Some remarks on the boolean algebra of nervous nets in mathematical biophysics

Recent demonstration by the author has shown that the fundamental equations of the mathematical biophysics of the central nervous system can be considered as describing the behavior of very large numbers of neurons, of which each one follows discontinuous laws, such as discussed by W. S. McCulloch and W. Pitts. In that light some of the old problems are discussed. The comparative merits of the “microscopic” and “macroscopic” approaches are discussed for the problem of the point to point correspondence between the retina and the cortex, with the number of connecting fibers much less than the number of cells. Some aspects of discrimination of intensities are also discussed. Finally, a few generalizations of the McCulloch-Pitts treatment are suggested, and a nervous network is constructed which illustrates some aspects of the perception of numbers.     

Some bio-sociological aspects of the mathematical theory of communication

In the first part of the paper a general discussion of the transmission of information through neural chains is given in terms of the Shannon-Weaver theory. It is pointed out that with the all-or-none law a single chain of neurons connected in series transmits one bit of information per signal. A set ofN independent parallel chains transmitsN bits per signal. If, however, the chains are interconnected, the amount of information is reduced. At the same time, however, the degree of coordination of the final neuromuscular reaction is increased. A relation between the maximum possible speed of a reaction and its degree of coordination is derived, and possible applications to spoken language are suggested. A general quantitative discussion of the relation between amount of information and amount of knowledge which an individual may obtain when confronted with the external world is made and a possible connection with new trends in logical thinking is pointed out. In the second part transmission of information through “social chains” is discussed under certain special assumptions. An expression for the “social channel noise” in terms of the length of the channel is derived. Finally an expression is given for the amount of information transmitted from one individual to another in a social group of uniform density as a function of the physical distance between the two individuals.     

Coupled Spike Activity in Micropopulations of Motor Cortex Neurons in Rats

Using eight-channel metal microelectrodes (diameter of a separate channel 12 μm), we extracellularly recorded the impulse activity of 186 single neurons or their small groups (usually, pairs) localized in the motor cortex of rats anesthetized with ketamine. In 60 cases (32.3%), action potentials (APs) of two single neurons were generated in a parallel manner and demonstrated fixed time relations with each other. This is interpreted as being a result of excitation of two neighboring functionally connected (coupled) cells. These AP pairs could be recorded via one and the same or two neighboring microelectrode channels. Second APs in the pair were elicited exclusively in the case where an AP was preliminarily generated by another neuron, while APs of the latter in some cases could arrive independently. Therefore, “leading” and “accompanying” cells could be identified in such neuronal pairs. The coupling coefficient in the generation of APs by an accompanying unit with respect to APs generated by a leading cell was close to 100%, with no dependence on the discharge frequency in the latter. Intervals between APs of two neurons in different coupled pairs varied from about 1.0 to 22-23 msec. In the case of minimum values of these interspike intervals, APs generated by coupled neurons overlapped each other; this resulted in the formation of spikes looking like “complex APs.” Within some time intervals, interspike intervals could increase, and such APs began to be decomposed. The above-described data are considered electrophysiological proof of the existence of tight functional coupling between a significant part of cortical neurons spatially close to each other, i.e., members of a micropopulation, which was obtained in an in vivo experiment.     

Discussion: McCulloch-Pitts and related neural nets from 1943 to 1989

The McCulloch-Pitts paper “A Logical Calculus of the Ideas Immanent in Nervous Activity” was published in theBulletin of Mathematical Biophysics in 1943, a decade before the work of Hodgkin, Huxley, Katz and Eccles. The McCulloch-Pitts neuron is an extremely simplified representation of neural properties, based simply on the existence of a threshold for the activation of an action potential. This work has been supported in part by Grants from the University of Chicago Brain Research Foundation, and the U.S. Department of the Navy, Office of Naval Research (Grant No. N 00014-89J-1099).     

Morphological characterization of single fan-shaped body neurons in <Emphasis Type="Italic">Drosophila melanogaster</Emphasis>

The fan-shaped body is the largest substructure of the central complex in Drosophila melanogaster. Two groups of large-field neurons that innervate the fan-shaped body, viz., F1 and F5 neurons, have recently been found to be involved in visual pattern memory for “contour orientation” and “elevation” in a rut-dependent manner. The F5 neurons have been found to be responsible for the parameter “elevation” in a for-dependent manner. We have shown here that the F1 neuron also affects visual memory for “contour orientation” in a for-dependent way. With the help of Gal4/UAS and FLP-out techniques, we have characterized the morphological features of these two groups of neurons at single neuron resolution. We have observed that F1 or F5 neurons are groups of isomorphic individual neurons. Single F1 neurons have three main arborization regions: one in the first layer of the fan-shaped body, one in the ventral body, and another in the inferior medial protocerebrum. Single F5 neurons have two arborization regions: one in the fifth layer of the fan-shaped body and the other in the superior medial protocerebrum. The polarity of the F1 and F5 neurons has been studied with the Syt-GFP marker. Our results indicate the existence of presynaptic sites of both F1 and F5 neurons located in the fan-shaped body and postsynaptic sites outside of the fan-shaped body. This work was supported by the “973 Program” (2005CB522804 and 2009CB918702), the National Natural Sciences Foundation of China (30621004, 30625022, and 30770682), and the Knowledge Innovation Program of the Chinese Academy of Sciences (KSCX2-YW-R-28).     

