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Control in Physiological System Simulation and Modelling
CONTROL IN PHYSIOLOGICAL SYSTEM SIMULATION AND MODELLING E. Biondi* and L. Divieti** • IsWuto di Elettrotecnica ed Elettronica del Politecnico di Milano, Italy ··Centro di Teoria dei Sistemi del C.N.R.
Abstract. This paper is concerned with the fundamental role of regulation and control of biological systems and the means of modelling them. Attention is drawn to the methods of modelling with the elementary components of a system. An example of clinical application of a model is also given. Keywords. Biocontrol; biomedical; models; neural nets; physiological models. Anatomo-physiological models present a complete correspondence between the structure and parameters of the model and the anatomophysiology of the modelled biological system. All the parameters appearing in the mathematical relationships should be specified in terms of anatomic or chemo-physical properties of the biological system.
MATHEMATICAL MODELLING OF BIOLOGICAL SYSTEMS Several are the aims of mathematical modelling in medicine: a) improvement of the system knowledge (both in terms of settling the acquired data and in terms of progranuning future experiments); b) improvement of therapeutic and diagnostic methodologies (possibly finding out new methodologies); c) improvement of diagnostic and therapeutic devices of prostheses and artificial organs (possibly designing new devices ).
Artificial models are mathematical descriptions of either artificial devices or theoretically predicted behaviours which perform tasks similar to those carried out by a specific biological system. A trivial example is given by considering the heart constituted by two pumps: the mathematical relationships governing the functioning of the two pumps provide an artificial model of the heart. More significant examples are given by sensory organs when one tries to describe their functioning referring to the properties of the devices designed for automatic . pattern recognition. It is important to note that also these models can be very useful in medicine and often they are the only available models on account of the lack of knowledge in anatomy and physiology.
This is not the place where to spend words to justify these claims; anyhow at the end a model application will be shown. Generally speaking the methodologies to be used for the determination of these models are the same used by system engineers, taking into account the experimental difficulties due, for example, to: a) system complexity (bearing in mind the 10 10 neurons which constitute , the nervous system, experiments performed on some thousands of neurons can be a non-significant "sample" of the system under consideration); b) difficulties in doing experiments on the "man system" while the results obtained on "animals" (which are the most ancient natural model of man) cannot always be transferred to the "man system"); c) difficulties due to the fact that we do not know the "designing criteria" followe d in man implementation (see foreward) .
Finally it is worth noting that also in the modelling of biological systems one or more of the following aspects must be taken into account: a) chemical and physical balance; b) energy balance; c) information processing. Taking into account that the code is often unknown, most times this leads to artificial models of all systems where the information processing plays an important role.
Many of the criteria followed in the classification of the mathematical models in other fields (often depending only on the mathematical "structure" of the model) are also used for the modelling of biological systems,while a typical criterion for the classification of these systems points out the correspondence of the model with the anatomy and physiology of the system. In what follows two different types of models must be kept in mind (Biondi, Schmid, 1972): a) anatomo-physiological models; b) artificial models. In between, there are several other types of models.
PHYSIOLOGICAL CONTROL SYSTEMS The main task is to study the whole "man system" because every division into subsystems may give unsatisfactory results; mor~ over, keeping in mind the difficulties we find in modelling sub-systems, a realistic approach is to study first the sub-systems and then, possibly, look for their mutual interactions.
2237
E. Biondi and L. Divieti
2238
In this paper we shall subdivide the man sys.tem into sub-systems. A functional cri terion will point out those sub-systems where amacr~ scopic quantity is controlled: from here on, these sub-systems will be referred to as systems. We shall not treat partially unknown single-cells control systems; cells will be taken into account as elementary components of the control systems. In biological control systems (as in any other control system) we can point out the following elements: a) transducers of the controlled variables, constituted by the receptors; b) networks for information processing and trans-. mission, constituted by elementary components named neurons; c) actuators which can be of two types: muscles or glands; d) set points: their interpretation in biological systems is quite doubtful, as it will be shown later.
