On parsing the neural code in the prefrontal cortex of primates using principal dynamic modes.

Nonlinear modeling of multi-input multi-output (MIMO) neuronal systems using Principal Dynamic Modes (PDMs) provides a novel method for analyzing the functional connectivity between neuronal groups. This paper presents the PDM-based modeling methodology and initial results from actual multi-unit recordings in the prefrontal cortex of non-human primates. We used the PDMs to analyze the dynamic transformations of spike train activity from Layer 2 (input) to Layer 5 (output) of the prefrontal cortex in primates performing a Delayed-Match-to-Sample task. The PDM-based models reduce the complexity of representing large-scale neural MIMO systems that involve large numbers of neurons, and also offer the prospect of improved biological/physiological interpretation of the obtained models. PDM analysis of neuronal connectivity in this system revealed "input-output channels of communication" corresponding to specific bands of neural rhythms that quantify the relative importance of these frequency-specific PDMs across a variety of different tasks. We found that behavioral performance during the Delayed-Match-to-Sample task (correct vs. incorrect outcome) was associated with differential activation of frequency-specific PDMs in the prefrontal cortex.

Multiple-input nonlinear modelling of cerebral haemodynamics using spontaneous arterial blood pressure, end-tidal CO2 and heart rate measurements.

In order to examine the effect of changes in heart rate (HR) upon cerebral perfusion and autoregulation, we include the HR signal recorded from 18 control subjects as a third input in a two-input model of cerebral haemodynamics that has been used previously to quantify the dynamic effects of changes in arterial blood pressure and end-tidal CO2upon cerebral blood flow velocity (CBFV) measured at the middle cerebral arteries via transcranial Doppler ultrasound. It is shown that the inclusion of HR as a third input reduces the output prediction error in a statistically significant manner, which implies that there is a functional connection between HR changes and CBFV. The inclusion of nonlinearities in the model causes further statistically significant reduction of the output prediction error. To achieve this task, we employ the concept of principal dynamic modes (PDMs) that yields dynamic nonlinear models of multi-input systems using relatively short data records. The obtained PDMs suggest model-driven quantitative hypotheses for the role of sympathetic and parasympathetic activity (corresponding to distinct PDMs) in the underlying physiological mechanisms by virtue of their relative contributions to the model output. These relative PDM contributions are subject-specific and, therefore, may be used to assess personalized characteristics for diagnostic purposes.

General methodology for nonlinear modeling of neural systems with Poisson point-process inputs.

This paper presents a general methodological framework for the practical modeling of neural systems with point-process inputs (sequences of action potentials or, more broadly, identical events) based on the Volterra and Wiener theories of functional expansions and system identification. The paper clarifies the distinctions between Volterra and Wiener kernels obtained from Poisson point-process inputs. It shows that only the Wiener kernels can be estimated via cross-correlation, but must be defined as zero along the diagonals. The Volterra kernels can be estimated far more accurately (and from shorter data-records) by use of the Laguerre expansion technique adapted to point-process inputs, and they are independent of the mean rate of stimulation (unlike their P-W counterparts that depend on it). The Volterra kernels can also be estimated for broadband point-process inputs that are not Poisson. Useful applications of this modeling approach include cases where we seek to determine (model) the transfer characteristics between one neuronal axon (a point-process 'input') and another axon (a point-process 'output') or some other measure of neuronal activity (a continuous 'output', such as population activity) with which a causal link exists.

Compartmental and Data-Based Modeling of Cerebral Hemodynamics: Linear Analysis.

Compartmental and data-based modeling of cerebral hemodynamics are alternative approaches that utilize distinct model forms and have been employed in the quantitative study of cerebral hemodynamics. This paper examines the relation between a compartmental equivalent-circuit and a data-based input-output model of dynamic cerebral autoregulation (DCA) and CO2-vasomotor reactivity (DVR). The compartmental model is constructed as an equivalent-circuit utilizing putative first principles and previously proposed hypothesis-based models. The linear input-output dynamics of this compartmental model are compared with data-based estimates of the DCA-DVR process. This comparative study indicates that there are some qualitative similarities between the two-input compartmental model and experimental results.

