System analysis of Phycomyces light-growth response: madC, madG, and madH mutants.

The light-growth response of Phycomyces has been studied further with the sum-of-sinusoids method in the framework of the Wiener theory of nonlinear system identification. The response was treated as a black box with the logarithm of light intensity as the input and elongation rate as the output. The nonlinear input-output relation of the light-growth response can be represented mathematically by a set of weighting functions called kernels, which appear in the Wiener intergral series. The linear (first-order) kernels of wild type, and of single and double mutants affected in genes madA to madG were determined previously with Gaussian white noise test stimuli, and were used to investigate the interactions among the products of these genes (R.C. Poe, P. Pratap, and E.D. Lipson. 1986. Biol. Cybern. 55:105.). We have used the more precise sum-of-sinusoids method to extend the interaction studies, including both the first- and second-order kernels. Specifically, we have investigated interactions of the madH ("hypertropic") gene product with the madC ("night blind") and madG ("stiff") gene products. Experiments were performed on the Phycomyces tracking machine. The log-mean intensity of the stimulus was 6 x 10(-2) W m-2 and the wavelength was 477 nm. The first- and second-order kernels were analyzed in terms of nonlinear kinetic models.(ABSTRACT TRUNCATED AT 250 WORDS)     

System analysis of Phycomyces light-growth response. Wavelength and temperature dependence.

The light-growth response of the Phycomyces sporangiophore was studied further with the sum-of-sinusoids method of nonlinear system identification. The first- and second-order frequency kernels, which represent the input-output relation of the system, were determined at 12 wavelengths (383-529 nm) and 4 temperatures (17 degrees, 20 degrees, 23 degrees, and 26 degrees C). The parametric model of the light-growth response system, introduced in the preceding paper, consists of nonlinear and linear dynamic subsystems in cascade. The model parameters were analyzed as functions of wavelength and temperature. At longer wavelengths, the system becomes more nonlinear. The latency and the bandwidth (cutoff frequency) of the system also vary significantly with wavelength. In addition, the latency decreases progressively with temperature (Q10 = 1.6). At low temperature (17 degrees C), the bandwidth is reduced. The results indicate that about half of the latency is due to physical processes such as diffusion, and the other half to enzymatic reactions. The dynamics of the nonlinear subsystem also vary with wavelength. The dependence of various model components on wavelength supports the hypothesis that the light-growth response, as well as phototropism, are mediated by multiple interacting photoreceptors.     

System analysis of Phycomyces light-growth response with gaussian white-noise test stimuli

The light-growth response of the Phycomyces sporangiophore is a transient change of elongation rate in response to changes in ambient blue-light intensity. The white-noise method of nonlinear system identification (Wiener-Lee-Schetzen theory) has been applied to this response, and the results have been interpreted by system analysis methods in the frequency domain. Experiments were performed on the Phycomyces tracking machine. Gaussian white-noise stimulus patterns were applied to the logarithm of the light intensity. The log-mean intensity of the broadband blue illumination was 0.1 W m^-2 and the standard deviation of the Gaussian white-noise was 0.58 decades. The results, in the form of temporal functions called Wiener kernels, represent the input-output relation of the light-growth response system. The transfer function, which was obtained as the Fourier transform of the first-order kernel, was analyzed in the frequency domain in terms of a dynamic model that consisted of a first-order high-pass filter, two secondorder low-pass filters, a delay element, and a gain factor. Parameters in the model (cutoff frequencies, damping coefficients, latency, and gain constant) were evaluated by nonlinear least-squares methods applied to the complex-valued transfer function. Analysis of the second-order kernel in the frequency domain suggests that the residual nonlinearity of the system lies close to the input.     

