Biological Cybernetics

Biol. Cybernetics34, 125-135 (1979)

9 by Springer-Verlag 1979

Auxiliary Spinal Networks for Signal Focussing in the Segmental Stretch Reflex System U. Windhorst PhysiologischesInstitut der Universit~it(LehrstuhlII), G6ttingen, FRG

Abstract. In continuation of a previous paper, the auxiliary signal focussing properties of more complicated spinal neuronal networks are considered here. Special emphasis is put on the distributive function of the recurrent feedback system of e-motoneurones, but also the inhomogeneous distribution of excitatory and inhibitor input to motoneurones is taken into account as an essential prerequisite for signal focussing. Simple hypothetical calculations for steady-state conditions yield a more vivid insight into the interaction of the two types of neuronal circuitry contributing to signal focussing.

Renshaw cells are also interesting insofar as they seem to influence correlations between motoneurone discharge patterns (Adam et al., 1978 ; see also Gel'fand et at., 1963) which in turn pertains to the problem of oscillations and hence stability in motor control. Thus recurrent inhibition via Renshaw cells has only recently been discussed as a possible source of the phenomenon known as physiological tremor (Elble and Randall, 1976; Allum and Buetler, 1978). in this paper, emphasis is put on some aspects of the diverse functions that have been hypothetically attributed to recurrent inhibition (see Haase et al., 1975).

2. The Recurrent Feedback System of ~-Motoneurones I. Introduction

In a previous paper (Windhorst, 1978b), hypothetical mechanisms were considered that might focus motor activity so as to achieve a localization of reflex effects. These mechanisms were based on the fastest reflex pathway: the monosynaptic connections between Ia fibres from primary spindle endings and e-motoneurones, in conjunction with correlations between Ia fibres. Both elements were assumed to be subject to a topographical order. It may be worthwhile looking for supplementary, more complicated spinal circuitries that, at the expense of time, might help focussing motor output. We shall consider two additional mech-' anisms suggested by recent experimental results (see below) : one on the input, the other on the output side of motoneurones. The latter refers to the recurrent collateral system of motoneurones and thus also takes into account recurrent inhibition exerted by Renshaw cells. These have already been mentioned previously (Windhorst, 1978b) as possibly providing a supplementary mechanism for focussing motor activity such as was first suggested by Brooks and Wilson (1958, 1959).

The system to be considered subsequently is depicted semi-schematically in Fig. 1A. For simplicity, the considerations deal with homonymous connections, i.e. those associated with one muscle only, but can principally be extended to heteronymous synergistic muscles. It should first be recalled that homonymous emotoneurones are organized in columns extending rostro-caudally in the anterior horn of the spinal gray matter (cf. Romanes, 1951 ; Thomas and Wilson, 1967 ; Burke et al., 1977). Moreover, at least in some extensor muscles such as the gastrocnemius medialis (Swett et al., 1970; Burke et al., 1977) and the rectus femoris (Markee and L6wenbach, 1945a, b), the location of a motoneurone within this column is roughly related to the location of its motor unit territory in the muscle. The recurrent motor axon collateral system has recently been re-investigated with a new technique by Cullheim and coworkers (Cullheim et at., 1977; Cullheim and Kellerth, 1978a, b). The most interesting results can be summarized as follows (cf. Fig. 1A). The axons of an average c~-motoneurone gives off an extensive tree of profusely branching recurrent col0340-1200/79/0034/0125/$02.20

126

_,=

x

A/

[]

22

(

Eli)Is)

/

L2"--'221

~ ' ~ PAIllls

Kl(s)

\

---I-- ~m E E

EI~)(~J: -" i C'~) ~ pc31(=] LD'=~AI~II~'=',-~

7

:=,,.

i K3'''c') r

- A'31')

=~

Fig. 1. A Simplified sketch of the physiological system under consideration (middle column) and distribution of strength of synaptic coupiings (left and right columns). The homonymous c~-motoneurone column is partitioned into three compartments as indicated by horizontal dashed lines, each compartment being represented by a numbered a-motoneurone (MN; soma represented by open circle). For simplicity, the distribution of synaptic effects has only been drawn from compartment (t) onto itself and others. Excitatory synapses are indicated by bifurcations, inhibitory ones by black dots. Interneurones, i,e. Renshaw cells (RC) and Ia-inhibitory interneurones (Ia-IN), are represented by closed circles. The small horizontal pointed lines in compartment (1) encompass the area of distribution of recurrent motor axon collaterals within the e-motoneurone column. The axons of Renshaw cells excited by such collaterals may project to c~-motoneurones in the same, the adjacent (2) and eventually (as indicated by the vertical dashed line) the third compartment. The strength of these inter-motoneuronal couplings is likely to vary smoothly with distance from the input-giving motoneurone as symbolized by the two (short-range and long-range) bell-shaped curves in the left diagram. For ease of computation, these continuous lines have been discretized to staircase functions with amplitudes X~ 1i) (for excitatory inter-motoneuronal coupling) and y~(ii), y~(21),and ys l) (for inhibitory connections between motoneurones through Renshaw cells). The same procedure has been followed for the distribution of excitatory input from Ia fibre 1 (Ial) to e-motoneurones (right-column: ~4(z) A-10~ u(2) 1110, and Lr(3)~ / / 1 0 7 ' The Ia-inhibitory input is assumed to be homogeneously distributed (J0). B Block-diagram of the physiological system in A. The part of the diagram left to and including the summing points X1 corresponds to the motoneurone system and its interconnections (for simplicity only connections between adjacent compartments are taken into account corresponding to one class of cases considered in the text). The part right to and including 2:2 represents the input side, i.e. the Ia fibre-motoneurone connections. The summing points themselves represent the soma-dendritic membrane system; S 1 and $2 are actually identical but have been separated conceptually for ease of treatment. For fuller description see text

