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 nonlinear behavior in an insect sensory neuron: Kernel and cascade approaches

A functional expansion was used to model the relationship between a Gaussian white noise stimulus current and the resulting action potential output in the single sensory neuron of the cockroach femoral tactile spine. A new precise procedure was used to measure the kernels of the functional expansion. Very similar kernel estimates were obtained from separate sections of the data produced by the same neuron with the same input noise power level, although some small time-varying effects were detectable in moving through the data. Similar kernel estimates were measured using different input noise power levels for a given cell, or when comparing different cells under similar stimulus conditions. The kernels were used to identify a model for sensory encoding in the neuron, comprising a cascade of dynamic linear, static nonlinear, and dynamic linear elements. Only a single slice of the estimated experimental second-order kernel was used in identifying the cascade model. However, the complete second-order kernel of the cascade model closely resembled the estimated experimental kernel. Moreover, the model could closely predict the experimental action potential train obtained with novel white noise inputs.     

存在睡眠呼吸异常的慢病患者呼吸影响心率变异性的初步分析*

目的: 基于整体整合生理学医学理论提出的呼吸引起循环指标变异的假说,分析研究存在睡眠呼吸异常的慢病患者睡眠期间呼吸和心率变异之间的相关关系。方法: 纳入存在睡眠呼吸异常且呼吸暂停低通气指数（AHI）≥15次/小时的慢病患者11例,签署知情同意书后完成标准化症状限制性极限运动的心肺运动试验（CPET）和睡眠呼吸监测,计算分析病人睡眠期间波浪式呼吸（OB）期与正常平稳呼吸期的呼吸鼻气流、心电图R-R间期心率变异的规律。结果: 存在睡眠呼吸异常的慢病患者CPET峰值摄氧量（Peak VO₂）和无氧阈（AT）为（70.8±13.6）%pred和（71.2±6.1）%pred;CPET有5例存在运动诱发的波浪式呼吸（EIOB）,6例为呼吸不稳定,提示整体功能状态低于正常人。本组慢病患者AHI为每小时（28.8±10.0）次,睡眠呼吸异常总时间占睡眠总时间的比值为（0.38±0.25）;OB周期的平均时间长度为（51.1±14.4）s。本组慢病患者正常平稳呼吸期的呼吸周期数与心率变异周期数的比值（B-n/HRV-B-n）为1.00±0.04,每个呼吸周期节律的心率变异平均幅度（HRV-B-M）为（2.64±1.59） bpm,虽然低于正常人（P<0.05）,但却与无睡眠呼吸异常的慢病患者相似（P>0.05）;HRV-B-M的变异度CV（HRV-B-M的SD/x）为( 0.33±0.11）,期间血氧饱和度（SpO₂）虽略低,但并无明显规律性下降与上升。本组慢病患者的OB期间呼吸周期数与心率变异周期数（OB-B-n/OB-HRV-B-n）比值为（1.22±0.18）,OB期每个呼吸周期节律的心率变异平均幅度（OB-HRV-B-M）为（3.56±1.57）bpm及其变异度（OB-CV =OB-HRV-B-M的SD/x）为（0.59±0.28）,每个OB周期节律的心率变异平均幅度（OB-HRV-OB-M）为（13.75±4.25）bpm,OB期间低通气时SpO₂出现明显的下降,OB期间SpO₂平均变异幅度（OB-SpO₂-OB-M）为（4.79±1.39）%,OB期的OB-B-n/OB-HRV-B-n比值、OB-HRV-OB-M比其正常平稳呼吸期对应指标显著增大（P<0.01）。OB-HRV-B-M虽然与正常平稳呼吸期HRV-B-M相比差异无统计学意义（P>0.05）,但其变异度OB-CV却显著增大（P<0.01）。结论: 睡眠呼吸异常的慢病患者OB期的心率变异幅度大于其正常平稳呼吸期,当呼吸模式发生改变时心率变异也发生明显改变,其平稳呼吸期的呼吸周期数与心率变异周期数的比值与正常人以及无睡眠呼吸异常的慢病患者相同,证实心率变异为呼吸源性;而其OB期间心率变异周期数相对于呼吸周期减少直接源于此时的低通气或者呼吸暂停,心率变异也是呼吸源性。     

Induction of oscillatory ventilation pattern using dynamic modulation of heart rate through a pacemaker