System analysis of Phycomyces light-growth response in single and double night-blind mutants

The sum-of-sinusoids method of nonlinear system identification has been applied to the light-growth response of the Phycomyces sporangiophore. Experiments were performed on the Phycomyces tracking machine with the wild-type strain with single and double mutants affected in genes madA, madB, and madC. The sum-of-sinusoids test stimuli were applied to the logarithm of the light intensity. The log-mean intensity level was 10^-1 Wm^-2 and the wavelength was 477 nm. The system identification results are in the form of first- and second-order frequency kernels, which are related to temporal kernels that appear in the Wiener functional series. The first-order kernels agree well with those obtained previously by the white noise method. In particular, the madA madB and madB madC double mutants show very weak responses. With the superior precision of the sum-of-sinusoids methods, we have achieved sufficient resolution to measure and analyze their second-order kernels. The first- and second-order frequency kernels were interpreted by system analysis methods involving a nonlinear parametric model. In addition a nonparametric hypothesis concerning interactions of gene products was tested. Results from the interaction tests confirm the earlier conclusion that the madB and madC gene products interact. In addition, with the enhanced precision and with the extension to nonlinear analysis, we have found evidence of interaction of the madA gene product with the madB and madC gene products. Thus all three genes appear to have mutual interactions, presumably because of their close physical association in a photoreceptor complex.     

System analysis of Phycomyces light-growth response. Photoreceptor and hypertropic mutants.

The light-growth responses of Phycomyces behavioral mutants, defective in genes madB, madC, and madH, were studied with the sum-of-sinusoids method of system identification. Modified phototropic action spectra of these mutants have indicated that they have altered photoreceptors (P. Galland and E.D. Lipson, 1985, Photochem. Photobiol. 41:331). In the two preceding papers, a kinetic model of the light-growth response system was developed and applied to wild-type frequency kernels at several wavelengths and temperatures. The present mutant studies were conducted at wavelength 477 nm. The log-mean intensity was 6 X 10(-2)W m-2 for the madB and madC night-blind mutants, and 10(-4)W m-2 for the madH hypertropic mutant. The prolonged light-growth responses of the madB and madC mutants are reflected in the reduced dynamic order of their frequency kernels. The linear response of the hypertropic mutant is essentially normal, but its nonlinear behavior shows modified dynamics. The behavior of these mutants can be accounted for by suitable modifications of the parametric model of the system. These modifications together support the hypothesis that an integrated complex mediates sensory transduction in the light responses and other responses of the sporangiophore.     

Response dynamics and receptive-field organization of catfish ganglion cells [published erratum appears in J Gen Physiol 1995 Aug;106(2):following 388]

Responses from catfish retinal ganglion cells were evoked by a spot or an annulus of light and were analyzed by a procedure identical to the one used previously to study catfish amacrine cells (Sakai H. M., and K.-I. Naka, 1992. Journal of Neurophysiology. 67:430-442.). In two- input white-noise experiments, a response evoked by simultaneous stimulation of the center and surround was decomposed into the components generated by the center and surround through a process of cross-correlation. The center and surround responses were also decomposed into their linear and nonlinear components so that the response dynamics of the linear and nonlinear components could be measured. We found that the concentric organization of the receptive field was determined by linear components, i.e., the first-order kernels generated by the center and surround were of opposite polarity. Both the center and surround generated second-order kernels with similar signatures, i.e., the second-order components formed a monotonic receptive field. The peak response time of the first- and second-order kernels from the surround was longer by approximately 20 ms than that of the center. Except for the DC potential present in the intracellular responses, almost identical first- and second-order kernels for the center and surround were obtained from both the intracellular response and spike discharges. Thus, information on concentric organization of a receptive field is translated into spike discharges with little loss of information. A train of spike discharges carries, simultaneously, at least four kinds of information: two linear and two nonlinear components, which originate in the receptive field center and the surround. A spike train is not a simple signaling device but is a carrier of complex and multiple signals. Victor, J. D., and R. M. Shapley (1979. Journal of General Physiology. 74:671-687.) discovered similarly that, in the cat retina, static second-order nonlinearity is encoded into spike trains. Results obtained in this study support the thesis that signals generated by the preganglionic cells are translated into spike discharges without major modification and that those signals can be recovered from the spike trains (Sakuranaga, M., Y. Ando, and K.-I. Naka. 1987. Journal of General Physiology. 90:229-259.; Korenberg, M. J., H. M. Sakai, and K.-I. Naka. 1989. Journal of Neurophysiology. 61:1110-1120.). Current injection studies have shown that such signal transmission is possible (Sakai, H. M., and K.-I. Naka, 1988a. Journal of Neurophysiology. 60:1549-1567.; 1990. Journal of Neurophysiology. 63:105-119.).     