laterals p r o j e c t i n g b a c k to the m o t o n e u r o n e p o o l s a n d to the R e n s h a w cell a r e a of the s p i n a l g r a y m a t t e r .

2.1. Excitatory Connections between Motoneurones T h e collaterals of the m o t o n e u r o n e s c a n m a k e m o n o synaptic, very p r o b a b l y e x c i t a t o r y c o n n e c t i o n s o n t h e m s e l v e s a n d o n o t h e r m o t o n e u r o n e s , so t h a t there is a positive c o u p l i n g b e t w e e n m o t o n e u r o n e s . W i t h i n the h o m o n y m o u s m o t o n e u r o n e pool, the " s y n a p t i c swellings" of the collaterals (from o n e a x o n ) o n the a v e r a g e d i s t r i b u t e over a l i m i t e d r o s t r o - c a u d a l a r e a of r o u g h l y

300 g m e x t e n t (cf. Fig. 1 A), so t h a t o n l y m o t o n e u r o n e s w i t h i n this a r e a c a n be c o n t a c t e d o n their s o m a t a . M o r e r e m o t e m o t o n e u r o n e s c o u l d be c o n t a c t e d via d e n d r i t i c s y n a p s e s such t h a t the m o r e r e m o t e (rostrally or c a u d a l l y ) the m o t o n e u r o n e is situated, the w e a k e r a n d slower w o u l d p r o b a b l y b y the s y n a p t i c c o u p l i n g . Since d e n d r i t i c trees of c ~ - m o t o n e u r o n e s m a y e x t e n d u p to ca. 1 0 0 0 g m f r o m the s o m a [ d e p e n d i n g o n m o t o n e u r o n e size; see B a r r e t t a n d Crill (1974)], the e x c i t a t o r y influence f r o m o n e m o t o n e u r o n e o n o t h e r s seems to be restricted to a r a n g e of r o u g h l y 1 m m in either direction, r o s t r a l or c a u d a l , w h i c h is o n l y p a r t of

127 the overall longitudinal extent of a m o t o n e u r o n e colu m n ( ~ 6 - 1 0 m m ; cf. Burke et al., 1977). Moreover, this influence very p r o b a b l y undergoes a decrimental gradient, both in gain and dynamics.

2.2. Inhibitory' Connections between Motoneurones The circuitry involving Renshaw cells is quite complex because these receive m a n y excitatory and inhibitory inputs from segmental and supraspinal sources in addition to those from m o t o r axon collaterals, and in turn exert inhibitory influences on other neurones than merely m o t o n e u r o n e s (for a review see Haase et al., 1975). We again concentrate on the connections between h o m o n y m o u s m o t o n e u r o n e s via Renshaw cells and also neglect the "mutual inhibition" between these interneurones (Ryall, 1970) which by the way seems to be modest (Haase et al., 1975). Recurrent inhibition via Renshaw cells extends over larger ranges than just shown for the m o n o s y n a p tic coupling between c~-motoneurones. The rostrocaudal, nearly Gaussian frequency distribution of synaptic swellings originating from one m o t o r axon is b r o a d e r outside the m o t o r nucleus than within (ca. 500-600 g m ; with a slight caudal displacement of the mean). The radial dendrites of Renshaw cells have lengths of several h u n d r e d g m (up to 1 m m according to v a n K e u l e n , 1971; but see J a n k o w s k a and Lindstr6m, 1971), so that again one m o t o r axon collateral tree excites Renshaw cells within an area of limited rostro-caudal extent of perhaps 1 ram. (This situation is not exactly reproduced in Fig. 1A, since it is of m i n o r importance.) Whereas Renshaw cells thus receive input from m o t o n e u r o n e s in a restricted region, they in turn distribute their o u t p u t over quite large distances of m a n y millimeters (e.g. T h o m a s and Wilson, 1967; Ryall et al., 1971; J a n k o w s k a and Lindstr6m, 1971 ; J a n k o w s k a and Smith, 1973). This distribution seems to depend on two main factors: proximity of coupled m o t o n e u r o n e s (Eccles et al., 1961; T h o m a s and Wilson, 1967) and on the functional relation between these m o t o n e u r o n e s (Kuno, 1959; Wilson et al., 1960; T h o m a s and Wilson, 1967). With regard to the first factor, recent evidence indicates that axonal branches of Renshaw cells projecting to m o t o n e u r o n e areas appear to extend up to ca. 4 m m longitudinally, with a frequency distribution declining with distance (Jankowska and Smith, 1973). The synaptic contacts on m o t o n e u r o n e s are p r o b a b l y located on or near the soma (Smith et al., 1967 ; Burke et al., 1971), so that the spread of inhibitory influence from one m o t o n e u r o n e on rostral or caudal (homonymous) m o t o n e u r o n e s does not exceed 5 - 6 m m and in this region is subject to a rather quick decrease in strength (see also T h o m a s and Wilson, 1967).