Biosemiosis and the Cellular Basis of Mind

The human brain is a complex organ made up of neurons and several other cell types, and whose role is processing information for use in elicitation of behaviors. To accomplish this, the brain requires large amounts of energy, and this energy is obtained by the oxidation of glucose (Glc). However, the question of how the oxidation of Glc by individual neurons in brain results in their collective ability to rapidly generate feats of cognition that allow them to recognize the nature of the universe in which they live and to communicate this information remains unclear. In this article, insights into this process are provided by first considering the brain’ s homeostatic “operating system” for supply of energy to stimulated neurons, and how this system defines the basic unit of brain “structure”. This is followed by consideration of the brain’s “two-cell” neuronal communication mechanism which defines the basic unit of brain “function”. Finally, an analysis of the nature of frequency-encoded “neuronal languages” that enable ensembles of neurons to translate energy derived from the oxidation of Glc into a collective “mind”, the aggregate of all brain processes including those involving perception, thought, insight, foresight, imagination and behavior.     

Applications of matrix algebra to communication nets

A “generic” problem amenable to matrix algebraic treatment is outlined. Several examples are given and one, a communication system, is studied in some detail. A typical structure matrix is used to describe the channels of communication and a “status” matrix is used to describe the distribution of information in the system at any time. A theorem is proved relating the status matrix at any timet to thetth power of the structure matrix. The elements of the communication system are interpreted as individuals who can send messages to each other. For the individuals attempting to solve a “group problem” certain relations are derived between the structure and status matrices and time of solution. The structure of the communication system is permitted to vary with time. A general theorem is proved relating the status matrix to the matrix product of the series of structure matrices representing the changing structure of the system. Some suggestions are made for further generalizations. In particular, it is suggested that so-called “higher order” information transmission can be similarly treated.     

Contribution to the probabilistic theory of neural nets: II. Facilitation and threshold phenomena

The output curve of a single neuron with a threshold of response with respect to the frequency of the stimuli is derived. If the stimuli are regularly spaced in time, the output curve has discontinuities. If the threshold and/or refractory period are sufficiently large, the output curve approaches the “all-or-none” curve. In the case of completely randomized stimuli, the output curve is sigmoid. The equation of this curve is derived and some properties are studied. Threshold and “all-or-none” effects can be achieved by “pyramiding” neurons of this type to converge on neurons of higher order.     

Contribution to the theory of random and biased nets

The probabilistic theory of random and biased nets is further developed by the “tracing” method treated previously. A number of biases expected to be operating in nets, particularly in sociograms, is described. Distribution of closed chain lengths is derived for random nets and for nets with a simple “reflexive” bias. The “island model” bias is treated for the case of two islands and a single axon tracing, resulting in a pair of linear difference equations with two indices. The reflexive bias is extended to multiple-axon tracing by an approximate method resulting in a modification of the random net recursion formula. Results previously obtained are compared with empirical findings and attempts are made to account for observed discrepancies.     

Contribution to the probabilistic theory of neural nets: IV. Various models for inhibition

Input-output formulas are derived for a neuron upon which converge single axones of two other neurons, which are subjected to a Poisson shower, where a number of different assumptions are made concerning the mechanism of inhibition. In one assumption so-called “bilateral pre-inhibition” is considered. That is to say, both neuronsN ₁ andN ₂ may exciteN ₃, but if the stimulus of one of them follows within a certain interval σ of the other, the second stimulus is not effective. This model is essentially no different from that involving two excitatory neurons acting upon a neuron having a refractory period. Another mechanism considered involves so-called “pre-and-post” inhibition, in which if two stimuli fromN ₁ andN ₂ fall within σ,both are ineffective. This case being mathematically much more involved than the preceding, an approximation method is used for deriving the input-output formula. Previous papers of this series are denoted by I, II, and III in this paper.     

On the spread of information with time and distance

The transmission of some information or behavior pattern is treated as a flow of “particles” which execute random motions over a population of individuals and which may multiply or disappear. Equations are derived for the number density of these “particles” and from this is calculated the number of individuals through which the “particles” have passed. The results are applied to a number of situations such as 1) uniform spatial distribution with multiplication factor decreasing with time because of loss of interest or confusion of the information, 2) multiplication factor constant but the rate of spreal decreasing with multiple hearings, 3) one-dimensional region with a small starting region with or without an absorbing barrier 4) two-dimensional region with absorbing barrier, 5) continous sources of information within a small region in one dimension, 6) uniform spatial distribution in which individuals do not respond to more than one hearing.     