Use of a random forcing for high-frequency ventilation.

Previous applications of high-frequency oscillatory ventilation (HFOV) have used cyclic forcings with the frequency of oscillation considered to be a fundamental parameter. A question that is addressed in the present study is whether or not periodicity is an essential requirement for this mode of ventilation to occur. It was found possible to adequately ventilate anesthetized and paralyzed cats with volume excursions below the dead-space level using a random band-limited forcing. Experimental conditions were close to a constant flow variance (VARF) state, and arterial CO2 tension varied linearly as a function of the ratio of noise bandwidth and VARF. Periodicity per se did not appear to be a requirement for HFOV to occur, a result consistent with predictions of Taylor dispersion theory.

Signal transformation and coding in neural systems.

The subject of signal transformation and coding in neural systems is fundamental in understanding information processing by the nervous system. This paper addresses this issue at the level of neural units (neurons) using nonparametric nonlinear dynamic models. These models are variants of the general Wiener-Bose model, adapted to this problem as to represent the nonlinear dynamics of neural signal transformation using a set of parallel filters (neuron modes) followed by a binary operator with multiple real-valued operands (equal in number to the number of modes). The postulated model constitutes a reasonable compromise between mathematical complexity and current neurophysiological evidence. It incorporates nonlinear dynamics and spike generation mechanisms in a fairly general, yet parsimonious manner. Although this study has objectives limited to a single unit and represents a small contribution in a vast and complex research area, it is hoped that it will facilitate progress in the systematic study of the functional organization of neural systems with multiple units.

Modeling of neural systems by use of neuronal modes.

A methodology for modeling spike-output neural systems from input-output data is proposed, which makes use of "neuronal modes" (NM) and "multi-input threshold" (MT) operators. The modeling concept of NM's was introduced in a previously published paper in order to provide concise and general mathematical representations of the nonlinear dynamics involved in signal transformation and coding by a class of neural systems. This paper presents and demonstrates (with computer simulations) a method by which the NM's are determined using the 1st- and 2nd-order kernel estimates of the system, obtained from input-output data. The MT operator (i.e., a binary operator with multiple real-valued operands which are the outputs of the NM's) possesses an intrinsic refractory mechanism and generates the sequence of output spikes. The spike-generating characteristics of the MT operator are determined by the "trigger regions" defined on the basis of data. This approach is offered as a reasonable compromise between modeling complexity and prediction accuracy, which may provide a common methodological framework for modeling a certain class of neural systems.

On the efficacy of linear system analysis of renal autoregulation in rats.

In order to assess the linearity of the mechanisms subserving renal blood flow autoregulation, broad-band arterial pressure fluctuations at three different power levels were induced experimentally and the resulting renal blood flow responses were recorded. Linear system analysis methods were applied in both the time and frequency domain. In the frequency domain, spectral estimates employing FFT, autoregressive moving average (ARMA) and moving average (MA) methods were used; only the MA model showed two vascular control mechanisms active at 0.02-0.05 Hz and 0.1-0.18 Hz consistent with previous experimental findings [Holstein-Rathlou et al., Amer. J. Physiol., vol. 258, 1990.]. In the time domain, impulse response functions obtained from the MA model indicated likewise the presence of these two vascular control mechanisms, but the ARMA model failed to show any vascular control mechanism at 0.02-0.05 Hz. The residuals (i.e., model prediction errors) of the MA model were smaller than the ARMA model for all levels of arterial pressure forcings. The observed low coherence values and the significant model residuals in the 0.02-0.05 Hz frequency range suggest that the tubuloglomerular feedback (TGF) active in this frequency range is a nonlinear vascular control mechanism. In addition, experimental results suggest that the operation of the TGF mechanism is more evident at low/moderate pressure fluctuations and becomes overwhelmed when the arterial pressure forcing is too high.

Nonlinear system analysis of renal autoregulation in normotensive and hypertensive rats.

