw trajectory during locomotion

Abstract

We investigated which of cat limb kinematic variables during swing of regular walking and accurate stepping along a horizontal ladder are stabilized by coordinated changes of limb segment angles. Three hypotheses were tested: 1) animals stabilize the entire swing trajectory of specific kinematic variables (performance variables); and 2) the level of trajectory stabilization is similar between regular and ladder walking and 3) is higher for forelimbs compared with hindlimbs. We used the framework of the uncontrolled manifold (UCM) hypothesis to quantify the structure of variance of limb kinematics in the limb segment orientation space across steps. Two components of variance were quantified for each potential performance variable, one of which affected it (“bad variance,” variance orthogonal to the UCM, V_ORT) while the other one did not (“good variance,” variance within the UCM, V_UCM). The analysis of five candidate performance variables revealed that cats during both locomotor behaviors stabilize 1) paw vertical position during the entire swing (V_UCM > V_ORT, except in mid-hindpaw swing of ladder walking) and 2) horizontal paw position in initial and terminal swing (except for the entire forepaw swing of regular walking). We also found that the limb length was typically stabilized in midswing, whereas limb orientation was not (V_UCM ≤ V_ORT) for both limbs and behaviors during entire swing. We conclude that stabilization of paw position in early and terminal swing enables accurate and stable locomotion, while stabilization of vertical paw position in midswing helps paw clearance. This study is the first to demonstrate the applicability of the UCM-based analysis to nonhuman movement.

one of the central problems of motor control is the problem of motor redundancy (Bernstein 1967). It reflects the fact that, in any analysis of the neuromotor system, the number of elemental variables (those produced by system's elements, i.e., body limbs, joints, muscles, etc.) is higher than the number of constraints associated with typical motor tasks. For example, the same movement of the hand or foot can be performed with different combinations of joint angles of the limb because the number of degrees of freedom (DOF) in the upper and lower extremities exceeds the number of space dimensions [two- (2D) or three-dimensional (3D)] in which the endpoint of the limb is constrained to move. Given an unlimited choice of combinations of elemental variables (e.g., joint angles) to move the limb endpoint along a trajectory, how does the nervous system select specific combinations of elemental variables to organize the movement? Recently, an approach to solving this problem has been developed based on the principle of abundance (Gelfand and Latash 1998; Latash 2012). According to this principle, the central nervous system (CNS) often does not look for single optimal solutions to problems of motor redundancy, but facilitates families of solutions equally capable of solving the task. A signature of this control strategy is a particular pattern of variance in the space of elemental variables. A quantitative method for analysis of motor variance has been developed within the uncontrolled manifold (UCM) hypothesis (Latash et al. 2007; Scholz and Schoner 1999). Within this method, variance across consecutive trials or cycles (for a cyclic task) is partitioned into two components. One of them (variance within the UCM, V_UCM or “good variability”) keeps a potentially important performance variable unchanged, while the other (variance orthogonal to the UCM, V_ORT or “bad variability”) leads to changes in that variable. Neural mechanisms responsible for variance distributions characterized by an inequality V_UCM > V_ORT have been referred to as “synergies.”

The term “synergy” has been used in the literature in at least three meanings. In clinical literature, this word has a strong negative connotation and implies pathological, stereotypical patterns of muscle activation (e.g., after stroke) interfering with voluntary movements (Bobath 1978; Dewald et al. 1995). In a more traditional meaning, synergy means a group of variables that scale together during changes in task parameters and/or over time; methods of matrix factorization have been commonly used to identify such groups of variables (reviewed in Ting and McKay 2007). This definition follows the traditions set by Bernstein (1967), who viewed such grouping as a method of reducing the number of variables a hypothetical neural controller has to manipulate. Our definition links the notion of synergies to stability of movements, which is paramount for successful performance in the changing environment.

There is no agreed-upon hypothesis on the origin of motor synergies (in our definition). Synergies have been described as products of an optimal feedback control scheme, a feed-forward scheme, a scheme with central back-coupling loops, and a hierarchical scheme based on ideas of equilibrium-point control (Goodman and Latash 2006; Latash 2010; Latash et al. 2005; Martin et al. 2009; Todorov and Jordan 2002). Most of these schemes make no claims regarding potential roles of different brain regions in synergies. Studies of patients with cortical and subcortical disorders (Park et al. 2012, 2013; Reisman and Scholz 2003) point at subcortical loops, possibly those involving the basal ganglia and the cerebellum, as crucial for synergy formation in humans.

As of now, all of the experimental evidence in favor of the principle of abundance and the idea of synergies has come from studies on people. A number of studies of animals, including reduced animal preparations, suggest that synergies may be organized at the spinal cord level (Berkinblit et al. 1986; Boyce and Lemay 2009; Hultborn et al. 2004; Mussa-Ivaldi et al. 1994). However, only indirect animal studies of the structure of variance have been available (Bauman and Chang 2013; Chang et al. 2009). One of the main goals of the current study has been to provide evidence for synergies stabilizing potentially important performance variables during a natural behavior of intact cats. We selected locomotion as the behavior of interest because, first, the central role of the spinal cord in the production of locomotion has been well established; second, one can formulate reasonable hypotheses with respect to kinematic performance variables that may be stabilized by a multijoint synergy; and third, comparing synergies in the forelimbs and hindlimbs allows the exploration of the potential role of vision in such synergies.

It is well known that animals, including humans, can voluntarily modify the location of foot placement during locomotion when they step on specific support surfaces (Beloozerova and Sirota 1993a, 1993b; Metz and Whishaw 2002), circumvent or step over obstacles (Lavoie et al. 1995; Patla and Greig 2006), walk along a prescribed path (Galvez-Lopez et al. 2011; McAndrew Young et al. 2012), or alter stride length and stance width when stability of locomotion is threatened (Dingwell et al. 2008; MacLellan and Patla 2006; Marigold and Patla 2008). Potentially, several motor strategies are available to provide accurate foot placement to a selected location. An entire foot trajectory could be planned before initiation of swing (Hollands and Marple-Horvat 1996, 2001) to enable safe foot clearance over the ground, and any deviation from the trajectory is corrected. Alternatively, foot trajectory could be stabilized only in the vicinity of a foot placement location (Reynolds and Day 2005) or in early swing of locomotion.

During quadrupedal locomotion, the animal can see only the final part of forepaw trajectory prior to paw contact with the ground, whereas the initial part of the forepaw trajectory and the entire hindpaw trajectory cannot be seen. Nevertheless, displacements and symmetric bell-shaped velocity profiles of fore- and hindpaws look essentially identical during swing of regular walking and skilled accurate stepping on a horizontal ladder (Beloozerova et al. 2010; Prilutsky et al. 2005), suggesting that, in these tasks, paw placements are planned in advance and are not corrected during swing (Beloozerova et al. 2010). Alternatively, a stereotypic paw trajectory of each limb during swing could result from a continuous or intermittent stabilization of limb endpoint position by coordinated small changes in joint angles such that they reduce variability of the paw based on a combination of visual and proprioceptive feedback. This stabilization of paw trajectory could potentially explain substantial changes in the modulation of neural activity from limbs' representation in the motor cortex during swing of skilled accurate stepping compared with the activity of the same cortical cells during regular walking, despite virtually identical limb kinematics (Beloozerova et al. 2010).

