Explanation of this vector notation? In a journal article I'm reading, a vector is written as:
\begin{gather*}
R_{ON} = \Sigma_{\varphi_{k}} \langle x_{\varphi_{k}} \rangle_{ON} \cdot e^{j \varphi_{k}}
\end{gather*}
What's really confusing me is how the angle brackets are being used here, and how that's working with the summation. Any tips? Can someone write out the whole vector if 
$\varphi_0 = 0, \varphi_1 = \pi, \varphi_2 = 2 \pi, x_{\varphi_0} = 1.1, x_{\varphi_1} = 1.2, x_{\varphi_2} = 1.3$ ? 
The article also mentions
\begin{gather*}
TR_{ON} = \frac{|R_{ON}|}{\Sigma_{\varphi_k} \langle x_{\varphi_k} \rangle_{ON}}
\end{gather*}
TR is described as the "tuning ratio," which makes me think that the top and bottom of that fraction are both scalars? What's up with that?
Any help would be much appreciated! 
P.S.
All of this is from the Journal of Neuroscience, published in 2011. The article is: http://www.jneurosci.org/content/31/29/10689.long
You can find the exact notation in the "Direction selectivity calculation" paragraph, two paragraphs below "Figure 1." (I some light substitutions for simplicity's sake).
 A: Here are some minor observations:


*

*First, the denotion with angle brackets is a scalar by its nature. In article it's said that it was get via averaging of some statistical information from detectors. Possibly it's a voltage or something else that is usually measured at neurons. I think, you understand that part better than me. 

*If you want an example for this summation, here it is:
$$ R_{ON} = 1.1 * e^{j\cdot 0} + 1.2 * e^{j \cdot \pi} = 1.1 - 1.2 = -0.1 $$
It's a complex number $-0.1 + 0 \cdot j$ that can be viewed as a vector $(-0.1,\; 0)$. I've omitted the $2\pi$-term, because article doesn't include it in sum.
What's completely obscure to me here is that data obtained only from $8$ directions, but sum uses all directions from $0$ degrees to $315$ (why $315$? why not $359$?).

*As for $TR_{ON}$ -- it's scalar too. The numerator is an absolute value of vector $R_ON$, so it's scalar. Denominator is a sum of responses along all $\varphi_k$-directions.


Hope that helps. Ask more if something is still vague.
