The Journal of Neurophysiology Vol. 78 No. 3 September 1997, pp. 1531-1549
Copyright ©1997 by the American Physiological Society

Simulated Recruitment of Medial Rectus Motoneurons by Abducens Internuclear Neurons: Synaptic Specificity vs. Intrinsic Motoneuron Properties

Paul Dean

Department of Psychology, University of Sheffield, Sheffield S10 2TP, United Kingdom

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Dean, Paul. Simulated recruitment of medial rectus motoneurons by abducens internuclear neurons: synaptic specificity vs. intrinsic motoneuron properties. J. Neurophysiol. 78: 1531-1549, 1997. Ocular motoneuron firing rate is linearly related to conjugate eye position with slope K above recruitment threshold . Within the population of ocular motoneurons K increases as  increases. These differences in firing rate between motoneurons might be determined either by the intrinsic properties of the motoneurons, or by differences in synaptic input to them, or by a combination of the two. This question was investigated by simulating the input signal to medial rectus motoneurons (MR-MNs) from internuclear neurons of the abducens nucleus (INNs). INNs were represented as input nodes in a two-layer neural net, each with weighted connections to every output node representing an MR-MN. Individual simulated MR-MNs were assigned parameters corresponding to an intrinsic current threshold I_R and an intrinsic frequency-current (f-I) slope . Their firing rates were calculated from these parameters, together with the effective synaptic current produced by their synaptically weighted INN inputs, with the use of assumptions employed in computer simulations of spinal motoneuron pools. The experimentally observed firing rates of MR-MNs served as training data for the net. The following two training conditions were used: 1) synaptic weights were fixed and the intrinsic parameters of the MR-MNs were allowed to vary, corresponding to the situation in which each MR-MN receives a common synaptic drive and 2) intrinsic MR-MN properties were fixed and synaptic weights were allowed to vary. In each case, the varying quantities were trained with a form of gradient descent error reduction. The simulations revealed the following three problems with the common-drive model: 1) the recruitment of INNs produced nonlinear responses in MR-MNs with low s; 2) the range of I_Rs required to reproduce the observed range of  were generally larger than those measured experimentally for cat ocular motoneurons; and 3) the intrinsic f-I slope  increased with I_R. Experimental data from cat indicate that  decreases with I_R. When synaptic weights were allowed to vary, all three problems with the common-drive model were overcome. This required MR-MNs receiving selective input from INNs with similar firing rate thresholds. These results suggest that the differences in firing rate properties among MR-MNs in relation to steady-state eye position cannot be derived from their intrinsic properties alone but result at least partly from differences in their synaptic inputs. An MR-MN's individual set of synaptic inputs constitutes, in effect, a premotor receptive field.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

The firing rates of ocular motoneurons (OMNs) in relation to steady-state eye position have been measured in a number of species (reviewed by, e.g., Carpenter 1988). An individual OMN only begins to fire when the position of the eye reaches a threshold value  in the ON direction of the relevant muscle. Above this threshold the firing rate varies linearly with eye position with slope K. Within populations of motoneurons (MNs),  and K are related: the higher the threshold , the bigger the slope K (e.g., Van Gisbergen and Van Opstal 1989). Although these properties are well known, their origin is obscure. The projection from abducens internuclear neurons (INNs) to medial rectus MNs (MR-MNs) offers a unique opportunity for studying this problem by virtue of the extensive data available on position-related firing rates for both INNs and MR-MNs (see below).

The three main possibilities for the origin of OMN position-related discharges are as follows.

1) Each member of an OMN pool receives identical afferent input. Differences between the resultant firing rates are caused by differences in the intrinsic properties of the OMNs, for example current threshold for spike initiation (cf. Heckman and Binder 1990). This arrangement has been proposed for skeletal MN pools (Henneman et al. 1965) and termed the "common-drive" mechanism (De Luca and Erim 1994).

2) Differences between the firing rates of OMNs are determined solely by differences in their afferent input to OMNs. The intrinsic properties of OMNs do not vary systematically.

3) Both intrinsic OMN properties and afferent input contribute to differences between OMN firing rates, a combination that has been invoked to explain the effects of stimulating the mesencephalic locomotor region on the discharges of medial gastrocnemius MNs (Tansey and Botterman 1996).

One reason for trying to decide between these possibilities is that the eye position command to OMN pools is the result of outputs from prior stages in oculomotor signal processing, including the neural integrator (Robinson 1989). Characterizing the inputs to the OMN pool therefore helps to characterize the outputs of these prior stages, a step necessary for understanding how they work. For example, many models of oculomotor function treat the eye position command to ocular MN pools as a single, lumped variable that is linearly related to eye position (cf. Dean 1996). This treatment would give cause for concern if the common-drive hypothesis of the origin of OMN firing rates turned out to be incorrect.