Nonlinear analysis of the human visual evoked response

Using time-domain correlation techniques, the first- and second-order Wiener kernels have been calculated for the system mediating the human visual evoked response. The first-order kernels indicate the linear element is a resonant one, with a natural frequency near 20 Hz, and a memory of approximately 250 ms. The transport delay associated with this element is approximately 56 ms. The second-order kernels indicate a quadratic nonlinear element with a memory less than 20 ms. The analytic form of this element can be approximated by a parabola shifted to the right of the origin. A close correspondance between the spectrum of the first-order kernel and the spectrum of the main diagonal of the second-order kernel suggests the nonlinear element preceeds the linear one. Tests of reproducibility on the first-order kernel and the main diagonal of the second-order kernel suggest they are reliable describing functions for the system mediating the human visual evoked response.     

White noise analysis of Phycomyces light growth response system. I. Normal intensity range.

The Wiener-Lee-Schetzen method for the identification of a nonlinear system through white gaussian noise stimulation was applied to the transient light growth response of the sporangiophore of Phycomyces. In order to cover a moderate dynamic range of light intensity I, the imput variable was defined to be log I. The experiments were performed in the normal range of light intensity, centered about I0 = 10(-6) W/cm2. The kernels of the Wierner functionals were computed up to second order. Within the range of a few decades the system is reasonably linear with log I. The main nonlinear feature of the second-order kernel corresponds to the property of rectification. Power spectral analysis reveals that the slow dynamics of the system are of at least fifth order. The system can be represented approximately by a linear transfer function, including a first-order high-pass (adaptation) filter with a 4 min time constant and an underdamped fourth-order low-pass filter. Accordingly a linear electronic circuit was constructed to simulate the small scale response characteristics. In terms of the adaptation model of Delbrück and Reichardt (1956, in Cellular Mechanisms in Differentiation and Growth, Princeton University Press), kernels were deduced for the dynamic dependence of the growth velocity (output) on the "subjective intensity", a presumed internal variable. Finally the linear electronic simulator above was generalized to accommodate the large scale nonlinearity of the adaptation model and to serve as a tool for deeper test of the model.     

White noise analysis of temporal properties in simple receptive fields of cat cortex

We studied the linear and nonlinear temporal response properties of simple cells in cat visual cortex by presenting at single positions in the receptive field an optimally oriented bar stimulus whose luminance was modulated in a random, binary fashion. By crosscorrelating a cell's response with the input it was possible to obtain the zeroth-, first-, and second-order Wiener kernels at each RF location. Simple cells showed pronounced nonlinear temporal properties as revealed by the presence of prominent second-order kernels. A more conventional type of response histogram was also calculated by time-locking a histogram on the occurrence of the desired stimulus in the random sequence. A comparison of the time course of this time-locked response with that of the kernel prediction indicated that nonlinear temporal effects of order higher than two are unimportant. The temporal properties of simple cells were well represented by a cascade model composed of a linear filter followed by a static nonlinearity. These modelling results suggested that for simple cells, the nonlinearity occurs late and probably is a soft threshold associated with the spike generating mechanism of the cortical cell itself. This result is surprising in view of the known threshold nonlinearities in preceding lateral geniculate and retinal neurons. It suggests that geniculocortical connectivity cancels the earlier nonlinearities to create a highly linear representation inside cortical simple cells.This work comprises a portion of a PhD thesis submitted by the first author. This study was supported in part by NIH Grant EY04630 and EY06679 to R.C.E., and EY01319 (Core Grant) to the Center for Visual Science     

