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It is found that for a simple circuit of neurons, if this contains an odd number of inhibitory fibers, or none at all, or if the product of the activity parameters is less than unity, then the stimulus pattern always determines uniquely the steady-state activity. For circuits not of one of these types, it is possible to classify exclusively and exhaustively all possible activity patterns into three types, here called “odd”, “even”, and “mixed”. For any pattern of odd type and any pattern of even type there always exists a stimulus pattern consistent with both, but in no other way can such an association of activity patterns be made.

By introducing a parameter which vanishes when the accommodation is normal the modifications in theoretical predictions when accommodation is non-normal are discussed, particularly in the case of stimulation by alternating currents. It is found that non-normality always has the effect of lowering the optimal amplitude of the alternating current, measured in units of the observed rheobase.

As an essay towards the determination of the effect of structural relations among nerve fibers upon the character of their activity, preliminary consideration is given to the steady-state activity of some simple neural structures. It is assumed as a first approximation that while acted upon by a constant stimulus, each fiber reaches a steady-state activity whose intensity is a linear function of the applied stimulus. It is shown by way of example that for a simple two-fiber circuit of inhibitory neurons knowledge of the stimuli applied to the separate fibers does not necessarily suffice to determine uniquely the activity that will result. On the other hand, there are deduced certain restrictions on the possible types of activity that may be consistent with a given pattern of applied stimulation.

Rashevsky's equations for describing the joint variation of cell shape and concentration of a metabolite are discussed. Conditions for the existence of non-spherical equilibria and the location of these are obtained and involve only two parametersa andA. Sufficient (but not necessary) conditions for the stability of these equilibria can also be expressed in terms of these parameters alone. Necessary conditions involve in some cases a third parameterB. Quasi-periodic fluctuations about a stable nonspherical equilibrium may occur, but only in caseB lies on a certain finite range which can be defined in terms ofa andA.

With the view of suggesting a possible mechanism, it is shown that sporulation may be synchronous, as with the malarial parasite, given the following condition: A metabolite which is essential to the growth of the parasite is periodically released from storage organs and dissipated from the blood stream of the host.

Continuing the investigation previously introduced, it is shown here that when the product of the activity parameters of the circuit is not exceeded by unity (algebraically) a steady state is not possible in which all fibers of the circuit are active, whereas when this product is exceeded by unity, any stimulus pattern which is consistent with such a state of complete activity is inconsistent with any state of partial activity of the circuit.

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 quantitative formulation of Ogle's induced size effect is carried out on the assumption that vertical equality of the retinal images acts as a cue for localing the subjective median plane as opposed to the horizontal retinal dispartities which serve for localizing the subjective frontal plane; that this cue operates, independently of such cues as are provided by ocular versions, and predominates when the versions are not great. The equation which embodies the theoretical predictions involves no parameters not directly measurable.

Conditions under which either of two distinct activity patterns may arise from the same stimulus pattern are deduced for the case of a net-work which consists ofN simple circuits all jointed at a common synapse. If the product of the activity parameters of all the fibers in any circuit is called the activity parameter of the circuit, or, more briefly, the circuit parameter, then the condition for the existence of such mutually consistent activity patterns is that there be a sum of circuit paramaters which is not less than unity.

O. Fischer's dynamical equations for unbranched systems in two dimensions are extended to the branched systems exhibited by a quadruped.

A theoretical central mechanism for the discrimination of intensities as previously developed, together with plausible assumptions concerning the receptors, are employed for the derivation of the discriminable difference between lifted weights as a function of the smaller of these weights. The function so derived depends upon three parameters, one parameter being the weight of the supporting member. Some empirical data are compared with the theoretical predictions, and a few remarks are added to describe the physiological significance of the parameters.

The isometric length-tension diagram for individual fibers and for whole muscle is considered, and it is proposed that the tensionp may be represented for any muscle whose fibers are parallel and not in series, in the form $$p = f\left( x \right) + \beta \phi \left( {\alpha ,l,x} \right),$$ where the form of ϕ is known and the same for all muscles, the parameters α and l are experimentally measurable,x represents the length, β the degree of activity of the muscle, and the form off(x) varies from muscle to muscle.