We approached the analysis of multijoint synergies potentially stabilizing the paw trajectory or other limb kinematic variables using the method developed within the UCM hypothesis. Namely, for each time sample of the swing phase, V_UCM and V_ORT were computed across consecutive strides, both quantified per DOF in the corresponding spaces, and then a “synergy index” (ΔV) was computed reflecting the relative difference between V_UCM and V_ORT. V_UCM > V_ORT (ΔV > 0) is interpreted as a reflection of a purposeful neural strategy stabilizing the performance variable. In some motor behaviors, like blacksmith hammering (Bernstein 1923) or pistol shooting (Scholz et al. 2000), the performance variables can be reasonably assumed a priori, and their stabilization tested using the UCM analysis. In locomotor behaviors, hypothetical performance variables are not obvious, and their selection for the UCM analysis requires careful considerations.

During skilled walking on a horizontal ladder, i.e., accurate stepping, the cat walks by placing paws on 5-cm-wide crosspieces of the ladder (Beloozerova et al. 2010; Beloozerova and Sirota 1993a). This locomotor behavior is trained for about 1 mo using operant conditioning and food rewards (see methods), and once the task is learned the cat never fails to place paws on the ladder crosspieces during walking. Thus in this locomotor behavior, paw position during swing can be hypothesized to be a performance variable. Recent studies on dynamic stability of regular walking in humans (Hof et al. 2005, 2007) and cats (Farrell et al. 2014) have also demonstrated the importance of accurate foot or paw placement in front of the extrapolated center of mass of the body to maintain dynamic stability in the frontal and sagittal planes. Therefore, it is reasonable to assume that paw position during cat regular walking could be a performance variable. Other potential performance variables during cat regular walking that appear to be stabilized by multijoint kinematic synergies even after injury of major ankle extensors are the limb length and orientation (Bauman and Chang 2013; Chang et al. 2009), whose values may be encoded in the activity of the dorsal spinocerebellar tract neurons (Bosco et al. 2000).

Our more specific hypotheses related to possible differences in stabilization of the potential fore- and hindlimb performance variables during swing and between regular walking and accurate stepping on crosspieces of a horizontal ladder. Three hypotheses were tested: 1) animals stabilize the entire trajectory of the performance variable during swing [based on human studies (Domkin et al. 2005; Krishnan et al. 2013)]; 2) the index of performance variable trajectory stabilization during walking on a horizontal ladder is similar to that during regular walking [based on studies showing no correlation between indexes of stabilization and accuracy of performance (Gorniak et al. 2008; Shapkova et al. 2008)]; and 3) the index of performance variable trajectory stabilization is higher for forepaws compared with hindpaws [based on the importance of visual information for accurate stepping (Marigold and Patla 2008; Reynolds and Day 2005)].

METHODS

Subjects and Experimental Procedures

Locomotor kinematics of hind- and forelimbs were obtained from recordings of five adult cats (3 males and 2 females), participants of a larger study; animal characteristics are shown in Table 1. All experimental procedures were conducted in accordance with the US Public Health Service Policy on Humane Care and Use of Laboratory Animals and with the approval of the Institutional Animal Care and Use Committees of both Georgia Institute of Technology and Barrow Neurological Institute.

Table 1.

Individual and mean cat characteristics

	Cat
Parameter	BU	BL	AG	C8	FM	Mean ± SD
Upper arm length, mm	93	88	110	103	103	99 ± 9
Forearm length, mm	93	89	125	113	110	106 ± 15
Carpals + digits length, mm	35	36	35	40	30	35 ± 4
Thigh length, mm	98	100	110	110	105	105 ± 6
Shank length, mm	98	100	115	133	116	112 ± 14
Tarsals + digits length, mm	72	78	75	68	67	72 ± 5
Mass, kg	3.1	3.0	4.6	4.5	4.0	3.8 ± 0.8

Cats were trained for at least 1 mo to walk along a walkway in a Plexiglas-enclosed chamber on a flat surface and on a horizontal ladder (crosspieces 5 cm wide) using food reward (for details see Beloozerova et al. 2010; Gregor et al. 2006; Prilutsky et al. 2005). Food reward, several dry food pellets, was given each time the cat walked across the walkway with or without horizontal ladder with a steady speed. The cat consumed pellets during short breaks of 5–15 s before initiating the next walking trial and continued walking until it lost interest in food and stopped eating after completing 50–80 trials. After training, high-speed (sampling frequency 112–120 Hz) motion capture systems Vicon (Vicon Motion Systems) or Visualeyez System (Phoenix Technologies) were used to record 3D coordinates of markers attached to shaved skin over bony landmarks of the cat body using double-sided adhesive tape (Fig. 1) (Beloozerova et al. 2010; Prilutsky et al. 2005). Marker locations used in this study included the greater trochanter (hip joint), approximate knee joint center, lateral malleolus (ankle joint), base of the fifth metatarsal (metatarsophalangeal joint) of one or both hindlimbs and the greater tubercle (shoulder joint), approximate elbow joint center, ulna styloid process (wrist joint), base of the fifth metacarpal (metacarpophalangeal joint) of one or both forelimbs. The animals performed regular walking and accurate stepping within one experimental session, either intermittently by continuously crossing interconnected walkways with flat surface and a horizontal ladder or sequentially by performing multiple trials of one locomotion task before switching to the other task; the order of tasks was balanced between cats. The distance between crosspieces of the horizontal ladder was set for each cat to be approximately equal to the mean cat's step length of regular walking with self-selected speed (on average, about 25 cm).

Fig. 1.

Kinematic model of fore- and hindlimbs used for uncontrolled manifold (UCM) analysis. A: kinematics of each limb is described by 5 generalized coordinates (elemental variables): Cartesian x- and y-coordinates of suspension point [hip (H) or shoulder (S)] and 3 segment angles with the horizon [thigh (T) or upper arm (UA), shank (S) or forearm (FA), hindfoot (HF) or forefoot (FF)] (hindfoot comprises tarsals and hind digits, forefoot consists of carpals and fore digits). B: potential performance variables for UCM analysis are the two-dimensional (2D) position of the limb endpoint, i.e., Cartesian x- and y-coordinates of forepaw (FP) or hindpaw (HP), vertical or horizontal position of the paw, limb length (L) and limb orientation (θ). The open circles indicate positions of reflective markers for motion capture.

Length of each segment of the fore- and hindlimb (Table 1) was measured using a caliper while the animal was sedated (dexmedetomidine, 40–60 μg/kg sc).

Data Processing

All recorded walking trials were screened to ensure that only cycles in which cats walked with a constant, steady speed were used for further analysis. Recorded marker displacement data of the main joints of left and/or right fore- and hindlimbs (Fig. 1) in the sagittal plane were low-pass filtered (fourth-order, zero-phase lag Butterworth filter, 10-Hz cutoff frequency). To minimize displacement errors caused by skin motion near the knee and elbow joints (Miller et al. 1975), coordinates of these joints were recalculated using recorded joint marker positions of the adjacent joints and measured length of the segments forming the knee and elbow joints. For example, for each instant of time, the position of the knee joint center in the sagittal plane was determined trigonometrically as an intersection point between the two circles with the centers located at the ankle joint marker and the hip joint marker positions and with the circles' radii corresponding to the shank length and thigh length, respectively. Out of two intersection points, the one corresponding to the anatomical constraint on maximum knee extension (<180°) was selected. The same procedure was used to determine the position of the elbow joint in the sagittal plane.

The relative position of the limb endpoint with respect to the limb most proximal joint (hip or shoulder) was used to identify the swing phase onset (paw-off, PO) and offset (paw contact, PC), as described elsewhere (Pantall et al. 2012). Each analyzed stride cycle was defined as the period between consecutive PO instances; the swing phase was defined as the period between PO and PC. The smoothed marker displacements were time normalized within each walking cycle and swing phase.

