INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Classically, motoneurons were divided into two groups:  motoneurons that innervate extrafusal muscle and receive monosynaptic excitation from group Ia muscle spindle afferents, and  motoneurons that innervate intrafusal muscle fibers and do not receive group Ia excitation (see Burke and Rudomin 1977). This simple dichotomy became more complicated with the discovery that some motoneurons, referred to as skeletofusimotor, or , innervate both intra- and extrafusal muscle fibers (Bessou et al. 1965; Emonet-Dènand and Laporte 1976). The fragmentary evidence available suggests that  motoneurons receive monosynaptic group Ia excitation comparable to that in  motoneurons (Burke and Tsairis 1977), suggesting that they represent a source of positive feedback. Because  motoneurons are relatively frequent in many motor nuclei (Barker et al. 1970; McWilliam 1975; Scott et al. 1995), it is of interest to investigate the possible functional consequences of the interactions between  and  fusimotor drive.

The functional role played by  motoneurons is unknown because  motoneurons are difficult to identify experimentally, and their influence cannot be removed without disrupting other feedback loops (Grill and Rymer 1985). Therefore it is useful to explore the possible consequences of  feedback using quantitative models as a guide to design further experiments. The simplest possible model of the  loop is a single muscle spindle that receives fusimotor input proportional to its own Ia afferent firing. This required development of a muscle spindle model capable of accurately representing the response of a group Ia afferent to combinations of stretch during mixed dynamic fusimotor and static fusimotor drives. We also wanted a spindle model simple enough to be used as a component in large-scale simulations of many motor units and proprioceptors.

To our knowledge, there are only two published models of muscle spindle behavior that provide for fusimotor modulation of spindle afferent behavior (Hasan 1983; Schaafsma et al. 1991), but, for reasons given in the Discussion, neither was suitable for our purposes. In the first part of this paper, we develop a model of group Ia spindle afferent behavior that meets our requirements and offers a useful trade-off between range of physiological behavior and computational efficiency. The conceptual elements are adapted from published studies on the velocity and position sensitivity of group Ia spindle afferents. The model matched data from a variety of experimental studies of Ia firing behavior to a degree well within the range of variation found in the literature. The model runs quickly in MATLAB and can be implemented in a graphical systems simulator such as SIMULINK. The second part uses the spindle model to demonstrate that positive feedback through  motoneurons can be large enough to be meaningful in controlling muscle length. The model suggests a feasible experimental design that could provide a direct test of  loop effects. A preliminary account of some of this work has appeared in abstract form (Maltenfort and Burke 2001).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Division of spindle response into position and velocity sensitivities

As described in earlier studies, the response of cat group Ia afferents to a ramp muscle stretch can be described as the sum of four components: a pure velocity sensitivity, a pure position sensitivity, a mixed velocity and position sensitivity, and baseline afferent firing at the initial length of the muscle (Hasan 1983; Lennerstrand and Thoden 1968a; Prochazka and Gorassini 1998). All of these components are modulated by fusimotor bias. In a series of studies, Lennerstrand and colleagues (Andersson et al. 1968; Lennerstrand 1968a; Lennerstrand and Thoden 1968b,c), identified these components of spindle afferent response as separate functions of muscle stretch and fusimotor drive. Their underlying view of spindle afferent response R to muscle length, x, and velocity, v, is summarized in the following equation, where  denotes fusimotor drive

(1)

and R = instantaneous Ia firing rate, S_ss() = a velocity-independent position sensitivity, S_v(,v) = a velocity-dependent position sensitivity, Q(,v) = a pure velocity sensitivity, and B() = the baseline firing of the spindle at the initial length.

Figure 1A shows an idealized representation of how the spindle afferent response to ramp-and-hold stretch arises from Eq. 1.

