Dynamic behaviour of α motoneurone sub-pools subjected to inhomogeneous Renshaw cell inhibition

Biol. Cybern. 46, 217-228 (1983) Biological Cybernetics �9 Springer-Verlag 1983

Dynamic Behaviour of �9 Motoneurone Sub-Pools Subjected to Inhomogeneous Renshaw Cell Inhibition

U. Windhorst and W. Koehler Zentrum Physiologie und Pathophysiologie der Universitiit, Abteilung Neuro- und Sinnesphysiologie, G6ttingen, Federal Republic of Germany

Abstract. The spinal e-motoneurone-Renshaw cell system was simulated by a meshed system of three principal negative feedback loops interconnected via "cross"-feedback pathways. Three types of e-motoneurone (MN): S-type, FR-type, and FF-type MNs, and their differing connections to and from Renshaw cells (RCs) were taken into account. The dynamic behaviour of RCs was taken from data provided by Cleveland and Ross (1977) and assumed to be given by a transfer function with one zero and two poles whose time constants z~ depended on the overall amount of excitatory input to RCs. Also, the static gain of recurrent inhibition was taken to decrease with increasing excitatory input from c~-MN axon collaterals (Cleveland et al., 1981) and to be depressed by spinally descending motor command signals. S-type MNs as well as F-type MNs were assumed to have high-pass characteristics though with slightly different cut-off frequencies. The closed-loop frequency responses of each sub-pool of MNs, S, FR, and FF, at three different levels of recruitment of these sub-pools, were calculated and shown to change signi- ficantly with recruitment level. These changes were essentially due to two reasons: firstly, to the general reduction of static gains within the recurrent inhibitory pathways with increasing motor output (recruitment), and secondly, to the increasing complexity of the whole network by recruitment of each new MN type. The particularly strong effect of the latter factor could easily be demonstrated by a comparison of the frequency responses of the MN types when these were, firstly, integrated into the network at their particular level of recruitment, and when they were, secondly, hypothetically assumed "isolated" from the remaining network, i.e., when subjected only to "self-inhibition', the cross-inhibitory links to other MN types being cut. These results illustrate that the dynamic behaviour of c~-MNs submitted to an inhomogeneously distributed recurrent and variable inhibition are not invariant, but

depend upon the variable characteristics of a complex M N - R C network. This suggests that an important physiological function of recurrent inhibition via Renshaw cells, particularly of its inhomogeneous distribution, may be to adjust the dynamic MN sensitivity to the particular requirements prevailing at different motor output levels.

1. Introduction

The operation and function of spinal recurrent inhibition via Renshaw cells (RCs) are still largely unknown. This is due to the fact that these interneurones are part of an intricate neuronal network. Thus, even their main excitatory input from g-motoneurone (g-MN) axons stems not only from one defined MN pool but from a number of different pools (for a recent survey see Ryall, 1981). There are merely some statistical tendencies such as "proximity" of ~-MN pools within the spinal cord and "specificity", i.e., synergistic function of ~-MN pools, which govern RC input (see review by Haase et al., 1975). Moreover, RCs receive excitatory and inhibitory inputs from other segmental and supraspina 1 sources (for a review of earlier work see Haase et al., 1975 ; also Koehler et al., 1978; Windhorst et al., 1978), and "mutual" inhibition mostly, but not exclusively, from RCs associated predominantly with antagonist g-MNs (Ryall, 1970, 1981). On the other hand, RCs inhibit not only ~-MNs but also ~-MNs (see Haase et al., 1975 ; Ellaway and Murphy, 1981), Ia inhibitory interneurones (over- views : Hultborn, 1972; (Lindstroem, 1973 ; also Benecke et al., 1975), and cells of origin of the ventral spinocerebellar tract (Lindstroem and Schomburg, 1973; Lindstroem, 1973). This multitude of connections, inhomogeneities of recurrent inhibition among

218

different types of e-MNs (see Sect. 2.3), and the plain impossibility of observing the whole system working under somewhat natural conditions have led to a diversity of hypotheses concerning the function of RCs. In spite of this complexity, the early and simplest possible concept envisaging the e -MN-RC system as a single-loop negative feedback system, in which MNs as well as RCs are lumped together into single input- single output systems, has survived until today. Time and again, it was supported by pieces of experimental evidence (Ross et al., 1975 ; van Keulen, 1981), indicat- ing that even single c~-MNs could inhibit themselves via RCs. This scheme has served as the basis for many model studies (Wenstoep and Rudjord, 1971; Ross, 1976; Tuckwell, 1978; Cleveland, 1977, 1980). Since Cleveland and Ross (1977) studied the dynamic transfer characteristics of RCs, Cleveland (1977, 1980) has used this single-loop scheme to theoretically inves- tigate the implications for the dynamic behaviour of e-MNs submitted to recurrent self-inhibition.

However, RCs receive an inhomogeneously distributed input from different types of c~-MN and in turn distribute their inhibitory output inhomogeneously to the latter types (see Sect. 2.3). This non-homogeneity can be expected to affect the dynamic characteristics of ~-MNs. Hence, the lumping referred to in the preceding paragraph would not appear appropriate in any attempt to describe RC operation in a more detailed and precise manner. This is the general theme of this paper. Again, many of the above-mentioned connections of RCs will be neglected here. But it will become apparent that even at the "simple" level of homonymous e-MN-RC interactions there is still a good deal of ignorance to be removed. To expose the lack of data by trying to model the system and hence to stimulate further experimental work is therefore another objective of this paper.

