Aufgabe 5: Integrate and fire versus Spike response model

[8+6* Punkte, ausgegeben am 26.4.2005, Abgabe bis 10.05.2005, pdf, ps.gz]

An Integrate-and-fire neuron of the form

$\displaystyle \tau_m \frac{d u_i}{dt} = -u_i(t)+R \sum_j w_{ij} \sum_f \alpha(t-t_j^{(f)})+R I_i^{ext}(t)$    

can be rewritten in the form of a spike response model as

$\displaystyle u(t) = \eta(t-\hat t_i)+ \sum_j w_{ij} \sum_f \epsilon(t-\hat t_i, t-t_j^{(f)}) + \int_0^{\infty} \kappa(t-\hat t_i, s) I_i^{ext}(t-s) ds .$    

A) Kernels(8 Points)

You can use the formulas derived so far in the lecture. Derive the kernels $ \eta$, $ \epsilon$, and $ \kappa$ accordingly.

B) Response Kernel(6* Points)

Consider a synaptic current of the form

$\displaystyle \alpha(s)=\frac{1}{\tau_s} exp(-s/\tau_s) \Theta(s) .$    

Derive the response-kernel $ \epsilon$ in this case. Interprete the result.

Neuronale Netzwerke B (SS05), Rober Legenstein, 2005