Silicon Brains, Photonics, etc.

The semester is officially over, which is exciting: I finally get to get back into the swing of research (with the occasional GRE study break, of course). As such, blogging will tend to be lighter; but before I lock myself in the lab, here are a few things that came to pass while I was busy finishing up the semester…

Building Brains in Silicon
Among other things, I wrote a paper for my computational neuroscience class on – you guessed it – some really cool work coming out of Kwabena Boahen’s group (formerly here at Penn, now at Stanford) on silicon-based artifical neural systems. This is sometimes classed as ‘neuromorphic engineering’, a term (coined by Carver Mead in the 1980’s) which has come to refer to a relatively recent interdisciplinary paradigm dealing with the development and study of artificial neural systems, drawing on principles from such fields as physics, biology, and computer/electrical engineering to design electronic-based analogues of biological systems. A number of people are using this to try to design new VLSI-based systems based on biological systems.

Some others are trying to reverse this scenario: while ‘real’ neural systems are experimentally studied by neurobiologists while grossly simplified ones are modeled by computational neuroscientists, groups like Boahen’s are trying to bridge these modes of inquiry by exploiting similarities between electronic and neural circuits. Mahowald and Douglas wrote a seminal paper in 1991 describing the first ‘silicon neuron’, and a good deal of work has gone on since then. For example, a number of ‘thermodynamic’ models of ion channels have been developed, building on concepts like Hodgkin/Huxley-type models. Anyway, by exploiting the beautiful similarity between ion channels and metal-oxide-semiconductor (MOS) transistors as two-state systems (simplistically, ion channels are either open or closed, with the energy barrier – and hence the transition rate – between the two states being modulated via, for example, a voltage; on the other hand, a voltage applied across the source and the drain of a MOSFET causes charges to diffuse through the ‘conduction channel’, with the effective barrier to this diffusion being modulated by a gate voltage), Boahen and his graduate student Kai Hynna have recently taken an important step toward ‘building a brain in silicon’. Using an approach that combines the advantages of experiment and artificial modeling, they have developed a simple electronic circuit that replicates the nonlinear dynamics of the gating particles of voltage-dependent ion channels.

References:
- Hynna and Boahen’s recent paper: K. M. Hynna and K. Boahen, Neural Computation 19, 327 (2007).
- 1991 silicon neuron paper: M. Mahowald and R. Douglas, Nature 354, 515 (1991).
- Thermodynamic models of ion channels: A. Destexhe and J. R. Huguenard, J. Comput. Neurosci. 9, 259 (2000).

Update: I guess Tech Review thought this stuff is cool, too: the latest issue has an article on Boahen’s work. It takes a broader view of his work than I have above - I just focused on one particular aspect.

Quasicrystals and Complex Materials as 3D Photonic Structures
I wrote another paper for my modern optics class, based on this recent experimental paper by Man, Megens, Steinhardt and Chaikin on three-dimensional quasicrystals as complete photonic bandgap materials. Here’s the deal: since Schrödinger’s wave equation and the electromagnetic wave equation are formally similar (neglecting spin statistics), it isn’t all that surprising that a number of analogies exist between electronic waves and light. In particular, electromagnetic waves can propagate in structures of periodic dielectric constant, and interference due to multiple Bragg reflections from these interfaces leads to directional-dependent energy band gaps. A major goal is to try to develop artificial structures to act as complete, omnidirectional photonic bandgap (PBG) crystals with bandgaps in the visible regime (wavelength ~ 400-700nm), and a lot of effort has gone into this. Interestingly, recent innovations in materials science and the study of complex materials – such as quasicrystals (QC), liquid crystals (LCs), and colloidal self-assembly – have breathed new life into this quest.

References:
- Experimental confirmation of the almost-spherical effective Brillouin zone (and hence the potential of developing a 3D PBG structure) of a macroscopic 3D icosahedral photonic QC: W. Man, M. Megens, P. J. Steinhardt and P. M. Chaikin, Nature 436, 993 (2005).
- Experimental approach towards assembling 3D analogues of the QC structures studied by Man et al. on a smaller scale using holographic optical trapping: Y. Roichman and D. G. Grier, Opt. Exp. 13, 5434 (2005).
- Another experimental approach, using a novel 7-beam optical interference holography technique: W. Y. Tam, Appl. Phys. Lett. 89, 251111 (2006).
- Using nematic liquid crystals in ‘inverse opal’ structures as PBG materials (tuned by parameters such as an external electric field) – for example, since liquid crystals are birefringent, modulating their orientational order using a field can influence their optical properties (a principle on which liquid crystal displays are based): K. Busch and S. John, Phys. Rev. Lett. 83, 967 (1998).
- Recent computational work has indicated a feasible method of fabricating 3D visible PBG crystals with two different types of lattice structure using self-assembly of a mixture of colloidal spheres of two different sizes: A. P. Hynninen, J. H. Thijssen, E. C. Vermolen, M. Dijkstra, and A. van Blaaderen, Nature Mater. 6, 202 (2007).