Modeling techniques have long proved fruitful for studying the interaction between synaptic drive and intrinsic MN properties for skeletal MN pools (e.g., Rall 1955). The basis for recent studies has been the assumptions that, for steady-state conditions, 1) an MN fires when the total effective synaptic current I_N entering the cell exceeds its current threshold for spike initiation and 2) above threshold, firing rate is linearly related to I_N. These assumptions have been used to construct computer simulations of the firing rates of MN pools in response to synaptic drive from different afferent pathways (Binder et al. 1993; Heckman 1994; Heckman and Binder 1990, 1991, 1993a,b; Powers and Binder 1995a,b; Powers et al. 1992). The present study applies this framework to the responses of MR-MNs to conjugate eye-position-related inputs from INNs. One reason for choosing this system is the evidence suggesting that INNs are the major if not sole source of eye position signals in MR-MNs. INN axons travel rostrally from the abducens nucleus in the medial longitudinal fasciculus to their target MR-MNs in the oculomotor nucleus. Clinical studies of internuclear ophthalmoplegia and experimental studies in monkeys (e.g., Cogan 1970; Evinger et al. 1977; Gamlin et al. 1989b) have shown that interruption or inactivation of this pathway "essentially paralyses the ipsilateral medial rectus except for vergence" (Pola and Robinson 1978, p. 254).

Because OMNs are located within the skull, it is technically easier to record movement-related discharges from them than from MNs in the spinal cord (e.g., Robinson 1986). Thus, there are good data for position-related firing rates for both INNs and MR-MNs (Fuchs et al. 1988; Gamlin and Mays 1992; Gamlin et al. 1989a). This advantage was exploited in the present computer simulation by treating the INN and MR-MN pools as the input and output layers of a linear net (Widrow and Stearns 1985). On a given "training trial" the outputs of the simulation were compared with the "desired outputs," i.e., the real firing rates of MR-MNs. The difference between the actual and desired outputs was used as an error signal to change either the input weights from INNs to MR-MNs or properties of the MR-MNs (intrinsic thresholds and gains) so that the simulation produced firing rates closer to those observed experimentally. This is an example of the use of artificial neural nets to estimate the properties needed for real neurons to produce experimentally observed behavior without prejudice toward the issue of how those properties are derived biologically (Churchland and Sejnowski 1992; Dean 1996; Zipser 1992).

The neural net model of INNs and MR-MNs was trained under various constraints to investigate the following questions.

1) For identical INN input to every MR-MN (common drive), how do estimates of intrinsic OMN thresholds and gains from the simulation compare with experimental findings (e.g., Grantyn and Grantyn 1978)?

2) How are these estimates affected when the connecting weights between INNs and MR-MNs are allowed to vary?

Parts of this work have appeared previously in abstract form (Dean 1995).

	METHODS

Abstract Introduction Methods Results Discussion References

The methods are described in two parts: 1) the modeling of individual MNs and their incorporation into an artificial neural net and 2) the selection of experimental data for INNs and MR-MNs that were used both as input and training data for the net and for comparison with the results of the simulations.

Simulation: individual MNs

The method for calculating the output of an MN given its synaptic inputs and intrinsic properties was taken from Heckman and Binder (1990) and Binder et al. (1993). It is based on a simple neuronal model that neglects dendritic geometry, treats synapses as current sources and not conductance changes, and deals only with steady-state conditions (how far these simplifications apply to OMNs is considered in the DISCUSSION). The method assumes that an MN starts to fire, i.e., is recruited, when the rheobase current I_R (current threshold for spike initiation) is exceeded by the effective synaptic current I_N.

1) I_R is an intrinsic property of the neuron that derives from the neuron's voltage threshold for spike initiation V_T and its input resistance R_N (Eq. 1)

(1)

The R_N itself depends on the specific membrane resistance of the neuron averaged over the relevant membrane area. For skeletal MNal pools, it appears that the ~10-fold range in I_R derives mainly from 2- to 3-fold variance in both soma size and membrane resistivity, with a relatively small contribution from voltage threshold (e.g., Gustafsson and Pinter 1984; Heckman and Binder 1990; Pinter et al. 1983).

2) I_N (defined as the total current that reaches the soma) is primarily extrinsic. If each of a set of synapses on a simplified MN (Fig. 1A) has an input firing rate F_i and a weight w_i, the total synaptic current delivered to the soma is the sum of the current delivered by each synapse (Eq. 2). The present simulation treats I_N as identical

(2)

to I_TOT, i.e., variables such as dendritic geometry that influence current transfer to the soma are neglected (Heckman and Binder 1990, p. 185-186).

View larger version (8K): [in this window] [in a new window]   View larger version (7K): [in this window] [in a new window]   FIG. 1. A: diagram of model ocular motoneuron (OMN), indicating its intrinsic current threshold IR and frequency-current (f-I) slope . MN receives input from F1, Fi, and Fm afferent pathways, each of which fires at Fi Hz and results through its synaptic weight wi nA/Hz in a postsynaptic current Fi·wi nA. B shows how the sum effective synaptic current IN (nA) of these currents (Eq. 2) is related to the output firing rate FR (Hz) of the model OMN as a function of IR and (Eq. 3).

