A Cubic Millimeter of Your Brain

Are there more connections in a cubic millimeter of your brain than there are stars in the Milky Way? We are going to answer that question in a moment, but first take a look at this image of hippocampal neurons in a mouse's brain. It is an actual color image from a transgenic mouse in which fluorescent protein variations are expressed quasi-randomly in different neurons. This kind of image is known as a brainbow and is aesthetically awesome further it may be one way to empirically examine a cubic millimeter of the brain (neuron tomography).

In reality mapping even an entire cubic millimeter of the brain is an extremely daunting task, but we can still answer my original question. First, I know that there are different kinds of neurons that vary in size and that some neurons can have a soma (the big part that has the nucleus from which the dendrites extend) spanning a millimeter in size. Thus if you picked a random cubic millimeter of brain you could run right into the heart of a neuron and you would find very few connections. Given this fact, we can very easily answer this question with a resounding no, however, this seems like an unsatisfactory trite approach. So I looked up some numbers on how many neurons are in the brain, how many connections are in the brain, and how many stars are in the Milky Way. Lets answer the question using the 'average' number of connections per cubic millimeter.

How many neurons and connections there are in the brain? This is kind of a tricky question and I am not a nuerobiologist so I have gone to several resources for the answer. Professor of Computational Neuroscience at MIT Sebastung Seung says in a TED talk

your brain contains 100 billion neurons and 10,000 times as many connections

Professor of Molecular Cellular Physiology at Stanford Stephen Smith says in a press release on brain imaging that

In a human, there are more than 125 trillion synapses just in the cerebral cortex alone

René Marois from the Center for Integrative and Cognitive Neurosciences at Vanderbilt Vision Research Center states in a recent paper ^[1]

The human brain is heralded for its staggering complexity and processing capacity: its hundred billion neurons and several hundred trillion synaptic connections can process and exchange prodigious amounts of information over a distributed neural network in the matter of milliseconds.

I have enough expert sources now to confidently say these experiments agree that the human brain has some 100 billion neurons (10¹¹). The number of connections seems less precise, but it is at least several 100 trillion connections (10¹⁴) as judged by Marios and Smith and as much as 10¹⁵ as judged by Seung.

The number of connections in the brain is tricky to define. We may define a synaptic connection as each place the neuron touches another neuron and a synapse is present. It doesn't seem to make sense to simply count incidental contact. Further, there is the question of whether we should count redundant contacts between neurons. We can obtain an upper bound on the number of connections in the brain by considering the case in which every neuron is connected to every other neuron. Coincidentally the operation of connecting every node in a network with every other node is a process I am familiar with from cross correlating radio signals. Anyways, the equation we are looking for is N(N-1)/2 where N is the number of nodes in the network. Thus, for our N=10¹¹ neurons the maximum number of non-redundant connections is about 10²². This maximum bound is huge! But how huge is it really? Hilariously, while searching for an answer to my original question I found a message board pondering the grand statement

There are more connections in the brain than atoms in the Universe.

A really clever person pointed out that

Theoretically, if we took all the atoms in the universe; wouldn't that include the atoms within the brain?

People have this feeling that the number of connections between items can be much larger than the number of actual items in the collection and while this intuition is true the idea that there are more connections in the brain than there are atoms in the universe is absurd. Lets put it in perspective that a few grams of any substance, like water, is measured units of moles. A mole is standard unit of measurement corresponding to the absolute 6.02 x 10²³. Thus even a drop of water contains more atoms than there are connections in the brain.

Now we need to know how many neurons and connections are in an average cubic millimeter of the brain. How big is the brain? John S. Allen of the Department of Neurology at University of Iowa stated in a recent paper that^[2]

The mean total brain volumes found here (1,273.6 cc for men, and 1,131.1 cc for women) are very comparable to the results from other high-resolution MRI-volumetric studies.

We can take the volume of the brain as 1000cc as a low estimate (which will only over estimate the density of connections).

The final thing we need to know to answer the question at hand is the number of stars in the Milky Way. Like every other number we have been working with it is rather uncertain. Even if we define a star as only those spheres of gas which are large enough to fuse hydrogen at some point in their lifetime we don't know the answer because we can't see the multitudes of dim stars. There are probably at least 500 billion star like objects in the Milky Way. Lets take 100 billion as the number to be conservative.

