“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.     

The smallest value of the axon density for which ‘ignition’ can occur in a random net

As shown by A. Rapoport (1952), when a very brief stimulation or “instantaneous input” is applied to a random net, the subsequent events are determined by the parameters of the net as follows: If the axon densitya is sufficiently large and the fraction γ of the neurons initially stimulated exceeds a certain value γ₁ (theover-all threshold of the net for instantaneous stimulation), excitation will spread through the net until a steady state is reached in which a fraction γ₂ ⩾ γ₁ of the neurons is firing (“ignition phenomenon”). If γ < γ₁ the activity in the net dies out. However, if the axon density is too small, the activity will ultimately die out, no matter how large the fraction of initially stimulated neurons. Thus there exists a limiting valueA of the axon density below which the net cannot “ignite”. ThisA is a function ofh, theindividual threshold of the neurons constituting the net (we assume hereh≥2, since forh=1 the situation is essentially different). Geometrically γ₁ and γ₂ are determined as the two intersection points of a straight line with a sigmoid curve. Whena<A the two curves do not intersect and fora=A they are tangent. In this paper the “tangency case” is investigated and the general features of the functionA(h) are determined. It is shown thatA increases monotonically withh (as one would expect). For all values ofh>1 we haveA(h)>h, but the fractionA(h)/h and the derivativedA(h)/dh approach unity ash increases. An analytical expression of the functionA(h) valid for very large values ofh is derived.     

Steady states in random nets: I

A neural net is taken to consist of a semi-infinite chain of neurons with connections distributed according to a certain probability frequency of the lengths of the axones. If an input of excitation is “fed” into the net from an outside source, the statistical properties of the net determine a certain steady state output. The general functional relation between the input and the output is derived as an integral equation. For a certain type of probability distribution of connections, this equation is reducible to a differential equation. The latter can be solved by elementary methods for the output in terms of the input in general and for the input in terms of the output in special cases.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

A neural model for conditioned avoidance learning

A mathematical model for learning of a conditioned avoidance behavior is presented. An identification of the net excitation of a neural model (Rashevsky, N., 1960.Mathematical Biophysics. Vol. II. New York: Dover Publications, Inc.) with the instantaneous probability of response is introduced and its usefulness in discussing block-trial learning performances in the conditioned avoidance situation is outlined for normal and brain-operated animals, using experimental data collected by the author. Later, the model is applied to consecutive trial learning and connection is made with the approach of H. D. Landahl (1964. “An Avoidance Learning Situation. A Neural Net Model.”Bull. Math. Biophysics,26, 83–89; and 1965, “A Neural Net Model for Escape Learning.”Bull. Math. Biophysics,27, Special Edition, 317–328) wherein lie further data with which the model can be compared.     

Some remarks on the mathematical bio-sociology of the relativity of injustice

The author's theory of the adoption of certain types of behavior patterns (Rashevsky, N., 1957, “Contributions to the Theory Initiative Behavior”.Bull. Maths. Biophysics,19, 91–119; 1968,Looking at History through Mathematics, Cambridge, Massachusetts: M.I.T. Press) consisting of elementary behaviors for each of which there is an opposite one and the two are mutually exclusive, is applied to describe the changes in the general type of behavior of a society. The elementary acts of which the whole problem consists may be either overt activities or beliefs or opinions. The general behavior patternsadopted by the society are considered as the “proper” or “just” ones. Any deviation from it in either one or more of the component elementary behaviors is considered as “unjust” and is subject to some punitive action. The total number of possible mutually exclusive behavior patterns is very large but finite. Within this very large range of possible patterns, we find that this notion of justice is relative, because changes from any behavior pattern to any other may occur. It is further shown that the amount of punishment for the deviation from the accepted pattern in order to be effective as well as efficient must be applied in different ways to different individuals even for the same transgression.     

