Wave propagation in a fitzhughnagumotype model with. In this section, firstly, we introduce excitable media and fitzhughnagumo. Epstein1, 1department of chemistry, brandeis university, ms 015, waltham, massachusetts 02454, usa 2computing centre of the russian academy of sciences, vavilova 40, 119333 moscow, russia received 19 february 2010. This equation arises in genetics, biology, and heat and mass transfer. The fitzhughnagumo model dynamics with an application to the. Overview fitzhugh nagumo fitzhugh 1969 open source brain. Fitzhugh nagumo, 1960s hindmarshrose, 1980s networks effective numerical simulation allow for most common features excitability spiking, different time scales integrateandfire models stochastic models hopfield network, 1980s onoff neuron, learning, stat. The fitzhugh nagumo model is a twodimensional simplification of the hodgkinhuxley model of spike generation in squid giant axons. To overcome this limitation would require analytic extension of the fitzhughnagumo dynamics to higher dimensionality.
Parameters analysis of fitzhughnagumo model for a reliable. Fitzhughnagumo fhn model 1,2 has been since long time one of the basic models to study the cardiac neuron dynamics, due to its simplicity 2 variables, 3 parameters and relation to ionic models. Hodgkinhuxley and fitzhughnagumo models differential. This model is a simpler version of the hodgkinhuxley model which demonstrates the spiking potentials in neurons and emulates the potential signals observed in a living organisms excitable nerve cells. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined lyapunov functional. Fast and slow waves in the fitzhughnagumo equation core reader. Impulses and physiological states in theoretical models of nerve membrane. Dynamics of the fitzhughnagumo neuron model zechariah thurman 619 abstract in this paper, the dynamical behavior of the fitzhughnagumo model is examined. Fast and slow waves in the fitzhughnagumo equation core. The present monograph analyses the fitzhughnagumo fn model le. The initial value problem p0, in all of the space, for the spatio temporal fitzhugh nagumo equations is analyzed. A theoretical and computational study on the estimation of the parameters of a single fitzhughnagumo model is presented. The difference of this work from a conventional system identification is that the measured data only consist of discrete and noisy neural spiking spike times data, which contain no amplitude information. Analytical properties of the perturbed fitzhughnagumo.
Abstracttime optimal control governed by the internally controlled linear fitzhughnagumo equation with pointwise control constraint is considered. The forms of the functions and values for the parameters were selected to fit data from a large neuron. Fitzhughnagumo java applet single cell mathematical modeling of cardiac dynamics and arrhythmias. Fitzhugh nagumo java applet single cell mathematical modeling of cardiac dynamics and arrhythmias. Fitzhughnagumo the fitzhugh nagumo model is a twodimensional simplification of the hodgkinhuxley model of spike generation in squid giant axons. Contribute to rocappfitzhughnagumopy development by creating an account on github.
The fitzhughnagumo model dynamics with an application to. Nagumo equations include only two variables, the membrane. Existence and uniqueness for this system is given in 1978 by rauch and smoller 4, in which they showed that small solutions decay to 0 as and large pulses produce a traveling wave. On dynamical behaviour of fitzhughnagumo systems filoso e licentiatavhandling by matthias ringqvist to be presented on the 8th of september 2006 abstract in this thesis a class of fitzhughnagumo system is studied. We characterize the values of the parameters for which a zero. Multipeak solutions for the neumann problem of an elliptic. Fitzhugh nagumo fhn is the canonical model of excitable. Analytical properties of the wellknown fitzhughnagumo model are studied. Travelling pulse solutions for the discrete fitzhughnagumo system h. It is shown that the standard fitzhughnagumo model does not pass the painleve test in the general case and does not have any meromorphic solutions because there are not any expansions of the general solution in the laurent series.
State university of new york college at geneseo, geneseo, ny 14454 stability analysis of fitzhugh nagumo with smooth. Numerical simulation of the fitzhughnagumo equations. We suggest employing the firstorder stable rc filters, based on a single capacitor, for control of unstable fixed points in an array of oscillators. For a problem, i need to implement the fitzhughnagumo model with spatial diffusion via cranknicolsons scheme. Fitzhughnagumo equations, a caricature of the hodgkin and huxley model. Various coupling schemes are also examined, and their e ects on the dynamics of the. This simulation allows the user to set parameter values as well as see the phase space in a separate window. Dynamics of the fitzhughnagumo neuron model zechariah thurman 619 abstract in this paper, the dynamical behavior of the fitzhugh nagumo model is examined. We study several aspects of fitzhughnagumos fhn equations without diffusion. This point of view has been suggested for the treatment of certain symptoms of parkinsons disease, which is thought to be caused by synchronising neurons in the. For a problem, i need to implement the fitzhugh nagumo model with spatial diffusion via cranknicolsons scheme. The two algorithms are illustrated by studying an initial value problem. Feb 26, 2012 the initial value problem p0, in all of the space, for the spatio temporal fitzhugh nagumo equations is analyzed.
