Precise
computation of mean firing rate in roughness
RF model:
Mean firing rate of the cortical neuron (=
positive area of E –I) at zero threshold is:
where x is
instantaneous firing rate, μ is mean of E-I, and p is probability of finding particular firing rate
x.
The first and second
integrals defined as I1 and I2 can be
calculated as follows:
(More detailed calculation
of I1 is described below.)
One needs to find derivation of g(t)(i.e.,
d/dt(g(t)) that is equal to the content inside the integral (i.e., f(t)).
From the equation above,
For the second
integral I2 :
where G(μ)
is Gaussian function. Mean μ is 0 because we made a substitution for
x – μ to be t. Combining the solutions of two
integrals, mean rate is going to be:
There are three situations
as the solutions for this equation.
(1) When E = I, μ
= 0.
(2) When E >>
I, μ >> 0. This
leads to I1 = 0 and G(μ) = 1 because μ
= infinity. Therefore,
(3) When I <<
E, there will be no spikes.
In
a realistic situation E and I are fairly balanced, and mean rate (positive area
of E-I) is proportional to the standard deviation of E and I as the situation
(1) shows. (It should also be noted that the variance of spikes are
correlated with mean firing rate.
