Measurement Techniques, Vol. 40, No. I, 1997

STANDARDIZING MEASUREMENTS OF THE PARAMETERS OF MATHEMATICAL

MODELS OF NEURAL NETWORKS

UDC 517.9:621.317:573

O. E. Filatova

This article examines the problem of standardizing the states of complex dynamic objects - dynamic biological systems (DBIs). The resolution of this problem reduces to using mathematical models of these and analyzing the dynamic processes in DBIs in response to specially identified optimum parameters (amplitude, duration) of external (electrical, mechanical, chemical) stimuli. The potential for the use of such approaches in medicine and biology is also discussed.

The formalization of biomedical research is a process that is still far from completion. The use of mathematical methods in such investigations along with various mathematical models of the biological systems being studied is still in the developmental stage. The mathematical methods that are used are usually based on statistical analyses of data and the determination of numerical characteristics of the expected distribution function of a given random variable - moments of different orders (mathematical expectation, variance, modes) or correlation coefficients [1, 2]. However, it is still relatively rare for researchers to attempt to construct mathematical models of the dynamics of medical-biological processes, and when this is done the model is sometimes regarded as an exception to the rule and a mathematical oddity. This situation is first of all the result of the lack of a unified approach to the mathematical representation of biomedical processes, as well as to the cautious attitude of specialists in biology and medicine as regards the use of math - specialists who tend to be advocates of pure empiricism. Despite advances in mathematics and biophysics and the work of researchers in neurocomputing and systems analysis, no significant strides are being made in medical schools or university biology programs in the teaching of physico-mathematical methods of studying living things. The efforts here are weak and disorganized. The importance of the use of rigorous methods in research and the standardization of biomedical measurements is not being stressed, and students are not being fully exposed to the possibilities of mathematical methods of studying the functional systems of humans and animals. Graduates in the biomedical fields finish their schooling with the firm conviction that the issues of precision in measurement and metrological evaluation of the equipment they use are the concerns of other specialists. The blame for poor-quality measurements in biology and medicine must be laid at the foot not of these specialists, but those working in biology and medicine. The "consumer's" attitude of doctors and biologists toward technical systems obviously can only be changed through the efforts of specialists in metrology and physics together with mathematicians and engineers. The disciplines mentioned above must be better incorporated into the training programs for doctors and biologists (in the form of special courses on metrology, neurocomputing, and mathematical methods). Otherwise, the gap between these two areas of intellectual activity will only widen, and the last opportunity to formalize biomedical research will be lost. The Biophysics and Neurodynamics Laboratory at Samarskii Pedagogical University has placed an emphasis on enhancing the role of physico-mathematical methods in the study of the health and pathology of humans and animals and formalizing such instruction. In connection with this effort, the lab has devised standardized methods for mathematically modeling the functional systems of organisms (FSOs) and methods of classifying the states of FSOs within the framework of mathematical modeling in order to standardize biomedical research. The concern here is standardizing the functional state of a biological specimen, not just metrologically certifying the electronic equipment used to study the specimen. Using studies of neural networks of the respiratory center of mammals as an example, we have devised a unified methodological approach

Translated from Izmeriternaya Tekhnika, No. 1, pp. 37-40, January, 1997. 0543-1972/97/4001-0055518.00 9