View larger version (27K): [in this window] [in a new window]   Fig. 1. Structure of the muscle spindle model. A: idealized diagram of group Ia afferent firing rate, R, to ramp-and-hold stretch as described by the equation: R = B + Q + x *(Sv +Sss). Before the ramp stretch begins, the Ia firing rate is at baseline B. At stretch onset, R instantly increases with velocity sensitivity Q. During the ramp, R continues to increase by the position sensitivity terms, x *(Sv +Sss). At the end of the ramp, the firing rate reaches B + Q + x *(Sv +Sss). In the idealized case shown, R then instantaneously decreases by Q + x*Sv so the steady-state firing rate of the spindle afferent is B + x* Sss. In the present model, the adaptation observed in actual Ia responses during the hold period is produced by filtering the velocity transitions (see text). B: schematic flow diagram for the muscle spindle model (see Eq. 2A). Position input is received by each of the 3 blocks that represent the passive, static, and dynamic components of the spindle response. Each block implements the idealized diagram in Fig. 1. The filter in Eq. 3 produces a realistic decay of spindle afferent response over time. The outputs of the static and dynamic blocks are summed with the output of the passive block, and the effect of occlusion is subtracted. In practice, the 2 fusimotor input lines can be of the same type.

Division of spindle responses into passive and fusimotor components

The present model, described in Eqs. 2A-2C, is an extension of Eq. 1. The terms B, S_ss, Q, and S_v are subdivided into the response of the passive spindle (no  input) to stretch and additive responses to stretch modulated by dynamic (_d) and static (_s) fusimotor inputs. It should be noted that  is used a generic variable for fusimotor drive from either  or  motoneurons, which are assumed to be identical in their influence on Ia afferent responses (see following text).

The basic model structure is sketched in Fig. 1B. The afferent firing rate associated with the passive spindle (R_passive) sums with the additional afferent firing produced by dynamic (R_dynamic) and/or static (R_static) fusimotor inputs. The total Ia afferent firing rate, R(t) is modeled as

(2A)

where f_occlusion is defined in Eq. 4 and the passive response is given by

(2B)

The portion of spindle response to stretch that is modulated by fusimotor drive is described by

(2C)

where the subscript g represents either static or dynamic drive, and x is muscle length.

The term v_f refers to an estimate of velocity based on high-pass filtering of the position input. Such filtering was suggested by Lennerstrand and Thoden (1968a) to smooth transitions between ramp stretch and ramp shortening

(3A)

where the time constant   1.0 s. Similar filtering may take place in the physiological spindle due to mechanical interactions between the sensory and contractile regions of the intrafusal fiber (Hasan 1983). In our model, this transfer function was implemented by digitally filtering the position x to produce x_lag

(3B)

(3C)

where t is the time step of simulation (1 ms).

Values of  were selected so that the phase of the modeled Ia afferent response to sinusoidal stretch would be consistent with data from Hulliger et al. (1977a). For passive and static components of the model spindle afferent response,  was set to 20 ms. For dynamic components,  had to be a larger value, 100 ms, to produce an appropriate phase advance. This longer value of  can be justified by the intrafusal mechanics underlying "creep" in the bag₁ fiber (Hulliger 1984).

Nonlinear summation of simultaneous fusimotor effects: occlusion

In experimental studies, group Ia afferent firing that results from simultaneous activation of two separate  motoneurons innervating the same spindle is less than the sum of the rates produced by either  motoneuron individually (Banks et al. 1997; Carr et al. 1998; Hulliger et al. 1977b; Lennerstrand 1968b; Schafer 1974). This occlusion is probably due to competition between branches of the sensory axon that innervate different intrafusal fibers within the same spindle (Banks et al. 1997; Hulliger and Noth 1979). However, mechanical interactions between intrafusal fibers may also contribute (Carr et al. 1998). Schafer (1974) demonstrated that the degree of occlusion depended on how strongly each fusimotor input increased the firing of the spindle afferent above the firing rate of the passive spindle. A mathematical relation that could reasonably reproduce Schafer's data were found to be

(4)

where R_g1 and R_g2 are the increments in Ia firing rate (over passive) caused by fusimotor inputs ₁ and ₂ individually. Equation 4 applies whether ₁ and ₂ are both static, both dynamic or of different types. If either fusimotor drive component was zero or negative (due to shortening), then the nonlinear correction in Eq. 4 was omitted.

Using Schafer's data for pairs of individual fusimotor axons, the predicted summation from Eq. 4 was very similar to the net increases actually reported for the two axons stimulated simultaneously, as shown in Fig. 2. Banks et al. (1997) showed that the degree of nonlinear summation was roughly the same whether spindle length was increasing, decreasing, or held constant. Therefore we assume that our nonlinear summation rule holds for spindles undergoing either lengthening or shortening; note that this assumption is based on a population average [Banks et al. (1997) suggested that there may be differences between individual Ia afferents]. If the passive Ia afferent firing is large compared with either fusimotor component, then strong occlusion can be seen (Hulliger and Noth 1979; Lennerstrand 1968a).

