Scientology on psychiatry

Cruising the web, I ran into the official Application for Enrollment in Scientology Religious Services. Surprisingly, it reads much more like a legal contract than a standard application. I found the paragraph that discusses psychiatry quite interesting. It certainly helps explain some of Tom Cruise's behavior. Below is the full paragraph, with comments on the main points.

Scientology is unalterably opposed, as a matter of religious belief, to the practice of psychiatry, and espouses as a religious belief that the study of the mind and the healing of mentally caused ills should not be alienated from religion or condoned in nonreligious fields. I am in full agreement with this religious belief. [italics added]

Here they stake an interesting claim. Basically they are like Christian Scientists who restrict their refusal of medical help to one specialty. I can't say I'm completely unsympathetic: because of its subject matter, psychiatry is not nearly as well-developed or objective as other branches of medicine. That doesn't mean psychiatry doesn't help a lot of people (it does) or that it is pseudoscience (it isn't always).

They go much further than this, though. They add that any study of the mind, with "mentally caused" ills as a special case, should only be performed within a religious context. Any purely secular study of the mind is permanently opposed by Scientology.

Even the most kooky right-wing Creationist hacks wouldn't go this far. It would imply that psychology (other than behaviorism) is a taboo science. Psychophysics, cognitive tests on Alzheimer's patients, and the study of the visual capacities of people with different types of brain damage all seem fairly vanilla secular studies of the mind. So, the Scientologists really should clarify the subset of questions that can appropriately be targeted in a secular context.

This connects with another interesting issue. While they will not defer to psychiatrists in the case of "mentally caused" ills, what about physically caused mental ills? E.g., if I lose a chunk of my cortex in an accident, that will certainly cause some mental ills. I'll need to see a neurologist, if not a psychiatrist. Also, what is not mentally caused in their framework? Is sleepiness mentally caused? What about cancer? Given the emphasis on personal power and responsibility in their philosophy, we need to ask.

I do not believe in or subscribe to psychiatric labels for individuals. It is my strongly held religious belief that all mental problems are spiritual in nature and that there is no such thing as a mentally incompetent person--only those suffering from spiritual upset of one kind or another dramatized by an individual. I reject all psychiatric labels and intend for this Contract to clearly memorialize my desire to be helped exclusively through religious, spiritual means and not through any form of psychiatric treatment, specifically including involuntary commitment based on so-called lack of competence.

The above reads almost word-for-word like the beliefs of Christian Scientists, who believe that all "illness" is really an invasion by evil spirits, not something that can be cured by medicine.

Why this animosity toward psychiatry? This could be an attempt (not necessarily conscious) to kill a direct competitor to their method: traditional methods of therapy within a secular context. This may sound crazy, but consider that L Ron Hubbard's first Scientology text, Dianetics, is subtitled 'The Modern Science of Mental Health.' It was published in 1950, when psychiatry was a fairly young science. The theory of the mind (and mental problems) advanced by Dianetics reads like warmed over psychoanalytic theory. The subject is directed to bring to consciousness certain buried memories ('engrams') that contribute to your 'reactive mind' that, unbeknownst to you, controls your behavior and emotions in the present. This is done via a process of talking to an 'auditor'. If interested, see the page How the mind works at the Scientology web site. This sounds a lot like early clinical psychiatry.

Since Dianetics is basically a reheated model of the mind, with its own set of exotic categories ('Clear' when your 'engrams' are properly 'audited'), this psychiatry hate is sort of amusing. The Scientologists clearly consider Dianetics a science, and science is self-conscious about its fallibility, open-ended nature, and openness to new information and data. Scientology, or at least this document, aspires to none of these scientific ideals.

There are, of course, interesting issues in the underbrush when the say they reject psychiatric categories. If you read the source of the categories, the Diagnostic and Statistical Manual of Mental Disorders (DSM), it isn't a perfect document. Some of the categories seem ill-defined or subjective, and there is a good deal of overlap among symptoms that makes it hard to come up with definitive labels. Historically it has included some silly cultural prejudices (like homosexuality) in its list of disorders. It still includes 'female hypoactive sexual desire disorder', which is controversial (see link above for other controversies).

All told, I rather like the DSM because it is self-consciously fallible, open-ended, and evolving. It reflects the tentative classifications formed by those working with patients, doctors that have noticed many different suits of symptoms tend to cluster together. If nothing else, it provides a consistent language for practitioners to use to talk with each other. While you could describe a long list of symptoms to a colleague ("Subject experiences auditory hallucinations, nervous and paranoid, prone to violent outbursts"), it is easier to say '"paranoid schizophrenic".

Under no circumstances, at any time, do I wish to be denied my right to care from members of my religion to the exclusion of psychiatric care or psychiatric directed care, regardless of what any psychiatrist, medical person, designated member of the state or family member may assert supposedly on my behalf. If circumstances should ever arise in which government, medical, or psychiatric officials or personnel or family members or friends attempt to compel or coerce or commit me for psychiatric evaluation, treatment or hospitalization, I fully desire and fully expect that the Church or Scientologists will intercede on my behalf to oppose such efforts and/or extricate me from that predicament so my spiritual needs may be addressed in accordance with the tenets of the Scientology religion free from psychiatric intervention.

This is also an interesting paragraph. I sympathize with its likely motivation: they don't want Scientologists to be forced to receive psychiatric treatment against their will. I'm sure this is a serious concern partly because of their oddball beliefs, such as their putative beliefs about Xenu the alien dictator.

On the other hand, to sign a contract in which you agree to never see a psychiatrist regardless of the situation seems reckless and dangerous. Instances of postpartum depression, with clear physiological causes and treatments, don't respond particularly well to psychotherapy, and do respond well to antidepressant drugs. It seems silly to apply a screwdriver to a nail, and the Scientologists are saying there is only one tool you need to cure all ills with any mental dimension.

The clause that you want Scientology to intercede and oppose any attempts to provide you with psychiatric care is particularly worrisome. Does this mean Scientologists should actively try to prevent someone from seeking such care, even if they want it? Probably not. However, one practical consequence is that the social pressure to avoid psychiatry is very strong within their subculture. This will lead to many people not receiving adequate care for easily treatable conditions.

These are not idle worries. We have the tragic case of Linda McPherson, who died after her tortured 'Introspection Rundown' with the Scientologists. She was likely psychotic, but they refused to send her to a hospital, even after a Scientologist doctor told them they should send her, for fear that she would be taken to see psychiatrists.

What good is large-scale oscillatory activity?

As shown below in EEG traces recorded during different stages of sleep, when we record neuronal activity on a large scale (either local field potentials or EEG signals) things do not typically look 'flat.' Since the signals recorded at these electrodes are basically the linear sum of the voltage fields generated by dendritic activation in a little sphere around the electrode, if the voltages from different neurons was random, they would be fairly flat. That they are rarely flat suggests there is some degree of synchronous fluctuation in the voltage fields from the cells near the electrode.

What is the function, if any, of such large-scale "oscillatory" activity? Recently I read an argument that such activity is just epiphenomenal, that it has no biologically useful causal role. I am something of an atheist turned agnostic about the importance of such signals, so in what follows I try my best to step into the role of advocate for the claim that such oscillations have interesting functional consequences.

