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INTRODUCTION |
System identification with the use of Gaussian white noise (GWN) is a "black box" modeling approach that can define the transfer characteristics of neural systems (Marmarelis and Marmarelis 1978
; Marmarelis and Naka 1972
). Unlike traditional approaches in which sinusoidal, ramp, or pulse inputs are used, the white noise approach provides a complete and systematic description of a system. The resulting system characterization is statistically averaged and describes the dynamic linear and nonlinear characteristics, and it provides a characterization that is, in many cases, free of contaminating noise. The GWN signal is an information-rich stimulus because it contains, theoretically, every possible frequency and amplitude, with a Gaussian distribution of signal amplitude. Given the limited life of an experimental preparation, this method of analysis provides an ideal means to test a system effectively and rapidly. With the use of these methods we have been investigating the dynamics of the responses of neurons in a neuronal network that produces and controls a negative feedback resistance reflex of the hind leg of an insect, the desert locust (Kondoh et al. 1995
; Newland and Kondoh 1997
).
In the locust hind leg a femoral chordotonal organ (FeCO) monitors the position and movements of the tibia about the femorotibial joint, and contains ~90 sensory neurons. These sensory neurons respond to either extension or flexion movements of the tibia and have been shown to encode position, velocity, and acceleration, or combinations of these parameters (Kondoh et al. 1995
; Matheson 1990
; Matheson and Field 1990
; Usherwood et al. 1968
; Zill 1985
). Recently we have shown with the use of white noise analysis that the coding properties of the FeCO sensory neurons vary systematically across the entire population (Kondoh et al. 1995
). Thus at one end of the spectrum of responses sensory neurons are position sensitive, whereas at the other extreme they are acceleration sensitive. Between these two extremes afferents code position, velocity, and acceleration to varying degrees.
These sensory neurons make direct synaptic connections, and form feedback loops, with either flexor or extensor tibiae motor neurons that control the muscles that move the tibia about the femorotibial joint (Burrows 1987
). When a leg is extended the flexor tibiae muscles are activated to return the leg to its initial position, and when the leg is flexed the extensor tibiae muscles are activated. The flexor motor neurons receive a barrage of synaptic inputs from the FeCO when its apodeme is relaxed (Field and Burrows 1982
), and the majority of these inputs are dependent on the velocity of movement of the stimulus (Newland and Kondoh 1997
). Field and Coles (1994)
, however, demonstrated that there is also a position dependency of the resistance reflex. Although we know much about the dynamic coding properties of the FeCO afferents and flexor tibiae motor neurons, we know little about the extensor motor neurons, except that they do receive both position- and velocity-dependent inputs (Field and Burrows 1982
).
The aim of this study was twofold: first, to analyze the position dependency of the feedback pathway described by Field and Coles (1994)
; and second, to analyze in detail the response properties of the extensor tibiae motor neurons. Are their response properties similar to those of their antagonists, the flexor tibiae motor neurons (Newland and Kondoh 1997
), or can the coding of movement of a single leg segment about a particular joint be different for antagonistic pools of motor neurons? Moreover, within the extensor motor neuron pool, are there any difference in coding properties between the fast and slow extensor tibiae motor neurons (FETi and SETi, respectively)?
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METHODS |
Adult male and female locusts, Schistocerca gregaria (Forskål), taken from our crowded laboratory culture in Cambridge were used for all experiments. Locusts were mounted ventral surface uppermost in Plasticine, with a hind leg fixed anterior surface uppermost at a femorotibial angle of 60°. A small window was cut in the ventral thorax to expose the meso- and metathoracic ganglia, which were then stabilized with pins on a wax-covered silver platform. The sheath of the ganglion was treated with protease (Sigma type XIV) for 45 s before recording. Microelectrodes filled with potassium acetate, and with DC resistances of 50-80 M
, were driven through the sheath and into the somata or neuropilar processes of extensor and flexor tibiae motor neurons. Intracellular recordings were made with the use of an Axoclamp 2A amplifier (Axon Instruments). A pair of 50-µm copper wires, insulated except at the tips, were implanted in the extensor muscle of a hind leg and were used to stimulate the extensor muscle and evoke antidromic spikes in FETi. The motor neurons were identified on the basis of the positions of their somata, their responses to imposed leg movements (Fig. 1A), the speed of tibial movement they evoke when stimulated with depolarizing current, and their responses to antidromic stimulation of the extensor muscle. The results are based on 34 successful recordings from FETi and 24 from SETi in 58 locusts.

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| FIG. 1.
