Understanding the mechanisms involved in oscillatory biological regulation is a fundamental issue to analyze living systems. Oscillators convert direct current dc from a power supply to an alternating current ac signal. Designing oscillators it is important, however, to identify the two functions and assure that they are working properly under influence of each other including the ability to start oscilla ting at the required frequency. Continuous, deterministic simulations show similar behavior to that of the toy model. Synchronization of pulsecoupled biological oscillators. A simple model for synchronous firing of biological oscillators based on peskins model of the cardiac pacemaker mathematical aspects of heart physiology, courant institute of mathematical sciences, new york university, new york, 1975, pp. We show that stochastic model checking verification of biological systems can. Modelling biological oscillations modelling neural oscillation fitzhughnagumo oscillators i proposed by richard fitzhugh in 1961 and implemented by nagumo using electronic circuit in 1962. Analysing biochemical oscillation through probabilistic model. The harmonic, or linear, oscillator produces a sinusoidal output. To maintain the conditions for an undamped resonator, the model of the circuitry that delivers power must contain a conductance in parallel to the load of same size but with negative sign.
I fitzhughnagumo model is a simpli ed version of the famous. Biological oscillators periodic fluctuations in biological processes are found at all levels of life and are frequently the result of changes to gene expression. For the case of identical oscillators the subject of this paper, these. Biological oscillators have been studied extensively as they form the core of many crucial biological processes such as circadian rhythms and the cell cycle. Biological and biochemical oscillators 1st edition. Interaction with extremely low frequency fields, radio sci. The threegenes oscillation model was extended to an n coupled genetic model by strelkowa and barahona, 11 hori, hara and kim, 12 zeiser, muller and liebscher, and hori and hara. An electronic oscillator is an electronic circuit that produces a periodic, oscillating electronic signal, often a sine wave or a square wave or a triangle wave. Stochastic bifurcations in a birhythmic biological model.
As outlined in the introduction, probabilistic model checking algorithms can either be exact e. The most common form of linear oscillator is an electronic amplifier such as a transistor or operational amplifier connected in a feedback loop with its output fed back into its input through a frequency selective electronic filter to provide positive feedback. Biological and biochemical oscillators compiles papers on biochemical and biological oscillators from a theoretical and experimental standpoint. Dynamic modelling is a core element in the systems biology approach to understanding complex biosystems. Model checking biological oscillators sciencedirect. Biological oscillators ieee control systems society. However, inhibitory or repulsive coupling is very common in biological systems. A comparative analysis of synthetic genetic oscillators. Abstractwe define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. University of camerino model checking biological oscillators hscb 2009. This book discusses the oscillatory behavior, excitability, and propagation phenomena on membranes and membranelike interfaces. The sorts of experimental data which are needed to.
Detecting synchronisation of biological oscillators by model. Experimentally driven verification of synthetic biological. We also introduce a logic, biological oscillators synchronisation logic bosl, that is able to describe collective synchronisation properties of a population of coupled oscillators. A bayesian approach to model checking biological systems 219 checking 15 as a powerful tool forformally reasoning about the dynamic properties of such models e. The function ij represents how other oscillators actions affect oscillator iand the scalar 0 is a. The application of mathematical modelling to molecular cell biology is not a new endeavour. Probabilistic model checking can be used to formally specify performance. Generating a biological oscillator science signaling. In sections 3, 4 and 5, we consider three examples to demonstrate both how to model biological pathways in the prism language and how probabilistic model checking can be used to analyse such pathways. Feedback and control mechanisms, hodgkinhuxley theory for nerve membrane. In recent years, model checking has become a familiar tool in software and hardware industries. The meacham 9, 10 and the hewlett oscillators are bridge oscillators.