Let's consider a few experimental facts about such large-scale synchronous activity (not an exhaustive list!):

1. Neuronal activity can be directly influenced by weak external electric fields. These are known as "ephaptic interactions" in the literature. The question of the function, extent, and importance of ephaptic interactions (relative to synaptic interactions) is an open question that has received little attention. Google "ephaptic interactions" if you want to see some of the research that has been done. It could be that, for a neuron near threshold, an ephaptic signal may nudge it up into a spiking regime (or push it down away from threshold, making it less likely to spike).

2. The efficacy with which plasticity is induced in rat barrel cortex is synchrony-dependent. In the study from Diamond's group, more synchrony implied higher probability of inducing cortical plasticity with their protocol. Note in this paper they measured synchrony among spike trains using extracellular recordings, not LFP or EEG.

[Link to paper]

3. Gilles Laurent's group pharmacologically disrupted "synchronous oscillations" in the honeybee CNS and showed this disruption impaired behavioral discrimination of similar odorants (but not discrimination of clearly distinct odorants). Similar pharmacological manipulations have been done in vertebrate systems (see Buzsaki paper below).

Note these pharmacological manipulations are not clean, which makes it hard to interpret the results.

[Laurent paper] [Buzsaki paper]

The second and third results mentioned are likely indicative of some very cool and interesting synaptic processing, which puts the first line of evidence in a unique class. Ephaptic interactions are the key for those who think oscillations actually do something. Unfortunately, there isn't a lot of good work on the role of ephaptic interactions in nervous systems.

My guess is that it will turn out that ephaptic interactions are important for the functioning of some parts of some systems, but settling which parts of which systems isn't a question that can be resolved a priori. Unfortunately, right now we are stuck mostly with correlation studies, which keeps the ratio of inference to data unacceptably high.

Stepping back from specific data suggesting that oscillations are important, Buzsaki's book Rhythms of the Brain is likely a good place to start to get the theist's side of things with respect to the importance of oscillations. I should admit I haven't read it, but only seen him speak and have read some of his papers, so caveat emptor. I can say his experimental work is wonderful, much of it driven by questions about the function of oscillations in the CNS, so his book seems like a natural place to start for someone interested in this question.


References:
Irina A. Erchova and Mathew E. Diamond (2004) Rapid Fluctuations in Rat Barrel Cortex Plasticity. The Journal of Neuroscience, June 30, 2004, 24(26):5931-5941

Stopfer, M., Bhagavan, S., Smith, B. H. and Laurent, G. (1997). “Impaired odour discrimination on desynchronization of odour-encoding neural assemblies.” Nature 390(6655): 70-4.

Robbe D, Montgomery SM, Thome A, Rueda-Orozco PE, McNaughton BL, Buzsaki G. Cannabinoids reveal importance of spike timing coordination in hippocampal function. Nat Neurosci. 9:1526-33 2006.

Anscombe's Quartet

Anscombe's Quartet is a group of four data sets that provide a useful caution against blindly applying statistical methods to data. Each data set consists of ten x- and y-values such that the mean and variance of x and y, the correlation coefficient, regression line, and error of fit using the line are the same. But as you can see, they are clearly quite different data sets:

The x- and y-values are included at the end of this post in a Matlab-friendly format. The Quartet provides a stark lesson on how useful it can be to simply look at one's data before diving in with all sorts of statistical ninja. For instance, Set 2 can be modelled with linear regression to yield the same mean-squared-error between the regression line and the data as the other plots, but it ain't a linear relationship.

This is old hat for most of us, but I like the Quartet for its simplicity and visual impact.

These data and graphs were first presented by F.J. Anscombe in 1973 in his paper Graphs in Statistical Analysis. It is quite fun reading over the paper, which ends:

Unfortunately, most persons who have recourse to a computer for statistical analysis of data are not much interested either in computer programming or in statistical method, being primarily concerned with their own proper business. Hence the common use of library programs and various statistical packages. Most of these originated in the pre-visual era. The user is not showered with graphical displays. He can get them only with trouble, cunning, and a fighting spirit. It's time that was changed.

Thank goodness for Matlab.



The data (coded for Matlab)
x1=[10 8 13 9 11 14 6 4 12 7 5];
x2=x1;
x3=x1;
x4=[8 8 8 8 8 8 8 19 8 8 8];

y1=[8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68];
y2=[9.14 8.14 8.74 8.77 9.26 8.1 6.13 3.1 9.13 7.26 4.74];
y3=[7.46 6.77 12.74 7.11 7.81 8.84 6.08 5.39 8.15 6.42 5.73];
y4=[6.58 5.76 7.71 8.84 8.47 7.04 5.25 12.5 5.56 7.91 6.89];

Dude, where's my 'dudes?

Neurodudes, one of the first neuroscience blogs I ever saw, seems to have gone 404. Hopefully it is just temporary.

Note added 4/14/08: it seems to be back up, with a sleek new look.

Getting intimate with rat whiskers (2): Ritt et al.

In the previous post, I briefly discussed the background and methods behind this paper by Ritt et al.. In this post I summarize and discuss the main results. I just noticed that Neuron added the apt banner, "What the rat's vibrissa tell the rat's brain" to the article.

Ex vivo whiskers
As I discussed in the previous post, the resonance hypothesis is the claim that the resonance frequencies of whiskers provide an ethologically important channel of information about what is going on in the world of the rat. Subsequent papers, however, have argued against the resonance hypothesis, as they were unable to reproduce whisker resonance in a more ethologically realistic preparation in which a whisker is fixed at a base and moved across a fixed surface (this is a little odd, since Hartmann already showed that resonance occurs in whiskers of awake behaving rats).

The first set of experiments in the paper addressed these issues, monitoring whisker movement as a whisker was brushed against a fixed surface at different speeds. They did observe resonance, as expected, but found an interesting twist--resonance wasn't prominent at every speed, but at a relatively narrow range of speeds, a 'hot spot' (see Figure 1B of the paper, shown below, which shows whisker angular velocity traces when the whisker is moved across the same surface at different speeds). This hot spot is thought to be within the range of speeds that whiskers move in ethological contexts, so it could be that rats exploit resonance during certain tasks by moving their whiskers at the appropriate speed.
In situ whiskers
The majority of the paper examined the micromotions of whiskers while rats performed a texture discrimination task. The task was relatively simple: the rat runs into a chamber that contains two different textures along the wall--a rough texture and a smooth texture (think of it as rough and smooth sandpaper). The rats are trained to get reward by running to a door that corresponds to a particular texture (e.g., when the smooth texture is on the right, enjoy a reward when you move to the right).