View larger version (17K): [in this window] [in a new window]   Fig. 2. Comparison of observed vs. simulated afferent firing rates during combined fusimotor drive at constant muscle length. The observed increases in firing rate in 17 group Ia afferents during simultaneous activation of pairs of  axons (Fig. 3 in Schafer 1974) are plotted on the abscissa for muscle spindle afferents. The  axon pairs were dynamic-dynamic (), static-static (), or dynamic-static () as marked on plot. Schafer's data for the responses of the same spindles to stimulation of the fusimotor axons individually were entered into Eq. 4, which defines occlusion during combined fusimotor drive in the spindle model. The resulting predictions for combined fusimotor drive (foccl; ordinate) fitted the observed data quite well (note, - - -, unity).

Specification of model parameters

The equations used to implement the four model components were developed by fitting experimental data empirically. We made no attempt to associate any equation with either intrafusal muscle fiber mechanics or afferent encoder properties.

Baseline firing (B)

A spindle held at constant length and receiving constant  drive will fire at a constant rate (baseline B) (Andersson et al. 1968; Lennerstrand 1968b). The observed relationship between either type of  drive and B was roughly linear for  efferent firing rates 200 pps. Measurements of baseline firing from two other studies (Crowe and Matthews 1964; Lennerstrand and Thoden 1968c) fell along similar lines for increasing _d and _s bias. Fits to all data by the line y = kx were used to define initial firing rate before the onset of stretch, assuming that B = 0 at  = 0. The resulting equations were

(5A)

(5B)

Position sensitivity (S_ss)

Assuming that measurements from the cat soleus are typical, the steady-state position sensitivity (S_ss) for the passive spindle was set to 3.9 pps/mm (Brown et al. 1969; Lennerstrand 1968a). We assumed that S_ss (passive plus _s -mediated) increased during _s stimulation as was seen during stimulation of more than one _s axon (Brown et al. 1969), although no change or even a decline has been seen in the response to single _s axons (Lennerstrand 1968a). To reconcile these results for use in the present multiple input model, the data of Brown et al. (1969) was scaled by 0.2 to approximate the average effect of single static  axon activity on the spindle

(6A)

We assumed that the decrease in S_ss with _d stimulation (Lennerstrand and Thoden 1968b) follows a similar sigmoidal relation

(6B)

Velocity-dependent terms (Q and S_v) during lengthening

The empirical equations used to simulate the velocity-dependent components of the model are listed in Table 1. Functions used to describe the relationship between Q and S_V and velocity were selected so that the velocity-dependent terms would be zero for a velocity of zero and that they would not increase without bound. Curve fits were used to parameterize each function for the mean values in the data available (Lennerstrand 1968a; Lennerstrand and Thoden 1968b,c) for the conditions of  = 35, 70, and 200 pps. Finally, similarly constrained curves were used to model each parameter as a function of  bias. Our goal was not to precisely fit any one set of data but to create a spindle afferent model whose behavior was a reasonable match to the published experimental results, which exhibit a great deal of scatter.

There are conflicting data on how Q varies during static  stimulation. Group Ia afferents are reported to become less sensitive to ramp velocity as static  drive increases (Crowe and Matthews 1964). Increasing static  also reduces the modulation of afferent firing during small sinusoidal stretches (Hulliger et al. 1977a,b). Both sets of data imply that _s drive reduces Q from its passive value, Q_passive. Lennerstrand and Thoden (1968c) found that their estimated Q(, v) for static  of 200 pps was smaller than for 70 pps, but both values were larger than their estimate of Q_passive. We reconciled these conflicting data by reducing the Lennerstrand and Thoden estimates for static-biased Q by 35 pps, making them smaller than Q_passive. This produced a realistic response of the static-biased spindle afferent to both ramp stretches and sinusoidal stretches (see RESULTS).

Velocity-dependent terms (Q and S_v) during shortening

With no fusimotor bias, data of Lennerstrand (1968a) suggest that Ia firing rate decreases more with velocity during muscle shortening than it increases with the same velocity of lengthening. This asymmetry was implemented in the model by decreasing the coefficient of Q_passive during shortening to 5.88 from 2.94 for lengthening (Table 1).

