11.Renewal Theory

Renewal Theory

Review of three key functions:

1. $P_I$: 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 $[\hat t, t_f]$.
2. $S_I$: 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 $[\hat t, t]$.
3. $\rho_I$: 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: $$ P_I(t\mid \hat t) =\rho_I(t\mid \hat t) S_I(t\mid \hat t) $$

$$ S_I(t\mid \hat t) = \exp\left( - \int_{\hat t}^t \rho_I(t’\mid \hat t) dt’ \right) $$

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:

1. 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

   $$\langle s\rangle = -\int_0^\infty s \cdot \mathrm dS_0(s) =- \int_0^\infty s S_0(s)ds - \left( -\int_0^\infty S_0(s)ds \right) $$

2. 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

   $$

3. 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.

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   Note that Wiener-Khinchin 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 $$ C^0(s) = C(s) -\nu_i^2 $$ Autocorrelation for $s>0$ $$ \nu_i C_+(s) = C(s)\Theta(s) $$

Some Questions and Comments about The Book

1. 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.

2. Dirac delta function has an integral form $$ \delta(\omega) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{itx} dt $$ ​