2. The �9 Motoneurone-Renshaw Cell System - Basic Data and Assumptions

In considering signal transmission lines in the e-motoneurone-Renshaw cell system, we shall always be dealing with more or less large groups (or pools) of cells, even if their transfer characteristics are sometimes derived from data on single cells. This reduces the problem of non-linearity (cf . Windhorst, 1979). Moreover, a small-signal analysis will be performed insofar as the input-output relation of the respective systems within small regions around specified operating points is considered.

Most of the data used in this paper relate to the medial gastrocnemius (MG) muscle and the related motoneurone pool of the cat.

2.1. Motor Unit Types

It has become widely accepted to divide the motor unit (MU) 1 population of most skeletal muscles into four categories: 1) slow-twitch, fatigue-resistant (type "S") units, 2) fast-twitch, fatigue-resistant (type "FR' ) units, 3) fast-twitch, intermediate [-type "F(int)"] units, 4) fast-twitch, fatigable (type "FF") units (for a compre- hensive survey see Burke, 1980). In the cat medial gastrocnemius muscle, S units make up about 25 %, FR 25%, FOnt) 5%, and FF units 45% of the entire population. 2

The overall output of a MN pool determines the muscle force by variation of two variables : the number of temporarily active (recruited) ct-MNs (recruitment variable) and the discharge frequency of each recruited MN (frequency variable). At low contraction force levels, small S units are first recruited at low initial discharge rates. With increasing force, successively bigger motor units (from FR to FF) are recruited whereas previously recruited smaller units increase their firing rate (e.g., Milner-Brown et al., 1973; Gydikov and Kosarov, 1974; Freund et al., 1975; Kukulka and Clamann, 1981; de Luca et al., 1982).

In the following, we shall consider three motor output levels. The first (I) is characterized by almost total recruitment of all S units, the second (II) by nearly complete recruitment of FR units (plus the previously recruited S units). Finally, the third level (III) includes about half of the FF units, whereby the three MU sub-pools of different type contain about the same number of MUs (for the cat medial gastrocnemius; see above).

For simplicity, the small-signal variation of motor output around each of these levels (or operating points) is assumed to be predominantly due to MN firing rate variation.

Let us define the input to MNs as synaptie current (or current of other origin, e.g. intracellularly injected via microelectrodes), and the output as firin 9 frequency (in the case of single MNs). [This relatively general definition of MN input circumvents the problem of presupposing any specific synaptic organization which might depend upon MN types (see e.g. Burke et al., 1976; Burke, 1979).]

By injecting current of ramp-and-hold shape via microelectrodes into single (cat) e-MNs, Baldissera et al. (1982) showed that slow as well as fast MNs were

1 By definition, a MU consists of an c~-MN (including its axon) and the skeletal muscle fibres innervated by the MN. The group of muscle fibres innervated by one MN axon is sometimes called "muscle unit". The properties of MNs and those of the muscle units innervated by them are interrelated (see Burke. 1980) 2 In the following, F(int) units will not be considered separately, since, firstly, they contribute only a small percentage to the MN pool, and, secondly, in some studies there are no data for them

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sensitive to both the steady-state and the velocity component of the current shape. Slow MNs (which we shall here presume to correspond to S-type MNs) were even slightly more sensitive to dynamic current transients than fast (FR and FF) MNs. We therefore assume that, as compared to the transfer function of fast MNs (see below), the transfer characteristics of small (S) MNs are given by

M~ = M(ol~(1 + 0.1 s), (1)

where M(o 1) is the static gain, and the superscript (1) refers to SMNs. [M(o 1) will be used to denote static gains of single MNs as well as of groups (of equal numbers) of MNs.]

Fast c~-MNs excited by sinusoidally modulated intracellularly injected current were shown by Baldissera et al. (1979) to exhibit increasing phase advance of their firing frequency with respect to input current at frequencies above about 3 Hz. For the latter MN type (probably FR and FFMNs), Cleveland (1980) therefore assumed a transfer function of the following form

M(i)(s) ----= M~)(1 +0.05 s), (2)

where M~ ) is the static gain with i= 2 for FR and i= 3 for FF MNs.

In part of the calculations, we shall also take account of a MN membrane time constant z u of the order of 5 ms (Barrett and Crill, 1974), whereby Eqs. (1) and (2) change to

m~ = m(0~)(1 + 0.1 s)/(1 + 0.005 s) (3)

and

M(~ = M~)(1 + 0.05 s)/(1 + 0.005 s), (4)

respectively. For the value of M(~ ), i= 1, 2, 3, see next section. Frequency response functions for (2), (3), and (4) with M~)= 1 (i= 1, 2, 3) are shown in Fig. 2A-C (left column), respectively. Thick continuous lines represent gain curves, thin dotted lines phase curves. These frequency responses are assumed to reflect the "intrinsic" dynamic characteristics of MNs, i.e., those characteristics uninfluenced by recurrent inhibition via RCs. This assumption seems warranted because, in the pentobarbitone anaesthetized preparations used by Baldissera and coworkers (1979, 1982), recurrent inhibition (particularly "self-inhibition" of single MNs) can be expected to have been weak.