Responses of extensor and flexor tibiae motor neurons to movements of the femoral chordotonal organ (FeCO) apodeme. A: intracellular recordings from the slow and fast extensor tibiae motor neurons (SETi and FETi, respectively) and 3 posterior flexor motor neurons. Relaxations of the apodeme (equivalent to extensions of tibia) lead to hyperpolarization of the extensor tibiae motor neurons and depolarization of the flexor motor neurons, the fast (PFFlTi), intermediate (PIFlTi), and slow (PSFlTi) posterior motor neurons. First- (B) and 2nd-order (C) kernels from synaptic responses in FETi recorded intracellularly from the soma. The FeCO apodeme was stimulated mechanically with Gaussian white noise with cutoff frequency (fc) = 27 and 58 Hz. First-order kernels from 20 locusts at fc = 27 Hz (Bi) and 17 locusts at fc = 58 Hz (Bii) were normalized and superimposed. Second-order kernels were averaged from the same locusts at fc = 27 Hz (Ci) and fc = 58 Hz (Cii).
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Stimulation and data analysis
The apodeme of the metathoracic FeCO was exposed by cutting a small window of cuticle in the distal anterior femur (see Field and Burrows 1982
), grasping with fine forceps attached to a vibrator (Ling Altec, type 101), and cutting distal to the forceps. The forceps holding the apodeme were moved with conventional ramp wave stimuli or with GWN with cutoff frequencies (fcs) of 27 or 58 Hz. Details of this procedure, the properties of the GWN signal, and the analysis of synaptic potentials in motor neurons are given in detail in Newland and Kondoh (1997)
, Sakuranaga and Naka (1985a)
, and Sakai and Naka (1987)
. Briefly, the GWN used for both stimulation and cross-correlation was based on a binary sequence generated with the use of a random binary generator (CG-742N, NF Circuit Design Block) band limited from DC to 27 or 58 Hz by a low-pass filter (SR-4BL, NF Circuit Design Block) with a decay of 24 dB/octave. This band-limited white noise was used for both stimulation and cross-correlation. The amplitude of the GWN with fc = 27 Hz covered the entire linear range of movement about the femorotibial joint from full flexion (0° joint angle) of the hind leg to the full extension (120°), whereas the GWN with fc = 58 Hz covered the range of 15-105°.
Data were stored on a digital audio tape recorder (RD-101T, TEAC). The synaptic responses of the extensor tibiae motor neurons were then fed to a 16-bit personal computer (PC-9800VX, NEC) through a 12-bit analog-to-digital converter (ADX-98H,Canopus Electronics) at a sampling rate of 1-2 kHz for 20 s. Spike trains in SETi were also analyzed with the use of the cross-correlation technique by first converting each SETi spike to a unitary pulse 2 ms in duration with the use of a Schmidt trigger circuit. Data were also sampled through a 12-bit analog-to-digital converter at a sampling frequencies of 2 kHz, again for 20 s. All data were then transferred to a VAX 4000 computer (Digital Equipment) on which the software for white noise analysis, STAR, was run. To compute the first- and second-order Wiener kernels, the stimulus and synaptic response signals were digitally filtered at 0.5-200 Hz. First- and second-order Wiener kernels, which represent the linear and nonlinear components of their responses, characterize the response dynamics of the extensor tibiae motor neurons. We used the cross-correlation method of Lee and Schetzen (1965)
to compute kernels, and for estimation and convolution we used algorithms described by Sakuranaga and Naka (1985a
,b
). The first-order kernel is the first-order cross-correlation between the input, the white noise modulated position of the tibia, and the output, the evoked synaptic response of an extensor motor neuron. The second-order kernel defines the multiplicative interaction between two parts of the input in the past, and is therefore a function of two time lags, t1 and t2. The magnitude of the nonlinear response is represented by contour lines on a two-dimensional plot. Solid lines indicate positive or depolarizing peaks and dashed lines indicate negative or hyperpolarizing valleys. The detailed algorithms for computing the kernels, model responses, power spectra, and mean square errors (MSEs) have been described in Sakuranaga and Naka (1985a
,b
) and Sakai and Naka (1987)
.
Model responses of the extensor motor neurons were predicted by convolving the white noise input with the first- and second-order Wiener kernels. A quantitative measure of the agreement of the model with the actual response is the MSE reduction. This is given as a percentage by computing the ratio of the deviation between the response and the model, and indicates the accuracy of the model prediction (Marmarelis and Marmarelis 1978
). For example, the MSE for the linear model represents the degree of linearity, i.e., the ratio of the linear component to the total response.
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RESULTS |
SETi and FETi are inhibited by relaxations of the FeCO apodeme (Fig. 1A) that are equivalent to extensions of the tibia about the femorotibial joint, and excited by stretches equivalent to flexion movements of the tibia. The same movements produce opposite responses in flexor tibiae motor neurons. If the leg were free to move these responses would therefore produce resistance reflexes that would counteract the imposed tibial movement.
Linear and nonlinear responses of FETi
When stimulated with white noise having fc = 27 Hz, the first-order kernel of the synaptic response of FETi had a positive peak at time,
= 16.4 ± 2.6 (SD) ms (n = 21) that was followed by a smaller negative undershoot (Fig. 1Bi). Because the first-order Wiener kernel is the best approximation of a motor neuron's response to an impulse (in this case, a sudden change of the femorotibial joint angle consisting of a brief flexion of the tibia followed by an extension), then a monophasic first-order kernel with a depolarizing peak indicates that the motor neuron is depolarized when the leg is flexed, and that the magnitude of the depolarization is proportional to the degree of the flexion, i.e., the position of the leg. Thus the responses of FETi are primarily dependent on the position of the tibia, but with a small velocity component (seen as the small negative undershoot of the kernel).
