On some problems of random nets

The probability problems connected with random nets are restated as probabilities of drawings from an urn containing black and white balls. A partial difference equation is obtained and its solution is given. For large nets a series expression is obtained for the connectivity γ, and this is shown to be equivalent to the transcendental equation obtained by R. Solomonoff and A. Rapport (1951).     

Mesoscopic neurodynamics.

An overview is given on the history, theory, and applications of mesoscopic neural models. It is shown why and how we can describe the behavior of a large population of neurons by the method of statistical neurodynamics. Normal and epileptic propagation of cortical activity have been simulated.     

Statistics of RNA secondary structures

A statistical reference for RNA secondary structures with minimum free energies is computed by folding large ensembles of random RNA sequences. Four nucleotide alphabets are used: two binary alphabets, AU and GC, the biophysical AUGC and the synthetic GCXK alphabet. RNA secondary structures are made of structural elements, such as stacks, loops, joints, and free ends. Statistical properties of these elements are computed for small RNA molecules of chain lengths up to 100. The results of RNA structure statistics depend strongly on the particular alphabet chosen. The statistical reference is compared with the data derived from natural RNA molecules with similar base frequencies. Secondary structures are represented as trees. Tree editing provides a quantitative measure for the distance d_t, between two structures. We compute a structure density surface as the conditional probability of two structures having distance t given that their sequences have distance h. This surface indicates that the vast majority of possible minimum free energy secondary structures occur within a fairly small neighborhood of any typical (random) sequence. Correlation lengths for secondary structures in their tree representations are computed from probability densities. They are appropriate measures for the complexity of the sequence-structure relation. The correlation length also provides a quantitative estimate for the mean sensitivity of structures to point mutations. © 1993 John Wiley & Sons, Inc.     

Molecular Dynamics Simulations With First Order Coupling to a Bath of Constant Chemical Potential

Abstract

In molecular dynamics simulations the temperature or pressure can be controlled by applying a weak first-order coupling to a bath of constant temperature or pressure. This weak coupling technique to control system properties using a first-order relaxation equation is analyzed from a statistical mechanics point of view. It is shown, how the weak coupling scheme can be generalized and applied to a bath of contstant chemical potential. The presented method, to which in the following will be referred to as chemical potential weak coupling, is applied and tested on a Lennard-Jones fluid. The thermodynamic quantities known from the literature are accuratly reproduced.

The temperature and chemical potential weak coupling methods aim to sample the canonical and grand canonical ensembles respectively. By analyzing the fluctuations in energy and number of particles, the tight relation between the ensembles and the distributions obtained from the weak coupling simulations is demonstrated. The influence of the choice of the coupling parameters on the quality of the approximation of the ensemble distribution is discussed.     

Correlations in spiking neuronal networks with distance dependent connections

Can the topology of a recurrent spiking network be inferred from observed activity dynamics? Which statistical parameters of network connectivity can be extracted from firing rates, correlations and related measurable quantities? To approach these questions, we analyze distance dependent correlations of the activity in small-world networks of neurons with current-based synapses derived from a simple ring topology. We find that in particular the distribution of correlation coefficients of subthreshold activity can tell apart random networks from networks with distance dependent connectivity. Such distributions can be estimated by sampling from random pairs. We also demonstrate the crucial role of the weight distribution, most notably the compliance with Dales principle, for the activity dynamics in recurrent networks of different types.     

Bond Type and Discretization of Nonmuscle Myosin II Are Critical for Simulated Contractile Dynamics

