Colored noise .
(i) Calculate the noise spectrum of the colored noise defined by Eq. ( 8.15 ) which we repeat here:
where is white noise with mean zero and variance
(ii) Calculate the membrane potential fluctuations caused by the colored noise in Eq. ( 8.55 ), using the differential equation
(iii) Show that the limit process of balanced excitatory and inhibitory input with synaptic time constant leads to colored noise.
Autocorrelation of the membrane potential Determine the autocorrelation of the Langevin equation ( 8.7 ) where is white noise.
Membrane potential fluctuations and balance condition . Assume that each spike arrival at an excitatory synapse causes an EPSP with weight and time course for . Spike arrival at an inhibitory synapse causes an IPSP with weight and, for a time course where and .
The membrane potential is
Excitatory and inhibitory spike arrival are generated by Poisson processes rate and , 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 without changing the mean or the variance of the membrane potential. Does this limit exist for all combinations of parameters and or do you have to impose a specific relation ? Interpret your result.
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