The group structure of some neural nets

A method is given for using a set of disjoint cycles as the main body of a neural net that will given rise to more than one temporal response pattern for different afferent stimuli. The method arises out of considering the correspondence between this type of net and certain Abelian groups of finite order. The consideration also gives rise to a possible definition of the “complexity” of this type of neural net.     

Non-linear excitation theory: Non-accommodative,sub-threshold effects

The standard two factor excitation theories should be called “preexcitation” theories since they apply only to those events occurring just up to excitation. A true phenomenological excitation theory which describes thewhole excitation cycle must involve non-linear equations. The nature of these non-linearities is suggested by B. Katz's subthreshold response data. From this data is constructed a “local phenomenological characteristic” which is analogous to the current-voltage characteristic of a non-linear electrical or mechanical system capable of displaying relaxation oscillations. Excitation by constant currents is shown to occur where the slope of the characteristic changes sign. The variation of the time constant of excitation with degree of response, explained by W. A. H. Rushton in terms of a liminal length, is described here in purely formal terms. The theory as presented explicity treats only those events in the excitation cycle up to and a little beyond excitation; the complete excitation cycle (including recovery and repetition) is mentioned as being amenable to mathematical treatment by an extension of the present theory.     

An exact method for the computation of the connectivity of random nets

The problem of finding the “weak connectivity” of a random net is reduced to one involving a Markov process. This provides a mathematically exact treatment of the problem which had previously been treated by an approximation, whose justification was not rigorous. The exact method allows in principle not only the calculation of the “weak connectivity”, but also of the “strong connectivity”, and, in general, the probability that from a randomly selected neuron in the net there exist paths to a specified number of neurons. The computations become exceedingly involved for large nets.     

A quantum model for controlled enzymic reactions: II. Controlled biochemical networks

Two control units, the switching and the two factor discriminating net are described. They are derived as a consequence of the enzymic oscillatory behavior induced by substrate “perturbation”. A complex network encompassing long sequences of metabolic reactions is constructed and the organization of cellular metabolic activities in well defined “regimes” and “states” inferred.     

Source-sink landscape theory and its ecological significance

Exploring the relationships between landscape pattern and ecological processes is the key topic of landscape ecology, for which, a large number of indices as well as landscape pattern analysis model were developed. However, one problem faced by landscape ecologists is that it is hard to link the landscape indices with a specific ecological process. Linking landscape pattern and ecological processes has become a challenge for landscape ecologists. “Source” and “sink” are common concepts used in air pollution research, by which the movement direction and pattern of different pollutants in air can be clearly identified. In fact, for any ecological process, the research can be considered as a balance between the source and the sink in space. Thus, the concepts of “source” and “sink” could be implemented to the research of landscape pattern and ecological processes. In this paper, a theory of sourcesink landscape was proposed, which include: (1) In the research of landscape pattern and ecological process, all landscape types can be divided into two groups, “source” landscape and “sink” landscape. “Source” landscape contributes positively to the ecological process, while “sink” landscape is unhelpful to the ecological process. (2) Both landscapes are recognized with regard to the specific ecological process. “Source” landscape in a target ecological process may change into a “sink” landscape as in another ecological process. Therefore, the ecological process should be determined before “source” or “sink” landscape were defined. (3) The key point to distinguish “source” landscape from “sink” landscape is to quantify the effect of landscape on ecological process. The positive effect is made by “source” landscape, and the negative effect by “sink” landscape. (4) For the same ecological process, the contribution of “source” landscapes may vary, and it is the same to the “sink” landscapes. It is required to determine the weight of each landscape type on ecological processes. (5) The sourcesink principle can be applied to non-point source pollution control, biologic diversity protection, urban heat island effect mitigation, etc. However, the landscape evaluation models need to be calibrated respectively, because different ecological processes correspond with different source-sink landscapes and evaluation models for the different study areas. This theory is helpful to further study landscape pattern and ecological process, and offers a basis for new landscape index design. __________ Translated from Acta Ecologica Sinica, 2006, 26(5): 1444–1449 [译自: 生态学报]     

