Impulse Responses of Identical Transit Stages

Why an Impulse Response Is Useful

A short input can produce a delayed output when a signal passes through a sequence of transit stages. The cascade does not simply shift the input by a fixed amount: each first-order stage contributes its own delay and spreads the signal over time. An impulse response describes the output produced by an idealized, instantaneous input. It provides a compact way to summarize the cascade’s timing and shape.

Here the stages are serial, have the same first-order rate, and are linear. The response at the end of the chain has a gamma-shaped form. Its mean is the average transit time, its variance describes the spread of transit times, and its peak time identifies when the output flux is greatest. These quantities are related, but they are not interchangeable.

Model and Definition

Consider \(n\) stages, where \(n\) is a positive integer. Let \(x_i(t)\) be the amount in stage \(i\) at time \(t\), and let \(k>0\) be the common first-order rate constant, with units of inverse time. An amount leaves each stage at rate \(k x_i(t)\); for stages before the last, this amount enters the next stage. An external input \(u(t)\) enters stage 1.

The model equations are

The final-stage output flux is \(y(t)=k x_n(t)\). For an impulse of amount \(A\), written \(u(t)=A\delta(t)\), the input causes \(x_1\) to jump by \(A\) at time zero, while \(x_i(0^+)=0\) for \(i\geq 2\). Here \(\delta(t)\) denotes the ideal impulse, whose integral over time is 1. The response per unit input amount is \(h(t)=y(t)/A\).

This setup assumes a linear, time-invariant serial chain; the same rate \(k\) applies in every stage; and there is no feedback, branching, or loss other than transfer through the stages. The impulse is idealized as instantaneous, and the stages begin empty. If the measured output is the amount in the last stage rather than its exit flux, it is \(x_n(t)\), which differs from \(h(t)\) only by a constant factor \(A/k\) for this input. Its normalized time profile has the same shape.

Deriving the Response

The first stage receives the impulse and then empties exponentially. Each subsequent stage receives the preceding stage’s output and itself loses material at rate \(k\). Applying this recursion gives the amount in stage \(i\); the final output flux follows by multiplying the last-stage amount by \(k\).

The recursion can be checked directly. Suppose \(x_{i-1}(t)=A e^{-kt}(kt)^{i-2}/(i-2)!\). The integrating-factor solution for stage \(i\) is

This verifies the stage formula from the known response of stage \(i-1\); stage 1 supplies the base case. Consequently, the final-stage output flux and the unit-area impulse response are

This is an Erlang density: a gamma density with integer shape \(n\) and rate \(k\). Its area is 1, so an impulse of amount \(A\) produces a total output flux integrated over time of \(A\). The normalization follows from substituting \(z=kt\) in the integral and using \(\int_0^\infty z^{n-1}e^{-z}\,dz=(n-1)!\).

Mean Delay, Variance, and Peak

The response can be treated as a probability density over transit time \(T\): \(h(t)\) gives the fraction of the impulse output emerging per unit time. For any nonnegative integer \(r\), its \(r\)th moment is obtained by integrating \(t^r h(t)\). Substituting \(z=kt\) gives

In particular, \(E[T]=n/k\) and \(E[T^2]=n(n+1)/k^2\). Subtracting the squared mean from the second moment gives the variance. The peak is found by differentiating the log of the response for \(t>0\), which has the same maximizing time as the response itself:

For \(n>1\), setting the derivative to zero gives \(t_{\text{peak}}=(n-1)/k\); the derivative is positive before this time and negative after it. For \(n=1\), \(h(t)=ke^{-kt}\) decreases from its maximum at \(t=0\), so the peak time is zero. Thus the mean occurs later than the peak for every \(n>1\). The units also provide a useful check: the mean and peak time have units of time, while the variance has units of time squared.

Assumptions, Diagnostics, and Alternatives

The gamma-shaped result depends on the model, not merely on the appearance of a delayed curve. It assumes a single serial path, identical first-order rates, linear transfer, and an instantaneous input into an initially empty chain. With these assumptions, the response is nonnegative and has unit area after normalization. Its mean, variance, and peak must also obey the formulas above.

When evaluating whether the response is plausible, inspect the time course as well as the summary statistics. A substantially negative response is incompatible with this impulse response unless the measured quantity or preprocessing allows negative values. A response with a long rise-and-fall pattern may be qualitatively compatible, but an apparent peak before zero, multiple peaks, or a timing relationship inconsistent with the model deserves investigation. Finite observation windows can miss late output and make the observed area too small; sparse sampling can also obscure the true peak. Measurement noise and a non-impulsive input can alter the observed profile, so disagreement does not by itself identify the cause.

A common mistake is to call \(n/k\) the peak time because it is the cascade’s “delay.” That reasoning confuses the mean output time with the mode, or peak. Differentiating the response gives \(t_{\text{peak}}=(n-1)/k\) when \(n>1\), whereas integration gives the mean \(n/k\). For example, in the three-stage example the peak is at 4 min, not 6 min.

A single first-order stage is the special case \(n=1\): its response peaks immediately and then decays exponentially. A fixed pure delay instead shifts an input without the progressive spread implied by multiple first-order stages. The identical-stage cascade lies between these simplified descriptions by representing a distributed delay with a specific shape. If stage rates differ, the identical-rate Erlang formula does not apply; that case requires a different response. Likewise, feedback, nonlinear transfer, branching, and time-varying rates violate assumptions used in this derivation.

The model’s simplicity is useful when a smooth, unimodal delayed response is a reasonable approximation and a small set of timing summaries is informative. Its limitations matter when interpreting parameters: \(n\) is constrained to a positive integer in this model, and \(k\) is common to every stage. Real systems may not satisfy either constraint. A good fit to one response curve is not, on its own, evidence that the underlying biological process literally contains \(n\) identical compartments.