During dynamic  drive, the position sensitivity (S_V + S_SS) of Ia firing during shortening is roughly equal to the steady-state position sensitivity, S_SS (Lennerstrand and Thoden 1968b); this implies that both passive and dynamic S_V are small during shortening. The total velocity sensitivity to shortening with dynamic  bias is also likely to be smaller than the sensitivity to stretching. Therefore both Q and S_V were set to zero during shortening (Table 1).

In contrast, the Ia afferent response to length change is roughly equal for lengthening or shortening during static  stimulation (Hulliger et al. 1977a; Lennerstrand and Thoden 1968c). Because Q and S_V are asymmetric in the passive spindle, the equations used to describe the changes with static  input include terms that essentially cancel these asymmetries during shortening (Table 1). Ideally, these compensation terms should be contingent on a minimum level of static  drive but there is currently no data available to determine an appropriate value for such a minimum.

Effects of mixing static  and  bias on Ia afferents

Static  motoneurons appear to innervate only chain intrafusal fibers (Jami et al. 1985; Kucera and Hughes 1983), while static  motoneurons most often activate both bag₂ and chain fibers (Emonet-Dènand et al. 1997). Therefore static  and  drives can be expected to have different effects on Ia spindle afferent responses to stretch (Emonet-Dènand et al. 1997). The existing experimental data are insufficient to constrain possible differences between static  and  effects, and so they are treated identically out of necessity. Static  input to the chain fiber should set the minimum firing rate of a real Ia afferent response during shortening (Hulliger et al. 1977b) and similar behavior was seen in our model during combined dynamic and static fusimotor drive (Fig. 5). These differences between static  and  drives raised the question of whether simulation of simultaneous drive by two static fusimotor inputs would require that the additive compensation used to produce symmetric afferent output during shortening or stretch (Table 1) be applied to both inputs. We know of no experimental data that would inform this choice. In the interests of caution, we decided that in such cases the compensation for passive shortening would be done only once. This decision does not affect calculations of " loop gain" because increasing  motoneuron drive would not increase the amount of the compensation.

Less-than-linear summation (occlusion) resembling that for two  inputs simultaneously driving a single spindle afferent (see following text) has been observed between dynamic  and dynamic  inputs (Emonet-Dènand and Laporte 1983; Hulliger and Noth 1979), and we assume it holds for other combinations. However, it is possible that bag₂ and chain fiber effects from different static fusimotor fibers may sum linearly on the same spindle (Celichowski et al. 1994).

Simulation of  loop feedback

The simplest arrangement for simulating  feedback on a muscle spindle afferent is a single spindle receiving a fusimotor drive that is proportional to the instantaneous firing rate of that afferent (Fig. 3). In the present simulations, the term "" implies a source of fusimotor drive that receives a fraction of Ia afferent output, whereas "" indicates a fusimotor drive source with no such feedback. Each fusimotor input ( or ) is assumed to be either static or dynamic because convergence of static and dynamic  innervation onto the same spindle is relatively uncommon in tenuissmus (Jami et al. 1978) and triceps surae (Grill and Rymer 1987), although it is more frequent in peroneus brevis and peroneus tertius (Emonet-Dènand et al. 1992). Our model assumes that the effect on spindle afferent firing of  motoneuron activity has the same strength and time course as the effect from a  motoneuron of the same type (Bessou et al. 1965). The 5.3-Hz low-pass filtering ( = 30 ms) and 11-ms delay indicated in Fig. 3 represent how the Ia afferent firing responds to changes in  stimulation (Andersson et al. 1968).

View larger version (23K): [in this window] [in a new window]   Fig. 3. Input-output diagram of the feedback loop model driven by  and  fusimotor drives and triangular stretch.  feedback is implemented as a delayed, filtered fraction of Ia afferent output. The Ia afferent firing rate is fed back to the  fusimotor input through a gain (G = 0.2), a delay (d = 11 ms), and a low-pass filter (cut-off frequency, 5.3 Hz) based on experimental observations of spindle and motoneuron responses (Andersson et al. 1968; Grill and Rymer 1985). The  loop and  drives may be each be either static or dynamic, depending on the simulation.

