A mathematical model of the gate control theory of pain

The first test which any theory of pain must pass is that it must be able to explain the phenomena observed in acute pain in humans. This criterion is used to test the major theory of pain at present, the gate control theory of Melzack & Wall (1965, 1982). The theory is explicit enough to be cast in mathematical terms, and the mathematical model is shown to explain the observations considered. It also points up a common misconception on the consequences of the theory, and thus demolishes an argument which has been used against it. A hypothesis of the origin of rhythmic pain is then made, and consequent testable predictions given. This is the first time that the gate control theory has been used to explain any quality of pain. It has important consequences for the treatment of such pain. Finally, the applicability of the gate control theory as an explanation for chronic pain is discussed.     

A mathematical theory of size distributions in tissue culture

A model for growth of a tissue culture consisting of cell clumps is given. A set of equations for following the size distribution of clumps is used to determine total biomass accumulation. Existence and uniqueness of a solution to the equations is proved, and estimates of the biomass growth is given in a number of situations.     

A new mathematical continuum theory of axoplasmic transport.

The hypothesis is advanced that successive waves of apparent contraction-relaxation (due perhaps to filament sliding) propagate along the filamentous proteins that pierce axoplasm oriented parallel to the axon length. A mathematical continuum model is developed to characterize the flow that could result in the viscous fluid bathing the moving filamentous proteins. This flow is complicated and oscillatory in time and space, but, on the average, it yields a bi-directional drift of fluid. It would transport various substances riding in the fluid, soluble and particulate, at various distinct speeds both up and down the axon. The model can thus account qualitatively for many observed features of axoplasmic transport.     

Studies in the mathematical theory of excitation

The general linear two-factor nerve-excitation theory of the type of Rashevsky and Hill is discussed and normal forms are derived. It is shown that in some cases these equations are not reducible to the Rashevsky form. Most notable is the case in which the solutions are damped periodic functions. It is shown that in this case one or more—in some cases infinitely many—discharges are predictable, following the application of a constant stimulusS. The number of discharges increases withS, but the frequency is a constant, characteristic of the fiber and independent ofS.     

A theory of drug tolerance and dependence II: the mathematical model

The preceding paper presented a model of drug tolerance and dependence. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The present paper discusses the mathematical model in terms of its design. The model is a nonlinear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus-the drug intake-and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes.     

Biological adaptation and the mathematical theory of information

Adaptation is viewed as a tendency maximizing the Shannon entropy of an ecosystem, where the ecosystem is considered as two interacting subsystems, namely, the biota and its environment. We derive theadapted structures starting from three fundamental hypotheses and we apply this result to an ecological topic: the cryptic and aposematic behaviour.     

A note on the mathematical theory of vibrations of walls of blood vessels

Periodic vibrations of the walls of a distensible elastic tube through which a fluid is flowing are studied by the method used by Lord Rayleigh in his theory of vibrations of jets. The results are found to conform with those obtaine previously by a more general but approximate method.     

On a mathematical theory of embolism

A theory of embolism based on an optimization model of blood flow is proposed and used to explain the topographic distribution of emboli in arterial trees.     