Finally, lets bring all the numbers together. One cubic millimeter is 1/1000 of a cubic centimeter and 1/1000000 (10^-6) of the entire volume of the brain. We can scale the total number of connections in the brain (using the high estimate of 10¹⁵ connections in the brain) then we find that there are 10⁹ connections in a cubic millimeter of the brain. The 10⁹ connections in a cubic millimeter of the brain is two orders of magnitude smaller than a low estimate of the number of stars in the Milky Way. No, on average there are not more connections in a cubic millimeter of your brain than there are stars in the Milky Way. 

My first response to this question was bullshit! This question (or rather statement) is made by David Eagleman here at a TEDx talk and here on the Colbert Report. Colbert also called out Eagleman when he dropped this factoid, but it didn't stop the interview. I have also contacted some actual neuroscientists to see what they thought of this statement and they agree with me that it is not true. Maybe there is special part of the brain particularly more dense in connections than the brain on average, but that would be misleading like saying the density of the Milky Way is that of water because, you know, certain parts of the Milky Way are water. The better statement would be to say that there are are more connections in the brain than there are stars in the Milky Way. As Colbert would say, I am putting you on notice Eagleman.

While we are on the subject I want to mention my favorite talk about the brain which mixes just the right amount of wonder and fact. It is the TED talk I mentioned earlier by Sebastian Seung on what he calls the connectome - the network of connections in your brain between neurons which physically dictates how you think. In the video he discusses another volume tomography technique in the brain using a cube of mouse brain tissue just 6 microns on a side. It is another great visualization for what is actually in a cubic millimeter of your brain.

References

[1] Marois, R., & Ivanoff, J. (2005). Capacity limits of information processing in the brain Trends in Cognitive Sciences, 9 (6), 296-305 DOI: 10.1016/j.tics.2005.04.010

[2] Allen, J., Damasio, H., & Grabowski, T. (2002). Normal neuroanatomical variation in the human brain: An MRI-volumetric study American Journal of Physical Anthropology, 118 (4), 341-358 DOI: 10.1002/ajpa.10092

On Replications

Repetition is ubiquitous and has many different meanings in education, art, literature, science, and life Ideas replicate and mutate; cultural memes spread through culture seamlessly. Manufactured goods are produced as nearly identical as possible. Deviations from the mold are discarded and parts are interchangeable. Digital data is almost limitlessly replicable. Any data or idea committed to the digital world is perfectly copied (sparing the occurrence of a flipped bit) until it is intentionally modified. This characteristic of digital ideas presents a unique challenge for creators of content, distributors, and bored people on the internet. And of course animals and plants on Earth have the ability to self replicate themselves with minor variations. What do we make of all of this?

I am keen on the intersection of art and science on this matter. I like making collages and have highlighted repeated images before with 35 images of space helmet reflections and 100 images of macchiatos. Through repetition and distortion images may be amplified or diminished. It depends on perspective. Generally in artistic endeavors, as in life, the slight variations of a repeated theme are aesthetically pleasing. On the other hand technical work such as engineering, data analysis, or manufacturing requires precise replication. I work in radio astronomy where each radio telescope in the array is nearly identical and the need for precision trumps all other considerations. I find that randomness is never particularly interesting, but neither is absolute order. Somewhere in between these extremes we have something really beautiful.

Looking back and looking forward

This photo was taken on the last US manned space flight in 1975 before the first shuttle launch in 1981. Portrayed is the historic handshake between Tom Stafford and Alexey Leonov through the open hatch between the American Apollo and Russian Soyuz ships. Today the Atlantis shuttle lifted off for the the last shuttle mission. It is the end of an era. Alas, all good things must end.

Mars Rover Curiosity

This animation depicts what will happen in August 2012 if all goes as planned for Curiosity, NASA's next Mars rover. This rover is much larger and and more competent than the previous rovers. It is about the size of a small car and has an entire suite of experiments on board. During entry it uses a series of thrusters to maneuver to the designated landing area. Once the ship has slowed down to Mach two (keep in mind that the atmospheric pressure on the surface of Mar's is of the order .05% that of Earth's) a parachute is deployed. As the vehicle slows the heat shield comes off and a radar detects how close the surface is approaching in order to slow for a smooth landing. The last daring step is a so called 'sky crane' which lowers the rover with a long cable from the rocket thrusted ship above. Eventually Curiosity will begin roving, but it won't be limited to roving only during the day by solar panels as the previous rovers were. The large tilted box on the back of the rover contains 4.8 kg of plutonium dioxide which emits heat serving as the power source of the rover. The power should keep flowing for much longer than the minimum specked science mission of two Earth years. The rover will seek out rough rocks such as ancient Martian riverbeds or canyons where evidence of early environments on Mars can be found. The ability to navigate to these areas is an important science requirement for the rover and is one of the reasons for the rover's large size and nuclear battery which should allow it to travel at least 20 kilometers during its lifetime. Geologists and astrobiologists also want to know if certain conditions such as those necessary for organic molecules are present. In the video a laser and a drill are shown performing experiments. The laser is ChemCam which will project onto hard to reach rocks and detect the reflected light in order to discern the chemical composition of rocks. The drill is about a centimeter in diameter and will extract the dust from the holes it creates to run experiments in mineralogy (the laser device inside the rover shown in the video) or detecting organic molecules. All of these experiments aim to answer the question, could Mars have had an environment capable of supporting life at one time?