Galois Theory In A Nutshell

I have an abstract algebra exam coming up, and so I’ve been looking over everything we’ve covered so far. The subject and I haven’t really gotten along this past year. In particular, unlike in physics courses (where things are driven by the ‘big picture’), the point of the subject material was often quickly obscured. John Baez puts it perfectly:

“I used to hate this subject: it seemed like a massive waste of time. Newspapers, magazines and even lots of math books seem to celebrate the idea of people slaving away for centuries on puzzles whose only virtue is that they’re easy to state but hard to solve. For example: are any odd numbers the sum of all their divisors? Are there infinitely many pairs of primes that differ by 2? Is every even number bigger than 2 a sum of two primes? Are there any positive integer solutions to for ? My response to all these was: who cares?!

Sure, it’s noble to seek knowledge for its own sake. But working on a math problem just because it’s hard is like trying to drill a hole in a concrete wall with your nose, just to prove you can! If you succeed, I’ll be impressed - but I’ll still wonder why you didn’t put all that energy into something more interesting.

Now my attitude has changed, because I’m beginning to see that behind these silly hard problems there lurks an actual theory, full of deep ideas and interesting links to other branches of mathematics, including mathematical physics. It just so happens that now and then this theory happens to crack another hard nut.”

Indeed: in particular, now that we’ve reached the section on Galois theory, I’ve developed a new appreciation for the subject (and the way it fits into everything else). The connection to physics is still somewhat fuzzy to me; but intuitively, it’s a beautiful way of looking at things. In particular, Baez’s post on the subject (where the above quote comes from) is a very nice introduction, as well as the first reference by Stewart that he cites.

Random 3am Thoughts

I typed this up pretty late last night, but I guess I never hit ‘post’. So…

• I just finished a 20-page lab report on using the Mössbauer effect to study the isomer shift and hyperfine splitting of ⁵⁷Fe, and I must say, it’s an incredible experiment. You move a thin foil of isotopically-enriched absorber (e.g. stainless steel/Fe metal) at extremely, extremely non-relativistic speeds of several millimeters per second relative to a ⁵⁷Co source. You record the number of 14.4keV gamma-rays detected. You analyze the heck out of your data. And somewhere along the way, somehow, you figure out how to deduce the magnitude of these splittings (sometimes as small as ~10^-9eV), among other things, from this mess of data. It works like magic, and apparently other people think so too (with applications from testing GR to looking at car exhaust/hemoglobin to the Mars Rover).

• Speaking of physics lab, I’ve just realized something: although classes, talks, homework, et cetera are fun (for very broad definitions of fun in some cases), I find that I’m happiest either when I’m actually doing an experiment, or trying to dissect my data. There’s this thrill involved with trying to learn something new about a system by playing with it, probing it, trying to force it to reveal its secrets in a systematic and careful manner; and either actually learning something new about it, or more commonly finding out why your approach is flawed. But debugging an experiment or trying to find a nugget of signal in a sea of noise can be fun, too. It’s like being a detective (cliché, I know, but it’s true). I really love experimental science, and I’m lucky because all of the experiments I get to do for this modern physics lab class are really beautiful. I’ve learned a ton from this class - and not just physics, too, but things like statistics, or more methodological things like really thinking things through and being careful and systematic. Which is perhaps why this post (via Chad Orzel) pissed me off, although I have better things to do than rant about it.

• And speaking of things of a curricular nature, I really, really hate in-class midterms for upper-level classes (the ones that you actually have to think deeply about). It just doesn’t make sense to compress the thought process involved in solving problem sets (an intense process of deep thought, trial, error, et cetera spread out over a week) into an hour-long block, and I find that when faced with such a situation, I’m so scared of screwing up that, well, I screw up. If I were in charge, I’d give really, really hard take-home midterms, or something of that sort, I think.

• I recently ordered a poster-size version of the ‘map of science’ (from here for $10), featured in Nature several months ago. I’ve always had the desire to map out the sciences, particularly the ones I’m interested in. It always struck me as a kid how throughout the history of science, hot new fields always seemed to emerge by drawing connections between fields that otherwise hadn’t been connected, and if you had a map of it all, identifying places to draw new connections would be a breeze. I’m not sure if I think quite so simplistically anymore, but I still agree with the general philosophy to a certain extent. That being said, the people who constructed this ‘map’ of science did a lot of work, and it shows: the only way to actually read the thing is by squinting.

Quantum Information, et cetera

Today’s been a pretty exciting day. Among other things, we finally used all these abstract concepts relating to modules that we’ve been developing in my algebra class to derive some really neat results: in particular, rational canonical form and Jordan canonical form for matrices that can’t be diagonalized. There’s just something about taking all this seemingly useless theory and deriving something nice (and not so obvious) that you can actually use from it that’s very satisfying. Hey, I may even blog about it at some point. Another interesting math idea today was the subject of a colloquium that I wasn’t able to attend (but found out a good deal about from those who did): can we hear the shape of a drum? (Among other things, the question ties in with work done by our Dean, NASA’s WMAP project/this PRL, and the general notion of inverse problems such as those people deal with in things like MRI). And of course, my abstract algebra recitation session turned into me arguing with my TA (an algebraic geometer) and a computer scientist about why statistical mechanics is The Coolest Thing ever. What can I say? Never get me started on statistical mechanics - it’s just such a gorgeous subject, and I can’t get enough of it.

