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The Lagrangian integral time is the area under a normalized velocity autocorrelation along a Lagrangian trajectory. If this correlation is integrable, Taylor turbulent dispersion has asymptotic mean-square displacement $2R_L(0)t_{L}t$. The integral time need not equal every short-time decorrelation scale, particularly when the autocorrelation oscillates.

For stationary statistics along a Lagrangian trajectory $\mathbf{X}$, the velocity autocorrelation depends on the lag $s$. Integrating velocity twice gives the mean-square displacement $2{\int }_0^t(t-s)R_L(s)ds$. If $R_L$ is integrable, the long-time displacement is diffusive. Its normalized one-sided area is the Lagrangian integral time.

Several chains from dispersed initial states, trace plots, autocorrelation, effective sample sizes, and the rank-normalized split potential scale reduction factor $\hat{R}$ assess mixing and disagreement between chains. An $\hat{R}$ close to one is useful evidence but cannot prove convergence or prove posterior propriety. Comparing chains that explore different modes is especially important; monitor each scientifically important observable and its Monte Carlo uncertainty. A long apparent plateau can also occur when an improper-target sampler is drifting toward a boundary.

The moment of inertia about a specified axis measures how a mass distribution resists changes of angular velocity about that axis. For point masses $m_i$ at perpendicular distances $r_{\perp i}$, or a continuous distribution with mass density $\rho$,Its SI unit is $\mathrm{kg}{\mathrm{m}}^2$. The distance is to the entire axis, not to one point on it: moving a point mass along the axis leaves its contribution unchanged. The moment of inertia depends on both the distribution and the chosen axis; it is not determined by total mass alone. About an axis fixed in an inertial frame, a rigid body with larger moment of inertia has smaller angular acceleration under the same net axial torque, and greater rotational kinetic energy at the same angular speed.^[ref]

For a simple model, put four equal point masses of total mass $M$ on light rigid spokes at radius $a$. Their moment of inertia is $M{a}^2$. Doubling $a$ quadruples the moment of inertia without changing $M$. The illustrated placements $a=R/3$ and $a=R$ give $M{R}^2/9$ and $M{R}^2$. Actual spokes and finite-sized weights contribute their own moments of inertia; the point mass model neglects them. Repositioning weights during rotation also breaks the fixed-distribution assumption, so it requires a separate work and angular momentum calculation.

The moment of inertia of a uniform disc follows by dividing it into concentric annuli: $dm=(2M/R^2)rdr$ and $I={\int }_0^R{r}^2(2M/R^2)rdr=M{R}^2/2$. For an ideal thin ring every mass element is at $R$, so $I=M{R}^2$. A ring of finite radial thickness instead uses the moment of inertia of a uniform annulus, $I=M(R^2+b^2)/2$, with inner radius $b$. A moment of inertia is thus a compact summary of radial placement rather than a complete description of shape. For several joined pieces, their moments of inertia add only when each is calculated about the same axis.^[ref]

A physical pendulum makes the moment of inertia measurable through timing. Martín Monteiro, Cecilia Stari, Cecilia Cabeza and Arturo C. Martí used a metal bar with suspension holes $1.0\mathrm{cm}$ apart and an attached smartphone. They report bar length $1.199\mathrm{m}$, bar mass $0.2518\mathrm{kg}$ and phone mass $0.1311\mathrm{kg}$. Changing the suspension hole changes the pivot axis. Their readouts were phone gyroscope angular velocity records, analyzed by autocorrelation or a fit to a sinusoid, and an independent photogate; the logged records for the sinusoid fits lasted $10\mathrm{s}$. They observed agreement between the three timing methods and the geometry-based model, including a minimum of oscillation period as the suspension point varied. These are reported apparatus details and observations, not measurements made for this article.^[ref]

