Slow Manifold of a Neuronal Bursting Model

2007

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Biophysical Journal, 1976

Modifications based on experimental results reported in the literature are made to the Hodgkin-Huxley equations to describe the electrophysiological behavior of the Aplysia abdominal ganglion R15 cell. The system is then further modified to describe the effects with the application of the drug tetrodotoxin (TTX) to the cells' bathing medium. Methods of the qualitative theory of differential equations are used to determine the conditions necessary for such a system of equations to have an oscillatory solution. A model satisfying these conditions is shown to predict many experimental observations of R15 cell behavior. Numerical solutions are obtained for differential equations satisfying the conditions of the model. These solutions are shown to have a form similar to that of the bursting which is characteristic of this cell, and to predict many results of experiments conducted on this cell. The physiological implications of the model are discussed.

Journal of Neurophysiology, 2005

Synaptic modulation of the interspike interval signatures of bursting pyloric neurons. J Neurophysiol 89: 1363Neurophysiol 89: -1377Neurophysiol 89: , 2003 10.1152/jn.00732.2002. The pyloric network of the lobster stomatogastric nervous system is one of the best described assemblies of oscillatory neurons producing bursts of action potentials. While the temporal patterns of bursts have been investigated in detail, those of spikes have received less attention.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2012

Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model AIP Advances 2, 042125 Nonlinear multimode dynamics and internal resonances of the scan process in noncontacting atomic force microscopy J. Appl. Phys. 112, 074314 (2012) Secondary nontwist phenomena in area-preserving maps Chaos 22, 033142 (2012) Bifurcation theory for the L-H transition in magnetically confined fusion plasmas Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the twofast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow. V C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4766943]

The 2006 IEEE International Joint Conference on Neural Network Proceedings

In vitro neuronal culture is a potentially powerful tool for investigation of information processing of neurons. However, in a mature dissociated neuronal culture, synchronized bursting is often observed. Bursting strongly drives the dynamics of the network, thus making it difficult to investigate the computational properties. A quantitative modeling of the dynamics would provide a means for understanding the processes that produce bursts and a means to manipulate the dynamics to control them. A simulated network composed of leaky integrate and fire neurons and dynamic synapses is used to model the phenomenon. We investigate the variability of burst dynamics according to parameters such as connectivity, recovery time constant, and noise.

International Journal of Bifurcation and Chaos, 2006

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 part...

Journal of Mathematical Biology, 1981

In this paper we consider a model for the phenomenon of bursting in nerve cells. Experimental evidence indicates that this phenomenon is due to the interaction of multiple conductances with very different kinetics, and the model incorporates this evidence. As a parameter is varied the model undergoes a transition between two oscillatory waveforms; a corresponding transition is observed experimentally. After establishing the periodicity of the subcritical oscillatory solution, the nature of the transition is studied. It is found to be a resonance bifurcation, with the solution branching at the critical point to another periodic solution of the same period. Using this result a comparison is made between the model and experimental observations. The model is found to predict and allow an interpretation of these observations.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021

Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neurotransmitters and hormones and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations or, in the case of so-called “pseudo-plateau bursting,” canards that are induced by a folded node singularity. In this article, we show an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a mu...

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009

The paper considers the neuron model of Hindmarsh-Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a passage occurs in a chaotic region and different explanations have been given in the literature to represent the process, generally based on a slow-fast decomposition of the neuron model. This paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of the equilibrium point. An analytical approximation is developed for the related crucial elements and a subsequent numerical analysis signifies the properness of the suggested conjecture.

Physica A: Statistical Mechanics and its Applications, 2004

We study a simple map as a minimal model of excitable cells. The map has two fast variables which mimic the behavior of class I neurons, undergoing a sub-critical Hopf bifurcation. Adding a third slow variable allows the system to present bursts and other interesting biological behaviors. Bifurcation lines which locate the excitability region are obtained for different planes in parameter space.