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The setup of this problem is basically to model a neuron that spikes randomly according to some Poisson distribution over some time interval.

Suppose I have a time interval $[0,T]$ for some fixed $T$ (we're treating $T$ as some large number). Suppose we have a random variable $X \sim \mathcal{Pois}(\lambda)$ over that time interval. Let $t_1,t_2,\dots, t_n$ be the times in $[0,T]$ that my event occurs. Define a function:

$$\phi(t) = 2\pi k + \frac{t-t_k}{t_{k+1}-t_{k}}$$

where $k$ enumerates the spikes. The numerator is basically the time since last spike, and the denominator is the time between spikes. The order parameter $R(t)$ is defined as $|e^{i \phi(t)}|$. Is there any way to compute $E\left[R(t) \right]$ as a function of $\lambda$, or at least get an estimate of it non-numerically?

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    If $r$ is a real number then $e^{ir}$ lies on the unit disc, i.e. $|e^{ir}|=1$. – 2017-01-31
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    @nullUser Hmmm yes that's true. I'm attempting to look at this problem (eqs. 13 and 14): https://arxiv.org/pdf/1503.02213v1.pdf, except I'm assuming the $N$ neurons in my network are i.i.d so I assumed that averaging over $N$ i.i.d ones is the same as just solving the equation above for one of them. – 2017-01-31

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