Feng J, Brown D
Biomathematics Laboratory, Babraham Institute, Cambridge, UK.
Biol Cybern. 1998 May;78(5):369-76. doi: 10.1007/s004220050441.
We consider how the output of the perfect integrate-and-fire (I&F) model of a single neuron is affected by the properties of the input, first of all by the distribution of afferent excitatory and inhibitory postsynaptic potential (EPSP, IPSP) inter-arrival times, discriminating particularly between short- and long-tailed forms, and by the degree of balance between excitation and inhibition (as measured by the ratio, r, between the numbers of inhibitory and excitatory inputs). We find that the coefficient of variation (CV; standard deviation divided by mean) of efferent interspike interval (ISI) is an increasing function of the length of the tail of the distribution of EPSP inter-arrival times and the ratio r. There is a range of values of r in which the CV of output ISIs is between 0.5 and 1. Too tight a balance between EPSPs and IPSPs will cause the model to produce a CV outside the interval considered to correspond to the physiological range. Going to the extreme, an exact balance between EPSPs and IPSPs as considered in [24] ensures a long-tailed ISI output distribution for which the moments such as mean and variance cannot be defined. In this case it is meaningless to consider quantities like output jitter, CV, etc. of the efferent ISIs. The longer the tail of the input inter-arrival time distribution, the less is the requirement for balance between EPSPs and IPSPs in order to evoke output spike trains with a CV between 0.5 and 1. For a given short-tailed input distribution, the range of values of r in which the CV of efferent ISIs is between 0.5 and 1 is almost completely inside the range in which output jitter (standard deviation of efferent ISI) is greater than input jitter. Only when the CV is smaller than 0.5 or the input distribution is a long-tailed one is output less than input jitter [21]. The I&F model tends to enlarge low input jitter and reduce high input jitter. We also provide a novel theoretical framework, based upon extreme value theory in statistics, for estimating output jitter, CV and mean firing time.
我们研究单个神经元的理想积分发放（I&F）模型的输出如何受到输入特性的影响，首先是传入兴奋性和抑制性突触后电位（EPSP、IPSP）到达间隔时间的分布，特别区分短尾和长尾形式，以及兴奋与抑制之间的平衡程度（通过抑制性和兴奋性输入数量的比率r来衡量）。我们发现，传出峰间间隔（ISI）的变异系数（CV；标准差除以均值）是EPSP到达间隔时间分布的尾部长度和比率r的增函数。存在一个r值范围，其中输出ISI的CV在0.5到1之间。EPSP和IPSP之间过于紧密的平衡会导致模型产生的CV超出被认为对应于生理范围的区间。极端情况下，如文献[24]中所考虑的EPSP和IPSP之间的精确平衡确保了长尾ISI输出分布，其均值和方差等矩无法定义。在这种情况下，考虑传出ISI的输出抖动、CV等数量是没有意义的。输入到达时间分布的尾部越长，为了引发CV在0.5到1之间的输出脉冲序列，对EPSP和IPSP之间平衡的要求就越低。对于给定的短尾输入分布，传出ISI的CV在0.5到1之间的r值范围几乎完全在输出抖动（传出ISI的标准差）大于输入抖动的范围内。只有当CV小于0.5或输入分布是长尾分布时，输出才小于输入抖动[21]。I&F模型倾向于放大低输入抖动并减小高输入抖动。我们还基于统计学中的极值理论提供了一个新颖的理论框架，用于估计输出抖动、CV和平均发放时间。