The response of FETi was independent of the upper fc of the stimulus so that a similar predominantly monophasic waveform was obtained when the FeCO apodeme was stimulated at fc = 58 Hz (Fig. 1Bii). The time-to-peak of the first-order kernel was 19.9 ± 2.1 ms (n = 17).
The second-order Wiener kernel predicts the nonlinear component of the response. That of FETi had a depolarizing peak on the diagonal (for
1 =
2 = 14-16 ms) at fc = 27 Hz with two small off-diagonal valleys (Fig. 1Ci), a configuration characteristic of a low-pass component. Thus the nonlinearity in FETi is also position dependent. The peaks and troughs of the second-order kernel were extended parallel to the diagonal when the FeCO apodeme was stimulated at fc = 58 Hz, with an elongated peak on the diagonal for
1 =
2 = 15-30 ms (Fig. 1Cii). The times to positive peak of the second-order kernels were similar to those of the first-order kernels, indicating that a depolarizing response in the linear term is enhanced by the nonlinear second-order component, whereas a hyperpolarization in the linear term is reduced. This is a compression nonlinearity.
More quantitative data for the linearity of response in FETi were obtained by predicting the linear and nonlinear components of the responses of FETi by convolving the first- and second-order kernels with the same stimulus input (Fig. 2A). The linear components were large, but a summation of the linear and second-order models (L + 2nd) provided a more accurate prediction of the actual response. The MSE for the linear model of FETi was 35-45%, whereas that for the second-order, nonlinear model was 8-15% atfc = 27 Hz (Fig. 2B).

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| FIG. 2.
Response nonlinearity in FETi. A: intracellular recording from the soma of FETi during white noise stimulation at fc = 27 Hz, and corresponding linear and 2nd-order models. Mean square errors (MSEs) for the linear model and summation of the linear and 2nd-order models (L + 2nd) were 50.4% and 39.6%, respectively. B: the accuracy of model predictions by the 1st- and 2nd-order kernels in FETi expressed as MSEs for the linear and 2nd-order models against the response at fc = 27 and 58 Hz. Average improvements of MSEs for linear (open bars) and 2nd-order (filled bars) models against the response (which were from 21 locusts at fc = 27 Hz, and from 17 locusts at 58 Hz, respectively) are shown. Linear and 2nd-order models predicted 35-45% and 8-15% of the total response, respectively.
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Frequency and gain characteristics of the FETi response
Because a GWN signal contains, in theory, every possible frequency and all amplitudes with a Gaussian distribution, the first-order kernel contains the frequency and transfer characteristics (gain and phase) of a system. The gain and phase characteristics of the FETi response during FeCO stimulation were obtained by fast Fourier transformation of the averaged first-order kernel shown in Fig. 3A and are shown in Fig. 3, B and C. There was no phase shift in the frequency range of 0.5-8 Hz, but above 8 Hz phase was negative (delayed) (Fig. 3B). This phase delay increased with an increase in stimulus frequency and reached
180° at 20-30 Hz.

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| FIG. 3.
Transfer (gain and phase) characteristics of FETi response. A: averaged 1st-order kernels from FETi responses recorded intracellularly from the soma at fc = 27 Hz ( ) and 58 Hz (- - -), from which gain and phase curves were produced by fast Fourier transformation. B and C: phase and gain curves produced from the 1st-order kernels in A, at fc = 27 Hz ( ) and 58 Hz (- - -). B: no significant phase shift was found in the frequency range of 0.5-8 Hz. Phase became delayed above 8 Hz, and reached 180° at 30 Hz. C: gain increased slowly across the frequency ranges examined.
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Gain increased only slightly across all frequencies examined (from 0.5 Hz to the fc of the stimulus, 27 or 58 Hz), suggesting that the response in the FETi motor neuron was essentially constant gain and low passed, i.e., position sensitive, with only a minor velocity dependence (Fig. 3C). In the frequency range of 0.5-20 Hz, the gain at fc = 58 Hz was similar to that observed at fc = 27 Hz, although the first-order kernel at fc = 58 Hz was larger in amplitude than that at fc = 27 Hz (Fig. 3A). Thus the increased amplitude of the first-order kernel observed at fc = 58 Hz is probably due to the addition of the response to the high-frequency components of the stimulus above 27 Hz.
Filter properties of the primary neurite of FETi
Because the somata of motor neurons are often a few hundred micrometers distant from their sites of integration in the neuropil, the signals in neuropilar segments will be delayed and reduced in amplitude according to the passive cable properties of the membrane. Thus we must address the problem of whether the Wiener kernels obtained from our analysis from the soma recordings represent the actual response dynamics or whether they are the result of signal transformation along the primary neurite.
