8 Noisy Input Models: Barrage of Spike Arrivals


  1. 1.

    Colored noise .

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

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

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

    ξ(t)ξ(t)=σ2τsδ(t-t),\langle\xi(t)\,\xi(t^{\prime})\rangle=\sigma^{2}\,\tau_{s}\,\delta(t-t^{\prime% })\,, (8.56)

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

    τm   d      d   tu(t)=-u(t)+RIdet(t)+RInoise(t),\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 τs\tau_{s} leads to colored noise.

  2. 2.

    Autocorrelation of the membrane potential Determine the autocorrelation u(t)u(t)\langle u(t)\,u(t^{\prime})\rangle of the Langevin equation ( 8.7 ) where ξ(t)\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 wexc=+ww^{\rm exc}=+w and time course ϵexc(t)=(t2/τexc3)exp(-t/τexc)\epsilon^{\rm exc}(t)=(t^{2}/\tau_{\rm exc}^{3})\,\exp(-t/\tau_{\rm exc}) for t>0t>0 . Spike arrival at an inhibitory synapse causes an IPSP with weight -bwexc-b\,w^{\rm exc} and, for t>0t>0 a time course ϵinh(t)=(t/τinh2)exp(-t/τinh)\epsilon^{\rm inh}(t)=(t/\tau_{\rm inh}^{2})\,\exp(-t/\tau_{\rm inh}) where τinh>τexc\tau_{\rm inh}>\tau_{\rm exc} and b>1b>1 .

    The membrane potential is

    u(t)=wexct(f)ϵexc(t-t(f))-bwinht(f)ϵinh(t-t(f))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 νexc=ν1\nu^{\rm exc}=\nu_{1} and νinh=βν1\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 ν1\nu_{1} without changing the mean or the variance of the membrane potential. Does this limit exist for all combinations of parameters bb and β\beta or do you have to impose a specific relation b=f(β)b=f(\beta) ? Interpret your result.