# Exercises

1. 1.

Colored noise .

(i) Calculate the noise spectrum of the colored noise defined by Eq. ( 8.15 ) which we repeat here:

 $\tau_{s}{{\text{d}}I^{\rm noise}(t)\over{\text{d}}t}=-I^{\rm noise}(t)+\xi(t)$ (8.55)

where $\xi(t)$ is white noise with mean zero and variance

 $\langle\xi(t)\,\xi(t^{\prime})\rangle=\sigma^{2}\,\tau_{s}\,\delta(t-t^{\prime% })\,,$ (8.56)

(ii) Calculate the membrane potential fluctuations $\langle(\Delta u(t))^{2}\rangle$ caused by the colored noise in Eq. ( 8.55 ), using the differential equation

 $\tau_{m}\frac{{\text{d}}}{{\text{d}}t}u(t)=-u(t)+R\,I^{\rm det}(t)+R\,I^{\rm noise% }(t)\,,$ (8.57)

(iii) Show that the limit process of balanced excitatory and inhibitory input with synaptic time constant $\tau_{s}$ leads to colored noise.

2. 2.

Autocorrelation of the membrane potential Determine the autocorrelation $\langle u(t)\,u(t^{\prime})\rangle$ of the Langevin equation ( 8.7 ) where $\xi(t)$ is white noise.

3. 3.

Membrane potential fluctuations and balance condition . Assume that each spike arrival at an excitatory synapse causes an EPSP with weight $w^{\rm exc}=+w$ and time course $\epsilon^{\rm exc}(t)=(t^{2}/\tau_{\rm exc}^{3})\,\exp(-t/\tau_{\rm exc})$ for $t>0$ . Spike arrival at an inhibitory synapse causes an IPSP with weight $-b\,w^{\rm exc}$ and, for $t>0$ a time course $\epsilon^{\rm inh}(t)=(t/\tau_{\rm inh}^{2})\,\exp(-t/\tau_{\rm inh})$ where $\tau_{\rm inh}>\tau_{\rm exc}$ and $b>1$ .

The membrane potential is

 $u(t)=w^{\rm exc}\sum_{t^{(f)}}\epsilon^{\rm exc}(t-t^{(f)})-b\,w^{\rm inh}\sum% _{t^{(f)}}\epsilon^{\rm inh}(t-t^{(f)})$ (8.58)

Excitatory and inhibitory spike arrival are generated by Poisson processes rate $\nu^{\rm exc}=\nu_{1}$ and $\nu^{\rm inh}=\beta\nu_{1}$ , respectively.

(i) Determine the mean membrane potential.

(ii) Calculate the variance of the fluctuations of the membrane potential.

(iii) You want to increase the rate $\nu_{1}$ without changing the mean or the variance of the membrane potential. Does this limit exist for all combinations of parameters $b$ and $\beta$ or do you have to impose a specific relation $b=f(\beta)$ ? Interpret your result.