This was a test render to demonstrate a new feature, in re. Fitzhughnagumo, 1960s hindmarshrose, 1980s networks effective numerical simulation allow for most common features excitability spiking, different time scales integrateandfire models stochastic models hopfield network, 1980s onoff neuron, learning, stat. This became known as the fitzhughnagumo fn model of mathematical neuroscience and is a simpler version of the hodgkinhuxley hh model, which demonstrates the spiking currents in neurons. Although similar to the fitzhughnagumo model, the hindmarshrose model hindmarsh and rose, 1982 was developed from first principles with the assumptions that the rate of change of membrane potential dxdt depends linearly on z an externally applied current and y an intrinsic current. The fitzhughnagumo system is a fastslow system with two fast variables ac was supported by the foundation for polish science under the mpd programme \geometry and topology in physical models, co nanced by the eu european regional development fund. Richard fitzhugh in 1961 1, as a simpli cation of the noble prize winning hodgkinhuxley model 2, which very successfully modelled the initiation and propagation of neural action potential using a squid axon. The present monograph analyses the fitzhugh nagumo fn model le. Nonclassical symmetry solutions to the fitzhugh nagumo. For periodic initial data are possible damped travelling waves and their speed of propagation is evaluated. The neuronal transmembrane voltage is assumed to be observed, hidden ion currents and input signals to the model neuron are estimated from the observed voltage. A single capacitor is sufficient to stabilize an entire array, if the oscillators are coupled strongly enough. The variational iteration method and adomian decomposition method are applied to solve the fitzhughnagumo fn equations.
Fitzhugh nagumo system is a wellstudied object see e. For systems of elliptic equations of fitzhughnagumo type on bounded domains and with small diffusion in one equation, we construct solutions with multiple sharp peaks close to each other and close to, but not on, the boundary. The presence of the three parameters leads to a large. Iv fitzhughnagumo model 1961, 1962 iva motivation ignore conductance based modeling for now try phenomenological modeling i. Stabilization of a network of the fitzhughnagumo oscillators. In this thesis, i revisit the issue of the utility of the fitzhugh nagumo fhn model for capturing neuron firing behaviors. Travelling pulse solutions for the discrete fitzhugh. Sandstede abrown university division of applied mathematics abstract the existence of fast travelling pulses of the discrete fitzhugh nagumo equation is obtained in the weak. The role of time delay in the fitzhughnagumo equations. A study of wave propagation in the fitzhugh nagumo system. By using the theory of lyapunov coe cient to analyze hopf and bautin bifurcation it is shown that at most. Mar 24, 2015 the fitzhughnagumo fhn model is a mathematical model of neuronal excitability developed by richard fitzhugh as a reduction of the hodgkin and huxleys model of action potential generation in the squid giant axon fitzhugh 1955. This paper is devoted to the study of the fitzhugh nagumo fn system.
For the fn equations, the existence of such solutions has been proved by conley 4, carpenter 2, and hastings lo, provided f, u, and y satisfy appropriate conditions. Nonclassical symmetry solutions to the fitzhugh nagumo equation a thesis presented to the faculty of the department of mathematics east tennessee state university in partial ful. Comparison of the hodgkinhuxley with the fitzhughnagumo models highlight the quantitative limitations of the latter in the region of the minimum characterizing the back portion of the pulse. We investigate a possibility of realization of structurally stable chaotic dynamics in neural systems. Using matlab, the following function can be given to i. Find, read and cite all the research you need on researchgate.
Fitzhughnagumo model an overview sciencedirect topics. Dynamics of the fitzhugh nagumo neuron model zechariah thurman 619 abstract in this paper, the dynamical behavior of the fitzhugh nagumo model is examined. The fitzhugh nagumo system is a fastslow system with two fast variables ac was supported by the foundation for polish science under the mpd programme \geometry and topology in physical models, co nanced by the eu european regional development fund. The relationship between neuron input current and the ring frequency of the neuron is characterized. In this thesis, i revisit the issue of the utility of the fitzhughnagumo fhn model for capturing neuron firing behaviors. It is of mathematical interest to know the number of limit cycles for a polynomial. Analytical properties of the perturbed fitzhughnagumo model. Newellwhitehead equation a special case of the fitzhugh. Stability analysis of fitzhughnagumo with smooth periodic. The obtained results show that only few terms are required to deduce approximated solutions which are found to be accurate and efficient. The original model respects basically the conditions.
