# 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