A 2D, 5 DOF kinematic model of a limb was used to describe kinematics of the fore- and hindlimb (Fig. 1) and derive the Jacobian matrix (see below). Generalized coordinates describing model kinematics included the Cartesian horizontal and vertical coordinates of the limb suspension point (the hip joint for hindlimb and the shoulder joint for forelimb) and three segment angles with respect to the horizon (Fig. 1: thigh, shank, and foot angles for the hindlimb and upper arm, forearm and fore foot angles for the forelimb). Since UCM analysis, as described in the Introduction, involves calculations of variance of kinematic variables at the same normalized time instant across walking cycles, it is important to ensure that each elemental kinematic variable has a similar periodic pattern within walking cycles. All generalized coordinates of the limb model are periodic functions of walking cycle time, except for the horizontal displacement of the hip and shoulder joints (e.g., Prilutsky et al. 2005). To make these latter variables periodic, a linear trend (a linear regression between the horizontal joint displacement and normalized cycle time) was computed and then subtracted from the horizontal displacement of the joint at each percent of the cycle time. This detrending procedure is analogous to determining horizontal coordinates of a joint in a coordinate frame moving with a constant speed corresponding to the average speed of walking in a given cycle. In other words, the obtained detrended horizontal coordinates of the hip and shoulder can be considered displacements, as observed during walking on a treadmill operating at a constant speed.

Uncontrolled Manifold Analysis

We used the framework of the UCM analysis (Latash et al. 2007; Scholz et al. 2000; Scholz and Schoner 1999) to partition variance of elemental kinematic task variables (limb generalized coordinates) at each time instant across multiple walking cycles into two subspaces. One subspace (UCM) consists of variance of elemental variables that does not affect a performance variable, V_UCM (“good variance”); the second subspace is orthogonal to the first one and contains variance that affects a performance variable, V_ORT (“bad variance”). Both variance components, V_UCM and V_ORT, were computed for each instance of the normalized walking cycle time and then normalized per DOF of the corresponding subspace. To examine whether a performance variable was stabilized over walking cycles by coordinated changes in elemental variables, the V_UCM and V_ORT were compared by analyzing their normalized difference (index of synergy, ΔV, e.g., Klous et al. 2011).

Potential performance and elemental variables.

As discussed in the Introduction, five limb kinematic variables could be assumed to be stabilized by multijoint kinematic synergies during swing of regular and ladder walking: 1) 2D paw position in the global coordinate frame related to the ground; 2) horizontal paw position in the global coordinate frame; 3) vertical paw position in the global coordinate frame; 4) limb length (the distance between the most proximal joint and the paw); and 5) limb orientation.

To examine stabilization of potential performance variable 1, 2D paw position, the global Cartesian horizontal x and vertical y coordinates of the paw were expressed as functions of five elemental task variables (generalized limb coordinates; Fig. 1):

$$\{\begin{array}{c}x=q_1+L_3\hspace{0.17em}\cos \hspace{0.17em}{q}_3+L_4\hspace{0.17em}\cos \hspace{0.17em}{q}_4+L_5\hspace{0.17em}\cos \hspace{0.17em}{q}_5\\ y=q_2-L_3\hspace{0.17em}\sin \hspace{0.17em}{q}_3-L_4\hspace{0.17em}\sin \hspace{0.17em}{q}_4-L_5\hspace{0.17em}\sin \hspace{0.17em}{q}_5\end{array}$$ (1)

where generalized limb coordinates are as follows: q₁ and q₂, horizontal and vertical coordinates of hip (shoulder) joint; q₃, q₄ and q₅, angles formed by the thigh (upper arm), shank (forearm), and tarsals (carpals) with the horizon; L₃, L₄, L₅, lengths of the thigh (upper arm), shank (forearm), and tarsals (carpals), respectively (Table 1). In contrast to some previous UCM analyses (Auyang et al. 2009; Kapur et al. 2010; Latash et al. 2007; Scholz and Schoner 1999), we used heterogeneous elemental variables with different dimensions, i.e., angles and Cartesian linear coordinates, to analyze stabilization of paw 2D position or paw's vertical and horizontal positions separately. Note that similar approaches were used earlier (e.g., Yang and Scholz 2005). To eliminate dependence of computed variance of elemental variables on their dimensions, the variables were substituted by new variables using the range of variables' changes (R_i) across analyzed cycles (R_i = q_i^max − q_i^min, i = 1, . . ., 5; where q_i^max and q_i^min are maximum and minimum values, respectively, of ith elemental variable across walking cycles for a given cat and limb, Table 2). For horizontal coordinate q₁, the range was determined after the detrending procedure. By introducing new elemental variables q̂_i = (q_i − q_i^min)/R_i that change between 0 and 1, system (Eq. 1) can be rewritten as:

$$\{\begin{array}{c}x=(q_1^{\min }+R_1{\stackrel{̑}{q}}_1)+L_3\hspace{0.17em}\cos (q_3^{\min }+R_3{\stackrel{̑}{q}}_3)+L_4\hspace{0.17em}\cos (q_4^{\min }+R_4{\stackrel{̑}{q}}_4)+L_5\hspace{0.17em}\cos (q_5^{\min }+R_5{\stackrel{̑}{q}}_5)\\ y=(q_2^{\min }+R_2{\stackrel{̑}{q}}_2)-L_3\hspace{0.17em}\sin (q_3^{\min }+R_3{\stackrel{̑}{q}}_3)-L_4\hspace{0.17em}\sin (q_4^{\min }+R_4{\stackrel{̑}{q}}_4)-L_5\hspace{0.17em}\sin (q_5^{\min }+R_5{\stackrel{̑}{q}}_5)\end{array}$$ (1a)

The Jacobian of system (Eq. 1a) for each time instant is

$${\mathit{J}}_{\mathit{x}\mathit{,}\mathit{y}}=\left[\begin{array}{ccc}{R}_1& 0& -L_3{R}_3\hspace{0.17em}\sin \hspace{0.17em}{q}_3-L_4{R}_4\hspace{0.17em}\sin \hspace{0.17em}{q}_4-L_5{R}_5\hspace{0.17em}\sin \hspace{0.17em}{q}_5\\ 0& R_2& -L_3{R}_3\hspace{0.17em}\cos \hspace{0.17em}{q}_3-L_4{R}_4\hspace{0.17em}\cos \hspace{0.17em}{q}_4-L_5{R}_5\hspace{0.17em}\cos \hspace{0.17em}{q}_5\end{array}\right]$$ (2)

The Jacobian J_x,y was used for computing the V_UCM and V_ORT of the five elemental variables describing the 2D paw position (see below).

Table 2.

Ranges of elemental variables describing paw position in the global coordinate system for fore- and hindlimbs during locomotor tasks

Cat	Limb	Rq1, mm	Rq2, mm	Rq3, °	Rq4, °	Rq5, °
AG	Fore	63	47	81	85	125
	Hind	61	70	77	79	112
BL	Fore	75	37	84	117	137
	Hind	55	41	96	91	96
BU	Fore	81	45	104	108	152
	Hind	42	53	98	92	95
C8	Fore	72	63	95	118	169
	Hind	62	57	77	63	109
FM	Fore	73	60	73	113	106
	Hind	44	45	69	76	84

The Jacobians J_x and J_y for computing V_UCM and V_ORT for paw horizontal and vertical positions (potential performance variables 2 and 3) in the space of the same five elemental variables corresponded to the first and second rows of matrix (2).

