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).

Talks Part 1: Modeling Cells

I always enjoy attending good talks, and we’ve had quite a few lately: they’ve either been relevant to my current research, interesting applications of concepts I’ve encountered before, or just plain cool. (And of course, going to talks - or blogging, for that matter - is always a good way of taking a break from working all the time i.e. staying sane.) The material presented is always a good springboard for learning more. Here are a few summaries and references for the talks that stuck out the most, mainly because they dealt with things that I hadn’t directly encountered before.

Modelling Cell Motion and Morphogenesis: Mark Alber (Notre Dame)
Although this was billed as an applied math talk, it felt more like a physics talk: the speaker focused on trying to convey what was actually going on and less on the mathematical details involved in modeling it, which was refreshing. I’ve been to too many math talks where the speaker gets hung up on mathematical details, and the big picture somehow gets left out. Anyway, Prof. Alber is an applied mathematician at Notre Dame who spends a good deal of time modeling cells and biological processes at various scales (hence the term multiscale modeling) using ideas from statistical mechanics. The point is this: different modeling methods have their advantages and disadvantages. For example, macroscopic continuum methods abstract away many (often crucial) things and can miss different kinds of phenomena, while microscopic cell-level models - in which stochasticity is very important - can be computationally very intensive. This is particularly important in biological systems, where important processes take place at pretty much all scales; ideally, one would be able to construct a 3-dimensional model of a system and be able to zoom in and out with ease. This is what Alber et al. have been working on, in two different ways.

The first method is a 3D stochastic model of myxobacteria dynamics based on a lattice-gas cellular automata model, and using this, they’re able to study experimentally-observed phenomena like rippling, the formation of things like ’streams’ and ‘traffic jams’, and cell swarming/aggregation. The second method treats the cells as extended objects and goes off the philosophy that “while individual organisms and organs have very different structures and behaviors, many of the underlying interactions and components are the same.” In particular, it is an implementation of the Cellular Potts Model (CPM) of statistical mechanics, a non-equilibrium variant of the Ising model, coupling this to a continuum reaction-diffusion model for morphogen production/diffusion and a set of conditions dictating how genes are regulated. In particular, each cell is represented as a cluster of pixels in the CPM (with a multidimensional index indicating the type of cell) and interacts with other cells via a pairwise adhesion, and the cool thing is that they can use this to model - to a certain extent - limb formation in things like growing chicken embryos.

Further reading…
- Modeling myxobacteria dynamics: O. Sozinova et al., “A Three-Dimensional Model of Myxobacterial Aggregation by Contact-mediated Interactions“, PNAS 102 11308 (2005) and D. Kaiser, “Coupling Cell Movement to Multicellular Development in Myxobacteria“, Nature Reviews Microbiol. 1 45 (2003).
- Modeling limb formation: R. Chaturvedi et al., “On Multiscale Approaches to 3-Dimensional Modeling of Morphogenesis“, J. R. Soc. Interface 2 237 (2005).
- Somewhat related: D. A. Beysens et al., “Cell Sorting is Analogous to Phase Ordering in Fluids“, PNAS 97 9467 (2000) and W. Zeng et al., “Non-Turing Stripes and Spots: a Novel Mechanism for Biological Cell Clustering“, Physica A 341 482 (2004). In particular, I find the similarities between figure 2 of the latter reference and this picture of stripe formation in ferrofluids in one of my previous posts on electronic liquid crystals are striking.

High-Energy Materials Science

Someone I know recently asked me about this recent New Scientist article, and honestly, I’m not sure what to make of it. To summarize the article, ‘string-net condensation‘ (a model in which “electrons are not really elementary, but are formed at the ends of long ’strings’ of other, fundamental particles”) predicts interesting new phases of matter in certain spin models, and recent experiments on the amusingly-named Herbertsmithite may be a signature of one such phase of matter.

The subject matter is definitely interesting, although I am in no position to comment on the actual science (especially the theory, because I don’t understand it - my very little exposure to renormalization group theory so far has been in the context of statistical mechanics). On the experimental side, there is no doubt that recent work studying material properties of this system (such as looking at the specific heat or the temperature-dependence of the magnetic susceptibility, as well as using inelastic neutron scattering) is yielding some very cool physics. And of course on the theory side of things, it is clear that being able to come up with a unified theory from which electrons and photons are emergent is a Big Deal. The problem I have is that while the New Scientist article makes it sound like these measurements are a clear signature of a new phase of matter as predicted by string-net condensation, I can’t really discern the degree to which the link between the theory and experiments is scientifically rigorous, at least from reading the relevant papers. Rather, the article strikes me as being yet another example of really bad sensationalist journalism (see this post by Doug Natelson for more).

