Compendium of Analyses, Part III: Advanced Single Cell

I wrote the first two compendiums of analyses last year covering single cell and population analysis. Since then, I've been keeping track of the more esoteric analyses I've encountered, and also found a short perspective from Mitra that covers many spike train analyses. Today, I'll cover three analysis techniques: sparseness, ROC analysis, and entropy.

Sparseness:

The simplest, first. I originally found the sparseness calculation in Poo and Isaacson, 2009, where they calculated the sparseness of odor responses in piriform cortex. Sparseness simply measures how much of a stimulus space a neuron responds to, between 0 and 1 (1 being highly sparse). The equation is quite simple:

Sparseness equation.Where r_i is the firing rate of the neuron to stimuli i, from the set of stimuli n. From Rolls and Tovee, 1995.

If a neuron responds to single stimuli in a set, and does not fire for the rest, the numerator and denominator are equal, and the sparseness is 1. If, however, a neuron fires at 1Hz to each of 10 stimuli, the sum of the numerator would be 1/100 * 10 = 0.1; and the denominator would be 1/10 * 10 = 1; and the sparseness would be 0.1. Poo and Isaacson found that the sparseness of piriform cortex neurons was 0.88, indicating most cells there responded selectively to a single odor. You can also look at population sparseness, using the response of a cell to a specific stimuli, in a population of cells n.

A more interesting example comes from vision, from Vinje and Gallant. They recorded from visual cortex while presenting movies; the movies varied in size, with the smallest only stimulating the classical receptive field (CRF), while the largest movies covered an area 4x the CRF. For the small movies, the firing rate was moderate, and increased following some frames (below, left). For the large area movies, however, the firing rate was near zero except for specific frames.

Stimulation of larger areas increases sparseness of V1 neurons. Left. Recordings from macaque V1 during playback of movie in the CRF or an area 4xCRF. The firing rate was higher during the CRF stimulation, and more sparse for the 4xCRF stimulation. Right. Sparseness calculation for populations of neurons for stimulation of different size CRFs. As the stimulus area increased, the sparseness also increased.
From Vinje and Gallant, 2000.

To look at this for the population, they calculated the sparseness of the response, where each stimuli was an individual frame. For stimulation of only the CRF, the average sparseness was 41%; stimulation of 4xCRF yielded a sparseness of 62%.

Sparseness seems like a quick and dirty measure, but of limited application to olfaction or taste. First, chemosensory stimulus spaces are generally small enough that you can simply quantify the percent of stimuli a neuron responds to. Second, olfactory responses are highly phasic (or tonic-phasic as one guy at my poster at ISOT insisted), which makes measuring firing rate less useful. To adapt it to olfaction, you would probably need to substitute a different metric for responsiveness, such as "difference in spikes from control breath."

Receiver Operator Characteristic (ROC) analysis:

ROC analysis is used to tell to what degree, and under what conditions, two responses are discriminable. I first came across this in Cury and Uchida, and they have a nice supplemental figure that shows how it's useful in OB coding.

They recorded from the OB of rats while presenting various odors, and wanted to know: 1. whether a neuron fired differently during odor presentation than pre-odor breaths; 2. when the firing was most different; and 3. what time window allowed the best discrimination. To do this, they compared the firing rates between control an odor breaths during defined windows during the breath (red and blue areas in panels A/B below). For each trial, they counted the number of spikes in the epoch (panel C), and then plotted the distributions of spike counts for each trial (panel E).

Copied from Cury and Uchida:

"(A) PETH of an example M/T cell, aligned by the first odor inhalation onset. Black, odor; gray, blank. The red (1) and blue (2) shaded areas indicate two example analysisepochs. This neuron and these epochs are used in all subsequent panels.

(C) Raster plot of spike trains over multiple trials, aligned by the first odor inhalation onset. Odor trials (above) are separated from blank trials (below) by the horizontal black line.

(D) Response reliability (area under the ROC curve, auROC), calculated for varying epochs (bin size: 5 ms to 160 ms.; bin center: t = 0 to t = 160 ms), as in Figure 2D. auROC values are indicated using the color scale at right, with red and blue signifying increased and decreased spike counts, respectively. The black circles indicate the three example windows plotted in (B). Selection of the optimal epoch was restricted to occur within the 0 to 160 ms response window, the bounds of which are indicated by the diagonal dotted lines.

(E) Distribution of single trial spike counts for both odor(black) and blank (gray) for the same three example epochs.
(F) The corresponding ROC curves for the same three example epochs, comparing the hit rate to the false alarm rate as a discrimination threshold is slid across the distributions. The resulting auROC value is listed to the right."