2.3. Finer Features of Excitatory-Inhibitory Interactions between Motoneurones It may be worthwhile to briefly consider the possible interactions of recurrent excitatory and inhibitory cross-influences between motoneurones. Assume that motoneurone c~(1) of Fig. 1A (being a representative specimen) discharges an impulse. In neighbouring motoneurones contacted monosynapticaIly via recurrent synapses on the soma, this would lead to short-latency, relatively large-amplitude EPSPs of quick rise and short duration (Rali, 1967) followed by a longer-latency disynaptic IPSP due to Renshaw cell discharge, lit is known that single discharges of single motoneurones can make a Renshaw cell fire (Ross et al., 1975) although the convergence of correlated discharges of several motoneurones would of course be more effective.] Within this mixed effect, the earlier excitation would possibly predominate. In slightly more remote motoneurones contacted monosynaptically via dendritic synapses, discharge of motoneurone ~(1)would evoke a longer-latency smaller EPSP of slow rise and decay (Rall, 1967; Munson and Sypert, 1978), superimposed on a disynaptic IPSP from Renshaw cells [the small conduction delay of Renshaw cell axons having a conduction velocity of the order of 30 m/s (Jankowska and Smith, 1973) would hardly contribute to the overall delay]. This Renshaw cell IPSP might thus well be able to cancel the remote excitatory effects. A focussing of lateral excitatory coupling between motoneurones would result. In very remote motoneurones, e.g. c~(2), only inhibitory effects via Renshaw cells are left.

2.4. ModeUing The excitatory and inhibitory influences of a representative c~-motoneurone on h o m o n y m o u s ones are complex (see previous section) and p r o b a b l y change smoothly with distance (cf. Fig. 1A, left continuous curves). F o r an easy mathematical treatment, however, we shall lump m o t o n e u r o n e s and their crossconnections together as follows. A m o t o n e u r o n e colu m n of say 7-8 m m length is divided into three compartments of approximately equal length ( ~ 2 . 5 ram), each c o m p a r t m e n t being represented by an average m o t o n e u r o n e at medium height. Such a m o t o n e u r o n e (let's say of the most rostral c o m p a r t m e n t ; cf. Fig. 1 A) exerts excitatory and inhibitory influences on m o t o neurones of its own compartment, solely inhibitory effects on m o t o n e u r o n e s of the adjacent (middle) compartment and eventually inhibitory effects( see below) on the most remote (caudal) compartment. As in a previous paper (Windhorst, 1978b), we assume that signal transmission t h r o u g h the outlined system can be described in terms of linear transfer functions which here are given in Laplace transform notation. This may be justified as follows. First it is assumed that the system is excited by small signals. Under these conditions even the major nonlinear elements, i.e. the neuronal encoders, can be described by linear transfer functions (cf. Poppele and Chen, 1972). Second, although the harmonic distortion of the output of a single motoneurone driven by large-amplitude sinusoidal muscle stretch is increased by recurrent inhibition (Windhorst et al,, 1978), the decorrelating effects of Renshaw cells may suppress distortion of motoneurone population responses (Adam et al., 1978). The latter

128 are actually considered by our lumping procedure. Third, the linear input-output relations of Renshaw cells have recently been investigated (Cleveland and Ross, 1977).

and

2.4.1. Notation. The signals are denoted as follows (cf.

To simplify the notation, define

Fig. 1B) :

X '(ID= X (ij). C ()J) and

E(k)(s) (efferent) output

signal of the k-th emotoneurone (cf. Windhorst, 1978b); P~k)(s) aggregate membrane potential fluctuations of the k-th motoneurone; P~)(s) afferent input signal of the k-th motoneurone. The transfer elements are denoted as follows:

C(k)(S) transfer function of the k-th motoneurone encoder (cf. Windhorst, 1978b); Z(iD(s) transfer function of the recurrent (feedback) element describing the transformation of the output of the j-th motoneurone into membrane fluctuations of the i-th motoneurone. These functions are assumed to be made up of two additive components