However, I find this to be a nice example of how things like high-energy physics theories of string-net condensation or quantum electrodynamics are becoming increasingly important in the study of materials. The ‘hot’ material for this kind of thing these days is obviously graphene, the “new bridge between condensed matter physics and quantum electrodynamics“. Another surprising example of this, for example, is this recent paper connecting SU(2) Yang-Mills theory in the low-temperature phase to nematic liquid crystals (who’d have thought? Certainly not me - I found it amusing that the only equations in the paper I actually understood were 18-20, the ones relating to liquid crystals).

And of course, this Herbertsmithite stuff is an example of another big thing in condensed matter physics - namely, ‘discovering’ new states of matter (at least theoretically). A lot of interesting physics is coming out of this effort, such as this recent work by Shoucheng Zhang at Stanford.

Tonks-Girardeau Gas

I had the chance to kick back and read a few papers over spring break, including a number of experimental biophysics and atomic/molecule/optical (AMO) physics papers for well-roundedness. Some dealt with the Tonks-Girardeau gas, something I first encountered in my liquid crystals class last semester (I’m taking a few of the equations below from my notes from that class). Among other things, this system is interesting because it was one of the first exactly-solvable systems in statistical mechanics, of which there still remain very few (the Ising model in 1D/2D being another). From a soft condensed-matter physics point of view, the Tonks gas serves as a good starting point for exploring the notion of excluded volume, a surprisingly important concept (for example, leading to ideas like the depletion attraction and entropic attraction/organization - see this book for a nice introduction.)

Working at the research laboratory of the General Electric company, John Bradshaw Taylor and Irvin Langmuir spent a good deal of time in the early 30’s developing very precise means of determining the number of caesium atoms adsorbed on tungsten and using these to study the properties of such monoatomic films. Motivated by this, Lewi Tonks derived the equation of state not just for a two-dimensional gas, but for a one-dimensional and three-dimensional gas as well, ignoring the nature of the forces between them and treating the simplest case of hard elastic spheres. (Twenty-four years later, Marvin Girardeau established a rigorous one-to-one correspondence between this system and a 1D system of spinless fermions. This ‘fermionization’ helps explain why such a system of bosons doesn’t undergo condensation.)

For example, one can calculate the equation of state for a dilute gas of strongly-interacting impenetrable bosons confined to the one dimensional space quite simply - since we’re treating the particles as hard spheres of radius , the associated potential is given by for and for . Then the -sphere partition function is just:

(There’s a constant prefactor involving and what Kittel & Kroemer call the quantum concentration, but it’s not important here since we’ll be taking derivatives.) A slick way of solving this is by changing variables to :

and since and , where in this case the ‘volume’ , one finds that

It is interesting to note that including a simplistic two-body attractive interaction regains the van der Waals equation of state exactly (see for example section 4.9 of Mattis’ somewhat misleadingly-titled book on statistical mechanics). This can also be extended to three dimensions by expanding the expression for P and considering the virial expansion, and a lot of interesting concepts, like the depletion interaction, fall out very nicely (see the first chapter of the soft condensed-matter physics book I mentioned above).

Another soft-matter application of the Tonks gas that I came across a while back is this paper by Tom Chou at UCLA. He’s developed an exact one-dimensional theory of histone adsorption and wrapping on DNA by considering each histone as a Tonks gas particle, which I think is pretty neat; after all, histones are pretty much hard spheres confined to a line of sorts - as wikipedia puts it, they “act as spools around which DNA winds”.

(This model can be theoretically extended in other ways, too. For example, Takahashi added in the effect of an arbitrary bounded interaction potential between going from the zero potential to the infinite hard-core potential; see another book by Mattis for more on this.)

But the Tonks gas is important in other areas of physics, as well - in particular, in the study of atomic gases at low temperature and particle density. During the summer of 2004, two groups published papers detailing their experiments using ultracold rubidium-87 atoms trapped using optical lattices in which they observed a transition to the strongly-correlated 1D Tonks gas regime (although they verified this in different ways: the Penn State group studied the energy and size of their system while the European group looked at the momentum distribution, both comparing their results to relevant theoretical predictions). And the research has been continuing ever since: for example, the Penn State group has extended these measurements to the study of a quantum Newton’s cradle, and other groups have studied 1D Bose gases as both Mott insulators and Luttinger liquids (the latter being a particularly nice connection to carbon nanotubes).