Seed-coat anatomy of the non-pleurogrammic seeds in the tribe Ingeae (Leguminosae,Mimosoideae)

An anatomically complex structure of the seed-coat is a general characteristic of all members of the Leguminosae. Nonetheless, certain genera exhibit a particular type of seed, the “overgrown” seen that has been defined by a developmental criterion and by more simple anatomical features of the seed-coat. Only the last criterion seems suitable for unambiguously distinguishing this type of seed, which is here referred to as “overgrown-like.” This type appears to be apomorphic in the Ingeae (and probably in all the Leguminosae) and likely results from a heterochronic loss of tissue differentiation. Variations in this character may be useful at the generic level, and detailed anatomical observations reveal the occurrence of three distinct patterns. The high degree of correlation with other characters suggests that overgrown-like seeds have evolved separately at least three times in the Ingeae and thatPithecellobium s.s. may be polyphyletic. The overgrown-like seeds are likely to be an adaptive response to wet tropical climates.     

A canonical form for neural nets without circles

By “neural net” will be meant “neural net without circles.” Every neural net effects a transformation from inputs (i.e., firing patterns of the input neurons) to outputs (firing patterns of the output neurons). Two neural nets will be calledequivalent if they effect the same transformation from inputs to outputs. A canonical form is found for neural nets with respect to equivalence; i.e., a class of neural nets is defined, no two of which are equivalent, and which contains a neural net equivalent to any given neural net. This research was supported by the U.S. Air Force under Contract AF 49(638)-414 monitored by the Air Force Office of Scientific Research.     

Spread of information through a population with socio-structural bias: I. Assumption of transitivity

A previously derived iteration formula for a random net was applied to some data on the spread of information through a population. It was found that if the axon density (the only free parameter in the formula) is determined by the first pair of experimental values, the predicted spread is much more rapid than the observed one. If the successive values of the “apparent axon density” are calculated from the successive experimental values, it is noticed that this quantity at first suffers a sharp drop from an initial high value to its lowest value and then gradually “recovers”. An attempt is made to account for this behavior of the apparent axon density in terms of the “assumption of transitivity”, based on a certain socio-structural bias, namely, that the likely contacts of two individuals who themselves have been in contact are expected to be strongly overlapping. The assumption of transitivity leads to a drop in the apparent axon density from an arbitrary initial value to the vicinity of unity (if the actual axon density is not too small). However, the “recovery” is not accounted for, and thus the predicted spread turns out to beslower than the observed.     

Reflex responses of the spinal cord under conditions of the spinal “superreflexia” evoked by pharmacological agents increasing its excitability

We studied monosynaptic reflex discharges (MRD) recorded from the ventral roots (VR) of rats subjected to systemic administration with thyroliberin, thyroxin, or 4-aminopyridine (4-AP). Under such conditions, in some of the experiments the MRD amplitude reached values sufficient to excite non-active VR fibers. In these cases, immediately after the MRD peak had been reached, abnormally increased responses (AIR) developed, whose amplitude was 2–2.5 times higher than the amplitude of highly facilitated MRD. Proofs are presented that AIR reflect excitation of “neighboring” VR fibers, which were not involved in the reflex response, and MRD plays the role of a “stimulus” exciting these fibers. Therefore, we demonstrate the possibility of transmission of excitation from “active” to “silent” fibers within a nerve trunk under conditions of the development of “superreflexia”. This state can be considered a model of excitation spreading via a non-synaptic pathway under conditions of hyperexcitability of the CNS structures, in seizure states in particular. neirofiziologiya/Neurophysiology, Vol. 32, No. 2, pp. 120–127, March–April, 2000.     

