Random Loops Research Articles

Adaptive closed-loop modulation of cortical theta oscillations: Insights into the neural dynamics of navigational decision-making

Navigational decision-making tasks, such as spatial working memory (SWM), rely highly on information integration from several cortical and sub-cortical regions. Performance in SWM tasks is associated with theta rhythm, including low-frequency oscillations related to movement and memory. The interaction of the ventral hippocampus (vHPC) and medial prefrontal cortex (mPFC), reflected in theta synchrony, is essential in various steps of information processing during SWM. We used a closed-loop neurofeedback (CLNF) system to upregulate theta power in the mPFC and investigate its effects on circuit dynamics and behavior in animal models. Specifically, we hypothesized that enhancing the power of the theta rhythm in the mPFC might improve SWM performance. Animals were divided into three groups: closed-loop (CL), random-loop (RL), and OFF (without stimulation). We recorded local field potential (LFP) in the mPFC while electrical reward stimulation contingent on cortical theta activity was delivered to the lateral hypothalamus (LH), which is considered one of the central reward-associated regions. We also recorded LFP in the vHPC to evaluate the related subcortical neural changes. Results revealed a sustained increase in the theta power in both mPFC and vHPC for the CL group. Our analysis also revealed an increase in mPFC-vHPC synchronization in the theta range over the stimulation sessions in the CL group, as measured by coherence and cross-correlation in the theta frequency band. The reinforcement of this circuit improved spatial decision-making performance in the subsequent behavioral results. Our findings provide direct evidence of the relationship between specific theta upregulation and SWM performance and suggest that theta oscillations are integral to cognitive processes. Overall, this study highlights the potential of adaptive CLNF systems in investigating neural dynamics in various brain circuits.

Effect of active loop extrusion on the two-contact correlations in the interphase chromosome.

The population-averaged contact maps generated by the chromosome conformation capture technique provide important information about the average frequency of contact between pairs of chromatin loci as a function of the genetic distance between them. However, these datasets do not tell us anything about the joint statistics of simultaneous contacts between genomic loci in individual cells. This kind of statistical information can be extracted using the single-cell Hi-C method, which is capable of detecting a large fraction of simultaneous contacts within a single cell, as well as through modern methods of fluorescent labeling and super-resolution imaging. Motivated by the prospect of the imminent availability of relevant experimental data, in this work, we theoretically model the joint statistics of pairs of contacts located along a line perpendicular to the main diagonal of the single-cell contact map. The analysis is performed within the framework of an ideal polymer model with quenched disorder of random loops, which, as previous studies have shown, allows us to take into account the influence of the loop extrusion process on the conformational properties of interphase chromatin.

Coexistence, enhancements and short loops in random walk loop soups

Coexistence, enhancements and short loops in random walk loop soups

Statistical modeling and significance estimation of multi-way chromatin contacts with HyperloopFinder.

Recent advances in chromatin conformation capture technologies, such as SPRITE and Pore-C, have enabled the detection of simultaneous contacts among multiple chromatin loci. This has made it possible to investigate the cooperative transcriptional regulation involving multiple genes and regulatory elements at the resolution of a single molecule. However, these technologies are unavoidably subject to the random polymer looping effect and technical biases, making it challenging to distinguish genuine regulatory relationships directly from random polymer interactions. Here, we present HyperloopFinder, a method for identifying regulatory multi-way chromatin contacts (hyperloops) by jointly modeling the random polymer looping effect and technical biases to estimate the statistical significance of multi-way contacts. The results show that our model can accurately estimate the expected interaction frequency of multi-way contacts based on the distance distribution of pairwise contacts, revealing that most multi-way contacts can be formed by randomly linking the pairwise contacts adjacent to each other. Moreover, we observed the spatial colocalization of the interaction sites of hyperloops from image-based data. Our results also revealed that hyperloops can function as scaffolds for the cooperation among multiple genes and regulatory elements. In summary, our work contributes novel insights into higher-order chromatin structures and functions and has the potential to enhance our understanding of transcriptional regulation and other cellular processes.