Anyway, while the math colloquium was going on, I was off at today’s physics colloquium by Prof. Charles Marcus of Harvard University, something I’ve been looking forward to for a good deal of time now. And what a talk it was: although I would have preferred more technical details, the talk catered to a pretty general audience, and it was perhaps the clearest physics talk I’ve been to in a long time. He started off by reviewing the history of computation (from the Antikythera mechanism of 150 BC to the first integrated circuit, 50-ish years ago), noting that quantum information processing is really a new paradigm in this history (to paraphrase, the parallelism in computation implied by the multiplicity of states inherent in quantum mechanics is something that hasn’t really been possible till now) and surveying recent developments in solid-state implementations of controllable qubits from his lab. Most of the relevant papers are on his lab webpage (linked above), and a lot of the technical details are presented in this talk he gave at KITP in 2006, although he did present some very recent data pertaining to this paper. All in all, a very cool talk - I especially like the terminology in this field, what with the ‘Zamboni’ effect and ‘bucket brigades’.

I can’t wait for next month’s colloquium - David Nelson will be speaking!

Magneto-what?

Today’s post is a quick one, since quantum mechanics and abstract algebra (in particular) and classes (in general) have kind of taken over my life. The subject is something that I got interested in about a year ago, in an advanced applied math class of all things: Magnetohydrodynamics, or MHD for short.

The basic idea behind MHD is relatively straightforward. One of the basic principles of electromagnetism is that electric and magnetic fields are intimately linked in a manner that is encapsulated quite elegantly in Maxwell’s equations; for example, a changing magnetic field can give rise to an electric field and induce a current in a conductor. One of the basic principles of fluid dynamics is that, well, fluids move in an interesting way, and the starting point for understanding this is the Navier-Stokes equations; for example, turbulence manifests itself in myriad ways, from smoke rising to ocean currents to the atmospheres of stars, and is still not very well understood.

Very well then: what happens when you have a charge (e.g. ionized) fluid whose dynamics are describable by the Navier-Stokes equations in the presence of a time-varying magnetic field? For example, the field affects the overall macroscopic motion of the fluid, which in turn gives rise to further electromagnetic effects since it is charged. How does one understand this kind of system? This question is the heart of MHD, a subject initiated by Hannes Alfven - for which he received the Nobel Prize in 1970 - and one that has been the subject of research in many, many fields, including applied mathematics, plasma physics, astrophysics, and geophysics.

The sun is an excellent example of a rich MHD system (the image above is from here, by the way): everything from sunspots to the solar wind to the Parker spiral involves magnetohydrodynamical effects. (That last one is particularly fascinating: did you know that the spiral shape of the Sun’s magnetic field “as it extends through the solar system… is similar to the pattern produced by a spinning lawn sprinkler, for similar reasons”? I didn’t before I came across MHD.) Another system that seems to be under quite intensive study at the moment is the magnetorotational insability (MRI), which is apparently a means of explaining anomalous viscosity in accretion discs.

Anyway, the mathematical details (which I had to work out for said applied math class and were surprisingly fun) get slightly complicated, but the key principles of ‘ideal’ MHD are that:

• the charged fluids are continuous, so one can ignore mean free path effects
• the charged fluids are perfect conductors and flow without drag
• thermal effects don’t matter
• the speeds under consideration are small enough to ignore relativistic effects

Now clearly these assumptions are whopping ones, and do indeed break down in many circumstances - but that’s ok. A good deal of hard work by a number of people has led to immense progress in understanding ‘non-ideal’ MHD, but it’s surprising how far you can get with these simplifying assumptions. One could, for example, start with the Navier-Stokes continuity (basically a way of saying that mass is conserved) and force (a way of saying that momentum is conserved) equations, plugging in expressions for electric and magnetic fields, and throwing in the energy equation (the first law of thermodynamics, which is a way of saying that energy is conserved) and the induction equation (one of Maxwell’s equations, which specifies how the magnetic field changes with time). But wait, we’re not done: to ‘close’ this system of equations, one can throw in two more equations: another of Maxwell’s equations (the divergence-free condition for the magnetic field to make sure you have no magnetic monopoles floating around) and the ideal gas equation of state. Turn the crank, and out pops a new field of study - MHD. So there you have it.

As an aside, I want to briefly point out that one of the reasons why this and other things of a fluid dynamics nature are of interest to applied mathematicians is because most of the time, one can’t solve these equations exactly and the analysis can be very tough (in math-speak, these equations constitute a nonlinear hyperbolic system, which means that finite-difference schemes tend to give solutions that blow up, among other things). Mathematicians can be quite good at coming up with ways of coping with this - for example, coming with means of simulating them numerically (such as Godunov’s scheme).