To infer a moment of inertia from such a measurement, determine the total mass $M_{\mathrm{tot}}$ including every attached part, locate the assembly's centre of mass $G$, and measure its distance $h>0$ below the horizontal pivot $O$. Record successive crossings in the same direction, or the elapsed time $t_N$ for $N$ complete oscillations, giving oscillation period $P=t_N/N$. The physical pendulum model with a fixed pivot, a uniform gravitational field, negligible friction and the small-angle approximation givesThe crossing times, masses and geometric distances are the measured inputs; $I_{O,\mathrm{inferred}}$ is an inference conditional on this model, while an integral over the assumed shape supplies an independent model prediction. The parallel axis theorem converts the inferred pivot moment of inertia to one about a parallel axis through $G$. The controls and interpretation of that comparison are described there. An axial measurement determines one scalar moment of inertia, not the complete inertia tensor needed for general rotation.

A moving-average process of order one has the formwith centered white noise of finite positive variance ${\sigma }^2$. Each disturbance enters the current observation with weight $1$ and the next observation with weight $\theta$, then leaves the filter. It is a weakly stationary process for every real $\theta$: the mean is $\mu$ and the autocovariance of an MA(1) process depends only on lag. Strong white noise additionally makes it a strictly stationary process. No $|\theta |<1$ restriction is needed for stationarity.

For a worked observation calculation, take $\mu =10$, $\theta =0.7$, and consecutive noise values ${\epsilon }_{t-1}=-1$, ${\epsilon }_t=2$, ${\epsilon }_{t+1}=0$. Then $X_t=11.3$ and $X_{t+1}=11.4$; the same disturbance $2$ appears in both. If ${\sigma }^2=1$, the variance is $1.49$, the lag-one autocovariance is $0.7$, and the lag-one autocorrelation is $70/149\simeq 0.4698$. All higher-lag autocorrelations vanish, as shown in autocorrelation.

An MA(1) is useful for brief dependence caused by overlapping disturbances or measurements. Its name refers to a weighted moving sum of noise; its weights need not be positive or sum to one, and it differs from the data smoother in seasonal extraction by a centred moving average. Under strong white noise, observations more than one interval apart depend on disjoint independent noise families and are independent. Under weak white noise, the same cutoff establishes only uncorrelated random variables.

Invertibility of a moving-average model is a separate question: can the specified driving noise be recovered by a stable linear filter of present and past observations? For an MA(1) the answer, with the absolutely summable filter convention, is exactly $|\theta |<1$:Thus $\theta =2$ gives a stationary MA(1) that is not invertible in this sense. At $\theta =\pm 1$ it is still stationary, but the inverse coefficients are not absolutely summable. The case $\theta =0$ is simply shifted white noise. See Bartlett, lecture 5: MA(1) and stable invertibility.

Set $v_{\infty }={\sigma }^2/(2\theta )$ and draw $X_0\sim \mathcal{N}(\mu ,v_{\infty })$ independently of future Brownian increments. The Ornstein-Uhlenbeck process is then a strictly stationary process, withEvery time has the same normal distribution, and the covariance function depends only on the time lag. For a Gaussian process these properties also make all its finite-dimensional distributions invariant under a common time shift. The stationary distribution balances the continual random forcing against the contracting drift.^[ref]

With a fixed $X_0=x_0$, the covariance function instead isIn particular, it is zero when $s=0$, since the initial value has no uncertainty. The second term records the fixed start; it is the general scaling of the zero-start Ornstein-Uhlenbeck covariance. The pure exponential covariance function displayed above assumes stationary initialization. For a fixed start, that function emerges as the late-time limit at fixed lag, while the one-time probability distributions approach $\mathcal{N}(\mu ,v_{\infty })$ by convergence in distribution.

For the stipulated $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$, stationary initialization has variance one from the outset. Its autocorrelation is $e^{-|h|}$, so the correlation time is one: a lag of one reduces the autocorrelation to $e^{-1}\approx 0.3679$. Its raw second moment $\mathbb{E}[X_s{X}_t]=4+e^{-|t-s|}$ includes the squared expected value; it must not be confused with the covariance.

