layout | title | date | author | summary | weight |
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11.Renewal Theory |
2016-07-05 |
ErbB4 |
Renewal Theory |
11 |
Review of three key functions:
-
$P_I$ : probability density of finding spikes. Also called hazard function. Thus$\int_{\hat t}^{t_f} P(t\mid \hat t)dt$ is the probability of finding spikes during$[\hat t, t_f]$ . -
$S_I$ : survivor function. Defined as$S_I(t\mid \hat t) = 1 - \int_{\hat t}^t P_I(t'\mid \hat t) dt'$ . The probability of staying quite during$[\hat t, t]$ . -
$\rho_I$ : rate of decay, defined as$\rho_I(t\mid \hat t) = - \frac{d}{dt} S_I(t\mid \hat t) \big / S)I(t\mid \hat t)$ .
Relations between the three: $$ P_I(t\mid \hat t) =\rho_I(t\mid \hat t) S_I(t\mid \hat t) $$
Stationary input? Not easily realized in experiments (for in vivo experiments). Reasoning: in put to a neuron by other neurons in vivo is not necessarily constant.
in vitro experiments: impose constant input current.
Three important quantities:
-
mean firing rate,
$\nu = 1/\langle s\rangle$ , where the mean interval$\langle s\rangle = \int_0^\infty s P_0(s) ds$ . Since$P_0=-dS_0(s)/ds$ , we have$P_0(s) ds= -dS_0(s)$ , which leads to$$\langle s\rangle = -\int_0^\infty s \cdot \mathrm dS_0(s) =- \int_0^\infty s S_0(s)ds - \left( -\int_0^\infty S_0(s)ds \right) $$ -
autocorrelation function, $$
C(s) = \langle S_i(t) S_i(t+s) \rangle_t = \frac{1}{T} \int_{-T/2}^{T/2} S_i(t) S_i(t+s)dt
$$
-
power spectrum, which is defined as $$ \mathscr P_T(\omega) = \frac{1}{T} \left \vert \int_{-T/2}^{T/2} S_i(t) e^{-i\omega t} \right \vert^2 $$ The importance of it, is that we could find out which frequency mode has the most important amplitude.
Note that Wiener-Khinchin theorem says
$P(\omega) = \hat C(\omega) = \mathscr{F}(C(s))$ .(Proof is straightforward.)
signal to noise ratio
Define normalized autocorrelation
$$
C^0(s) = C(s) -\nu_i^2
$$
Autocorrelation for
-
On page 160,
$\nu \Delta t$ should be the number of spike during$\Delta t$ . IF we think of it as the probability of spikes, this is not normalized. -
Dirac delta function has an integral form $$ \delta(\omega) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{itx} dt $$