Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

Filter characteristics of cercal afferents in the cockroach

Summary The response dynamics of cercal afferents in the cockroach, Periplaneta americana, were determined by means of a cross-correlation technique using a Gaussian white noise modulation of wind as a stimulus. The white noise stimulus could evoke sustained firing activity in most of the afferents examined (Fig. 1). The spike discharges were unitized and then cross-correlated with the stimulus to compute 1st- and 2nd-order Weiner kernels. The Ist-order kernels from a total of 28 afferents were biphasic and closely matched the time differential of a pulse (Figs. 1, 3 and 4). The amplitude and waveform of the kernels depended on the stimulus angle in such a way that the kernels were the mirror image of those on the polar opposite side (Figs. 2 and 3). The 2nd-order kernels were also differential. They had 2 diagonal peaks and 2 off-diagonal valleys in a 2-dimensional plot with 2 time axes (Figs. 1, 5 and 6). This 4-eye configuration was basically invariant irrespective of the stimulus angle, although the kernels varied in amplitude when the stimulus angle was changed. The time between the peak and a following trough of the 1st-order kernel was constant and had a mean of 4.6±0.1 ms, whereas the time between 2 diagonal peaks of the 2nd-order kernels was 4.7±0.1 ms (Figs. 4 and 6), suggesting that wind receptors (filiform sensilla) on cerci act as a band-pass filter with a peak frequency of about 106 Hz. The peak time, however, varies from 2.3 to 6.9 ms in both kernels, which may reflect the spatial distribution of the corresponding hairs on the cercus. The summation of the 1st- (linear) and 2nd-order (nonlinear) models precisely predicted the timing of the spike firing (Fig. 8). Thus, these 2 lower-order kernels can totally characterize the response dynamics of the wind receptors. The nonlinear response explains the directional sensitivity of the sensory neurons, while the differentiating 1st-order kernel explains the velocity sensitivity of the neurons. The nonlinearity is a signal compression in which one of the diagonal peaks of the 2nd-order kernel always offsets the downward phase of the 1st-order kernel (Fig. 7) and obviously represents a half-wave rectification property of the wind receptors that are excited by hair movement in only one direction and inhibited by hair movement in the polar opposite direction.     

Reverse correlation analysis of the stretch response of primary muscle spindle afferent fibers

The nonlinear responses of deefferented primary muscle spindle afferent fibers to muscle stretching consisted of a train of action potentials which was analyzed when random changes in muscle length (band-limited gaussian white noise) were applied in cats. The upper cutoff frequency of the applied noise (the source stimulus) was varied between 1.6 and 570 Hz; the amplitude of the random input was varied between 0.002 and 1.2 mm. In a previous report the reverse correlation of 1st and 2nd order was studied for its ability to analyze data of a continuous input signal and pulsatile events in the output. Computations of the Wiener kernelsh ₁ andh ₂ or their equivalents, the perispike averages of the 1st and 2nd order, were computed from the random stretch responses of muscle-spindle afferents. Then the 1st- and the 2ndorder predictions and the summation of both to random muscle stretch was estimated. A general finding was that the 1st-order component was approximately 10 times that of the 2nd-order component, when both were combined in approximation procedures to give the closest prediction of observed responses to random test stimuli. The approximation was poor when the source stimulus was less than 0.03 mm and improved when it was greater. With the increase in the upper cutoff frequency of the random source input, the approximation worsened continuously. Predictions to ramp-and-hold stimuli were computed, as well as responses to random stimulation. Limiting the upper cutoff frequency did not diminish the value of the techniques applied.     

