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1 Centre for Neuroscience and Department of Physiology, University of Alberta, Edmonton, Alberta, Canada
2
Department of Bioengineering, University of Utah, Salt Lake City, UT, USA
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Abstract |
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(Received 24 May 2004;
accepted after revision 25 August 2004;
first published online 26 August 2004)
Corresponding author R. B. Stein: Centre for Neuroscience, University of Alberta, Edmonton, Alberta T6G 2S2, Canada.Email: richard.stein{at}ualberta.ca
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Introduction |
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Most studies on the role of afferents in proprioception have examined the encoding properties of single receptors in isolation. In some studies, ensembles have been compiled from separate recordings, allowing correlations between firing rate and variables such as joint position and velocity or a muscle's length and force (Loeb et al. 1985; Prochazka & Gorassini, 1998b; Jones et al. 2001; Cordo et al. 2002; Ribot-Ciscar et al. 2003). The implicit assumption is that the central nervous system assembles the sensory activity into a useful representation of variables, such as joint angle, and the representation of each joint is eventually combined to give a sense of the position of our limbs with respect to the body. Since this representation may require a combination of many neurones, single-unit recording techniques are not adequate to study proprioception directly.
In this study we recorded simultaneously from populations of neurones in the L6 and L7 dorsal root ganglia (DRG) of anaesthetized cats. This enabled us to study directly how sensory information is encoded into the firing rates of a population of neurones and how these firing rates may be decoded to predict the position of the limb in space. Neural recordings were made while the hindlimb was passively moved through a variety of trajectories, including random, cyclical and centre-out paths. We used multivariate, linear regressions to model the relationship between hindlimb kinematics and sensory activity. The results demonstrate that limb trajectories can be accurately reconstructed from less than 10 selected neurones. The Appendix presents a model that demonstrates how even a few muscle receptors can provide accurate information about the end-point of a limb in a physiologically plausible way, without the need for complex trigonometric calculations from individual joint angles. The Discussion considers some methodological limitations, as well as the functional implications of our results for the normal sense of position.
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Methods |
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Multichannel neural recording technique
The electrodes used in these experiments were arranged in a rectangular configuration with 5 rows of 10 electrodes, 1.5 mm in length and spaced 400 µm apart. In addition to providing many sites for recording action potentials, this dense arrangement of electrodes serves to anchor the implanted array among the densely packed cell bodies within the ganglion. The electrode arrays were connected to a 100- channel amplifier. The gain of the amplifiers was 5000 (bandwidth 250–7500 Hz) and signals from each electrode were sampled at 30 kHz. A Pentium class computer recorded and saved the signals in conjunction with a Neural Signal Acquisition System (NSAS; Cyberkinetics Inc.). This system required thresholds to be set on each channel and only saved brief (1 ms) segments of the signal around the time that the threshold was crossed (Guillory & Normann, 1999).
Single units were discriminated offline from the set of recorded waveforms on each electrode using a Matlab-based algorithm (Shoham et al. 2003). The waveforms were first projected onto their principle components (PC), and an expectation-maximization clustering algorithm then identified the number of clusters and their parameters (see Fig. 1).
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Sensory afferents were activated by palpation and manipulation of the hindlimb. The response properties were used to categorize each unit (Aoyagi et al. 2003). Briefly, the hip, knee, ankle and toes were moved manually to identify muscle and joint receptors. A hand-held vibrator (
140 Hz) was generally applied over the tendon or muscle belly to identify primary spindle afferents. Golgi tendon organs may have been missed, because the animals were deeply anaesthetized and the muscles were completely flaccid. Cutaneous receptors were identified by palpation (touch, pressure, pinch and vibration). Gentle blowing or focal touch was used to identify hair receptors. During each manipulation, 10 s recordings were made to document the waveform and response for each unit.
After the units on each electrode were categorized, various movements were applied to the foot manually or with a robotic manipulator. The manipulator had two DC servomotors (BE233DJ; Parker Hannifin, Rohnert Park, CA, USA) and was programmed to deliver repeatable movements. For example, to generate random movements, the manipulator moved through a series of positions selected at random from a rectangular grid of points in the sagittal plane and the velocity of each movement was also chosen at random over a range of speeds. The movements continued until all points in the grid had been reached so there was a uniform coverage of the workspace. In several experiments, the identification and application of movements were repeated several hours later. For example, in one experiment in which 60 units were initially recorded 22 of them were still present in a second series of movements applied more than 4 h later. Thus, over a third of the units could be recorded for at least 4 h.
