Brunel Nicolas, Latham Peter E
CNRS, NPSM, Université Paris René Descartes, 75270 Paris Cedex 06, France.
Neural Comput. 2003 Oct;15(10):2281-306. doi: 10.1162/089976603322362365.
We calculate the firing rate of the quadratic integrate-and-fire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, tau(s), to the neuronal time constant, tau(m). We calculate the firing rate exactly in two limits: when the ratio, tau(s)/tau(m), goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O(tau(s)/tau(m)), which is qualita tively different from that of the leaky integrate-and-fire neuron, where the correction is O( radical tau(s)/tau(m)). The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O(tau(m)/tau(s)). By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of tau(s)/tau(m) in the suprathreshold regime-that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when tau(s) becomes large compared to tau(m).
我们计算了二次积分发放神经元对有色噪声输入电流的发放率。由于尖峰的随机轰击，这样的输入电流是对噪声的一个很好近似，噪声的相关时间对应于突触的衰减时间。决定发放率的关键参数是有色噪声的相关时间τ(s)与神经元时间常数τ(m)的比值。我们精确计算了两种极限情况下的发放率：当比值τ(s)/τ(m)趋于零时（白噪声）以及当它趋于无穷大时。对短相关时间极限的修正为O(τ(s)/τ(m))，这与泄漏积分发放神经元的修正定性不同，在泄漏积分发放神经元中修正是O(√(τ(s)/τ(m)))。这种差异是由于神经元膜电位在发放阈值处概率密度函数的边界条件不同。对长相关时间极限的修正为O(τ(m)/τ(s))。通过结合短相关时间极限和长相关时间极限，我们推导出一个表达式，该表达式在阈值以上区域——即平均电流足以使细胞发放的区域——对整个τ(s)/τ(m)范围内的发放率提供了一个很好的近似。在阈下区域，当τ(s)相对于τ(m)变得很大时，该表达式会有些失效。