P=Z.E+PA=Z.C.P+P

A.

y,(~s)= y(ij). C(jj ) ;

then Z' = Z. C. Rearranging (2.1) yields P = ( I - Z)- ~-PA,

(2.2)

where I is the unit matrix. Provided that the determinant I I - Z ' l # 0 p _ Adj(I- Z')

II-zG

'PA,

(2.3)

where Adj(I- Z') is the adjoint matrix of ( I - Z'). 2.4.3. Preliminary Remarks on Stability. If there are no inputs to the outlined system, i.e. PA=0, the latter is submitted to its own unforced dynamics as expressed by (I- Z'). P = 0.

z(iD(s) =X(iD(s)- y(iJ)(s), where X(~J)(s) describes the excitatory crosscoupling between motoneurones via recurrent collaterals and monosynaptic contacts, and - Y"J)(s) describes the inhibitory crosscoupling via recurrent collaterals and Renshaw cells. For brevity, the arguments, s, are mostly omitted in the following.

2.4.2. Signal Flow. Define the following signal vectors

(2.1)

(2.4)

It can be assumed that P4=0 on account of the ever-present spontaneous activity occurring in the considered physiological system. Then (2.4) being a set of simultaneous homogeneous equations has a non-trivial solution only if I__Zelll)

__ Z'(12)

2'(13) I

(2.5) This is the characteristic equation of the system. Using Sarrus' rule (2.5) results in (1 - Z '( tl )) (1 -

Z '(az)) (1 -

Z '(33)) _ (1 - Z '(11)). Z , ( 2 3 ) . Z ' (32)

-- (1 -- Z'(22)) 9Z t(l 3). Zt (3 l) - ( 1 - Z'(33)). Zt(12). Z t(21)

/ P=/P<2)/ PA=/eT>/ \P
_ Zr(12). Z,(2 3). Z t ( 3 1 ) _ Z,(13 ). Z,(21) 9 Z'~

\PT/

and the "recurrent signal transmission matrix"

1

(2.6)

Equation (2.6) has been written out in order to stress its complexity as compared to the characteristic equation of a single-line (uncoupled) feedback system which in our notation would be of the form

(i-z')=0. \Z(3~)

0

0

Z(32) Z(33)/

X ~33

With the diagonal matrix C II)

=I~ we get E=C.P

C (22) 0

C (33)

\yr

y~32) y~33)/

Since the recurrent feedback system constituted by ~-motoneurones and Renshaw cells has recently again attracted attention as a possible source of oscillations in the neuromuscular control system (see Introduction), it seems appropriate to emphasize that its complexity as expressed by (2.6) should not be neglected when considering this problem. Equation (2.6) is so complicated because it takes into account those loops established by cross-connections between compartments. That complex multi-neuronal loops might function as reverberating circuits in the spinal cord under certain conditions, is suggested by results of Hultborn and coworkers (Hultborn et al., 1975 ; Hultborn and Wigstr6m, 1978). Perhaps the cross-connections between compartments form the basic for the slow spread of correlated neuronal activity (like from a focus) that underlies the equally slow build-up process of physiological tremor (Mori, 1973, t975; Smith et al., 1977) during quiet stance.

129

2.4.4. Focussin9 Features. At this stage of discussion we shall restrict ourselves to considering steady-state conditions which will demonstrate some important features. (Actually there is still a considerable lack of dynamic data, this lack urging further experimental work.) To evaluate (2.3), we first determine A d j ( I - Z'). For simplicity, let us introduce the following "symmetry" properties : X,(11) = X , ( 2 2 ) = X , ( 3 3 )

This has been investigated particularly well for the Renshaw cell feedback which is subject to supraspinal modulations (e.g. Haase and Vogel, 1971; Koehler et al., 1978) besides segmental influences (see Introduction). The efficacy of the positive feedback loops (X~ 11)) could also be changed by altering the "bias" of the motoneurone depolarization level. From a series of computations, the following five cases have been selected to illustrate some important, features :

y,(11) : y,(22) = y~(33) y,(12) = y,(21) = y,(23) = y,(32)

y,(13) ~___y,(31). This may be justified by the striking regularity of the neuronal network. The matrix Adj(I- Z') is then given by

1 2 3 4 5

0.3 0.3 0.3 0.3 0.3

0.4 3.2 3.2 3.2 3.2

0.1 0.8 0.8 1.28 1.28

0.0 0.0 0.32 0.0 0.32

(l -- Z ,(11))Z'(12) q_ Z,(12)Z,(13)

(1- z'(la))Z'(13) +(Z'(12))2 1

(1 - Z '(11))Z'(12) _~ Z,(12)Z,(13)

(1 -- Z '(11))2- (2,(13))2

(1 - Z '(11))Z'(12) _}_Z,(121Z,(13) I .