rad et al., 1974; Holden, 1976). Tpere are different criteria to choose the proper model. In our opinion the following ones are the most significant (they are reported here for the neuron, because there are plenty of models, but they are in common also for the other components) : a) correspondence between model and real system; b) complexity of the single neuron model dealing with simulation difficulties. For this purpose one must keep in mind that often networks with some hundreds or thousands of neurons have to be simulated; c) complexity of the single neuron model in dealing with theoretical difficulties to evidence general problems of neuronal networks (when the simulation results are not sufficient, but a general analytical solution of the problem is required). Neuronal networks
Biological control systems can be classified following various criteria taking into account: a) the nature of the actuator (muscle or gland); b) the dependence or non-dependence of the control action on the subject's will. Some cases can be classified in between, such as the breathing control, which is usually non-voluntary, but can be modified by voluntary actions by the subject. Taking into account the limits of the authors' knowledge, only the skeletal muscle control systems will be considered; anyhow it is to be noted that the chemical and physical quantity control systems are very important ones (see for example: Iberall, Guytond, 1973). As an example, an inventory of the skeletal muscle control systems is given: a) limb position and movement control systems; b) spine posture and movement control systems; c) eyeball position and movement control systems; d) breathing control systems, and several others. MATHEMATICAL MODELS OF CONTROL SYSTEM COMPONENTS For every component, as already said in the previous section, we shall illustrat~ first the eleQentary component (generally a single cell) and then the set of cells constituting the control system components.
The leading problem has already been pointed out speaking about artificial models and about difficulties due to the fact that the code used for the signal transmission and processing is often unknown. In the study of neuronal networks of biological control systems it is useful to keep in mind the following general problem, which is often found in the applications: it is assigned a neuronal network of N neurons, with S input signals and A output signals. Very often the set of S input signals can be considered as provoked by the same cause and the set of A output signals can be regarded as a single effect. Then it is possible to determine en equivalent neuron with a single input (cause) and a single output signal which represents the whole effect of the neuronal network (Fig. 1). This equivalence can be imposed following different criteria as for instance: a) voltage variations in the surrounding space are equal both in the real case and in the simulation where the equivalent neuron takes the place of neuronal network (Biondi et al., 1975); b) the information processed and transmitted by the neuronal network to the successive neuronal nucleus is the same given at the output of the equivalent neuron. Transducers
Information Processing and Transmission Network Elementary component: the neuron The neuron is the structural unit of the nervous system. It can be seen as a dynamic system with multi-inputs (several thousands) and a single output; the neuron generates an output signal (spike) whenever the pool of these inputs (generally outputs of other neurons) is such that an internal variable (state variable) of the neuron reaches and gets over a threshold. As a first approximation, the spike can be considered an on-off signal which is fed to other neurons or actuators by its axon. The literature about neuron models is surely wider than that regarding any other biological system (see for example: Con-
Elementary component: the receptor In the human body there are several transducerE measuring a lot of variables which can be classified, for instance, taking into account the transduced variable: a) chemoreceptors; b) mechanoreceptors, etc. In the determination of their model, the following aspects of their functioning must be kept in mind: a) in many cases the static characteristic input signal (transduced variable)-output signal is sufficiently approximated by a logarithmic curve. This type of characteristic can explain many physiological phenomena. The output signal is often identified (though not always correctly) with the output frequency of the spikes of the first
Control in Physiological System Simulation and Modelling neuron connected with the receptor. b) \.]hen the receptor has a linear characteristic (or linearizing the characteristic for small changes about some operating conditions), one can obtain the transfer function which defines the output-input ratio in the s domain (see for example: Weed, 1973). c) Many receptors have efferent fibers coming from the Central Nervous System, but the action of these fibers is not completely known (this may happen also for afferent fibers); therefore we are compelled to make arbitrary hypotheses in the determination of the mathematical model. d)As already mentioned, the mean frequency of the output spikes is not the only significant variable. For some receptors the spike probability density function greatly varies in presence of a stimulus; probably this is due to a particular information code of the central nervous system. The fact that we can only make reasonable hypotheses on this code implies that the model might not correspond to the real system.