Their high-resolution monitoring of the whisker motions revealed a very interesting pattern to the movements. A whisker will slip along the surface, and then stick to a particularly abrasive feature, staying in place as the rat continues to move its head. Then, after the tension grows enough, the whisker whips forward (slips) before sticking at another spot further along (for example, see the clip from Figure 3 below: the top panel shows the trace of a whisker during the trial, and the bottom panel shows the angle (red) and position (blue) of the whisker). This is often accompanied by a kind of 'ringing' of the whisker as it sticks in place. Perhaps not surprisingly, this slip-n-stick pattern occurrs more often when it sweeps along the rough surface.
In the frequency domain, the mean frequency of movement of each whisker was inversely proportional to whisker length. This suggests that the elasticity that confers different resonance frequencies upon whiskers of different lengths also influences the frequency of whisker vibrations during the present discrimination task (the figure below, clipped from the paper, shows an example of the likelihood of different frequency components in two whiskers of different length).
While the frequencies of oscillation were consistent with resonance frequencies observed ex vivo, the rats didn't seem to sample the textures with all of their whiskers, which they might do if their goal was to maximize the information about frequency available from whiskers of different lengths (e.g., the "whiskers as cochlea" picture may need to be tweaked). Indeed, the largest two arcs of whiskers rarely touched the textured surface.

Among other things, their analysis also revealed a unique signature of rough-surface contact: a set of high velocity, large amplitude whisker deflections that were not seen in previous studies. Figure 8 of the paper provides a scatter plot that includes the rise time (roughly, time from trough to peak of a whisker movement) and velocity of whisker movements, and those events that were both high velocity and longer in duration (signifying higher amplitude) were much more likely to occur during contact with the rough surface.

Where are we?
This paper, a heroic effort that has established a great method and some intriguing results, is only the beginning. The whisker coding literature is finally catching up to vision when it comes to understanding the inputs to this system. Many more experiments are needed, not just to examine the role of resonance in transmitting information about the world, but to determine the role of different types of whisking patterns the rat employs as it scurries about.

Apropos of the resonance hypothesis, it would be interesting to see how rats fare when their whiskers are cut so they are all the same length. This should decrease the differences in resonance frequencies in different whiskers. Hence, to the extent that resonance is important for a task, we would expect to see their performance compromised (note, though, things are a little more complicated: resonance frequency depends on other factors than length). It will also be crucial to measure which micro-features of whisker movement are tracked neurally in awake behaving rodents.

This paper has opened up possibilities beyond just the resonance hypothesis. It reveals the slippery, sticky, messiness that is present when rats use their wonderful keratinous appendages to explore their world.


Acknowledgments:: Thanks to Jason Ritt for clarifying an important point about the velocity 'hot spot' mentioned in the ex vivo experiments.

Getting intimate with rat whiskers (1): Ritt et al.

This is the first of a two-part post on the recent Neuron paper Embodied information processing: vibrissa mechanics and texture features shape micromotions in actively sensing rats by Jason Ritt and others in Christopher Moore's group. They provide a much-needed high-resolution look at the the movement of whiskers in freely-moving rats as they perform a discrimination task.

Where's the neuroscience?
Before describing the main results, I'd like to discuss why, as neuroscientists, we should care about what whiskers are doing. "Where is the neuroscience?" someone may ask of a paper that just examines whisker movements. To that I would respond, "What is the stimulus?" The first rule of psychophysics and sensory coding is, "Know thy stimulus" (it was EJ Chichilnisky that hammered this into my mind). Students of psychophysics and sensory coding try to quantify the relationship between the stimulus and some other variable (either behavior (psychophysics) or neuronal activity (sensory coding)), so we need to precisely quantify the stimulus on each trial to make sure we have all the relevant information contained in the stimulus.

Because of such concerns, the question we should be asking is why researchers studying coding in the whisker system know so little about what the whiskers are doing, given the importance of understanding the stimulus in sensory coding and psychophysics. In the present paper, Ritt et al. provide a much-needed corrective to this deficit.

The resonance hypothesis
For a few years now, Moore (and Mitra Hartmann at Northwestern) have been pointing out that whiskers are not simple passive detectors of what goes on in the world. Rather, whiskers have interesting intrinsic properties that shape their responses as the rat scurries about palpating the world.

For example, each whisker has a resonance frequency. If you attach a stimulator to the end of a whisker and move it at frequencies, there is a 'hot spot' at which the whisker will vibrate with a significantly higher amplitude. The following figure (Figure 2 from Niemark et al) shows the amplitude of a whisker's movement as a function of stimulation frequency. The breakaway images show data from individual frequencies (gray line: stimulus, black line: whisker movement downstream from the stimulus):


This whisker in the figure has a resonance frequency just under 400 Hz. The resonance frequency of a whisker depends on that whisker's length (longer whiskers have lower resonance frequencies). Interestingly, the length of whiskers increases quite significantly as you move from the front to the back of a rat's face.

As an interesting historical aside, the possiblity that whiskers have resonance frequencies was initially noticed by Niemark while she was still an undergraduate working with John Hopfield. Then the idea was fleshed out experimentally by Niemark, Moore and Mark Andermann (leading to this paper) and by Mitra Hartmann and others at Cal Tech (leading to this paper). The present paper is the latest in a series of papers from Christopher Moore's group exploring the consequences of whisker resonance.

Getting back to the science, what is the significance of all this resonance business? Is it possible that whiskers with different resonant frequencies are induced to resonate by different types of stimuli? This is part of what has become known as the resonance hypothesis (review here). As described in the original paper by Niemark and others, "vibrissa resonance is optimally positioned to increase the range of detection and the specificity of discrimination, and may provide the ability to represent complex stimuli through a compact, somatotopically distributed code" (p 6500).

Many of us think of the resonance hypothesis as the 'whiskers as cochlea hairs' hypothesis. It has been a quite productive idea for experimentalists, and it is interesting to note that it wasn't even on our radar until Neimark asked a simple question: what are the properties of the sensor used by the rat?

Resonant oscillations have been observed in detached whiskers (image above) and in the whiskers of anesthetized rats. It has also been shown that cortical neurons respond differently when a whisker is stimulated at its resonance frequency (see the above review of the resonance hypothesis). But is whisker resonance ethologically important?

Does resonance really matter? Why it is hard to answer this question
A key question is whether resonance is important in the awake freely moving rat performing a discrimination task. One step toward anwering this question is to determine whether resonance is observed in the whiskers in a particular task. If not, then resonance cannot be used so we'd have evidence that resonance is irrelevant for that task.

So, what's the problem? Measure the whiskers in awake freely moving rats and get on with it. Unfortunately, it turns out to be extremely difficult to track tiny whisker movements in a rat that is running around moving its body, head, and whiskers any old place it wants. You need to get the optics just right to see many of these wispy whiskers in focus, use a high-speed camera (the resonance frequencies can be higher than 500 Hz), and then set up a system to automate the tracking of the whiskers (no way you are going to code all that data by hand with the required large fields of view and fast frame rates). It would be a herculean effort.

Mitra Hartmann, presently at Northwestern, chipped away at this problem a bit. She analyzed some interesting video data from the awake freely moving rat in this paper where whisker movements were sampled between 250 and 1000 Hz. She observed that whiskers often exhibit resonant oscillations after contacting an object, but not when the rat is simply whisking in open air.

But what about when a rat is performing a stimulus discrimination task? Could resonant oscillations actually be used to discriminate the features of an object such as its texture? To get at such questions, Ritt et al have pushed things much further. They have developed algorithms to automatically track the entire length of individual whiskers during a texture discrimination task in freely moving rats. They capture images at the impressive frame rate of 3000 Hz. Here's one example from Figure 3 of their paper:
Here they show the results when a single whisker is tracked (it shows the shape of the whisker over multiple frames into the past in red) as the rat moves its head across a surface that changes from rough to smooth texture (you can think of it as moving from rough to smooth sandpaper). To someone interested in sensory coding and psychophysics in the whisker system, this image is simply a symphony.

