A Two-Variable Model of Somatic–Dendritic Interactions in a Bursting Neuron

Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, ON, Canada, K1N 6N5.
Bulletin of Mathematical Biology (Impact Factor: 1.39). 10/2002; 64(5):829-60. DOI: 10.1006/bulm.2002.0303
Source: PubMed


We present a two-variable delay-differential-equation model of a pyramidal cell from the electrosensory lateral line lobe of a weakly electric fish that is capable of burst discharge. It is a simplification of a six-dimensional ordinary differential equation model for such a cell whose bifurcation structure has been analyzed (Doiron et al., J. Comput. Neurosci., 12, 2002). We have modeled the effects of back-propagating action potentials by a delay, and use an integrate-and-fire mechanism for action potential generation. The simplicity of the model presented here allows one to explicitly derive a two-dimensional map for successive interspike intervals, and to analytically investigate the effects of time-dependent forcing on such a model neuron. Some of the effects discussed include 'burst excitability', the creation of resonance tongues under periodic forcing, and stochastic resonance. We also investigate the effects of changing the parameters of the model.

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    • "A special type of phenomenological models is based on low-dimensional maps. There have been proposed only few explicit maps capable of generating essential aspects of bursting dynamics (for instance, see [19] [20] [3] and [10]). The models are designed with the aim of gaining a deeper understanding of the mathematical structure underlying the oscillations. "
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    ABSTRACT: There are numerous families of Neural Networks (NN) used in the study and development of the field of Artificial Intelligence (AI). One of the more recent NNs involves the Bursting neuron, pioneered by Rulkov. The latter has the desirable property that it can also be used to model a system (for example, the “brain”) which switches between modes in which the activity is excessive (“bursty”), to the alternate case when the system is “dormant”. This paper, which we believe is a of pioneering sort, derives some of the analytic properties of the Bursting neuron, and the associated NN. To be more specific, the model proposed by Rulkov [11] explains the so-called “bursting” phenomenon in the system (brain), in which a low frequency pulse output serves as an envelope of high frequency spikes. Although various models for bursting have been proposed, Rulkov’s model seems to be the one that is both analytically tractable and experimentally meaningful. In this paper, we show that a “small” scale network consisting of Bursting neurons rapidly converges to a synchronized behavior implying that increasing the number of neurons does not contribute significantly to the synchronization of the individual Bursting neurons. The consequences of such a conclusion lead to a phenomenon that we call “behavioral synchronization”.
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