(1 - Z ,(11))Z'(13) Av (Z,(12))2

(1 - Z '(11))Z*(12) AVZt(12)Z,(13)

(1 - Z'O 1))2- (Z'(12)) 2

(1 - Z '(11))2 _

(Z,(12))2

Let us also assume that all the transfer functions X '~~ and Y'(iJ) are of the following form

[I (1+ sT,)

(I (i + s~) k-1

9e x p ( - sz(iY)), m < n, (2.8) where X~ is the dc gain and the exponential function accounts for the (nearly negligible) synaptic and conduction delays z (~y). If the system described by (2.3) is excited by a set of step functions such that A =~I

AO/

~AO/~

(T denoting transposition), the steady-state response, Po(t) (where P(t)=Sf-*[P(s)]), is obtained by use of the final value theorem: P0(t) = lim P(t) = Jim s.P(s) j ( ! Z'l - Z') = !i+rn A dIIPAO -- (p(1) p(2) p(3)] r --'~--AO--AO--AO]

PAO with

"

Under these steady-state conditions, the Z'(~ in Z' are replaced by the respective dc gains, Zo(~J)'s(and Z' is written Zo). In order to visualize the "focussing" properties of the matrix (I-Z{))-1 [see Eq. (2.2)] let us tentatively assign numeric values to the Z{){~a). These must be chosen somewhat arbitrarily since they have not yet been experimentally determined with appropriate methods. This applies especially to the X{}~) because of the novelty of the corresponding connections9 Moreover, the gains of the recurrent paths are variable.

/

/

For these combinations of values, the elements of the matrix ( I - Z o ) - * have been computed and represented as vertical bars in matrix order in the left column of Fig. 2 (labelled "recurrent network"). The main results can be summarized as follows. 1) Irrespective of the precise values of X~ 11) and 170o 1), the general appearance of the plots is similar. There is a high-gain positive transmission from P~)0 to P~), a low-gain positive or negative transmission from P(1) Ao to P~03)and vice versa, and a low- to medium-gain negative transmission from P~)o to pg_+l). These general features will be referred to as "focussing" properties of the respective neuronal circuitry which is characterized by a variably strong "lateral depression" of signal spread. 2) The absolute transmission gains are largely variable by changing the relation between X{~1~) and Yo(11). But also the relative weights of high-gain positive transmission (principal diagonal) and negative transmission (first parallel off-diagonal) is alterable (compare Case 1 with Case 2 or 4). This implies a modifiability of focussing properties, which would seem to be important in muscle control.

3. The Segmental Motoneurone Input System

Only input from homonymous muscle stretch receptors will be considered. The topographical organization of this system is much more hypothetical than the recurrent motor axon collateral system dealt with above. Let us first concentrate on Ia fibres from primary muscle spindle endings. The monosynaptic Ia

130 Jo = 0.0

0

pot3L_~

u

[po(~._~,~_ ~,~

~o = 0.I

l 0 = 0,3

v

Cos

o

~.

0

0.5

0

1.0-

i

Cose 1

0.5-

f recurrent network

total network

Fig. 2. Steady-state gain matrices for signal transmission through the motoneurone-recurrent collateral system (left column: "recurrent network") and through the total circuitry including Ia input (right three columns : "total network"). Consider the upper left matrix (Case 5, recurrent network). A particular solid bar, say in row 2, Column 1, represents the gain factor by which the respective input signal, in this case P~t0~, is multiplied to yield its contribution to the respective output signal, in this case P~02).The matrix columns are specified by the input signals (and so labelled) and the rows by the output signals. Equivalent specifications hold for the total network matrices (see matrix in first row, second column). Bars above the matrix plane indicate positive factors, bars below the plane negative ones (i.e. negative coupling). In the total network matrices (right three columns) the open bars symbolize gains for the case of the recurrent network working whereas solid bars represent gains without the recurrent network active (see text). The right three columns have been computed for three different gains of Iainhibitory background (J0=0.0, 0.1, and 0.3 from left to right), For the specifications of Cases 1-5 see text

fibre-motoneurone connections, i.e. the fastest pathways, have been treated in a previous paper (Windhorst, 1978b) where it has been hypothetized that there may be a topographical order such that a Ia fibre from a certain muscle region makes strong contacts with a motoneurone having its motor unit territory in that region, but weaker contacts with more "remote" motoneurones. A similar weighted coupling of Ia fibres to homonymous c~motoneurones has been proposed by Binder and Stuart (1978) (cf. also Botterman et ah, 1978). However, these authors seem to suppose an even finer (sharper) pattern without the smooth long-range gradation of synaptic connectivity assumed in the previous paper (Windhorst, 1978b), such that a Ia fibre from a certain muscle region would establish strong monosynaptic contacts with "its" topog-

raphically related motoneurone(s) but would distribute about equal though weaker effects to others. Indeed, for a single Ia fibre there does not appear, on the average, to be such a long-range gradation of synaptic effects on longitudinally arranged motoneurones, at least for certain muscles (Binder et al., 1977b). Nonetheless, we shall retain the scheme proposed in Fig. lA, with the specification that it describes the distribution of effects of a 9roup offi~nctionatty related (correlated) Ia fibres from a muscle area on homonymous motoneurones. This grouping will probably produce some smearing of the very sharp pattern suggested by Binder and Stuart (1978), the smearing being due to the somewhat complicated projection of a three-dimensionally dispersed set of spindles onto the virtually onedimensional array of motoneurones. The exact form of weighted Ia fibre-motoneurone coupling is by the way not that material for the conclusions to be drawn from the following as long as there does exist a topographically ordered inhomogeneity. And for this there is some evidence (Letbetter and Wolf, 1973).