For I_Ns that are above I_R, the firing rate FR of the MN is assumed to be linearly related to the difference with slope  (Eq. 3, Fig. 1B). If

(3)

injected and synaptic currents are equivalent,  corresponds to the slope of what has been termed the f-I relation, where f denotes MN firing rate and I the magnitude of the injected current (Binder et al. 1993). As with spinal MNs (Binder et al. 1993), the current threshold for repetitive firing (I_o) is "slightly higher" than the I_R for OMNs (Grantyn and Grantyn 1978, p. 263) and OMNs do not fire below a minimal firing rate (f_o) of 10-20 Hz (Fuchs et al. 1988). It is assumed here that the point (f_o,I_o) lies on the line of Eq. 3. If f_o ~ 15 Hz and  ~ 30 Hz/nA (see below), this assumption puts I_o at ~0.5 nA greater than I_R, probably in agreement with the remark of Grantyn and Grantyn (1978). The effects of possible nonphysiological low firing rates in the model on the fit between model output and data are considered in the DISCUSSION.

Simulation: neural net

The properties of the simulated individual MNs outlined above are similar to those of model neurons used in linear artificial neural nets (e.g., Anderson 1995). Accordingly, a two-layer artificial neural net was constructed (Fig. 2) with simulated MR-MNs as the output layer and INNs as the input layer. The main features of the net are as follows.

View larger version (13K): [in this window] [in a new window]   FIG. 2. Diagram of the artificial neural net used in the simulations. The m input units, representing abducens internuclear neurons (INNs), each project to all n output units, representing medial rectus MNs (MR-MNs) via the weights w11 to wmn. Firing rates Fi of the INNs in response to the (unknown) fixation command  are given as a function of the eye position  that results from  with the use of the firing rate threshold T and slope s (Eq. 4). Output firing rates oj of the model MR-MNs in terms of their inputs, bias terms Bj, and slopes Gj are given in Eq. 5 and 6. Model outputs can then be compared with real firing rates of MR-MNs to train the neural network to give accurate outputs.

1) As is conventional, the artificial neurons in the input layer do no intrinsic processing but merely convey desired patterns of input to the output layer. In this case the desired patterns are the firing rates of INNs. Thus the method by which these firing patterns are generated from the fixation command  need not be specified provided that the actual firing rates of INNs as a function of  are known. In fact this relation is not known directly. However, in the properly calibrated system, the fixation command  always produces the desired eye position  (cf. Dean 1996). It is therefore possible to use the experimentally observed relations of INN firing rates to eye position, which are approximated by Eq. 4. T_i is the threshold at which the ith INN begins firing

(4)

and s_i is the slope of the straight line relating firing rate to eye position.

2) All input layer neurons connect to all output layer neurons via weights (w₁₁, etc.), which in the present simulation are constrained to be positive or zero. The activation in the jth output neuron a_j produced by a particular input pattern is given by Eq. 5. Because both Fs and ws are

(5)

positive or zero, so too are the activations. The activation term in Eq. 5 corresponds to the total synaptic current of Eq. 2 and thus, by the assumptions of the model, to I_N.

3) All the output neurons have a bias term B_j and a gain term G_j. Their outputs are calculated from Eq. 6

(6)

B is constrained to be negative or zero, which corresponds to a positive I_R in Eq. 3, and G corresponds to the f-I slope  (this dual set of symbols is used to emphasize the distinction between empirically estimated quantities and their counterparts, which are manipulated within the model).

4) To train the network to produce the firing rates displayed by real MR-MNs, these were used as desired outputs of the model (Eq. 7). t_j denotes the actual output of

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the jth MR-MN in response to the fixation command  that produces the eye position . _j is the firing rate threshold of the jth MR-MN and K_j is the slope of its firing rate with respect to eye position. It should be emphasized that these are observed firing rate thresholds and slopes, not the intrinsic properties described above.

Comparison of desired and actual model outputs for a given eye position  yields an error signal e_j for the jth MR-MN (Eq. 8). Rules for altering model parameters that used this error signal

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were derived with the use of gradient-descent methods for fully linear nets (cf. Widrow and Stearns 1985), with adjustments for the nonlinearities in the model (Eq. 9-11)

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(10)

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_G, _B, and _w are learning rate constants whose values were assigned by a two-stage procedure. In the first stage, those parameters required by the overall design of the simulation to be kept constant were assigned learning constants of zero. For example, in the common-drive condition, _w = 0. Second, values of nonzero learning constants were determined by trial and error as those producing rapid learning without instability. The function of the learning rules was to minimize the mean square error E_j for each MR-MN over the training set of eye positions (Eq. 12)

(12)

The values in the training set were chosen at random from the oculomotor range of ±50°.

Data: INNs

The firing rates of INNs FSK88 and GGM89 with respect to steady-state eye position have been described by Fuchs et al. (1988) and Gamlin et al. (1989a). In both studies, the firing rates of individual INNs could be approximated by Eq. 4, with the values of the slopes s_i not significantly correlated with the thresholds T_i. The two estimates of the mean value of the slopes were similar [n = 36, mean value of s = 4.6 Hz/deg (Fig. 5 in Fuchs et al. 1988) and n = 25, mean value of s = 5.3 Hz/deg (Table 1 in Gamlin et al. 1989a)]. However, the distributions of thresholds in the two studies were different (Fig. 3). This is reflected in the 13.4° gap between the mean thresholds of the two samples (29.2° in Fuchs et al. 1988; 15.8° in Gamlin et al. 1989a) and in the proportions of neurons with thresholds greater than 20° (5 of 36 in Fuchs et al. 1988; 16 of 25 in Gamlin et al. 1989a) (², P < 0.0002).