The course of hydrolysis of β-glycerophosphate catalyzed by a group of different enzyme extracts, both with and without the addition of Mg, with and without preincubation of the enzyme, has been studied and the results discussed on the basis of a mathematical analysis. In all the extracts, it appears that two distinct and independently acting constituent enzymes—or perhaps “principles” of the same enzyme—are present, one acting much more rapidly but also more rapidly inactivated than the other. Storage in the refrigerator changes markedly the behavior of both constituents, though in different ways. There is evidence that in some cases an enzyme is limited in its hydrolytic “capacity” in the sense that after an enzyme molecule has decomposed a definite number of substrate molecules, it thereafter becomes entirely passive. Further, there is evidence, in the case of one extract, that the roles of catalytically more and less active constituents in the absence of Mg are reversed in its presence. Finally, a damped periodicity is found which indicates the presence of two factors of an unknown sort which influence and are influenced by the inactivation of the enzyme.

Electric impedance measurements on systems consisting of nonuniform elements (e.g. nerve bundles, cell suspensions, imperfect dielectrics)_can be interpreted if the impedance characteristics of the individuals and their impedance distribution are known. Conversely, given the observed overall impedance,z(p), the impedance distribution of the individuals is not uniquely determined in both phase angle and static capacitance. If all individual phase angles are equal, the distribution,D(τ), is the solution of Stieltjes' integral equation $$F[z(p)] = \int_0^\infty {\frac{{D(\tau )d\tau }}{{1 + p\tau }}} ,$$ F being a known function of the geometry of the system. It is shown that the phase angle of the equivalent suspension approaches that of the original suspension as the dispersion ofD(τ) decreases.

A previously discussed neural mechanism for the discrimination of intensities is here applied to the judgment of visual lengths and distances on the assumption that the “intensity” associated with the magnitude being perceived is the intensity of innervation of the appropriate eye muscles necessary for scanning and fixating. Comparison with experimental data is made in the case of the judgment of lengths.

A general, necessary and sufficient condition for the occurrence of anodal excitation is given on the basis of Rashevsky's two-factor theory of excitation, expressions for the anodal rheobase and the minimal shock intensity following as simple special cases.

In experiments of Kluver on monkeys and of Hertz on bees, observed reactions indicate the presence of a neural mechanism which serves to order geometric figures in a linear sequence, so that any given figure occupies a definite place in a one-dimensional array. In a theoretical mechanism described by Rashevsky, presentation of a visual stimulus-figure results in an excitation at a given center whose intensity is a function upon the curve is examined in some special cases, and the mode of variation is further described qualitatively.

By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.

A somewhat simpler solution is given to the problem previously discussed by R. R. Williamson relating to the diffusion of a metabolized electrolyte whose ions have different mobilities.

Without assuming a spheroidal shape for the cell, more general approximate deformation equations are derived which allow for non-uniformities in the external diffusion field and within the cell itself. It is found that in the absence of such non-uniformities, however, the only stable shapes are those in which the cell is spheroidal. Moreover, in conformity with previous results derived by somewhat different approximations, it is found that the spherical equilibrium is always stable and the approach monotonic unless, for a sufficiently large volume or a sufficiently high metabolic rate, there exist two non-spherical equilibria, one of prolate type and one of oblate type.

New and direct proofs of the equivalence of the three theories of nervous excitation are developed, and by the introduction of theHauptnutzzeit as the unit of time and a special unit for the measurement of the excitatory state, a normal form for Rashevsky's equations is obtained which exhibits clearly the relations among the various independent parameters.

The equation for growth by intussusception involves the product of the internal concentrations ofn metabolites. This products is shown, in general, to satisfy a certain algebraic equation of then-th degree. Approximate solutions are exhibited for a somewhat wider class of cases than are considered by Rashevsky.

Experimental evidence obtained by various research workers [see Sonneborn, (1947) and (1949)] has provided interesting information about the existence and behavior of macronuclear units in Paramecium. All paramecia of the same stock probably have about the same number of macronuclear units. Before division takes place each macronuclear unit doubles, and each daughter cell presumably selects one-half of the macronuclear units present at the time of division. In this way the number of units per animal is kept constant in a manner analogous to the behavior of chromosomes. Unlike chromosomes, however, the macronuclear ullits do not separate into homologous sets and as yet very little is known about the nature of the macronuclear unit selection process. The purpose of this paper is to test one hypothesis about the selection process by incorporating it into a stochastic model and by comparing predicted results with experimental data. In section 2, a state is defined and the transition probabilities are derived. Section 3 deals with the theory which underlies the algebraic treatment of finite Markov chains. Numerical results for relatively small numbers of macronuclear units are given in section 4, and these results are compared with experimental data in section 5.