Finite Dynamical Systems Research Articles

Modeling and Analysis of Germ Layer Formations Using Finite Dynamical Systems

Modeling and Analysis of Germ Layer Formations Using Finite Dynamical Systems

An optimal bound on the number of moves for open mancala

We determine the optimal bound for the maximum number of moves required to reach a periodic configuration of open mancala (also called open owari), inspired by a popular African game. A mancala move can be interpreted as a map from the set of compositions of a given integer in itself, thus relating our result to the study of the corresponding finite dynamical system.

Dichotomy results for fixed point counting in boolean dynamical systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.

Limit cycle structure for dynamic bi-threshold systems

In this paper we determine the limit cycle structures of synchronous and sequential finite dynamical systems governed by dynamic bi-threshold functions over non-uniform networks. So far, work in this area has been concerned with static threshold values, and our results generalize these. In particular, this includes the celebrated result by Goles and Olivos on synchronous neural networks (1981) [5], the work by Kuhlman et al. (2012) [12] on static bi-threshold systems, and the work by Chang et al. (2014) [2] on dynamic standard threshold systems.In our work, the state of each vertex v includes the usual binary state xv∈{0,1}, the dynamic up-thresholds kv↑ and down-thresholds kv↓ whose values change only upon a transition of xv. We show that the projection of the periodic orbits onto the Boolean x-dynamics have maximal size 2 under the synchronous update method regardless of how the up- and down-thresholds evolve. The proof is a careful extension of the technique originally developed in the proof by Goles and Olivos (1981) [5] and which was later modified by Kuhlman et al. (2012) [12] to cover bi-threshold systems. We also derive sufficient conditions on the evolution of the up- and down-thresholds that ensures that the sequential systems only have fixed points as limit sets. The results should be relevant for modeling and analyzing a large range of social and biological systems where agent adaptation occurs and needs to be accounted for.

On a variational sequential bargaining pricing scheme

We propose a minimization problem as a model for the interaction between two agents trading a contingent claim in an incomplete discrete-time multiperiod financial market. The agents personal valuations for the contingent claim are assumed to depend on probability measures representing their beliefs concerning the future states of the world. The agents’ goal is to achieve a common price for the contingent claim to be traded, while deviating as litle as possible from their initial beliefs. Under appropriate conditions, we prove that the minimization problem under consideration admits at least one solution. Furthermore, we provide a detailed description for the minimizers – orbits of a finite horizon discrete-time dynamical system on the space of probability measures representing the agents beliefs.

AND-NOT logic framework for steady state analysis of Boolean network models

In this paper we propose the class of AND-NOT networks for modeling biological systems and show that it provides several advantages. Some of the advantages include: Any finite dynam ical system can be written as an AND-NOT network with similar dynamical properties. There is a one-to-one correspondence between AND-NOT networks, their wiring diagrams, and their dynamics. Results about AND-NOT networks can be stated at the wiring diagram level without losing any information. Results about AND-NOT networks are applicable to any Boolean network. We apply our results to a Boolean model of Th-cell differentiation.

Rigidity and non-recurrence along sequences

AbstractWe study two properties of a finite measure-preserving dynamical system and a given sequence$({n}_{m} )$of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some weakly mixing system or more generally for some ergodic system. The main focus is on weakly mixing systems. For example, we show that for any integer$a\geq 2$the sequence${n}_{m} = {a}^{m} $is a sequence of rigidity for some weakly mixing system. We show the same for the sequence of denominators of the convergents in the continued fraction expansion of any irrational$\alpha $. We also consider the stronger property of IP-rigidity. We show that if$({n}_{m} )$grows fast enough then there is a weakly mixing system which is IP-rigid along$({n}_{m} )$and non-recurrent along$({n}_{m} + 1)$.