Then, they asked, if you had to classify a trial as "blank" or "odor" based on the number of spikes in a trial, how successful would various thresholds be? For example, in the top part of panel E, a threshold of 0.5 spikes would correctly classify most odor trials (a "hit"), while mis-classifying a few blank trials as odor ("false alarms"). By contrast, for the bottom part of panel E, no threshold could discriminate blank from odor.

For each threshold, you get a proportion of hits vs false alarms, and can plot this as the ROC curve (panel F). If two populations are easy to discriminate, you will get curves like those shown in blue or red; if the populations are hard to discriminate, you get a curve like that shown in black. You can collapse the ROC curve into a single measure of discriminability by integrating the area under the curve (auROC); values near 0 or 1 show strong discrimination for that epoch.

So that's how you get the auROC for a specific epoch. You can then repeat the procedure for different epoch lengths and onsets (panel D). This panel shows that there are two good epoch to discriminate odor from blank: ~60ms, and ~140ms, and that the optimal epoch size is 60-80ms. In general, they found that excitatory responses could be found throughout the sniff, with epoch sizes of ~40-60ms; in contrast, inhibitory responses tended to be between 80-120ms, and have epoch sizes of ~60-80ms.

ROC analysis is kinda funky, so I recommend playing with this demonstration to get a better feel for ROC curves.

Entropy/ information / bits!

I must admit, I don't completely grok entropy. I (think I) understand the equations, but I don't understand what it's useful for. Calculating how much information is in a neuron's firing without decoding the stimulus seems like calculating the optimal bit rate for an mp3 without listening to the song.

In any case, the entropy of a firing rate can intuitively be understood as the unpredictability of the firing rate, and by inference how much information is in the firing rate. The relationship between unpredictability and information is easy to understand: a neuron that fires at exactly 10Hz in response to all stimuli is conveying no information; in contrast, a neuron that fires at 10Hz to one stimuli, but not to others, contains some information; more subtly, a neuron that fires on average at 1Hz, but with bursts and silent periods, depending on the stimulus, also contains information with each spike.

So how do you calculate the entropy of a spike train? (The best review of this I found was from Bhumbra and Dyball). First, you need to bin the spike train into a histogram where each bin can contain only one spike. This histogram has three features: the number of bins, the bin size, and the number of spikes in the histogram. The likelihood of a given spike train is defined by how often that pattern occurs out of all possible spike trains. If a neuron uses all possible spike trains, the entropy of all spike trains is simply log₂(# of spike trains possible). Since calculating the potential number of spike trains involves factorials, and is unwieldy for large numbers of bins, this equation can be approximated as (skipping all the algebra here) log₂(e/(M*dt)), where M is the mean firing rate, and dt is the bin size.

Of course, neurons don't use all possible spike trains possible, but only a subset of them. In that case, you simply multiply the probability of a spike train times that train's information:

The entropy of a spike train is the probability of a spike train i, times the log of that probability. From Wikipedia entry for entropy.

"Ok, fine, I can look up the details of how entropy is calculated later. What's it good for?" I searched a bit for papers using entropy, and they seem to fall into three categories: comp neuro papers about how to calculate it; papers that use it for a panel in one figure; and vision papers.

To get a good measurement of entropy, you need to record from a neuron for a long time to get a large set of possible spike trains, preferably including many stimuli and trial repeats. This makes entropy ill suited for olfaction, where the unit of measure is one breath (~320ms), and odor presentation can take ten seconds for odor loading and clearance. Vision, however, does not have these problems: you can play a movie to a neuron, yielding 30Hz of stimuli, and can repeat this hundreds of times.

One of the earliest, most cited entropy papers in vision is Nirenberg et al, 2001. They recorded from many neurons from mouse retina while presenting movies 300 times. Some sets of cells had correlated firing, which they calculated as the correlated spikes that would appear above chance, the excess correlated fraction (ECF). To see whether these correlated spikes were informative, they calculated the entropy of the neuron pairs' spike train both including the correlated spikes, and excluding them (or rather, treating them independently). They found that even for neuron pairs with high ECF, the correlated spikes only contained <10% of the total information.

Retinal ganglia act independently. They calculated the information in a neuron pairs' spike train both including and excluding correlated spikes (y-axis), and compared this to neuron pairs' correlation (ECF). While higher ECF pairs' correlated firing did include some information, most information was independent of correlated firing.
From Nirenberg et al, 2001.

ISOT 2012, dag fem

(For previous entries, see days one, two, three, and four.)


Kristin Scott:

The Scott lab focuses on taste receptors and circuits in drosophila, and today she presented two new stories.