2.2. Static Motoneurone Gain M~ )

The static gain M~ ) of the various MN types (not the contractile power of the related muscle units), is best defined as the slope of the relationship between a MN's discharge .frequency (once the MN has been recruited)

and the depolarizing current (of whatever origin) injected into the cell. This definition is based on the definition of a MN's input and output as given in the preceding section. Within the "primary range" of motoneuronal discharge rates (cf. Granit, 1970), the above slope as determined by current intracellularly injected into MNs, depends upon the cell size as measured by its input resistance (Kernell, 1966). This slope is about twice as great for cells with the lowest input resistance [i.e., for the largest cells, presumably of FF-type; cf. Fleshman et al. (1981)] as for those with the highest input resistance (presumably of S-type3). We shall assume (conservatively) that the intrinsic gains M(~ ) of MNs of different type are as follows (normalized with respect to S MNs) :

S FR FF

Rel. intrinsic MN gain M~ ) 1 1.3 1.5

This array will be called "standard MN gain vector", M 0. These static gains (and those to be defined later; see next section) will henceforth be taken as dimensionless.

We shall also assume that with increasing excitatory input and, hence, successive recruitment of the MN types, the intrinsic motoneuronal gains do not change, but that, instead, the linear current-frequency relationship (in the primary range) is shifted upwards [Granit's "algebraical summation": Granit (1970, pp. 141 ff.)].

However, the gain of the overall MN-RC system may be altered through segmental and spinally descending influences acting upon RCs (see Sect. 2.5).

2.3. Connections Between Motoneurone Types and Renshaw Cells

The overall static gain of the recurrent pathways is more complex than the gains M~ ) defined for MNs in the preceding section. It is not only given by the slope of the current-frequency relation [which has only rarely been measured in RCs: Ross (1976) and Hultborn and Pierrot-Deseilligny (1979b)] but is the product of several factors. One among several possible ways of isolating and hence defining these factors is as follows. The first factor is the gain of synaptic transmission from MN axon collaterals to RC synaptic current (i.e., the gain of a system with the two MN output variables as input variables and RC synaptic current as output variable), the second the above current-frequency relation (now for RCs), and the third

3 The range of motoneuronal input resistances measured is however greater in Kernell's (1966) work [ca. 0.5-4.5 M ~ ; see also Kernel1 and Zwaagstra (1981)] than in the cited study of Fleshman et al. (ca. 0.25-2.7 Mf~)

220

A

I

L . _ . . . . . . . . . . . .

L - - -

B

o o o .

m

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A(ll(s)

Fig. IA and B. Schematic representation of the inhomogeneous distribution of recurrent inhibition of a-motoneurones. A Sketch modified from Friedman et al. (1981), their Fig. 6. Excitatory synapses are symbolized by open circles, inhibitory ones by solid dots. The different amounts of excitation delivered to the RC pool by the different types of e-MN (denoted by FF, FR, and S) are represented by various numbers of open circles, the amounts of RC inhibition received by the MNs are represented by different numbers of solids dots. B Block diagram representing the differing quantitative interrelations between e-MN types via RCs. The M")(s)'s denote the transfer functions of MNs with i= 1 for S-type, i=2 for FR-type, and i=3 for FF-type MNs. The R(~)(s)'s denote the transfer functions of all possible recurrent pathways. The At~ (with i = 1, 2, 3) represent input signals to the respective MNs, the E(~ output signals (in Laplace notation)

the strength of synaptic transmission from RC axons to MNs. The exact gains in terms of these definitions (with the appropriate dimensions) are hardly known. We shall therefore define arbitrary and dimensionless gain factors. This section deals with the first (in part) and third of the above factors. The second factor representing the static RC behaviour is subject to modulating influences from segmental and supraspinal sources and will be introduced and dealt with as a general (varying) scaling factor in Sect. 2.5.

It has long been claimed on more or less indirect physiological evidence that the input from different types of a -MN to Renshaw cells and from the latter to the former are inhomogeneously distributed (cf. Fig. 1A).

Ryall et al. (1972), Hellweg et al. (1974), and Pompeiano et al. (1975a, b) provided evidence that large phasic MNs excite RCs more easily than small tonic MNs do. Cullheim and Kellerth (1978) described an anatomical basis for these physiological findings. On average, FF-type MNs produce more recurrent collaterals and "synaptic swellings" (on average 92.9 swellings per M N presumed to form synapses with postsynaptic neuronesr than F R M N s (45.3*) and S MNs (32.2*) do. Let us assume that these figures reflect the relative amounts of input to an "average" RC (or to the RC pool) delivered by the three types of MN. [In Fig. 1A, this assumption is symbolized by the number of synaptic boutons (open circles) at the end of

4 These figures are purified by those synaptic swellings found in the MN area of the spinal cord (cf. Table 1 of Cullheim and Kellerth, 1978)

recurrent collaterals (dashed lines) given off by MNs denoted S, FR, and FF.] In relative units normalized with respect to S MNs these figures read:

s FR FF

Rel. RC input gain 1 1.16 2.37

These figures would thus represent the relative gains of synaptic transmission from MNs (of different type) to RCs (the first gain factor defined above).

The inhibitory effect of RCs on various types of c~-MN has also been known to be different, i.e., small tonic a-MNs are subject to more powerful recurrent inhibition than large phasic ones (see Haase et al., 1975). Friedman et al. (1981) showed that recurrent inhibitory postsynaptic potentials (RIPSPs) were larger in S than in F R MNs and larger in the latter than in F F M N s . Normalized to the RIPSP amplitudes of S MNs, the amplitudes in the other M N types were as follows (determined in medial gastrocnemius MNs by supramaximal antidromic stimulation of the heteronymous lateral gastrocnemius-soleus nerve or by slightly suprathreshold stimulation of the homonymous M G nerve):

ReI. RC output gain S FR FF

Heteronymous 1 0.58 0.24 Homonymous 1 0.51 0.15 Averaged (rounded) 1 0.55 0.2

We assume that the averaged figures of the last row represent the relative strength of inhibition exerted by

221

an "average" RC (or by a given pool of RCs) on the different types of a-MN, and hence the third gain factor defined above. [In Fig. 1A, this assumption is symbolized by the different numbers of synaptic boutons (closed circles) on S, FR, and FF MNs given off by the RC pool.]