Plenum Publishing Corporation

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to studying FSOs based not only on mathematical theories, but also on computer programs that allow the automatic construction of mathematical models (MMs) of FSOs and investigation of the conditions under which they function. Below, we examine one of the developments in this area - the identification of steady-state regimes of functioning of respiratory neural networks (RNNs). The identification methods used are based on the modern compartmental theory of RNNs, which considers the bullet (block) construction of the complex dynamic system - the RNN - and the existence of exciting (energetic) and inhibiting (informational) linkages between individual compartments (bullets) [2, 3]. These linkages are represented by a matrix of weighted linkages A, the elements of which aid (i -- j) also quantitatively characterize the degree of interaction between the ioth and j-th elements of the RNN. The state vector x of the system as a whole contains components x i that represent the state of the i-th compartment (in the given case, the bioelectric activity of the i-th neuron bulle0. In the general case, the components of the vector x can represent the energy characteristics of the i-th block (compartment) or its electrical characteristics (conductivity, impedance, etc.). In other words, the sense of the components of x may also be purely physical (technical), depending on the type of system being modeled. In our case, we are interested only in the biological meaning of the vector x - the state vector of a compartmental RNN. As in any natural system, allowance is made for the possibility of energy dissipation (in the given case, excitation, i.e., bioelectric activity of the RNN). This is represented by a negative term ( - b x ) in the right side of the system of differential equations or difference equations modeling the behavior of the RNN being studied. We further assume that the system is open and that certain energetic factors (in the given case, external excitation) can influence the system from outside it by affecting the rate of change in the vector - *. The influx of external stimuli is represented by the vector ud, where u is a scalar, while the vector d accounts for the component-by-component (block-by-block) effect of external excitation on the rate of change in the vector - x. Considering all of the above, the mathematical model that describes the dynamics of the circulation of a stimulus in an RNN appears as follows in vector-matrix form = Ax -- bx + ud --

(1)

the mathematical model appears as follows in the form of a system of differential equations y = crx.

where y is the output of the system, or x..t=Ax

--bx

+ud.

(2)

the mathematical model appears as below in the form of a system of difference equations y,~ "= C r X l

If we consider the effect of feedback, then the feedback matrix P(y) also appears in (1) and (2) and the model as a whole takes the form of systems of nonlinear equations. For example, (1) is transformed into = A P (y) x - -

b x + ud.

(3)

y = CTX,

where P(y) is a certain matrix containing nonlinear functions expressing the feedback from the output of the system y. A thorough investigation of compartmental mathematical models of RNNs (their stability, the conditions under which the formation of new cycles becomes a branching process, and aspects of direct and continuous control in such systems) was made in [2]. Our research allows for the possibility of identifying such models by using actual respiratory neural networks as examples. In particular, we used such mathematical models to identify and standardize the functional regimes of biological dynamic systems - RNNs. Here, we specifically address the problem of stabilizing these regimes and on this basis identifying the functional states of any biological dynamic system. There is no doubt as to the importance of the formulation of the problem, since until now 56

