9 Noisy Output: Escape Rate and Soft Threshold


  1. 1.

    Integrate-and-fire model with linear escape rates . Consider a leaky integrate-and-fire neuron with linear escape rate,

    ρI(t|t^)=β[u(t|t^)-ϑ]+\rho_{I}(t|\hat{t})=\beta\,[u(t|\hat{t})-\vartheta]_{+}\, (9.38)

    (a) Start with the non-leaky integrate-and-fire model by considering the limit of τm\tau_{m}\to\infty . The membrane potential of the model is then

    u(t|t^)=ur+1Ct^tI(t)   𝑑   t;u(t|\hat{t})=u_{r}+{1\over C}\int_{\hat{t}}^{t}I(t^{\prime}){\text{d}}t^{% \prime}\,; (9.39)

    Assume constant input, set ur=0u_{r}=0 and calculate the hazard and the interval distribution.

    (b) Consider the leaky integrate-and-fire model with time constant τ\tau and constant input I0I_{0} . Determine the membrane potential, the hazard and the interval distribution.

  2. 2.

    Likelihood of a spike train . In an in-vitro experiment, a time-dependent current I(t)I(t) was injected into a neuron for a time 0<t<T0<t<T and four spikes were observed at times 0<t(1)<t(2)<t(3)<t(4)<T0<t^{(1)}<t^{(2)}<t^{(3)}<t^{(4)}<T .

    (a) What is the likelihood that this spike train could have been generated by a leaky integrate-and-fire model with linear escape rate defined in Eq. ( 9.38 )?

    (b) Rewrite the likelihood in terms of the interval distribution and hazard of time-dependent renewal theory.