Bressloff Paul C, Newby Jay M
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
Phys Biol. 2014 Feb;11(1):016006. doi: 10.1088/1478-3975/11/1/016006. Epub 2014 Jan 29.
Following recent advances in imaging techniques and methods of dendritic stimulation, active voltage spikes have been observed in thin dendritic branches of excitatory pyramidal neurons, where the majority of synapses occur. The generation of these dendritic spikes involves both Na(+) ion channels and M-methyl-D-aspartate receptor (NMDAR) channels. During strong stimulation of a thin dendrite, the resulting high levels of glutamate, the main excitatory neurotransmitter in the central nervous system and an NMDA agonist, modify the current-voltage (I-V) characteristics of an NMDAR so that it behaves like a voltage-gated Na(+) channel. Hence, the NMDARs can fire a regenerative dendritic spike, just as Na(+) channels support the initiation of an action potential following membrane depolarization. However, the duration of the dendritic spike is of the order 100 ms rather than 1 ms, since it involves slow unbinding of glutamate from NMDARs rather than activation of hyperpolarizing K(+) channels. It has been suggested that dendritic NMDA spikes may play an important role in dendritic computations and provide a cellular substrate for short-term memory. In this paper, we consider a stochastic, conductance-based model of dendritic NMDA spikes, in which the noise originates from the stochastic opening and closing of a finite number of Na(+) and NMDA receptor ion channels. The resulting model takes the form of a stochastic hybrid system, in which membrane voltage evolves according to a piecewise deterministic dynamics that is coupled to a jump Markov process describing the opening and closing of the ion channels. We formulate the noise-induced initiation and termination of a dendritic spike in terms of a first-passage time problem, under the assumption that glutamate unbinding is negligible, which we then solve using a combination of WKB methods and singular perturbation theory. Using a stochastic phase-plane analysis we then extend our analysis to take proper account of the combined effects of glutamate unbinding and noise on the termination of a spike.
随着成像技术和树突刺激方法的最新进展，在兴奋性锥体神经元的细树突分支中观察到了活跃的电压尖峰，而大多数突触就发生在这些细树突分支中。这些树突棘的产生涉及钠离子通道和N-甲基-D-天冬氨酸受体（NMDAR）通道。在对细树突进行强刺激期间，会产生高水平的谷氨酸，它是中枢神经系统中的主要兴奋性神经递质，也是一种NMDA激动剂，会改变NMDAR的电流-电压（I-V）特性，使其表现得像电压门控钠离子通道。因此，NMDAR可以引发再生性树突棘，就像钠离子通道在膜去极化后支持动作电位的起始一样。然而，树突棘的持续时间约为100毫秒，而不是1毫秒，因为它涉及谷氨酸从NMDAR缓慢解离，而不是超极化钾离子通道的激活。有人提出，树突状NMDA尖峰可能在树突计算中发挥重要作用，并为短期记忆提供细胞基础。在本文中，我们考虑了一种基于电导的树突状NMDA尖峰随机模型，其中噪声源于有限数量的钠离子和NMDA受体离子通道的随机开闭。由此产生的模型采用随机混合系统的形式，其中膜电压根据分段确定性动力学演化，该动力学与描述离子通道开闭的跳跃马尔可夫过程相耦合。在谷氨酸解离可忽略不计的假设下，我们将树突棘的噪声诱导起始和终止表述为首次通过时间问题，然后使用WKB方法和奇异摄动理论相结合的方法来求解。然后，我们使用随机相平面分析扩展我们的分析，以适当考虑谷氨酸解离和噪声对尖峰终止的综合影响。