Nonlinear signal transmission between second- and third-order neurons of cockroach ocelli

Transfer characteristics of the synapse made from second- to third-order neurons of cockroach ocelli were studied using simultaneous microelectrode penetrations and the application of tetrodotoxin. Potential changes were evoked in second-order neurons by either an extrinsic current or a sinusoidally modulated light. The synapse had a low-pass filter characteristic with a cutoff frequency of 25-30 Hz, which passed most presynaptic signals. The synapse operated at an exponentially rising part of the overall sigmoidal input/output curve relating pre- and postsynaptic voltages. Although the response of the second-order neuron to sinusoidal light was essentially linear, the response of the third-order neuron contained an accelerating nonlinearity: the response amplitude was a positively accelerated function of the stimulus contrast, reflecting nonlinear synaptic transmission. The response of the third-order neuron exhibited a half-wave rectification: the depolarizing response to light decrement was much larger than the hyperpolarizing response to light increment. Nonlinear synaptic transmission also enhanced the transient response to step-like intensity changes. I conclude that (a) the major function of synaptic transmission between second- and third-order neurons of cockroach ocelli is to convert linear presynaptic signals into nonlinear ones and that (b) signal transmission at the synapse between second- and third-order neurons of cockroach ocelli fundamentally differs from that at the synapse between photoreceptors and second-order neurons of visual systems so far studied, where the synapse operates in the midregion of the characteristic curve and the transmission is essentially linear.     

The dynamic nonlinear behavior of fly photoreceptors evoked by a wide range of light intensities.

Fly photoreceptor cells were stimulated with steps of light over a wide intensity range. First- and second-order Volterra kernels were then computed from sequences of combined step responses. Diagonal values of the second-order Volterra kernels were much greater than the off-diagonal values, and the diagonal values were roughly proportional to the corresponding first-order kernels, suggesting that the response could be approximated by a static nonlinearity followed by a dynamic linear component (Hammerstein model). The amplitudes of the second-order kernels were much smaller in light-adapted than in dark-adapted photoreceptors. Hammerstein models constructed from the step input/output measurements gave reasonable approximations to the actual photoreceptor responses, with light-adapted responses being relatively better fitted. However, Hammerstein models could not account for several features of the photoreceptor behavior, including the dependence of the step response shape on step amplitude. A model containing an additional static nonlinearity after the dynamic linear component gave significantly better fits to the data. These results indicate that blowfly photoreceptors have a strong early gain control nonlinearity acting before the processes that create the characteristic time course of the response, in addition to the nonlinearities caused by membrane conductances.     

Nonlinear neuronal mode analysis of action potential encoding in the cockroach tactile spine neuron

 Neuronal mode analysis is a recently developed technique for modelling the behavior of nonlinear systems whose outputs consist of action potentials. The system is modelled as a set of parallel linear filters, or modes, which feed into a multi-input threshold. The characteristics of the principal modes and the multi-input threshold device can be derived from Laguerre function expansions of the computed first- and second-order Volterra kernels when the system is stimulated with a randomly varying input. Neuronal mode analysis was used to model the encoder properties of the cockroach tactile spine neuron, a nonlinear, rapidly adapting, sensory neuron with reliable behavior. The analysis found two principal modes, one rapid and excitatory, the other slower and inhibitory. The two modes have analogies to two of the pathways in a block-structured model of the encoder that was developed from previous physiological investigations of the neuron. These results support the block-structured model and offer a new approach to identifying the components responsible for the nonlinear dynamic properties of this neuronal encoder. Received: 30 December 1994/Accepted in revised form: 25 April 1995     