The upper panel shows three simulated sequences with independent stationary initial values and exact transitions at spacing $0.05$. Joining their points only guides the eye; it does not simulate a continuous sample path between those points. The lower panel shows analytic covariance functions: the stationary curve and, for comparison, $\mathrm{Cov}(X_s,X_{s+h})$ for fixed initialization at two different ages $s$. These are simulated samples and model curves, not measured trajectories or empirical covariance estimates.

The partial autocorrelation at lag $k$ is the correlation of $X_t$ and $X_{t-k}$ after their linear projections onto the intervening observations are removed. A causal autoregressive model of order $p$ has zero partial correlations after lag $p$. This complements the finite cutoff of the autocorrelation function for a moving-average model.

Assume ${\sigma }_1,{\sigma }_2>0$ and $|\rho |<1$, as required for the displayed nonsingular probability density function. Put $z_j=(x_j-{\mu }_j)/{\sigma }_j$. Its quadratic exponent isHolding $x_2$ fixed and normalizing the term depending on $x_1$ givesThe analogous completion of the square gives

Using independent draws ${\eta }_{1,r},{\eta }_{2,r}$ from the standard normal distribution, a sweep isUse the newly drawn first coordinate in the second update. Recording the pairs after each sweep gives the dependent sample.

The Gaussian Gibbs sweep autocorrelation makes the dependence explicit. In standardized coordinates, substitution yieldsThe new noise has variance $1-{\rho }^4$, so the stationary variance is one and the lag-$h$ sweep autocorrelation is ${\rho }^{2h}$. Thus the sampler converges geometrically for $|\rho |<1$, but becomes slow near perfect correlation. When $\rho =0$, complete sweeps yield independent pairs. The singular cases $|\rho |=1$ do not have the stipulated two-dimensional probability density function.

Let ${\sigma }_v^2=⟨v(0{)}^2⟩$, assumed finite. The displacement is $\Delta X(t)={\int }_0^{t}v(s)ds$. By stationarity, its two-time covariance is $⟨v(s)v(r)⟩={\sigma }_v^2\rho (s-r)$, with $\rho$ even for a real scalar process. Integrating over the two triangles of the square gives the Taylor turbulent dispersion identityA sufficient condition for ordinary long-time variance growth is ${\int }_0^{\infty }|\rho (s)|ds<\infty$ with ${\int }_0^{\infty }\rho (s)ds>0$. The dominated convergence theorem then givesMore generally the necessary asymptotic condition for a finite positive variance-based diffusion coefficient is convergence of ${\sigma }_v^2{\int }_0^t(1-s/t)\rho (s)ds$ to that coefficient. Absolute integrability is sufficient, not necessary. A zero integral does not give nondegenerate ordinary diffusion. Moreover a second-moment calculation alone does not establish a normal distribution or an entire diffusive scaling limit; those require additional probabilistic assumptions.

For the specified positive autocorrelation, direct integration gives, when $\alpha \ne 1,2$,Thus the power-law velocity-correlation dispersion has the following regimes. If $\alpha >1$, the Lagrangian integral time is $1/(\alpha -1)$ and ordinary variance growth holds, with $D={\sigma }_v^2/(\alpha -1)$. At $\alpha =2$ the exact expression is $2{\sigma }_v^2[t-\log (1+t)]$. If $\alpha =1$,For $0<\alpha <1$,The last two regimes spread faster than ordinary diffusion and have no finite long-time diffusion coefficient. Their slowly decaying velocity memory is responsible. At short times all these correlations instead give the ballistic law $⟨\Delta X^2⟩\sim {\sigma }_v^2{t}^2$.

