 Solution

For a constant-rate Poisson process, let $T$ be the waiting time from a spike to the next spike. By independent increments, there is no spike in an interval of length $u$ with probability $e^{-ru}$. Hence, for $r>0$,Its density is $r{e}^{-ru}$, so the interspike intervals are independent exponential distributions with mean $1/r$. If $U$ is uniform on $(0,1)$, inverse transform sampling gives $T=-\log U/r$. Repeatedly add independent draws to the current spike time to simulate a constant-rate spike train.

For a prescribed time-dependent nonnegative rate $r(t)$, use an Inhomogeneous Poisson process. Conditional on the present event at $t_i$, its next waiting time has survival probabilityThus its conditional CDF is $1$ minus that expression for $u\ge 0$. Define the cumulative intensity $\Lambda (t)={\int }_0^{t}r(s)ds$. For each new independent uniform draw, set $E_i=-\log U_i$ and solveThis transforms independent unit-rate exponential intervals in integrated-intensity time into physical spike times. Use a generalized inverse when the rate vanishes on an interval: no spikes are placed in a flat part of $\Lambda$. On a finite simulation window, stop when the next intensity target exceeds the available $\Lambda$. If the total future intensity is finite, a target beyond it means that no further spike occurs.

Numerically, integrate the known rate and invert the accumulated integral, or integrate exactly across piecewise-constant rate bins. Drawing an exponential interval using only $r(t_i)$ is not exact when the rate changes during the interval. The Poisson model also omits the neuronal refractory period and other spike-history effects.