Action spectra of the light-growth response of Phycomyces

The light-growth response of Phycomyces blakesleeanus (Burgeff) is a transient change in elongation rate of the sporangiophore caused by a change in light intensity. Previous investigators have found that the light-growth response has many features in common with phototropism; the major difference is that only the light-growth response is adaptive. In order to better understand the light-growth response and its relationship to phototropism, we have developed a novel experimental protocol for determining light-growth-response action spectra and have examined the effect of the reference wavelength and intensity on the shape of the action spectrum. The null-point action spectrum obtained with broadband-blue reference light has a small peak near 400 nm, a flat region from 430 nm to 470 nm, and an approximately linear decline in the logarithm of relative effectiveness above 490 nm. The shape of the action spectrum is different when 450-nm reference light is used, as has been shown previously for the phototropic-balance action spectrum. However, the action spectrum of the light-growth response differs from that for phototropic balance, even when the same reference light (450 nm) is used. Moreover, for the light-growth response, the relative effectiveness of 383-nm light decreases as the intensity of the 450-nm reference light increases; this trend is the opposite of that previously found for phototropic balance. The dependence of the lightgrowth-response action spectrum on the reference wavelength, its difference from the phototropic-balance action spectrum, and the reference-intensity dependence of the relative effectiveness at 383 nm may be attributable to dichroic effects of the oriented photoreceptor(s), and to transduction processes that are unique to the light-growth response.I dedicated to Masaki Furuya on the occasion of his 65th birthdayThis work was supported by a grant from the National Institutes of Health (GM29707) to E.D. Lipson. Anuradha Palit, Promod Pratap, and Benjamin Horwitz participated in the early phases of this work. We thank Leonid Fukshansky and Benjamin Horwitz for helpful discussions, David Durant for computer programming, and Steven Block for providing us with a C-language program of Reinsch's procedure for cubic spline interpolation. One of us (R.S.) gratefully acknowledges a junior faculty fellowship leave from the Department of Physics at Yale University.     

System analysis of Phycomyces light-growth response: Single mutants

The white-noise method of system identification has been applied to the transient light-growth response of a set of seven mutants of Phycomyces with abnormal phototropism, affected in genes madA to madG. The Wiener kernels, which represent the input-output relation of the light-growth response, have been evaluated for each of these mutants and the wild-type strain at a log-mean blue-light intensity of 0.1 W m^-2. Additional experiments were done at 3x10^-4 and 10 W m^-2 on the madA strain C21 and wild-type. In the normal intensity range (0.1 W m^-2) the madA mutant behaves similarly to wild-type, but, at high intensity, the madA response is about twice as strong as that of wild-type. Except for C21 (madA), the first-order kernels of all mutants were smaller than the wild-type kernel. The first-order kernels for C111 (madB) and L15 (madC) show a prolonged time course, and C111 has a longer latency. The kernels for C110 (madE), C316 (madF), and C307 (madG) have a shallow and extended negative phase. For C68 (madD), the latency and time course are shorter than in the wild-type. These features are also reflected in the parameters estimated from fits of the anlytical model introduced in the previous paper to the experimental transfer functions (Fourier transforms of the kernels). The kernel for L15 (madC) is described better by a model that lacks one of the two second-order low-pass filters, because its response kinetics are dynamically of lower order.     

A spatiotemporal white noise analysis of photoreceptor responses to UV and green light in the dragonfly median ocellus