For disease states characterized by oscillatory ventilation, an ideal dynamic therapy would apply a counteracting oscillation in ventilation. Modulating respiratory gas transport through the circulation might allow this. We explore the ability of repetitive alternations in heart rate, using a cardiac pacemaker, to elicit oscillations in respiratory variables and discuss the potential for therapeutic exploitation. By incorporating acute cardiac output manipulations into an integrated mathematical model, we observed that a rise in cardiac output should yield a gradual rise in end-tidal CO2 and, subsequently, ventilation. An alternating pattern of cardiac output might, therefore, create oscillations in CO2 and ventilation. We studied the effect of repeated alternations in heart rate of 30 beats/min with periodicity of 60 s, on cardiac output, respiratory gases, and ventilation in 22 subjects with implanted cardiac pacemakers and stable breathing patterns. End-tidal CO2 and ventilation developed consistent oscillations with a period of 60 s during the heart rate alternations, with mean peak-to-trough relative excursions of 8.4 +/- 5.0% (P < 0.0001) and 24.4 +/- 18.8% (P < 0.0001), respectively. Furthermore, we verified the mathematical prediction that the amplitude of these oscillations would depend on those in cardiac output (r = 0.59, P = 0.001). Repetitive alternations in heart rate can elicit reproducible oscillations in end-tidal CO2 and ventilation. The size of this effect depends on the magnitude of the cardiac output response. Harnessed and timed appropriately, this cardiorespiratory mechanism might be exploited to create an active dynamic responsive pacing algorithm to counteract spontaneous respiratory oscillations, such as those causing apneic breathing disorders.     

Classification of oscillatory states of blood pressure after bleeding in cats (author's transl)]

Blood pressure and heart rate responses were recorded on bleeding of blood volume of 1 ml/kg body weight (test input) from the abdominal aorta of cat. Forward path gain, backward path gain and minification were determined from blood pressure responses to test input. Oscillations in blood pressure were elicited spontaneously or artificially after bleeding of blood volume of 5 ml/kg body weight (conditioning input). The oscillatory states are classified into latent state, underdamped oscillatory state and harmonic oscillatory state. Open loop gain decreased in the latent state but increased in the other states. Therefore, oscillation was hard to elicit in the latent state. Minification, however, enlarged in the latent state, making the system irritable to disturbances and reducing its control accuracy. In the latent state, an insensible disturbance increased backward path gain and open loop gain to induce the oscillations. There was an increase of heart rate at an elicitation of oscillation in any state and a decrease of heart rate made it damped. Therefore, the heart rate regulatory system plays a main role in the feedback path of the blood pressure regulatory system and an increase of heart rate by an insensible disturbance or test input increases backward path gain.     

Competition in the presence of a lethal external inhibitor

The study considers two organisms competing for a nutrient in an open system in the presence of an inhibitor (or toxicant). The inhibitor is input at a constant rate and is lethal to one competitor while being taken up by the other without harm. This is in contrast to previous studies, where the inhibitor decreases the reproductive rate of one of the organisms. The mathematical result of the lethal effect, modeled by a mass action term, is that the system cannot be reduced to a monotone dynamical system of one order lower as is common with chemostat-like problems. The model is described by four non-linear, ordinary differential equations and we seek to describe the asymptotic behavior as a function of the parameters of the system. Several global exclusion results are presented with mathematical proofs. However, in the case of coexistence, oscillatory behavior is possible and the study proceeds with numerical examples. The model is relevant to bioremediation problems in nature and to laboratory bio-reactors.     

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 with sum-of-sinusoids test stimuli.

The light-growth response of Phycomyces has been studied with the sum-of-sinusoids method of nonlinear system identification (Victor, J.D., and R.M. Shapley, 1980, Biophys. J., 29:459). This transient response of the sporangiophore has been treated as a black-box system with one input (logarithm of the light intensity, I) and one output (elongation rate). The light intensity was modulated so that log I, as a function of time, was a sum of sinusoids. The log-mean intensity was 10(-4) W m-2 and the wavelength was 477 nm. The first- and second-order frequency kernels, which represent the linear and nonlinear behavior of the system, were obtained from the Fourier transform of the response at the appropriate component and combination frequencies. Although the first-order kernel accounts for most of the response, there remains a significant nonlinearity beyond the logarithmic transducer presumed to occur at the input of the sensory transduction chain. From the analysis of the frequency kernels, we have derived a dynamic nonlinear model of the light-growth response system. The model consists of a nonlinear subsystem followed by a linear subsystem. The model parameters were estimated from a combined nonlinear least-squares fit to the first- and second-order frequency kernels.     