Kinematic recording technique
Walking-like, centre-out movements (from a central point to eight points in the periphery) and random movements were studied, all of which were largely confined to the sagittal plane. For example, the random movements (Fig. 2) covered most of the physiological range of the cat's hindlimb in the anterior–posterior plane (30 cm) and in the vertical direction (20 cm), but only 1–2 cm in the medio-lateral plane. A U- shaped holder made of dental acrylic was fitted around the cat's paw, proximal to the metatarsophalangeal (MTP) joint. The top of the U was tied so that the paw was held securely. Any pressure on the skin was distributed widely and direct contact with the skin by the experimenters or the manipulator was minimized.
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The sampling rate of the 6D- Research system was 30 Hz and was well above the highest frequency components applied to the cat's paw (5–10 Hz). For the magnetic recordings we ensured that all instruments near the sensors, including sections of the spinal frame, contained no metal to avoid distorting the signals from the electromagnetic sensors. A synchronization pulse was used to align the neural and motion data offline.
A high-speed digital video camera (120 fields s–1, GRDV9800R, JVC Corp.) recorded the limb movements produced by the robotic manipulator. A light-emitting diode (LED) was used to synchronize the video with the neural data. White markers were glued to the skin over the iliac crest, and the joint centres of the hip, knee, ankle and MTP joints. The centroid of the marker was automatically located in each image of the video using custom Matlab (Mathworks, Inc.) software. The camera plane was parallel to the sagittal plane of the leg. Calibration markers were spaced 10 cm apart in the horizontal and vertical planes and used to calibrate the camera view. Parallax errors were compensated by scaling the segment vectors by the measured separation distance between the ankle and MTP markers (i.e. foot length, which is constant).
Hip, knee and ankle joint angles were computed from the digitized marker positions, extension corresponding to a positive angular displacement. The knee marker was not used, because the skin overlying the knee tends to slide over the joint. Instead, the knee-joint angle was calculated using eqn (1), which follows from the law of cosines.
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| (1) |
, orientation). Neural encoding
A multivariate linear regression was used to model the firing rate of each neurone as a function of kinematic variables of the hindlimb (neural encoding). The full procedure included three processing steps.
(1) The neural and kinematic data were aligned at the LED onset time. Neural firing rates were calculated using the filter in eqn (2).
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| (2) |
t, ti + t], and t is the sampling interval. Essentially, a contribution to the rate is added for the two nearest sample times for the kinematic variables in a way that all spikes are equally weighted and the mean time of the weights is the actual time of the spike. This method is similar to the partial binning methods previously described (Richmond et al. 1987; Schwartz, 1992; Stein et al. 2004). (2) The rate function was filtered with a critically damped, second-order, low-pass filter (Stein et al. 2004). The impulse response of this filter is an EPSP-like waveform (Jack et al. 1975). Rate constants between 15 and 30 rad s–1 were used, corresponding to time constants of 67–33 ms. Different filters and other time constants for the EPSP-like filter were also applied using the Matlab function ‘filt’. In general, longer time constants (more filtering) gave better fits, as expected. However, if the time constant was extended beyond the values cited, very little improvement was seen. The same filtering was also applied to the kinematic variables to avoid introducing relative time delays. Filtering was done after step (1) above to ensure that all spikes were given equal weight.
