Principles and methods of the graph-theoretical analysis of natural binary arborescences

This paper studies the fundamental bases for the topological analysis of natural objects branched by means of bifurcations. The first part analyses and exploits the available concepts and methods. They allow the determination of the general topological model (that of “single-stemmed, binary arborescences”) and of its mathematical properties. Conversely, they only offer a limited access to more refined analyses. There are two different approaches; one attempts to measure topological parameters and the other to identify trees. The level of analysis of the first is too general, while that of the second is too precise. A desirable level of analysis (dealing for instance with the comparison between observed patterns or their classifications in sets of sufficient size) actually lacked adapted tools.The second part of the study bridges this gap from both directions: by combining various parameters and by widening the methods of identification. The first direction makes it possible to characterize certain patterns governed by precise mathematical laws and to show that two of these (the linear and exponential patterns) are two theoretical extremes between which one may quantitatively locate other patterns. The second direction leads to broader classes. A continuous gamut of methods is offered. The third part extends the study to other observable patterns differing from the general model and shows that they all belong to one single topological category. A comparison with other topological patterns permits their interpretation.     

How do glutamatergic and GABAergic cells contribute to synchronization in the medial septum?

The medial septum-diagonal band (MSDB) complex is considered as a pacemaker for the hippocampal theta rhythm. Identification of the different cell types, their electro-physiological properties and their possible function in the generation of a synchronized activity in the MSDB is a hot topic. A recent electro-physiological study showed the presence of two antiphasically firing populations of parvalbumin containing GABAergic neurons in the MSDB. Other papers described a network of cluster-firing glutamatergic neurons, which is able to generate synchronized activity in the MSDB. We propose two different computer models for the generation of synchronized population theta oscillation in the MSDB and compare their properties. In the first model GABAergic neurons are intrinsically theta periodic cluster-firing cells; while in the second model GABAergic cells are fast-firing cells and receive periodic input from local glutamatergic neurons simulated as cluster-firing cells. Using computer simulations we show that the GABAergic neurons in both models are capable of generating antiphasic theta periodic population oscillation relying on local, septal mechanisms. In the first model antiphasic theta synchrony could emerge if GABAergic neurons form two populations preferentially innervate each other. In the second model in-phase synchronization of glutamatergic neurons does not require specific network structure, and the network of these cells are able to act as a theta pacemaker for the local fast-firing GABAergic circuit. Our simulations also suggest that neurons being non-cluster-firing in vitro might exhibit clustering properties when connected into a network in vivo. Action Editor: David Golomb     

Experimentelle Untersuchung,Modellierung und optimale Berechnung eines mehrstufigen Säulenfermentors II. Experimentelle Bestimmung der Parameter und Koeffizienten des mathematischen Modells

Using a system analysis for the investigation of all processes which occur in a biochemical reactor on the micro and macro level, a mathematical model was worked out. It characterizes the model of the kinetics and stoichiometry of the growth of microorganisms and the rules of hydrodynamics and mass transfer in form of blocks. Relating to the discussed mathematical total model [11], experimental data on which the calculation of the model parameters is based are described in this second part of the paper. They were determined not only directly from the cultivation process, but also from experiments with model media.     

Biomathematical aspects of numerical evaluation and mathematical modelling of non-monotonous exponential measuring processes

Measurements of endocrinological and pharmacological processes often yield courses of time series with exponentially saturated increasing first part followed by an exponentially decreasing part. Such measured courses may be mathematically modelled by the so-called BATEMAN function type, an expression consisting of 2 e-function terms. In this paper, the method of locally adjusted functional approximation for model-free quantitative evaluation of measured time series is sketched. By means of 2 real examples of measured data, it will be demonstrated how the results of the model-free evaluation may serve for internal regression to estimate starting parameter values for an iterative fitting of a BATEMAN function to measured data courses. Furthermore, it is shown that the model-free approach of data evaluation may give substantial hints for the mathematical model building process and for model verification.     