Phase-locking, quasiperiodicity and chaos in periodically driven noisy neuronal models: a spectral approach

In auditory experiments neurons are often driven by periodic or periodically modulated inputs. The response of the neurons is often periodic and phase-locked to the stimulus. However, this is not always the case. It is well-known that even simple deterministic models can exhibit quasiperiodic behavior, with no clear relationship between the phases of the stimulus and response. Moreover, the biological situation is further complicated by presence of noise in the periodic inputs and intrinsic cellular properties, e.g. firing rate adaptation. We develop analytical tools for detection of parameter regimes corresponding to different locking behaviors in more detailed stochastic biophysical models, so that these regimes can be avoided or chosen as necessary. In an effort to develop a mathematical theory applicable to the above biologically-motivated project on phase-locking and related phenomena, we have developed a spectral approach to stochastic circle maps. A stochastic circle map is defined as a Markov chain on the circle. This abstract class of objects includes a wide range of models for firing times of periodically forced noisy neuronal models. We define and analyze some dynamic properties of the system such as stochastic phase locking and stochastic bifurcation cascades. The main tool is analysis of the spectrum and eigenspaces of the transition operator of the Markov chain. We relate the dynamics of the stochastic system to those of a few identifiable finite state deterministic dynamical systems, the dynamics of which can in turn be deduced from the spectral properties of the corresponding matrices. We also show that the firing regime of the stochastic system can be determined by the shape of its eigenvalue “cloud.”

An Algebraic Approach to Reverse Engineering Finite Dynamical Systems Arising from Biology

Finite dynamical systems have been used successfully in modeling biological processes. When certain regulatory mechanisms of a biological system or a model are unknown it is important to be able to identify the best model with the available data. In this context, reverse engineering of finite dynamical systems from partial information is an important problem. While this problem has been studied in the past, there are currently no algorithms that can predict the signs of the interactions. In this paper we propose a framework and algorithms to reverse engineer the possible signed wiring diagrams of a finite dynamical system from data. The algorithm consists of encoding all possible wiring diagrams using ideals and algebraic sets and choosing those that are minimal using the primary decomposition and the irreducible components.

Stability Analysis for Discrete Biological Models Using Algebraic Methods

This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Grobner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.

ON THE STABILITY OF SELF-PROPELLED BODIES WITH RESPECT TO THEIR SHAPE MOTION

This paper studies the stability of the trajectories of self-propelled bodies immersed in a fluid at low Reynolds number, with respect to their dynamic shape deformation. We consider both adherence and perfect slip boundary conditions on the moving body. Using shape derivative arguments in association with recent higher order regularity results for the Bogovskiĭ operator, we bring the shape stability question to a finite dimension dynamical system analysis.

Dynamics of boolean networks

Boolean networks are special types of finite state time-discrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a Boolean network's dynamical information via its disjunctive normal form is explained. For a given Boolean network, a matrix with entries 0 and 1 is associated with the polynomial function that represents the network, then the information on the fixed points and the limit cycles is derived by analyzing the matrix. The described method provides an algorithm for the determination of the fixed points from the polynomial expression of a Boolean network. The method can also be used to construct Boolean networks with prescribed limit cycles and fixed points. Examples are provided to explain the algorithm.

Update sequence stability in graph dynamical systems

In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability.

Self-excited oscillations of a string on an elastic foundation subject to a nonlinear feed-forward force

We investigate the self-excited oscillations of a string on an elastic foundation that is subject to a nonlinear feed-forward force. The feed-forward follows that of a model first proposed by Steele and Baker [1] for an active cochlear, and is due to the gain factor profile which depends on the string displacement. In order to determine the bifurcation structure induced by the nonlinear feed-forward mechanism, we formulate a taut string initial-boundary-value problem with periodic boundary conditions which is reduced to a finite order modal dynamical system. We employ an asymptotic multiple-scales method to obtain slowly varying evolution equations that enable an analytical derivation of the periodic system response and analysis of its orbital stability. The resulting bifurcation structure includes multiple regions of both stable and unstable coexisting periodic solutions defined by primary and secondary Hopf stability thresholds. Numerical verification of the bifurcation structure determines the accuracy of the analytically predicted periodic self-excited response and reveals the existence of quasiperiodic combination-tone solutions and complex nonstationary solutions that emerge in a range of the asymptotically predicted unstable solutions. This analysis enables construction of a comprehensive analytical bifurcation structure and may shed light on mechanisms governing complex multi-component spectra that have been documented for spontaneous otoacoustic emissions in the mammalian inner ear.