Biological oscillators, ordinary di erential equations, markov models, stochastic model checking, simulation 1 introduction the quantitative analysis of the dynamics of biological systems is a fundamental task in systems biology, a new research eld that focuses on the systematic. Measuring direction in the coupling of biological oscillators. Pdf detecting synchronisation of biological oscillators by. Biological oscillators have been studied extensively as they form the core of. We define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. Formal verification using model checking researchgate. The correctness of the designs is established by systematic analysis of their inputoutput behaviour using simulation and probabilistic model checking for uniform reaction rates and low molecular count. This approach is known as systems biology where dynamical systems and control theory play crucial roles 1. Oscillation is the repetitive variation, typically in time, of some measure about a central value often a point of equilibrium or between two or more different states the term vibration is precisely used to describe mechanical oscillat. A feedback quenched repressilator produces turing pattern. I used to describes how action potentials in neurons are initiated and propagated. A simple model for synchronous firing of biological oscillators based on peskins model of.
Then, we introduce a logic, kuramoto synchronization logic ksl, and a model checking algorithm in order to verify collective synchronization properties of a population of coupled oscillators. Oscillators that sync and swarm nature communications. Analyzing resilience properties in oscillatory biological. A probabilistic bounded ltl pbltl formula is a formula of the form p. Pulsecoupled oscillators in the canonical model of pulsecoupled oscillators 21, an oscillator j2nsends out a pulse whenever it crosses zero j 0. Coates 200720 the actual value of m depends on how effectively the two inductors are magnetically coupled, which among other factors depends on the spacing between the inductors, the number of turns on each inductor, the dimensions of each coil and the material of the common core. Lecture notes how to build a biological oscillator. Bridge oscillators seem to be candidates for realizing good approximations to linear oscillators. Electronic notes in theoretical computer science 229 2009 4158. Designing oscillators it is important, however, to identify the two functions and assure that. Automated analysis of biological oscillator models using mode decomposition. Modeling, analysis, and control of biological oscillators.
They are widely used in many electronic devices ranging from simplest clock generators to digital instruments like calculators and. E model checking of the stochastic systems by resampling and. Modeling the avian circadian system with coupled nonlinear. A model checking algorithm is proposed for the defined logic and it is implemented in a model checker. In section 3 we start with a simple set of reactions. Formal methods for biological systems carnegie mellon university. A bayesian approach to model checking biological systems.
Sales tax will be calculated at checkout price includes vatgst. Purchase biological and biochemical oscillators 1st edition. The x and y oscillators are indirectly coupled via the woscillator, that is they form a. Jul 01, 2014 statistical model checking smc is a highly scalable simulationbased verification approach for testing and estimating the probability that a stochastic system satisfies a given linear temporal property. Feedback and oscillators tlt8016 basic analog circuits 20052006 2 9. Pdf model checking biological oscillators luca tesei. Time delays play a major role in the sustainability and control of oscillations, as shown for example in phenomena related to the mammalian circadian clock leloup and goldbeter, 2003. Pdf detecting synchronisation of biological oscillators.
The kuramoto model mnsim modelling, numerical simulation. Biological and biochemical oscillators 1st edition elsevier. Several decades later, a three component loop helped to inspire the field of synthetic biology, when michael elowitz and stanislas leibler in 2000 built a cycle of three repressors, called the repressilator fig 10. Then, we introduce a logic, kuramoto synchronization logic ksl, and.
Design of synthetic genetic oscillators using evolutionary. Analysing biochemical oscillation through probabilistic. In other cases, the biological role of synchronization is still under discussi. Feb 28, 2009 model checking biological oscillators ezio bartocci, 1 flavio corradini, 2 emanuela merelli, 3 luca tesei 4 dipartimento di matematica e informatica university of camerino via madonna delle carceri, 9, camerino mc, 62032, italy abstract we define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. It is not a comprehensive description of all known biological oscillators, since this would require a much bigger volume as well as a different. Stochastic model checking of genetic circuits acm journal. The hewlett oscillator is the wellknown wien bridge oscillator 11, 12. To prove that the cube c is forward invariant, we need to check the sign of the vector field gx. The model consists of a population of identical integrateandfire oscillators. Parameter estimation in models of biological oscillators.