For a weakly stationary process with finite second moments, let $\mu =\mathbb{E}{X}_t$ and $\gamma (h)=\mathrm{Cov}(X_{t+h},X_t)$, independent of $t$. If $\gamma (0)>0$, its autocorrelation function is $\rho (h)=\gamma (h)/\gamma (0)$. Given observations $x_1,\dots ,x_N$, one usual sample autocovariance function isThe correlogram plots $\hat{\rho }(h)$ against lag $h$. The partial autocorrelation function at lag $h$ is the correlation between $X_t$ and $X_{t-h}$ after linearly projecting each onto the intervening observations. Its sample version can be calculated by solving the Yule-Walker regression equationsthen taking the last coefficient ${\hat{\varphi }}_{h,h}$. Equivalently, use the corresponding residual correlation from the sample regressions, with a consistent covariance convention.

For a minimal causal autoregressive model of order $p$, the population partial autocorrelation function is zero after lag $p$, while the autocorrelation function follows an exponentially decaying or damped oscillatory tail. For a moving-average model of order $q$, its autocovariance and hence autocorrelation are zero after lag $q$, because observations farther apart share no driving-noise terms; its partial correlations usually tail off. These order identification by autocorrelation cutoffs patterns suggest AR or MA candidate orders. Sample cutoffs are approximate and should be judged with sampling uncertainty and residual diagnostics; the familiar white-noise bands near $\pm 1.96/\sqrt{N}$ are a null approximation, not universal confidence limits for a dependent series.

For the trend problem, write a symmetric estimator asReproducing arbitrary constant and quadratic trends requires $2a+2b+c=1$ and $8a+2b=0$; the linear-trend condition holds automatically by symmetry. These constraints follow by expanding $T_{t+j}=T_t+j(b_0+2c_{0}t)+c_0{j}^2$ for a trend $T_t=a_0+b_{0}t+c_0{t}^2$. Thus $b=-4a$, $c=1+6a$. Since the errors are uncorrelated and the estimator is unbiased, its mean squared error isThe strictly convex quadratic is minimized at $a=-3/35$, giving the five-point symmetric quadratic trend smootherThis weighted moving average allows negative coefficients; a convex average could not cancel the quadratic moment.

For three points, symmetry gives weights $(a,b,a)$. Unbiasedness for every quadratic requires $2a+b=1$ and $2a=0$, so the only solution is $(0,1,0)$. Thus there is no nontrivial three-point quadratic-preserving smoother. The literal claim that there is no unbiased estimator at all is false, since ${\hat{T}}_t=X_t$ is unbiased with error variance ${\sigma }^2$. This is the three-point quadratic reproduction forces the identity filter distinction.

For seasonal extraction by a centred moving average, use the additive decomposition $X_t=T_t+S_t+{\epsilon }_t$, where $S_t$ has period $m$ and its values over one period sum to zero. An average covering a full period removes the seasonal component and approximately retains a slowly changing trend. If $m$ is odd, take the centred length-$m$ average; if $m$ is even, centre it by averaging two adjacent length-$m$ averages, giving half weights at the two endpoints. Subtract this trend estimate from the data, average residuals within each seasonal phase over different cycles, and centre the resulting phase estimates to sum to zero. For a multiplicative decomposition, use ratios and normalize the phase estimates to average one.

Successive moving averages compose by convolution of their weights. Even an initially uncorrelated noise series becomes correlated: for weights $w_j$, the filtered noise has covariance ${\sigma }^2{\sum }_j{w}_j{w}_{j+h}$. Further smoothing changes this covariance again and can turn random variation into apparently persistent patterns while obscuring genuine short-scale features. Spectrally, repeated use of a transfer function $H$ multiplies the spectrum by $|H{|}^{2r}$ after $r$ applications, so repeated smoothing is not harmless repetition of the same estimate.