Computing complete solution sets for approximate four-bar path synthesis

The approximate path synthesis of four-bar linkages has been framed and solved with many different optimization techniques. Here we present a polynomial objective that is invariant to the number of approximate design positions selected, and a solution technique capable of finding all minima. The invariance property caps compute time despite increasing the size of input task specification data. This is performed by collecting a variable amount of task data into an invariable number of polynomial coefficients, called moments, before numerical optimization begins. The minima are found by applying the method of random monodromy loops to the zero gradient polynomial system of the aforementioned objective. This procedure finds all critical points, including the local and global minimum, and provides an in-process estimate of the percentage of critical points found. We applied our methodology to four-bar path synthesis problems of various computational scales by altering dimensional pre-specifications. The most general case was estimated to have 1,820,238 ± 3810 critical points, while pre-specification of one or two ground pivots yielded 26,052 and 503 roots, respectively, as validated by a trace test. The results are applied to a variety of examples.

Cover Times of the Massive Random Walk Loop Soup

We study cover times of subsets of Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^2$$\\end{document} by a two-dimensional massive random walk loop soup. We consider a sequence of subsets An⊂Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n \\subset {\\mathbb {Z}}^2$$\\end{document} such that |An|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|A_n| \\rightarrow \\infty $$\\end{document} and determine the distributional limit of their cover times T(An)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}(A_n)$$\\end{document}. We allow the killing rate κn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _n$$\\end{document} (or equivalently the “mass”) of the loop soup to depend on the size of the set An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n$$\\end{document} to be covered. In particular, we determine the limiting behavior of the cover times for inverse killing rates all the way up to κn-1=|An|1-8/(loglog|An|),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _n^{-1}=|A_n|^{1-8/(\\log \\log |A_n|)},$$\\end{document} showing that it can be described by a Gumbel distribution. Since a typical loop in this model will have length at most of order κn-1/2=|An|1/2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _n^{-1/2}=|A_n|^{1/2},$$\\end{document} if κn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _n^{-1}$$\\end{document} exceeded |An|,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|A_n|,$$\\end{document} the cover times of all points in a tightly packed set An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n$$\\end{document} (i.e., a square or close to a ball) would presumably be heavily correlated, complicating the analysis. Our result comes close to this extreme case.

Footprints of loop extrusion in statistics of intra-chromosomal distances: An analytically solvable model.

Active loop extrusion-the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes-is a firmly established mechanism responsible for chromatin spatial organization at different stages of a cell cycle in eukaryotes and bacteria. The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model. Here, we consider the simplest analytically solvable model of an interphase chromosome, which is treated as an ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci, which is valid beyond the one-loop approximation in diagrammatic representation. In addition, we analyze the loop-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector. The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.

Intrachain Distances in a Crumpled Polymer with Random Loops

Intrachain Distances in a Crumpled Polymer with Random Loops

Macroscopic Loops in the Bose Gas, Spin O(N) and Related Models

We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model,random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in {mathbb {Z}}^d, d ge 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate {mathbb {Z}}^d by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.

Scalar Conformal Primary Fields in the Brownian Loop Soup

The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity \(\lambda >0\), with central charge \(c=2 \lambda \). Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary “layering vertex operators” \(\mathcal {O}_{\beta }\) with dimensions \((\Delta , \Delta )\), with \(\Delta = \frac{\lambda }{10}(1-\cos \beta )\), that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions \((\Delta + k/3, \Delta + k'/3)\), for all non-negative integers \(k, k'\) satisfying \(|{k-k'}| = 0 \mod 3\). In this paper we introduce the edge counting field \({\mathcal {E}}(z)\) that counts the number of loop boundaries that pass close to the point z. We rigorously prove that the n-point functions of \({\mathcal {E}}\) are well defined and behave as expected for a conformal primary field with dimensions (1/3, 1/3). We analytically compute the four-point function \(\left\langle {\mathcal {O}_{\beta }(z_1) \mathcal {O}_{-\beta }(z_2) \mathcal {E}(z_3) \mathcal {E}(z_4)}\right\rangle \) and analyze its conformal block expansion. The operator product expansions of \(\mathcal {E} \times \mathcal {E}\) and \(\mathcal {E} \times \mathcal {O}_{\beta }\) contain higher-order edge operators with “charge” \(\beta \) and dimensions \((\Delta + k/3, \Delta + k/3)\). Hence, we have explicitly identified all scalar primary operators among the new set mentioned above. We also re-compute the central charge by an independent method based on the operator product expansion and find agreement with previous methods.

