Neuronal refractory period

{title2=$f\leq1/t_{\mathrm{abs}}$}

If every <interspike interval> is at least $t_{\mathrm{abs}}>0$, a sustained <spike train> has mean firing rate at most $1/t_{\mathrm{abs}}$. Across a window of duration $T$, the event count satisfies $N(T)\leq1+T/t_{\mathrm{abs}}$; dividing by $T$ and taking the long-time limit gives the bound. The extra one accounts for a possible first event. The bound permits much slower firing and says nothing about whether stronger drive can maintain the limiting rate in a real <neuron>.

For the chosen <phenomenological paired-pulse recovery>, the hold alone gives $500\,\mathrm{Hz}$. Add the further assumption that every successful event resets precisely the same recovery curve, with no accumulation from earlier events. Then $0.35\,\mathrm{nA}$ onset-timed pulses can sustain at most $1000/4.942488\approx202.327\,\mathrm{Hz}$ in that model. This repeatable reset is an extra assumption: a two-pulse result alone cannot establish it. A real <neuron>'s sustained rate also depends on <neural adaptation>, input delivery and evolving <ion-channel inactivation>. The existing <leaky integrate-and-fire model> similarly adds integration time to its imposed hold.

A homogeneous <Poisson process> of rate $\lambda$ has <interspike intervals> with an <exponential distribution>, so
$$
\Pr(S<t_{\mathrm{abs}})=1-e^{-\lambda t_{\mathrm{abs}}}>0.
$$
With the purely illustrative choices $\lambda=100\,\mathrm{Hz}$ and $t_{\mathrm{abs}}=0.002\,\mathrm{s}$, this is $1-e^{-0.2}\approx0.181269$. Thus about $18.13\%$ of intervals predicted by that model violate this chosen hold. Matching a mean <spike train> count does not make a stimulus-only <Poisson process> respect recent firing.

A minimal alternative is a <renewal process> with <independent and identically distributed> intervals $S=t_{\mathrm{abs}}+X$, where $X$ has an <exponential distribution> of rate $\lambda$. Its <hazard function> is zero during the hold and $\lambda$ afterwards. Its mean interval is $t_{\mathrm{abs}}+1/\lambda$ and long-run firing rate is
$$
f=\frac{\lambda}{1+\lambda t_{\mathrm{abs}}}.
$$
The same $100\,\mathrm{s}^{-1}$ rate after a $2\,\mathrm{ms}$ hold gives $83.333\,\mathrm{Hz}$, not $100\,\mathrm{Hz}$. This separate stochastic example has no gradual relative recovery; it illustrates why an available-state rate differs from an observed mean firing rate. Smooth dependence on time since the last event can extend the <hazard function>, while longer histories require more than this simple <renewal process>. <Time-rescaled spike-train diagnostics> can test the resulting history-dependent predictions against recorded <spike trains>.


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 Incoming links: Refractory limits on neuronal firing