基于混沌降噪的神经元放电峰峰间期序列分析

围绕如何来消除神经元峰峰间期序列中随机噪声影响从而提取出决定不规则性的确定性动力学关系这个问题,本文首先简要介绍峰峰间期序列样本的制备,然后着重讨论一个简单可行的混沌时间序列降噪方法的原理和算法实现,最终将该方法运用到神经元放电活动数值模拟和实验记录到的峰峰间期时间序列样本分析中。本文分析结果再次证明神经放电活动中确实存在着不规则混沌运动,而且降噪结果进一步揭示了神经电生理实验中决定混沌放电的不连续但分段光滑的单峰函数关系     

State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data

Precise spike coordination between the spiking activities of multiple neurons is suggested as an indication of coordinated network activity in active cell assemblies. Spike correlation analysis aims to identify such cooperative network activity by detecting excess spike synchrony in simultaneously recorded multiple neural spike sequences. Cooperative activity is expected to organize dynamically during behavior and cognition; therefore currently available analysis techniques must be extended to enable the estimation of multiple time-varying spike interactions between neurons simultaneously. In particular, new methods must take advantage of the simultaneous observations of multiple neurons by addressing their higher-order dependencies, which cannot be revealed by pairwise analyses alone. In this paper, we develop a method for estimating time-varying spike interactions by means of a state-space analysis. Discretized parallel spike sequences are modeled as multi-variate binary processes using a log-linear model that provides a well-defined measure of higher-order spike correlation in an information geometry framework. We construct a recursive Bayesian filter/smoother for the extraction of spike interaction parameters. This method can simultaneously estimate the dynamic pairwise spike interactions of multiple single neurons, thereby extending the Ising/spin-glass model analysis of multiple neural spike train data to a nonstationary analysis. Furthermore, the method can estimate dynamic higher-order spike interactions. To validate the inclusion of the higher-order terms in the model, we construct an approximation method to assess the goodness-of-fit to spike data. In addition, we formulate a test method for the presence of higher-order spike correlation even in nonstationary spike data, e.g., data from awake behaving animals. The utility of the proposed methods is tested using simulated spike data with known underlying correlation dynamics. Finally, we apply the methods to neural spike data simultaneously recorded from the motor cortex of an awake monkey and demonstrate that the higher-order spike correlation organizes dynamically in relation to a behavioral demand.     

Spatially localized interactive neural populations—II: Stability and dynamics of excitatory sets

The stability characteristics and dynamical behavior of a system of mutually excitatory neurons in close spatial proximity are investigated with a mathematical model. The model predicts the existence of uniform, intermediate levels of activity other than those of no activity and maximal activity. The model also, yeilds a good explanation of data obtained from periglomerular neurons in the olfactory bulb of the cat. This paper is based on a dissertation submitted by the author to Rensselaer Polytechnic Institute in partial fulfillment of the requirements for the degree of Doctor of Philosophyy. This work was supported in part by a Fellowship sponsored by the IBM Corporation and in part by the National Science Foundation under Grant MCS75-08328.     

The morphofunctional characteristics of the neurons in the sensorimotor cortex of old rabbits during the trace assimilation of rhythm]

Extracellular neuronal activity was recorded from 460 neurons from alert young (5-7 months), middle-aged (54-65 months) and old (66-85 months) rabbits. Trace rhythmic activity of sensorimotor cortical neurons was examined after long-lasting (10-20 min) rhythmic (0.5-2 Hz) electrocutaneous stimulation of the contralateral forelimb. Spectral analysis of spike activity showed age-related differences in capability of producing a rhythm of previous stimulation in spontaneous neuronal activity. In young animals propriate rhythmic fluctuations of firing rate appeared after the first or second sessions of stimulations (on the first experimental day), in middle-aged ones--after 2-4 sessions (on the second or third days); cortical neurons in old rabbits did not exhibit trace rhythmic activity. Significant morphological changes in glial and neuronal cells were observed in sensorimotor cortex of old rabbits. It is proposed that morphological deteriorations may be the reason of the impairement of trace processes during aging.     