~
! 5
SYlAPSES
1AXONS
~
e:~3~~·~E~UA~o~.s~~§~~B ~
+-=-r.
I
---::----.t. EQUIVALENT •
Fig. 1
IEUAOM
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•
Equivalent neuron
Receptive field The measure of a variable does not depend upon a single receptor but upon a pool of receptors constituting a receptive field. In simulation and modelling of a receptive field two points must be remembered: a) receptor characteristics in the same receptive field are often different; for instance the functioning range is different and this may be due to the width of the range which is necessary for the measure of the variable; b) there is a reciprocal action among several receptors whose output depends on the stimuli coming from many other receptors of the same field: a typical example is provided by the retina of the eye (see for example: Lettvin et al., 1959). The studies carried out on animal retina are in some cases so developped that the results can allow the determination of anatomo-physiological models. For other receptive fields, only artificial models are available.
2239
!'!us c les as Actuators Elementary component: the sarcomere The skeletal muscle is made up of bundles of fibers, each fiber being a single cell. Skeletal muscle fibers may be arranged in a parallel or pennate fashion or in various combination of these arrangements. Every fiber contains a bundle of myofibrils of the same length of the fiber. Myofibrils are made of a serially repeated unit, the sarcomere.Typical relations of the whole skeletal muscle are muscle tension as function of length and maximum velocity of shortening as function of load or force. The same characteristic relations for the sarcomere can be easily obtained from the ones of a skeletal muscle with parallel fibers (Muller, Pedotti, 1972). The same kind of relations for pennate or other types of muscles can be calculated, starting from those of the sarcomere by a proper integration on the whole muscle. Thus the typical characteristics of various types of muscles such as the pennate, whose experimental characteristics are only partially known, can be obtained. It was confirmed that pennate muscles are designed for small powerful movements whereas muscles with parallel fibers can move over small distances more rapidly. Motor unit A motor unit consists of a motor-neuron, its axon and all the muscle fibers innervated by the axon. The motor unit is the smallest functional unit involved in normal neuromuscular activity. The muscle tension as a whole is the sum of all motor unit contributions; it depends upon the number of firing motor-neurons and their firing patterns as well as upon length and shorteninf, velocity. Most of the current models, with the firing patterns as input and muscle tension as output, can be considered anatomo-physiological models. In fact the muscle dynamics is obtained by motor unit dynamics; this in turn is obtained adding up the contributions of the single fibers generally provided by their mechanical models consisting of a number of active contractile components properly connected with elastic and viscous components (Mountcastle V.B.,1968; Jewell and IvUkie, 1958; Gordon and Huxley, 1964). Set Points It is quite difficult to localize particular parts of the body constituting the set points; moreover one can wonder whether these organs are really necessary or whether we look for them, wishing to find a strict correspondence between our artificial models and the biological system. We could also think, in terms of artificials models, that the automatic control systems optimize performance indices. The problem becomes more difficult, taking into account that different control systems have different performance criteria. Therefore we should think of a mul tigoal and mul tilevel control system. Though we think that this is a fundamental step for the complete knowledge of the biological control system behaviour, we decided better not to dwell on
E. Biondi and L. Divieti
2240
this item, on account of the lack of physiological knowledge. Plants The modelling of the controlled plants does not lead to a unifying approach because the plants are very different from one another. It should be noted that, if we do not take into account the problems of information coding, themo delling of the plants constitutes very often the easiest part or, at least, the less complicated part of the work.
internal parameters of the system (gain, time constant, etc.). In this way a quantitative measure of the illness can be also given and in some cases the lesion can be localized (Mira et al., 1975a and 1975b).
• nor",.1 subjtct.
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AN EXAMPLE OF CLINICAL APPLICATION The IOOdel approach has been used for the identification of some parameters of the vestibular system.