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     

Third-order reverse correlation analysis of muscle spindle primary afferent fiber responses to random muscle stretch

The response of primary muscle spindle afferent fibers to muscle stretch is nonlinear. Now spindle responses (trains of action potentials) to band-limited Gaussian white noise length perturbations of the gastrocnemius muscles (input signal) are described in cats. The input noise upper cutoff frequency was clearly above the frequency range of physiological length changes in cat hindleg muscles. The input-output relation was analyzed by means of peri-spike averages (PSAs), which could be shown to correspond to the kernels of Wiener's white noise approach to systems identification. The present approach (the reverse correlation analysis) was applied up to the third order. An experiment consisted of two recordings: one (the source recording) to determine PSAs and the other (the test recording) to provide an input signal for predicting responses. The predictions of different orders were compared with the actual neuronal response (the observation) of the test recording. Four different approximation procedures were developed to adapt prediction and observation and to determine weighting factors for the predictions of different orders. The approximations also yielded the value of the power density P of the input noise signal: at a variety of stimulus parameters, P from approximations had the same magnitude as P determined directly from the input signal amplitude spectrum. The prediction of a sequence of action potentials improved the higher the order of components. 37 of 42 action potentials of a test recording (the observation) could be confidently predicted from PSAs or kernels. Compared with the size of the linear first-order prediction curve, the relative sizes of the second and third-order prediction curves were: 1.0∶0.47∶0.26.     

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     

Third-order reverse correlation analysis of muscle spindle primary afferent fiber responses to random muscle stretch

 The response of primary muscle spindle afferent fibers to muscle stretch is nonlinear. Now spindle responses (trains of action potentials) to band-limited Gaussian white noise length perturbations of the gastrocnemius muscles (input signal) are described in cats. The input noise upper cutoff frequency was clearly above the frequency range of physiological length changes in cat hindleg muscles. The input–output relation was analyzed by means of peri-spike averages (PSAs), which could be shown to correspond to the kernels of Wiener’s white noise approach to systems identification. The present approach (the reverse correlation analysis) was applied up to the third order. An experiment consisted of two recordings: one (the source recording) to determine PSAs and the other (the test recording) to provide an input signal for predicting responses. The predictions of different orders were compared with the actual neuronal response (the observation) of the test recording. Four different approximation procedures were developed to adapt prediction and observation and to determine weighting factors for the predictions of different orders. The approximations also yielded the value of the power density P of the input noise signal: at a variety of stimulus parameters, P from approximations had the same magnitude as P determined directly from the input signal amplitude spectrum. The prediction of a sequence of action potentials improved the higher the order of components. 37 of 42 action potentials of a test recording (the observation) could be confidently predicted from PSAs or kernels. Compared with the size of the linear first-order prediction curve, the relative sizes of the second and third-order prediction curves were: 1.0 : 0.47 : 0.26. Received: 15 November 1994/Accepted in revised form: 23 May 1995     

The analysis of nonlinear synaptic transmission

In order to characterize synaptic transmission at a unitary facilitating synapse in the lobster cardiac ganglion, a new nonlinear systems analysis technique for discrete-input systems was developed and applied. From the output of the postsynaptic cell in response to randomly occurring presynaptic nerve impulses, a set of kernels, analogous to Wiener kernels, was computed. The kernels up to third order served to characterize, with reasonable accuracy, the input-output properties of the synapse. A mathematical model of the synapse was also tested with a random impulse train and model predictions were compared with experimental synaptic output. Although the model proved to be even more accurate overall than the kernel characterization, there were slight but consistent errors in the model's performance. These were also reflected as differences between model and experimental kernels. It is concluded that a random train analysis provides a comprehensive and objective comparison between model and experiment and automatically provides an arbitrarily accurate characterization of a system's input-output behavior, even in complicated cases where other approaches are impractical.     

