11 Encoding and Decoding with Stochastic Neuron models


  • Linear filter as optimal stimulus . Consider an ensemble of stimuli 𝐱x with a ‘power’ constraint |𝒙|2<c|\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 𝒙=𝒌\mbox{\boldmath\(x\)}=\mbox{\boldmath\(k\)} .

    Hint: Use Lagrange-multipliers to implement the constraint |𝒙|2=c|\mbox{\boldmath\(x\)}|^{2}=c .

    (ii) Assume that the a spatially localized time-dependent stimulus x(t)x(t) is presented in the center of the positive lobe of the neurons receptive field. Describe the neuronal response as

    ρ(t)=ρ0+0Sκ(s)x(t-s)   𝑑   s\rho(t)=\rho_{0}+\int_{0}^{S}\kappa(s)\,x(t-s){\text{d}}s (11.18)

    where ρ0\rho_{0} is the spontaneous firing rate in the presence of a gray screen and SS the temporal extent of the filter κ\kappa . What stimulus is most likely to cause a spike under the constraint 0S[x(t-s)]2   𝑑   s<c\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 𝐱x with a radially symmetric distribution p(𝒙)=q(|𝒙|)p(\mbox{\boldmath\(x\)})=q(|\mbox{\boldmath\(x\)}|) , then reverse correlation measurements provide an unbiased estimate linear filter 𝐤k under a LNP model

    ρ(t)=f(𝒌𝒙t);\rho(t)=f(\mbox{\boldmath\(k\)}\cdot\mbox{\boldmath\(x\)}_{t}); (11.19)

    i.e., the expectation of the reverse correlation is parallel to 𝐤k .

    Hint: Write the stimulus as

    𝒙=(𝒌𝒙)𝒌+(𝐞𝒙)𝐞\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.