The inhomogeneities of recurrent inhibition are visualized in Fig. 1A (modified from Friedman et al., 1981, Fig. 6).

A block diagram representing the facts visualized in Fig. 1A is shown in Fig. lB. Here the MNs are denoted by transfer functions M(~ where i= 1, 2, 3 refers to S-, FR, and FF-type MNs, respectively (cf. previous section). The RC pool represented by a single block in Fig. 1A is split into separate signal lines and blocks in Fig. lB. For example, the recurrent inhibition exerted by S MNs upon FR MNs is represented by the block denoted R(21)(s), and so forth. There is probably no exact anatomical and physiological ana- logue for this splitting since Kato and Fukushima (1974) found that most of the (single) Renshaw cells received more or less homogeneous input from MNs of different size. So there would be a certain lumping of some of the pathways separated in Fig. lB. However, the rest of the RCs were preferentially activated by either small or large MN axons (Kato and Fukushima, 1974; see also Hultborn et al., 1979). This at least partially justifies the separation of recurrent pathways in Fig. 1B from a physiological viewpoint.

The distribution of gains of recurrent inhibition (due to the first and third factors mentioned above) within the recurrent network of Fig. 1B, i.e., the static gains, R~ 3), of the transfer functions, RtiJ)(s) i, j = 1, 2, 3, can be summarized in the following "recurrent gain matrix", Ro, resulting from the proper multiplication of the above two relative gain vectors : (1 t. Ro= 0.55 0.67 1.37~.

\0.2 0.28 0.57/

Let us call this matrix the "standard" recurrent gain matrix. Note that it is strongly asymmetrical.

2.4. Renshaw Cell Dynamic Behaviour

A detailed quantitative analysis of RC behaviour has been performed within the past few years by Cleveland and coworkers (Cleveland and Ross, 1977; Ross, 1976; Cleveland, 1980; Cleveland et al., 1981).

Cleveland and Ross (1977) demonstrated that the response of (single) RCs to frequency-modulated stimulation of ventral roots (or parts of them) could be described by linear frequency response functions whose parameters varied with mean stimulation rate

(i.e., carrier frequency). They fitted transfer functions of the following form to the data:

R(iJ)(s)=R(~j) (1 + zls)(1 +z2s ) (1 + %s)(1 + z4s)(1 + zss ) exp(-- 6s), (5)

where R~ j~ is the static gain, and the superscript (ij) refers to the block of the same notation in Fig. 1B (see below).

The values for the time constants zi at various carrier frequencies v 0 (of antidromic stimulation) were (in s) 5 :

v 0 (Hz) % z 2 % z4 z 5

30 O. 15 2.45 0.003 0.09 2.0 40 0.16 2.65 0.003 0.09 2.0 50 0.09 2.65 0.002 0.04 2.0 60 0.14 2.8 0.002 0.03 2.1

Since "[2 and z 5 are of about the same magnitude and relate to frequencies below 0.1 Hz, which are of no interest here, we neglect them and simplify the transfer function (5) to:

R(iJ)(s)=R~ ~ (1 +%s) exp( - 0.0015 s). (6) (1 + z3s)(1 + "c4s )

This transfer function is assumed to describe the dynamic characteristics of any of the entire recurrent pathways represented by blocks in Fig. lB. The synapses from RC axons to MNs are thus supposed to introduce no dynamics of their own. The exponential function represents a (uniform) delay of 0.0015 s in each recurrent pathway. There are no similarly complete data on the effect exerted upon the above time constants by the second RC input variable, namely the number of converging exciting MN axon collaterals (recruitment variable). However, it has been shown (Cleveland, 1980) that changes in the number of converging exciting collaterals and changes in the frequency of discharge upon those collaterals (i.e., changes in the two MN output variables; see Sect. 2.1) are in principle interchangeable and yield equivalent results for the dynamic RC behaviour.

Assuming three levels (I-III ; see Sect. 2.1) of motor output (RC input), each level will be associated with the following set of time constants zi:

Recruitment level z a % z 4

I 0.14 0.003 0.09 II 0.14 0.002 0.04 III 0.14 0.002 0.025

Frequency response functions of the form of Eq. (6) with the above three sets of time constants and R~ J) = 1

5 From Cleveland and Ross (1977) supplemented by data from Cleveland (1980)

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~ 0 "".. - 90 r~

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~ - 9 0 . , " 1-180

o.1 i io 1oo o.1 I 1o 1oo

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Fig. 2A-F. Assumed dynamic behaviour of ~-MNs and RCs. The left column displays frequency responses of e-MNs, graphs A through C corresponding to, respectively, Eqs. (2) through (4) in the text with M~ ) = 1 (for i = 1, 2, 3). The right column shows frequency responses of RCs, graphs D through F corresponding to Eq. (6) in the text with R~ J~ = 1 and the ~z's related to motor output (recruitment) levels I through III, respectively (see Sect. 2.4). Gain functions in this and the following figures are represented by thick continuous lines, phase functions by thin dotted lines. For the RC phase curves, the inhibitory RC action, i.e., the sign inversion, was not taken into account by a 180 ~ phase shift. [The frequency responses in this and the following figures were calculated on a PDP-11/03 (Digital Equipment) computer]

are shown in Fig. 2D-F (right column). The phase curves (thin dotted lines) do not reflect the 180 ~ phase shift induced by the negative signs corresponding to the inhibitory action of RCs.