'UT' lr .'Dtl
t
''''ttfl(1I
DIGitAL COMPiHl
Fig. 2
Rotatory test set up for vestibulooculomotor reflex
As sketched in Fig. 2, rotatory tests were adopted. On account of the vestibulo-ocular reflex, these tests produce eyeball movements, which can be detected by means of proper electrodes and thus an electronystagmogramm is obtained. The vestibular system can be schematized as follows: 1) a mechanical part (semici~cular canals) for each side; 2) a peripheral nervous part for each side; 3) a central nervous part. The correspondence of the model to the anatomy and physiology of the system were tested with a continous improvement, before its clinical use. Among the various results obtained concerning this point (Mira et al., 1975a and 1975b), the one we report in Fig. 3 has the advantage that it needs only few comments. Two parameters of the IOOdel system are reported along the axes: the parameter values are shown for several normal and pathological subjects, divided into these two groups according to traditional tests: it is clear that the model reproduces the physiological system in such a way as to distinguish normal subjects from pathological ones. In order to point out the potential possibilities of the model in clinical applications, we give two examples: 1) traditional medical test of the nystagmogramm is carried out qualitatively, determining only some IOOrphological parameters without any precise correlation with the system parameters. The use of the model allows a precise determination of some
o
o
o
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Fig. 3
h
subjects
Comparison between normal and pathological subjects in model parameter space
2) the model allows to test hypotheses on the system behaviour in pathological cases. For example, in the case of patients with unilateral Meniere disease (mechanical part diseas e), changes of the parameters of the involved canal surely occur. Instead, clockwise and counterlockwise post rotational nystagmogramms show small differences between the two parameter sets obtained by these tests; these parameters are much reduced in comparison of normal ones. Since the parameters of the intact semicircular canal cannot be varied, its apparent decrease can only be explained as the effect of a process of a central compensation taking place through a reduction of the intact side function (Stefanelli et al., 1978) • REFERENCES Biondi, E., and R. Schmid (1972). Mathematical models and prostheses for sense organs. In R.R. Mohler and A. Ruberti (Ed.), Theory and Applications of Variable Structure Systems, Academic Press, New York and London, pp. 183-211. Biondi, E., G.F. Dacquino, and F. Grandori (1975). Compound action potential and single fiber activity generation. An equivalent neuron approach, Int. J. Bio-Medical Comp •.~ ~, pp. 157-167. Conrad M., W. Guttinger, and M. Dal Cin (Ed.), (1974). Physics and Mathematics on the Nervous System, Springer~Verlag, Berlin, Heidelberg and New York. Gordon A.M., Huxley (1964). The leneth-tension diagram of single vertebrate striated muscle fibers, J. Physiol., 171. Holden A.V. (1976). Models of the Stochastic Activity of Neuronen, Springer Verlag, Berlin Heidelberg and New York. Iberall A.S., and A.C. Guytond (Ed.), (1973), Regulation and Control in Physiological Systems, I.F.A.C. Jewell B.R., and D.R. Wilkie (1958). An analysis of the mechanical components in frog's striated muscle, J. Physiol.143.
Control in Physiological System Simulation and Modelling Lettvin J.Y., H.R. Maturana, W.H. Pitts, and W.S. I1cCulloch (1959). Two Remarks on the Visual System of the Frog. In A. WaIter and A. Rosenblith (Ed.), Sensory Communications, M.I.T. Press, Cambridge, pp. 757 776. Mira E., R. Schmid, and M. Stefanelli (1975 a). Application clinique d'un modele mat hematique du systeme vestibulo-oculomoteur. Arta oto-rhinolaryngol belg, ~ pp. 29. Mira E., R. Schmid, and M. Stefanelli (1975b). Clinical analysis of vestibularly induced eye movements based on a mathematical model of the vestibulo-ocular reflex. In G. Lennerstrand and P. Bach-y-Rita (Ed.) Basic mechanisms of ocular motility and their clinical implications, Pergamon
2241
Press, Oxford, pp. 501-504. MountcaStle V.B., (1968). Medical physiology Mosby. Muller A., and A. Pedotti (1972). Untersuchungen mit Hilfe der EDV-Methoden der strukturellen Muskeleigenschaften, Congress Medizin-Technik, Stuttgart. --Stefanelli M., E. Mira, R. Schmid, and R. Lombardi (1978). Quantification of vestibular compensation in unilateral Meniere disease, Acta Otolyng. (Stochk),
weedift.~:e~i973), Physiological Sensor System. In A.S. Iberall and A.C. Guy ton (Ed.), Regulation and Control in PhysioloCical Systems, I.F.A.C., pp. 165 - 178.
