What does Weber’s law tell us about spike statistics?
© Shouval et al; licensee BioMed Central Ltd. 2012
Published: 16 July 2012
where K=(±β(1-ρ)/α)n/τ and n=1/(1-ρ). Figure 1B below shows parts of the derivation, and Figure 1C examples of log-power tuning curves. All the parameters of this function except one are determined by the spike statistics, and the free parameter does not alter the scaling. More complex exact calculations confirm the validity of the results obtained using the linear approximation. We also compare our predictions to available data.
This analysis can also be applied in the temporal domain to the scalar timing law using spike rates that vary in time with a log-power profile. Our linear approximation demonstrates the principles of our theory and a discrete series approximation captures almost exactly the results of stochastic simulations.
We also test the optimality of these scaling laws using use variational calculus to find a joint differential equation for the tuning curves and the probability distribution of the measured variable. In a surprisingly general result, this analysis shows that Weber’s law is optimal only if the estimated variable has a scale invariant power-law distribution with an exponent of n=-2.
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