Modelling of cortical and thalamic 600 Hz activity by means of oscillatory networks

The purpose of this study is to investigate information processing in the primary somatosensory system with the help of oscillatory network modelling. Specifically, we consider interactions in the oscillatory 600 Hz activity between the thalamus and the cortical Brodmann areas 3b and 1. This type of cortical activity occurs after electrical stimulation of peripheral nerves such as the median nerve. Our measurements consist of simultaneous 31-channel MEG and 32-channel EEG recordings and individual 3D MRI data. We perform source localization by means of a multi-dipole model. The dipole activation time courses are then modelled by a set of coupled oscillators, described by linear second-order ordinary delay differential equations (DDEs). In particular, a new model for the thalamic activity is included in the oscillatory network. The parameters of the DDE system are successfully fitted to the data by a nonlinear evolutionary optimization method. To activate the oscillatory network, an individual input function is used, based on measurements of the propagated stimulation signal at the biceps. A significant feedback from the cortex to the thalamus could be detected by comparing the network modelling with and without feedback connections. Our finding in humans is supported by earlier animal studies. We conclude that this type of rhythmic brain activity can be modelled by oscillatory networks in order to disentangle feed forward and feedback information transfer.     

Separation of Respiratory Influences from the Tachogram: A Methodological Evaluation

The variability of the heart rate (HRV) is widely studied as it contains information about the activity of the autonomic nervous system (ANS). However, HRV is influenced by breathing, independently of ANS activity. It is therefore important to include respiratory information in HRV analyses in order to correctly interpret the results. In this paper, we propose to record respiratory activity and use this information to separate the tachogram in two components: one which is related to breathing and one which contains all heart rate variations that are unrelated to respiration. Several algorithms to achieve this have been suggested in the literature, but no comparison between the methods has been performed yet. In this paper, we conduct two studies to evaluate the methods'' performances to accurately decompose the tachogram in two components and to assess the robustness of the algorithms. The results show that orthogonal subspace projection and an ARMAX model yield the best performances over the two comparison studies. In addition, a real-life example of stress classification is presented to demonstrate that this approach to separate respiratory information in HRV studies can reveal changes in the heart rate variations that are otherwise masked by differing respiratory patterns.     

Modelling strategies for minimizing the impact of an imported exotic infection

The global epidemic of severe acute respiratory syndrome (SARS) in 2003 demonstrated the need to determine control strategies for exotic infections. The prior determination of such strategies, and the use of mathematical models to assist this, is hampered by the obvious lack of data. We propose an integral equation model of Kermack-McKendrick type that may be used to compare strategies based on the isolation of infectious individuals. The model structures the incidence of infection according to the location of an infected individual at exposure, and requires knowledge of the infectivity kernel and the initial rate of exponential increase of cases. The model's use in the design of strategies to minimize the risk of SARS in a previously unexposed community is demonstrated.     

The effect of thoracic pressoreceptors on heart rate in man

Sinus arrhythmia (SA) is a physiological event related to the respiratory activity. The aim of this study was to find a relation between SA and different deep breathings. A transductor monitored the chest respiratory movements and a cardiotachometer calculated and plotted the instantaneous heart rate, synchronously during different breathing efforts. In 10 normal subjects a good linear correlation was found between the breathing depth and SA quantifying the relation between the chest receptor activity and SA also with limited chest expansion.     

Diffusion along intercellular spaces: An analysis of a transient situation

We present an analysis of the diffusion of a tracer in a model of a cell-intercellular space system. The problem reduces to the resolution of a system of a linear partial differential equation and of a linear integral differential equation. The mathematical results have been obtained in terms of their Laplace transforms, which have been inverted by a numerical procedure for some parameter values. The importance of considering gradients of concentrations in intercellular spaces instead of lumping them with the external mediums has been discussed together with the possibility of extending Ussing's relation to transient cases, in order to detect active transports. Some possible implementations of the model to take into account more general situations have been considered.     

Membrane potential changes of skeletomotor neurons in response to random stretches of the triceps surae muscles in decerebrate cats

 The properties of membrane potential changes of skeletomotor neurons (S, FR, and FF) innervating triceps surae muscles during pseudorandom stretching of these muscles were studied in decerebrate cats. Peak amplitudes of pseudorandom muscle stretches ranged from 119 μm to 4.15 mm peak-to-peak. Sequences of ten identical stretching periods were applied for averaging. Shapes of membrane potential changes and probability density distribution of amplitudes of the input and output signals and power spectra suggest that the skeletomotor neuron membrane has nonlinear properties. First- and second-order Wiener kernels were determined by applying the cross-correlation (Lee-Schetzen) method. The results suggest that the transfer function between muscle stretches and subthreshold membrane potentials is a Wiener-type cascade. This cascade is consistent with a linear, second-order, underdamped transfer function followed by a simple quadratic nonlinearity [linear (L) system followed by nonlinear (N) system, or LN cascade]. Including the nonlinear component calculated from the second-order Wiener kernel improved the model significantly over its linear counterpart, especially in S-type motoneurons. Qualitatively similar results were obtained with all types of motoneurons studied. Received: 1 April 1993/Accepted in revised form: 24 March 1994     

