Neural Mechanism for a Cognitive Timer

To examine a possible biological mechanism for a cognitive timer, the stochastic dynamics of a network of neurons possessing two stable states (“on” and “off” states) is studied. The fraction of on neurons existing at $t=0\mathrm{}$ remains large for an extended interval, and then abruptly falls. The distribution of the lengths of the interval is scale invariant in the following sense: The ratio $\sqrt[\mathrm{A}\mathrm{P}\mathrm{S}::\mathrm{X}\mathrm{M}\mathrm{L}::\mathrm{G}\mathrm{e}\mathrm{n}\mathrm{P}\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{e}=\mathrm{H}\mathrm{A}\mathrm{S}\mathrm{H}\left(0\mathrm{x}1\mathrm{b}633\mathrm{c}\right)\mathrm{}]{{\mu }^{\left(k\right)}}/m$, with $m$ and ${\mu }^{\left(k\right)}$ being the mean and the $k$th central moment, respectively, is invariant under scale transformations of $m$ and ${\mu }^{\left(k\right)}$. In the special case $k=2\mathrm{}$, this gives Weber's law, a hallmark of cognitive timing.