Theoretical Neuroscience UE 1

Exercise1-0.pdf

What is a firing rate?

The Number of Action Potential Spikes per Second

What physical unit does a firing rate, its variance, and standard deviation have?

• firing rate: Hz
• Varianz Hz
• Standard deveration: Hz

Consider an experiment in which you record multiple trials from a neuron in an impatient animal. Which trade-off do you have to consider if you want to estimate a time-varying rate as best as possible?

Length of record vs result variance over time because of the animal getting stressed

If you are able to obtain only one trial, how can you nevertheless estimate a (continuous) firing rate even for times where you did not observe a spike? Which downside does your method have?

Use mean firing rate ???

2

a)

There is on spike per i so there are n spikes.

b)

$a(r)$

$$\begin{array}{rl}V& =2r_0{a}_0=1\\ a_0& =\frac{1}{2r_0}\end{array}$$

$b(r)$

$$\begin{array}{rl}V& =2b_1{r}_0-\frac{2b_0{r}_0}{2}\\ & =2b_1{r}_0-b_0{r}_0\\ & =1\\ \frac{1}{r_0}& =2b_1-b_0\\ b_0& =-\frac{1}{r_0}+2b_1\\ b_1& =\frac{\frac{1}{r_0}+b_0}{2}\\ & =\frac{\frac{1}{r_0}}{\frac{2}{1}}+\frac{b_0}{2}\\ & =\frac{1}{2}{r}_0+\frac{b_0}{2}\\ & =\frac{1}{2}(r_0{b}_0)\\ b_1-b_0|r_0|& \ge 0\end{array}$$

Spike count is preserved because:

The unbounded Integral / Volume of the kernel function is one.
So in the unbounded Integral over a Spiketrain is a sum of $n$ one kernels with are one. ⇒ unbounded Integral over a Spiketrain is $=n$