Effect of doublet impulse sequences in the crayfish claw opener muscles and the computer-simulated neuromuscular synapse

The effects of doublet impulse sequences of the excitatory motor axon on the movement of the claw opener muscles in the crayfish were examined. The excitatory motor axon was stimulated electrically with various patterns of doublet impulse sequences generated by a digital computer. Doublet impulse sequences of stimulation produced a larger sustained movement than an uniform impulse sequences at the same mean rate of stimulation. The movement was largest when the interval between the impulses of a doublet was about 5 ms. This interval generated a movement amplitude 25% greater than that for the uniform impulse sequence. A simple model was formulated to stimulate the neuromuscular synapse of the claw opener muscle. The relationship between stimulation sequences with alternating long and short intervals and responses (firing probabilities) of the neuromuscular synapse at the same mean rate was investigated. The responses was classified into two typical types which are noneffective Type I and effective Type II to the absolute refractory period (ARP). The characteristics which are larger responses with short intervals in Type I and reduction of responses in the ARP region of Type II formed a plateau peak of the experimental results. By incorporating the reduction of end-plate potential (EPP) as a property of nonlinear rule for temporal summation into the model, it was shown that Type I response is maximal with a plateau peak at short interval, agreeing well with the experimental results from the claw opener muscles.     

Probabilistic Firing of Neurons Considered as a First Passage Problem

A formalized neuron receiving unitary excitatory impulses at random is considered. Each impulse provokes an effect of equal magnitude and of a duration not constant for each impulse, but which varies according to an exponential distribution. The effects sum until a threshold is reached when a response occurs. The distribution of intervals between successive responses is computed and compared with those obtained from a model in which the effects decay exponentially with time. Upon introducing inhibitory impulses also, the theory is applied to data on discharge characteristics of driven and spontaneously active thalamic neurons reported in the literature.     

Analysis of nonlinear physiological systems with single or multiple spike inputs and analogue or spike outputs

This paper outlines an extension of Krausz' (1975) approach to nonlinear analysis of physiological systems with random impulse train inputs. This extension concerns multiple (here two) inputs and spike train outputs in addition to analogue outputs. The respective theoretical formula are given, and the approach is illustrated by experimental data relating to the nonlinear interaction of cat motor units activated with random pulse trains. As outputs are considered muscle tension (as an example for an analogue signal) and discharge frequency of a Ia fibre (as an example for a spike output).     

Determination and analysis of antigenic epitopes of prostate specific antigen (PSA) and human glandular kallikrein 2 (hK2) using synthetic peptides and computer modeling.

Prostate specific antigen (PSA) and human glandular kallikrein 2 (hK2), produced essentially by the prostate gland, are 237-amino acid monomeric proteins, with 79% identity in primary structure. Twenty-five anti-PSA monoclonal antibodies (Mabs) were studied for binding to a large array of synthetic linear peptides selected from computer models of PSA and hK2, as well as to biotinylated peptides covering the entire PSA sequence. Sixteen of the Mabs were bound to linear peptides forming four independent binding regions (I-IV). Binding region I was localized to amino acid residues 1-13 (identical sequence for PSA and hK2), II (a and b) was localized to residues 53-64, III (a and b) was localized to residues 80-91 (= kallikrein loop), and IV was localized to residues 151-164. Mabs binding to regions I and IIa were reactive with free PSA, PSA-ACT complex, and with hK2; Mabs binding to regions IIb, IIIa, and IV were reactive with free PSA and PSA-ACT complex, but unreactive with hK2; Mabs binding to region IIIb detected free PSA only. All Mabs tested (n = 7) specific for free PSA reacted with kallikrein loop (binding region IIIb). The presence of Mabs interacting with binding region I did not inhibit the catalytic activity of PSA, whereas Mabs interacting with other binding regions inhibited the catalysis. Theoretical model structures of PSA, hK2, and the PSA-ACT complex were combined with the presented data to suggest an overall orientation of PSA with regard to ACT.     