If the sky crane works we may soon know the answer to this question. Curiosity has a launch window from November 25 to December 18, 2011 from Kennedy Space Center in Florida. And in other news NASA's James Webb Space Telescope is being threatened with the axe in budget bill in the U.S. House of Representatives today. NASA will never run out of adversaries pulling it down: Gravity and the budget.

Gödel's Proof

There is an idea of reason in the Universe. It is an abstraction which mathematicians have never been content with. Given that scientists exclusively use logic (or mathematical reasoning) for theories and experiments it is of incredible importance to know the limits of logic. It turns out that the study of math itself, metamathematics, has amazing insights on what is knowable.

In 1931 an unassuming paper was published in a German mathematics journal, the title of the paper (translated to English) was 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems I'. It is a confusing title and the kind of paper which I would not understand. The author was a 26 year old Austrian named Kurt Gödel and he had just created a revolutionary idea, but as with so many great ideas it was not simple and it took great minds to fully appreciate it.

The ideas he put forth have been extremely influential and the collective name for the theories that grew from it are known as Gödel's incompleteness theorem. This theorem is a revolutionary outcome in mathematical logical that has implications for not only the philosophy of mathematics, but philosophy in general. It is thus surprising how relatively unknown the theorem is to the general public and even many scientists. I recently read the book Gödel's Proof by Nagel and Newman in just a few sittings at a coffee shop. It is a short and concise explanation of the proof that incrementally brought me closer to understanding the intricacies of Gödel's works.

Gödel's incompleteness theorem is a massive mountain of ideas that I will not attempt to conquer, but I think it is important that everyone at least gets a view. Gödel basically found that no solid guarantee is possible that mathematics is entirely free of internal contradiction. However, Gödel was not out to trash mathematics, contrarily he used mathematics itself to temper the reach of mathematics and place constraints on what is possible to known through mathematics much like a physicist theorizing that a black hole's event horizon places a limit on spaces which the physicist could actually go and measure. Gödel created a new technique of analysis and introduced new probes for logical and mathematical investigation.

The specifics of Gödel's proof even as outlined on wikipedia are extremely complicated (the entire proof is long and there are on the order of 200 links in the article so by the time you were done reading all of the prerequisite mathematical definitions you would have read thousands of pages) for anyone without (or with) extensive mathematical training, so I must admit that I don't understand it completely and not have I attempted to read the actual paper. I want to present here the shortest definition of Gödel's theorem not the the most rigorous.

The key to Gödel's incompleteness theorem is a concept of mapping. In the information age the concept of mapping or coding is familiar to many as in the case of mapping Morse code dots to letter characters. In the explanation that follows take it as a given that it can be shown that all logic systems are equivalent to or mappable to the operators we will be using; this assumption is vital, and I can't quite explain it without detail so I refer the inquisitive mind to read Nagel and Newman's book.

Let us construct a simple logic system using the arbitrary operators P, N, ⊃, and x that have certain properties which we take as given by the table defined below. In the left column a combination of operators is given and in the right column a definition in English of the operators meaning is given.

P⊃x	'print x'
NP⊃x	'never print x'
PP ⊃x	'print xx'
NPP⊃x	'never print xx'

We can combine these statements and create more complicated statements. For example P⊃y where y=P⊃x would mean 'print P⊃x' (note that implicitly I am also using the equals operator). Crucially then NPP⊃x would mean 'never print xx', and this statement could also be written NP⊃xx.

Next ponder what the last statement used on itself means. The statement NPP⊃y where y=NPP⊃ would mean 'never print NPP⊃NPP⊃', but this strange statement could also be written NPP⊃NPP⊃.

So either our system prints NPP⊃NPP⊃ or it never prints NPP⊃NPP⊃. It must do one or the other. If our system prints NPP⊃NPP⊃ then it has printed a false statement because the statement contradicts itself by self reference once it is printed. On the other hand if the system never prints NPP⊃NPP⊃ then we know that there is at least one true statement our system never prints.