Estimating the parameters of fitzhughnagumo neurons from. The fourdimensional hodgkinhuxley equations are considered as the prototype for description of neural pulse propagation. An array, composed of 24 to 30 meanfield coupled fitzhughnagumo fhn type asymmetric oscillators, is considered as a case study. Fitzhughnagumo equations with generalized diffusive coupling.
Unscented kalman filter ukf applied to fitzhughnagumo neuron dynamics. Understand the dynamics of the fitzhughnagumo model with an. See chapter 4 and especially chapter 4 section 3 for background knowledge on phase plane analysis. We show the existence of both supercritical and subcritical hopf bifurcations. Reduction of a twocomponent fitzhugh nagumo model to a. It is also the second most cited cardiac neuron model. When \i\ is weak or zero, the equilibrium intersection of nullclines is on the left stable branch of \v\nullcline, and the model is resting. Turing patterns in a networkreduced fitzhughnagumo model. The following system of pdes is the fitzhugh nagumo caricature of the hodgkinhuxley equations modelling the nerve impulse propagation along an axon. The model is able to reproduce many qualitative characteristics of electrical impulses in cardiac tissues, e.
The considered model of interacting neurons consists of a pair of coupled fitzhugh nagumo systems, with the parameters being periodically modulated in antiphase, so that the neurons undergo alternate excitation with a successive transmission of the phase of oscillations from one. Now the problem lays withing the spatial diffusion. The model consists of a voltagelike variable having cubic nonlinearity that allows regenerative selfexcitation via a positive feedback, and. Their mathematical complexity and sophistication prompted a simplified twodimensional model, the fitzhughnagumo equations, which display many of the formers dynamical features. The fitzhughnagumo model fhn, named after richard fitzhugh 19222007 who suggested the system in 1961 and j. When the re action kinetics of the model can be outlined by means of piecewise linear approximations, then the solution of vq is explicitly obtained.
Travelling pulse solutions for the discrete fitzhughnagumo. Huxley model which is the motivation for the fitzhugh nagumo equation that extract the essential behavior in a simple form. Renno abstract the initial value problem vq, in all of the space, for the spatio temporal fitzhugh nagumo equations is analyzed. Nagumo fhn type asymmetric oscillators, is considered as a case. Travelling pulse solutions for the discrete fitzhugh nagumo system h. When the reaction kinetics of the model can be outlined by means of piecewise linear approximations, then the solution of p0 is explicitly obtained. A large class of biological systems from cardiac tissue to slime mold are examples of excitable media. Sandstede abrown university division of applied mathematics abstract the existence of fast travelling pulses of the discrete fitzhughnagumo equation is obtained in the weak. Due to its modular structure, applications to similar problems should be easy to accomplish.
State university of new york college at geneseo, geneseo, ny 14454 stability analysis of fitzhughnagumo with smooth. The fitzhugh nagumo model has only a few parameters and two coupled differential equations for the membrane potential, v. The fitzhughnagumo model alan hodgkin and andrew huxley developed the rst quantitative model of the propagation of an electrical signal the action potential along a squid giant axon, but their ideas have since been extended and applied to a wide variety of excitable cells. Phase plane and bifurcation analysis book chapters. Fitzhughnagumo reactiondiffusion by jesper a fork of sketch. Making use of ekelands variational principle, we obtain pontryagins maximum principle for a time optimal control problem. The considered model of interacting neurons consists of a pair of coupled fitzhughnagumo systems, with the parameters being periodically modulated in antiphase, so that the neurons undergo alternate excitation with a successive transmission of the phase of. The fitzhughnagumo fhn model is a mathematical model of neuronal excitability developed by richard fitzhugh as a reduction of the hodgkin and huxleys model of action potential generation in the squid giant axon fitzhugh 1955. Wave propagation in a fitzhughnagumotype model with modi. Model and then state the objective of this paper and how the chapters. Nagumo proposed a model for emulating the current signal observed in a living organisms excitable cells.
Learn more about matlab, ode, ode45, fitzhugh, biophysics matlab. Fitzhugh nagumo fhn is the canonical model of excitable media. Huxley model which is the motivation for the fitzhughnagumo equation that extract the essential behavior in a simple form. Types of bifurcations, periodical forcing and stability regions by a lyapunov functional tanya kostova lawrence livermore national laboratory l561, livermore, ca 94550, usa email. The motivation for the fitzhughnagumo model was to isolate conceptually the essentially mathematical properties of excitation and propagation from the electrochemical properties of sodium and potassium ion flow. Jan 27, 2012 the fitzhugh nagumo model explains the excitation block phenomenon, i. Existence and stability of traveling pulse solutions of the fitzhughnagumo equation.
396 1216 552 1246 660 277 131 1415 1419 1623 875 1298 567 72 1144 981 73 1253 1234 1309 1192 761 1067 930 1161 718 1413 474 574 1216