Adult dragonflies augment their compound eyes with three simple eyes known as the dorsal ocelli. While the ocellar system is known to mediate stabilizing head reflexes during flight, the ability of the ocellar retina to dynamically resolve the environment is unknown. For the first time, we directly measured the angular sensitivities of the photoreceptors of the dragonfly median (middle) ocellus. We performed a second-order Wiener Kernel analysis of intracellular recordings of light-adapted photoreceptors. These were stimulated with one-dimensional horizontal or vertical patterns of concurrent UV and green light with different contrast levels and at different ambient temperatures. The photoreceptors were found to have anisotropic receptive fields with vertical and horizontal acceptance angles of 15 degrees and 28 degrees, respectively. The first-order (linear) temporal kernels contained significant undershoots whose amplitudes are invariant under changes in the contrast of the stimulus but significantly reduced at higher temperatures. The second-order kernels showed evidence of two distinct nonlinear components: a fast acting self-facilitation, which is dominant in the UV, followed by delayed self- and cross-inhibition of UV and green light responses. No facilitatory interactions between the UV and green light were found, indicating that facilitation of the green and UV responses occurs in isolated compartments. Inhibition between UV and green stimuli was present, indicating that inhibition occurs at a common point in the UV and green response pathways. We present a nonlinear cascade model (NLN) with initial stages consisting of separate UV and green pathways. Each pathway contains a fast facilitating nonlinearity coupled to a linear response. The linear response is described by an extended log-normal model, accounting for the phasic component. The final nonlinearity is composed of self-inhibition in the UV and green pathways and inhibition between these pathways. The model can largely predict the response of the photoreceptors to UV and green light.     

White noise analysis of Phycomyces light growth response system. II. Extended intensity ranges.

By means of white gaussian noise stimulation, the Wiener kernels are derived for the Phycomyces light growth response for a variety of intensity conditions. In one experiment the intensity I, rather than log I, is used as the input variable. Under the very limited dynamic range of that experiment, the response is fairly linear. To examine the dependence of the kernels on dynamic range, a series of experiments were performed in which the range of log I was halved and doubled relative to normal. The amplitude of the kernels, but not the time course, is affected strongly by the choice of dynamic range, and the dependence reveals large-scale nonlinearities not evident in the kernels themselves. In addition kernels are evaluated for experiments at a number of absolute intensity levels ranging from 10(-12) to 10(-3) W/cm2. The kernel amplitudes are maximal at about 10(-6) W/cm2. At 10(-12) W/cm2, just above the absolute threshold, the respond is very small. The falloff at high intensity, attributable to inactivation of the photoreceptor, is analyzed in the framework of a first-order pigment kinetics model, yielding estimates for the partial extinction coefficient for inactivation epsilonI455 = (1.5 +/- 0.2) X 10(4) liter/mol-cm and a regeneration time constant of tau = (2.7 +/- 0.6) min. A model is introduced which associates the processes of adaptation and photoreceptor inactivation. The model predicts that the time constants for adaptation and pigment should be identical. This prediction is consistent with values in this and the preceding paper. The effects of pigment inactivation are simulated by a linear electronic analog circuit element, which may be cascaded with the linear simulator circuit in the preceding paper.     

White noise analysis of graded response in a wind-sensitive,nonspiking interneuron of the cockroach

1. A novel approach using a Gaussian white noise as stimulus is described which allowed quantitative analysis of neuronal responses in the cercal system of the cockroach, Periplaneta americana. Cerci were stimulated by air displacement which was modulated by a sinusoidal and a white noise signal. During the stimulation, intracellular recordings were made from a uniquely identifiable, nonspiking, local interneuron which locates within the terminal abdominal ganglion. The white noise stimulation was cross-correlated with the evoked response to compute first- and second-order kernels that could define the cell's response dynamics. 2. The interneuron, cell 101, has an exceptionally large transverse neurite that connects two asymmetrical dendritic arborizations located on both sides of the ganglion. 3. The first-order Wiener kernels in cell 101 were biphasic (differentiating). The waveforms of the kernels produced by the ipsilateral and contralateral stimulations were roughly mirror images of each other: the kernels produced by wind stimuli on the side ipsilateral to the cell body of the interneuron are initially depolarized and then hyperpolarized, whereas those on the other side are initially hyperpolarized. The polarity reversal occurred along the midline of the animal's body, and no well-defined kernel was produced by a stimulus directed head on or from the tail. 4. Mean square error (MSE) between the actual response and the model prediction suggests that the linear component in cell 101 comprises half of the cell's total response (MSEs for the linear models were about 50% at preferred directions), whereas the second-order, non-linear component is insignificant. The linear component of the wind-evoked response was bandpass with the preferred frequency of 70-90 Hz. 5. Accounting for a noise, we reasonably assumed that at high frequencies the graded response in cell 101 is linearly related to a modulation of the air displacement and sensitive to the rate of change of the signal (i.e., wind velocity) and the direction of its source. It is suggested that the dynamics of the first-order kernel simply reflect the dynamics of sensory receptors that respond linearly to wind stimulation.     