So what did they observe? To give away the punchline, it seems resonance is present during the task. But there is enough cool stuff in the paper that I'll put off the discussion of the results and interesting open questions to Part 2.

Acknowledgements Many thanks to Mitra Hartmann and Christopher Moore for explaining some of the history of the resonance hypothesis and for clarification of some key points.

Large-scale thamamocortical model

While the Blue Brain folk want to construct an incredibly detailed model of a single cortical column, a recent paper by Izhikevich and Edelman (Large-scale model of mammalian thalamocortical systems) reports on a less detailed model of the entire human thalamocortical system.

Some of the details of their model (roughly from large-scale to lower scale) include:
1. The cortical sheet's geometry was constructed from human MRI data.
2. Projections among cortical regions were modeled using data from diffusion tensor MRI of the human brain (above image is Figure 1 of the paper showing a subset of such connections).
3. Synaptic connectivity patterns among neurons within and between cortical layers are based on detailed studies of cat visual cortex (and iterated to all of cortex).
4. Individual neurons are not modelled using the relatively computationally intensive Hodgkin-Huxely models, but a species of integrate-and-fire neuron that included a variable threshold, short-term synaptic plasticity, and long term spike-timing dependent plasticity.
5. The only subcortical structure included in the model is the thalamus, but the model does include simple simulated neuromodulatory influences (dopamine, acetylcholine).

Their model exhibited some very interesting behavior. First, larger-scale oscillatory activity that we see in real brains emerged in the model (e.g., as you would observe via EEG). Also like real brains, the model exhibited ongoing spontaneous activity in the absence of inputs (note this only occurred after an initial 'setup' period in which they simulated random synaptic release events: the learning rule seemed to take care of the rest and push the brain into a regime in which it would exhibit spontaneous activity). Quite surprisingly, they also found that when a single spike was removed from a single neuron, the state of the entire brain would diverge compared to when that spike was kept in the model. There is a lot more, so if this sounds interesting check out the paper. They also mention in the paper that they are currently examining how things change when they add sensory inputs to the model.

Of course, a great deal of work is yet to be done, a great deal of thinking through the implications (and biological relevance) of some of the model's behavior (especially its global sensitivity to single spikes, which to me sounds biologically dubious). However, I find it quite amazing that by simply stamping the basic cortical template onto a model of the entire cortical sheet, and adding the rough inter-area connections, they observed many of the qualitative features of actual cortical activity. We tend to focus so much on local synaptic connections in our models of cortex, it is easy to miss the fact that the long-range projections could have similarly drastic influences on the global behavior of the system.

This paper is just fun. First, it is a great example of how to write a modeling paper for nonmathematicians. It had enough detail to give the modeler a sense for what they did, but not so much detail that your average systems neuroscientist would instinctively throw it in the trash (as is the case with too many modelling papers). Second, it provides a beautiful example of how people interested in systems-level phenomena can build biology into their model without making the model so computationally expensive that it would take fifty years to simulate ten milliseconds of cortical activity. It will be very interesting in the future as the hyper-realist Blue Brain style models make contact with these middle-level theories. I don't see conflict, but a future of productive theory co-evolution.

Visualizing the SVD

Warning: this post isn't directly about neuroscience, but a mathematical tool that is used quite a bit by researchers. 

One of the most important operations in linear algebra is the singular value decomposition (SVD) of a matrix. Gilbert Strang calls the SVD the climax of his linear algebra course, while Andy Long says, "If you understand SVD, you understand linear algebra!" Indeed, it ties about a dozen central concepts from linear algebra into one elegant theorem.

The SVD has many applications, but the point of this message is to examine the SVD itself, to massage intuitions about what is going on mathematically. To help me build intuitions, I wrote a Matlab function to visualize what is happening in each step of the decomposition (svd_visualize.m, which you can click to download). I have found it quite helpful to play around with the function. It takes in two arguments: a 3x3 matrix (A) and a 3xN 'data' matrix in which each of the N columns is a 'data' point in 3-D space. The function returns the three matrices in the SVD of A, but more importantly it generates four plots to visualize what each factor in the SVD is doing.

To refresh your memory, the SVD of an mxn matrix A is a factorization of A into three matrices, U, S, and V' such that:

A=USV'

where V' means the transpose of V. Generally, A is an mxn matrix, U is mxm, S is mxn, and V' is nxn.

One cool thing about the SVD is that it breaks up the multiplication of a matrix A and a vector x into three simpler matrix transformations which can be easily visualized. To help with this visualization, the function svd_visualize generates four graphs: the first figure plots the original data and the next three plots show how those data are transformed via sequential multiplication by each of the matrices in the SVD.

In what follows, I explore the four plots using a simple example. The matrix A is a 3x3 rank 2 matrix, and the data is a 'cylinder' of points (a small stack of unit circles each in the X-Y plane at different heights). The first plot of svd_visualize simply shows this data in three-space:



In the above figure, the black lines are the standard basis vectors in which the cylinder is initially represented. The green and red lines are the columns of V, which form an orthogonal basis for the same space (more about this anon).

When the first matrix in the SVD (V') is applied to the data, this serves to rotate the data in three-space so that the data is represented relative to the V-basis. Spelling this out a bit, the columns of V form an orthogonal basis for three-space. Multiplying a vector by V' changes the coordinate system in which that vector is represented. The original data is represented in the standard basis, multiplication by V' produces that same vector represented in the V-basis. For example, if we multiply V1 (the first column of V) by V, this rotates V1 so that V1 is represented as the point [1 0 0]' relative to the V-basis. Application of this rotation matrix V' to the above data cylinder yields the following:


As promised, the data is the exact same as in Figure 1, but our cylinder has been rotated in three-space so that the V-basis vectors lie along the main axes of the plot. The two green vectors are the first two columns of V, which now lie along the two horizontal axes in the figure (for aficianados, they span the row space of A, or the set of all linear combinations of the rows of A). The red vertical line is the third column of V (its span is the null space of A, where the null space of A is the set of all vectors x such that Ax=0). So we see that the V' matrix rotates the data into a coordinate system in which the null space and row space of A can be more readily visualized.

The second step in the SVD is to multiply our rotated data by the 'singular matrix' S, which is mxn (in this case 3x3). S is a "diagonal" matrix that contains nonnegative 'singular values' of A sorted in descending order (technically, the singular values are the square roots of the eigenvalues of A'*A that correspond to its eigenvectors, which are the columns of V). In this case, the singular values are 3 and 1, while the third diagonal elment in S is zero.

What does this mean? Generally, multiplying a vector x=(x1,....xn)' by a diagonal matrix with r nonzero elements on the diagonal s1,....sr simply yields b=(s1*x1, s2*x2, .... sr*xr, 0 .. 0). That is, it stretches or contracts the components of x by the magnitude of the the singular values and zeroes out those elements of x that correspond to the zeros on the diagonal. Note that S*V1 (where V1 is the first column of V) would yield b=(s1, 0 0 0 0 0). That is, it yields a vector whose first entry is s1 and the rest zero. Recall this is because S acts on vectors represented in the V-basis, and in the V-basis, V1 is simply (1,0, ..., 0).