So either there are logic statements which may be printed which are false statements, or there are true statements which are never printed. Our system must print some false statements if it is to print all true statements. Or our system will print only true statements, but it will fail to print some true statements.

In the example above I have taken arguments very similar to that in Raymond Smullyan's book Gödels Incompleteness Theorems in order to create an extremely concise, but hopefully accurate description of what lies at the core of Gödel's insight. In Nagel and Newman's book they explain Gödel's proof in much more detail by working out the details of mapping. For example in the explanation above I mapped mathematical statements to the idea of printing, but print could be equivalently be existence. Further, Nagel and Newman argue as Gödel did that all formal axiometric systems can be mapped in some way such that even the most complicated mathematical systems using the common operators of +,-,=, x,0,(,),⊃ and so on can be shown to be incomplete.

Gödel's incompleteness theorem has many forms and implications. Briefly I will demonstrate an analogous, but weaker form of Gödel's incompleteness theorem by analogy to the halting problem. I believe this demonstration is of importance to those of us immersed in the information age and perhaps easier to grasp or at least more applicable than Gödel's work.

The halting problem is to decide whether given a computer program and some input, whether the program will ever stop or will it continue computing infinitely. The key to the halting problem is the concept of computation and algorithms. In the original proof by the enduring Alan Turing specific meanings to the concepts of algorithm and computation were defined. He used a computational machine now known as a Turing complete computer, or a Turing machine. The definition of what constitutes a computer is to the halting problem what the mapping of symbols is to Gödel's theorem. It is at the heart of the problem, and thus actually one of the harder points to define so I will again leave that task as an exercise to the reader.

So lets look at two psuedocode programs and lets imagine that we also have a very special program written by a genius scientist which is called Halting. The scientist claims that Halting can correctly tell your own code B(P,i) whether a program halts.

Program B(P,i)
  if Halting(P,i)==true then
    return true // the program halts
  else
    return false // the program does not halt


Now here is the important part. The genius scientist claims we can analyze any kind of computer program, this is indeed the crux of the halting program, we want to know if any and every program stops. Now imagine a program E that takes X, which is any program, as an argument.

Program E(X)
  if B(X,X)==true then
    while(true) //loops forever
  else
   return true


The first thing E does is take B and passes it X for both arguments. Program E will get back from B either true or false. If it receives back true it will enter an infinite loop and if it receives back false it will terminate.

So suppose I take B and feed it E for both arguments. What answer will B(E,E) give? Think about it.

We will be running our special Halting program on E(E) which will then run the program B(E,E). The answer to B(E,E) will either be either true or false. If the result is false the program E actually returns true and halts immediately; if the result is true then Halting thinks our program does halt, but the program E throws itself into a loop upon this condition and will never halt. Either way program E lies. E was written very craftily to break B on purpose, but nonetheless the damage is done. E cannot be made reliable even in principle. It matters not how clever you are and or how powerful your computer is. There is simply no reliable computer program that can determine whether another program halts on an arbitrary input. The incompleteness problem may have seemed a little bit distant and philosophical, but if you have read this far it should be evident that the halting problem has deep implications computing.

What does Gödel's theorem mean for the real word, experimental verification, and deductive sciences? Well take for example the Banach-Tarski paradox which states that a solid ball in three dimensional space can be split into a finite number of non-overlapping pieces, and then be put back together in a different way to yield two identical copies of the original ball. This process violates sensible physic notions of conservation of volume and area. It turns out that Banach and Tarski came to this conclusion based upon deductions from the Axiom of choice. Now, whether we know anything at all about the axiom of choice we do know that the deductive conclusions drawn from it are in violation of physics. Thus, a physicist could argue that the axiom of choice is not a valid axiom for our Universe. Within mathematics it is unknown, unprovable Gödel says, whether or not we should accept the axiom of choice, because it is after all an axiom. The argument for whether a given axiom is to be accepted must be discussed outside the confines of the logic structure one is arguing about. It turns out that the axiom of choice is important for many other really important mathematical proofs which are used in physics all the time. I don't know what to make of it really, perhaps a mathematician out their should weigh in on this question.

Another important theorem that goes along with Gödel's theorem is Tarski's undefinability theorem. Tarski's undefinabtliy theorem makes a more direct assertion about language and self referential systems. Basically any language sufficiently powerful to be expressively satisfying is limited. In summation we have two vital points to the concept of incompleteness.

1.  If a system is consistent, it cannot be complete and is limited.
2. The consistency of the assumptions or axioms cannot be proven entirely within the system.