Stimulus coding in crayfish antennal mechanoreceptors estimated by noise analysis

Two different kinds of mechanoreceptive hairs (smooth and feathered) on the second antennae of the freshwater crayfish, Orconectes virilis, have been investigated for their stimulus coding propertics. These mechanoreceptors show a great deal of non-linear behaviour both in threshold and in directionality. An effective appraoch for the investigation of such systems is noise analysis in the frequency domain. This method has been used here to calculate zero-, first- and second-order kernels. Sensory cells reveal different first- and second-order kernels, depending on which type of hair is being stimulated. The first-order kernel has a pronounced peak in the frequency response at 110 Hz if a feathered hair is stimulated and at 60 Hz if a smooth hair is stimulated. The second-order kernel shows a number of pronounced peaks in the frequency response between 40 and 110 Hz, but only if a feathered hair is stimulated. Smooth hair stimulation results in less sharp peaks but in higher gain for the same range of stimulus frequencies.     

Spike discharges of skeletomotor neurons during random noise modulated transmembrane current stimulation and muscle stretch

 Spike discharges of skeletomotor neurons innervating triceps surae muscles elicited by white noise modulated transmembrane current stimulation and muscle stretch were studied in decerebrated cats. The white noise modulated current intensity ranged from 4.3 to 63.2 nA peak-to-peak, while muscle stretches ranged from 100 μm to 4.26 mm peak-to-peak. The neuronal responses were studied by averaging the muscle length records centered at the skeletomotor action potentials (peri-spike average, PSA) and by Wiener analysis. Skeletomotor spikes appeared after a sharp peak in PSA of the injected current, preceded by a longer-lasting smaller wavelet of either depolarizing or hyperpolarizing direction. The PSA amplitude was not related to the injected current amplitude nor showed any differences related to the motor unit type. The PSA amplitudes were virtually independent of the stretching amplitude σ, after an initial increase with stretching amplitudes in the range of 15–40 μm (S.D.), or 100–270 μm peak-to-peak.Analyses of cross-spectra indicated a small or absent increase in gain with frequency in response to injected current, but about 20 dB/decade in the range 10–100 Hz in response to muscle stretch. The peaks of both Wiener kernels in response to current injection appear to decrease with the amplitude of injected current, but this decrease was not statistically significant. The narrow first-order kernels suggest that the transfer function between the current input and spike discharge is lowpass with a wide passband, i.e. there is very little change in dynamics. The values of the second-order kernels appear to be nonzero only along the main diagonal. This is characteristic of a simple Hammerstein type cascade, i.e. a zero memory nonlinearity followed by a linear system. Small values of second-order kernels away from the origin and narrow first-order kernels suggest that the linear cascade contributes very little to the overall dynamic response.In contrast to Wiener kernels found in response to current injection, the Wiener kernels in response to stretch showed a decreasing trend with stretch amplitude. The size of the second-order kernels decreased to a somewhat larger extent with input amplitude than that of the first-order kernels, indicating an amplitude-dependent nonlinearity. Overall, the transformation between length and spike output was described as an LNNL cascade with second-order nonlinearities. Received: 1 April 1993/Accepted in revised form: 24 March 1994