Abstract. This paper is concerned with the fundamental role of regulation and control of biological systems and the means of modelling them. Attention is drawn to the methods of modelling with the elementary components of a system. An example of clinical application of a model is also given. Keywords. Biocontrol; biomedical; models; neural nets; physiological models. Anatomo-physiological models present a complete correspondence between the structure and parameters of the model and the anatomophysiology of the modelled biological system. All the parameters appearing in the mathematical relationships should be specified in terms of anatomic or chemo-physical properties of the biological system.
MATHEMATICAL MODELLING OF BIOLOGICAL SYSTEMS Several are the aims of mathematical modelling in medicine: a) improvement of the system knowledge (both in terms of settling the acquired data and in terms of progranuning future experiments); b) improvement of therapeutic and diagnostic methodologies (possibly finding out new methodologies); c) improvement of diagnostic and therapeutic devices of prostheses and artificial organs (possibly designing new devices ).
Artificial models are mathematical descriptions of either artificial devices or theoretically predicted behaviours which perform tasks similar to those carried out by a specific biological system. A trivial example is given by considering the heart constituted by two pumps: the mathematical relationships governing the functioning of the two pumps provide an artificial model of the heart. More significant examples are given by sensory organs when one tries to describe their functioning referring to the properties of the devices designed for automatic . pattern recognition. It is important to note that also these models can be very useful in medicine and often they are the only available models on account of the lack of knowledge in anatomy and physiology.
This is not the place where to spend words to justify these claims; anyhow at the end a model application will be shown. Generally speaking the methodologies to be used for the determination of these models are the same used by system engineers, taking into account the experimental difficulties due, for example, to: a) system complexity (bearing in mind the 10 10 neurons which constitute , the nervous system, experiments performed on some thousands of neurons can be a non-significant "sample" of the system under consideration); b) difficulties in doing experiments on the "man system" while the results obtained on "animals" (which are the most ancient natural model of man) cannot always be transferred to the "man system"); c) difficulties due to the fact that we do not know the "designing criteria" followe d in man implementation (see foreward) .
Finally it is worth noting that also in the modelling of biological systems one or more of the following aspects must be taken into account: a) chemical and physical balance; b) energy balance; c) information processing. Taking into account that the code is often unknown, most times this leads to artificial models of all systems where the information processing plays an important role.
Many of the criteria followed in the classification of the mathematical models in other fields (often depending only on the mathematical "structure" of the model) are also used for the modelling of biological systems,while a typical criterion for the classification of these systems points out the correspondence of the model with the anatomy and physiology of the system. In what follows two different types of models must be kept in mind (Biondi, Schmid, 1972): a) anatomo-physiological models; b) artificial models. In between, there are several other types of models.
PHYSIOLOGICAL CONTROL SYSTEMS The main task is to study the whole "man system" because every division into subsystems may give unsatisfactory results; mor~ over, keeping in mind the difficulties we find in modelling sub-systems, a realistic approach is to study first the sub-systems and then, possibly, look for their mutual interactions.
2237
E. Biondi and L. Divieti
2238
In this paper we shall subdivide the man sys.tem into sub-systems. A functional cri terion will point out those sub-systems where amacr~ scopic quantity is controlled: from here on, these sub-systems will be referred to as systems. We shall not treat partially unknown single-cells control systems; cells will be taken into account as elementary components of the control systems. In biological control systems (as in any other control system) we can point out the following elements: a) transducers of the controlled variables, constituted by the receptors; b) networks for information processing and trans-. mission, constituted by elementary components named neurons; c) actuators which can be of two types: muscles or glands; d) set points: their interpretation in biological systems is quite doubtful, as it will be shown later.