Synaptic responses of FETi recorded in the soma were always smaller in amplitude than those recorded simultaneously in a neuropilar segment (Fig. 4A). Moreover, simultaneous recordings, one from the soma and the other from a neuropilar process of FETi, produced first-order Wiener kernels with that from the soma 30-40% smaller than that from a neuropilar process (Fig. 5A). Additionally, the rise time of the first-order kernels from the soma was slower, indicating the low-pass nature of the signal propagation from the neuropilar process. The power spectrum of the response recorded from a neuropilar process had a peak at ~35 Hz (Fig. 4B, arrowhead), which was absent from the soma recording, suggesting that the reduction in amplitude of the response in the soma was due, in part, to a loss of a high-frequency component that was present in the neuropil.

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| FIG. 4.
Higher-frequency components have greater power in neuropilar processes. A: simultaneous recordings from the soma (top trace) and a neuropilar process (middle trace) of FETi during stimulation of the FeCO with a Gaussian white noise (bottom trace) with fc = 58 Hz. The response recorded from the soma had a significantly lower amplitude compared with that recorded in the neuropilar process. B: power spectra of the responses from the soma ( ) and a neuropilar segment (- - -) of FETi at fc = 58 Hz. Note the higher power of high-frequency signals in the neuropil recording (arrowhead).
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| FIG. 5.
Comparison of transfer characteristics of the linear and nonlinear components of FETi responses recorded in the soma and in a neuropilar process. A: averaged 1st-order kernels of FETi from a neuropilar process (- - -) and soma( ) at fc = 58 Hz. Note the time-to-peak was shorter and the amplitude greater in the recording from a neuropilar process. Fast Fourier transformation of the averaged 1st-order kernels produced the gain and phase curves shown in B and C. Phase was delayed at higher frequency, whereas gain was flat at all frequencies. Di and Dii: 2nd-order kernels of the responses from the soma (Di) and a neuropilar process (Dii), averaged from 4 locusts. No differences were found in the configuration of the 2nd-order terms at different recording sites. Both kernels had positive peaks at 1 = 2 = 20 ms with very small negative valleys on off-diagonal areas, which were elongated diagonally.
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An examination of the frequency responses showed that phase was delayed above a stimulus frequency of 2 Hz, reaching
30° at 20 Hz at fc = 58 Hz (Fig. 5B) and increased with increasing stimulus frequency. Above 5 Hz, phase delay was greater in the soma compared with the neuropilar process, probably because of membrane capacitance between the two recording sites. Gain, however, was always larger in recordings from neuropilar processes at all frequencies tested (Fig. 5C). In particular, at fc = 58 Hz, the gain in the soma recording was much lower in the frequency range of >20 Hz (Fig. 5C). This clearly represents a signal reduction due to the passive cable properties of the membrane of the primary neurite.
Although there were small differences in the gain and phase characteristics, the basic configurations of the first and second-order kernels were similar from both recording sites (Fig. 5, A and D), indicating little transformation of the signal. Both second-order kernels had positive peaks at
1 =
2 = 20 ms with very small hyperpolarizing valleys on the off-diagonal area.
The transfer characteristics of signal propagation along the primary neurite were examined by injecting white noise modulated current into a neuropilar segment and recording the response in the soma (Fig. 6A). A cross-correlation between the injected current and recorded soma synaptic potentials produced well-defined monophasic first-order kernels (Fig. 6B). The linear model from this kernel predicted the current-evoked response with an MSE of 0.9%. The primary neurite of FETi was thus equivalent to a simple low-pass filter.

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| FIG. 6.
Filter properties of FETi primary neurite. A: intracellular recording from the soma of FETi and its linear prediction evoked by white noise modulated current injected via a 2nd electrode into a neuropilar process. The MSE for the linear prediction was 0.9%. B: 1st-order kernel of FETi synaptic response produced by a cross-correlation between the soma response and injected white noise modulated current. C: simultaneous recording from a neuropil process and soma of FETi. Model response was predicted by convolving the 1st-order kernel shown in B with the recording from a neuropilar process during FeCO stimulation. Note that the predicted model response closely matches the true response recorded simultaneously in the soma.
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Convolving the response of FETi evoked in a neuropilar process during FeCO stimulation (Fig. 6C) with the first-order kernel calculated from the above white noise modulated current experiment (Fig. 6B) produced a model response that closely matched the actual soma response that was recorded simultaneously (Fig. 6C).
