Renewal Theory
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 $S_I(t\mid \hat t) = 1  \int_{\hat t}^t P_I(tâ€™\mid \hat t) dtâ€™$. 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:
Stationary Renewal Theory
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

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 The importance of it, is that we could find out which frequency mode has the most important amplitude.
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Note that WienerKhinchin theorem says $P(\omega) = \hat C(\omega) = \mathscr{F}(C(s))$.(Proof is straightforward.)
signal to noise ratio
Derive Autocorrelation function of stationary renewal process
Define normalized autocorrelation Autocorrelation for $s>0$
Some Questions and Comments about The Book

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 â€‹