rad et al., 1974; Holden, 1976). Tpere are different criteria to choose the proper model. In our opinion the following ones are the most significant (they are reported here for the neuron, because there are plenty of models, but they are in common also for the other components) : a) correspondence between model and real system; b) complexity of the single neuron model dealing with simulation difficulties. For this purpose one must keep in mind that often networks with some hundreds or thousands of neurons have to be simulated; c) complexity of the single neuron model in dealing with theoretical difficulties to evidence general problems of neuronal networks (when the simulation results are not sufficient, but a general analytical solution of the problem is required). Neuronal networks
Biological control systems can be classified following various criteria taking into account: a) the nature of the actuator (muscle or gland); b) the dependence or non-dependence of the control action on the subject's will. Some cases can be classified in between, such as the breathing control, which is usually non-voluntary, but can be modified by voluntary actions by the subject. Taking into account the limits of the authors' knowledge, only the skeletal muscle control systems will be considered; anyhow it is to be noted that the chemical and physical quantity control systems are very important ones (see for example: Iberall, Guytond, 1973). As an example, an inventory of the skeletal muscle control systems is given: a) limb position and movement control systems; b) spine posture and movement control systems; c) eyeball position and movement control systems; d) breathing control systems, and several others. MATHEMATICAL MODELS OF CONTROL SYSTEM COMPONENTS For every component, as already said in the previous section, we shall illustrat~ first the eleQentary component (generally a single cell) and then the set of cells constituting the control system components.
The leading problem has already been pointed out speaking about artificial models and about difficulties due to the fact that the code used for the signal transmission and processing is often unknown. In the study of neuronal networks of biological control systems it is useful to keep in mind the following general problem, which is often found in the applications: it is assigned a neuronal network of N neurons, with S input signals and A output signals. Very often the set of S input signals can be considered as provoked by the same cause and the set of A output signals can be regarded as a single effect. Then it is possible to determine en equivalent neuron with a single input (cause) and a single output signal which represents the whole effect of the neuronal network (Fig. 1). This equivalence can be imposed following different criteria as for instance: a) voltage variations in the surrounding space are equal both in the real case and in the simulation where the equivalent neuron takes the place of neuronal network (Biondi et al., 1975); b) the information processed and transmitted by the neuronal network to the successive neuronal nucleus is the same given at the output of the equivalent neuron. Transducers
Information Processing and Transmission Network Elementary component: the neuron The neuron is the structural unit of the nervous system. It can be seen as a dynamic system with multi-inputs (several thousands) and a single output; the neuron generates an output signal (spike) whenever the pool of these inputs (generally outputs of other neurons) is such that an internal variable (state variable) of the neuron reaches and gets over a threshold. As a first approximation, the spike can be considered an on-off signal which is fed to other neurons or actuators by its axon. The literature about neuron models is surely wider than that regarding any other biological system (see for example: Con-
Elementary component: the receptor In the human body there are several transducerE measuring a lot of variables which can be classified, for instance, taking into account the transduced variable: a) chemoreceptors; b) mechanoreceptors, etc. In the determination of their model, the following aspects of their functioning must be kept in mind: a) in many cases the static characteristic input signal (transduced variable)-output signal is sufficiently approximated by a logarithmic curve. This type of characteristic can explain many physiological phenomena. The output signal is often identified (though not always correctly) with the output frequency of the spikes of the first
Control in Physiological System Simulation and Modelling neuron connected with the receptor. b) \.]hen the receptor has a linear characteristic (or linearizing the characteristic for small changes about some operating conditions), one can obtain the transfer function which defines the output-input ratio in the s domain (see for example: Weed, 1973). c) Many receptors have efferent fibers coming from the Central Nervous System, but the action of these fibers is not completely known (this may happen also for afferent fibers); therefore we are compelled to make arbitrary hypotheses in the determination of the mathematical model. d)As already mentioned, the mean frequency of the output spikes is not the only significant variable. For some receptors the spike probability density function greatly varies in presence of a stimulus; probably this is due to a particular information code of the central nervous system. The fact that we can only make reasonable hypotheses on this code implies that the model might not correspond to the real system.
~
! 5
SYlAPSES
1AXONS
~
e:~3~~·~E~UA~o~.s~~§~~B ~
+-=-r.