A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue

It is proposed that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional. On this view, the third (radial) dimension of cortical and thalamic structures is associated with a redundancy of circuits and functions so that reliable signal processing obtains in the presence of noisy or ambiguous stimuli.A mathematical model of simple cortical and thalamic nervous tissue is consequently developed, comprising two types of neurons (excitatory and inhibitory), homogeneously distributed in planar sheets, and interacting by way of recurrent lateral connexions. Following a discussion of certain anatomical and physiological restrictions on such interactions, numerical solutions of the relevant non-linear integro-differential equations are obtained. The results fall conveniently into three categories, each of which is postulated to correspond to a distinct type of tissue: sensory neo-cortex, archior prefrontal cortex, and thalamus.The different categories of solution are referred to as dynamical modes. The mode appropriate to thalamus involves a variety of non-linear oscillatory phenomena. That appropriate to archior prefrontal cortex is defined by the existence of spatially inhomogeneous stable steady states which retain contour information about prior stimuli. Finally, the mode appropriate to sensory neo-cortex involves active transient responses. It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.List of Symbols (t) Post-synaptic membrane potential (psp) - Maximum amplitude of psp - t Time - The neuronal membrane time constant - Threshold value of membrane potential - r Absolute refractory period - Synaptic operating delay - v Velocity of propagation of action potentil - x Cartesian coordinate - _jj (x) The probability that cells of class j are connected with cells of class j a distance x away - b _jj The mean synaptic weight of synapses of the jj-th class at x - _jj The space constant for connectivity - _e Surface density of excitatory neurons in a one-dimensional homogeneous and isotropic tissue - _i Surface density of inhibitory neurons in a one-dimensional homogeneous and isotropic tissue - E(x, t) Excitatory Activity, proportion of excitatory cells becoming active per unit time at the instant t, at the point x - I(x, t) Inhibitory Activity, proportion of inhibitory cells becoming active per unit time at the instant t, at the point x - x A small segment of tissue - t A small interval of time - P(x, t) Afferent excitation or inhibition to excitatory neurons - Q(x, t) Afferent excitation or inhibition to inhibitory neurons - N _e (x, t) Mean integrated excitation generated within excitatory neurons at x - N _i (x, t) Mean integrated excitation generated within inhibitory neurons at x - _e [N _e ] Expected proportion of excitatory neurons receiving at least threshold excitation per unit time, as a function of N _e - _i [N _i ] Expected proportion of inhibitory neurons receiving at least threshold excitation per unit time, as a function of N _i - G( _e ) Distribution function of excitatory neuronal thresholds - G( ₁ ) Distribution function of inhibitory neuronal thresholds - ₁ A fixed value of neuronal threshold - h(N _e ; ₁) Proportion per unit time of excitatory neurons at x reaching ₁ with a mean excitation N _e - 1[ ] Heaviside's step-function - R _e (x, t) Number of excitatory neurons which are sensitive at the instant t - R _i (x, t) Number of inhibitory neurons which are sensitive at the instant t - R _e Refractory period of excitatory neurons - r _i Refractory period of inhibitory neurons - E(x, t) Time coarse-grained excitatory activity - I(x, t) Time coarse-grained inhibitory activity - Spatial convolution - Threshold of a neuronal aggregate - v Sensitivity coefficient of response of a neuronal aggregate - E(t) Time coarse-grained spatially localised excitatory activity - I(t)> Time coarse-grained spatially localised inhibitory activity - L ₁,L ₂,L,Q See § 2.2.1, § 2.2.7, § 3.1 - Velocity with which retinal images are moved apart - Stimulus width - E _o, I _o Spatially homogeneous steady states of neuronal activity - k _e ,k _ij S _e S _ij See § 5.1     

A theory of contraction and a mathematical model of striated muscle

A theory of contraction and an associated model of striated muscle are presented, based on the assumption that chemical energy is being converted into electrical energy which, in turn, is being converted into mechanical energy and heat.The model, set up for the frog sartorius muscle, is able to predict the “rowing” motion of the cross-bridges, the force-velocity relation, the tension-length curve, the isometric force, all energy rates (heat and work rates), the metabolic rates and all known features of the stretched, stimulated muscle (no ATP-splitting, stretching tension higher than isometric tension, etc.). It also offers an alternative explanation for Hill's thermoelastic effect. The significance of Hill's force-velocity equation in the context of this theory is also discussed in detail.     

A suggestion for a new approach to the mathematical theory of imitative behavior

The theory of imitative behavior as developed hitherto by the author was based on the assumption that each individual has a natural preference for one of the two mutually exclusive behaviors. The endogenous fluctuations in the central nervous system then result in the individual’s exhibiting the two behaviors alternately with a relative frequency determined by the natural preference. Imitation shifts the natural preference towards one or the other of the two mutually exclusive behaviors. In the present approach it is suggested that the relative frequency of the two mutually exclusive behaviors exhibited alternately is determined by maximizing the “satisfaction function” of the individual, that is by hedonistic factors rather than by purely random fluctuations. Corresponding equations are developed. It is shown that in certain cases, even when the imitation effect is absent, a sort of “pseudoimitation” may occur. Another situation leads, in the case of two individuals only, to a complete “division of labor” between them, with respect to the two behaviors. Each one exhibits only one behavior. After that imitation is introduced explicitly by assuming that imitation by one individual or another increases the satisfaction function of the imitating individual. Results thus obtained show similarities to the results of the old theory.     

A note on a possible mathematical approach to the theory of individual freedom

As suggested in previous publications, freedom may be defined quantitatively as a restriction upon the choice of a number of activities. If the choice is determined by maximizing the satisfaction function, it is suggested that freedom may be defined in terms of the satisfaction function. If an individual is isolated and no physical restrictions limit his choice of activities, he is free to choose any activity in an amount which maximizes his satisfaction. This isolated state may be considered therefore as that of maximum freedom. If the individual interacts with another, he will choose different amounts of his object of satisfaction depending on whether he behaves egoistically or altruistically. But in either case the value chosen will not maximize his satisfaction function considered alone. A simple analytical expression is suggested as a measure of freedom in this case, and some problems which arise from this suggestion are mentioned.