In simple exponential smoothing, choose $0<\alpha <1$ and update ${\ell }_t=\alpha X_t+(1-\alpha ){\ell }_{t-1}$. Iteration gives ${\ell }_t=\alpha {\sum }_{j=0}^{t-1}(1-\alpha {)}^j{X}_{t-j}+(1-\alpha {)}^t{\ell }_0$, so older observations receive exponentially decreasing weights. Forecast ${\hat{X}}_{t+h\mid t}={\ell }_t$ for every $h\ge 1$. This is appropriate for a locally stable level with no persistent trend or seasonality.

For a local linear trend, double exponential smoothing supplies level and slope. Holt's linear trend method usesAnother standard convention is Brown's double exponential smoothing: smooth twice with the same gain, $s_t^{(1)}=\alpha X_t+(1-\alpha )s_{t-1}^{(1)}$ and $s_t^{(2)}=\alpha s_t^{(1)}+(1-\alpha )s_{t-1}^{(2)}$, then put $a_t=2s_t^{(1)}-s_t^{(2)}$, $b_t=\alpha (s_t^{(1)}-s_t^{(2)})/(1-\alpha )$ and forecast $a_t+h{b}_t$. These variants handle a nonseasonal trend; seasonal effects need separate modelling, and a long-horizon linear extrapolation assumes the current local slope persists.

For a weakly stationary process with mean $m$ and nonzero variance, put $\gamma (h)=\mathrm{Cov}(X_{t+h},X_t)$ and $\rho (h)=\gamma (h)/\gamma (0)$. The function $\rho$ is its autocorrelation function. For observations $x_1,\dots ,x_N$, one common sample autocorrelation function isA correlogram plots these estimates against lag; the population analogue plots $\rho (h)$.

The lag-$h$ partial autocorrelation function measures the correlation between $X_t$ and $X_{t-h}$ after removing their linear projections on the intervening observations. Its sample counterpart can be calculated by the Yule-Walker equations: solveand take $\hat{\alpha }(h)={\hat{\varphi }}_{h,h}$, assuming the matrix is nonsingular. This defines a standard sample partial autocorrelation function.

For a minimal causal autoregressive model of order $p$, the autocorrelation typically decays geometrically, possibly with damped oscillations, whereas the partial autocorrelation function is zero beyond lag $p$: after those $p$ preceding values have been projected out, the innovation is orthogonal to older values. For a moving-average model of order $q$, nonoverlapping sets of driving white noise give $\gamma (h)=0$ for $|h|>q$; its partial autocorrelation function generally tails off when the representation is invertible. Thus an ACF cutoff suggests an MA order; a PACF cutoff suggests an AR order. An ARMA model generally has neither cutoff. These are population properties; sample noise makes cutoffs approximate, so fitted residuals and uncertainty should also be inspected. Under independent white noise, fixed-lag sample correlations have approximate standard error $N^{-1/2}$, giving a rough diagnostic band rather than a universal simultaneous test.

Small random-walk proposal variance gives many accepted tiny moves and strong autocorrelation; excessive variance proposes mostly low-density points, giving low acceptance and long rejection plateaus. Tune each coordinate or block during a pilot run to balance movement and acceptance, and inspect effective sample size per unit computation. Rescale or transform parameters with very different units, and account for posterior correlation rather than adjusting only diagonal variances.

There is no universal optimal acceptance percentage for every target or update dimension. Keep the chosen proposal fixed in the production run, or use an adaptive method with a proved diminishing-adaptation and containment justification. The direction of the trace-based changes for this particular run is given in the parent solution.

Convergence means that the chain's distribution approaches the intended posterior, independently of its initial state, under the relevant ergodicity assumptions. It is separate from obtaining enough effectively independent draws for a precise answer. Start several chains at dispersed plausible points, examine trace plots and stability of posterior summaries, compare between-chain and within-chain variation, and inspect autocorrelation and effective sample size of a Markov chain. Discard an initial transient when appropriate and report Monte Carlo uncertainty.