2.5. Variable Static Gain of Recurrent Inhibition

The static gains within the recurrent inhibitory network as summarized in the "standard recurrent gain matrix", R 0, of Sect. 2.3, depend upon the motor output level. This dependency has several reasons which are briefly treated now, but which are yet too complicated to be totally quantifiable.

I ra group of MN axons sending collaterals to a RC recorded from is stimulated electrically at various constant frequencies, the static discharge frequency fRC of the RC varies nonlinearly with input (antidromic stimulation) frequency fAD (frequency variable), such that this dependency can be described by a saturating curve exhibiting a declining slope with increasing input frequency (Cleveland et al., 1981). The reason for this

nonlinearity can probably be found in the synaptic transmission process, i.e., in the first factor defined in Sect. 2.3 (Cleveland et al., 1981). Whereas this decline of static RC gain (being equivalent to the above slope) was well documented by the above authors, the same was not done regarding a similar dependency on the recruitment variable. In view of the exchangeability of the two RC input variables with respect to their effect on RC dynamic behaviour (see previous section), it seems most reasonable to assume the same for the current issue of static gains. One might therefore simply scale down the standard recurrent gain matrix by some necessarily arbitrary scalar factors < 1, which are smaller the more one proceeds from the lowest to the highest recruitment level. However, two points complicate this matter. Firstly, the number of activated excitatory synapses on a RC increases more rapidly than linearly with the number of recruited MNs sending collaterals to the RC (see Sect. 2.3). Secondly, with continuing recruitment of MNs, new RCs which were silent before are recruited (Kato and Fukushima, 1974; Hultborn et al., 1979) and may increase recurrent inhibition. For these reasons and a further one given in the next paragraphs there is no (known) simple relation between the overall activity of a pool of RCs and its static gain (this overall activity again being composed of two variables: the number of active RCs and the discharge frequencies of the latter).

A further complication results from the fact that the efficacy of recurrent inhibition is modifiable by spinally descending command signals related to the prevailing motor output level.

In experiments on normal human subjects, Hultborn and Pierrot-Deseilligny (1979a) employed an indirect H-reflex technique to estimate the amount of recurrent (Renshaw) inhibition under the conditions of phasic ("ramp") and tonic muscle (triceps surae) contractions. They concluded from their results that during the course of linearly increasing muscle contractions, RCs are initially facilitated and then pro- gressively inhibited, probably by the same spinally descending signals causing the muscle contraction by e-MN activation. The same may hold for steady-state contractions of various forces, i.e., the gain within the recurrent inhibitory pathway is high at low force levels and low at high levels. There is supporting evidence from animal studies, since the electrical stimulation of various supraspinal centres has been shown to alter, mostly suppress, the RC excitability (for older references see Haase et al., 1975; also Koehler et al., 1978; Windhorst et al., 1978). The site of action of these effects is probably, first, the number of recruited RCs and, second, the "intrinsic" gain (see Sect. 2.2; factor two in Sect. 2.3) of each single active RC. From these results one might again infer that the gains within

223

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frequency (Hz) Fig. 3A-H. Dynamic input-output relations of the c~-MN sub-pools (columns from left to right relating to S-, FR-, and FF-type MNs, respectively, see labels on top) subjected to inhomogeneous recurrent inhibition at three different motor output (recruitment) levels (rows from top to bottom with labels I through III to the left, respectively). The MN membrane time constant is neglected here, zM=0 , i.e. S-type MNs have frequency characteristics as expressed by Eq. (1), and F-type MNs a frequency response as in Eq. (2) and Fig. 2A. The various gains are "standard", i.e., M 0 and R o are as given in the text. Thus the strength of recurrent inhibition is R 0 at level I, 0.7 x R 0 at level II and 0.4 x R 0 at level IlL The dynamics of recurrent (RC) inhibition at levels I through III are like those in Fig. 2D through F, respectively. The middle and right plots D and G in the first row are frequency responses of FR MNs D and FF MNs G at levels II and III, respectively, when subjected only to "self-inhibition', i.e., with the recurrent cross-links to other MN types being cut

the recurrent pathways generally decrease with increasing motor output level.

Weighing all these items, it may seem justified to presume that, on the whole, relative to the lowest motor output level (I), the gains within the various recurrent pathways are uniformly scaled down by factors < 1 at the higher motor output levels (II, III). We shall assume the following arbitrary scaling factors for the recurrent gain matrix at the three motor output levels :

Level I Level II Level III

Rel. scaling factor 1 0.7 0.4

Thus, at each level of motor output, the gains of all the recurrent pathways (see Fig. 1B) will be uniformly scaled down by the respective factors given above. This amounts to multiplying the "standard" recurrent gain matrix R o by 0.7 at level II and by 0.4 at level III.

3. Frequency Response of �9 Motoneurone Sub-Pools

The main feature to be demonstrated here is the change in the frequency responses of sub-pools of e-MNs when subjected to an inhomogeneously distributed recurrent inhibition at different motor output levels. We are thus interested in the dynamic characteristics of signal transmission between inputs A(~ and out- puts E(~ (cf. Fig. 1B) for i = 1, 2 or 3, which numbers refer to S-, FR- or FF-type MNs, respectively. The frequency responses assumed for the different MN types are those represented by Eqs. (1) and (2) in Fig. 3, and those represented by Eqs. (3) and (4) in Figs. 4 and 5.