131

Besides monosynaptic there are polysynaptic excitatory connections from Ia fibres to motoneurones (e.g. Tsukahara and Ohye, 1964; Kanda, 1972), but the latter will be neglected here. More important in the present context are inhibitory interneurones which have very recently been demonstrated experimentally (Fetz et al., 1978 ; Jankowska et al., 1978). The physiological properties, i.e. strength and distribution, of this inhibitory system and hence its function have not yet been established. It seems likely that the synapses of the inhibitory interneurones are located on or near the motoneurone somata (cf. Fig. 1A) as suggested by similar findings for Ia inhibitory interneurones responsible for reciprocal ("antagonist") inhibition (Burke et al., 1971). These synapses thus occupy strategically favourable sites, even on remote motoneurones. That is, if there were a specificity of connections such that for example those Ia interneurones illustrated in Fig. 1A received a topographically ordered input preferentially from certain functionally related Ia fibres (Ial), the latter would exert a powerful control in remote motoneurones over the monosynaptic excitation originating from the same sources. Nothing is known at present of the density distribution of homonymous Ia inhibition within the homonymous motoneurone pool. There might of course be a similar decrease with distance as demonstrated for Renshaw cell inhibition (cf. Sect. 2.2). But it should be noted that, as compared to Renshaw cell inhibition, the Ia inhibitory interneurones receiving input from a group of functionally related Ia fibres need not be situated within an area of limited rostro-caudal extent since the input-giving Ia fibre branches distribute over a large longitudinal distance (cf. Fig. 1A). Thus a decriment, if present, would probably not be due to a "thinning out" of axonal branches of the inhibitory interneurones with distance from the soma. There might of course also be the reverse connectivity patter, i.e. an increase of inhibitory power with distance from the inputgiving Ia fibres, although there would at present not seem to be a physiological sense associated with such an arrangement (but see next paper). This specific "reverse" pattern could be supported by recurrent inhibition via Renshaw cells if it assumed that the latter inhibit homonymous Ia inhibitory interneurones as well as those mediating reciprocal inhibition (Hultborn, 1972) and crossed inhibition (Jankowska et aI., 1978). If the strength of recurrent inhibition decreases with distance from the input-giving motoneurones (e.g. c~(1); cf. Sect. 2.2), this might result in less inhibition of "remote" Ia inhibitory interneurones [in compartment (2) or more so in (3)] than of related Ia interneurones [in (1)]. Thus, even if a priori a group of functionally related Ia fibres (Ial) distributed its inhibitory effects equally strongly, i.e. homogeneously, to all homonymous c~motoneurones, the inclusion of recurrent inhibition via Renshaw cells could result in an inhomogeneous distribution such that homonymous Ia inhibition increases with distance from its source. However, these considerations are at present highly hypothetical and warrant experimental verification.

For simplicity, let us assume a uniform distribution of homonymous Ia inhibition throughout the homonymous motoneurone pool which does not even require any specificity, as illustrated in Fig. 1A.

3.1. Modelling The input signals are

A'i(s)

signal on the i-th Ia fibre at spinal level (cf. Windhorst, 1978b).

The transfer elements are denoted as follows:

K}i)(s)

the transformation of Aj(s) into P~)(s) and is composed of the excitatory component HJ)(s) (cf. Windhorst, 1978b) and the inhibitory component J~i)(s)" describes

K}/)(S) = n}i)(s) - J}0(s).

With A ' = (A i A i A;) r and / g (1,

K (1,

1(3' l

K:(KI K2) K'+2,] \K?) we get PA =K-A'.

(3.1)

Substituting (3.1) into (2.3) yields e : (I -- Z') -1. K-A'

-

Adj(I - Z'). K. A'.