Molecular motors drive cytoskeletal rearrangements to change cell shape. Myosins are the motors that move, cross-link, and modify the actin cytoskeleton. The primary force generator in contractile actomyosin networks is nonmuscle myosin II (NMMII), a molecular motor that assembles into ensembles that bind, slide, and cross-link actin filaments (F-actin). The multivalence of NMMII ensembles and their multiple roles have confounded the resolution of crucial questions, including how the number of NMMII subunits affects dynamics and what affects the relative contribution of ensembles’ cross-linking versus motoring activities. Because biophysical measurements of ensembles are sparse, modeling of actomyosin networks has aided in discovering the complex behaviors of NMMII ensembles. Myosin ensembles have been modeled via several strategies with variable discretization or coarse graining and unbinding dynamics, and although general assumptions that simplify motor ensembles result in global contractile behaviors, it remains unclear which strategies most accurately depict cellular activity. Here, we used an agent-based platform, Cytosim, to implement several models of NMMII ensembles. Comparing the effects of bond type, we found that ensembles of catch-slip and catch motors were the best force generators and binders of filaments. Slip motor ensembles were capable of generating force but unbound frequently, resulting in slower contractile rates of contractile networks. Coarse graining of these ensemble types from two sets of 16 motors on opposite ends of a stiff rod to two binders, each representing 16 motors, reduced force generation, contractility, and the total connectivity of filament networks for all ensemble types. A parallel cluster model, previously used to describe ensemble dynamics via statistical mechanics, allowed better contractility with coarse graining, though connectivity was still markedly reduced for this ensemble type with coarse graining. Together, our results reveal substantial tradeoffs associated with the process of coarse graining NMMII ensembles and highlight the robustness of discretized catch-slip ensembles in modeling actomyosin networks.     

Dynamical coding of sensory information with competitive networks.

Based on experiments with the locust olfactory system, we demonstrate that model sensory neural networks with lateral inhibition can generate stimulus specific identity-temporal patterns in the form of stimulus-dependent switching among small and dynamically changing neural ensembles (each ensemble being a group of synchronized projection neurons). Networks produce this switching mode of dynamical activity when lateral inhibitory connections are strongly non-symmetric. Such coding uses 'winner-less competitive' (WLC) dynamics. In contrast to the well known winner-take-all competitive (WTA) networks and Hopfield nets, winner-less competition represents sensory information dynamically. Such dynamics are reproducible, robust against intrinsic noise and sensitive to changes in the sensory input. We demonstrate the validity of sensory coding with WLC networks using two different formulations of the dynamics, namely the average and spiking dynamics of projection neurons (PN).     

Effect of Structure on Function in Model Nerve Nets

A theoretical analysis has been made on the effect of the pattern of interneuronal connectivity in model nerve nets on the activity of these nets. Two types of nets have been investigated: one in which the likelihood of a connection between a given neuron and any other element in the net is given by a Poisson probability distribution, and a second type in which the pattern of interconnection follows a Gaussian distribution. An analytical treatment is presented of the equations for noiseless nets in these two conditions. The principal result is that nets with Poisson connectivity law are activated by extraneous firing of a single neuron and continue in spontaneous activity indefinitely. On the other hand, similar nets in which the connections are, however, distributed according to a normal connectivity law, exhibit a definite threshold and produce spontaneous activity only subsequent to extraneous activation of a substantial fraction of the population. Moreover, spontaneous activity in Gaussian nets, but not in Poisson nets, becomes extinguished if the number of active neurons falls below the critical threshold. Some neuroanatomical implications are discussed which suggest that the pyramidal system of the cerebral cortex and other neuronal systems histologically characterized by large numbers of synapses per neuron may incorporate a Gaussian connectivity law, whereas a Poisson law may be characteristic of these cortical layers and nuclei primarily containing granule cells.     

Emergence of Complex Haplotypes from Microevolutionary Variation in Sequence and Structure of <Emphasis Type="Italic">Colias</Emphasis> Phosphoglucose Isomerase

A molecular evolutionary explanation of natural genetic variation requires analysis of specific variants’ evolutionary dynamics. To pursue this for phosphoglucose isomerase (PGI) of Colias butterflies, whose polymorphism is maintained by strong natural selection, we assembled a large data set of wild haplotypes, highly variable at the amino acid and DNA levels. The most common electrophoretic, i.e., charge macrostate, allele class, 3, is conserved in its pattern of charged amino acid residues. The next most common macrostate, 4, has multiple patterns of charge, i.e., microstates, while less common (1, 2, 5, 6) macrostates are very diverse. Macrostate 4 shows significant linkage disequilibrium (LD) among its variants, especially for two groups of five haplotypes each. We find extensive intragenic recombination among all haplotypes except the two high-LD groups of macrostate 4, which display none. Phyletic relations among haplotypes are largely reticulate, again except for the high-LD groups of macrostate 4, which form clades with strong bootstrap support. Charge-changing and linked charge-neutral amino acid variants occur in diverse parts of PGI’s sequence. Homology-based modeling of PGI’s structure shows that these regions are related spatially in ways suggesting functional interaction. The high-LD groups of macrostate 4 display parallel amino acid variation in these regions. This pattern of haplotype clades with high LD among multiple varying sites, emerging from chaotically recombining variation, may be a “signature” of refinement of complex adaptive sequences by recombination and selection. It should be tested further in this study system and others as a possibly general feature of the evolution of living complexity.     