Application of our singular matrix to the above data yields the following:



This 3-D space represents the outputs space (range) of the A transformation. In this case, the range happens to be three-space, but if A had been Tx3, the input data in three-space would be sent to a point in a T-dimensional space. The figure shows the columns of the matrix U (in green and red) are aligned with the main axes: so the transform S returns values that are in the range of A, but represented in the orthogonal basis set in U. The green basis vectors are the first two columns of U (and they span the column space of A), while the red vector is the third column of U (which spans the null space of A').

Since the column space of A (for this example) is two dimensional, any point in 3-D space in the input space (the original data) is constrained to be projected onto a plane in the output space.

Notice that the individual circles that made up the cylinder have all turned into ellipses in the column space of A. This is due to the disproportionate stretching action of the singular values: the stretching is maximum for the vectors in the direction of V1. Also note that in the U-basis, S*V1 lies on the same axis as U1 (U1, in the U-basis, is of course (1, 0, 0)), but s1 units along that axis for reasons discussed in the text after Figure 2.

One way to look at S is that it implements the same linear transformation as the matrix A, but with the inputs and outputs represented in different basis sets. The inputs to S are the data represented in the V-basis, while the outputs from S are the data represented in the U-basis. That makes it clear why we first multiply the data by V': this changes the basis of the input space to that which is appropriate for S. As you might guess, the final matrix, U, simply transforms the output of the S transform from a representation in the U-basis back into the standard basis.

Hence, we shouldn't be surprised that the final step in the SVD is to apply the mxm (in this case, 3x3) matrix U to the transformed data represented in the U-basis. Just like V', U is a rotation matrix: it transforms the data from the U-basis (above picture) back to the standard basis. The standard basis vectors are in black in the above picture, and we can see that the U transformation brings them back into alignment with the main axes of the plot:



Pretty cool. The SVD lets you see, fairly transparently, the underlying transformations implicitly lurking in any matrix. Unlike many other decompositions (such as a diagonalization), it doesn't require A to have any special properties (e.g., A doesn't have to be square, symmetric, have linearly independent columns, etc). Any matrix can be decomposed into a change of basis (rotation by V'), a simple scaling (and "flattening") operation (by the singular matrix S), and a final change of basis (rotation by U).

Postscript: I realize this article uses a bit of technical jargon, so I will post a linear algebra primer someday that explains the terminology. For the aficianados, I have left out some details that would have complicated things and made this post too long. In particular, I focused on how the SVD factors act on "data" vectors, but little on the properties of the SVD itself (e.g., how to compute U, S, and V; comparison to orthogonal diagonalization, and tons of other things).

If you have suggestions for improving svd_visualize, please let me know in the comments or email me (thomson ~at~ neuro [dot] duke -dot- edu).

Edit in 2022: it has been almost 15 years since I made this post. It has held up fairly well as one Gilbert-Strang-like take on SVD, but I wrote it as if V and U are always rotation matrices when you perform SVD. This is not right: they can be either rotation or reflection matrices (the latter have a determinant of -1, while the former have a determinant of 1). For the example used in the code here, the determinants are 1 so we have rotation matrices, but this will not always be the case. Note a reflection is when certain axes flip (e.g., like going from a right to a left-handed coordinate system), so is similar to a rotation, but is mathematically not the same.

Sensory processing in mouse motor cortex

Over at Nature's neuroscience group, I wrote up a summary and discussion of the excellent paper Spatiotemporal Dynamics of Cortical Sensorimotor Integration in Behaving Mice by by Ferezou et al.. You can find the original paper here, and my summary is here.

Here is the conclusion paragraph of my summary:

Ferezou et al. showed that subthreshold responses to whisker stimulation can be quite broadly distributed, often extending into M1. This suggests that M1 does not have a purely motor function, but serves also to process sensory information. While M1 projects directly to the brain stem and spinal cord to coordinate motor activity, its tight link with S1 opens up interesting questions about its role in sensory processing and sensorimotor transformations. Also, the sensory response in S1 and M1 depends on the behavioral state of the animal, suggesting that sensory processing isn’t a stationary process, but is sensitive to the context in which a stimulus is delivered. So, when someone asks how a mouse’s cortex would respond to a given stimulus, you probably have to ask, “What is the critter doing?”

Frontiers in Neuroscience

A new neuroscience journal, Frontiers in Neuroscience, recently published its inaugural issue. It has a big-name editorial board including Larry Abbot,Henry Markram, and my postdoctoral advisor Miguel Nicolelis.

They are taking a big risk with this journal, as it flouts the traditional business model of the big journals like Nature (expensive, for profit journals with access only to paid subscribers). Just like its successful cousin, PLOS, Frontiers is open access, authors have copyright control and can distribute the article as they see fit. With these guys, you won't be directed to any annoying web pages asking you to pay $50.00 for an article you need.

What makes the Frontiers journals even more interesting is their novel policy for article reviewers. Reviewers are not anonymous, but rather "the Referee remains anonymous only during the review period. After the review, the screen is lifted and the Referees are disclosed and acknowledged on the published paper." No more annoying reviews by lazy referees who obviously haven't read the paper closely.

Also, if you review a paper that is ultimately accepted for publication you will have the option of writing up "a one-page summary of the paper co-authored by all participating Referees. These commentaries are referenced and citable and are major incentive for Referees because the current trend is for readers to read more meta-papers before going to the deeper original studies."

This is a very interesting experiment in publication practices. On one hand, writers are almost guaranteed to get constructive and helpful criticisms rather than half-thought-out potshots. On the other hand, if you are a small fish reviewing the paper of a "big name" lab, you might be tempted to hold your punches so as not to incur the wrath of someone with a lot of power in your subfield. Also, referees might be tempted to accept publications so they can get their summary published, thereby packing their CV.

Time will tell whether this radical experiment in open access journals has legs. The first issue has some very interesting articles. The first four are:
1.Shaul Druckmann, Yoav Banitt, Albert A. Gidon, Felix Schürmann, Henry Markram and Idan Segev A Novel Multiple Objective Optimization Framework for Constraining Conductance-Based Neuron Models by Experimental Data.
2.Alex Thomson and Christophe M. Lamy Functional maps of neocortical local circuitry.
3.Sidarta Ribeiro, Xinwu Shi, Matthew Engelhard, Yi Zhou, Hao Zhang, Damien Gervasoni, Shih-Chieh Lin, Kazuhiro Wada, Nelson A. Lemo and Miguel A. Nicolelis Novel experience induces persistent sleep-dependent plasticity in the cortex but not in the hippocampus.
4.Nestor Parga and Larry Abbott Network model of spontaneous activity exhibiting synchronous transitions between up and down states.

Cortical computation: some complications

Developing Intelligence provided a helpful summary of a fun and thought provoking paper by Douglas and Martin that addresses the question, "What is the basic cortical computation?" with special emphasis on the traditional answer: whatever it is that a column does.