The aim of this work is to generalize and systematize the model validation and invalidation approach based on the frequency. Coates 200720 sweep oscillators a sweep waveform is another name for a sawtooth wave. The technique has been applied to discrete and continuous time markov chains, stochastic timed automata and most recently hybrid systems using the tool uppaal smc. This article describes how stochastic model checking, a method for determining the likelihood that certain events occur in a system, can by applied to models of genetic circuits by translating them into continuoustime markov chains ctmcs and analyzing them using markov chain analysis to check continuous stochastic logic csl properties. Emanuela merelli joint work with university of camerino italy. Stochastic bifurcations in a birhythmic biological model with. Feedback and oscillators tlt8016 basic analog circuits 20052006 16 the effects of voltage feedback on output impedance the effects of current feedback on output impedance figure 9. A simple model for phase locking of biological oscillators. Detecting synchronisation of biological oscillators by. As in the soliton story, these numerical observations then inspired a series of theoretical advances. Repulsive synchronization in an array of phase oscillators. This paper presents a new model checking algorithm that is wellsuited for verifying properties of very. A statistical approach reveals designs for the most robust.
Model checking biological oscillators ezio bartocci, 1 flavio corradini, 2 emanuela merelli, 3 luca tesei 4 dipartimento di matematica e informatica university of camerino via madonna delle carceri, 9, camerino mc, 62032, italy abstract we define a subclass of timed automata, called oscillator timed automata, suitable to model biological oscillators. Identical phase oscillators with global sinusoidal coupling. Oscillating systems also serve as a model for the understanding and engineering of complex and emergent phenomena. Detecting synchronisation of biological oscillators by model checking. The x and y oscillators are indirectly coupled via the woscillator, that is they form a 3oscillator chain. These rhythms play key roles in a variety of important processes, including circadian regulation, metabolism, embryo development, neuron firing, and cardiac rhythms. The term vibration is precisely used to describe mechanical oscillation.
Biological oscillators, ordinary differential equations, markov models. Biological oscillators are amenable to qualitative analysis even before they have been described exhaustively in quantitative terms. Two essential elements of a biological oscillator are 1 an inhibitory feedback loop, which includes one or. Electronic notes in theoretical computer science 229 2009 319. Familiar examples of oscillation include a swinging pendulum and alternating current oscillations occur not only in mechanical systems but also in. Thesis proposal qinsi wang march 28, 2016 computer science department school of computer science carnegie mellon university pittsburgh, pa 152 thesis committee. Nov 15, 2017 kuramotos model in turn has been generalized to other large systems of biological oscillators, such as chorusing frogs 6, firing neurons 7,8,9,10,11, and even human concert audiences clapping. Model checking in biology jasmin fisher and nir piterman abstract model checking is a technique to check whether programs and designs satisfy properties expressed in temporal logic. Biological waves for single species model and multiplespecies model. They linked three repressors in e coli, with a green fluorescent gene gfp as a readout. Synchronization of pulsecoupled biological oscillators siam. Model checking 1,8 is a formal method for verifying if an abstract represen tation of a. By ezio bartocci, flavio corradini, emanuela merelli and luca. Oscillation is the repetitive variation, typically in time, of some measure about a central value often a point of equilibrium or between two or more different states.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Pdf model checking biological oscillators emanuela. Analytical, numerical and topological considerations are used to discuss the patterns of phase locking as a function of the amplitude of the sinusoidal stimulus and the relative frequencies of the oscillator and the sinusoidal stimulus. In this paper, we use electroreceptors of paddlefish, a welldefined sensory system comprising two distinct and synaptically coupled oscillators as a test model to.
Qualitative analysis can identify the elements essential for generating the oscillations and can enhance our understanding of underlying oscillator mechanisms. Coupled biological oscillators may synchronise, as emerging behaviour, after a. Note that the signals are denoted as xi, xf, xo, and so on. Rc network is basically used to determine the resonant frequency of the oscillator, whilst resistor divider circuit is used to provide attenuated feedback. Statistical model checking for biological systems springerlink. Their mathematical analysis introduces the main features of the dynamic properties of biological oscillators and the mathematical techniques necessary for their investigation.
Emanuela merelli joint work with university of camerino. This wave shape is useful for changing sweeping the. Oscillators principles of oscillators requirements 1 the lo must be able to supply sufficient power to drive the mixer at the required level. In biology, oscillators coordinate events only at the coarsest level of. The 3dosc model represents an electronic oscillator model. A mathematical model is presented for phase locking of a biological oscillator to a sinusoidal stimulus. Figure 4 shows the configuration of a generalized nstage gene oscillator.
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