The Brownian loop soup stress-energy tensor

The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity λ > 0. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is c = 2λ. In the case of domains with boundary we identify a boundary operator that has properties consistent with the boundary stress-energy tensor. We show that this operator generates local deformations of the boundary and that it is related to a boundary operator that induces a Brownian excursion starting or ending at its insertion point.

Interacting Loop Ensembles and Bose Gases

We study interacting Bose gases in thermal equilibrium on a lattice. We establish convergence of the grand-canonical Gibbs states of such gases to their mean-field (classical field) and large-mass (classical particle) limits. The former is a classical field theory for a complex scalar field with quartic self-interaction. The latter is a classical theory of point particles with two-body interactions. Our analysis is based on representations in terms of ensembles of interacting random loops, the Ginibre loop ensemble for Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, our results also hold in infinite volume. The results of this paper were previously sketched in Fröhlich et al. (J Stat Phys 180(1–6):810–831, 2020).

Bioinformatics analysis and pharmacological effect of Stevia rebaudiana in the prevention of type-2 diabetes

Different groups of enzymes and regulatory elements are involved in the synthesis of Ribodioside A, which is one of the most important sweetening compounds in stevia. The UGT family (UDP-glycosyltransferase) is a group of regulatory genes that are very effective in converting steviol glycoside to Ribodioside A. Bioinformatics analyses on this gene family, which included the UGT74G1, UGT76G1, and UGT85C2 genes, showed that the protein encoded by these genes had a UDPGT protected protein domain. Also, the study of the secondary structure of these proteins showed that the total corrosion of these proteins is mainly from alpha-helix and random screws or loops that are connected with linear strands. Also, the study of the secondary structure of these proteins showed that the total corrosion of these proteins is mainly from alpha-helix and random screws or loops that are connected with linear strands. The results of studying the three-dimensional structure of the studied proteins confirmed the previous findings of high genomic similarity between these proteins. The results of the ProtScale program showed that the abundance of amino acids with negative hydropathicity in the sequence of these proteins is high, which is effective in creating plant resistance to drought stress. Finally, the codon preference trend of these proteins was investigated using the sequence manipulation suite database. This information can be used for other research, including the transfer of these proteins.

Topological expansion in isomorphism theorems between matrix-valued fields and random walks

We consider Gaussian fields of real symmetric, complex Hermitian or quaternionic Hermitian matrices over an electrical network, and describe how the isomorphisms between these fields and random walks give rise to topological expansions encoded by ribbon graphs. We further consider matrix-valued Gaussian fields twisted by an orthogonal, unitary or symplectic connection. In this case the isomorphisms involve traces of holonomies of the connection along random walk loops parametrized by boundary cycles of ribbon graphs.

Image encryption-then-transmission combining random sub-block scrambling and loop DNA algorithm in an optical chaotic system

In this paper, an image encryption scheme based on random sub-block scrambling and loop deoxyribonucleic acid algorithm in optical chaotic system is proposed. A main laser drives four twin slaver laser pairs to generate high random and complex chaos sequences, which are employed for encryption and transmission. Multiplying eight control parameters (stemming from Hash algorithm between the external key and plain image) by the three groups of optical chaotic sequences, we yield seven pseudo-random sequences. While the fourth optical chaotic carrier is used for transmitting the encrypted image data. Utilizing the random sub-block scrambling and loop deoxyribonucleic acid (DNA) encryption algorithm, we achieve the shuffling and diffusion of image. Furthermore, we carry out a quantitative analysis aimed at the security of image encryption algorithm, and the results show that the image encryption method can offer better security and encryption efficiency. Moreover, we numerically simulate the image data transmission using a chaotic carrier, and decode the information by monitoring the synchronization error between emitter and receiver successfully. The fact implies that the security is further enhanced.