Spindle afferent firing was fed back equally to either type of  motoneuron. The input-output function for the  motoneurons was assumed to be a simple static gain. Grill and Rymer (1985) estimated the steady-state position sensitivity for Ia spindle afferents (6.1 pps/mm) and for  motoneuron firing (1.3 pps/mm) for increasing stretch. Assuming that motoneuron firing was dependent only on Ia input, the gain for simulations was simply

(7)

We assumed that the maximum rate at which  motoneurons can fire is 45 pps. This represents a realistic maximum for  (and presumably ) motoneuron instantaneous firing under transient conditions (Hoffer et al. 1987). It should be emphasized that the gain term G is not the same as the  loop gain described by Grill and Rymer (1985).  loop gain (discussed in the following text; cf. Fig. 10) is the product of G and another gain term that describes the response of Ia afferents to increased fusimotor drive. It measures how much an increase in synaptic drive to a  motoneuron will be further increased by the positive feedback loop. The MATLAB codes that were used in the present study and a SIMULINK implementation of the spindle afferent model are available from Dr. Maltenfort.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Verification of the spindle model: response to ramp stretch

Figure 4, A and B, shows examples of simulated Ia afferent discharge during ramp-and-hold stretches, with the time axis scaled to the duration of different ramps. These examples were similar in amplitude and time course to the published experimental data of Crowe and Matthews (1964; their Fig. 2). Figure 4, C-F, shows the peak firing rates and the dynamic index values (peak rate minus rate 0.5 s later, as in Fig. 4A, arrow labeled DI) found by Crowe and Matthews (1964) (C and D) and in the model responses (E and F) to ramp stretches of increasing velocity. Although our model did not perfectly reproduce the published data, especially at higher velocities of stretch, it did exhibit comparable behaviors given different levels and types of fusimotor drive

View larger version (27K): [in this window] [in a new window]   Fig. 4. Model responses to 6-mm ramp-and-hold stretches, with comparison to experimental data. A and B: simulated responses of spindle afferent to ramp stretches of 5 and 30 mm/s with different levels of fusimotor drive: thicker black line, passive spindle; thin dashed line, spindle with 50 pps static fusimotor; thick gray line, spindle with 50 pps dynamic drive. The model behavior was comparable to the behavior of the real spindle in Fig. 2 of Crowe and Matthews (1964; not shown). Arrow in A indicates how the dynamic index (DI) was calculated, as the difference between the maximum spindle afferent firing rate and the firing rate 0.5 s later. C and D: experimental data from Crowe and Matthews (1964) for peak firing rates and dynamic indices of Ia afferents during ramp stretches. E and F: corresponding behavior of the spindle model for identical conditions. Solid lines denote spindle afferent responses with static drive; dashed lines denote behavior during dynamic drive; and the thick black line indicates the passive response.

Systematic variation of dynamic and static bias during sinusoidal stretches

In simulating sinusoidal stretches, the starting length of the spindle was 4 mm longer than that used for the ramp trials. This was done to compare the results with data of Hulliger and colleagues (1977a,b) for sinusoidal stretches at a length of 1-2 mm less than maximum physiological length. The plots in Fig. 5, A-D, show the model response to one cycle of a 1-Hz sinusoidal stretch at 0.7-mm peak amplitude for steady fusimotor bias (either static or dynamic) at four rates between 0 and 125 pps. A and B show that simulation of _s drive alone produced a marked reduction in peak Ia firing variation during the stretch cycles, while _d bias reduced the response during shortening and increased it during lengthening, as in actual spindles (Hulliger et al. 1977a,b). Figure 5, C and D, shows that the model produced appropriate summation of static and dynamic contributions to spindle afferent firing. In Fig. 5C, the response of the spindle is plotted for different amounts of _s bias superimposed on a tonic background of 125 pps _d bias. The peaks of the simulated data were close together during lengthening, but the valleys during shortening were spread apart in relation to the rate of the static fusimotor bias. The opposite resulted when variations in _d bias were pitted against a tonic background of 70 pps _s; the valleys were close together while the peaks varied with dynamic fusimotor bias (Fig. 5D). Note also that the peak firing rates were slightly increased over the corresponding peaks in Fig. 5B.

View larger version (32K): [in this window] [in a new window]   Fig. 5. Simulated afferent responses to sinusoidal stretch during combined dynamic and static fusimotor drive. Seven hundred micrometer peak amplitude, 1-Hz sine wave. The simulation conditions were designed to match those used experimentally by Hulliger et al. (1977; their Fig. 8). Phase angle of 360° corresponds to maximum length. A: increasing static fusimotor drive alone. C: increasing static fusimotor drive against a constant background of 125 pps dynamic drive. B: increasing dynamic drive alone. D: increasing dynamic drive against a constant background of 70 pps static drive (bottom). See text for discussion.