We compared the dynamic characteristics in renal autoregulation of blood flow of normotensive Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR), using both linear and nonlinear systems analysis. Linear analysis yielded only limited information about the differences in dynamics between SDR and SHR. The predictive ability, as determined by normalized mean-square errors (NMSE), of a third-order Volterra model is better than for a linear model. This decrease in NMSE with a third-order model from that of a linear model is especially evident at frequencies below 0.2 Hz. Furthermore, NMSE are significantly higher in SHR than SDR, suggesting a more complex nonlinear system in SHR. The contribution of the third-order kernel in describing the dynamics of renal autoregulation in arterial blood pressure and blood flow was found to be important. Moreover, we have identified the presence of nonlinear interactions between the oscillatory components of the myogenic mechanism and tubuloglomerular feedback (TGF) at the level of whole kidney blood flow in SDR. An interaction between these two mechanisms had previously been revealed for SDR only at the single nephron level. However, nonlinear interactions between the myogenic and TGF mechanisms are not detected for SHR.

Laser induced fluorescence attenuation spectroscopy: detection of hypoxia.

The development of a new laser-induced fluorescence (LIF) spectroscopy technique for the measurement of the attenuation spectrum of tissue is described. The technique, termed laser-induced fluorescence attenuation spectroscopy (LIFAS), has been applied to study the effects of hypoxia on the in vivo optical properties of renal and myocardial tissue in the 350-600-nm band. Excimer laser (Xe-Cl) is used to excite a small volume of the tissue (rabbit model, N = 20) and induce autofluorescence. The emitted LIF is monitored fiberoptically at two locations that are unevenly displaced about the fluorescing volume. The optical attenuation of the tissue is calculated from the dual LIF measurements by assuming an exponential decay of the fluorescence with distance. The results indicate that hypoxia modulates the attenuation spectrum leading to characteristic changes in its shape. Primarily, the spectral profile becomes more concave between 455 nm and 505 nm and two spectral peaks at about 540 and 580 nm disappear leaving in their place a single peak at about 555 nm. The attenuation spectra of normoxic and hypoxic tissue are used to train partial least squares multivariate model for spectral classification. The model detected acute renal and myocardial hypoxia with an accuracy greater than 90% (range: 90%-96%) and 74% (range: 74%-90%), respectively.

A novel network for nonlinear modeling of neural systems with arbitrary point-process inputs.

This paper address the issue of nonlinear model estimation for neural systems with arbitrary point-process inputs using a novel network that is composed of a pre-processing stage of a Laguerre filter bank followed by a single hidden layer with polynomial activation functions. The nonlinear modeling problem for neural systems has been attempted thus far only with Poisson point-process inputs and using cross-correlation methods to estimate low-order nonlinearities. The specific contribution of this paper is the use of the described novel network to achieve practical estimation of the requisite nonlinear model in the case of arbitrary (i.e. non-Poisson) point-process inputs and high-order nonlinearities. The success of this approach has critical implications for the study of neuronal ensembles, for which nonlinear modeling has been hindered by the requirement of Poisson process inputs and by the presence of high-order nonlinearities. The proposed methodology yields accurate models even for short input-output data records and in the presence of considerable noise. The efficacy of this approach is demonstrated with computer-simulated examples having continuous output and point-process output, and with real data from the dentate gyrus of the hippocampus.

Volterra models and three-layer perceptrons.

This paper proposes the use of a class of feedforward artificial neural networks with polynomial activation functions (distinct for each hidden unit) for practical modeling of high-order Volterra systems. Discrete-time Volterra models (DVMs) are often used in the study of nonlinear physical and physiological systems using stimulus-response data. However, their practical use has been hindered by computational limitations that confine them to low-order nonlinearities (i.e., only estimation of low-order kernels is practically feasible). Since three-layer perceptrons (TLPs) can be used to represent input-output nonlinear mappings of arbitrary order, this paper explores the basic relations between DVMs and TLPs with tapped-delay inputs in the context of nonlinear system modeling. A variant of TLP with polynomial activation functions-termed "separable Volterra networks" (SVNs)-is found particularly useful in deriving explicit relations with DVM and in obtaining practicable models of highly nonlinear systems from stimulus-response data. The conditions under which the two approaches yield equivalent representations of the input-output relation are explored, and the feasibility of DVM estimation via equivalent SVN training using backpropagation is demonstrated by computer-simulated examples and compared with results from the Laguerre expansion technique (LET). The use of SVN models allows practicable modeling of high-order nonlinear systems, thus removing the main practical limitation of the DVM approach.