(3) The filtered firing rates were fitted to a weighted sum of position and velocity variables in each of three coordinate systems: Cartesian (x, y) and polar (r, ) coordinates for the toe sensor, and joint angles (hip, knee and ankle) for the limb. This allowed a comparison of the predictions in Cartesian, polar and joint angular coordinates. For example, the predicted firing rate (gi) for the ith neurone can be written in Cartesian coordinates:
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| (3) |
i n). Corresponding forms of eqn (3) were used to accommodate kinematics expressed in polar and joint angular coordinates. In joint coordinates, intersegmental angles (extension was taken as positive) were used to describe the limb position in the sagittal plane. In polar coordinates, the toe position and velocity were also expressed with respect to an origin at the hip, which was fixed in space. The variance accounted for (VAF) expressed as a percentage was used to evaluate the goodness of fit for each coordinate. Prediction of position in the sagittal plane requires combining coordinates and the root mean square (r.m.s.) error for the predictions was calculated. The coefficients of the linear encoding model (eqn (3)) describe the sensitivity of the neural response to each kinematic variable and linear correlation coefficients were also calculated for the relation between each kinematic variable and the firing rates. Neural decoding
A linear filter model was used to reconstruct the hindlimb trajectories from the ensemble of neural firing rates f. Equation (4) shows the form of the model for decoding the horizontal (x) position of the toe in Cartesian coordinates:
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| (4) |
is the predicted value of x at the time point j. This is the prediction from the filtered firing rates of n neurones, based on the present (j) and one previous (j–1) time point. In general, for L previous time points and n neurones, the decoding model takes the form:
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| (5) |
Equation (5) gives estimates of variables such as the x position and the velocity dx/dt. Integrating the velocity gives an independent estimate of the position. Using a weighted average of these two variables (eqn (6)) substantially improved the fit:
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| (6) |
is the new estimate using a weight of w. To start the process, 
. The weight w was varied between 0 and 1 to find the best estimate of 
. Note that for w = 0, only position information is used and for w = 1, only velocity information is used, except for the initial condition. Predictions were made for the other variables in each of the coordinate systems. Although eqn (6) is somewhat ad hoc, eqn (7) and the Appendix describe a related method for combining velocity and position information, based on the properties of muscle spindles, which respond to both length and velocity. Decoding with a leaky integrator (eqn (8)) having properties close to those found for muscle spindles gave equally good fits. |
Results |
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Figure 2A shows the path of the paw as it was moved pseudorandomly (dotted line). The movements were applied manually and covered most of the passive range of motion (about 15 cm forward, backward and upward). In Fig. 2B a robot applied movements as a series of point-to-point movements over a more restricted grid of points in the sagittal plane; see Methods and Stein et al. (2004). Firing rates were computed at the sampling times for the kinematics (see Methods). The response surface maps in Fig. 3 illustrate firing rates as a function of limb position (x, y) for the movements of Fig. 2A. Figure 3A shows the firing rate of a slowly adapting cutaneous afferent with a receptive field on the front of the knee (trends are highlighted by colour-coding). When the toe was raised, for example from 20 to 15 cm below the hip, its firing rate increased because the skin around the knee was stretched. Similarly, it fired faster when the toe was moved forward. Figure 3B shows the firing rate of a hamstring muscle spindle afferent that responded to knee extension and hip flexion. These movements were correlated with downward and forward movements of the toe. Thus, the firing rate of single sensory neurones can be correlated with movement of the toe with respect to the hip.
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and the direction is 
= tan–1(ai2/ai1). Figure 5 shows the distributions in one data set of all cells for which the VAF was more than 40%. There are a wide range of amplitudes and preferred directions.
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In contrast to the encoding of kinematic information in the firing rates of neurones, decoding involves the prediction of limb position from the firing patterns of a population of neurones. Various decoding methods have been proposed (Bialek et al. 1991; Salinas & Abbott, 1994; Schwartz, 1994; Wessberg et al. 2000; Scott et al. 2001; Serruya et al. 2002; Taylor et al. 2002), and we have chosen a linear filter similar to Wessberg et al. (2000) and Serruya et al. (2002). Figure 7 illustrates the results of decoding toe position in polar coordinates from the firing rates of 30 neurones. Polar coordinates were used since they gave rather better predictions than Cartesian coordinates. Firing rates and kinematic data (positions and velocities) from the first half of a trial were used to identify the coefficients of the model (training set). Data from the second half of the trial were used to test the model's ability to predict toe position (test set). Figure 7A illustrates the fit of the model to the training set. The VAF (training) was 99 and 98% for the distance and orientation variables, respectively, in polar coordinates using 30 units in the calculation. The root mean square (r.m.s.) errors were a few millimetres and <0.1 rad, respectively, so the actual (continuous lines) and predicted (dots) positions superimpose for the most part. From these values the x and y position of the paw could be calculated to an accuracy of about 1 cm. Figure 7B demonstrates the ability of the model to predict movements for the second half of the data (test set). The VAF was reduced (82 and 93%, respectively) and the accuracy in prediction was correspondingly poorer. The test predictions used the 5 neurones with the best correlation to the kinematic variables.