View larger version (10K): [in this window] [in a new window]   FIG. 5. Results from the equal-weight version of the common-drive model (details in text) plotted against the firing rate threshold  of the simulated MR-MNs for the artificial INN distribution. A: error scores. B: estimated MR-MN rheobase or IR, corresponding to the model variable B (Eq. 6). C: estimated MR-MN intrinsic f-I slope , corresponding to the model variable G (Eq. 6).

View larger version (36K): [in this window] [in a new window]   FIG. 3. Histogram to show distribution of firing rate thresholds from 2 studies of INNs (Fuchs et al. 1988; Gamlin et al. 1989a). Thresholds are plotted in 10°-wide bins whose center values are shown on the abscissa. Data from Table 1 of Gamlin et al. (1989a), and replotted from Fig. 5 of Fuchs et al. (1988) with the use of Flexitrace software.

It is apparent that obtaining a representative sample of the INN population is difficult (possible reasons are considered in the DISCUSSION). In the present simulation, the effects of sampling were investigated by comparing three distributions: Fuchs et al. (1988), Gamlin et al. (1989a), and an artificial distribution with 25 neurons, the thresholds of which were evenly spread between the limits of 65 and +25° (giving a lowest threshold neuron of 61.5° and a highest of +21.5°) and the slopes of which were fixed at 5 Hz/deg.

Data: MR-MNs

1) The firing rates of a sample of 74 MR-MNs have been described by Gamlin and Mays (1992, their Table 1). As with INNs, their behavior could be approximated by Eq. 4, only in this case the slopes for conjugate eye position K_j varied with the thresholds _j as is typical for OMNs (e.g., Van Gisbergen and Van Opstal1989). The relation between the two is given in Eq. 13, taken from p. 67 of Gamlin and Mays (1992)

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The range of thresholds in Table 1 of Gamlin and Mays (1992) is -41 to +4°, which is somewhat smaller than the range of 60 to +25° given for OMNs in two reviews (Keller 1981; Van Gisbergen and Van Opstal 1989). Possible reasons for this discrepancy are again considered in the DISCUSSION. For the purposes of the simulations, an artificial distribution of MR-MN thresholds was used, identical to that described for INNs. The slope values were assigned from Eq. 13. Because the behavior of any individual MR-MN in the model is independent of the behavior of the other MR-MNs (Fig. 2), this distribution allows the properties of the Gamlin and Mays distribution to be assessed by interpolation.

2) Although the intrinsic properties of primate MR-MNs appear not to have been measured, data are available for OMNs in cat. One set of in vivo measurements was carried out in the cat abducens nucleus by Grantyn and coworkers (Grantyn and Grantyn 1978; Grantyn et al. 1977). R_Ns for identified MNs ranged from 1.2 to 6.7 M (n = 47) (Grantyn and Grantyn 1978, p. 256). I_Rs of OMNs had values roughly consistent with those of the R_Ns, assuming a threshold depolarization of ~10 mV in Eq. 2. Thus a mean R_N = 2.3 M corresponded to mean I_R = 4.7 nA (n = 15). Neurons with a higher R_N (3-6 M) had a lower average I_R (2.2 nA) than neurons with a lower R_N (1.2-2.9 M), which had an average I_R of 6.3 nA. This again is consistent with Eq. 2, which suggests that the I_R range is similar to the R_N range of five- to sixfold, i.e., from ~1.5 to 8.3 nA, assuming a depolarization threshold of 10 mV. However, Grantyn and Grantyn (1978) do not explicitly report an I_R range. Nelson et al. (1986) obtained an I_R range of 1-13 nA from trochlear MNs (n = 21, R_N range 1.95-7.17 M, correlation between I_R and R_N = 0.73). It is this larger value for the I_R range that is used here for comparison with the results of the simulations.

Measurement of the steady-state firing rates in cat abducens MNs (Grantyn and Grantyn 1978) produced by injected currents gave a mean value for the f-I slope  of 27 Hz/nA (n = 11). Measurement of  and R_N in the same cells showed that  varied with R_N (Fig. 4) (plotted from Table 2 in Grantyn and Grantyn 1978) with a correlation coefficient of r = 0.713 (P = 0.018; although it should be noted that the number of data points is small). The equation for the line of best fit is given in Eq. 14

(14)

Equation 14 together with Eq. 1 allows estimation of the f-I slopes for the abducens MNs with the highest and lowest current thresholds, giving the following valueslowest-current-threshold abducens MN: I_R = 1.5 nA, R_N = 6.7 M,  = 46.3; highest-current-threshold abducens MN: I_R = 8.3 nA, R_N = 1.2 M,  = 21.1. As a consequence of the inverse relation between I_R and R_N, OMNs with lower current thresholds have higher intrinsic gains than OMNs with high thresholds. This relation has not been observed in spinal MNs (see DISCUSSION). If Eq. 14 is also true for cat trochlear OMNs (Nelson et al. 1986), then the range for  would be correspondingly greater than for abducens MNs.