Respiratory influences on non-linear dynamics of heart rate variability in humans

 The goal of our study was to determine whether evidence for chaos in heart rate variability (HRV) can be observed when the respiratory input to the autonomic controller of heart rate is forced by the deterministic pattern associated with periodic breathing. We simultaneously recorded, in supine healthy volunteers, RR intervals and breathing volumes for 20 to 30 min (1024 data point series) during three protocols: rest (control), fixed breathing (15 breath/min) and voluntary periodic breathing (3 breaths with 2 s inspiration and 2 s expiration followed by an 8 s breath hold). On both the RR interval and breathing volume series we applied the non-linear prediction method (Sugihara and May algorithm) to the original time series and to distribution-conserved isospectral surrogate data. Our results showed that, in contrast to the control test, during both fixed and voluntary periodic breathing the variability of breathing volumes was clearly deterministic non-chaotic. During all the three protocols, the RR-interval series’ non-linear predictability was consistent with one of a chaotic series. However, at rest, no clear difference was observed between the RR-interval series and their surrogates, which means that no clear chaos was observed. During fixed breathing a difference appeared, and this difference seemed clearer during voluntary periodic breathing. We concluded that HRV dynamics were chaotic when respiration was forced with a deterministic non-chaotic pattern and that normal spontaneous respiratory influences might mask the normally chaotic pattern in HRV. Received: 7 August 1995 / Accepted in revised form: 20 March 1997     

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.     

Time series prediction of lung cancer patients' breathing pattern based on nonlinear dynamics

This study focuses on predicting breathing pattern, which is crucial to deal with system latency in the treatments of moving lung tumors. Predicting respiratory motion in real-time is challenging, due to the inherent chaotic nature of breathing patterns, i.e. sensitive dependence on initial conditions. In this work, nonlinear prediction methods are used to predict the short-term evolution of the respiratory system for 62 patients, whose breathing time series was acquired using respiratory position management (RPM) system. Single step and N-point multi step prediction are performed for sampling rates of 5 Hz and 10 Hz. We compare the employed non-linear prediction methods with respect to prediction accuracy to Adaptive Infinite Impulse Response (IIR) prediction filters. A Local Average Model (LAM) and local linear models (LLMs) combined with a set of linear regularization techniques to solve ill-posed regression problems are implemented. For all sampling frequencies both single step and N-point multi step prediction results obtained using LAM and LLM with regularization methods perform better than IIR prediction filters for the selected sample patients. Moreover, since the simple LAM model performs as well as the more complicated LLM models in our patient sample, its use for non-linear prediction is recommended.     

Effects of muscle model parameter dispersion and multi-loop segmental interaction on the neuromuscular system performance

The effects of parameter dispersion among motor units on the neuromuscular system performance as well as interaction between muscle segments and spinal cord mechanisms are investigated. Elementary components of the system are modeled to simulate with simple models their input-output characteristics. A leaky SS-IPFM encoder with a time-dependent threshold simulates the motor-neuron encoding characteristics. An amplitude and time dependent nonlinear model represent the motor unit mechanical output to neuronal input relationship. The dispersion of parameters in the components of the whole muscle control model is investigated in the open loop mode. It is shown that the dispersion of parameters in the multi-efferent channels converging on a common tendon provides a spatial filtration generating a smoother muscle force in addition to extending the linear dynamic range compared to a similar system having identical motor units. Muscle segmental interaction is investigated in this distributed model by closing the loop through a coupling matrix, representing afferent-motorneuron interaction on the spinal cord level. A diagonal matrix represents no segmental interaction and a uniform matrix represents a uniform interaction between segments through the muscle spindles and Golgi tendon feedback elements. The close loop simulation studied shows that (a). The type of segmental interaction has little effect on the overall system performance, i.e., range of linerity and stability, which is the result of having a muscle system with a large number of motor units. (b) There are only minor differences in results between the uniform and normal parameter distributions tested. (c) A loop gain of 4 divided by 8 in the distributed model can provide linearity through the full physiological force range. (d) Type of segmental interaction has significant effects on the individual segment. A uniform matrix provides a more stable segment due to the spatial filtration resulting from the segmental interaction, while the diagonal noninteracting matrix shows instabilities on the local segmental level despite global stability. The more realistic exponentially decaying spatial interaction matrix yields both global neuromuscular and local segmental stability with the same linear dynamic range generated with the uniform or diagonal matrices.