# Exercises

1. 1.

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.

2. 2.

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 and four spikes were observed at times $0 .

(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.