II-Z'l

(3.2)

4. Steady-state Behaviour of the Overall System Equation (3.2) describes the dynamic behaviour of the overall system outlined in Fig. 1A. In order to estimate the focussing properties of this system, let us again proceed to steady-state conditions, i.e. apply the same kind of analysis as in Sect. 2.4.4. The H j(i), s and Jj(i),s are assumed to be of the same form as the X'tU)'s [see (2.8)], and the dc gains are supposed to have the following properties and values: H(1) ~(2)_14(3) 10 ~--'1120 - - ~ 3 0 : 1.0 H(1) _ ~(2) =

20 - - ~ 1 0

H(3~_ T4(3)= 0.6 --~20

H(1)_u(3) =0.1 30--~110

j(i) j0 : j o =cons t for all i,j (J0 =0.0 or 0.i or 0.3). The absolute values of the dc gains are arbitrary, but for focussing respects only their relative magnitudes are of importance since they can be scaled by any constant factor without changing these features. Three different levelsof afferent inhibitory input are assumed, and for these the "gain matrices" of the overall system

132 were computed for the five cases considered in Sect. 2.4.4 (cf. Fig. 2, right three columns labelled "total network"). The numeric values of the elements of these matrices are represented by the open bars. In order to see the efficacy of the recurrent feedback system in enhancing signal focussing, consider the special example given by the "total gain matrix" for Case 4, J0 =0.1 : 0.2088

0.0668 - 0.0220

0.0668

0.1868

0.0668

- 0.0220

0.0668

0.2088

which means p(ol)= 0.2088 A'I o + 0.0668 A~o - 0.0220 A~o and so forth. The special structure of this matrix is due to the post-multiplication of ( I - Z ~ ) -a by K o [cf. (3.2)], where 0.2916-0.1073 ( I - Z ~ ) a=

-0.1073

0.0352\

0.3268 -0.1073

0.0352 -0.1073

0.2916

and

Ko=

0.9

0.5

0.0)

0.5

0.9

0.5 .

\0.0

0.5

0.9

If there were no cross-coupling between compartments through the recurrent system, i.e. if ( I - Z ; ) - 1 were a diagonal matrix ( I - Z')- 1 = (6ijc(iJ)), (i,j = 1, 2, 3), where 6i~ is the Kronecker symbol, then

p(ol)=c(l~l.(O.9A'ao +O.5 A'2o +O.OA3o) and so forth. Adjusting 0.9c (a~ to 0.2088 and 0.9C (22) to 0.1868 would yield the following total gain matrix 0.2088

0.1160

0.0

\

0.1038

0.1868

0.1038/.

0.0

0.1160

0.2088/

The difference of this matrix to the above total gain matrix is then due to the absence of focussing by recurrent feedback. This difference is also noticeable in Fig. 2 where the values of the last matrix have been represented by solid bars. It is thus obvious that recurrent feedback enhances "lateral depression" of gains (lateral with respect to the principal diagonal). Before proceeding, it should be emphasized that the total neuronal network can lead to a negative coupling of an afferent input (e.g. A;o ) into a remote moto-

neurone (see P(o1~ above), even if the net input itself is solely excitatory. The possible functional significance of this result, if there is any, can at present only be speculated on and will be dealt with in a subsequent paper. In order to enable a somewhat quantitative comparison and relation between matrices, the following normalized "matrix profile index" was arbitrarily defined. For each matrix, the absolute differences of the eight margin elements to the centre element were added, and the sum divided by eight and the numeric value of the centre element. Such an index provides a very simple measure of the depth of modulation in a matrix being equal to zero if all the matrix elements are equal, i.e. the matrix is "fiat". To visualize the dependence of the profiles of the total gain matrices upon those of the recurrent network matrices, the respective indices were plotted against each other in Fig. 3, with the afferent inhibitory input, Jo, as an additional parameter. In Fig. 3A are plotted the absolute values for the two cases of total gain matrices (open and filled bars in Fig. 2), and in Fig. 3B are plotted the relative changes (in %) of the profile indices for total gain matrices with and without recurrent feedback focussing active. Particularly from Fig. 3B it is evident that the efficacy of the recurrent network in enhancing "focussing features" of the total gain matrices increases with the profile index of its own gain matrices. But this increase is not smooth since there are steeper than average rises for transitions from Case 2 to 3 and from Case 4 to 5 (indicated by arrows). These transitions are characterized by increases of the lateral extent of recurrent inhibition (i.e. Yo(13)=0 in Cases 2 and 4, but Yo(13) =0.32 in Cases 3 and 5). Thus the overall system is obviously more sensitive to changes of the lateral spread of Renshaw inhibition than changes of other parameters. Self-evidently these results are preliminary and somewhat crude and require experimental verification. But they are suggestive of the focussing potency of spinal cord mechanisms which are necessary to provide for localization of stretch reflexes.