Efficient RMSD measures for the comparison of two molecular ensembles. Root-mean-square deviation

Quantitative measures are presented for comparing the conformations of two molecular ensembles. The measures are based on Kabsch's formula for the root-mean-square deviation (RMSD) and the covariance matrix of atomic positions of isotropically distributed ensembles (IDE). By using a Taylor series expansion, it is shown that the RMSD can be expressed solely in terms of the IDE matrices. A fast approximate method is introduced for the pairwise RMSD determination whose computational cost scales linearly with the number of structures. A similarity measure for two structural ensembles that is based on the trace metric of the differences of powers of the IDE matrices is presented. The measures are illustrated for conformational ensembles generated by a molecular dynamics computer simulation of a partially folded A-state analog of ubiquitin.     

Neurobiologie und System-Theorie eines visuellen Muster-Erkennungsmechanismus bei Kröten

Quantitative behavioral experiments have shown that the toad uses mainly two types of gestalt information in prey/enemy discrimination: pattern extension in the direction of movement promotes, in general, the signal value prey, while extension perpendicular to the direction of movement promotes that of enemy. Registrations from single fibers and single cells at different stages on the visual path showed that the object extension perpendicular to the direction of movement is chiefly analysed by means of the retinal and thalamus pretectal nerve nets, whereas the extension in the direction of movement is analysed mostly by certain tectal nerve nets. Further neurobiological findings indicated that the prey/enemy discrimination is the result of subtractive interaction between the tectal and thalamus pretectal nerve nets. The system answers given by the retina, the retina-pretectum and the retinatectum to the input patterns used in the neurobiological experiments were determined for relevant space and time parameters on the basis of two dimensional neuron network models. The experimental results agree well with the theoretical ones. If the subtractive interaction between the model networks hypothesized from the neurophysiological results is applied, the resulting system answer describes the behavioral findings very well. So it is shown that the networks investigated would suffice in principle for the behavioral interpretations of the key stimulus prey/enemy — so far as these are known.

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Mit Unterstützung der Deutschen Forschungsgemeinschaft Ew 7/6 u. Forsch.-Gr. Az. 741,29.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

On the mathematical theory of rumor spread

The applicability of the theory of random nets to the theory of rumor spread is shown. In particular the “weak connectivity” of the net appears as the saturation fraction of “knowers” in a thoroughly mixed population through which a message diffuses where each knower tells the message to a finite average number of individuals. Further it is shown how the time course equation of rumor spread, where time is measured by the number of “removes” from the starters, can be translated into an ordinary continuous time course equation if the distribution of the telling intervals is known.     

Data Mechanics and Coupling Geometry on Binary Bipartite Networks

We quantify the notion of pattern and formalize the process of pattern discovery under the framework of binary bipartite networks. Patterns of particular focus are interrelated global interactions between clusters on its row and column axes. A binary bipartite network is built into a thermodynamic system embracing all up-and-down spin configurations defined by product-permutations on rows and columns. This system is equipped with its ferromagnetic energy ground state under Ising model potential. Such a ground state, also called a macrostate, is postulated to congregate all patterns of interest embedded within the network data in a multiscale fashion. A new computing paradigm for indirect searching for such a macrostate, called Data Mechanics, is devised by iteratively building a surrogate geometric system with a pair of nearly optimal marginal ultrametrics on row and column spaces. The coupling measure minimizing the Gromov-Wasserstein distance of these two marginal geometries is also seen to be in the vicinity of the macrostate. This resultant coupling geometry reveals multiscale block pattern information that characterizes multiple layers of interacting relationships between clusters on row and on column axes. It is the nonparametric information content of a binary bipartite network. This coupling geometry is then demonstrated to shed new light and bring resolution to interaction issues in community ecology and in gene-content-based phylogenetics. Its implied global inferences are expected to have high potential in many scientific areas.     