The paper points out that the traditional functional columns (as seen in rat whisker barrels and ocular dominance columns in V1) don't align neatly with minicolumns, which are often taken to be the fundamental computational unit (minicolumns are 'vertical cylinders' of cells about about 50 microns in width (the above image is an example from this review)). Columns, on the other hand, are about 500 microns in diameter and the cells within a column are typically taken to have similar functional properties (e.g., similar receptive fields).

Douglas and Martin say, regarding this mismatch between minicolumns and columns, 'This means that even in a highly specialized “column” such as those evident in the rodent barrel cortex, one cannot simply clip out a cylinder of tissue that contains the whole local circuit, for later reconstruction “in silico”'. They may be overstating things a bit: while the minicolumn and the column don't align neatly, that doesn't mean the columns aren't the fundamental unit of computation. So, the fundamental unit isn't the minicolumn, it's a whole bunch of minicolumns organized into a column via horizontal connections. What's the problem?

On the other hand (and Douglas and Martin might agree) even larger-scale structures like barrels are suspect as the basic computational units. When you stimulate an individual whisker in awake rats, often ten or more whisker barrels are activated, on average more than five (review here). A recent study by Ferezou et al. showed that when a single mouse whisker is stimulated in awake mice, about ten milliseconds after the initial localized response most of the barrel cortex is activated, and this activity sweeps even to motor cortex (image to left is taken from Figure 3 of this paper). While it would certainly be inconvenient if cortical representations, even of simple spatially punctate stimuli, were much more distributed than previously thought, it is starting to look like an inconvenient truth.

I don't want to overstate these results. After all, the initial response is fairly localized in that study! However, the above results are at least consistent with the claim that the simplest anatomical (i.e., columns) and functional (i.e., receptive fields determined in anesthetized animals) maps may not generate an accurate functional decomposition of the cortex in all its complicated glory.

To make matters worse, some think that we shouldn't even talk about the basic functional decomposition of the cortex, but given the tight feedback between cortical and thalamic systems, we should think more about the fundamental computations being carried out by thalamocortical loops (for example, see this review).

These speculations about thalamocortical loops mostly apply to the "back" of the brain. In the frontal lobe, we have to consider the importance of the role of the basal ganglia, which then speaks back to the cortex via the thalamus (for example see this review). For all we know, we may end up having to describe two functional basis sets, the thalamocortical computational units and the striato-thalmo-cortical units. One loop geared toward processing things sensory, the other geared toward processing things motor. Wouldn't that be convenient.

Review of Seung's talk at SFN

It's been a few weeks since SFN. It was exceptionally well organized, San Diego was beautiful, and the scientific content overwhelming as always. Sebastian Seung's entertaining Presidential Lecture, The Once and Future Science of Neural Networks, generated more chatter and emotion than any I've seen at SFN.

There were two main neuroscience topics in Seung's talk, the hedonistic synapse and connectomics. I'll briefly review each topic, and then discuss why the talk generated so much discussion.

The hedonistic synapse
The hedonistic synapse (review article in Neuron here) refers to a hypothetical learning rule in which synaptic weights are adjusted in proportion to a global reward signal broadcast through the CNS. For example, if a presynaptic spike is followed immediately by a postsynaptic spike, that synapse will be strenghtened if dopamine is present. If the same series of spikes occurs without dopamine, there is no change in the weights. Technically, the weight change is a product of the global reward signal and an 'eligibility trace' which describes the relevant relations among spikes in the pre- and postsynaptic neurons.

Seung's group has shown that a network of neurons using the hedonistic rule will tend to maximize reward in the system. So, hedonistic synapses can in theory maximize animal reward.

Seung then discussed the model in the context of birdsong learning, hypothesizing that when a fledgling's birdsong closely matches the stored template, a reward signal is broadcast to the appropriate nucleus, inducing hedonistic synaptic changes that will eventually push the actual song into alignment with the template. For that work, see this paper.

I am interested to see what happens when experimentalists vary reward signals while performing the typical experiments to evoke synaptic weight change. Will the rules change depending on reward? That would be very cool.

I haven't studied Seung's model in enough detail to understand how it relates to the well-known experimental demonstrations of spike-timing dependent plasticity (STDP: for a review, see this paper). STDP studies show that the sign of a synaptic weight change depends on the relative timing of spikes in pre- and post-synaptic neurons. Many of these experiments were done in slice or culture, and to my knowledge did not modulate dopamine or other putative reward signals in the preparation.

Connectomics
Connectomics is the study of the distribution and abundance of synaptic connections among neurons in the brain, a kind of synaptic ecology. Seung described an exciting collaboration with Winfried Denk, in which they are reconstructing, at nanometer scales, the structure of relatively large chunks of neural tissue (for a review, see this paper). The technique is known as serial block-face scanning electron microscopy (SBFSEM). SBFSEM will yield not only the wonderful dendritic and axonal morphology in the tissue, but will provide a full description of the distribution of synaptic connections as well.

You can view a beautiful movie that shows an example of such a reconstruction carried out in the retina (inner plexiform layer) of the rabbit. This movie received spontaneous applause from much of the audience.

Of course, simply having a circuit diagram of a chunk of tissue is not sufficient to reconstruct its functional architecture. The anatomy doesn't tell you the biophysical properties of the individual neurons and synapses, and network properties can be dramatically influenced by the neuromodulatory mileau in which the circuits are bathed. However, ultimately we want to wed the detailed anatomical and functional stories, and this work is a promising step in that direction.

Incidentally, 'connectomics' is a cheesy mouthful of a word. Some words just weren't meant to be biologized with the -omics suffix, and 'connect' is one of them. 'Genomics,' cool. 'Proteomics,' a little annoying and not very creative. 'Connectomics,' stop the bus I want to get off. At the very least let's reserve the -omics for molecular biology. Thank goodness Bialek didn't try to pull us into the world of 'infomics.'

The hubub
The scientific content of Seung's talk was cool. You'd think the reactions would be mostly positive. What generated the meta-talk was the style of his presentation. It was peppered with dramatic pauses (e.g., the above movie was prefaced with, "Ladies and gentleman, I present...the retina."), lots of jokes, and big-picture musings on neuroscience.

Many people sitting near us at the talk were scowling much of the time; some said his theatrics were out of proportion to any truly novel scientific content. It wasn't like he was presenting something on a par with General Relativity.

In contrast, my reaction was quite positive (and I think the split is about 50/50 on this: very few were neutral). I found it to be a fun, informative, and inspiring talk. At the end of a long day of staring at posters and concentrating on technical talks, to have a seasoned theoretician from physics (the Bell Labs cult no less) give a conversational, approachable, big-picture perspective on the field of systems neuroscience was refreshing and quite entertaining. I found his excitement infectious and wanted to go and start doing experiments right when the talk ended. Sure, it required that I tone down my usual enraged skeptic demeanor (as a good sport will do in a horror movie), but at the end of a day at SFN, I had no trouble with this.

Back in the swing

Grants are done, things are manageable so I'm firing this thing up again. I notice there are lots of new neuro-themed blogs out there so I'll have to update my sidebar over the next couple of weeks.

As for the blog, my initial experiment gave a negative result: a blog isn't a good place to simply provide dry and detailed summaries of a technical work such as Hille's. My plan with that is to finish the summaries through Chapter 3 (which means, finish the Boltzmann Distribution stuff), and then compile it into a kind of 'User's Guide to Hille.'