All of these simulated afferent responses were consistent with experimental data in which the same stretch parameters and fusimotor bias mixtures were used to drive single Ia afferents (Fig. 8 in Hulliger et al. 1977b) with two caveats. First, the observed minimum Ia firing rates with for _d bias alone were smaller (~10 Hz) and more clustered than in the simulation (Fig. 5B). Second, with any combination of fusimotor drive that included _d bias (Fig. 5, B-D), the observed peak Ia afferent firing rates were ~70 pps smaller than in the corresponding model results. These differences were comparable to the observed physiological variability of afferent firing modulation during sinusoidal stretch (SD of 40 pps half-peak-to-peak) (Hulliger et al. 1977a).

The nonzero portions of the simulation results shown in Fig. 5, A-D, were fitted with sine functions to describe their amplitudes and phases in relation to the stretch cycle. The half-peak-to-peak amplitudes of these fits are depicted in Fig. 6, A and C, as functions of fusimotor stimulation rates. Figure 6, B and D, shows the corresponding phase relations from the sine fits. These plots again were comparable to the experimental results reported by Hulliger and coworkers (1977b). The largest difference between model and experiment was that the modulation of Ia firing rate with increasing _s bias was smaller in the model (8-24 vs. 25-45 pps, respectively) in the absence of _d. In contrast, modulation with increasing _d bias was higher in the simulations than observed (24-108 vs. 45-75 pps) in the absence of _s. In the absence of any fusimotor bias, the range of sinusoidal modulation of afferent firing was ~50% (15-20 pps) smaller in the model than in the physiological spindle. This might be because the model does not include a component that would represent stiffness in passive intrafusal muscle fibers due to persistent cross-bridges (Schaafsma et al. 1991).

View larger version (23K): [in this window] [in a new window]   Fig. 6. Amplitude and phase of spindle response to sinusoidal stretch during combinations of static and dynamic fusimotor drive. The effect on modulation of spindle firing is shown as the amplitude P of sine wave fits, y = P * sin(2t + phase) + mean firing, to the simulation results in Fig. 5 (as in Hulliger et al. 1977a,b). A: modulation due to increasing static drive by itself or against a background of 125-pps dynamic drive. B: corresponding effect on phase of sinusoidal fit. C: effect on spindle behavior of increasing dynamic drive by itself or against a background of 70 pps static drive. D: corresponding effect on phase of sinusoidal fit.

Closing the  loop enhances spindle afferent firing during triangular stretches

ENHANCEMENT DEPENDS ON DIRECTION OF STRETCH AND TYPE OF  MOTONEURON. The most direct demonstration of the effects of  feedback is the difference between Ia afferent firing rates with and without the presence of a  loop during muscle lengthening and shortening at constant rates (triangular stretches). Figures 7 and 8 illustrate the effects of closing  loops, either static (_s) or dynamic (_d), with superimposed  drive of the opposite type. These opposite combinations produced the largest  feedback effects (see Fig. 9). To put the enhancement of afferent firing in context, each 5-pps increment in afferent firing will increase  motoneuron firing by 1 pps (gain of 0.2; see METHODS), which could be significant particularly for slow-twitch motor units (Bobet and Stein 1998; Cope et al. 1986).

View larger version (40K): [in this window] [in a new window]   Fig. 7. Simulated Ia afferent responses from spindle model with tonic static  (s) bias (50 pps), with and without feedback through a dynamic  (d) loop. A: afferent firing with and without  feedback plotted as a time series during triangular stretches (thin gray lines; ramp velocity = ± 32 mm/s, peak-to-peak amplitude = 8 mm). Hatched region indicates the difference in spindle firing due to d feedback. B: same data as in A, plotted against the instantaneous spindle length. Note clockwise direction of plot of afferent response. C: enhancement of firing due to  feedback, defined as difference between curves in A and B (grayed regions), for different values of ramp velocity. Peak-to-peak stretch amplitude: 8 mm. D: enhancement of firing for different values of maximum amplitude of stretch. Ramp velocity ± 32 mm/s. Clockwise direction of enhancement loops in C and D is same as that in B.