The Jacobian J_L for computing V_UCM and V_ORT for limb length (potential performance variable 4) in the space of three limb segment angles q₃, q₄, and q₅ (Fig. 1) is:

$$\begin{gathered} {\mathit{J}}_{\mathit{L}}=[-\frac{L_3}{L}(x_L\hspace{0.17em}\sin \hspace{0.17em}{q}_3+y_L\hspace{0.17em}\cos \hspace{0.17em}{q}_3)-\frac{L_4}{L}(x_L\hspace{0.17em}\sin \hspace{0.17em}{q}_4+y_L\hspace{0.17em}\cos \hspace{0.17em}{q}_4) \\ -\frac{L_5}{L}(x_L\hspace{0.17em}\sin \hspace{0.17em}{q}_5+y_L\hspace{0.17em}\cos \hspace{0.17em}{q}_5)] \end{gathered}$$ (2a)

where L₃, L₄, and L₅ are lengths of three limb segments in the fore- or hindlimb (Fig. 1); L = $\sqrt{x_L^2\hspace{0.17em}+\hspace{0.17em}{y}_L^2}$ is the fore- or hindlimb length; x_L = L₃ cos q₃ + L₄ cos q₄ + L₅ cos q₅ is the horizontal hind- or forepaw coordinate with respect to the hip or shoulder joint; y_L = −(L₃ sin q₃ + L₄ sin q₄ + L₅ sin q₅) is the vertical hind- or forepaw coordinate with respect to the hip or shoulder joint.

The Jacobian J_θ for computing V_UCM and V_ORT for limb orientation (potential performance variable 5) in the space of three limb segment angles q₃, q₄, and q₅ (Fig. 1) is:

$$\begin{gathered} {\mathit{J}}_{\theta }=[-L_3(y_L\hspace{0.17em}\sin \hspace{0.17em}{q}_3-x_L\hspace{0.17em}\cos \hspace{0.17em}{q}_3)-L_4(y_L\hspace{0.17em}\sin \hspace{0.17em}{q}_4-x_L\hspace{0.17em}\cos \hspace{0.17em}{q}_4) \\ -L_5(y_L\hspace{0.17em}\sin \hspace{0.17em}{q}_5-x_L\hspace{0.17em}\cos \hspace{0.17em}{q}_5)] \end{gathered}$$ (2b)

For each potential performance variable, the corresponding Jacobian was used to linearize limb forward kinematics in the vicinity of the limb reference configuration at each time instant of jth walking cycle (Scholz and Schoner 1999):

$${\mathit{r}}_{\mathit{j}}-\overline{\mathit{r}}={\mathit{J}}_{\mathit{j}}({\mathit{q}}_{\mathit{j}}-\overline{\mathit{q}})$$ (3)

where r_j and q_j are vectors of performance variable components and elemental variables in cycle j and r̄ and q̄ are vectors of performance and elemental variables in the limb reference configuration, respectively; the reference limb configuration was set for each normalized cycle time instant as the mean limb configuration across all walking cycles within the task and animal (Figs. 2 and 3). For each time instant, the projection of deviations of the limb segment configuration vector from the limb reference configuration vector (Δq_j = q_j − q̄) onto the Jacobian's null-space is computed as:

$$\Delta {\mathit{q}}_{\parallel j}={\displaystyle \sum _{i=1}^3{\mathit{e}}_i\Delta {\mathit{q}}_{\mathit{j}}}$$ (4)

whereas the component of limb configuration deviations perpendicular to the null-space is

$$\Delta {\mathit{q}}_{\perp j}=\Delta {\mathit{q}}_{\mathit{j}}-\Delta {\mathit{q}}_{\parallel j}$$ (5)

where the basis vector e defines the null-space of the Jacobian (0 = Je), for which deviations of limb segment configurations in the vicinity of the reference limb configuration do not affect paw position. The amount of variance per DOF within the UCM (good variance) is

$$V_{\text{UCM}}=\frac{1}{N_{\text{UCM}}(N-1)}{\displaystyle \sum _{j=1}^N\Delta q_{\parallel j}^2}$$ (6)

where N is the number of cycles and N_UCM is the dimension of the UCM. The amount of variance per DOF in the space orthogonal to the null-space (bad variance) is

$$V_{\text{ORT}}=\frac{1}{N_{\text{ORT}}(N-1)}{\displaystyle \sum _{j=1}^N\Delta q_{\perp j}^2}$$ (7)

where N_ORT is the number of DOF for the space orthogonal to the UCM. The amount of total variance per DOF is

$$V_{\text{TOT}}=\frac{1}{N_{\text{TOT}}(N-1)}{\displaystyle \sum _{j=1}^N\Delta q_j^2}$$ (8)

where N_TOT is the number of DOF of the elemental variables.

Fig. 2.

Examples of forelimb elemental (left) and potential performance variables (right) during 30 cycles of ladder walking from a representative cat BL. Each individual cycle is represented by a thin black line. Vertical dashed line in each panel separates the swing and stance phase. Left (from top to bottom): Cartesian horizontal q_SX (after detrending, see text) and vertical q_SY coordinates of the shoulder, orientation angles of the upper arm q_UA, forearm q_FA and forefoot q_FF (for definition of the elemental variables, see Fig. 1). Right (from top to bottom): Cartesian horizontal q_FPX (after detrending) and vertical q_FPY coordinates of the forepaw, forelimb length L_FORELIMB and forelimb orientation θ_FORELIMB (for definition of the potential performance variables see Fig. 1). The gray line in each panel represents the mean of the linearized forward kinematics solutions computed for 30 walking cycles (see Eq. 3).

Fig. 3.

Examples of hindlimb elemental (left) and potential performance variables (right) during 30 cycles of ladder walking from a representative cat BL. Each individual cycle is represented by a thin black line. Vertical dashed line in each panel separates the swing and stance phase. Left (from top to bottom): Cartesian horizontal q_HX (after detrending, see text) and vertical q_HY coordinates of the hip, orientation angles of the thigh q_T, shank q_S and hindfoot q_HF (for definition of the elemental variables, see Fig. 1). Right (from top to bottom): Cartesian horizontal q_HPX (after detrending) and vertical q_HPY coordinates of the hindpaw, hindlimb length L_HINDLIMB and hindlimb orientation θ_HINDLIMB (for definition of the potential performance variables, see Fig. 1). The gray line in each panel represents the mean of the linearized forward kinematics solutions computed for 30 walking cycles (see Eq. 3).

The index of synergy ΔV was computed as

$$\Delta V=(V_{\text{UCM}}-V_{\text{ORT}})/V_{\text{TOT}}$$ (9)

The range of ΔV changes depends on the dimensions of V_UCM, V_ORT and V_TOT spaces that in turn depend on dimensions of the performance variable and the number of elemental variables. It follows from Eqs. 6–9 that values of ΔV for 2D paw position (potential performance variable 1) and five elemental variables (see above) can change between −2.50 and 1.67 (N_UCM = 3, N_ORT = 2, N_TOT = 5). For one-dimensional potential performance variables 2 and 3 (horizontal and vertical paw positions analyzed separately) and the same five elemental variables, ΔV can change between −5 and 1.25 (N_UCM = 4, N_ORT = 1, N_TOT = 5). For one-dimensional potential performance variables 4 and 5 (limb length and limb orientation) and three elemental variables of segment angles, ΔV can change between −3 and 1.5 (N_UCM = 2, N_ORT = 1, N_TOT = 3). ΔV values higher than zero at a given time instant indicate that the potential performance variable (e.g., paw position or limb length) is stabilized by covarying across cycles elemental variable changes at this time instant.