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1 cm. Predictions using only the one neurone, whose firing was best correlated to each kinematic variable, accounted for about 70% of the variance in position in both the training and the test sets in the two experiments shown. Interestingly, the best neurones for both r and in the two experiments were muscle receptors from the hip and knee (see Table 1). In the test set, as more neurones were added, the VAF increased and reached a plateau between 80 and 90% with 6–8 neurones. The VAF actually began to decline when more poorly correlated neurones were added. Similarly, the r.m.s. error decreased up to a point and then began to increase as more neurones were added. The r.m.s. error provides information related to, but not identical to the VAF. For example, if the predicted data were offset from the actual positions by 1 cm, but followed the variations in limb position perfectly, the VAF would be 100%, but the r.m.s. error would be 1 cm.
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How well does the linear decoder derived from one type of movement predict movements of a different type? In other words, is it a general model for a wide range of movements or only useful for the particular movements used to derive it? To answer these questions, we used data from pseudorandom movements as the training set (40 units, see Fig. 9A) and tested our predictions on walking-like (Fig. 9B) and centre-out movements (Fig. 9C). The centre-out task is a commonly used model in which movements are made from a central point to a number of positions around the periphery. In each part of the figure, the predicted distance and orientation (dots) fitted the actual data well (continuous lines), as indicated by the VAF and r.m.s. error. From the values in polar coordinates, the movement in the sagittal plane was again calculated and the predicted and actual trajectories are shown. The r.m.s. errors of the predictions from the actual positions were 1.1 cm (random), 1.7 cm (walking-like movements) and 2.3 cm (centre-out movements).
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Discussion |
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First, we found that decoding based on the activity of a selected subset containing less than 10 cells may provide an accurate representation of the position of the limb in the sagittal plane. Second, we have shown that these representations generalize across different kinds of movements (i.e. from pseudorandom to walking-like and centre-out movements). Third, our analysis provides a method of assessing the contribution of different receptor types to limb position coding. Muscle receptors contributed most to the determination of the limb position. Cutaneous afferents, particularly in skin overlying joints, were also important. Joint afferents may be involved, but many joint afferents are only active at the extremes of motion that were not explored fully. Under the passive, anaesthetized conditions studied, Golgi tendon organs are relatively difficult to activate and may be hard to distinguish from joint afferents since the tendon organs are very insensitive in an anaesthetized preparation.
Before discussing the nature of the coding process and its functional implications for the control of movement, several points should be stated. First, although even a few neurones can predict limb position accurately, this does not mean that recordings from only a few cells are needed or that the nervous system only uses a few cells. The cells that gave the optimal predictions in the test set were selected from the recorded neurones that were best correlated to the kinematics in the training set. Selecting much larger numbers at random did not give as good a prediction (our unpublished observations). Without having access to a substantial population of cells simultaneously, this result would probably not have emerged. Numerous previous studies, reviewed by Prochazka (1996), have recorded serially from sensory neurones in the leg, with a wide range of kinematic profiles (e.g. the locomotor step cycle) without exploring the possibility of predicting global variables such as the position of the toes in space. The deterioration of ‘test’ generalization as a result of including all neurones is a new finding, as well as the resulting suggestion that neural decoding studies should rely on a small number of selected neurones. Selecting a limited number of optimal neurones has not been tried in other systems, such as the motor cortex. If a similar result emerges, the number of cortical neurones needed for a reliable multiunit decoder (Wessberg et al. 2000; Nicolelis, 2003) may have been greatly over-estimated.
Second, our results do not necessarily imply that the somatosensory system has great redundancy. Large numbers of neurones are clearly needed to sense the precise location of a stimulus applied to a point on the leg or to discriminate between two closely spaced points. Neurones involved in these fine discriminations may contribute little or nothing to the global sense of limb position.
Third, the results may be different during normal behaviour. Varying fusimotor inputs will influence the discharge of the muscle spindles (Matthews, 1972; Prochazka, 1989). Cutaneous receptors, particularly from the skin of the paw, will fire in response to ground reaction forces and other forces that are applied to the paw during normal activities. Golgi tendon organs will be much more active during muscle contractions. Nonetheless, similar methods have been successful in relation to limb position in decerebrate animals and in animals walking freely on a treadmill (Weber et al. 2002; Poppele et al. 2003). Ensemble recordings in animals and recent human work suggest that muscle spindles basically function as stretch receptors, even during voluntary movements that include fusimotor activity (Prochazka & Gorassini, 1998b; Jones et al. 2001). Thus, these studies under anaesthesia are a good starting point for understanding coding in the freely moving animal. Finally, we have used a number of automated processing steps to extract and edit the spike trains from up to 100 electrodes. A detailed justification of the methods has been presented elsewhere (Shoham et al. 2003; Stein & Weber, 2004; Stein et al. 2004) and will not be discussed further here.