View larger version (39K): [in this window] [in a new window]   Fig. 8. Simulated Ia afferent responses from spindle model with tonic d bias (50 pps), with and without a s feedback loop. Format same as Fig. 7. Note that with the static  feedback, the enhancement loops in C and D flowed in a counter-clockwise direction because the loop effect appeared almost exclusively during the shortening phase of the triangular stretch.

View larger version (18K): [in this window] [in a new window]   Fig. 9. Peak enhancement of spindle afferent firing from closure of  feedback loop is linked to peak firing rate of spindle afferent that drives the  motoneuron. The maximum values of the enhancement curves from Figs. 7 and 8, plus the peak values for simulations where  and  were of the same type, are plotted (ordinates) against the maximum value of the spindle afferent firing under the same conditions of  bias, maximum amplitude and ramp velocity, but in the absence of  feedback (abscissae). The associated maximum firing rate of  motoneurons, calculated by multiplying the spindle afferent firing rate on the x axis by G = 0.2 (see METHODS and Eq. 7), is plotted on the upper abscissa. A: enhancement from s feedback vs. spindle afferent firing rate. B: enhancement from d feedback vs. spindle afferent firing rate.

Estimation of positive  feedback loop gain

The gain provided by a closed  feedback loop was assessed quantitatively by increasing  motoneuron firing from 5 to 10 pps under a variety of conditions during triangular stretch. This increment M = 5 pps, which ensured that  motoneuron firing never fell to zero, increased the spindle afferent firing rate by S, which varied with background fusimotor drive as well as the magnitude and velocity of stretch. The augmented  fusimotor input is M plus G times the spindle afferent rate (see Eq. 7). The  loop gain can therefore be directly measured as

(8)

This incremental approach was used because newly recruited motor units begin steady firing at rates greater than zero (Monster and Chan 1977). Because the maximum  motoneuron firing was assumed to be 45 pps, we used ramp stretches with 4 mm amplitude and 16 mm/s ramp velocity to avoid  firing saturation.

Figure 10 shows that when the simulations were repeated for constant M but different values of tonic  bias, the  loop gain estimates became smaller as  bias increased for all four combinations of fusimotor drive types. The relative ranking of  loop gains was the same as that shown in Fig. 9. The combination of a _s loop superimposed on _s bias produced the highest loop gain (Fig. 10A), whereas the combination of _s with _s was by far the least effective (Fig. 10D; note change in vertical scales). The counter-clockwise arrows in 10, A and D, indicate that, as in Fig. 8, the _s effects were prominent during shortening but occluded during lengthening. The remaining two combinations were rather similar and both involved _d loops. As in Fig. 7, these  loop effects were larger during lengthening (clockwise arrows).

View larger version (40K): [in this window] [in a new window]   Fig. 10. Estimation of  loop gain under 4 different combinations of fusimotor input types. Loop gains calculated from triangular stretches such as shown in Figs. 7 and 8, using Eq. 8. The triangular stretches used to drive the spindle afferent (4-mm peak-to-peak amplitude, ± 16 mm/s velocity) were selected so that  motoneuron firing remained below the maximum of 45 pps. Note that in all four cases, larger values of  bias resulted in smaller  loop gains because of the occlusion phenomenon (Eq. 4). A: the combination of dynamic  bias with static  feedback produced the largest peak gains. The counter-clockwise loop indicates that the static  effect was prominent during shortening but was smaller during lengthening because of occlusion (see Fig. 8). B:  loop gain for dynamic  feedback superimposed on static  bias was smaller than in A (note change in vertical scale) and also produced a clockwise loop because the dynamic  effect was prominent during lengthening (see Fig. 7). C: the clockwise loop gain of dynamic  bias with dynamic  feedback is only slightly smaller than that in B, although it dropped more rapidly with dynamic  bias at 100 pps because in this case  firing reached the saturation limit of 45 pps. D: the counter-clockwise  loop gain with static  feedback superimposed on static  bias was the lowest of the 4 possible input combinations (note vertical scale) but, as in A, the loops were mostly counter-clockwise.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

To our knowledge, this is the first attempt to simulate the effects of positive feedback through skeletofusimotor, or , motoneurons on group Ia afferent responses in mammalian muscle spindles. To this end, we developed an empirical spindle afferent model that permitted combinations of multiple fusimotor inputs. This model reproduced observed Ia afferent behavior with reasonable fidelity, given the range of variability in the available published data (Figs. 2 and 4-6). Our results suggest that, with certain combinations of  and  fusimotor drive,  feedback loops can produce significant increments in Ia afferent firing, either amplifying existing  fusimotor drive or perhaps compensating for insufficient  bias for the task at hand (Figs. 7-10). As discussed in the following text, these simulations can, in principle, be reproduced experimentally to test the validity of their predictions.