The ΔV, ΔV_UCM and ΔV_ORT were computed for every percentage of the normalized swing phase time of each walking task and limb for each cat and potential performance variable.

Statistics

To test the effects of locomotion task, limb and swing time on the ΔV, V_UCM and V_ORT, a three-way 2 (task: regular walking, ladder walking) by 2 (limb: forelimb, hindlimb) by 10 (time: 10% time bins of swing) repeated-measures ANOVA was performed for each dependent variable. Since absolute values of ΔV are constrained by a minimum and maximum value as described above, leading to a reduction of the ΔV variance when ΔV values approach their limits, the Fisher z transformation of ΔV was performed prior to ANOVA analysis. When ANOVA analysis indicated significant results, post hoc comparisons were performed with the Bonferroni test. To test whether ΔV was significantly different from zero and if V_UCM and V_ORT were different from each other, the nonparametric Wilcoxon matched-pairs test was used. Statistical analysis was performed using software STATISTICA 7 (StatSoft, Tulsa, OK). Significance level was set at 0.05.

RESULTS

General Kinematic Characteristics of Regular and Ladder Walking

The number of analyzed walking cycles per cat, limb and walking condition was 30. Walking speed, determined as the ratio of the horizontal displacement of the limb suspension point (hip or shoulder) over the cycle time, was not statistically different among the combinations of locomotor tasks and limbs; speed ranged from 0.71 ± 0.17 m/s for hindlimb cycles of ladder walking to 0.76 ± 0.22 m/s for forelimb cycles of ladder walking (repeated-measures ANOVA, F_1,4 = 0.07–0.83, P = 0.414–0.802, Table 3). Cycle times for different walking tasks and limbs were found to be between 685 ± 151 ms (forelimb, ladder walking) and 710 ± 37 ms (hindlimb, regular walking), and no significant difference in cycle time was detected among tasks and limbs (repeated-measures ANOVA, F_1,4 = 0.16–0.23, P = 0.66–0.71). Forelimb swing times were 253 ± 18 ms and 247 ± 40 ms for regular and ladder walking, respectively, whereas the corresponding hindlimb swing durations were longer, 285 ± 5 ms and 277 ± 47 ms (repeated-measures ANOVA, F_1,4 = 229.4, P < 0.05, Table 3). Duty factor (the ratio of stance time and cycle time) was accordingly lower for hindlimbs with the values of 0.60 ± 0.02 for regular and ladder walking than for forelimbs during regular and ladder walking: 0.64 ± 0.04 ms and 0.63 ± 0.03 ms, respectively (repeated-measures ANOVA, F_1,4 = 15.25, P < 0.05). Thus general timing characteristics of fore- and hindlimb movements during regular and ladder walking were in agreement with previously published data on cat locomotion (Beloozerova et al. 2010; Miller et al. 1975; Prilutsky et al. 2005).

Table 3.

General kinematic characteristics of fore- and hindlimbs during regular and ladder walking

Limb	Locomotor Task	Cycle Time, ms	Swing Time, ms	Duty Factor	Walking Speed, m/s
Forelimb	Regular	709 ± 67	253 ± 18	0.64 ± 0.04	0.72 ± 0.10
Hindlimb	Regular	710 ± 37	285 ± 5	0.60 ± 0.02	0.72 ± 0.10
Forelimb	Ladder	685 ± 151	247 ± 40	0.63 ± 0.03	0.76 ± 0.22
Hindlimb	Ladder	698 ± 124	277 ± 47	0.60 ± 0.01	0.71 ± 0.17

Values are means ± SD; n = 5 cats.

All cats demonstrated highly stereotypic kinematic patterns of the elemental variables: vertical and detrended horizontal coordinates of the shoulder and hip joints, fore- and hindlimb segmental angles (Figs. 2 and 3, left), as well as the potential performance variables: vertical and detrended horizontal coordinates of the fore- and hindpaw, limb length and limb orientation (Figs. 2 and 3, right) during both regular and ladder walking. Stereotypic kinematics across walking cycles within each cat were evident from the relatively small variability of the kinematic patterns and in similarity of the temporal changes in kinematic variables, i.e., their peaks and troughs were closely aligned across cycles (Figs. 2 and 3).

Linearization of limb forward kinematics in the vicinity of the limb reference configuration performed for each potential performance variable and the corresponding set of elemental variables (Eq. 3) gave close approximation of experimentally recorded 2D paw position, limb segment length and orientation for ladder walking (Figs. 2 and 3, right, gray lines) and regular walking (not shown).

Potential Performance Variables

2D paw position in 5D space of elemental variables.

Patterns of good variance of limb kinematics (V_UCMx,y), bad variance (V_ORTx,y), and synergy index (ΔV_x,y) were similar between regular and ladder walking (see a representative example for forelimbs of ladder walking in Fig. 4): V_UCM achieved its maximum values in midstance, V_ORT had one peak in midswing and values close to zero in stance, whereas ΔV_x,y had high positive values in early and late swing and during most of stance and was close to zero in midswing. Since this study focuses on paw trajectory during swing, the stance phase will not be considered further.

Fig. 4.

Representative patterns of index of synergy (ΔV_x,y, thick line), “good” variance (V_UCM, thin continuous line) and “bad” variance [variance orthogonal to the UCM (V_ORT), dashed line] calculated for forelimbs of one cat during a cycle of ladder walking. The left vertical axis is for V_UCM and V_ORT; the right vertical axis shows values of ΔV_x,y. The vertical dashed line separates the swing and stance phase. The top horizontal dotted line indicates the theoretical maximum of ΔV_x,y (1.67, see text). Cat BL, ladder walking, N = 30.

Synergy index ΔV_x,y patterns averaged across all cats within each task and limb had positive values throughout the whole swing phase for hindlimbs during regular walking and forelimbs during ladder walking (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05), indicating that the 2D paw position could be stabilized throughout the swing. Index of synergy ΔV_x,y for forelimb during regular walking and for hindlimbs during ladder walking was different from zero only in the beginning or first half of swing and late swing [Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05, Fig. 5, mean (black line) ± SE (gray shadow)]. Thus the 2D forepaw position during regular walking and hindpaw position during ladder walking were not stabilized in midswing. ANOVA performed on z-transformed ΔV_x,y values demonstrated a significant effect of swing time (F_9,36 = 13.7, P < 0.05, Fig. 6A): the index of synergy ΔV_x,y averaged across limbs and walking conditions for each time bin was higher at the beginning and end of swing than in midswing. There was a significant time-limb interaction effect on index of synergy ΔV_x,y (F_9,36 = 7.55; P < 0.05), and no significant effects of locomotion task or limb.

Fig. 5.

Index of synergy ΔV_x,y as a function of the normalized swing time [mean (black line) ± SE (gray shadow), 5 cats] computed for the 2D paw position in space of 5 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. The top horizontal dotted line indicates the theoretical maximum of ΔV_x,y (1.67); the bottom dotted line indicates the theoretical minimum of ΔV_x,y (−2.5, see text for details). Solid diamonds indicate 10% time bins of swing for which ΔV_x,y is significantly different from zero (P < 0.05).

Fig. 6.

Z-transformed index of synergy values averaged across limbs and walking conditions for each swing time bin (mean ± SE, 5 cats). A: index of synergy ΔV_x,y. B: index of synergy ΔV_y. C: index of synergy ΔV_x. D: index of synergy ΔV_L. E: index of synergy ΔV_θ. Asterisks and horizontal brackets indicate statistical difference between time bins (P < 0.05).