Linear and nonlinear responses of SETi
Stimulation of the FeCO with fc = 27 Hz produced a first-order kernel from the synaptic responses of SETi with a sharp positive peak at
= 19.8 ± 2.8 ms (n = 12) (Fig. 7Ai). When the FeCO apodeme was stimulated at fc = 58 Hz (Fig. 7Aii), the time-to-peak was 23.1 ± 3.1 m (n = 9). The second-order kernel of SETi had a large on-diagonal peak with two small off-diagonal valleys (Fig. 7Bi) andwas therefore similar in shape to that of FETi (Fig. 1C) for
1 =
2 = 0-50 ms. The time to the diagonal peak of the second-order kernel was ~20 ms at fc = 27 Hz and 26 ms at fc = 58 Hz in the averages shown in Fig. 7, Bi and Bii. However, the second-order kernel of SETi had an additional elongated negative, or hyperpolarizing valley on the diagonal (for
1 =
2 = 50-120 ms at fc = 27 Hz; for
1 =
2 = 50-100 ms at fc = 58 Hz), which was accompanied by two small off-diagonal depolarizing peaks at fc = 58 Hz. SETi was therefore primarily position sensitive, but also received a delayed low-passed inhibitory input.

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| FIG. 7.
Response dynamics of SETi. First- (Ai and Aii) and 2nd-order (Bi and Bii) kernels of the responses of SETi to FeCO stimulation recorded intracellularly from the soma. The FeCO apodeme was stimulated mechanically with Gaussian white noise with fc = 27 Hz (Ai and Bi) and 58 Hz (Aii and Bii). First-order kernels from 12 locusts at fc = 27 Hz and 9 locusts at fc = 58 Hz were normalized and superimposed. Each 2nd-order kernel was an average from 9 locusts.
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Phase was delayed in SETi (Fig. 8B) to a greater extent than in FETi (Fig. 3B), and was negative (phase delayed) at all frequencies tested (0.5-27 and 0.5-58 Hz). The phase delay increased with an increase in stimulus frequency, and reached
180° at 5-7 Hz and
360° at 20-30 Hz. This difference in the phase delay of SETi and FETi was, in part, due to the delayed inhibitory input to SETi, and also to differences in the cable properties of the respective neurons. Since SETi has a smaller-diameter primary neurite than FETi, the signals recorded in the neuropil could be reduced in amplitude and delayed in phase in the soma more than those in FETi.

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| FIG. 8.
Transfer (gain and phase) characteristics of the SETi response to FeCO stimulation. A: averaged 1st-order kernels from responses recorded intracellularly from the soma of SETi at fc = 27 Hz ( ) and 58 Hz(- - -), from which the gain and phase curves were produced by fast Fourier transformation. Kernels were both averaged from 8 locusts. B and C: gain and phase curves produced from the averaged 1st-order kernels of SETi shown in A, at fc = 27 Hz ( ) and 58 Hz (- - -). B: phase was negative (phase delayed) in all frequency ranges examined (0.5-27 and 0.5-58 Hz). Phase delay increased with an increase of stimulus frequency, and reached 180° at 5-7 Hz and 360° at 20-30 Hz. C: gain was constant in all frequency ranges examined and fell off at ~20-30 Hz.
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Gain in SETi again increased only slowly across all frequencies, indicating the low-passed nature of the response of SETi, i.e., position sensitive, with only a minor velocity dependence (Fig. 8C).
To assess the goodness-of-fit of the Wiener method of analysis and characterization, we predicted the response of SETi to white noise stimulation and compared it with the evoked response to the same stimulus. The MSE for the linear model of SETi was 36%, whereas that for the nonlinear model was 10%, and thus similar to those of FETi. The nonlinear component of the SETi response was relatively large (Fig. 9A) and its addition to the model improved it by 26% at fc = 27 Hz. The addition of the nonlinear component sharpened the peaks and rectified the negative deflections in the first-order model response.

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| FIG. 9.
Response nonlinearity in SETi. A: intracellular recording from the soma of SETi during white noise stimulation (fc = 27 Hz), and the corresponding linear and 2nd-order models. The MSE for the linear model and the summation of the linear and 2nd-order models (L + 2nd) was 60.3% and 49.4%, respectively. B: power spectra of the response shown in A (marked SETi), and the corresponding linear and 2nd-order models, and white noise stimulus with fc = 27 Hz.
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Power spectra of the models and the response of SETi show that their power falls off above 5 Hz (Fig. 9B), contrasting with FETi (Fig. 4B), in which the power falls off markedly only at the fc of the stimulus. At lower frequencies, however, the power of the SETi responses is greater than the power of those of FETi.
Interpretation of the nonlinear component of SETi
The second-order component of SETi was interpreted by taking cuts through the diagonal of the second-order kernels where
1 =
2. This "diagonal cut" corresponds to an impulse response in the second-order term. The diagonal cut of SETi shows an initial depolarization followed by a long hyperpolarization (Fig. 10A). Superimposing this on the first-order kernel (Fig. 10B) shows that the nonlinear component represents a half-wave rectification of both the initial position-sensitive depolarizing response and the delayed inhibitory response of SETi, i.e., when the tibia is flexed these two components sum (Fig. 10B), but they subtract when the tibia is extended.

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| FIG. 10.