A single compartment model of pacemaking in dissasociated Substantia nigra neurons

Spontaneous oscillations in the mid-brain dopaminergic neurons are an important feature of motor control. The degeneration of these neurons is involved in movement disorders, particularly Parkinson’s Disease. Modelling of this activity is an important part of developing an understanding of the pathogenic process. We develop a mathematical paradigm to describe this activity with a single compartment approach and a CellML version is made publicly available. The model explicitly describes the dynamics of the transmembrane potential with changes in the levels of important cations and is consistent with two major observations in the literature regarding its behaviour in the presence of channel blockers. Stability of the model behaviour is determined from the properties of its Monodromy matrix. We also discuss from the perspective of energy, a pharmacological intervention suggested in the treatment of Parkinson’s Disease.     

Mathematical models for estimating fluxes at the sediment-water interface in benthic chamber experiments

Experimental data collected by means of benthic chambers have been used to develop simple mathematical models in order to predict heavy metal and non-metal fluxes at the sediment-water interface. In particular, two mathematical models have been applied to Ba, Co, Fe, Pb, Mn and Cu.Both models estimate benthic fluxes by means of Fick's law but they differ in the modelling of partition between liquid and solid phases. The first model, which is appropriate for Ba and Co, assumes that sorption is the most important process in determining partition. The second model, which gives discrete results for Fe and Pb, assumes that precipitation is the most important process determining the time variation of concentrations. Neither model is suitable to explain experimental data for Mn and Cu.This paper demonstrates the benefit of the use of mathematical models in the interpretation of benthic chambers experiments and in the identification of the relevant processes occurring at the sediment-water interface.     

Effects of applying microelectrostimulation to different cell type zones in the caudate nucleus

The effects of applying electromicrostimulation to areas of the caudate nucleus with different neuronal activity patterns were investigated during chronic experiments on four cats. Caudate sites containing neurons responding to presentation of various sensory stimuli were selected for the first set of experiments and those where no neuronal activity manifested in the second series. Histological verification of electrolytic marker sites produced by electrical stimulation took place at the end of each experimental sequence and the cell types surrounding these lesions were examined. Microelectrostimulation consistently produced movement in the animal in the first set of experiments; markers were located along the surface of striosomes among large-sized cells, bundles of fibers, and blood vessels. In the second, electrical stimulation produced no alteration in naturally occurring animal behavior; markers were located within striosomes in accumulations of small- and medium-sized cells. A survey of the findings obtained would confirm our hypothesis that the neurons from which activity had been recorded by extracellular techniques in the caudate nucleus are large-sized cells with long axons.Institute for Research into Information Transmission, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 22, No. 2, pp. 162–171, March–April, 1990.     

Spatial organization and interneuronal relations in various areas of the neocortex in Cetacea

The data on pyramidal neurons joining in the parietal and temporal areas of the dolphin neocortex by means of apical dendrites fasciculi are presented. The fasciculi also contain dendrites of spindle-like and stellate cells. The vertical fasciculi of the dendrites unite neurons of the layer V and of the sublayer III2. In the sublayer III1, after dichotonic division of the apical dendrites, fasciculi of the second order are formed, to them the dendrites of the pyramidal neurons of the sublayer III1 and the layer II join. Several forms of the interneuronal contacts have been revealed: axo-dendritic, axo-spinous and dendro-dendritic. Synaptic complexes of the converged and divergent types have been shown. A suggestion is made on formation of larger neuronal modules++, having common afferent entrance and demonstrating selectivity to the stimulus properties.     