The obtained partition of synergy index ΔV_x,y into V_UCMx,y and V_ORTx,y showed that V_UCMx,y (not affecting 2D paw position) exceeded V_ORTx,y (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05) in those phases of swing in which ΔV_x,y was positive (Fig. 7). ANOVA performed on z-transformed V_UCM values revealed significant effect of swing time (F_9,36 = 6.48, P < 0.05), time-limb interaction (F_9,36 = 2.7, P < 0.05), and no significant effect of limb or walking condition. V_ORT was significantly affected by swing time (F_9,36 = 19.9, P < 0.05) and by time-limb (P < 0.05, F_9,36 = 5.25) and time-task (P < 0.05, F_9,36 = 2.63) interactions. The limb-task interaction effect was not significant (P = 0.05, F_1,4 = 7.68).

Fig. 7.

Patterns of good (V_UCM, dashed line) and bad variance (V_ORT, solid line) (mean ± SE, 5 cats) computed for the 2D paw position in space of 5 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. X's indicate 10% time bins of swing for which V_UCM is significantly greater than V_ORT (P < 0.05).

Paw vertical position in 5D space of elemental variables.

Mean synergy index ΔV_y computed across all cats for each task and limb had positive values (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05), approaching the theoretical maximum 1.25 throughout the entire swing for both limbs and walking tasks, except for hindlimb during midswing of ladder walking (Fig. 8). Thus forepaw and hindpaw vertical position was highly stabilized during swing phase of regular and ladder walking, whereas hindpaw vertical position was not stabilized in midswing (30–50% of swing time) of ladder walking. ANOVA performed on z-transformed ΔV_y values demonstrated a significant effect of swing time (F_9,36 = 6.45, P < 0.05); the index of synergy ΔV_y averaged across limbs and walking conditions for each time bin was higher in early swing (10%) than later in swing (30–100% of swing time, Fig. 6B). Locomotion task also significantly affected ΔV_y (F_1,4 = 12.2; P < 0.05), as the synergy index during regular walking was higher than during ladder walking. There were also significant time-limb (F_9,36 = 15.1; P < 0.05) and time-task (F_9,36 = 3.55; P < 0.05) interaction effects on ΔV_y.

Fig. 8.

Index of synergy ΔV_y as a function of the normalized swing time (mean ± SE, 5 cats) computed for the vertical paw position in space of 5 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. The top horizontal dotted line indicates the theoretical maximum of ΔV_y (1.25); the bottom dotted line indicates the theoretical minimum of ΔV_y (−5.0, see text for details). Solid diamonds indicate 10% time bins of swing for which ΔV_y is significantly different from zero (P < 0.05).

Patterns of V_UCM for the vertical paw position (Fig. 9) were generally similar to those of the 2D paw position (Fig. 7), which was not the case for V_ORTy values (Fig. 9) that were close to zero throughout most of swing, except for midswing of hindpaw during ladder walking.

Fig. 9.

Patterns of good (V_UCM, dashed line) and bad variance (V_ORT, solid line) (mean ± SE, 5 cats) computed for the vertical paw position in space of 5 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. X's indicate 10% time bins of swing for which V_UCM is significantly greater than V_ORT (P < 0.05).

Paw horizontal position in 5D space of elemental variables.

Values of synergy index ΔV_x were positive (horizontal paw position was stabilized) in early and late swing of ladder walking for both fore- and hindpaw; they were also positive in mid- and late swing of hindpaw during regular walking (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05, Fig. 10). Synergy index was not different from zero during the entire swing for forepaw of regular walking. Z-transformed ΔV_x values were significantly affected by swing time (ANOVA, F_9,36 = 14.88, P < 0.05); the index of synergy ΔV_x averaged across limbs and walking conditions for each time bin was higher in early (0–40%) and late swing (80–100%) than in midswing (50–70%, Fig. 6C). There was a significant time-limb interaction (F_9,36 = 8.54; P < 0.05) and time-task interaction effect (F_9,36 = 2.49; P < 0.05) on index of synergy ΔV_x, and no significant effects of locomotion task or limb. Patterns of V_UCMx and V_ORTx were generally similar to those determined for the 2D paw position with generally higher values of V_ORTx (not shown).

Fig. 10.

Index of synergy ΔV_x as a function of the normalized swing time (mean ± SE, 5 cats) computed for the horizontal paw position in space of 5 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. The top horizontal dotted line indicates the theoretical maximum of ΔV_x (1.25); the bottom dotted line indicates the theoretical minimum of ΔV_x (−5.0, see text for details). Solid diamonds indicate 10% time bins of swing for which ΔV_x is significantly different from zero (P < 0.05).

Limb length in 3D space of elemental variables.

Length of forelimb and hindlimb was stabilized during midswing of regular and ladder walking as index of synergy was positive (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05) and close to the maximum value 1.5 (Fig. 11). Forelimb length was also stabilized in late swing of regular and ladder walking. ANOVA performed on z-transformed ΔV_L values showed that limb length stabilization was significantly affected by swing time (F_9,36 = 12.74, P < 0.05, Fig. 6D); the index of synergy ΔV_L averaged across limbs and walking conditions for each time bin was higher in midswing (time bins 5 and 6) than in early or late swing. Limb also had a significant effect on index of synergy (F_1,4 = 34.19; P < 0.05): ΔV_L for forelimb was higher than for hindlimb. There was a significant time-limb interaction (F_9,36 = 4.57; P < 0.05) and time-task interaction effects on index of synergy ΔV_L (F_9,36 = 4.79; P < 0.05).

Fig. 11.

Index of synergy ΔV_L as a function of the normalized swing time (mean ± SE, 5 cats) computed for the limb length in space of 3 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. The top horizontal dotted line indicates the theoretical maximum of ΔV_L (1. 5); the bottom dotted line indicates the theoretical minimum of ΔV_L (−3.0, see text for details). Solid diamonds indicate 10% time bins of swing for which ΔV_L is significantly different from zero (P < 0.05).

High levels of limb length stabilization in midswing resulted from elevated V_UCML variance during that phase of movement, while V_ORTL values remained relatively constant throughout swing (Fig. 12).

Fig. 12.

Patterns of good (V_UCM, dashed line) and bad variance (V_ORT, solid line) (mean ± SE, 5 cats) computed for the limb length in space of 3 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. X's indicate 10% time bins of swing for which V_UCM is significantly greater than V_ORT (P < 0.05).

Limb orientation in 3D space of elemental variables.

Values of synergy index ΔV_θ computed for orientation of fore- and hindlimbs during regular and ladder walking were often negative (Wilcoxon matched-pairs test, Z = 2.02, n = 5, P < 0.05) except for short time periods in early swing (fore- and hindlimb during ladder walking) and late swing (forelimb during regular and ladder walking, Fig. 13). Negative synergy index indicates that leg orientation was not actively stabilized during most of swing. ANOVA performed on z-transformed ΔV_θ values demonstrated a significant effect of swing time (F_9,36 = 3.43, P < 0.05, Fig. 6E); the index of synergy ΔV_θ averaged across limbs and walking conditions for each time bin was higher at 10% of swing time than at 40–60% of swing phase (Fig. 6E). Although there was no significant effect of locomotor task or limb on ΔV_θ, there was significant time-limb interaction (F_9,36 = 5.5; P < 0.05) and time-task interaction effects (F_9,36 = 5.9; P < 0.05) on index of synergy ΔV_θ.

Fig. 13.

