25. The Significance of Single Spike
Single spike can have dramatic consequences on population activity.
Terms
- PSTH: peri-stimulus-time histogram, meaning the probability density of firing as a function of time, after the stimulus.
One Input Spike
- Some neuron takes constant input $I_0$ and noise $I_{\mathrm{noise}}$.
- We inject an extra input on to this neuron.
The factors of importance are
- amount of noise
- time course of PSP caused by the injection.
Relation between PSP and PSTH. Basically all well expalained in Fig 7.12:
- For large noise, PSTH is similar to PSP,
- For small noise, PSTH is the derivatives of PSP.
Amazing but why?
Read Fig 7.11. Consider two scenarios,
- with noise, basically noise will trigger a spike,
- without noise: related to the derivatives of psp because spike can only occur when the derivative is positive.
Understand the significance using homogeneous population model. Linearized equation is applied
$$ f_{\mathrm{PSTH}}(t) = \frac{d}{dt} \int_0^\infty \mathcal L (x) \epsilon_0(t-x) dx, $$
where
$$
\mathcal L(x) \sim \begin{cases}
\delta(x) & \qquad \text{low noise limit} \
\text{broad} & \qquad \text{high noise}
\end{cases}.
$$
Reverse Correlation
Reverse correlation: Record the input of the neuron just before it spikes, then average many spikes.
$$ C^{\mathrm{rev}}(s) = \langle \Delta I(t^{(f)-s}) \rangle_f, $$
where $\Delta I$ is the stimulus right before the spike at time $t^{(f)}$.
Reverse correlation is related to correlation function $C$ through
$$ \nu C^{rev}(s) = C(s), $$
where $\nu$ is the firing rate, $\nu=A_0$.
We will find the relation between this reverse correlation and transfer properties of a single neuron, which is described by
$$ \hat A(\omega) = \hat G(\omega) \hat I(\omega), $$
We derive the population activity using the transfer function $\hat G(\omega)$
$$ A(t) = A_0 + \int_0^\infty G(s) \Delta I(t-s) ds. $$
Fourier transform of multiplications leads to a convolution.
With the expression of $A(t)$, we could calculate reverse correlation
$$
\begin{align}
C(s) =& \lim_{T\to\infty} \frac{1}{T} \int_0^T A(t+s)\Delta I(t) dt \
=& \lim_{T\to\infty} \frac{1}{T} \int_0^T \int_0^\infty G(s’) \Delta I(t+s-s’) ds’ \Delta I(t) dt\
=& \int_0^\infty ds’ G(s’) \lim_{T\to\infty}\frac{1}{T}\int_0^T \Delta I(t+s-s’) \Delta I(t) dt\
=& \int_0^\infty ds’ G(s’) \langle \Delta I(t+s-s’)\Delta I(t)\rangle
\end{align}
$$
The reason we dropped the term $A_0$ is because we assume the input is stochastic i.e., $\langle \Delta I(t)\rangle=0$.
For white noise, we have
$$ \langle \Delta I(t’)\Delta I(t)\rangle=\sigma^2\delta(t’-t). $$
Then we find the relation between reverse correlation and transfer function,
$$ C^{rev}(s) = \frac{1}{\nu}C(s) = \frac{\sigma^2}{\nu} G(s). $$
Table of Contents
References:
Current Ref:
- snm/25.significance-of-single-spike.md