Induction of Tc2 cells with specificity for prostate-specific antigen from patients with hormone-refractory prostate cancer

Prostate-specific antigen (PSA) is a potentially useful antigen for targeted T-cell immunotherapy of prostate cancer (CaP). Our laboratory has identified a synthetic nonamer peptide (PSA 146-154) homologue of PSA, which binds to the prevalent human leukocyte antigen, HLA-A2, and elicits specific cytotoxic T-lymphocyte (CTL) responses from normal individuals of the HLA-A2 phenotype. In the present study, we report on the induction of CTL from peripheral blood mononuclear cells (PBMC) of patients with hormone-refractory CaP, which exhibit the same specificity. T-cell lines were established from two patients by stimulation of PBMC with PSA 146-154 peptide in vitro. The T-cell lines exhibited specific cytolytic activity against T2 cells pulsed with PSA 146-154 peptide, but not a control HLA-A2 binding peptide (HIV-RT 476-484) via chromium release assay (CRA). The T-cell lines also showed PSA 146-154 peptide-specific IL-4 responses, but no detectable interferon-gamma (IFN-gamma) responses via enzyme-linked immuno-spot assays. Magnetic immuno-selection studies of one of the T-cell lines demonstrated that both cytolytic and interleukin-4 (IL-4) responses were mediated by CD8(+), but not by CD4(+) T cells. This Tc2 line was further characterized for the ability to recognize endogenously processed PSA epitopes. The line specifically secreted IL-4 in response to HLA-A2(+) target cells transfected to express PSA and specifically lysed the PSA(+) target cells, but not control transfected cells. The results indicate that the PSA 146-154 peptide emulates a naturally processed and presented peptide epitope of PSA that is within the T-cell repertoire of HLA-A2(+)patients with CaP.     

Time domain analysis of oxygen uptake during pseudorandom binary sequence exercise tests.

Pseudorandom binary sequence (PRBS) exercise tests involve repeated switching between two work rates (WR) according to a computer-generated pattern. This paper presents an approach to analysis of O2 uptake (VO2) in the time domain. First, the autocorrelation function (ACF) of the input WR was recognized to be a triangular-shaped pulse that can be taken to be equivalent to a ramp increase followed by a ramp decrease in WR. Then the cross-correlation function of the input (WR) and the output (VO2) was treated as if it were the response to a triangular-shaped pulse. The cross-correlation function was analyzed by fitting a linear summation of the ramp form of a two-component exponential function to this triangular pulse. VO2 responses of eight subjects were obtained from two different PRBS tests, as well as step changes in WR. The first PRBS test consisted of 15 units, each 30 s in duration. Its ACF had a base width of 60 s. The ramp increase-ramp decrease model fit the data throughout the range of response. The second PRBS test had 63 units, each 5 s in duration; thus its ACF base width was 10 s. Again, the ramp model fit adequately. The data from the second PRBS test could be fit by the impulse form of the two-component exponential equation, although the fit in the first 30 s tended to be poorer. The time constants of VO2 dynamics estimated from step and PRBS tests were not significantly different. PRBS tests can be analyzed in the time domain, and the indicators of system dynamics reflect physiological properties similar to those investigated during step changes in WR.     

Gene duplication and divergence produce divergent MHC genotypes without disassortative mating

Genes of the major histocompatibility complex (MHC) exhibit heterozygote advantage in immune defence, which in turn can select for MHC‐disassortative mate choice. However, many species lack this expected pattern of MHC‐disassortative mating. A possible explanation lies in evolutionary processes following gene duplication: if two duplicated MHC genes become functionally diverged from each other, offspring will inherit diverse multilocus genotypes even under random mating. We used locus‐specific primers for high‐throughput sequencing of two expressed MHC Class II B genes in Leach's storm‐petrels, Oceanodroma leucorhoa, and found that exon 2 alleles fall into two gene‐specific monophyletic clades. We tested for disassortative vs. random mating at these two functionally diverged Class II B genes, using multiple metrics and different subsets of exon 2 sequence data. With good statistical power, we consistently found random assortment of mates at MHC. Despite random mating, birds had MHC genotypes with functionally diverged alleles, averaging 13 amino acid differences in pairwise comparisons of exon 2 alleles within individuals. To test whether this high MHC diversity in individuals is driven by evolutionary divergence of the two duplicated genes, we built a phylogenetic permutation model. The model showed that genotypic diversity was strongly impacted by sequence divergence between the most common allele of each gene, with a smaller additional impact of monophyly of the two genes. Divergence of allele sequences between genes may have reduced the benefits of actively seeking MHC‐dissimilar mates, in which case the evolutionary history of duplicated genes is shaping the adaptive landscape of sexual selection.     