Index of synergy ΔV_θ as a function of the normalized swing time (mean ± SE, 5 cats) computed for the limb orientation in space of 3 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. The top horizontal dotted line indicates the theoretical maximum of ΔV_θ (1. 5); the bottom dotted line indicates the theoretical minimum of ΔV_θ (−3.0, see text for details). Solid diamonds indicate 10% time bins of swing for which ΔV_θ is significantly different from zero (P < 0.05).

Large negative values of synergy index ΔV_θ during midswing of regular and ladder walking for both forelimbs and hindlimbs resulted from the peak of V_ORTθ variance in midswing, while V_UCMθ was not changing substantially during swing (Fig. 14).

Fig. 14.

Patterns of good (V_UCM, dashed line) and bad variance (V_ORT, solid line) (mean ± SE, 5 cats) computed for the limb orientation in space of 3 elemental variables. A: forelimb, regular walking. B: hindlimb, regular walking. C: forelimb, ladder walking. D: hindlimb, ladder walking. X's indicate 10% time bins of swing for which V_UCM is significantly smaller than V_ORT (P < 0.05).

DISCUSSION

This study is the first to demonstrate that cats organize the kinematics of locomotion based on the principle of abundance: they use variable time profiles of elemental variables (limb generalized coordinates) to ensure relatively reproducible trajectory of several limb kinematic variables during the swing phase. The indexes of synergy ΔV_x,y, ΔV_x, ΔV_y, ΔV_L, and ΔV_θ computed for each potential performance variable indicate that the vertical paw position was most consistently stabilized throughout the swing phase of regular and ladder walking for both fore- and hindpaw (Fig. 8). Since the horizontal paw position was also stabilized in initial and late swing of ladder walking and the horizontal hindpaw position was stabilized in late and midswing of regular walking (Fig. 10), it can be concluded that cats stabilized 2D paw position most consistently in early and late swing (see also Fig. 5 and Fig. 6, A and C). Furthermore, length of fore- and hindlimb was stabilized (in intrinsic, body-centered rather than in extrinsic, world-centered coordinates) in midswing of both walking tasks (Figs. 6D and 11). The limb length stabilization in midswing appears to occur at the expense of greater V_ORTθ variability of limb orientation (Figs. 6, D and E, 12, and 14). Taken together, the above results suggest that the nervous system might stabilize different sets of performance variables at different movement phases.

None of the three hypotheses formulated in the Introduction received unambiguous support for the three identified performance variables: 2D paw position, vertical paw position and limb length. Indeed, while the cats stabilized the vertical paw trajectory (hypothesis 1), they did so for the entire swing phase only in three out of four studied combinations of limbs and walking conditions; hindpaw vertical position during ladder walking was stabilized only over the early and late phases of swing, not during the midswing. Similar conclusions could be drawn about stabilization of 2D paw position that occurred in early and late swing in three out of four limb-walking task combinations, and about limb length stabilization observed in midswing. Since synergy index ΔV_y during regular walking exceeded that during ladder walking, hypothesis 2 was not supported for vertical paw position, although it was supported for 2D paw position and limb length. Finally, hypothesis 3 was supported for limb length stabilization because ΔV_L for forelimb was higher than for hindlimb; however, hypothesis 3 was not supported for 2D and vertical paw positions.

There are two major views on the problem of motor redundancy. One of them follows the classical formulation of Bernstein (1967) that the CNS “eliminates redundant degrees-of-freedom”; it assumes that single solutions are generated by the CNS, possibly based on an optimization principle (reviewed in Prilutsky and Zatsiorsky 2002; Seif-Naraghi and Winters 1990). The alternative view is that no DOF are ever eliminated, but they are all used in a covarying fashion to ensure stable performance with respect to important variables (principle of abundance, reviewed in Latash 2012). The UCM hypothesis has been developed based on the idea of abundance (Scholz and Schoner 1999) and applied to a variety of levels of analysis and tasks performed by different human populations (reviewed in Latash et al. 2007). The present study is the first one to show that the principle of abundance is not specific to human motor control, and that the method of UCM analysis can be successfully used to analyze movements of other animals. Animal models in combination with UCM analysis permit now the mechanistic studies of neural structures responsible for organizing motor synergies.

Several studies applied the method of analysis of joint covariation to whole body kinematics during human cyclic actions, such as rhythmic swaying (Freitas et al. 2006) and locomotion (Krishnan et al. 2013; Verrel et al. 2010). In particular, the latter two studies documented multijoint coordination stabilizing features of the step parameters and foot trajectory over the swing cycle.

The control of paw 2D position prior to paw placement during walking may be expected, given that cats [according to preliminary results (Rivers et al. 2010)] and humans (Patla and Vickers 2003) fix gaze at an intended stepping location two strides (in cats) or one stride (in humans) ahead of forepaw/foot placement. In addition, humans use the lower visual field to monitor foot trajectory during late swing of walking and use this information to improve accuracy of visually guided stepping (Marigold and Patla 2008; Reynolds and Day 2005). During accurate stepping, cats shift and rotate the head closer to the ground, presumably for a better view of the intended paw placement position (Beloozerova et al. 2010). Ensuring low variability of the paw 2D coordinates immediately prior to ground contact could also be expected because accurate paw placement is necessary for dynamically stable locomotion. Walking can be considered dynamically unstable if the vertical projection of the extrapolated center of mass of the body (that incorporates center of mass position and velocity) is beyond the base of support (Hof et al. 2005). To recover balance in this situation, the foot must be placed in front of the extrapolated center of mass and “in order to walk stable and in a straight path, the foot … has to be placed with an accuracy of a few millimetres” (p. 257: Hof et al. 2007). A recent study on dynamic stability in walking cats (Farrell et al. 2011, 2014) indicates that the cat is dynamically unstable in the sagittal plane (the extrapolated center of mass is in front of the base of support) during the short phases of double support by diagonal fore- and hindlimbs. The forward fall of the cat is prevented by placement of the swing forepaw in front of the extrapolated center of mass during both regular walking and walking with a constrained stance width (Farrell et al. 2011, 2014). Hindpaw accurate placement in front of the extrapolated center of mass in the frontal plane may also be important for maintaining body lateral stability at the transition from a double support period by ipsilateral fore- and hindlimbs to a three-legged support period (Farrell et al. 2014). Thus covarying limb segment angles (and possibly other elemental variables) to ensure reproducible paw position prior to paw placement seems important not only for accurate stepping, but also for regular walking.

The need to stabilize paw position during the first half of swing is less intuitive, especially because paw horizontal position is stabilized much less for hindlimb during regular walking (Fig. 10) or not at all (forelimb during regular walking, Fig. 10). This result can be explained by 1) considering that paw placement location is planned prior to swing onset (Hollands and Marple-Horvat 2001; 1996; Patla and Vickers 2003); and 2) assuming that accuracy of paw preplanned stepping depends to a large extent on accurate estimation of paw initial position, which is not under direct visual control for either fore- or hindlimb in early swing.

More vigorous stabilization of 2D paw position in final swing stages may be needed to adjust paw position prior to ground contact since the paw placement location estimated prior to swing initiation may be outdated. Feedback corrections of paw trajectory seem more useful in late swing than in midswing if no major motion perturbation or obstacle appearance is expected. In such a situation, humans start correcting foot trajectory for accurate target stepping in the last 120 ms of the swing phase (Reynolds and Day 2005), the period consistent with the reaction time to simple visual stimuli. In cats this type of reaction time is faster, 60–70 ms (Pettersson 1990), given shorter distances for propagation of neural signals. With reaction time of 65 ms, paw trajectory corrections would be expected to start at about 75–80% of swing time because, in our experiments, the swing durations were between 247 and 285 ms (Table 3). The obtained high positive values of the index of synergy ΔV_x,y for 2D paw position in late swing agree with these estimates (see Fig. 5).