At recruitment level I only S-type MNs are active. Hence they are subjected only to recurrent inhibition originating from themselves (i. e., to self-inhibition) and not from FR- or FF-type MNs. Thus there is only one feedback pathway in action, namely that denoted by

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f r e q u e n c y ( H z )

Fig. 4A-H. Same frequency responses as in Fig. 3 except that the MN membrane time constant was assumed to be equal to vg=5 ms, i.e., S-type MNs have frequency characteristics as in Eq. (3) and in Fig. 2B, and F-type MNs exhibit frequency responses as in Eq. (4) and in Fig. 2C. This figure has the same structure as Fig. 3

R(lt)(s) (the self-inhibitory pathway in Fig. 1B). In Fig. 3A (upper left plot), the closed-loop gain curves (thick continuous lines) and phase curves (thin dotted lines) for this single-loop feedback system with input A~ and output E(1)(s) are shown for the following parameters: M(1)(s)=(l+0.1s) [cf. Eq.(1)], i.e., M(ol)=l, m(o2)=m(o3)=O; R(~l)(s) is given by Eq. (6) with R~o 11)= 1 (the upper left element of the "standard recurrent gain matrix", Ro) and the "q's of recruitment level I (Sect. 2.4). Recall that the latter (RC) frequency response is depicted in Fig. 2D.

Let us now proceed to recruitment level II (second row of Fig. 3). FR-type MNs [represented by the block denoted by MC2)(s) in Fig. 1B] become active and thereby put into action three new feedback pathways denoted by R(21)(s), R(22)(s), and R(t2)(s) in Fig. lB. (M(0~)=l; M(oZ)=l.3; M(o3)=0). The distribution of static gains within the four feedback pathways now effective is given by the sub-matrix of the upper left four elements within the "standard recurrent gain matrix", R o (Sect. 2.3), multiplied uniformly by the scaling factor 0.7 reflecting the depression of recurrent inhibition at level II (Sect. 2.5). The dynamics of recurrent feedback have also changed from a change in

the zi's (cf. Sect. 2.4 and Fig. 2E). The frequency response of the FR MNs is given by Eq. (2) (cf. Fig. 2A) with M(o2)= 1.3. It can be seen from Fig. 3 that the frequency response curves of S-type MNs (first column labelled S; second row labelled II : Fig. 3B) have partly regained the original (intrinsic) high-pass characteristics of S-type MNs [cf. Eq. (1)], and that the gain has also increased at low frequencies (near 0.1 Hz) as compared to the previous situation (Fig. 3A). This gain increase is due to two reasons: firstly to the lowered self-inhibition (R(o 11)=0.7), and secondly to a positive feedback loop constituted by the blocks R(21)(s), MC2)(s), and R(lZ)(s), in this sequence. - The FR MNs whose frequency responses are shown in Fig. 3E (second column labelled FR, second row labelled II) cannot naturally be active in isolation, i.e., without S MNs being active concurrently. But in order to dis- close the effects of them being integrated into a complex network we have also, for comparison, calculated their frequency response for the hypothetical case of their being active alone at level II [i.e., R (I 2)(s ) ~ R (21)(s) ~ 0 ] . The resulting curves are plotted in Fig. 3D (which place would normally have been empty). Here, too, the gain curve in Fig. 3E displays

225

high-pass properties, quite in contrast to that in Fig. 3D. Compare these frequency response curves with those in Fig. 2A illustrating the intrinsic frequency response of F-type MNs not subjected to recurrent inhibition. It appears that, in the constel- lation of inhomogeneous recurrent inhibition as assumed for Fig. 3E, the FR MNs, too, have regained part of their intrinsic high-pass properties.

At recruitment level III all MN types are active (Fig. 3, lower row labelled III), which puts into action all the recurrent pathways (with the appropriate changes of parameters as explained above; cf. Sects. 2.3-2.5). The recruitment of FF MNs [transfer function see Eq. (2) with M(o3) = 1.5 ; Sect. 2.2] does not change the general tendency previously found for S MNs (Fig. 3, left column) and for FR MNs (middle column), although the gain increase with frequency above ca. 10 Hz is less steep for S- and FF-type MNs than for FR MNs. Again compare the curves in Fig. 3H to those in Fig. 3G (first row), which show the "isolated" frequency response of FF MNs subjected only to self-inhibition at level III. The curves particularly for the FR-type MNs (middle column, third row) closely resemble those of Fig. 2A, which are the open- loop characteristics of fast (FR and FF) MNs according to Eq. (2). One may thus infer that at motor output level III the chosen distribution of parameters in the meshed system nearly cancels the dynamic effect of recurrent inhibition as expressed in the respective plots of the first row of Fig. 3. But it should also be remembered that the recurrent gains R(~ ~) were scaled down (scaling factor 0.4) at level III and hence were low anyway.

While in Fig. 3 the MN membrane time constant ZM=5ms was not taken into account [-the transfer characteristics of S- and F-type MNs were those of Eqs. (1) and (2), respectively], it was in Fig. 4, i.e., the MN transfer functions were those of Eq. (3) for S MNs and Eq. (4) for F MNs. The other parameters were the same as in the treatment of Fig. 3 ; thus Figs. 3 and 4 have the same structure. It can be seen in Fig. 4 that the membrane time constant ZM=5 ms has a somewhat considerable effect on the frequency response only at higher frequencies, which is also evident from comparison of Fig. 2A and C. For the rest, the same general remarks made on Fig. 3 apply to Fig. 4.