View larger version (12K): [in this window] [in a new window]   FIG. 4. Relationship between f-I slope and input resistance for abducens MNs, taken from Table 2 of Grantyn and Grantyn (1978). Equation for the line of best fit is given in text.

	RESULTS

Abstract Introduction Methods Results Discussion References

Simulations were run under two conditions. In the first, the input weights from INNs to MR-MNs (Fig. 2) were held constant (_w = 0, Eq. 11) and the MR-MN variables bias B and gain G were allowed to vary. The purpose of this procedure was to identify values of MR-MN I_R and f-I slope necessary for the common-drive hypothesis (cf. INTRODUCTION) to account for the firing rate data. For the second condition, the variables B and G were fixed (_B = _G = 0, Eq. 9 and 10) and the weights were allowed to vary (the specific-synapse model) to see what connections between INNs and MR-MNs were consistent with the firing rate data.

This section describes the main results of the simulations. Relevant mathematical details are given in the APPENDIX.

Fixed input weights

An example of the results obtained from simulations under the fixed-weights condition is shown in Fig. 5. In this example, the artificial distribution of INN thresholds (described in METHODS) was used. The weight from each of the n = 25 INNs to each MR-MN was set at a value of 0.01, corresponding to a synaptic drive of 0.25 nA per Hz of averaged INN firing rates. In this version of the common-drive hypothesis each target MN receives the same synaptic drive irrespective of its intrinsic characteristics.

Figure 5A shows error scores, after training, as a function of MR-MN firing rate threshold . The errors were summed at 5° intervals over an oculomotor range of ±50° (Eq. 12). Figure 5A indicates that the error scores are higher for MNs with low s, even though the scores were not adjusted to take account of the expected values of the firing rates, which are higher in high- MR-MNs. Training with fixed weights was seemingly unable to produce a close fit between model and data for MR-MNs with  values less than about 25°.

Figure 5B shows values of the bias variable B in relation to . The bias term was intended to correspond to intrinsic MR-MN threshold current I_R. Two unrealistic features of the estimated I_R values are apparent. First, the values for high- MR-MNs correspond to I_Rs of up to 50 nA. Second, the values drop to 0 nA for MR-MNs with  values of less than about 35°.

Finally, Fig. 5C plots the values of the gain G against . The gain term was intended to correspond to the slope  of the straight line relating steady-state MR-MN firing frequency to input current (the f-I relation). The figure indicates that gain increases with increasing . This is unsurprising given that the slope K of firing rate versus eye position increases with  for OMNs. However, it is not consistent with the finding that for cat abducens MNs intrinsic gain  is inversely related to intrinsic threshold I_R (see METHODS).

Each of the three features of the simulation results illustrated in Fig. 5 indicates a failure of the simulation to reproduce experimental data. The origin of these failures, and their possible remediation by improved choice of model parameter, are described in the next three sections.

Errors

One possible cause of the poor fit between the model firing rates and those observed experimentally for low- MR-MNs (Fig. 5A) is inappropriate choice of values for the weights connecting simulated INNs to MR-MNs in the model. However, altering the weights by a factor of 10 had no effect on the error curve (not shown). Examination of the model output for a low- MR-MN showed the likely root of the problem (Fig. 6A). The output of the model did not vary linearly with eye position, so that a very close match between model and data (see METHODS) for a low- MR-MN was precluded. The output of the model varied with eye position in the same manner as the massed INN input (Fig. 6B), and in both cases the nonlinearity reflects the recruitment of new inputs as eye position changes (details in APPENDIX, Errors in common-drive model).

View larger version (11K): [in this window] [in a new window]   FIG. 6. A: comparison of experimental data and simulation results for MR-MN #8, with firing rate threshold  = 37.3°. B: summed firing rates of simulated INNs (n = 25, artificial distribution) as a function of eye position.

A prediction of this explanation for the errors shown in Fig. 5A is that the shape of the error curve should vary with the distribution of INN firing rate thresholds. Figure 7A shows that this is the case by comparing the error curve for the artificial distribution with the curves from the distributions of Fuchs et al. (1988) and Gamlin et al. (1989a). The total INN firing rate from these distributions (assuming n = 25 neurons) is shown in Fig. 7A. Because in the Fuchs et al. (1988) distribution many INNs are recruited in the range of 40 to 30° (Fig. 3), the slope is more uniform above 30° than that of the artificial distribution, and the errors of the model are correspondingly reduced. In contrast, in the Gamlin et al. (1989a) distribution most of the recruitment takes place above 30° (Fig. 3), which contributes toward higher error scores for MR-MNs with firing rate thresholds less than 30°.

View larger version (15K): [in this window] [in a new window]   FIG. 7. A: summed firing rates of simulated INNs (n = 25) for 3 different distributions as a function of eye position. Actual output of the Fuchs et al. (1988) distribution (FSK88), which was from n = 36 INNs, has been multiplied by 0.694 (= 25/36) for comparability with the other 2 distributions. B: error scores from the equal-weight version of the common-drive mode, as a function of MR-MN firing rate threshold , for 3 different INN distributions. GGM89, Gamlin et al. 1989a distribution.

MR-MNs: I_Rs

The unrealistic range of the B values apparent in Fig. 5B could have resulted from two deficiencies in the model rather than from any inadequacy of the common-drive mechanism.