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning tree constants are then given by T(z = 1). Here an extended T e (z) is constructed, relaxing the previous vt condition to q-regular lattices L and likewise leading to z L = T e (1). Further, in the vt case the method to derive T e yields a new integral formula for the lattice Green function. As examples, spanning tree generating functions for all the eleven vt Archimedean (surprisingly, easier to obtain from T e than from T) and two relevant non-vt, martini and the (4, 82) covering/medial, lattices are derived. The importance of T e (and T)—beyond just being a tool to calculate z L —is illustrated with the proof that the free energy of the random walk loop soup model (defined from closed random walks, the loops, over L) can be written in terms of T e (z). From such result, it is shown that the system critical point—existing only for d = 1 and d = 2—is directly related to z L .

Good speciation and endogenous business cycles in a constraint satisfaction macroeconomic model

We introduce a prototype agent-based model of the macroeconomy, with budgetary constraints at its core. The model is related to a class of constraint satisfaction problems (CSPs), which has been thoroughly investigated in computer science. The CSP paradigm allows us to propose an alternative price-setting mechanism: given agents’ preferences and budgets, what set of prices satisfies the maximum number of agents? Such an approach permits the coupling of production and output within the economy to the allowed level of debt in a simplified framework. Within our model, we identify three different regimes upon varying the amount of debt that each agent can accumulate before defaulting. In presence of a very loose constraint on debt, endogenous crises leading to waves of synchronized bankruptcies are present. In the opposite regime of very tight debt constraining, the bankruptcy rate is extremely high and the economy remains structure-less. In an intermediate regime, the economy is stable with very low bankruptcy rate and no aggregate-level crises. This third regime displays a rich phenomenology: the system spontaneously and dynamically self-organizes in a set of cheap and expensive goods (i.e. some kind of ‘speciation’), with switches triggered by random fluctuations and feedback loops. Our analysis confirms the central role that debt levels play in the stability of the economy. More generally, our model shows that constraints at the individual scale can generate highly complex patterns at the aggregate level.

DNA looping by protamine follows a nonuniform spatial distribution

DNA looping plays an important role in cells in both regulating and protecting the genome. Often, studies of looping focus on looping by prokaryotic transcription factors like lac repressor or by structural maintenance of chromosomes proteins such as condensin. Here, however, we are interested in a different looping method whereby condensing agents (charge ≥+3) such as protamine proteins neutralize the DNA, causing it to form loops and toroids. We considered two previously proposed mechanisms for DNA looping by protamine. In the first mechanism, protamine stabilizes spontaneous DNA fluctuations, forming randomly distributed loops along the DNA. In the second mechanism, protamine binds and bends the DNA to form a loop, creating a distribution of loops that is biased by protamine binding. To differentiate between these mechanisms, we imaged both spontaneous and protamine-induced loops on short-length (≤1 μm) DNA fragments using atomic force microscopy. We then compared the spatial distribution of the loops to several model distributions. A random looping model, which describes the mechanism of spontaneous DNA folding, fit the distribution of spontaneous loops, but it did not fit the distribution of protamine-induced loops. Specifically, it failed to predict a peak in the spatial distribution of loops at an intermediate location along the DNA. An electrostatic multibinding model, which was created to mimic the bind-and-bend mechanism of protamine, was a better fit of the distribution of protamine-induced loops. In this model, multiple protamines bind to the DNA electrostatically within a particular region along the DNA to coordinate the formation of a loop. We speculate that these findings will impact our understanding of protamine’s in vivo role for looping DNA into toroids and the mechanism of DNA condensation by condensing agents more broadly.

Synthesis of Watt-Type Timed Curve Generators and Selection From Continuous Cognate Spaces

Abstract Following recent work on Stephenson-type mechanisms, the synthesis equations of Watt six-bar mechanisms that act as timed curve generators are formulated and systematically solved. Four variations of the problem arise by assigning the actuator and end effector onto different links. The approach produces exact synthesis of mechanisms up to eight precision points. Polynomial systems are formulated and their maximum number of solutions is estimated using the algorithm of random monodromy loops. Certain variations of Watt timed curve generators possess free parameters that do not affect the output motion, indicating a continuous space of cognate mechanisms. Packaging compactness, clearance, and dimensional sensitivity are characterized across the cognate space to illustrate trade-offs and aid in selection of a final mechanism.

Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application, we derive site-monotonicity properties for the spin–spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates—improving previous positivity results which hold for the Cesàro sum. We also derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model and lattice permutations, thus extending the previous results of Lees and Taggi (2019).