Cerebral blood flow velocity response to induced and spontaneous sudden changes in arterial blood pressure

The influence of different types of maneuvers that can induce sudden changes of arterial blood pressure (ABP) on the cerebral blood flow velocity (CBFV) response was studied in 56 normal subjects (mean age 62 yr, range 23-80). ABP was recorded in the finger with a Finapres device, and bilateral recordings of CBFV were performed with Doppler ultrasound of the middle cerebral arteries. Recordings were performed at rest (baseline) and during the thigh cuff test, lower body negative pressure, cold pressor test, hand grip, and Valsalva maneuver. From baseline recordings, positive and negative spontaneous transients were also selected. Stability of PCO2 was monitored with transcutaneous measurements. Dynamic autoregulatory index (ARI), impulse, and step responses were obtained for 1-min segments of data for the eight conditions by fitting a mathematical model to the ABP-CBFV baseline and transient data (Aaslid's model) and by the Wiener-Laguerre moving-average method. Impulse responses were similar for the right- and left-side recordings, and their temporal pattern was not influenced by type of maneuver. Step responses showed a sudden rise at time 0 and then started to fall back to their original level, indicating an active autoregulation. ARI was also independent of the type of maneuver, giving an overall mean of 4.7 +/- 2.9 (n = 602 recordings). Amplitudes of the impulse and step responses, however, were significantly influenced by type of maneuver and were highly correlated with the resistance-area product before the sudden change in ABP (r = -0.93, P < 0.0004). These results suggest that amplitude of the CBFV step response is sensitive to the point of operation of the instantaneous ABP-CBFV relationship, which can be shifted by different maneuvers. Various degrees of sympathetic nervous system activation resulting from different ABP-stimulating maneuvers were not reflected by CBFV dynamic autoregulatory responses within the physiological range of ABP.     

Responses of human skin temperature and thermal sensation to step change of air temperature

1. 1. The purpose of this paper is to clarify the non-linearity of the human physiological and psychological responses to step change of air temperature by impulse response analysis using Discrete Fourier Transformation.

2. 2. Experiments were conducted to investigate the effect of thermal transients on human responses.

3. 3. Experimental conditions were as follows: lowering air temperature from 30 to 20°C and raising air temperature from 20 to 30°C.

4. 4. The responses of local skin temperature on lowering air temperature from 30 to 20°C are not necessarily opposite to the responses found on raising air temperature from 20 to 30°C.

5. 5. From impulse response analysis using Discrete Fourier Transformation, skin temperature responses to the opposite air temperature change do not necessarily coincide with each other whenever the same temperature stimulus is occurred.

The auditory aftereffects of radial sound source motion with different velocities

Auditory aftereffects were evaluated after short adaptation to radial sound source motion with different velocities. Approach and withdrawal of the sound source were simulated by means of rhythmical noise (from 20 Hz to 20 kHz) impulse sequences with an arising or diminishing amplitude. They were presented to an anechoic chamber through two loudspeakers placed at 1.1 and 4.5 m from the listener. The adapting stimulus velocities were 0.68, 3.43, 6.92, and 9.97 m/s with an adaptation duration of 5 s. At all motion velocities, the aftereffect manifested itself in divergence of psychometric functions upon approaching and withdrawing of adaptors. The direction of function displacements was opposite to that of the adaptor motion. Three parameters reflecting alteration of perception after motion adaptation were determined and compared with control data: the evaluation of stationary test stimuli; the velocity of moving test signal at the point of subjective equality (perceptually unmoving point); and the percentage of responses after averaging over all test signals. These parameters of auditory radial motion aftereffect similarly changed with the adaptor velocity. They demonstrated a significant effect at slow motion (0.68 and 3.43 m/s) and a small effect at a quick motion (6.92 and 9.97 m/s).