Modeling of nonlinear nonstationary dynamic systems with a novel class of artificial neural networks.

This paper introduces a novel neural-network architecture that can be used to model time-varying Volterra systems from input-output data. The Volterra systems constitute a very broad class of stable nonlinear dynamic systems that can be extended to cover nonstationary (time-varying) cases. This novel architecture is composed of parallel subnets of three-layer perceptrons with polynomial activation functions, with the output of each subnet modulated by an appropriate time function that gives the summative output its time-varying characteristics. The paper shows the equivalence between this network architecture and the class of time-varying Volterra systems, and demonstrates the range of applicability of this approach with computer-simulated examples and real data. Although certain types of nonstationarities may not be amenable to this approach, it is hoped that this methodology will provide the practical tools for modeling some broad classes of nonlinear, nonstationary systems from input-output data, thus advancing the state of the art in a problem area that is widely viewed as a daunting challenge.

Comparative nonlinear modeling of renal autoregulation in rats: Volterra approach versus artificial neural networks.

Volterra models have been increasingly popular in modeling studies of nonlinear physiological systems. In this paper, feedforward artificial neural networks with two types of activation functions (sigmoidal and polynomial) are utilized for modeling the nonlinear dynamic relation between renal blood pressure and flow data, and their performance is compared to Volterra models obtained by use of the leading kernel estimation method based on Laguerre expansions. The results for the two types of artificial neural networks (sigmoidal and polynomial) and the Volterra models are comparable in terms of normalized mean-square error (NMSE) of the respective output prediction for independent testing data. However, the Volterra models obtained via the Laguerre expansion technique achieve this prediction NMSE with approximately half the number of free parameters relative to either neural-network model. Nonetheless, both approaches are deemed effective in modeling nonlinear dynamic systems and their cooperative use is recommended in general, since they may exhibit different strengths and weaknesses depending on the specific characteristics of each application.

Spectro-temporal studies of Xe-Cl excimer laser-induced arterial wall fluorescence.

We report on spectro-temporal fluorescence studies of cadaver femoral arterial walls at different stages in the progression of atherosclerosis. After excitation with a Xe-Cl excimer pulse, the time course of the fluorescence spectrum was recorded over time, and time-resolved multispectral analysis was performed. Then, under the assumption of linearity, we derived a linear spectro-temporal kernel (a weighting function) which describes the temporal behavior of the fluorescence process independently of the pulse width of the photoexcitation. The data analysis revealed both static and dynamic fluorescence characteristics which exhibited a good correlation with histological findings.

Model-based physiomarkers of cerebral hemodynamics in patients with mild cognitive impairment.

In our previous studies, we have introduced model-based "functional biomarkers" or "physiomarkers" of cerebral hemodynamics that hold promise for improved diagnosis of early-stage Alzheimer's disease (AD). The advocated methodology utilizes subject-specific data-based dynamic nonlinear models of cerebral hemodynamics to compute indices (serving as possible diagnostic physiomarkers) that quantify the state of cerebral blood flow autoregulation to pressure-changes (CFAP) and cerebral CO2 vasomotor reactivity (CVMR) in each subject. The model is estimated from beat-to-beat measurements of mean arterial blood pressure, mean cerebral blood flow velocity and end-tidal CO2, which can be made reliably and non-invasively under resting conditions. In a previous study, it was found that a CVMR index quantifying the impairment in CO2 vasomotor reactivity correlates with clinical indications of early AD, offering the prospect of a potentially useful diagnostic tool. In this paper, we explore the use of the same model-based indices for patients with amnestic Mild Cognitive Impairment (MCI), a preclinical stage of AD, relative to a control subjects and clinical cognitive assessments. It was found that the model-based CVMR values were lower for MCI patients relative to the control subjects.