Pioneer neurons: A basis or limiting factor of lophotrochozoa nervous system diversity?

It has been demonstrated by us and other authors that first nervous cells in developing larvae from various trochozoan groups differentiate at the periphery. These pioneer neurons are distinguished by the set of characters. They are located outside the forming central ganglia; outgrowing fibers of central neurons use their processes as a “scaffolding” transmitter expression in these neurons is transient. On the one hand, pioneer neurons mark the “frame” of the adult nervous system and thus play a limiting role. On the other hand, pioneering navigation provides possible mechanisms for evolutional plasticity of the nervous system in adults. In addition, pioneer neurons can underlie functional adaptation of trochophore animals, which minimizes fitness decrease during the transition from the larval to the adult form during metamorphosis.     

Structure of the bank vole (<Emphasis Type="Italic">Myodes glareolus</Emphasis>) population cycles in the core and periphery of its species area

The results of long-term studies of two bank vole (Myodes glareolus) populations in stationary sites in the central part and periphery of its species area are described. Four phases of a multiannual population cycle and two of its structural parts have been detected for both populations. The first part of the cycle is “determined,” with the “peak” phase passing into a “depression” (population collapse). This transition is mainly determined by intrapopulation processes and is weakly dependent on the external conditions of each individual year. The second part is “stochastic,” starting from a stable point in the cycle in the depression phase. The duration of the second part is determined by the state of the population and its ability to increase its size, as well as by the weather and food factors, predation pressure, and location of the population within the species area. The transition from the peak phase to the depression phase (the determined part) for both populations takes place during one fall-winter-spring season and has no effect on the cycle duration. The duration of the stochastic part in the core of the species area (the period from depression phase to peak phase) is 1–3 years and in the periphery, 2–4 years.     

Forms of output distribution between two individuals motivated by a satisfaction function

Motivations of two individuals governed by a satisfaction function are assumed to determine their respective “efforts”, which result in the production of “output”, i.e., objects of satisfaction. In previous papers the sharing of output was prescribed in advance. In the present article, however, the sharing formula itself is determined to a certain extent by the satisfaction function. The rate of remuneration per unit of output for each individual is taken to be proportional to the derivative of the satisfaction of the other individual with respect to the effort of the first. The formulation of this condition leads to a partial differential equation whose solutions determine the sharing formula. Sharing determined in this way is referred to as sharing according to the Condition of Mutual Need (C.M.N.). Satisfaction resulting from five different situations are the computed and compared: (1) an individual producing and consuming alone; (2) two individuals sharing equally and neither taking the “initiative” to determine the optimum output; (3) sharing determined by C.M.N. with optimum output determined as in (2); (4) equal sharing but with one individual taking “initiative” in determining optimal output; and (5) sharing determined by C.M.N. and optiml output by the “initiative” of one individual. further considerations concern conditions imposed on the arbitrary function occurring in the solution of the above-mentioned partial differential equation.     

Chain processes and their biophysical applications: Part I. General theory

A mathematical theory applicable to the biological effects of radiations as chain processes is developed. The theory may be interpreted substantially as a “hit theory” involving the concepts of “sensitive volume” or “target area”. The variability of the sensitivity of the organism to the radiation and its capacity of recovery between single hits is taken into account. It is shown that in a continuous irradiation of a biological aggregate in which the effect of each single hit cannot be observed, recovery and variation of sensitivity are formally equivalent to each other so that a discrimination between these two phenomena is possible only by discontinuous irradiation or by using different radiation intensities. Methods for the calculation of the “number of hits” and for the determination of the kinetics of the processes from “survival curves” or similar experimental data are given. The relation between the recovery and the Bunsen-Roscoe law is discussed. The case in which the injury of the organism is dependent on the destruction of more than one “sensitive volume” is also considered.     

The pattern of sensory discharge can determine the motor response in young Xenopus tadpoles

Young Xenopus tadpoles were used to test whether the pattern of discharge in specific sensory neurons can determine the motor response of a whole animal. Young Xenopus tadpoles show two main rhythmic behaviours: swimming and struggling. Touch-sensitive skin sensory neurons in the spinal cord of immobilised tadpoles were penetrated singly or in pairs using microelectrodes to allow precise control of their firing patterns. A single impulse in one Rohon-Beard neuron (= light touch) could sometimes trigger “fictive” swimming. Two to six impulses at 30–50 Hz (= a light stroke) reliably triggered fictive swimming. Neither stimulus evoked fictive struggling. Twenty-five or more impulses at 30–50 Hz (= pressure) could evoke a pattern of rhythmic bursts, distinct from swimming and suitable to drive slower, stronger movements. This pattern showed some or all the characteristics of “fictive” struggling. These results demonstrate clearly that sensory neurons can determine the pattern of motor output simply by their pattern of discharge. This provides a simple form of behavioural selection according to stimulus. Accepted: 28 November 1996     