1) Because of the nonlinearity in the summed INN input, the model's attempts to find the best fit to the data produced impossible values of bias, i.e., B = 0 for MR-MNs with low firing rate threshold  (Figs. 5B and 6A). It is possible to overcome this problem with the use of the summed INN input (details in APPENDIX, Intrinsic current thresholds in common-drive model), which gives the estimates for I_R values shown in Fig. 8A.

View larger version (16K): [in this window] [in a new window]   FIG. 8. A: calculated MR-MN IR (corresponding to model variable B) from the equal-weight version of the common-drive mode, as a function of MR-MN firing rate threshold , for 3 different INN distributions. Synaptic weight w = 0.01 nA/Hz (artificial and Gamlin et al. 1989a distributions) and 0.069 nA/Hz (Fuchs et al. 1988 distribution, n = 36 INNs). For method of calculation see text. B: as in A, with synaptic weight adjusted, for each INN distribution, to give an IR value of 5 nA for the MR-MN with median firing rate threshold  = 20°. The ordinate is now logarithmic to show low values of IR more clearly. Vertical lines: range of MR-MN  values reported by Gamlin and Mays (1992), i.e., 40 to +5°. Weight values w = 0.0037 nA/Hz (artificial distribution), 0.015 nA/Hz (Gamlin et al. 1989a distribution), and 0.0028 nA/Hz (Fuchs et al. 1988 distribution).

2) The actual values of the I_R estimates are determined by the value of weight term, which was chosen arbitrarily as 0.01. It can be seen from Fig. 8A that, in comparison with data from cat abducens MNs (see METHODS), setting w = 0.01 produces I_R estimates that are generally higher than the 1-13 nA recorded experimentally. Appropriate choice of weights (details in APPENDIX, Intrinsic current thresholds in common-drive model) removes this mismatch and gives the I_R estimates shown in Fig. 8B (which uses an ordinate scaled logarithmically to show the values of I_R for low s more clearly).

It is apparent from Fig. 8B that, even after the two problems in the model have been removed, the range of I_R values required by the equal-weight version of the common-drive hypothesis still tends to be higher than that observed experimentally. The precise values depend on both the INN and MN distributions used in the simulation: for example, if the conservative range of 40 to +5° is taken for the MR-MN s (see METHODS), the value of the ratio is 44 for Fuchs et al. (1988), 226 for Gamlin et al. (1989a), and 8.3 for the artificial INN distribution. More importantly, these values are underestimates because the equal-weight version of the common-drive hypothesis needs to be altered to account for the experimental data on the f-I slopes of OMNs.

MR-MNs: f-I slopes

Measurements from cat abducens MNs indicate that the slope  of the f-I relation is inversely related to I_R (see METHODS). In contrast, Fig. 5C shows that the model's estimates for  increase with MR-MN firing rate thresholds, as do the estimates for I_R (Fig. 5B). The MR-MNs in the model, therefore, have s that increase with I_R, as can be seen in Fig. 9, which plots the  estimates directly against the (uncorrected) I_R estimates for the three INN distributions.

View larger version (20K): [in this window] [in a new window]   FIG. 9. Estimates of intrinsic MR-MN properties from the equal-weight version of the common-drive model, for 3 different INN distributions. Abscissa: IR (corresponding to model parameter B). Ordinate: f-I slope  (corresponding to model parameter G). For all 3 INN distributions,  increases with IR.

It is possible to avoid this problem by adopting a different version of the common-drive hypothesis in which the weights from INNs to MR-MNs are larger for MR-MNs with higher firing rate thresholds. For the simulations it was decided to choose weight values that kept G constant, rather than allowing it to vary inversely with B, for two reasons. One was that Eq. 14 is derived from a small number of points, and studies with larger numbers of spinal MNs suggest that I_R and  are largely independent (see DISCUSSION). The second reason was the practical difficulty of calculating weight values that gave appropriately covarying values of B and G. The values for the weights required for s that do not vary with MR-MN properties can be estimated (APPENDIX,f-I slopes in common-drive model) given the constraint that the weights to the MR-MN with the median firing rate threshold ( = 20°) are set to give an I_R of 5 nA (cf. Fig. 8B).

The results of training the model with the artificial INN distribution, and with the weights set to the required values (w = 0.0018 for MR-MN with  = 50, to w = 0.0062 for MR-MN with  = +20°), are shown in Fig. 10. Figure 10A compares the estimates of  with the new and old weights and shows that the new weights do abolish the increase of  with , although the nonlinearity of the INN input prevents perfect compensation at low s. The model's estimates of I_R (calculated as above to avoid the nonlinearity problem) are shown in Fig. 10B. The effect of altering the weights is to increase the estimated ratio of I_R values (APPENDIX, f-I slopes in common-drive model): for the range of MR-MN firing rate thresholds  = 40 to +5°, the estimates are 100 (Fuchs et al. 1988), 4,750 (Gamlin et al. 1989a), and 19 (artificial INN).