I
---::----.t. EQUIVALENT •
Fig. 1
IEUAOM
L I 0
•
Equivalent neuron
Receptive field The measure of a variable does not depend upon a single receptor but upon a pool of receptors constituting a receptive field. In simulation and modelling of a receptive field two points must be remembered: a) receptor characteristics in the same receptive field are often different; for instance the functioning range is different and this may be due to the width of the range which is necessary for the measure of the variable; b) there is a reciprocal action among several receptors whose output depends on the stimuli coming from many other receptors of the same field: a typical example is provided by the retina of the eye (see for example: Lettvin et al., 1959). The studies carried out on animal retina are in some cases so developped that the results can allow the determination of anatomo-physiological models. For other receptive fields, only artificial models are available.
2239
!'!us c les as Actuators Elementary component: the sarcomere The skeletal muscle is made up of bundles of fibers, each fiber being a single cell. Skeletal muscle fibers may be arranged in a parallel or pennate fashion or in various combination of these arrangements. Every fiber contains a bundle of myofibrils of the same length of the fiber. Myofibrils are made of a serially repeated unit, the sarcomere.Typical relations of the whole skeletal muscle are muscle tension as function of length and maximum velocity of shortening as function of load or force. The same characteristic relations for the sarcomere can be easily obtained from the ones of a skeletal muscle with parallel fibers (Muller, Pedotti, 1972). The same kind of relations for pennate or other types of muscles can be calculated, starting from those of the sarcomere by a proper integration on the whole muscle. Thus the typical characteristics of various types of muscles such as the pennate, whose experimental characteristics are only partially known, can be obtained. It was confirmed that pennate muscles are designed for small powerful movements whereas muscles with parallel fibers can move over small distances more rapidly. Motor unit A motor unit consists of a motor-neuron, its axon and all the muscle fibers innervated by the axon. The motor unit is the smallest functional unit involved in normal neuromuscular activity. The muscle tension as a whole is the sum of all motor unit contributions; it depends upon the number of firing motor-neurons and their firing patterns as well as upon length and shorteninf, velocity. Most of the current models, with the firing patterns as input and muscle tension as output, can be considered anatomo-physiological models. In fact the muscle dynamics is obtained by motor unit dynamics; this in turn is obtained adding up the contributions of the single fibers generally provided by their mechanical models consisting of a number of active contractile components properly connected with elastic and viscous components (Mountcastle V.B.,1968; Jewell and IvUkie, 1958; Gordon and Huxley, 1964). Set Points It is quite difficult to localize particular parts of the body constituting the set points; moreover one can wonder whether these organs are really necessary or whether we look for them, wishing to find a strict correspondence between our artificial models and the biological system. We could also think, in terms of artificials models, that the automatic control systems optimize performance indices. The problem becomes more difficult, taking into account that different control systems have different performance criteria. Therefore we should think of a mul tigoal and mul tilevel control system. Though we think that this is a fundamental step for the complete knowledge of the biological control system behaviour, we decided better not to dwell on
E. Biondi and L. Divieti
2240
this item, on account of the lack of physiological knowledge. Plants The modelling of the controlled plants does not lead to a unifying approach because the plants are very different from one another. It should be noted that, if we do not take into account the problems of information coding, themo delling of the plants constitutes very often the easiest part or, at least, the less complicated part of the work.
internal parameters of the system (gain, time constant, etc.). In this way a quantitative measure of the illness can be also given and in some cases the lesion can be localized (Mira et al., 1975a and 1975b).
• nor",.1 subjtct.
o pdtho(ogicdl -.;;
0
.,
4
.."
2
.."
8
~
AN EXAMPLE OF CLINICAL APPLICATION The IOOdel approach has been used for the identification of some parameters of the vestibular system.