On the receptor side, the Scott lab had identified the receptor for 3/4 of the cells in the fly taste ganglion, the SOG. Today she presented one channel for the last cell, ppk23. Another ppk, ppk25 was previously identified as a water/stretch receptor for flies. Ppk23 expression is sexually dimorphic: male flies' ppk25 cells project across the midline while females' do not; furthermore ppk23 is coexpressed with fruitless. They knocked ppk23 out, and found that male flies could no longer distinguish between males and females, and started courting male flies. Since ppk23 is in "taste" cells, they guessed it might be detecting sexually dimorphic surface hydrocarbons, and presented these hydrocarbons to the fly. Using calcium imaging they found two cells in the SOG that responded: one to male hydrocarbons, and the other to female. There are many other ppks expressed in those cells, so it's unclear whether ppk23 is the actual sensor or simply involved in signal transduction.

The second story concerned modulation of feeding behaviour. In a paper earlier this year, the found that dopamine activity  increases with hunger, and that dopamine activation can drive increased feeding behavior. Here they looked for neurons that drive satiety.  They generated GAL4 lines, and then measured how much sucrose the flies would accept. One line, 98, were "insatiable," and drank well beyond WT flies. In fact, they eagerly drank bitter fluid, or oil as well. The 98 line labelled ~30 cells in the SOG, so now they are trying to use mosaics to pin down the exact cells.

Robert Barretto:

Rob expanded on what Zuker previewed the first day: 2p endoscopy in the taste ganglion. He described seven populations: five single tastes, and two pairs, sweet/umami ans bitter/sour. Notably, umami was represented the least,  by far, with 30/1200 cells. In comparison, during the questions, he mentioned that 15-20 cells represented another pair, sour/salty (I think).
After his talk, a few people approached him with further questions. Since he did not publicly say them, I will refrain from repeating them. Suffice it to say that consideration of stimulus concentration, and crosstalk between receptors (e.g. artificial sweeteners and bitter) is essential to interpreting the results.
And with that the conference is over.

ISOT 2012, día cuatro

(For previous entries, see days one, two, and three.)

Very smelly morning, often about innate smell detection, but a significant portion of the works were already published. In the afternoon, a couple taste talks caught my eye.

Yuzo Ninomiya:

I'm a fan of Ninomiya's work, having covered his papers on leptin and cannabinoid modulation of sweet taste previously. Here, he provided an update to that work, concentrating on the relative strengths of leptin vs cannabinoids. In WT mice, cannabinoid antagonists are ineffective, while leptin antagonists are. However, in db/db mice (leptin knockouts) cannabinoid antagonists do reduce sweet responses in the CT nerve. Thus it appears that under normal conditions leptin is dominant. To verify this, they measured sweet responses in mice with varied blood leptin levels, and found that cannabinoid antagonists became more effective as leptin levels went down.

Sclafani:

Besides being sensed by the tongue, sugar is also detected by the stomach, which influences food intake over longer time scales. Sclafani's lab investigated this by directly injecting sucrose into the mouse gut (? Or IP) in T1R2 knockout mice (no canonical sweet receptor). Mice triggered the injection by licking a water spout. They employed a conditioning protocol,  where unsweetened cherry taste caused sugar water injection, while grape taste caused water injection. After conditioning, mice licked the cherry water more.

Sclafani mentioned three possible receptors: GLUT5 which can detect fructose but not galactose; and SGLT1/5 which can detect galactose but not fructose. To see which of these are involved, they switched the injection to fructose or galactose. Mice injected with fructose were not conditioned, showing GLUT5 is not responsible.  In contrast , galactose did work for conditioning.  To see whether metabolization is necessary they tried conditioning with MDG, a non-metabolizable galactose analog, and found conditioning still worked. Thus SGLT activation alone seems to be sufficient.

How could this signal downstream?  Scalfani noted that most gut hormones decrease food intake, while ghrelin, the one orexigenic hormone, is suppressed by glucose. Thus there is at the moment no clear pathway for the effect.

ISOT 2012, day 3

(For previous entries, see days one, and two.)

Lots of insect olfaction today, which is not my wheelhouse. Apparently some mosquitos are racist, and prefer to bite humans over guinea pigs, or vice versa. There were some interesting posters (e.g. ENaC knockout mice still retain some salt taste sensitivity), but it's I'm not sure how interesting brief poster summaries would be. Instead, two general points.

Zuker is well known for pushing the labelled line story from the periphery to insular cortex. In contrast every other taste researcher has found that the situation is more complicated: single cells can respond to multiple tastes on the tongue and in cortex; and some receptor knockouts can still respond to the tastes that should have been elided. So I've used this conference as an opportunity to see what other taste researchers think about Zuker's story. Their responses usually start with the phrase, "his data is beautiful but..." then give examples of results that can't fit the labeled line story. I haven't found another researcher who endorses Zuker's view, which is strange considering how high profile Zuker is. It is difficult to tell where clear-headed scientific thinking ends, and personal politics begins.