A relational theory of biological systems II

The general Theory of Categories is applied to the study of the (M, R)-systems previously defined. A set of axioms is provided which characterize “abstract (M, R)-systems”, defined in terms of the Theory of Categories. It is shown that the replication of the repair components of these systems may be accounted for in a natural way within this framework, thereby obviating the need for anad hoc postulation of a replication mechanism. A time-lag structure is introduced into these abstract (M, R)-systems. In order to apply this structure to a discussion of the “morphology” of these systems, it is necessary to make certain assumptions which relate the morphology to the time lags. By so doing, a system of abstract biology is in effect constructed. In particular, a formulation of a general Principle of Optimal Design is proposed for these systems. It is shown under what conditions the repair mechanism of the system will be localized into a spherical region, suggestive of the nuclear arrangements in cells. The possibility of placing an abstract (M, R)-system into optimal form in more than one way is then investigated, and a necessary and sufficient condition for this occurrence is obtained. Some further implications of the above assumptions are then discussed.     

A teleological explanation of Weber's law and the motor unit size law

It is shown that the Weber-Fechner law. which relates the response of a sensory biosystem to the intensity of the input stimulus, can be derived from a teleological principle of minimum transentropy (maximal noise reduction) provided the relative mean fluctuation (coefficient of variation) of the input intensity can be assumed to be (approximately) constant for all feasible mean input intensities. A law is then deduced from experimental results which quantifies the relationship existing between the relative amount of activated muscle mass and the “size” (which term is clearly defined) of a newly recruited motor unit. This law is found to be formally equivalent to the Weber-Fechner law when applied to motor unit recruitment. It is then shown that, in general, the ratio of the force increment upon recruitment, to the present force output does not obey Weber's law. Finally, it is proved that the “motor unit size law” as derived in this paper implies a fixed sequential order in the recruitment of motor units and that it may be viewed as the realization, by the mammalian neuromuscular system, of a general principle of maximum grading sensitivity.     

Responses of recurrent nets of asymmetric ON and OFF cells

A neural field model of ON and OFF cells with all-to-all inhibitory feedback is investigated. External spatiotemporal stimuli drive the ON and OFF cells with, respectively, direct and inverted polarity. The dynamic differences between networks built of ON and OFF cells (“ON/OFF”) and those having only ON cells (“ON/ON”) are described for the general case where ON and OFF cells can have different spontaneous firing rates; this asymmetric case is generic. Neural responses to nonhomogeneous static and time-periodic inputs are analyzed in regimes close to and away from self-oscillation. Static stimuli can cause oscillatory behavior for certain asymmetry levels. Time-periodic stimuli expose dynamical differences between ON/OFF and ON/ON nets. Outside the stimulated region, we show that ON/OFF nets exhibit frequency doubling, while ON/ON nets cannot. On the other hand, ON/ON networks show antiphase responses between stimulated and unstimulated regions, an effect that does not rely on specific receptive field circuitry. An analysis of the resonance properties of both net types reveals that ON/OFF nets exhibit larger response amplitude. Numerical simulations of the neural field models agree with theoretical predictions for localized static and time-periodic forcing. This is also the case for simulations of a network of noisy integrate-and-fire neurons. We finally discuss the application of the model to the electrosensory system and to frequency-doubling effects in retina.     

Dynamic analysis,with feedback characterization,of hybrid human musculoskeletal frameworks by the linegraph-flowgraph procedure

The dynamic response of human musculo-skeletal framework is treated by (i) idealization of the musculo-skeletal framework as hybrid structural networks possessing feedback characteristics and then (ii) employing linegraph-flowgraph procedures for the feedback characterization of the hybrid structural networks. Topological procedures are used in which a “tree” of a network furnishes the skeleton upon which the “linkage” (muscle representing) members provide interaction. Feedback characterization (representing the sensitivity of the skeletal members to the tensile forces) is defined, between the internal “linkage” and “tree” members, by means of the flowgraph. Mikusinski operational calculus is used to facilitate representation of inertia effects by dynamic feedback characterization, with inclusion of initial conditions.