researchers have not paid particular attention to the possibility or impossibility of identifying the conditions under which a given system functions. It has been assumed beforehand that if several phenotypic indicators (age, weight, breathing or heart rate) coincide along with specified external factors (the parameters of stimuli, doses of narcotics, etc.), then experimenters in different countries or different periods of time are dealing with the same object - such as an RNN. It is then concluded from this that the experimental conditions have been standardized. Thus, there are still no rigorous formal (mathematical) criteria for evaluating the degree to which test conditions and the states of specimens are identical, i.e., there is no standarization of FSOs. In our laboratory, we are working on a new conceptual approach to standardizing the conditions of biological specimens. It differs in important ways from existing approaches. First of all, the criteria for evaluating the concept of "standard" are clearly formalized (cast in mathematical form). Secondly, the state of the functional systems of an organism is evaluated by dynamic (stimulative) rather than statistical (observational) methods. The comparunental theory of RNNs [2-5] and the method of minimum realization (MMR), with simultaneous input from each method, are used to construct the first criterion for evaluating the standard state of an organism. We developed a program, based on the MMR, that makes it possible to identify mathematical models of RNNs in the form of model (1) or model (2) without allowance for the existence of feedback (i.e., with the exclusion of model (3)). Of course, using this approach first requires identification of the linear behavior of the RNN and proof that there is no nonlinear feedback (i.e., it is assumed that P(y) = 1 is a unitary matrix). The linear behavior of a RNN was substantiated experimentally by using a square pulse of the duration t or a series of n pulses of the total duration t. In the latter case, the period of the pulses T was equal to T = t/n. Sending a series of pulses to the input of the RNN is preferable, in light of the integrative and adaptive properties both of individual neurons and of the RNN as a whole. The fact is that an increase in the duration t of one pulse acnjal~y produces nonlinear effects even at the cell membrane level (rather than remaining constant, potential decreases exponentially due to polarization effects). Thus, to make an accurate quantitative evaluation, it is necessary to change n = t/T and thus also change t. We used a computer to check the changes in the amplitudes of the stimulating pulses Z and the output signal y of the RNN. As the output, we used the recorded bioelectric activity of efferent nerves. The latter were either diaphragm nerves or intercostal (generally external intercostal nerves in rats and both external and internal intercostal nerves in cats). In a number of cases, we recorded the bioelectrie activity of respiratory muscles. Taken as an average (with fairly high accuracy), this activity gave a good indication of the output characteristics (y = cTx) of the given RNN. For the periodic functioning of the RNN (normal respiration), the relationship between the input and output depended appreciably on the respiratory phase. This finding illustr,'ltes the nonlinear character of the behavior of the RNN. In particular, studies such as these allowed us to determine the character of the dependence of the elements of the matrix P(y) on y. It was found that the functions p(y) (the elements of the matrix P) have the form of functions with saturation, i.e. p(y) ~ 0 at y --, co. This was described in detail in [4]. We were interested in the linear and quasi-linear regimes of the RNN, when an increase in Z linearly increased the value of y. Such regimes were seen after hyperventilation, micro-injection of solutions of phenobarbital or earphedone into the medulla oblongata, or the use of other methods of stopping inhalation or exhalation [5, 6]. In the cases described above, there was a roughly linear relationship between the amplitudes of the input and output electrical signals in the RNN. It is important to note that we (with the aid of a computer) chose the range of Z and, thus, the range of y in such a way that the experimental error was within the limits of the error caused by the use of the method of minimum realization to identify the mathematical models. The measurement error was in the range (0-5)% in most of our studies. Indeed, the 5% value of the error of measurement of the change in amplitude and the eigenvalues of linear approximation matrix A calculated by the MMR-based program served as the criterion of accuracy of all of our measurements and calculations. The 5% interval in fact included the values of bioelectrie activity (BA) of the nerves or muscles, since the BA amplifiers and integrators that we used did not allow us to achieve a high measurement accuracy. Thus, the 5 % error for the measurements and calculations was actually our attainable standard error in all of the experiments. Most importantly, the optimum changes in the amplitudes of the input stimuli Z were measured with this error. The problem was that the elements that were excited (RNN, muscle fibers) are a priori nonlinear objects, since they all have threshold properties ("all or nothing ~ law). However, an approximately linear dependence of y on Z can be seen within a certain range of Z (Zmin ... Zmax), since an increase in Z leads to an increase in y (with a maximum error of 5 %, in our experiments). As an example, we present the regression equation y = 9.4Z + 1.6, with the indicated 5% error, for the stimulation of the ninth intercostal afferent nerve and recording of the response from the tenth efferent nerve. This is the socalled spinal-bulbar-spinal reflex, where a reflexive response is obtained after the transmission of a stimulus (stimulation 57

of the peripheral afferent nerve) through the respiratory neural network of the medulla obiongata and a response (at the level of the thoracic cavity) in efferent nerves and muscles. The same type of linear relation was also seen in the stimulation of other afferent nerves or individual iuspiratory and expiratory intrabulbar structures. In other words, roughly linear behavior y = y(Z) can be seen in many cases for different structures of the medulla oblongata. Thus, having a linear approximation in the modeling of the dynamics of the behavior of an RNN is not all that rare an oc~drrence when the RNN is in a certain steady state (one of those mentioned above) and the right side of Eq. (1) or Eq. (2) is equal to zero, i.e., s -- 0. The second problem that we will solve with the use of the MMR to construct mathematicaI models of RNNs and identify the parameters of these models concerns optimization of the duration of the external stimulus (such as the duration of the series of pulses) t. The solution should be obtained with the use of a computer as part of any electrophysiological experiment if the goal is to standardize measurements of the parameters of mathematical models of RNNs and standardize the state of the object (such as an RNN) as a whole. Until now, the values of Z and period t have been optimized subjectively by experimenters on the basis of observed output values y. Since we are concerned here with precise quantitative criteria for evalusting all of the parameters of external stimuli and the mathematical model as a whole, we used the new formalized approach first proposed in [3]. As is known, optimization of t involves deterrninin~ the eigenvalues X of the linear approximation matrix A of model (1) or (2) and verifying the constancy of the order m of the minimum realization constructed by the program for the observed RNN responses to changing values of t of the applied external stimulus [3-5]. It is important to emphasize here that the amplitude of the optimum stimulus Z* was chosen as the mean obtained from the interval (Zmin ... Zmax) in measuring the linear behavior of a RNN by the above-described method, i.e., Z* -- (Zmax + Zmin)/2. The upper and lower boundaries of t were chosen on the basis of an analysis of the spectral characteristics of A calculated with each multiple change in the initial value of t [3, 5]. If t increases by the factor q, then pairs of eigenvalues L! ..... L m of the new matrix Aq satisfy the following equality to within the reiabelins (and with allowance for the 5% error!) ( ~ ) q = L 1(i = 1, . . . ,