Use of microsatellite DNA and otolith Sr:Ca ratios to infer genetic relationships and migration history of four morphotypes of <Emphasis Type="Italic">Rhinogobius</Emphasis> sp. OR

The genetic diversity and relationship among four morphotypes of Rhinogobius sp. OR, Gobiidae (“Tōshoku,” “Shinjiko,” “Gi-tōshoku,” and “Shimahire”) were investigated with seven microsatellite DNA loci, and amphidromy of these morphotypes was verified by strontium (Sr) and calcium (Ca) deposition in the otolith. Samples of “Tōshoku,” “Shinjiko,” “Gi-tōshoku,” and “Shimahire” were collected from, respectively, three, three, two, and four locations in Japan. Microsatellite analysis detected high genetic diversity (based on the number of alleles, allelic richness, and average observed heterozygosity) in the “Tōshoku” and “Shinjiko” morphotypes relative to the “Shimahire” morphotype; the “Gi-tōshoku” morphotype had an intermediate level of variation. Almost all pairwise F _ST values were significantly different from zero (P < 0.001), except between two populations of “Tōshoku.” Clear genetic independence was observed between the “Shinjiko” and “Shimahire” morphotypes in the Maruyama River. A principle component analysis based on microsatellite data indicated that the “Tōshoku,” “Shinjiko,” and “Gi-tōshoku” morphotypes were genetically similar. Furthermore, the three populations of “Tōshoku” were closely related each other, and two of those collected from the Lake Biwa system were a single population. There was, however, a high degree of genetic differentiation between “Shimahire” and the other morphotypes; moreover, there was high genetic divergence among four populations of the “Shimahire” morphotype. Amphidromous migratory histories were indicated by Sr:Ca ratios in two of three populations of the “Shinjiko” morphotype and in one of two “Gi-tōshoku” morphotypes, whereas all populations of the “Shimahire” morphotype were freshwater residents. The large genetic divergence and low genetic diversity in “Shimahire” are likely related to migration history.     

Macroevolution via secondary endosymbiosis: a Neo-Goldschmidtian view of unicellular hopeful monsters and Darwin’s primordial intermediate form

Seventy-five years ago, the geneticist Richard Goldschmidt hypothesized that single mutations affecting development could result in major phenotypic changes in a single generation to produce unique organisms within animal populations that he called “hopeful monsters”. Three decades ago, Sarah P. Gibbs proposed that photosynthetic unicellular micro-organisms like euglenoids and dinoflagellates are the products of a process now called “secondary endosymbiosis” (i.e., the evolution of a chloroplast surrounded by three or four membranes resulting from the incorporation of a eukaryotic alga by a eukaryotic heterotrophic host cell). In this article, we explore the evidence for Goldschmidt’s “hopeful monster” concept and expand the scope of this theory to include the macroevolutionary emergence of organisms like Euglena and Chlorarachnion from secondary endosymbiotic events. We argue that a Neo-Goldschmidtian perspective leads to the conclusion that cell chimeras such as euglenids and dinoflagellates, which are important groups of phytoplankton in freshwater and marine ecosystems, should be interpreted as “successful monsters”. In addition, we argue that Charles Darwin had euglenoids (infusoria) in mind when he speculated on the “primordial intermediate form”, although his Proto-Euglena-hypothesis for the origin of the last common ancestor of all forms of life is no longer acceptable.     

An innovative LC/MS approach applied to the determination of zeralenone in maize: Alternate Isotope-coded Derivatization Assay (AIDA)

Zearalenone is a mycotoxin mainly produced by severalFusarium species, which are known to colonize grains in temperate climates. The purpose of the study is to provide a reliable isotope dilution method for the quantification of this mycotoxin. A derivative of the analyte to be used as standard is obtained by reaction with acetic anhydride, which is available in two pure isotopic forms, a protonated (“light”) and a hexadeuterated (“heavy”). The derivatized standards are added to the matrix split intwo parts. Then, the derivatization procedure is repeated on both matrices derivatizing the part containing the “heavy” labelled standard with the “light” acetic anhydride and the part containing the “light” labelled standard with the “heavy” acetic anhydride. Both extracted mixtures are analyzed by LC/MS, monitoring the “light” and the “heavy” labelled analytes and using the former as standard for the latter in one case and viceversa in the other case. The method allowed to obtain very good results, without the need of IAC purification. Presented at the 27^th Mykotoxin-Workshop, Dortmund, Germany, June 13–15, 2005. Financial support: The Italian Ministry of Health