No finite diagnostic proves convergence. Agreement among chains started in the same basin can miss another mode. If different starting points give incompatible summaries, extend runs or improve proposals rather than merely deleting more observations. Extra thinning does not repair poor exploration; retaining all post-transient draws usually preserves more information for estimating uncertainty.

The strain is an Ornstein-Uhlenbeck process. Multiplying its equation by the integrating factor $e^{t/\tau }$ and using the zero initial condition givesThe second expression is a stochastic integral against Brownian motion. A deterministic linear functional of Gaussian white noise is Gaussian, and its expected value is zero. For $t\ge t^{\prime }$, the noise covariance givesFor $t^{\prime }\gg \tau$, the initial-condition term is negligible. The stationary autocorrelation is thereforeIts normalized exponential decay has correlation time $\tau$. The zero-start Ornstein-Uhlenbeck covariance also shows how stationarity is approached.

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, $\hat{a}(u)=N^{-1}{\sum }_i[s(t_i-u)-\bar{s}]$. 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: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.

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.

The proposal must connect the whole positive posterior support by moves with nonzero reverse probability. Use the full Metropolis–Hastings acceptance probability for asymmetric proposals; dropping the reverse proposal factor generally targets the wrong distribution. Compute likelihoods and acceptance ratios on a logarithmic scale to avoid overflow and underflow, and handle zero weights explicitly. A lazy step prevents periodicity when distributional convergence is needed.

Finite-chain convergence does not guarantee a useful run length: separated modes and narrow bottlenecks can make mixing extremely slow. Tune proposals to explore efficiently, inspect several starting points and chains, and assess autocorrelation and Monte Carlo uncertainty rather than treating correlated draws as independent observations. Discarding an initial segment can reduce initialization effects, but diagnostics do not prove that every posterior mode has been visited. Thinning is not required for correctness and often throws away useful information. The model and prior must themselves be specified correctly; an exact sampler from an incorrect posterior cannot repair the model.

One useful starting point is reverse correlation, implemented as a spike-triggered average. Present a long stimulus record and retain the stimulus history before each observed spike. After subtracting the stimulus mean, estimatewhere $K$ is the number of spikes and $T_k$ is the history window. This identifies a candidate temporal receptive field. For white Gaussian stimulation and a one-filter encoding model, the average is proportional to the relevant filter, provided the response nonlinearity does not make that average vanish. For example an even response to the projection can have zero average despite strong stimulus dependence. Colored inputs require accounting for stimulus covariance, and more than one relevant stimulus direction may require a richer fitted model.

Use the candidate filter in an LNP model:Estimate the static nonlinearity $g$ by binning filter projections and dividing spike counts by exposure time in each bin. Equivalently, Bayes' rule gives $g(z)=\overline{r}p(z\mid \text{spike})/p(z)$ under this model. Refine the filter and nonlinearity by maximizing the conditional Poisson log-likelihoodwith suitable regularization. Simulate an Inhomogeneous Poisson process at the fitted rate using integrated-intensity inversion from the previous solution. This gives a statistical encoding model, not a claim to reconstruct the neuron's ion channels.

Use cross-validation: estimate parameters on one set of trials and test them on independent held-out trials. Under repeated identical stimuli, compare the observed and simulated trial-averaged firing rates, spike-count distributions, interspike-interval distributions and autocorrelations. Compare held-out likelihoods and predictive errors with simpler baseline models and with the variability between real repeats. Exact spike-by-spike agreement is not expected from two independent realizations of a stochastic model. Large rate agreement accompanied by incorrect short-interval or correlation statistics reveals missing dynamics. Adding a spike-history term, for example in a point-process generalized linear model, can account for refractoriness or bursting when the basic Poisson cascade fails.

A good model predicts held-out response distributions, not just its training spike train. Such a model summarizes what stimulus features affect the neuron, permits controlled simulated changes of those features, and tests whether the proposed computation is sufficient. Its failures can suggest further experiments or additional stimulus and history variables.