View larger version (15K): [in this window] [in a new window]   FIG. 10. Version of the common-drive model with bigger weights on MR-MNs with higher firing rate thresholds . A: estimates of the f-I slope  (corresponding to model parameter G) plotted against MR-MN firing rate threshold  for 3 different INN distributions. Weight values have been adjusted to give, after training, estimates of  that are roughly independent of  (details in text). B: estimates of IR (corresponding to model parameter B) as a function of MR-MN firing rate threshold  for the 3 different INN distributions shown in A. Vertical lines: range of MR-MN  values reported by Gamlin and Mays (1992), i.e., 40 to +5°.

The implication is that the version of the common-drive model that gives more realistic values for the f-I slope  produces estimates of the MR-MN I_R range that are greater than the 13-fold found experimentally for cat abducens MNs. In fact the estimates obtained were conservative: weight values were chosen to make the values for  independent of I_R (see above) rather than to vary inversely with them, and the range for MR-MN firing rate threshold  of 40 to +5° is almost certainly too low (see DISCUSSION). Even so, the estimates from the model were too high, and in the case of the experimental INN distributions very markedly so (100- and 4,750-fold vs. 13-fold). The extremely high value of 4,750 for the Gamlin et al. (1989a) distribution follows from the distribution containing only one INN with firing rate threshold T < 40°: but even for a reduced MR-MN  range of 30 to +5°, the estimated I_R range is still high at 64-fold.

Fixed weights: summary

When the neural network shown in Fig. 2 was trained under the constraint that the weights from simulated INNs to MR-MNs were fixed and independent of the properties of the parent INN, three problems were revealed.

1) The nonlinearity of the massed INN input to MR-MNs with low firing rate thresholds prevented a close fit between model output and experimental data for these MNs.

2) For the equal-weight condition, the estimated f-I slopes increased with I_R, contrary to measurements in cat.

3) Altering the model weights to counteract this problem produced a range of MR-MN I_Rs greater than that observed experimentally in cat trochlear MNs.

These problems were found with both experimentally obtained distributions of INN thresholds and with an artificial distribution. They are therefore unlikely to be the result of sampling artifacts.

Fixed intrinsic properties: synaptic weights variable

In this condition, the variables B and G were fixed (_B = _G = 0, Eq. 9 and 10), and the weights were allowed to vary, to see what connections between INNs and MR-MNs would be consistent with experimental measurements of INN and MR-MN firing rates. The results shown in Fig. 11 were obtained with the artificial INN distribution, with the bias term B set for all MR-MNs to -0.5 and the gain term G to 10. Figure 11A compares the error scores obtained after training the model in this condition with those obtained after training in the fixed-weights condition (Fig. 5A). Allowing weights to vary greatly reduces the error scores for MR-MNs with low s. The distributions of weights that achieved this result are shown for selected MR-MNs in Fig. 11B. The main feature of these distributions is that the effective inputs to a particular MR-MN only arise from INNs with similar firing rate thresholds. This restriction solves the nonlinearity problem associated with massed INN input. Two other features of the weight distributions are that the size of the weights tends to increase with MR-MN firing rate threshold  and that the shape of the weight distribution is different for MR-MNs with s near the top or bottom of the range than for MR-MNs with midrange s.

View larger version (14K): [in this window] [in a new window]   FIG. 11. A: error scores from the specific-synapse model plotted against firing rate threshold  of the simulated MR-MNs, for the artificial INN distribution. Error scores from the common-drive model (Fig. 5A) are shown for comparison. In the specific-synapse model the parameter B (corresponding to IR) was set to 0.5 nA and the parameter G (corresponding to the f-I slope ) was set to 10 Hz/nA. B: size of weights on different MR-MNs (specified by firing rate threshold ) plotted against firing rate threshold of their INN of origin. Weights of size 0 are not plotted.

Values of intrinsic MR-MN properties

The particular values chosen to represent intrinsic MR-MN properties used in the simulation of Fig. 5 (i.e., corresponding to I_R = 0.5 nA,  = 10 Hz/nA) enabled a good fit to be obtained between model output and experimental data. However, there are restrictions on the values that will do this. These are determined by the range of INN firing rate thresholds (see APPENDIX, Weight values and intrinsic OMN properties). For the two experimentally obtained INN samples used in the simulations, it is possible to estimate the range of MR-MN intrinsic properties required for a given range of MR-MN s. For the Fuchs et al. (1988) distribution, a BG value of 100 Hz covers the  range of 30 to +20°. A 10-fold range of BG (from 10 to 100 Hz) covers the  range of 50 to +20°. [For comparison, the mean value of I_R· reported for cat abducens MNs by Grantyn and Grantyn (1978) was 4.7 × 27 = 127 Hz.] For the Gamlin et al. (1989a) distribution, a BG value of 100 Hz covers the  range of 20 to +18°: a 13-fold range of BG (from 10 to 130 Hz) covers the  range of 40 to +20°.