This will become a more traditional science blog, with random musings and discussion of what is going on in the world of neuroscience. My first post will be about Sebastiun Seung's much discussed talk at SFN this year.

Update

Mid-November 2007, this blog restarts in earnest.

Still working on Boltzmann (well, experiments, really, but the Boltzmann is on my mind, and almost bloody done). Once I'm done with that, I'm gonna drastically change things here. Not sure how, yet!

How to count stuff

A new primer on combinatorics that I wrote can be found here. I first cover basic stuff like permutations and combinations. The rest focuses on a more complicated counting problem, which leads to a derivation of the multinomial coefficient. The multinomial coefficient is used in a lot of fields. For instance, it comes up in the derivation of the Boltzmann Distribution in statistical mechanics.

As usual, comments on the document are welcome.

Probability refresher

A PDF of my newly revised probability theory refresher can be downloaded here. It is a quick and informal review of concepts from elementary probability theory. If you want to refresh your memory about what a prior probability is, what marginal probabilities are, or what Bayes' Theorem is, it provides such information. As always, I am on the lookout for comments that will help me to improve it.

Now I am working on a tutorial on counting things (permutations, combinations, and such). It should be a useful reference for the derivation of the Boltzmann Distribution, which involves quite a bit of combinatorics.

It looks like the neuroscience blogosphere is starting to fill out nicely. I guess I should update my sidebar this weekend!

I'm not dead, just writing a grant

I expect to kick-start this puppy in August. I've been working on a grant, which I will send out next week.

As for the Boltzmann distribution post I keep promising, I luckily found a good text, Statistical Mechanics: A Survival Guide. It has an amazingly clear description of its derivation, and answers exactly the questions I was getting confused about (and this was after consulting a few physicsts and stat mech texts).

I will work up a brief summary of what I've learned about Boltzmann once I'm back. Also, I want to go back and edit some of my earlier entries to make them more clear and felicitious in their composition. I'd also like to change the format of the site, to make room for expansion in the future. Oh, and finish summarizing the first half of Hille and get on to more books!!! I'm thinking of something a bit more fun, and amenable to discussion, like Sejnowski's '23 Problems in Systems Neuroscience' or something.

See you all at SFN!

Chapter 3 (2): Pore it on!

Those of us who got our neuroscience education recently are raised from our undergraduate days to think that electric currents across the membrane are generated from ion flow through pores in channels. While true, it is easy to lose appreciation for the fact that this was hard-fought knowledge and things could have turned out quite differently. There are multiple lines of converging evidence for a pore which I'll review now.

TEA studies (Armstrong)
When TEA (a K-channel antagonist) is introduced intracellularly, it can be dislodged when the cell is hyperpolarized below E_k so K+ flows in. The fact that K+ can flow into, but not out of, TEA-blocked cells, suggests that TEA sticks in a cavity on the intracellular side, but can be dislodged by inflow of K+. [Ch 16]

Armstrong, based on further studies with TEA, also calculated that 600 K+ ions/ms flow out of a channel. Physically speaking, this seems too high for a carrier, too high for most enzymes, but well within possibilities for a pore. [Ch 11]

Ion selectivity of channels
Channels are permeable to ions only up to a certain size [Ch 14]. "Such size selection suggested a pore acting as an atomic mechanical sieve" (p. 72).

Pre-crystal structural evidence
Based on sequence information, which can be used to generate hydropathy plots (plots which show stretches of hydrophobic/hydrophilic protein) are used to infer the tertiary structure of channels in vivo. All voltage-gated channels consist of four domains (each containing six transmembrane spanning segments) that are either covalently linked (Na, Ca) or form by oligomerization of such subunits (K channels). This sequence information also allows us to study the effects of mutation on channel function (e.g., effects on ion flow, toxin-receptor interaction, and gating currents). Such molecular models suggest that each domain has a length of molecules that faces an internal pore. An excellent overview is provided in the following figure and text are from Catterall's group:

"Sodium-channel proteins in the mammalian brain are composed of a complex of a 260 kDa α subunit in association with one or more auxiliary β subunits (β1, β2 and/or β3) of 33-36 kDa [3] ([See accompanying figure: click on it for a larger version]). Nine α subunits (Nav1.1-Nav1.9) have been functionally characterized, and a tenth related isoform (Nax) may also function as a Na+ channel. The primary sequence predicts that the sodium channel α subunit folds into four domains (I-IV), which are similar to one another and contain six α-helical transmembrane segments (S1-S6). In each of the domains, the voltage sensor is located in the S4 segments, which contain positively charged amino-acid residues in every third position. A re-entrant loop between helices S5 and S6 is embedded into the transmembrane region of the channel to form the narrow, ion-selective filter at the extracellular end of the pore. The wider intracellular end of the pore is formed by the four S6 segments. Small extracellular loops connect the transmembrane segments, with the largest ones connecting the S5 or S6 segments to the membrane re-entrant loop. Larger intracellular loops link the four homologous domains. Large amino-terminal and carboxy-terminal tail domains also contribute to the internal face of the sodium channel. This view of sodium channel architecture has been largely confirmed by biochemical, electrophysiological, and structural experiments."

Crytallography
In the above paper, Catterall goes on to state that "A complete three-dimensional structure of the sodium channel is not yet available." The same is not true of the voltage-gated K channel. KcsA, a K channel found in the bacterium Streptomyces livedans has been successfully crystallized and visualized (Plates 2c and 2d). As expected, the channel shows a narrow pore through which K+ ions can flow.

Patch clamp
Hille's description of this seminal work introduced by Neher and Sakmann in 1976 is excellent, so I won't use much bandwidth summarizing it here. Essentially, they learned how to form high resistance seals with small areas of the cell membrane, even to the point where we can measure currents generated by single channels. The ability to record the activity of single proteins in vivo is extremely rare in biology. See figures 3.16 and 3.17 for traces of currents through Na- and K- channels. These studies have shown that the opening/closing of channels is extremly rapid, operating on time scales smaller than the microsecond. Also, it has been possible to calculate that individual ions traverse the length of the channel very quickly, on the order of nanoseconds. This fact alone makes all but the pore hypothesis untenable.

Chapter 3 (1): Toxins to the Rescue

Chapter 2 outlined the HH theory of the action potential: when they wrote their papers, nobody yet knew the mechanism of current flow. From p 61:

A variety of mechanisms were considered possible. These included permeation in a homogeneous membrane, binding and migration along charged sites, passage on carriers, and flow through pores...The pathways might be formed from phospholipid or from protein, or even from nucleic acid. Each of these ideas was seriously advanced and rationalized in published articles.

Between the early 1960s and 1976 (in 1976 Neher and Sakmann published the first single channel patch recording), the mystery of the currents was essentially solved. First, in the 1960s toxins were found that could selectively block I_na (TTX and STX) or I_k (TEA) while the other current remained unaltered (Figure 3.2). This, in addition to the results of HH's ion substitution experiments, was taken as evidence that there were two separate ion pathways, or channels, one for K and one for Na.

The discovery of neurotoxins has been a boon for neuroscience. In addition to letting us turn individual currents on and off at will, they have helped us count individual channels (channel density is an important parameter in realistic neuronal models), and have suggested structural models of channels.