'UT' lr .'Dtl
t
''''ttfl(1I
DIGitAL COMPiHl
Fig. 2
Rotatory test set up for vestibulooculomotor reflex
As sketched in Fig. 2, rotatory tests were adopted. On account of the vestibulo-ocular reflex, these tests produce eyeball movements, which can be detected by means of proper electrodes and thus an electronystagmogramm is obtained. The vestibular system can be schematized as follows: 1) a mechanical part (semici~cular canals) for each side; 2) a peripheral nervous part for each side; 3) a central nervous part. The correspondence of the model to the anatomy and physiology of the system were tested with a continous improvement, before its clinical use. Among the various results obtained concerning this point (Mira et al., 1975a and 1975b), the one we report in Fig. 3 has the advantage that it needs only few comments. Two parameters of the IOOdel system are reported along the axes: the parameter values are shown for several normal and pathological subjects, divided into these two groups according to traditional tests: it is clear that the model reproduces the physiological system in such a way as to distinguish normal subjects from pathological ones. In order to point out the potential possibilities of the model in clinical applications, we give two examples: 1) traditional medical test of the nystagmogramm is carried out qualitatively, determining only some IOOrphological parameters without any precise correlation with the system parameters. The use of the model allows a precise determination of some
o
o
o
0 0
ReFrdctor~ period [>eel
Fig. 3
h
subjects
Comparison between normal and pathological subjects in model parameter space
2) the model allows to test hypotheses on the system behaviour in pathological cases. For example, in the case of patients with unilateral Meniere disease (mechanical part diseas e), changes of the parameters of the involved canal surely occur. Instead, clockwise and counterlockwise post rotational nystagmogramms show small differences between the two parameter sets obtained by these tests; these parameters are much reduced in comparison of normal ones. Since the parameters of the intact semicircular canal cannot be varied, its apparent decrease can only be explained as the effect of a process of a central compensation taking place through a reduction of the intact side function (Stefanelli et al., 1978) • REFERENCES Biondi, E., and R. Schmid (1972). Mathematical models and prostheses for sense organs. In R.R. Mohler and A. Ruberti (Ed.), Theory and Applications of Variable Structure Systems, Academic Press, New York and London, pp. 183-211. Biondi, E., G.F. Dacquino, and F. Grandori (1975). Compound action potential and single fiber activity generation. An equivalent neuron approach, Int. J. Bio-Medical Comp •.~ ~, pp. 157-167. Conrad M., W. Guttinger, and M. Dal Cin (Ed.), (1974). Physics and Mathematics on the Nervous System, Springer~Verlag, Berlin, Heidelberg and New York. Gordon A.M., Huxley (1964). The leneth-tension diagram of single vertebrate striated muscle fibers, J. Physiol., 171. Holden A.V. (1976). Models of the Stochastic Activity of Neuronen, Springer Verlag, Berlin Heidelberg and New York. Iberall A.S., and A.C. Guytond (Ed.), (1973), Regulation and Control in Physiological Systems, I.F.A.C. Jewell B.R., and D.R. Wilkie (1958). An analysis of the mechanical components in frog's striated muscle, J. Physiol.143.
Control in Physiological System Simulation and Modelling Lettvin J.Y., H.R. Maturana, W.H. Pitts, and W.S. I1cCulloch (1959). Two Remarks on the Visual System of the Frog. In A. WaIter and A. Rosenblith (Ed.), Sensory Communications, M.I.T. Press, Cambridge, pp. 757 776. Mira E., R. Schmid, and M. Stefanelli (1975 a). Application clinique d'un modele mat hematique du systeme vestibulo-oculomoteur. Arta oto-rhinolaryngol belg, ~ pp. 29. Mira E., R. Schmid, and M. Stefanelli (1975b). Clinical analysis of vestibularly induced eye movements based on a mathematical model of the vestibulo-ocular reflex. In G. Lennerstrand and P. Bach-y-Rita (Ed.) Basic mechanisms of ocular motility and their clinical implications, Pergamon
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Press, Oxford, pp. 501-504. MountcaStle V.B., (1968). Medical physiology Mosby. Muller A., and A. Pedotti (1972). Untersuchungen mit Hilfe der EDV-Methoden der strukturellen Muskeleigenschaften, Congress Medizin-Technik, Stuttgart. --Stefanelli M., E. Mira, R. Schmid, and R. Lombardi (1978). Quantification of vestibular compensation in unilateral Meniere disease, Acta Otolyng. (Stochk),
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