Processing of the data in relation to sensory function
The firing rate of muscle receptors is linearly related to muscle length and velocity within a limited range (Terzuolo & Washizu, 1962; Matthews & Stein, 1969). Some have argued that the relationship to velocity is better described by a power function with an exponent less than 1, rather than a linear relationship (Houk et al. 1981; Prochazka & Gorassini, 1998a). However, a linear relationship is a good first approximation, and including velocity to a fractional power did not improve the fit significantly (our unpublished observations). Inclusion of acceleration terms did not lead to a statistically significant improvement in fitting the population of cells studied (Fig. 4). Cutaneous receptors have usually been studied in relation to precise stimuli applied to their receptive fields, rather than movements of a whole limb (Burke et al. 1988; Johnson, 2001). For consistency, the same processing was employed here to all units.
In addition, the spike rates were filtered with a second-order, critically damped, low-pass filter. This particular filter was chosen because its impulse response is a waveform that has often been fitted to EPSPs (Jack et al. 1975). The only free parameter is the rate constant that was optimal in the range of 15–30 rad s–1, which corresponds to a time constant of 33–67 ms. The appropriate rate constants for EPSPs in various pathways that receive inputs from primary sensory neurones are not known. Similar results were obtained with shorter time constants, but the VAF was somewhat reduced.
The linear decoding methods gave remarkably good predictions of the position and velocity of the toe in space. Such predictions have rarely been attempted because of the limitations of single-unit recording methods. The best attempt in the somatosensory system is the work of Bosco and Poppele (Bosco & Poppele, 2001, 2003; Poppele et al. 2001). Our study extended this work to a variety of continuous movements and demonstrated good predictions using only a few, selected neurones. Though our results only show that linear algebraic methods can predict limb kinematics, analogous methods in the nervous system are quite plausible. The linear weighting of the synaptic action of different neurones could be genetically ‘hard-wired’ and/or learnt by trial and error. In this way the sense of limb position we perceive would be matched to the knowledge of where our limbs are in space, derived from other sensory modalities such as vision.
How can sensory neurones predict global variables such as toe position?
Positions and velocities were initially predicted independently, but the two are obviously combined in the firing rate. If the firing rate, f(t), is a linear sum of position terms and velocity terms:
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| (7) |
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| (8) |
Figure 11 uses a model described in the Appendix, containing a few muscles with muscle receptors that obey eqn (7), where x(t) is now the length of the muscle. This figure also shows that muscle receptors in just a few muscles can give a good prediction of the position of the toe in space, using eqn (8). Figure 12 examines the temporal aspects of the same model. In the model we arbitrarily used a time constant of 100 ms, and Fig. 12A shows, as expected, that the least error occurs when a time constant of 100 ms is used in the calculations. In the biological system, selecting a time constant of 200 ms gave the most accurate predictions. In other words, to predict toe position optimally, a ‘leaky integrator’ with a time constant of 200 ms is required. Further work might test whether spinocerebellar cells, for example, provide the optimal time constant to predict limb position to the cerebellum and other higher structures.
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Appendix |
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Figure 12 gives one final example of interest. In Fig. 12A the r.m.s. error for the model is shown in the three coordinate systems as a function of the time constant. The minimum error occurs for a time constant of 100 ms in all coordinate systems. This result shows the self-consistency of the model, since 100 ms was the value used in the model. Figure 12B shows the same calculation for the experimental data. There is again a clear minimum, but experimentally the minimum occurs with a time constant near 200 ms. The predictions of Fig. 10 used this time constant.
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References |
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) were attached at the hip, near the knee and the ankle and on the paw near the metatarsophalangeal joint. From the positions of the sensors a stick figure of the cat's hindlimb in the sagittal plane was calculated. A, pseudorandom movement of the paw manually over its passive range of motion is shown as a dotted line. The position of the paw can be represented in terms of the forward (x) and vertical (y) position with respect to the hip (Cartesian coordinates). It can also be represented in polar coordinates as the distance (r) and the orientation angle (