Cleveland (1980) has presented closed-loop frequency responses of a single-loop feedback system similar to those shown in Fig. 3A, but assuming a flat frequency response for S-type MNs, i.e., M(1)(s)= M(o ~). The overall static open-loop gain (i.e., the product M(1) /~(11)~ o "~-o j was chosen to vary between 0.1 and 100 in order to cover a considerable range. The actual figures for this gain are not known, however. One might assume that they are relatively low as is also true for the spinal stretch reflex loop. The loop gain in Fig. 3A

was therefore selected to be low (= 1). In order to see the effects of higher loop gains, the standard MN gain vector was arbitrarily multiplied by 2 (Cleveland, 1977, suggested the intrinsic MN gains to be high) and the standard recurrent gain matrix by 5, amounting to a loop gain of 10 for the single-loop system at recruitment level I. Cleveland's frequency responses of the single-loop system (1980) did not show much further change when increasing the open-loop gain from 10 to 100 whereas there was a major change from 1 to 10). The resulting frequency responses are shown in Fig. 5, where all other parameters and the structure of the figure are equivalent to those of Fig. 4. It can be seen that the changes in the frequency responses taking place with increasing recruitment level differ somewhat from those in Fig. 4. Although in general the same conclusions may be drawn from Fig. 5 as from Fig. 4 regarding the high-pass characteristics of MNs, this does not apply for S-type MNs when passing from recruitment levels II to III (Fig. 5B and C), i.e., the increased gain at higher frequencies (above ca. 1 Hz) in Fig. 5B is reduced again in Fig. 5C. Moreover, for FF- type MNs the high-frequency cut-off in Fig. 5H is shifted to the right as compared to that in Fig. 4H.

We have also computed frequency responses like those above under the assumption that the S-type MNs display a flat frequency response [-i.e., M(1)(s)=M(o 1)-] as assumed by Cleveland (1980). Although, of course, the gain and phase curves differ in detail from those shown in Figs. 3-5, the same general conclusions can be drawn, i.e., with increasing recruitment level the dynamic behaviour of the different MN sub-populations subjected to recurrent inhibition approaches the intrinsic behaviour of the respective MNs without recurrent inhibition.

Whereas the frequency responses (particularly the gain functions) of all MN sub-pools are relatively flat below 1 Hz, they may vary considerably with frequency above this value. Indeed, it is this high-frequency part whose shape is highly dependent on the recruitment, i.e., motor output level, and on the integration of the respective MN sub-pools into the overall MN pool via inhomogeneously distributed recurrent inhibition (cf. the frequency response curves in the first rows with the respective ones in the second and third rows). This is of particular importance since it has frequently been claimed that RCs are "phasic" neurones in being especially sensitive to timevarying inputs {Hellweg et al., 1974: Pompeiano et al., 1975a, b; Cleveland and Ross, 1977; Cleveland, 1980) as is shown in Fig. 2D-F. It therefore seems justified to suggest that a prominent physiological function of recurrent inhibition via Renshaw cells, and particularly of its inhomogeneous distribution, may be the adjustment of dynamic MN sensitivity according to requirements which may change with motor output level.

226

w- 100-

A 10-

1

0.1-

O.Ol

S FR F F

__ J

D ../

~ J

100 ~ 180 i r I 01 . / , . . 90

II "~ t 0 0.1 - 9 0 a .

0.01 -180 F

G .,..."

,oo[ 1~ I

III ~ 1

0.1

0.01 0.1 1'0 100 0.1 1 1'0 100 0.1

f r e q u e n c y (Hz )

H

180

90 0

0~ 13

.g: - 9 0 o .

-180

1 : 5 0 x R o

II : 3.5 x R o

III : 2.0 x R 0

2.0 x Mo; ' t M = 0 .005sec

180

90 A O v

o~, 13 r

- 9 0 O.

-180 i 1~ loo Fig. 5A-H. Same frequency responses as in Fig. 4 except that the "standard" gains M o and R 0 are multiplied uniformly by 2 and 5, respectively. This figure has the same structure as Figs. 3 and 4

4. Discussion

The model calculations presented in this work were performed to give an idea of the possible changes that must be expected in the dynamic input-output relations of MN pools when subjected to a variable and inhomogeneous recurrent inhibition via RCs. We be- lieve that this goal has been achieved, although the frequency responses shown cannot be absolutely correct for several reasons. Firstly, many of the system parameters involved could only be estimated arbitrarily and roughly because there is a considerable lack of pertinent data. Secondly, these parameters almost certainly differ from one MN pool to the next. For instance; the recurrent inhibition of respiratory e-MNs has another distribution (Kirkwood et al., 1981), and some MN pools seem to lack recurrent inhibition at all, e.g., the short (cat) plantar, phrenicus and extrao- cular muscles (see Haase et al., 1975; Sasaki, 1963; Evinger et al., 1979 ; but see Kirkwood et al., 1981, for warnings), the reasons not being clear. A third important reason for the limited validity of the frequency responses shown is the inhomogeneous spatial distribution of recurrent inhibition (Windhorst, 1979; Kirkwood et al., 1981) that, for simplicity, was neglected here.