The repercussions of the meta analysis of logic are profound and subtle. Gödel really has thrown us for a loop. It is unclear if we should draw the line and say this is just a mere curiosity of mathematics or a deep truth about the Universe. It has been proposed by Douglas Hofstadter (author of Gödel, Escher, Bach) that consciousness itself comes from a kind of 'strange loop' induced by a self referential system in our minds. Primarily, I think we can conclude that Gödel's incompleteness theorem implies that in most situations the tools science has to analyze the world are more than adequate because the situations are not self referential. However, I do see a limit to what we can know about the Universe. As physicists forge forward, generally quite successfully, in understanding the Universe it appears that there really is some consistent mathematical basis for our Universe. Many physicists are searching for this mathematical basis to the Universe, it is the so called theory of everything. But does Gödel's result imply that this mathematical basis cannot be self consistent?

Consider this scenario. One day our most powerful, successful, and comprehensive theory ever will predict something that experiment cannot verify or worse an experiment will patently disagree with. Some will argue that the theory must be thrown out because classically the scientific method states that a theory disagreeing with experiment, or making nonsense predictions, is untenable. Another theory will be introduced that makes consistent and testable predictions, however this theory is not able to predict the most intricate traces of nature. Actually, that first case kind of sounds like string theory. Perhaps we will have to start talking about theories being incomplete instead of wrong one day.

G34.3

Wow, there really is something new to learn everyday.

I Hate Astrology

Perhaps it is cruel to snuff out the shinning gleam in the eyes of a person who upon hearing that I am an astronomer exclaims, "Oh, I love astrology!" and  I reply, "No, I study ASTRONOMY." But they don't understand it. The subtle differences in syllables of the words belies the vast gulf in empirical tendencies between the separate endeavors and it is too much to explain. I simply walk away.

I hate astrology and I hate when people get astronomy and astrology mixed up. I could be more understanding, but I have to choose my battles. I meet a lot of interesting people in coffee shops, bars, airplanes, parties and wherever else life takes me and when someone gets excited about the fact that I study astronomy it means they have a deep curiosity about the skies above. That curiosity is occasionally deeply misguided with astrology and their questions are so fundamentally misconceived I struggle to answer them with candor and accuracy (for example they ask, 'Do the planets affect our daily lives?' and I hesitate to answer honestly that we must consider their gravitational pull, so the answer must be yes). On the other hand I meet people who are genuinely interested in massive collections of gravitationally bound glowing gas and I am very happy to answer their questions.

There is a real danger when logic, or pseudologic, is applied to astrology. Recently there was an uproar about the shifting of the zodiac that made it into some news headlines. Briefly I shared the frustrated sentiments of astrologers because the shifted zodiac has been well known for some time, why is the public just now hearing about it? The book in the image above is from the seventies and claims right there on the cover that, 'Most astrology is unscientific and inaccurate', and goes on to explain the shifted zodiac and how to have a movie ending romance. The ideas in this book are the apotheosis of dangerous thought. A little bit of knowledge is a very dangerous thing when combined with pseudologic in the guise of rigorous proof. It has also not escaped my observation that many of the people I have known who believe in astrology also believe in God as if to demonstrate the utter confusion and inconsistency of their minds. I don't mean to badger defenseless people here. This is simply an honest expression of how I feel. I have summed up my sentiments into a paragraph which I think would be nice to place on a card with which to hand out to people who confuse astrology with astronomy:

I cannot rightly conceive of a logic which would allow one to study such disparate phenomena of love, planets, stars and come to see any connection. Perhaps, desperate for meaning people find it wherever they look; conclusions are forged before the data have been taken. Those who would apply science to astrology may as well attempt to apply science on whom to love and sociologists do study what makes a lasting relationship and neurobiologists study what chemicals are active in the brain during feelings of love, but no scientist will claim that Romeo shouldn't love Juliet. I believe science and an understanding of natural phenomena adds to the beauty life, but pseudologic and lies even when propagated with good intentions ultimately lead to pain and suffering. The human mind has the ability to find patterns anywhere, indeed often where they do not exist.

Making sense of a visible quantum object

Quantum mechanics predicts that nature is fundamentally uncertain. Particles are in multiple states at once; particles are here and there. As we extrapolate these properties of nature to macroscopic objects the results are counterintuitive. The counterintuitive predictions of quantum mechanics should be an observable phenomena, and indeed they are. In this talk the intuition is examined and in this paper by the same physicist, Aaron O'Connell, the physics is examined.

So I haven't been posting lately. I have been drowned in opportunities, hit by funding woes, and frankly it feels like my mind is melting. I will probably post more aggregated non-original content, like this post, but I also have lots of ideas and new things I am working on.