Linear filter as optimal stimulus . Consider an ensemble of stimuli with a ‘power’ constraint .
(i) Show that, under the linear rate model of Eq. ( 11.10 ) the stimulus that maximizes the instantaneous rate is .
Hint: Use Lagrange-multipliers to implement the constraint .
(ii) Assume that the a spatially localized time-dependent stimulus is presented in the center of the positive lobe of the neurons receptive field. Describe the neuronal response as
where is the spontaneous firing rate in the presence of a gray screen and the temporal extent of the filter . What stimulus is most likely to cause a spike under the constraint ? Interpret your result.
LNP model and reverse correlations . Show that, if an experimentalist uses stimuli with a radially symmetric distribution , then reverse correlation measurements provide an unbiased estimate linear filter under a LNP model
i.e., the expectation of the reverse correlation is parallel to .
Hint: Write the stimulus as
and determine the reverse correlation measurement by averaging over all stimuli weighted with their probability to cause a spike.
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