These estimates were checked by running simulations. Data for the Fuchs et al. (1988) distribution are shown in Fig. 12. In this simulation, the variable G was kept constant at a value of 10: B was made to vary linearly with  such that B = 1 when = 50° and B = 10 when  = +20° (positive B values generated by this procedure were replaced with B = 0.001). Figure 12A indicates that allowing the weights to vary with this set of B and G values markedly reduced errors scores in comparison with the fixed-weights condition. Figure 12B shows, for selected MR-MNs, the weight distributions that produced the good fit between model and data. As expected from the analysis in APPENDIX, Weight values and intrinsic OMN properties, the weights on the high- and low- MR-MNs came from a single INN. In contrast, a substantial number of INNs contributed to the midrange MR-MNs. Qualitatively similar results were obtained for the Gamlin et al. (1989a) distribution (Fig. 13). As with the Fuchs et al. (1988) simulation, the variable G was kept constant at a value of 10. B varied linearly with  such that B = 1 when  = 37° and B = 13 when  = +20° (positive B values generated by this procedure were replaced with B = 0.001). Again, error scores were much reduced in this simulation compared with those produced in the fixed-weights condition (Fig. 14A). (No condition eliminates errors for MR-MN  < 40.3°, because 40.3° is the lowest INN firing rate threshold in the Gamlin et al. 1989a sample.) As with the Fuchs et al. (1988) simulation, the shape of the weight distributions that produced the improved fit varied with MR-MN  (see next section).

View larger version (14K): [in this window] [in a new window]   FIG. 12. As in Fig. 11, for the INN distribution of Fuchs et al. (1988).

View larger version (14K): [in this window] [in a new window]   FIG. 13. As in Fig. 11, for the INN distribution of Gamlin et al. (1989a).

View larger version (15K): [in this window] [in a new window]   FIG. 14. Estimates from the common-drive and specific-synapse models of range of MR-MN IRs (corresponding to model parameter B) required to produce the observed range of MR-MN firing rate thresholds . A: for a restricted range of  (40  +5° for the artificial and Fuchs et al. 1988 distributions; 30  +5° for the Gamlin et al. 1989a distribution). Dotted line: IR range of 13, as found for cat trochlear MNs by Nelson et al. (1986). B: for a wider range of . Dotted line as in A.

The differences between the common-drive and specific-synapse models regarding required I_Rs are summarized in Fig. 14. This figure shows the required I_R range for different INN distributions and for a restricted (Fig. 14A) and wider (Fig. 14B) distribution of MR-MN firing thresholds . The restricted MR-MN  range is 40 to +5° for the artificial and Fuchs et al. (1988) distributions and 30 to +5° for the Gamlin et al. (1989a) distribution. The reduced  range for the Gamlin et al. (1989a) distribution was chosen because of the small number of INNs in that distribution that have firing rate thresholds less than 30°. The I_R ranges are based on the assumption that the value of the f-I slope is (roughly) constant and independent of the value of I_R (cf. Fig. 10). It can be seen that for each of the five parameter combinations illustrated, the common-drive model required an I_R range greater than the 13-fold found for cat abducens MNs, whereas the specific-synapse model required an I_R range less than or approximately equal to it.

Distributions of synaptic weights

Figures 11B, 12B, and 13B show that, in the specific-synapse model, MR-MNs with either high or low firing rate thresholds  receive inputs from a small number of INNs with similar firing rate thresholds. By analogy with the receptive fields of sensory neurons, these simulated MNs could be said to have very restricted receptive fields from the INN pool. The input stream from INNs to MR-MNs is chaneled rather than lumped at the extremes of the distribution of firing rate thresholds.

Figures 11B, 12B, and 13B also show that, for MR-MNs with firing rate thresholds closer to the middle of the distribution, the pattern of INN weights is more variable. The main reason for this is indicated in the APPENDIX, Weight values and intrinsic OMN properties. There are two equations that need to be satisfied for realistic MR-MN firing rates to be produced, and for MR-MNs with firing rate thresholds away from the edges of the distribution these equations have more than two unknowns (i.e., weights). The equations do not therefore specify a unique combination of weight values.

There is also a second reason for weight variability, which is illustrated in Fig. 15. Figure 15A shows the distribution of weights onto a midrange simulated MR-MN (firing rate threshold  = 20°) for different amounts of training. After 2,000 trials (error score 559), the distribution is broad and roughly symmetrical about INN firing rate threshold of 20°. As training proceeds, the distribution both narrows (so that the peak weights become larger) and becomes asymmetric, with weights from INNs with thresholds greater than 20° dropping out. Training was stopped after 7,000,000 trials (error score 0.014). An important point is that the performance of the simulated MR-MN after 2,000 trials was very close to that observed experimentally (Fig. 15B). As in the common-drive model, inputs from INNs with firing rate thresholds greater than that of the MR-MN do introduce nonlinearity to the response, but the effect is very small (cf. DISCUSSION).

View larger version (15K): [in this window] [in a new window]   FIG. 15. A: size of weights on a simulated MR-MN with firing rate threshold  = 20°, plotted against firing rate threshold of their INN of origin, after different numbers of training trials. Weights of size 0 are not plotted. Error scores: 2,000 trials, 559; 20,000 trials, 77.1, 200,000 trials, 12.3, 7,000,000 trials, 0.014. B: comparison of experimental data and simulation results for MR-MN firing rates. Model performance is shown after 2,000 and 7,000,000 trials of training.

For these two reasons there is a variety of receptive-field "shapes" possible for MR-MNs with firing rate thresholds toward the middle of the distribution.

	DISCUSSION

Abstract Introduction Methods Results D