Counting Channels
The action of TEA, TTX, and STX is well-described by a simple model in which a single molecule of the toxin T binds reversibly to a single receptor site R in an individual channel. This model is represented in Equation 3.1. The strength of binding is given by the equilibrium dissociation constant K_d. K_d is the toxin concentration at which half of the available receptors are occupied by the toxin. The model allows for a relatively simple algebraic derivation of the proportion of bound receptors, denoted with the variable y (see equation 3.2 and Figure 3.3).

If the above is unfamiliar, I highly recommend reading this site for a good introduction to the kinetics of such ligand-receptor interactions, which includes a discussion of K_d. As usual, Hille is leaving out lots of details in his exposition, but his terse summaries are very accurate.

The simple model of toxin-receptor interactions described above allows us to directly estimate the number of receptor sites, B_max in a preparation. This is typically done with radioactively labeled toxins denoted radioligands. After marinating the preparation with radioligand, the preparation is thoroughly washed out to remove all the radioligand that is not bound to the receptor. Often it is washed out with a specific competitor such as a nonradioactive form of the toxin. If the radioligand only bound to the receptor, then estimating B_max, the number of receptors in the tissue, would be relatively simple. We could apply different concentrations of radioligand and fit the data to the equation B_maxy, where y is the proportion of bound receptors as a function of toxin concentration (Equation 3.2), and B_max is the number of radioligand molecules at saturation. Note that B_maxy is the first term in Equation 3.4.

Unfortunately, reality isn't quite so simple: we are never able to wash out all of the radioligand that isn't bound to receptor. It binds "nonspecifically" to other molecules in the preparation. Indeed, there is evidence that certain drugs bind to the plastic out of which test tubes are made! Luckily, this general, nonspecific binding typically has very different properties than the toxin-receptor binding: it is a linear nonsaturating function (i.e., the radioligand remaining after washout continues to increase linearly without bound as the concentration of radioligand is increased). Because of such nonspecific binding, a second linear term must be added to the saturating function, as represented by the right-hand term a[T] in Equation 3.4. [T] is the toxin concentration and a is a free parameter that represents the slope of the linear relationship.

Figure 3.4 shows the various curves we have discussed in the context of a clever experiment. The amount of radiolabeled STX was measured at different concentrations before and after wash with a high concentration of TTX, which has a higher affinity for the Na-channel receptor than STX. The remaining radiolableed STX was due to nonspecific binding, and exhibits the expected linear relationship with STX concentration. The difference between the linear curve and the overall saturating curve observed before TTX wash reveals the amount of radioligand remaining due to specific binding of STX to the Na-receptor. Such data has shown us that the Na-channel density on membrane is about 100-400 channels per square micron of unmyelenated axon.

Initial channel models
In addition to helping us count receptors, toxins have helped generate predictions about channel structure. For instance, a model of K channels was suggested by the effects of intracellular TEA on I_k. Namely, intracellular TEA blocks I_k only in open channels (a phenomenon known as open channel block (Figure 3.5). This suggested that the channel is a pore with a wide cytoplasmic opening (large enough for a big toxin like TEA to block it), an opening which is typically blocked by a gate that moves out of the way when the channel opens, thus allowing TEA to enter and close the channel. Because the toxin does not simply flow out of the extracellular pore, the model also assumes that the pore narrows toward the extracellular side. A cartoon of this model of the voltage-gated channel is shown in Figure 3.6. I

Chapter 2 (3): Gating charges and gating currents

Hille discusses gating currents at greater length in later chapters (e.g., Chapters 9, 12, 18, and 19), and he will explicitly draw on the material from Chapter 2. Hence, I thought the topic deserved a few bits of bandwidth.

In the HH model of voltage-gated Na and K channels, there exist gating particles that respond to changes in voltage and thereby switch between the permissive and nonpermissive state (these are the m- and h-gates in the Na-channel and n-gates in the K-channel). The voltage-dependent changes in these molecular gates are thought to result from the interaction of V_m with charged amino acids in the channel, interactions that cause these miniature voltage sensors to move, or translocate, within the cell membrane. Since channels open during depolarization, we would expect positive gating charges to move in the extracellular direction when they enter the permissive state (equivalently, negative gating charges should move toward the inside of the cell).

Movement of a charge across the membrane is a current, the gating current, that should be observable (p. 57):

Hodgkin and Huxley pointed out that the necessary movement of charged gating particles within the membrane should also be detectable in a voltage clamp as a small electric current that would precede the ionic currents. At first the term "carrier current" was used for the proposed charge movement, but since we no longer think of channels as carriers, the term gating current is now universally used.

The empirical measurement of gating currents wasn't actually performed until the early 70s.

Hille provides a brief summary of how to use general energetic considerations to calculate the number of charges that the voltage sensor contains, the gating charge. First, recall from chemistry that a protein can exist in many conformational states, and the energy associated with each conformation forms a "conformation energy landscape", a bumpy terrain usually with lots of local minima separated by hills that represent the energy needed to make the transition between the conformations. So we can assume that even when a channel is not subject to a voltage across the membrane, the conformational energy of a channel in the closed state is X, its conformational energy in the open state is X+w, and hence the difference in energy between the two conformations is w.

It is important to note that the conformational change in the channel takes place in a voltage field V_m. By hypothesis, the gating particle has a charge Q (Q=z_gq_e where q_e is the elementary charge and z_g is the number of charges) and the sensor must move against V_m. Hence, in addition to the purely conformational energy change described above, at a fixed V_m, the work done (i.e., energy) to move the charge from the intracellular to the extracellular edge of the membrane is -QV_m (recall V is in J/Coulomb, so the units work out). This charge movement is exactly analogous to lifting a baseball above the ground and building up potential energy: similarly, a positive charge Q held just at the edge of the extracellular membrane will be pushed toward the intracellular side when the cell is at a hyperpolarized V_m. More explicitly, if the electrical potential in the closed state is Y, then the potential in the open state is Y-QV_m, and the change in the electrical potential is -QV_m. The sign is negative because we want a positive potential difference when the cell is hyperpolarized.

We can then examine both energies (conformational and electrical potential) simultaneously. The total energy associated with the closed state is E_closed=X+Y, that associated with the open state is E_open=(X+w)+(Y-QV_m). Hence, the change in energy associated with the transition from the closed to open state is E_open-E_closed=w-QV_m. Since we are describing a channel with only two states, and we know the energy difference between the states, we can apply the Boltzmann equation to describe the ratio of open to closed channels at thermodynamic equilibrium: it is a decaying exponential function of (w-QV_m) [see Equations 2.21 and 2.22].

If you substitute z_gq_e for Q in the Boltzmann distribution, you can fit the observed proportion of open channels to the Boltzmann distribution using different values of z_g, and thereby get an estimate of the number of charges in the voltage sensor (Figure 2.20). Technically, the best fit only provides a lower bound on the number of charges in the voltage sensor. This is because the above calculation assumes that z_g charges move completely from the inside of the membrane to the outside. However, the same electrical potential -QV_m could be generated by moving a greater number of charges partly across the membrane. Indeed, this is now thought to be exactly what happens.