Second-order, nonlinear component represents a half-wave rectification of both the position-sensitive depolarizing response and the delayed inhibitory response of SETi. A: the 2nd-order kernel produced from the response in SETi at fc = 27 Hz, on which the diagonal cut of the kernel (dashed trace) is superimposed. Solid lines: positive (or depolarizing) peaks. Dashed lines: negative (or hyperpolarizing) valleys. The kernel had a depolarizing peak on the diagonal at 1 = 2 = 25-40 ms with 2 very small hyperpolarizing valleys in the off-diagonal areas. B: 1st-order kernel (solid trace) from a response in SETi at fc = 27 Hz, on which the diagonal cut of the 2nd-order kernel (dashed trace) is superimposed.
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Dynamics of a spike train in SETi
Signal transformation within a neuron can be estimated by comparing the dynamics of the synaptic input with the spike response output. To analyze the spike response of SETi during FeCO stimulation (Fig. 11Ai, top and bottom traces, respectively) we first unitized the spike discharge with the use of a Schmidt trigger circuit (2nd traces). Cross-correlation between the white noise stimulus and the unitized spike discharge produced well-defined first- and second-order kernels (Fig. 11, B and C), in which positive peaks represent increases in spike density and negative peaks represent decreases. Spike and synaptic potential models could be predicted from the spike and synaptic potential kernels, respectively (Fig. 11, Aii-Aiv). Superimposing the averaged(n = 4) first-order kernels of the spike and synaptic response shows that they were similar in configuration, with initial positive low-pass components followed by delayed inhibition (Fig. 11B). Moreover, there was little difference in the configurations of the second-order components, with the exception of a loss of the delayed inhibition in the spike response. The similarity in the configurations of these kernels implies that although the spike production process itself is a highly nonlinear event, it does not cause significant change in the response properties of SETi, i.e., the spike response is also dependent on the position of the tibia about the femorotibial joint.

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| FIG. 11.
Comparison of response dynamics between the synaptic potential response (synaptic component) and the spike response from the soma of SETi. Ai: intracellular recording from the soma of SETi (marked Response) was segregated into 2 components: a synaptic potential produced by passing the signal though a low-pass filter with fc = 50 Hz to remove the spikes, and a train of spikes (Spike) produced by passing the response through a Schmidt trigger circuit. Aii-Aiv: linear, 2nd-order and their summation models predicted from the spike and synaptic potential kernels, respectively. Note that the linear models from the spike and synaptic potential kernels shown in B resembled each other, whereas 2nd-order nonlinear models differed. B: 1st-order kernels from the spike ( ) and synaptic potential(- - -) components of a response shown in Ai, which are similar in kernel configuration. Ci and Cii: 2nd-order kernels from the spike (Ci) and synaptic potential (Cii) components of the same response in B. Note that an elongated negative valley on the diagonal, characteristic of the 2nd-order kernel from synaptic potential response in SETi, is absent from the kernel of the spike component.
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DISCUSSION |
Response properties of extensor tibiae motor neurons
The extensor system of the negative feedback loop underlying the resistance reflex of the hind leg about the femorotibial joint of the locust is primarily dependent on the position of the apodeme of the FeCO, and therefore of the tibia itself.
The first-order Wiener kernels of FETi and SETi were primarily monophasic, indicating that both motor neurons received position-dependent synaptic inputs during FeCO stimulation. The gain of the responses of FETi and SETi were almost constant at all frequencies tested, and fell off at ~20 Hz for SETi and at ~50 Hz for FETi when the FeCO was stimulated at fc = 58 Hz. Thus the responses have constant gain and are low passed, providing further evidence that these motor neurons are primarily dependent on the position of the tibia about the femorotibial joint. In earlier studies, Field and Burrows (1982)
showed that both FETi and SETi received position- and velocity-dependent synaptic inputs during FeCO stimulation, whereas Field and Coles (1994)
showed that spike responses of the extensor muscles were also dependent on position.
Both FETi and SETi motor neurons are thought to receive inputs that are dependent on the velocity of FeCO stimulation. The small increases in gain their responses show with increasing stimulus frequencies (up to their respective fcs), coupled with the negative overshoot of the first-order kernels, supports the idea that both motor neurons receive velocity-dependent inputs from the FeCO, but these are small compared with the dominance of position-dependent inputs.
These results contrast with the responses of flexor tibiae motor neurons, which are primarily velocity dependent (Newland and Kondoh 1997
) with only a minor dependence on position. These differences in response dynamics of antagonistic motor neuron pools may reflect differing demands during different behaviors. For example, the resting position of the locust is with the tibiae of the hind legs flexed. Indeed, Field and Coles (1994)
showed for locusts standing on a horizontal surface that the femorotibial angle rarely exceeds 90°. Thus the body weight of the locust must be supported to a great extent by the extensor tibiae muscles, because their activity could regulate the height of the body above the ground. Thus a dependence of these motor neurons' responses on tibial angle (i.e. position sensitivity during FeCO stimulation) may be important for maintaining a particular joint angle (Field and Coles 1994
). Bush (1962)
also suggests that a position dependency of a leg resistance reflex in crabs may be important for maintaining a joint at a particular rest position.