Some multi-dimensional branching processes as motivated by a class of problems in mathematical genetics I

In this paper a class of branching processes applicable to populations reproducing by some asexual means or by a simple selfing system of mating is studied. The paper is divided into three parts. In part one the mathematical model is introduced, part two is a mathematical analysis of the model, and in part three concrete applications and examples are given. Many of the proofs of the theorems in part two are omitted but will appear in a subsequent issue of theBulletin. The research reported in this paper was supported in part by the United States Atomic Energy Commission, Division of Biology and Medicine Project AT(45-1)-1729.     

A method of analysis of the models of neural nets

Taking into account Caianiello's work of 1961 a model of a neuron quite similar to his is proposed and studied. For this model, where a temporal summation and a period of refractoriness are assumed, a mathematical approach and a simulation on computer were realized. Particular types of nets were used, namely: nets with topological structures, and fully random nets. The difference between the two types is that the first type has a two-dimensional square structure and depends on the rules of the formation of connection between the neurons, while the second type is realized by means of the probability distribution function governing the formation of the structure of the net.These types of neural nets are analysed by means of a method which permits to obtain various parameters which characterize their behaviour in time and space in terms of the trajectory of the system. Many experiments are also reported; the statistical analyses, made on them, show the great importance and influence of refractoriness on the behaviour of neural networks.In the last part of the work an interesting case is reported, in which the reaction of the net to a disturbance shows that a kind of adaptation takes place, although the structure of the net stays unchanged.On leave of absence from the Lithuanian Academy of Sciences, Vilnius, Lithuanian S.S.R.     

An empirical approach to the statistical analysis of mutagenesis data from the Salmonella test

A number of recent papers have suggested basing the statistical analysis of Salmonella (Ames) mutagenicity test results on a mathematical model of the complete dose-response curve. For most mutagens at low doses the curve increases linearly; then, as the dose increases, the curve may flatten and finally turn downwards due primarily to effects of toxicity. The exact mechanism underlying this shape is, however, not well understood and is likely to vary for different chemicals. A different approach is to assume that the initial part of the curve is linear and to base the statistical analysis solely on this region, reasoning that it contains most of the interpretable information about the mutagenesis dose response. In this paper a formal method of deciding which points are on the initial linear part of the curve is described, and a statistical method is proposed for analyzing these points. Computer simulations are used to examine the properties of the procedure and comparisons are made with a previously proposed mathematical model of the whole curve. It is concluded that the method suggested here provides a very satisfactory, robust method for the standard analysis of Salmonella data.     

Relationship between functional characteristics of cat visual cortex neurons,their response to novelty,and short-term memory

In the visual cortex of unanesthetized cats, the number and frequency of discharges in response to a new stimulus differed from the subsequent responses: the first response was more intensive in 34% of the neurons, but in 30% it was inhibited. The phenomenon of short-term memory was detected in 19% of the cells: it was expressed in regeneration of the configuration of response discharges after the cessation of rhythmic stimulation. These peculiarities can be linked with functional organization of the neurons. We divided them into two groups according to their response to photic stimuli. The first group includes short-latent neurons that respond with discharges of the phasic type and that virtually or totally lack spontaneous activity. The second group consists of long-latent neurons with the tonic type of discharges and distinct spontaneous activity. In the overwhelming majority of cases, response to novelty and short-term memory were discovered in neurons of the second group. It is hypothesized that the population of neurons of the first group — having narrower afferent connections — takes part mainly in analysis of properties of a photic stimulus; the population of neurons of the second group participates in information processing at the final and highest level, on which mechanisms of memory and attention are implicated.Institute of Higher Nervous Activity and Neurophysiology, Academy of Sciences of the USSR, Moscow. Translated from Neirofiziologiya, Vol. 2, No. 6, pp. 611–617, November–December, 1970.     