Nonetheless, the general tendencies described in Sect. 3 may hold. The most important result is that during continuing recruitment the dynamic behaviour of different MN types when submitted to a changing and inhomogeneously distributed recurrent inhibition approaches that of the respective MNs without recurrent inhibition, i.e., the intrinsic high-pass characteristics of the MNs are restored (in most cases). These high-pass characteristics are important for the fast build-up of tension by the related muscle units (see e.g. Baldissera and Parmiggiani, 1975). It should again be emphasised that this restoration of dynamic MN sensitivity is only achieved by the entire recurrent network and is thus not due to simple self-inhibition of MNs, i.e., to single-loop negative feedback.

A long list of functions have been proposed for recurrent inhibition via RCs (part of which are dis-" cussed in Haase et al., 1975) demonstrating the present inability of truly understanding this network. Cleveland (1977) suggested that MNs inherently pos- sess a high gain with a tendency towards instability, and that recurrent inhibition via RCs serves to stabil- ize MNs. However, he argued from a single-loop feedback model. The present results show that this may not be sufficient to deal with recurrent inhibition, for MNs, depending upon the prevailing conditions,

227

may or may not exhibit high dynamic sensitivity so as to produce overshoot and oscillations in response to step inputs. Essentially the same applies for the velocity of MN response to step inputs which has been suggested to increase by recurrent inhibition (Wenstoep and Rudjord, 1971; Cleveland 1977). - Cleveland (1980) also mentioned the fact that negative feedback could compensate and correct for possible differences in response properties of single elements (in this case of MNs). Whereas this may hold for MNs within a group of specified type, it does not do so invariantly for MNs of different type. This can be seen by comparing the frequency response (particularly the gain) curves of S-type and FR-type MNs at recruitment levels II and III in Figs. 3-5. The frequency response curves of S and FR MNs are relatively similar at level II but more different from each other at level III. The changes in frequency response at different recruitment levels can be quite considerable (Figs. 3-5). Although, for the sake of linearity and hence simplicity, we have applied a small-signal analysis around constant operating points (motor output levels I-III), it is as pertinent to regard dynamic transitions between these levels. Indeed, Hultborn and Pierrot-Deseilligny (1979a) found the most dramatic changes in recurrent inhibition during voluntary muscle contractions with ramp-like force increases at different velocities. One must therefore imagine the changes in M N dynamic behaviour (Figs. 3-5 : rows I through III) to take place in the course of movements in order to see which highly complicated and nonlinear system the recurrent inhibition of ~-MNs actually is. Indeed, the crude nonlinear behaviour of the whole system (as partly detailed in Sect. 2) would come to bear fully only during dynamic large-amplitude changes of motor output. It is feasible that recurrent inhibition may physiologi- cally be most important exactly under these conditions, and not under quasi-stationary conditions, under which it is often experimented and reflected upon. For example, from measurements of RC behaviour during fictive locomotion, Koehler etal. (1981) were led to the idea that recurrent inhibition might be involved in the switching of motor output patterns between agonist and antagonist muscles active in different phases of the locomotor cycle (see also McCrea et al., 1980). (However, this idea has never been clearly detailed and explained.) Similar, though modified, conclusions could he drawn from model studies of Miller and Scott (1977) on a spinal locomotor neuronal network in which RCs play an in- dispensible part. (This model, too, is debatable.) In the present context, however, it is most important to stress the change in dynamic behaviour of MNs. Recurrent inhibition changing with recruitment level has been proposed to serve as a "variable gain regulator" at the level of the "motor output stage" (Hultborn et al.,

1979). If this proposal were to be correct, it must not be restricted to a "static" interpretation of gain (as essentially done by the quoted authors), but would have to be extended to the full dynamic range, i.e., to the complete frequency response (in linear terms). However, to attempt this extension would much con- found the appealing simplicity of the hypothesis for- warded by Hultborn et al. (1979). Since RCs also inhibit other RCs, 7-MNs, and Ia inhibitory interneurones (see Introduction), the change in dynamic sensitivity of c~-MNs at different motor output levels is transmitted (via RCs) to these other neurones as well, and thus indeed affects the entire "motor output stage" considered by the above authors.

Thus the present work, although being limited to linear small-signal analysis, opens new nonlinear as- pects of the M N - R C system. Acknowledgements. This work was supported in part by a grant from the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 33 ("Nervensystem und biologische Information"), G/Sttingen. The authors are grateful to Prof. D.G. Stuart and his collaborators (R. Enoka, Z.Hasan, R. Reinking) for critical comments on the manuscript and to Mr. H. Schultens for scrutinizing the English text.

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Received: October 29, 1982

Dr. U. Windhorst Zentrum Physiologic und Pathophysiologie Universitgt G~Sttingen Abt. Neuro- und Sinnesphysiologie Humboldtallee 7 D-3400 Gbtt ingen Federal Republic of Germany

Verantwortlich fiir den Textteil: Prof. Dr. W. Reichardt. Max-Planck-lnstitut fiir biologische Kybernetik, Spemannstr. 38, D 7400 Tiibingen. Verantwortlich fiir den Anzeigenteil: E. Lticker- mann. G. Sternberg, Springer-Verlag, Kurf/irstendamm 237. D-1000 Berlin 15. Fernsprecher: (030)8821031, Telex: 01-85411. Springer-Verlag. Berlin.Heidelberg.New York.Tokyo. Druek der Briihlschen Universitiitsdruckerei, Giel3en. Printed in Germany. -- O Springer-Verlag GrnbH & Co KG Berlin Heidelberg 1983

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Dynamic behaviour of α motoneurone sub-pools subjected to inhomogeneous Renshaw cell inhibition