Integrate-and-fire models of biological neurons combine differential equations with discrete spike events. In the simplest case, the reset of the neuronal voltage to its resting value is the only spike event. The response of such a model to constant input injection is limited to tonic spiking. We here study a generalized model in which two simple spike-induced currents are added. We show that this neuron exhibits not only tonic spiking at various frequencies but also the commonly observed neuronal bursting. Using analytical and numerical approaches, we show that this model can be reduced to a one-dimensional map of the adaptation variable and that this map is locally contractive over a broad set of parameter values. We derive a sufficient analytical condition on the parameters for the map to be globally contractive, in which case all orbits tend to a tonic spiking state determined by the fixed point of the return map. We then show that bursting is caused by a discontinuity in the return map, in which case the map is piecewise contractive. We perform a detailed analysis of a class of piecewise contractive maps that we call bursting maps and show that they robustly generate stable bursting behavior. To the best of our knowledge, this work is the first to point out the intimate connection between bursting dynamics and piecewise contractive maps. Finally, we discuss bifurcations in this return map, which cause transitions between spiking patterns.

中文翻译：

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广义整合和激发神经元的局部收缩动力学。

生物神经元的积分触发模型将微分方程与离散尖峰事件相结合。在最简单的情况下，神经元电压重置为其静止值是唯一的尖峰事件。这种模型对恒定输入注入的响应仅限于强直尖峰。我们在这里研究了一个广义模型，其中添加了两个简单的尖峰感应电流。我们表明该神经元不仅表现出各种频率的强直尖峰，而且还表现出常见的神经元爆裂。使用分析和数值方法，我们表明该模型可以简化为适应变量的一维图，并且该图在一组广泛的参数值上是局部收缩的。我们推导出全局收缩地图参数的充分分析条件，在这种情况下，所有轨道都趋向于由返回图的固定点确定的强直尖峰状态。然后我们证明爆裂是由返回映射中的不连续性引起的，在这种情况下，映射是分段收缩的。我们对一类称为爆破图的分段收缩图进行了详细分析，并表明它们可以稳健地产生稳定的爆破行为。据我们所知，这项工作是第一个指出爆发动力学和分段收缩映射之间密切联系的工作。最后，我们讨论这个返回图中的分叉，它导致尖峰模式之间的转换。我们对一类称为爆破图的分段收缩图进行了详细分析，并表明它们可以稳健地产生稳定的爆破行为。据我们所知，这项工作是第一个指出爆发动力学和分段收缩映射之间密切联系的工作。最后，我们讨论这个返回图中的分叉，它导致尖峰模式之间的转换。我们对一类称为爆破图的分段收缩图进行了详细分析，并表明它们可以稳健地产生稳定的爆破行为。据我们所知，这项工作是第一个指出爆发动力学和分段收缩映射之间密切联系的工作。最后，我们讨论这个返回图中的分叉，它导致尖峰模式之间的转换。