Comparison of the current spindle afferent model to prior models

Prochazka and Gorassini (1998) have recently reviewed and tested a number of published muscle spindle models. They demonstrated that several nonlinear combinations of muscle length and velocity could reproduce the Ia afferent behavior during normal locomotion in cats. They also found that the resemblance between model behavior and experimental data could be improved by adding a component proportional to electromyography, which could represent either coactivation or the presence of a  feedback loop. Of the models they reviewed, only that of Hasan (1983) provided an explicit mechanism for adding fusimotor drive, either dynamic or static but not both. Unlike the model described in the present paper, Hasan's model is based explicitly on estimated intrafusal muscle properties and receptor dynamics. However, his breakdown of model responses into velocity and position sensitivities during ramp stretch has many similarities to the components used in the current model (see APPENDIX in Hasan 1983, Eq. I). In testing Hasan's model, we found that its response to a step change in dynamic fusimotor drive was much slower than expected for real muscle spindles. In addition, the high-pass dynamics of Hasan's model produced an initial burst of afferent activity at the start of each ramp stretch, while in the physiological spindle the initial burst is only visible in the first of a series of stretches (Matthews 1972). On the other hand, Hasan's model is intuitively clear and easy to implement using commercially available software. Schaafsma and colleagues (1991) have developed a much more detailed model of intrafusal muscle fiber properties that allows for multiple sources of changing fusimotor drive. However, it assumes that the afferent response will be completely controlled by the larger of the two fusimotor inputs, and such extreme occlusion is unrealistic (Fig. 3) (Schafer 1974; see also Banks et al. 1997; Carr et al. 1998; Fallon et al. 2001; Hulliger et al. 1977b). In addition, this model is difficult to implement in commercial software. Although Schaafsma's model was later extended to include more realistic estimates of occlusion between fusimotor inputs (Banks et al. 1997), the price was even greater computational complexity.

The present model possesses many of the advantages of these two models, while side-stepping their problems. The only differential equation is a first-order system (Eq. 3A), so the computational speed of the model is comparable to that of Hasan's model. Realistic responses to single and combined fusimotor inputs are represented simply (Eq. 4; Fig. 2). The formulation of the model also makes it convenient to apply engineering approaches to describing the reflex system (e.g., optimal control schemes).

Are model predictions relevant to expected physiological circumstances?

If the activation of  motoneurons were always tightly coupled to that of  motoneurons, then why would we need  motoneurons? One possibility is that the  system could provide additional modulation of Ia sensitivity if  motoneurons were near saturation during co-activation (see Grill and Rymer 1987). There is also a large body of evidence that  motoneurons, and indeed _s and _d groups, can be independently controlled by the CNS (reviewed in Prochaska 1996; see also Taylor et al. 2000). This leaves room for consideration of the possible differential effects of the various potential combinations of  and  fusimotor drives.

In a recent review, Prochazka (1996) provides a summary of experimental and behavioral circumstances in which static or dynamic fusimotor activity is either tied to  activity or independent of it (for example, during locomotion in the cat; see his Table 3.1). When - coactivation is inferred from Ia afferent activity (Hulliger 1984; Prochazka and Gorassini 1998), it is not necessarily clear whether the observations results from - coactivation, from  feedback, or possibly from both. In a more recent study of decerebrate cats during treadmill locomotion, Taylor and colleagues (2000) recorded directly from medial gastrocnemius (MG)  motoneurons. They found complex patterns of differential control of _d and _s motoneurons that included phasic linking to the step cycle in dynamic and some static  motoneurons during muscle shortening, and more tonic activity in other static  cells. Given these complexities, it seems possible that any of the combinations of fusimotor drive tested in the present study might occur during movements of different types.

It is evident from the results presented in Figs. 7-10 that the various possible combinations of  drive and  feedback have different efficacies. The largest loop gains were found for _s loops superimposed on _d background drive during simulated shortening (Figs. 9B and 10A). This situation would occur