5. Concluding Remarks

Two main points deserve some further comments, first, the possible significance of correlations between neuronal channels for dynamic system performance, and, second, the integration of other receptor types. Ad 1) This item refers to an essential aspect of the considerations presented in a previous paper (Windhorst, 1978b). In the present paper, the system was analyzed under steady-state conditions because of lack of dynamic data and simplicity of calculations. In

133

1.0-

0.5-

\

o v x e.-,

O_

I

0.80

,,g

150~

\ I

I

0.85

0.90

B

E

125-

J0

~

03 0.2 gl

100O.BO

0.85

0.90

matrix profile index(recurrent)

fact, however, essential parts of the system may display their full potency only under dynamic conditions. Renshaw cells are particularly sensitive to dynamic stimuli (Hellweg et al., 1974; Pompeiano et al., 1975a, b ; Cleveland and Ross, 1977), and the same appears to apply to the homonymous Ia-inhibitory interneurones which are easily excited by small-amplitude phasic muscle stretches evoking synchronized Ia fibre discharges (Fetz et al., 1978). The propagation and eventual reverberation of such "correlated discharges" through the system is likely to play a significant role for its excitability and stability (cf. Hultborn et al., 1975; Hultborn and Wigstr6m, 1978). There is some evidence that such a phenomenon as physiological tremor is generated as a build-up process by the gradual spread of correlated excitation from a "focus" perhaps in the motoneurone pool (Mori, 1973, 1975; Smith et al., 1977), and it is a challenging problem to investigate the possible involvement of the complex excitatory-inhibitory recurrent feedback system of motoneurones (see also Introduction). Ad 2) Group II fibres from secondary muscle spindle endings have now been established to exert a similar mixture of excitation and inhibition on homonymous ~-motoneurones (at least of extensors) as Ia fibres (Kirkwood and Sears, 1974; Stauffer et al., 1976).

Fig. 3A and B. Importance of the recurrent feedback for signal focussing in the spinal cord. Normalized "matrix profile indices" (MPI) have been defined as outlined in the text and computed for all the matrices in Fig. 2. The MPI's for the "recurrent network" matrices [MPI (recurrent)] have been plotted on the abscissae sucta that Cases 1 through 5 correspond to increasing indices (B). J0 as an additional parameter specifying columns in Fig. 2 has been taken into account by a third coordinate axis. On this basis, the absolute index values of the "total network" matrices [MPI (total)] have been plotted along the vertical ordinate in A distinguishing the two cases of open and solid bars in Fig. 2 by respective open and closed symbols. The percent increase of open symbols relative to closed symbols (reflecting an increase of signal focussing) is plotted in B. Here the closed circles represent Cases 3 and 5 which differ from Cases 2 and 4 (open circles), respectively, only by a spatial extension of recurrent inhibition onto non-adjacent compartments. The open circles lie almost on a straight line as indicated by the added dashed line for J0 =0.0. This enhances the divergence of the closed circles and consequently the significance of the lateral extension of recurrent inhibition in the spinal cord

To integrate them functionally into the presented model, it seems reasonable to assume that their connectivity with motoneurones exhibits a similar topographical pattern as that proposed for Ia fibres, at least with respect to the weighted distribution of their effects. They can receive input from several, probably neighbouring muscle spindles and discharge much more regularly than primary endings (Matthews and Stein, 1969b), at least partly because of their lower dynamic sensitivity (Matthews and Stein, 1969a). They would thus appear to convey more integrated information on steady state parameters of muscle as compared to primary spindle endings being sensitive to small local disturbances. Moreover, secondary endings have recently been postulated to play a much more significant part in the tonic stretch reflex than hitherto assumed (Kanda and Rymer, 1977). These facts would render them suitable to contribute significantly to the dc bias whereupon the Ia fibres from primary endings with their high dynamic sensitivity could act as phasic disturbance detectors responsible for subtle regulatory corrections and adjustments. This bias may be envisaged as a spatially inhomogeneously distributed set of conditional probabilities of excitation which, for example, codetermines the proper distribution of motor command signals. It is noteworthy that there is

134

evidence indicating that Group II muscle spindle afferents may have an inhibitory effect on Renshaw cells (Fromm et al., 1977). This might serve to alter the lateral spread of recurrent inhibition and thus the focussing properties of the spinal interneuronal system according to the present mechanical state of the muscle. Golgi tendon organs are quite obviously well designed to sensitively monitor the local activity of a small sample (10-15) of motor units (Binder et al., 1977a). As force transducers they would thus contribute to continuously signal the local "state variables" of muscle pertinent to an optimal motor performance (see next paper). It is interesting to note that Ib fibres from Golgi tendon organs converge on part of the same Iainhibitory interneurones excited by Ia fibres (Jankowska et al., 1978), which again raises the problem of an eventual topographical order. Finally, another possibility of providing an autogenetic inhibitory complement to the strong excitatory Ia fibre-motoneurone connection is given be presynaptic inhibition modulating the monosynaptic coupling. Interestingly, Thoden et al. (1972) suggested "... that the presynaptic inhibitory mechanism is activated particularly by the repetitive synchronous discharges of primary endings... (p. 103). The presumptive significance of this mechanism for correlations between motoneuronal excitability fluctuations has been extensively investigated by Rudomin and coworkers (Rudomin et al., 1969, 1975). Again, however, a possible topographical order of this system has yet to be established. Acknowledgements. I am grateful for comments on the manuscript to H.-D. Henatsch, W. Koehler, R. Schaumberg, and H. Schultens.

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