Second, while looking at posters,  I saw a number of people using channelrhodopsin by stimulating with long (>5 ms) pulses. This strikes me as incredibly imprecise, as you cannot know how the stimulated cell is firing, only that it is depolarized. Instead I'm a strong advocate of using pulsed stimulation at specific frequencies (up to 50 Hz!), so you can know exactly how the stimulated cells are firing. And to argue from authority, most high profile papers I can think of also used pulsed stimulation. People are certainly getting results with the long pulses, but I feel strange trying to interpret the results. I would be interested to know what other people who use ChR2 think.

ISOT 2012, deuxième jour

(First day's entry)


Daily blogging is rough.

This is the first medium size meeting I've been to (800 attendees, 2-3 sessions), and it is uniquely exhausting. While SFN is 40 times larger, and tiring in its sheer scope, there is something exhilarating about being surrounded by so many people, like walking through Shibuya or Times Square. If you get tired of taste or olfaction, you can take in a barrel cortex or retinal development talk. Six hours a day of chemosensory talks can be repetitive.

Since I went to so many talks today, I will briefly comment on two. The day started off with Cori Bargmann. I got in a little late, but the gist of the first part is that one of the C elegans neurons, AWC, is activated when an odor turns off (which sounds familiar to me), with a time constant of 10s of seconds. They looked at the mechanisms of this, but to be honest I got lost in the worm alphabet soup.

In the next section, they looked at behaviour. C elegans have simple motor behaviour: they can move forward, reverse, turn, or pirouette. Using a microfluidic device, they measured how the worms turned in response to odors turning on and off. They found that worms tend to move forward during odor presentation, and when the odor turned off, the worms started turning, with a time constant similar to AWC. They then tried imaging the AWC in the behaving worm, to see if AWC activity was correlated with turning or moving forward. However, that AWC was similarly active when the worm moved forward or turned, which means AWC neurons are simply sensory and not motor.

Next, she moved to the timescale of a few seconds. Worms move by wiggling back and forth in sinusoidal patterns. These sine waves in effect give the worm a metered sampling of the world, a sort of sniff. One worm behaviour that may rely on this sinusoidal sniffing is turning along odor edges; worms within an odor stripe will move parallel to the stripe, or turn inward, but not outward. Since AWC neurons fire in response to changes in odor, they tested how AWC neurons fired to stimuli at 1Hz, and found the AWC neuron could follow the stimulus. Then they performed reverse correlation on the calcium signal, using an L-N model, and found that AWC neurons could linearly follow integrate over a 1s window. (GCaMP's 0.5s non-linearity was also detected)

Jeff Isaacson:

Isaacson is looking at how experience an anesthesia can modify odor representations in the olfactory bulb. They now have 2p awake imaging working in the bulb, by injecting AAV-flex-GCaMP in Pcdh21-cre mice. In slices, they validated that GCaMP could encode APs linearly over 0-40Hz.

Using this, they imaged responses in the bulb. They found that in awake mice, individual cells responded to a relatively narrow range of odors (imaging over 4s, ignoring sniff phase). When they recorded in anesthetized mice, the responses increase in both magnitude, and broadness. They also lost inhibitory odor responses. They hypothesized that anesthesia might be selectively effecting GABAergic granule cells, and so applied gabazine in awake mice, and saw a similar effect to anesthesia. To look at this another way, they then imaged activity in the granule cells using GAD-cre mice with AAV-flex-GCaMP. The granule cells had sparse tuning. However, when they anesthetized these mice, the found the granule cells simply stopped responding to odors, which would explain the loss of inhibition.

In the 2nd half of the talk, he asked the question of how stable odor representations are. They imaged mice daily over 7 days, and found the response magnitudes decreased each day. Then they split their odor space in two; for group A, they imaged those odors every day; for group B, they imaged only on days 1 and 7. The group A responses all decreased, while the group B responses maintained their strength. Thus repeatedly presenting an odor selectively attenuates its response. This also held for single cell analyses. To look at glomerular inputs, they imaged OMP-SpH mice, and found that these responses were maintained over 7 days. They looked at how long it took for these responses to recover, and found that they began to recover after one week, but took a full 2 months to completely recover. Finally, they looked at how awareness effected this by imaging mice in both awake and anesthetized states. While the awake responses decreased, in the same mice, the anesthetized responses remained stable. Thus, the adaptation they observe in awake mice requires some sort of awareness.