Treutlein H, Schulten K
Eur Biophys J. 1986;13(6):355-65. doi: 10.1007/BF00265671.
The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.
单个脉冲的形状几乎相同且与频率无关；发放频率可在很宽的范围内变化；脉冲之间的时间间隔呈现出随机散射。基于确定性非线性动力学过程，例如邦霍费尔 - 范德波尔模型，无法理解这种行为。我们在本文中证明，给邦霍费尔 - 范德波尔模型添加一个噪声项可以很好地再现神经元的发放模式。为此，我们考虑了与随机邦霍费尔 - 范德波尔模型相对应的福克 - 普朗克方程。该方程已通过一种新的蒙特卡罗算法求解。我们证明，由此产生的分布函数仅代表基础力场的全局特征：吸引附近轨迹的零斜率线被证明是相空间中分布集中其幅度的区域。由于有两条这样的线，分布是双峰的，代表在两条零斜率线之间的重复波动。即使在确定性邦霍费尔 - 范德波尔模型不表现出极限环行为的情况下，随机系统也会产生一个极限环。这个环可以与神经脉冲的发放相联系。