Dynamics of the spike response of SETi
The generation of an action potential is a highly nonlinear process, with action potentials containing high-frequency components not related to information carried by the synaptic inputs to a neuron. Nevertheless it is still possible to analyze the information coded by spikes with the use of white noise analysis (Kondoh et al. 1991
, 1995
; Sakuranaga et al. 1987
). Cross-correlation between a white noise stimulus and a unitized spike discharge of SETi also produced well-defined first- and second-order kernels. The Wiener kernels of the synaptic component of SETi show a high degree of similarity with those of the spike response, indicating that the dynamics of both components are similar. A similar match between the synaptic and spike models was found in ganglion cells of the catfish retina (Sakuranaga et al. 1987
). In SETi, a depolarization in the linear model of the spike response corresponds to an increase in spike density, and a hyperpolarization to a decrease. These peaks and troughs in the SETi spike models match well with similar peaks and troughs in the synaptic models. These results show that the spike response of SETi is also primarily dependent on position, as was shown earlier by Field and Coles (1994)
. However, comparisons of the second-order kernels of the synaptic and spike models show that some signal transformation does occur within SETi itself. The characteristic elongated hyperpolarizing valley on the diagonal of the second-order kernel of the synaptic response in SETi was absent in the kernel from the spike component. This nonlinear component represented a delayed inhibitory synaptic input to SETi that is not coded in the spike response. Thus some transformation of information is occurring in the spike-producing process in SETi.
Most flexor tibiae motor neurons received delayed synaptic inputs in the form of depolarizing inputs during tibial flexion (Newland and Kondoh 1997
). These inputs must define an assistance reflex that opposes the primary resistance reflex expressed in these motor neurons when the FeCO is stimulated. Although we showed that these delayed inputs were of a much lower gain than those contributing to the negative feedback loop, it was difficult to understand their contribution to feedback during FeCO stimulation. Our finding that inputs can be transformed within the extensor motor neurons may mean that the delayed inputs to the flexor tibiae motor neurons may never be expressed at the muscles, because this information could be removed by the processing within the motor neurons themselves. Clearly, we must now examine the spike responses of the flexor tibiae motor neurons to determine what information they carry.
The final output in this reflex pathway, however, is movement of the tibia. Thus there is the potential for modification or transformation of information at points downstream from the motor neurons. The filter properties at the neuromuscular junction, structure and innervation patterns of the muscles, properties of force production, and mechanics of movement about the femorotibial joint itself will all contribute to the final movement of the tibia. In stick insects, for example, we know that the femorotibial muscle joint system acts as a low-pass filter, and recently Bässler and Stein (1996)
were able to show that properties of the extensor muscle, its innervation patterns, and co-contraction with the flexor muscles all help to maintain the stability of the feedback loop. We know also that not every spike evoked in SETi, when spiking at high frequency, will evoke a junction potential in the extensor muscle of Locusta migratoria (Bässler et al. 1996
). Moreover, we also know that the history of movement of the tibia itself can influence the membrane potential of leg motor neurons and nonspiking interneurons that control specific sets of motor neurons, and therefore influence the gain of the reflex movement (Siegler 1981
).
Now that we have detailed knowledge of the response dynamics of not only the extensor motor neurons but also many of the flexor tibiae motor neurons (Newland and Kondoh 1997
), we can begin to look at how these properties change depending on the state or behavior of an animal. For example, how are these reflex properties altered during walking or flight? Are these reflex movements simply gated out through presynaptic inhibition that is thought to alter the gain of leg movements (Burrows and Laurent 1993
; Burrows and Matheson 1994
), or are they modulated in phase with the step or wing beat cycle? Given the detailed knowledge we have of these networks of neurons controlling movements of the hind leg of the locust, we are now in a position to start addressing such questions.
Comparisons with other systems
The linear components of the responses of the fast and slow extensor motor neurons of stick insects have been examined in detail by Bässler (see review, 1993). The responses of both FETi and SETi during FeCO stimulation in the stick insect are strongly dependent on the velocity of movement of the FeCO (Bässler 1983
). In addition, both motor neurons receive a small position-dependent input. Because the resting potential of FETi is far from spike threshold, Bässler (1993)
suggests that these position-dependent inputs, which are only a few millivolts in amplitude when recorded in the soma, are unlikely to cause FETi to spike. Instead Bässler suggests that the spike response will be dependent on velocity-dependent synaptic inputs. Although this latter statement may be true, it is possible that the velocity-dependent response could be modulated by the position-dependent inputs. For example, Field and Burrows (1982)
showed in the locust that the amplitude of the response of SETi, and the number of spikes produced in it during a small ramp stimulus, are dependent on the set angle of the tibia. It is therefore possible that the small position-dependent component in FETi of the stick insect may have an influence on its velocity-dependent synaptic inputs.
