= Solution
Fit a stimulus-conditioned <linear-nonlinear-Poisson cascade model>. Choose a causal stimulus-history window, subtract the stimulus mean, and estimate a temporal filter $k$. A useful initial estimate is the <spike-triggered average>, $\widehat a(u)=N^{-1}\sum_i[s(t_i-u)-\bar s]$. For white Gaussian stimulation and a one-filter response, this estimates the filter direction. With correlated Gaussian stimulation it estimates a direction proportional to $Ck$, where $C$ is the history <covariance matrix>; use <stimulus-correlated spike-triggered averaging> to whiten or regularize this correction. An arbitrary stimulus or an even response nonlinearity does not guarantee that this average identifies the correct model.
A general fitting approach is to expand a causal filter in smooth basis functions, compute $a(t)=b+\int_0^U k(u)s(t-u)du$, and take a nonnegative rate $\lambda(t)=f(a(t))$. With $f(a)=e^a$, fit the coefficients by <Poisson spike-train likelihood fitting>:
$$
\boxed{\ell=\sum_i\log\lambda(t_i)-\int_0^{T_{\rm obs}}\lambda(t)dt}.
$$
The first term rewards rate at observed spikes and the second penalizes overprediction elsewhere. Include a filter-size or smoothness penalty, and choose complexity on held-out data. Another option estimates $f$ by stimulus-projection histograms: divide the density of projection values at spikes by their occupancy density, multiplied by the mean firing rate. Avoid interpreting poorly sampled bins as reliable rates. Temporal train/test splits should avoid leakage through the filter-history window.
Given the fitted rate and the same stimulus, generate an <Inhomogeneous Poisson process>. For small bins of width $\Delta t$, spike with probability $1-e^{-\lambda(t)\Delta t}$, choosing the bins sufficiently short that multiple events are negligible. More exactly, advance integrated intensity until it increases by an independent unit-exponential random variable, then place the next event. Repeating this generates different synthetic <spike trains> with the same fitted stimulus-dependent rate, not copies of the original event times.
Compare held-out log-likelihood, stimulus-conditioned firing-rate predictions, count distributions, interspike-interval distributions and <autocorrelation>. With repeated identical stimulus presentations, compare trial-averaged time courses and trial-to-trial count variance, including the <Fano factor>. A single recorded train does not by itself supply a reliable repeated-trial average or variance; use held-out temporal data, model-based uncertainty and further recordings instead. <Time-rescaled spike-train diagnostics> test whether integrated-intensity intervals look independently exponential, rather than comparing raw intervals to an exponential law under a changing rate.
\b[A successful model reproduces specified statistical features on new data; identical spike times are not the goal.] The basic Poisson model can miss refractoriness, bursting, adaptation, history dependence or stimulus-unexplained trial variability. If needed, add a spike-history filter to the conditional rate, including an absolute dead time, and fit the resulting history-dependent model. A <complex cell> may also require multiple filters or quadratic stimulus features: its spike-triggered mean can vanish despite strong tuning. These failures identify model limitations rather than evidence that a random spike generator should match every observed interval.
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