Direct measurement of the velocity of the metachronal wave in beating cilia

Recently a computerized electro-optical method was developed which enables one to simultaneously measure the frequency and the wavelength of the metachronal waves in beating cilia. The method is based on measurement of scattered light from two areas at a given distance apart. The distance between measured areas can be varied from zero to hundreds of microns. The relative ease of the measurement and data analysis of this method enable one to create large statistical ensembles in order to obtain reliable averages. In this work we show that in addition to the previously mentioned parameters this system can measure directly the velocity of the metachronal wave. It was found that the average velocity in the tissue culture from frog's palate epithelium at room temperature is approximately 270 micron/sec, about twice the average particle velocity at the frog's palate.     

Structure/thermodynamics relationships of lectin-saccharide complexes: the Erythrina corallodendron case.

Molecular dynamics (MD) simulations of Erythrina corallodendron lectin binding to a monosaccharide, alpha-galactose, and a disaccharide, N-acetyl lactosamine, have been performed in order to investigate the relationship between structure and thermodynamics. A simulated annealing protocol has been used to generate ensembles of structures for the two complexes, from which both qualitative and quantitative information on binding dynamics have been extracted. The ensembled averaged lectin-saccharide interaction enthalpy is equivalent for both sugars, whereas the calculation based on the X-ray structures does show a difference. Within large statistical errors, the calculated 'binding enthalpy' is also the same for the two systems. These errors arise largely from terms involving solvent and are a typical limitation of current MD simulations. Significant qualitative differences in binding between the two complexes are, however, observed over the ensembles. These could be important for unraveling the structure/thermodynamic relationship. Stated simply, there are a greater number of binding options available to the disaccharide compared to the monosaccharide. The implications of alternative binding states on thermodynamic parameters and the 'breaking of enthalpy-entropy compensation' are discussed. The role of solvent in lectin-saccharide complex formation is suggested to be significant.     

Visualization for understanding of neurodynamical systems

Complex neurodynamical systems are quite difficult to analyze and understand. New type of plots are introduced to help in visualization of high-dimensional trajectories and show global picture of the phase space, including relations between basins of attractors. Color recurrence plots (RPs) display distances from each point on the trajectory to all other points in a two-dimensional matrix. Fuzzy Symbolic Dynamics (FSD) plots enhance this information mapping the whole trajectory to two or three dimensions. Each coordinate is defined by the value of a fuzzy localized membership function, optimized to visualize interesting features of the dynamics, showing to which degree a point on the trajectory belongs to some neighborhood. The variance of the trajectory within the attraction basin plotted against the variance of the synaptic noise provides information about sizes and shapes of these basins. Plots that use color to show the distance between each trajectory point and a larger number of selected reference points (for example centers of attractor basins) are also introduced. Activity of 140 neurons in the semantic layer of dyslexia model implemented in the Emergent neural simulator is analyzed in details showing different aspects of neurodynamics that may be understood in this way. Influence of connectivity and various neural properties on network dynamics is illustrated using visualization techniques. A number of interesting conclusions about cognitive neurodynamics of lexical concept activations are drawn. Changing neural accommodation parameters has very strong influence on the dwell time of the trajectories. This may be linked to attention deficits disorders observed in autism in case of strong enslavement, and to ADHD-like behavior in case of weak enslavement.     

The dynamics of sparse random networks

Recurrent neural networks with full symmetric connectivity have been extensively studied as associative memories and pattern recognition devices. However, there is considerable evidence that sparse, asymmetrically connected, mainly excitatory networks with broadly directed inhibition are more consistent with biological reality. In this paper, we use the technique of return maps to study the dynamics of random networks with sparse, asymmetric connectivity and nonspecific inhibition. These networks show three qualitatively different kinds of behavior: fixed points, cycles of low period, and extremely long cycles verging on aperiodicity. Using statistical arguments, we relate these behaviors to network parameters and present empirical evidence for the accuracy of this statistical model. The model, in turn, leads to methods for controlling the level of activity in networks. Studying random, untrained networks provides an understanding of the intrinsic dynamics of these systems. Such dynamics could provide a substrate for the much more complex behavior shown when synaptic modification is allowed.