A recent study of kinematic synergies stabilizing the foot trajectory during human locomotion has produced a different time profile of the synergy index: the index showed a peak in midswing and dropped close to the toe-off and heel-strike phases (Krishnan et al. 2013). While there are major differences in the design of the two studies, given that only a handful of studies applied the UCM-based method for analysis of kinematic synergies, comparing the results is warranted. In particular, the Krishnan et al. study analyzed synergy indexes with respect to the mediolateral foot trajectory only, while our study explored the paw trajectory in two dimensions in the sagittal plane. In addition, the bipedal walking of humans is associated with higher demands for stability and a higher ability to rely on vision for guiding foot placement during stepping, in particular compared with motion of the hindlimbs in cats. Also, the differences in the synergy indexes may reflect different ecological values of accurate stepping in cats and humans. While cats frequently navigate complex terrains with highly constrained support surfaces, modern humans typically walk in conditions where accurate foot placement is not crucial. In contrast, in midswing, high stabilization of the mediolateral foot trajectory in humans may reflect the desire to avoid collision with the supporting leg.

One of the identified performance variables in this study, the limb length, was also stabilized mostly in midswing (Fig. 11). This stabilization could be related to the requirement of safe paw-ground clearance during midswing. High level of stabilization of vertical paw position throughout swing (Fig. 8) could contribute to both limb length stabilization in midswing and 2D paw position stabilization in early and late swing.

ANOVA revealed significantly higher index of synergy ΔV_y during regular walking than during ladder walking, whereas ΔV_L was found to be greater for forelimb than hindlimb. It has been demonstrated that, although forelimb kinematics during regular and skilled ladder walking are virtually identical (Beloozerova et al. 2010), the neural control mechanisms of these tasks are likely different. Regular walking can be performed without visual feedback (Beloozerova and Sirota 1993a), after lesion of motor cortex (Beloozerova and Sirota 1993a; Liddell and Phillips 1944) and in decerebrate (Shik et al. 1966) and spinal (Rossignol 2006) cats, whereas walking on a horizontal ladder is impossible in those conditions. Also, the modulation of neuronal activity from forelimb and hindlimb representations in the motor cortex is typically much higher during ladder walking (Armstrong and Marple-Horvat 1996; Beloozerova et al. 2010; Beloozerova and Sirota 1993a, 1993b; Drew et al. 2008). These data suggest that successful accurate stepping by the forelimbs requires visual input, which is integrated with motor cortical output. Thus different neural mechanisms involved in control of fore- and hindlimb movements during regular walking and accurate stepping appear to have differential effects on stabilization of paw position and limb length in several parts of the swing phase. The neural mechanisms responsible for hindpaw position stabilization may involve the asymmetric coupling between fore- and hindlimb locomotor pattern-generating networks (Juvin et al. 2012) and entrainment of hindlimb muscle activity by forelimbs during walking (Akay et al. 2006), also shown during postural corrections (Deliagina et al. 2006).

The combination of the nearly identical kinematics (Beloozerova et al. 2010) and modulation of indexes of synergy values (ΔV_y, ΔV_x,y, ΔV_L, the present study) for the fore- and hindlimb swing trajectories during regular and ladder stepping (Figs. 5, 8, 11) fit the scheme of the control of redundant systems that assumes the presence of two major groups of variables (Latash 2010; Latash et al. 2005). One of them defines the time evolution of important task-specific performance variables, likely by generating neural signals that translate into trajectories of referent values for those variables. The other defines stability properties of those variables. Several studies of both kinematic and kinetic variables in humans have shown that the same performance can be associated with significantly different indexes of covariation among elemental variables (Freitas and Scholz 2009; Klous et al. 2011; Olafsdottir et al. 2005). Our present study shows that the ability of the CNS to modulate stability properties without changing the overall movement pattern is also present during cat locomotion.

Finally, it is important to note that the revealed high stabilization of three performance variables during fore- and hindlimb swing, 2D paw position, vertical paw position and limb length, is caused by different contributions of V_UCM and V_ORT. For example, high stabilization of paw 2D position in early and late swing was caused by the low V_ORT rather than high V_UCM (cf., Figs. 5 and 7), largest positive values of index of synergy ΔV_x,y occur in phases similar to those in which V_ORT had the smallest values. This observation suggests that neural commands responsible for strong stabilization of paw trajectories during specific phases of swing tend to reduce deviations of limb segment configurations that affect paw position instead of increasing variability of segment angles that does not influence paw trajectory. This strategy seems to promote less variable, more stereotypic joint angle patterns in locomotion phases where stabilization of swing paw position is important. On the other hand, high values of index of synergy for limb length ΔV_L in midswing (Fig. 11) resulted from increased V_UCM and small changes in V_ORT (Fig. 12), indicating the opposite strategy of stabilization of this performance variable. Note that human experiments have provided evidence for increased V_UCM during movements to uncertain targets and practice in conditions of instability (de Freitas et al. 2007; Freitas and Scholz 2009; Wu et al. 2012). The differences in patterns of V_UCM and V_ORT for endpoint limb position between human movements and cat walking could reflect relatively high dynamic stability of quadrupedal locomotion (Farrell et al. 2011; Farrell et al. 2014).

In conclusion, we would like to acknowledge certain shortcomings of the present study. The first, and most obvious, one is the small number of animals that could potentially preclude us from reaching significance in some statistical tests. We would like to mention, however, that a large number of tests, in particular those directly related to testing the specific hypotheses, did lead to significant results. Hence, we can conclude that the sample sizes in our comparisons were sufficiently large. Second, we analyzed only a subset of kinematic variables potentially important for locomotion. For example, trajectory of the center of mass was not analyzed as a potential variable stabilized by multijoint synergies. This is partly due to problems in building a kinematic model that would link center of mass velocity to joint velocities, given that the center of mass location within the body migrates during locomotion and the spine kinematics cannot be easily reduced to a few fixed joint rotations. Third, it is possible that, due to temporal variability across strides, comparability of limb postures at specific time instances of swing could be compromised, particularly in midswing, when joint velocities are relatively high. This could have potentially had an impact on our analyses by adding spatial variance related to imprecise time alignment across strides. These effects, however, were likely small, because the temporal variability of elemental and potential performance variables was low (Figs. 2 and 3). Fourth, we used only one method of assessment of multijoint synergies. Other methods (for example, Muller and Sternad 2003; Verrel 2011; Yen and Chang 2010) could provide complementary information on the control of the kinematics of cat locomotion.

GRANTS

This work was partly supported by National Institutes of Health Grants HD-32571 and EB-012855 (to B. I. Prilutsky) and NS-058659 (to I. N. Beloozerova), and the Center for Human Movement Studies at Georgia Tech.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: A.N.K., B.J.F., I.N.B., and B.I.P. performed experiments; A.N.K. and B.J.F. analyzed data; A.N.K., I.N.B., M.L.L., and B.I.P. interpreted results of experiments; A.N.K. prepared figures; A.N.K. and B.I.P. drafted manuscript; A.N.K., B.J.F., I.N.B., M.L.L., and B.I.P. approved final version of manuscript; B.J.F., I.N.B., M.L.L., and B.I.P. edited and revised manuscript; I.N.B., M.L.L., and B.I.P. conception and design of research.