Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations

Abstract

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 preduct 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 preduct many results of experiments conducted on this cell. The physiological implications of the model are discussed.

Topological and phenomenological classification of bursting oscillations

“…Plant and Kim, 1976;Canavier et al, 1991;Bertram, 1993;Smolen and Keizer, 1992;Guckenheimer et al, 1994;Rinzel and Troy, 1982;Traub et al, 1991;Wang et al, 1991) and a systematic mathematical analysis of one such model was first performed by Rinzel (1985). In this analysis the system variables were classified as either "fast", if the variable changed significantly over the duration of a single spike, or "slow", if significant change occurred only over the duration of the burst.…”

Making a Swim Central Pattern Generator Out of Latent Parabolic Bursters

“…This bifurcation occurs in the fast 3D (V, h, n)-subspace of the model and is modulated by the 2D slow dynamics in the (Ca, x)-variables, which are determined by slow oscillations of the intracellular calcium concentration [Plant & Kim, 1975, 1976. The unfolding of this codimension-one bifurcation includes an onset of a stable equilibrium, which is associated with a hyperpolarized phase of bursting, and on the other end, an emergent stable periodic orbit that is associated with tonic spiking phase of bursting.…”

“…This observation indicates the type of neuronal models to be employed to describe network cores. Our model of choice for parabolic bursting is the Plant model [Plant & Kim, 1975, 1976Plant, 1981]. The Plant model has been developed to accurately describe the voltage dynamics of the R15 neuron in a mollusk Aplysia Californica, which has turned out to be an endogenous burster [Levitan & Levitan, 1988].…”

Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons

“…͑1͒ and ͑2͒, respectively. The slow subsystem can act independently, 21 be affected synaptically, 17 or interact locally with the spiking fast subsystem 5,[15][16][17] to produce alternating periods of spiking and silence in time. To examine the dynamical mechanism implicit to a bursting behavior, y is treated as a bifurcation parameter of the fast subsystem.…”

“…Autonomously bursting neurons are found in a variety of neural systems, from the mammalian cortex 1 and brainstem 2-4 to identified invertebrate neurons. 5,6 Multirhythmicity in a dynamical system is a specific type of multistability which describes the coexistence of two or more oscillatory attractors under a fixed parameter set. Multirhythmicity has been shown to occur in vertebrate motor neurons, 7 invertebrate interneurons, 8 and in small networks of coupled invertebrate neurons.…”