Bässler (1983)
showed that the response properties of SETi and FETi in stick insects were similar, having similarly shaped amplitude and phase plots and half-lives of rise and fall of depolarizations during ramp stimulation of the FeCO. Here we show that the same is true for FETi and SETi in the locust. Previous studies have shown that not only do members of particular motor neuron pools receive common spontaneous synaptic inputs (Burrows et al. 1989
), but they also, as is the case of many of the flexor tibiae motor neurons, have similar response properties when the FeCO is moved (Newland and Kondoh 1997
). Such a common drive is well known from studies on other neuromuscular systems, especially in vertebrates (De Luca and Erim 1994
), where it is thought that all members of a motor neuron pool receive the same net drive from the CNS. Differences in the response properties between individuals in a pool are then due primarily to the intrinsic properties of each individual member, such as spike threshold. Rhythmic swimmeret beating in the lobster (Davis 1969
) was modeled by Davis and Murphey (1969)
, who showed that the order of recruitment of swimmeret motor neurons within a pool was determined simply by differences in their threshold, which in turn was dependent on their size (Davis 1971
). Subsequently, Meyer and Walcott (1979)
showed in the cockroach that differences in leg motor neurons' membrane properties do effect their order of recruitment and spike frequency during walking movements.
Although FETi and SETi in the locust both receive primarily position-dependent synaptic inputs during FeCO stimulation, important differences exist in their response properties. SETi responds better at low frequencies of FeCO stimulation than does FETi. For example, the power of the response of SETi was >1 at all frequencies <8 Hz. The power of the response of FETi, on the other hand, never reached 1 at any frequency (compare Figs. 4B and 9B). Above 10 Hz, the power of the response of FETi is much greater than that of SETi. This difference is reflected in the fcs of the motor neurons. In SETi the power of the response begins to roll off gradually at 2 Hz, compared with 10 Hz in FETi. Thus SETi receives greater low-frequency inputs and FETi greater higher-frequency inputs. A similar relationship between fc and motor neuron type also exists in the flexor tibiae motor neurons of locusts (Field and Burrows 1982
; Newland and Kondoh 1997
) and stick insects (Debrodt and Bässler 1990
).
Filter properties of signal propagation in FETi
Recordings from the somata of neurons in the CNS of insects do not always reflect well their true inputs to fine neurites where integration occurs. It could be argued, therefore, that the filter properties of the motor neurons we describe are not due simply to their synaptic inputs but may also be due to substantial changes in the signals as they are conducted from the synaptic sites in fine neurites in the neuropil to the somata. To determine how propagation of a signal within a neuron might influence the signal recorded at a position distant from the integrative sites, we made simultaneous intracellular recordings from the soma and a neuropilar process of FETi during stimulation of the FeCO. Recordings from a neuropilar segment were of a greater amplitude and contained higher-frequency components compared with recordings from the soma. At a stimulus frequency of 20 Hz, responses recorded in the soma were delayed by ~30° and reduced in amplitude by some 30-40% compared with the neurite recordings. The responses to FeCO stimulation at low frequencies of stimulation, <10 Hz, were less affected by the filter properties of the primary neurite. Above 10 Hz, however, high-frequency inputs were removed. Thus the real responses of FETi, which contain high-frequency components, will be dependent on where they are recorded in the neuron. Although recordings from the somata of FETi give a reasonable approximation of its true responses, it must always be remembered that the filter properties of the primary neurite will reduce the amplitude of its signals and reduce its upper frequency level.
Second-order components
The linear models, produced by convolving the stimulus input with the first-order kernels, predicted poorly the actual responses of FETi and SETi, and only with the addition of the second-order (nonlinear) model did the predictions improve markedly. Thus the extensor tibiae motor neurons have substantial nonlinear properties, unlike the flexor tibiae motor neurons, which are almost linear (Newland and Kondoh 1997
). The second-order kernels of FETi and SETi were similar in shape: both kernels had a large on-diagonal peak with two small off-diagonal valleys. Additionally, the second-order kernel of SETi had an elongated valley on the diagonal that was accompanied by two off-diagonal depolarizing peaks at fc = 58 Hz. These nonlinear components represent half-wave rectification of the position-sensitive depolarizing response of FETi and SETi, and of the delayed inhibitory input to SETi.
In conclusion, this study shows that not only do individual members of a motor neuron pool receive common inputs and code similar features of a movement stimulus, but they may code different features of a stimulus than do members of an antagonistic motor neuron pool, i.e., the responses of the extensor tibiae motor neurons are primarily position dependent, whereas those of flexor tibiae motor neurons are primarily velocity dependent (Newland and Kondoh 1997
). Moreover, we have shown that signal transformation can occur not only between neurons, but also within motor neurons themselves. The local circuits controlling movements of the legs of locusts contain not only the FeCO sensory neurons and tibial motor neurons, but also spiking and nonspiking local interneurons whose outputs have profound effects on tibial motor neurons (Burrows 1987
; Burrows and Siegler 1982
; Burrows et al. 1988
). The next step in our analysis is to examine the dynamics of their responses and how they contribute to the resistance reflex of the locust hind leg.