Model-based quantification of cerebral hemodynamics as a physiomarker for Alzheimer's disease?

Previous studies have found that Alzheimer's disease (AD) impairs cerebral vascular function, even at early stages of the disease. This offers the prospect of a useful diagnostic method for AD, if cerebral vascular dysfunction can be quantified reliably within practical clinical constraints. We present a recently developed methodology that utilizes a data-based dynamic nonlinear closed-loop model of cerebral hemodynamics to compute "physiomarkers" quantifying the state of cerebral flow autoregulation to pressure-changes (CA) and cerebral CO2 vasomotor reactivity (CVMR) in each subject. This model is estimated from beat-to-beat measurements of mean arterial blood pressure, mean cerebral blood flow velocity and end-tidal CO2, which can be made reliably and non-invasively under resting conditions. This model may also take an open-loop form and comparisons are made with the closed-loop counterpart. The proposed model-based physiomarkers take the form of two indices that quantify the gain of the CA and CVMR processes in each subject. It was found in an initial set of clinical data that the CVMR index delineates AD patients from control subjects and, therefore, may prove useful in the improved diagnosis of early-stage AD.

Closed-loop dynamic modeling of cerebral hemodynamics.

The dynamics of cerebral hemodynamics have been studied extensively because of their fundamental physiological and clinical importance. In particular, the dynamic processes of cerebral flow autoregulation (CFA) and CO2 vasomotor reactivity have attracted broad attention because of their involvement in a host of pathologies and clinical conditions (e.g., hypertension, syncope, stroke, traumatic brain injury, vascular dementia, Alzheimer's disease, mild cognitive impairment etc.). This raises the prospect of useful diagnostic methods being developed on the basis of quantitative models of cerebral hemodynamics, if cerebral vascular dysfunction can be quantified reliably from data collected within practical clinical constraints. This paper presents a modeling method that utilizes beat-to-beat measurements of mean arterial blood pressure, cerebral blood flow velocity and end-tidal CO2 (collected non-invasively under resting conditions) to quantify the dynamics of CFA and cerebral vasomotor reactivity (CVMR). The unique and novel aspect of this dynamic model is that it is nonlinear and operates in a closed-loop configuration.

Design of optimal stimulation patterns for neuronal ensembles based on Volterra-type hierarchical modeling.

This paper presents a general methodology for the optimal design of stimulation patterns applied to neuronal ensembles in order to elicit a desired effect. The methodology follows a variant of the hierarchical Volterra modeling approach that utilizes input-output data to construct predictive models that describe the effects of interactions among multiple input events in an ascending order of interaction complexity. The illustrative example presented in this paper concerns the multi-unit activity of CA1 neurons in the hippocampus of a rodent performing a learned delayed-nonmatch-to-sample (DNMS) task. The multi-unit activity of the hippocampal CA1 neurons is recorded via chronically implanted multi-electrode arrays during this task. The obtained model quantifies the likelihood of having correct performance of the specific task for a given multi-unit (spatiotemporal) activity pattern of a CA1 neuronal ensemble during the 'sample presentation' phase of the DNMS task. The model can be used to determine computationally (off-line) the 'optimal' multi-unit stimulation pattern that maximizes the likelihood of inducing the correct performance of the DNMS task. Our working hypothesis is that application of this optimal stimulation pattern will enhance performance of the DNMS task due to enhancement of memory formation and storage during the 'sample presentation' phase of the task.

Diagnostic biomarkers for Alzheimer's disease using dynamic nonlinear models based on Principal Dynamic Modes.

Sensitive and robust diagnostic biomarkers for Alzheimer's disease (AD) were sought using dynamic nonlinear models of the causal interrelationships among time-series (beat-to-beat) data of arterial blood pressure, end-tidal CO(2) and cerebral blood flow velocity collected in human subjects (4 AD patients and 4 control subjects). These models were based on Principal Dynamic Modes (PDM) and yielded a reliable biomarker for AD diagnosis in the form of the "Effective CO(2) Reactivity Index" (ECRI). The results from this initial set of subjects corroborated the efficacy of the ECRI biomarker for accurate AD diagnosis.