An Analytic Picture of Neuron Oscillations

Phillipson, Paul E.; Schuster, Peter

doi:10.1142/s0218127404010151

Cited by 3 publications

(8 citation statements)

References 20 publications

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“…where f and g are real valued analytic functions on U × V × W ⊂ R 3 and that (0, 0, µ 0 ) is a fixed point of (11). Associated with (11) and this point is the system matrix, A, defined by …”

Section: Lyapunov Coefficientsmentioning

confidence: 99%

“…By Remark 3 and assumption (15) β(µ) > 0 for all µ in a small neighbourhood of µ 0 since β(µ) = 4∆(µ) − σ 2 (µ) and the determinant and trace functions are continuous. With these assumptions, the system defined by (11) can be put in the canonical form (see proof in [21])…”

Section: Lyapunov Coefficientsmentioning

confidence: 99%

“…1. for any ε and δ, 0 < ε ≤ ε 0 , 0 < δ ≤ δ 0 , and for any s, 1 ≤ s ≤ k, there exists a system (B) of class N (or respectively, analytical) which is δ-close to rank 2k + 1 to system (A) and has in U ε (O) precisely s closed paths. Table 1 Hopf bifurcation table for system (11).…”

Section: Definitionmentioning

confidence: 99%

“…in brain research, and to some extent in modelling cardiac movements; it is of great importance to get a good understanding of its dynamics. This is reflected in the number of articles written in this area, see for example [3][4][5][6][7][8][9][10][11] and the references therein. A short but rather complete description of the physiology behind the biological neuron and the corresponding derivation of the Hodgkin-Huxley model and its simplified version, the FitzHugh-Nagumo model, can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

“…Although the FitzHugh-Nagumo system is a well-studied object (see e.g. [12,13,11,14,14,15]), there are several other reasons that motivate the current study.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

Ringkvist

Zhou

2009

Nonlinear Analysis: Theory, Methods & Applications

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Section: Lyapunov Coefficientsmentioning

confidence: 99%

Section: Lyapunov Coefficientsmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Although the FitzHugh-Nagumo system is a well-studied object (see e.g. [12,13,11,14,14,15]), there are several other reasons that motivate the current study.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

Ringkvist

Zhou

2009

Nonlinear Analysis: Theory, Methods & Applications

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A Comparative Study of the Hodgkin–huxley and Fitzhugh–nagumo Models of Neuron Pulse Propagation

Phillipson

Schuster

2005

Int. J. Bifurcation Chaos

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The four-dimensional Hodgkin–Huxley equations are considered as the prototype for description of neural pulse propagation. Their mathematical complexity and sophistication prompted a simplified two-dimensional model, the FitzHugh–Nagumo equations, which display many of the former's dynamical features. Numerical and mathematical analysis are employed to demonstrate that the FitzHugh–Nagumo equations can provide quantitative predictions in close agreement with the Hodgkin–Huxley equations. The two most important parameters of a neural pulse are its speed c(T) and pulse height v max (T) and so numerical computations of these quantities predicted by the Hodgkin–Huxley equations are given over the entire temperature range T for stability of a neural pulse. Similarly, the FitzHugh–Nagumo equations are parameterized by two dimensionless quantities: a which determines the dynamics of the pulse front, and b whose departure from zero tailors the front to form the resultant pulse. Parallel computations are presented for the FitzHugh–Nagumo pulse whose relative simplicity permits analytic determination to close approximation of the dimensionless speed θ(a, b) and pulse height V max (a, b). It is shown that the two models are numerically identified by scaling according to c = 4904 θ cm/sec and v max = 115 V max mV where the numbers are a consequence of the experimental parameter values inherent to the Hodgkin–Huxley equations. With this connection, at a given temperature the Hodgkin–Huxley speed and pulse height determine unique values for the two FitzHugh–Nagumo parameters a and b. Approximate analytic solution for θ(a, b) allows construction of a three-dimensional [a, b, θ] state plot upon which a unique ridge defines, as a function of temperature, the speed and associated pulse height predicted by the Hodgkin–Huxley equations. The generality of the state plot suggests its application to other conductance models. Comparison of the Hodgkin–Huxley with the FitzHugh–Nagumo models highlight the quantitative limitations of the latter in the region of the minimum characterizing the back portion of the pulse. To overcome this limitation would require analytic extension of the FitzHugh–Nagumo dynamics to higher dimensionality.

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Analytical Dynamics of Neuron Pulse Propagation

Phillipson

Schuster

2006

Int. J. Bifurcation Chaos

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The four-dimensional Hodgkin–Huxley equations describe the propagation in space and time of the action potential v(z) along a neural axon with z = x + ct and c being the pulse speed. The potential v(z), which is parameterized by the temperature, is driven by three gating functions, m(z), n(z) and h(z), each of which obeys formal first order kinetics with rate constants that are represented as nonlinear functions of the potential v. It is shown that this system can be analytically simplified (i) in the number of gating functions and (ii) in the form of associated rate functions while retaining to close approximation quantitative fidelity to computer solutions of the exact equations over the complete temperature range for which stable pulses exist. At a given temperature we record two solutions (T < T max ) corresponding to a high-speed and a low-speed branch in speed-temperature plots, c(T), or no solution (T > T max ). The pulse is considered as composed of two contiguous parts: (i) a pulse front extending from v(0) = 0 to a pulse maximum v = V max , and (ii) a pulse back extending from V max through a pulse minimum V min to a final regression back to v(z → ∞) = 0. An approximate analytic solution is derived for the pulse front, which is predicted to propagate at a speed c(T) = 1203 Θ⅜ (T° C ) cm/sec, [Formula: see text] in close agreement with computer solution of the exact Hodgkin–Huxley equations for the entire pulse. These results provide the basis for a derivation of two-dimensional differential equation systems for the pulse front and pulse back, which predict the pulse maximum and minimum over the operational temperature range 0 ≤ T ≤ 25° C , in close agreement with the exact equations. Most neuron dynamics studies have been based on voltage clamp experiments featuring external current injection in place of self-generating pulse propagation. Since the behaviors of the gating functions are similar, it is suggested that the present approximations might be applicable to such situations as well as to the dynamics of myelinated fibers.

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An Analytic Picture of Neuron Oscillations

Cited by 3 publications

References 20 publications

On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

A Comparative Study of the Hodgkin–huxley and Fitzhugh–nagumo Models of Neuron Pulse Propagation

Analytical Dynamics of Neuron Pulse Propagation

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