Integrate-and-fire model with linear escape rates . Consider a leaky integrate-and-fire neuron with linear escape rate,
$\rho_{I}(t|\hat{t})=\beta\,[u(t|\hat{t})-\vartheta]_{+}\,$ | (9.38) |
(a) Start with the non-leaky integrate-and-fire model by considering the limit of $\tau_{m}\to\infty$ . The membrane potential of the model is then
$u(t|\hat{t})=u_{r}+{1\over C}\int_{\hat{t}}^{t}I(t^{\prime}){\text{d}}t^{% \prime}\,;$ | (9.39) |
Assume constant input, set $u_{r}=0$ and calculate the hazard and the interval distribution.
(b) Consider the leaky integrate-and-fire model with time constant $\tau$ and constant input $I_{0}$ . Determine the membrane potential, the hazard and the interval distribution.
Likelihood of a spike train . In an in-vitro experiment, a time-dependent current $I(t)$ was injected into a neuron for a time $0<t<T$ and four spikes were observed at times $0<t^{(1)}<t^{(2)}<t^{(3)}<t^{(4)}<T$ .
(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.
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