Excitatory inputs to spiny cells in layers 4 and 6 of cat striate cortex

The principal target of lateral geniculate nucleus in the cat visual cortex is the stellate neurons of layer 4. In previously reported work with intracellular recording and extracellular stimulation in slices of visual cortex, three general classes of fast excitatory synaptic potentials (EPSPs) in layer 4a spiny stellate neurons were identified. One of these classes, characterized by large and relatively invariant amplitudes (mean 1.7 mV, average coefficient of variation (CV) 0.083) were attributed to the action of geniculate axons because, unlike the other two classes, they could not be matched by intracortical inputs, using paired recording. We have examined in detail the properties of this synaptic input in twelve examples, selecting for study those EPSPs where there was secure extracellular stimulation of the single fibre input to a pair of stimuli 50 ms apart. In our analysis, we conclude that the depression that these inputs show to the second stimulus is entirely postsynaptic, since the evidence strongly suggests that the probability of transmitter release at the synaptic site(s) remains 1.0 for both stimuli. We argue that the most plausible explanation for this postsynaptic depression is a reduction in the average probability of opening the synaptic channels. Using a simple biochemical analysis (c.f. Sigworth plot), it is then possible to calculate the number of synaptic channels and their probability of opening, for each of the 12 connections. The EPSPs had a mean amplitude of 1.91 mV (+/- 1.3 mV SD) and a mean CV of 0.067 (+/- 0.022). The calculated number of channels ranged from 20 to 158 (59.4 +/- 48.7) and their probability of opening to the first EPSP had an average of 0.83 (+/- 0.09), with an average depression of the probability to 0.60 for the second EPSP. Geniculate afferents also terminate in layer 6. Intracellular recordings were also made in the upper part of this layer and a total of 51 EPSPs were recorded from pyramidal cells of three principal types. Amongst this dataset we sought EPSPs with similar properties to those characterized in layer 4a. Three examples were found, which is a much lower percentage (6%) than the incidence of putative geniculate EPSPs found in layer 4a (42%).     

Creating parts that allow for rational design: Synthetic biology and the problem of context-sensitivity

The parts-based engineering approach in synthetic biology aims to create pre-characterised biological parts that can be used for the rational design of novel functional systems. Given the context-sensitivity of biological entities, a key question synthetic biologists have to address is what properties these parts should have so that they give a predictable output even when they are used in different contexts. In the first part of this paper I will analyse some of the answers that synthetic biologists have given to this question and claim that the focus of these answers on parts and their properties does not allow us to tackle the problem of context-sensitivity. In the second part of the paper, I will argue that we might have to abandon the notions of parts and their properties in order to understand how independence in biology could be achieved. Using Robert Cummins’ account of functional analysis, I will then develop the notion of a capacity and its condition space and show how these notions can help to tackle the problem of context-sensitivity in biology.     

A theoretical approach to G-protein modulation of cellular responsiveness

 Structure and function of cells often depend critically on molecular signals arriving at their surface. There are universal mechanisms of signal transduction and signal processing across cell membranes. In this paper the mechanisms involving guanine-nucleotide regulatory proteins (“G-proteins”) and certain receptor-kinases are considered. On the basis of recent findings in molecular biology a mathematical model is developed taking into account all essential components in the biochemical network between first and second messenger. There are two coupled feedback loops inherent in this process. The model finally consists of three nonlinear equations, which are obtained from a system of originally ten equations by using conservation laws and quasi-steady state conditions. The second part of the paper contains a mathematical analysis of the model. Solutions describing the temporal development of the involved biochemical species are shown to be bounded, more specifically to remain, independent of the size of the input signal, in a bounded domain of the state space. For the situation of stationary input signals existence, uniqueness and asymptotic stability of steady states are derived. We also demonstrate biologically relevant stimulus-response properties like monotonicity and saturation effects. For temporally non-constant input signals we show numerically that the model is able to produce phenomena of hypersensitivity and desensitization which are important characteristics of cellular responsiveness. Received 18 March 1996; received in revised form 15 April 1996