m),

(4)

where I-,l are eigenvalues of the matrix A corresponding to the initial t, i.e., at q = 1. We should point out that in the experiments we took fractional values of q (but q > 1[), since it was not possible to greatly increase t. The values of q did not exceed 3-3.6, while the initial t was chosen from weU-known literature sources for the given type of afferent nerves. If q < 1, then the computer checked for the satisfaction of the following equality to within the relabeling (within the 5% error):

(~)llp = L1(i=l....,m )

(5)

with the condition that q -- lip (p = 2, 3 . . . . . k). In the automatic mode, the computer successively increased p and, at a certain value of k*, condition (5) ceased to be satisfied with a 5 % error in our tests. Then the computer determined the lowest value tmin = t/(k* - 1) at which the dynamic system (the RNN) remained in the standard state (standard relative to the initial state). We similarly chose the upper limit tmax = tq at which the eigenvalues of the linear approximation matrices A and Aq

sallied ~ . (4). Thus, the approach described above can be used to select standard values of the amplitude of the external stimulus Z* and the duration of this stimulus (series of electrical pulses) t* = (tmax + train)/2 at which the RNN being studied is in a certain steady (standard) state. If the RNN changes to another state - such as due to the injection of a drug or the application of a physical load - then both Z* and t* may change as well. In this case, we are dealing with a change in the standard state of the RNN or the organism as a whole. The above approach can be used to examine changes in the standard states of RNNs subjected to different external stimuli. At the same time, the standardization procedure can also be used in standardizing investigations of RNNs (for example). In fact, if identical intervals are obtained for Z and for t with the same error at different times and different points in space under identical conditions, then we can say that the functional states of different systems of animals or humans have been standardized. The use of such a clear mathematical (formal) criterion for evaluating the standard states of RNNs would undoubtedly resolve many of the conflicts that frequently arise when different researchers study supposedly identical objects. This conflict can often be seen at different conferences, when actual experimental results are compared.

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The problem of standar~iization is particularly important in medicine, when the health of a given individual is classified on the basis of purely external criteria (such as measurements of respiration and the cardiovascular system). It would be sufficient to introduce a stimulus into the RNN (an example would be stimulation by electrical pulses or the application of a physical load) and vary Z and t in order to confn'm earlier evaluations and even detect pathologies in a person who seems healthy based on outward appearance. On the whole, the procedure proposed here for analyzing the "standard" states of RNNs and the programs that have been developed should find application both in biological research and in studies of large populations (students, military inductees, etc.). Each specific use will obviously require some refinement and modification of the methodology to suit the given conditions. At the same time, it is clear that we have finally arrived at the point where we can envision the standardization of the states of complex dynamic systems - systems such as the respiratory neural networks of animals and humans.

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2. . .

5. 6.

A. Sachs, Statistical Evaluation [Russian translation], Statistika, Moscow (1976). V. M. Es'kov, Introduction to the Compartmental Theory of Respiratory Neural Networks [in Russian], Nauka, Moscow (1994). V. M. Es'kov and O. E. Filatova,Computer Identificationof Respiratory Neural Networks [in Russian], ONTI PNTs RAN, Pu.shcltino(1994). V. M. Es'kov and O. E. Filatova, lzmer. Tckh., No. I, 65 (1994). V. M. Es'kov, O. E. Filatova, and V. P. Ivashchcnko, Izmer. Tekh., No. 8, 65 (1994). V. M. Es'kov and O. E. Filatova, Neurophysiology (USSR), 2S, No. 6, 348 (1993).

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