Abstract Hopf bifurcation theorem and further extensions via second variation

Sufficient conditions of bifurcation stated in Arutyunov et al. (2009) [5] are investigated in order to reconsider celebrated Hopf bifurcation as the simplest bifurcation of Fredholm operators of zero index. In several examples abstract result is applied to both finite and infinite dynamical systems exhibiting classical Hopf bifurcation as well as double Hopf bifurcation.

Discretization of Time Series Data

An increasing number of algorithms for biochemical network inference from experimental data require discrete data as input. For example, dynamic Bayesian network methods and methods that use the framework of finite dynamical systems, such as Boolean networks, all take discrete input. Experimental data, however, are typically continuous and represented by computer floating point numbers. The translation from continuous to discrete data is crucial in preserving the variable dependencies and thus has a significant impact on the performance of the network inference algorithms. We compare the performance of two such algorithms that use discrete data using several different discretization algorithms. One of the inference methods uses a dynamic Bayesian network framework, the other-a time-and state-discrete dynamical system framework. The discretization algorithms are quantile, interval discretization, and a new algorithm introduced in this article, SSD. SSD is especially designed for short time series data and is capable of determining the optimal number of discretization states. The experiments show that both inference methods perform better with SSD than with the other methods. In addition, SSD is demonstrated to preserve the dynamic features of the time series, as well as to be robust to noise in the experimental data. A C++ implementation of SSD is available from the authors at http://polymath.vbi.vt.edu/discretization .

Reverse engineering time discrete finite dynamical systems: a feasible undertaking?

With the advent of high-throughput profiling methods, interest in reverse engineering the structure and dynamics of biochemical networks is high. Recently an algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is a top-down approach using time discrete dynamical systems. One of its key steps includes the choice of a term order, a technicality imposed by the use of Gröbner-bases calculations. The aim of this paper is to identify minimal requirements on data sets to be used with this algorithm and to characterize optimal data sets. We found minimal requirements on a data set based on how many terms the functions to be reverse engineered display. Furthermore, we identified optimal data sets, which we characterized using a geometric property called “general position”. Moreover, we developed a constructive method to generate optimal data sets, provided a codimensional condition is fulfilled. In addition, we present a generalization of their algorithm that does not depend on the choice of a term order. For this method we derived a formula for the probability of finding the correct model, provided the data set used is optimal. We analyzed the asymptotic behavior of the probability formula for a growing number of variables n (i.e. interacting chemicals). Unfortunately, this formula converges to zero as fast as , where and . Therefore, even if an optimal data set is used and the restrictions in using term orders are overcome, the reverse engineering problem remains unfeasible, unless prodigious amounts of data are available. Such large data sets are experimentally impossible to generate with today's technologies.

Linear dynamical systems over finite rings

The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring R such that | R | = q , where q is a positive integer, the algorithm determines whether a given linear system over R n is a fixed point system or not in time O ( n 3 log ( n log ( q ) ) ) .

Effects of nondenumerable fixed points in finite dynamical systems

The motion of a spinning soccer ball brings forth the possible existence of a whole class of finite dynamical systems where there may be a nondenumerably infinite number of fixed points. They defy the very traditional meaning of the fixed point that a point on the fixed point in the phase space should remain there forever, for, a fixed point can evolve as well! Under such considerations one can argue that a free-kicked soccer ball should be nonchaotic.

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes $${\mathcal{F}}$$ and graph classes $${\mathcal{G}}$$ , an ( $${\mathcal{F}},{\mathcal{G}}$$ )-system is a Boolean dynamical system such that all local transition functions lie in $${\mathcal{F}}$$ and the underlying graph lies in $${\mathcal{G}}$$ . Let $${\mathcal{F}}$$ be a class of Boolean functions which is closed under composition and let $${\mathcal{G}}$$ be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the planar graphs, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the graphs having vertex covers of size one, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.