Linear filter as optimal stimulus . Consider an ensemble of stimuli $x$ with a ‘power’ constraint $|\mbox{\boldmath\(x\)}|^{2}<c$ .
(i) Show that, under the linear rate model of Eq. ( 11.10 ) the stimulus that maximizes the instantaneous rate is $\mbox{\boldmath\(x\)}=\mbox{\boldmath\(k\)}$ .
Hint: Use Lagrange-multipliers to implement the constraint $|\mbox{\boldmath\(x\)}|^{2}=c$ .
(ii) Assume that the a spatially localized time-dependent stimulus $x(t)$ is presented in the center of the positive lobe of the neurons receptive field. Describe the neuronal response as
$\rho(t)=\rho_{0}+\int_{0}^{S}\kappa(s)\,x(t-s){\text{d}}s$ | (11.18) |
where $\rho_{0}$ is the spontaneous firing rate in the presence of a gray screen and $S$ the temporal extent of the filter $\kappa$ . What stimulus is most likely to cause a spike under the constraint $\int_{0}^{S}[x(t-s)]^{2}{\text{d}}s<c$ ? Interpret your result.
LNP model and reverse correlations . Show that, if an experimentalist uses stimuli $x$ with a radially symmetric distribution $p(\mbox{\boldmath\(x\)})=q(|\mbox{\boldmath\(x\)}|)$ , then reverse correlation measurements provide an unbiased estimate linear filter $k$ under a LNP model
$\rho(t)=f(\mbox{\boldmath\(k\)}\cdot\mbox{\boldmath\(x\)}_{t});$ | (11.19) |
i.e., the expectation of the reverse correlation is parallel to $k$ .
Hint: Write the stimulus as
$\mbox{\boldmath\(x\)}=(\mbox{\boldmath\(k\)}\cdot\mbox{\boldmath\(x\)})\,\mbox% {\boldmath\(k\)}+({\bf e}\cdot\mbox{\boldmath\(x\)})\,{\bf e}$ | (11.20) |
and determine the reverse correlation measurement by averaging over all stimuli weighted with their probability to cause a spike.
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