I gave the Bodian Seminar at the Zanvyl Krieger Mind/Brain Institute of Johns Hopkins today. I talked about cortical dynamics in the presence of conflicting stimuli. My slides are here. A summary of part of my talk can be found here. Other pertinent papers can be found here and here.
Archive for the ‘Computational neuroscience’ Category
Michael Buice and I have just published a review paper of our work on how to go beyond mean field theory for systems of coupled neurons. The paper can be obtained here. Michael and I actually pursued two lines of thought on how to go beyond mean field theory and we show how the two are related in this review. The first line started in trying to understand how to create a dynamic statistical theory of a high dimensional fully deterministic system. We first applied the method to the Kuramoto system of coupled oscillators but the formalism could apply to any system. Our recent paper in PLoS Computational Biology was an application for a network of synaptically coupled spiking neurons. I’ve written about this work multiple times (e.g. here, here, and here). In this series of papers, we looked at how you can compute fluctuations around the infinite system size limit, which defines mean field theory for the system, when you have a finite number of neurons. We used the inverse number of neurons as a perturbative expansion parameter but the formalism could be generalized to expand in any small parameter, such as the inverse of a slow time scale.
The second line of thought was with regards to the question of how to generalize the Wilson-Cowan equation, which is a phenomenological population activity equation for a set of neurons, which I summarized here. That paper built upon the work that Michael had started in his PhD thesis with Jack Cowan. The Wilson-Cowan equation is a mean field theory of some system but it does not specify what that system is. Michael considered the variable in the Wilson-Cowan equation to be the rate (stochastic intensity) of a Poisson process and prescribed a microscopic stochastic system, dubbed the spike model, that was consistent with the Wilson-Cowan equation. He then considered deviations away from pure Poisson statistics. The expansion parameter in this case was more obscure. Away from a bifurcation (i.e. critical point) the statistics of firing would be pure Poisson but they would deviate near the critical point, so the small parameter was the inverse distance to criticality. Michael, Jack and I then derived a set of self-consistent set of equations for the mean rate and rate correlations that generalized the Wilson-Cowan equation.
The unifying theme of both approaches is that these systems can be described by either a hierarchy of moment equations or equivalently as a functional or path integral. This all boils down to the fact that any stochastic system is equivalently described by a distribution function or the moments of the distribution. Generally, it is impossible to explicitly calculate or compute these quantities but one can apply perturbation theory to extract meaningful quantities. For a path integral, this involves using Laplace’s method or the method of steepest descents to approximate an integral and in the moment hierarchy method it involves finding ways to truncate or close the system. These methods are also directly related to WKB expansion, but I’ll leave that connection to another post.
The biggest news for neuroscientists in President Obama’s State of the Union Address was the announcement of the Brain Activity Map (BAM) project (e.g. see here and here). The goal of this project as outlined in this Neuron paper is to develop the technological capability to measure the spiking activity of every single neuron in the brain simultaneously. I used to fantasize about such a project a decade ago but now I’m more ambivalent. Although the details of the project have not been announced, people involved are hoping for 300 million dollars per year for ten years. I do believe that a lot will be learned in pursuing such a project but it may also divert resources for neuroscience towards this one goal. Given that the project is mostly technological, it may also mostly bring in new engineers and physicists to neuroscience rather than fund current labs. It could be a huge boon for computational neuroscience because the amount of data that will be recorded will be enormous. It will take a lot of effort just to curate this data much less try to analyze and makes sense of it. Finally, on a cautionary note, it could be that much of the data will be superfluous. After all, we understand how gases behave (at least enough to design refrigerators and airplanes, etc.) without measuring the positions and velocities of every molecule in a room. I’m not sure we would have figured out the ideal gas law, the Carnot cycle, or the three laws of thermodynamics if we just relied on an “Air Activity Map Project” a century ago. There is probably a lot of compression going on in the brain. If we knew how this compression worked, we could then just measure the nonredundant information. That would certainly make the BAM project a whole lot easier.
Michael Buice and I have finally published our paper entitled “Dynamic finite size effects in spiking neural networks” in PLoS Computational Biology (link here). Finishing this paper seemed like a Sisyphean ordeal and it is only the first of a series of papers that we hope to eventually publish. This paper outlines a systematic perturbative formalism to compute fluctuations and correlations in a coupled network of a finite but large number of spiking neurons. The formalism borrows heavily from the kinetic theory of plasmas and statistical field theory and is similar to what we used in our previous work on the Kuramoto model (see here and here) and the “Spike model” (see here). Our heuristic paper on path integral methods is here. Some recent talks and summaries can be found here and here.
The NAND (Not AND) gate is all you need to build a universal computer. In other words, any computation that can be done by your desktop computer, can be accomplished by some combination of NAND gates. If you believe the brain is computable (i.e. can be simulated by a computer) then in principle, this is all you need to construct a brain. There are multiple ways to build a NAND gate out of neuro-wetware. A simple example takes just two neurons. A single neuron can act as an AND gate by having a spiking threshold high enough such that two simultaneous synaptic events are required for it to fire. This neuron then inhibits the second neuron that is always active except when the first neuron receives two simultaneous inputs and fires. A network of these NAND circuits can do any computation a brain can do. In this sense, we already have all the elementary components necessary to construct a brain. What we do not know is how to put these circuits together. We do not know how to do this by hand nor with a learning rule so that a network of neurons could wire itself. However, it could be that the currently known neural plasticity mechanisms like spike-timing dependent plasticity are sufficient to create a functioning brain. Such a brain may be very different from our brains but it would be a brain nonetheless.
The fact that there are an infinite number of ways to creating a NAND gate out of neuro-wetware implies that there are an infinite number of ways of creating a brain. You could take two neural networks with the same set of neurons and learning rules, expose them to the same set of stimuli and end up with completely different brains. They could have the same capabilities but be wired differently. The brain could be highly sensitive to initial conditions and noise so any minor perturbation would lead to an exponential divergence in outcomes. There might be some regularities (like scaling laws) in the connections that could be deduced but the exact connections would be different. If this were true then the connections would be everything and nothing. They would be so intricately correlated that only if taken together would they make sense. Knowing some of the connections would be useless. The real brain is probably not this extreme since we can sustain severe injuries to the brain and still function. However, the total number of hard-wired conserved connections cannot exceed the number of bits in the genome. The other connections (which is almost all of them) are either learned or are random. We do not know which is which.
To clarify my position on the Hopfield Hypothesis, I think we may already know enough to create a brain but we do not know enough to understand our brain. This distinction is crucial. What my lab has been interested in lately is to understand and discover new treatments for cognitive disorders like Autism (e.g. see here). This implies that we need to know how perturbations at the cellular and molecular levels affect the behavioural level. This is an obviously daunting task. Our hypothesis is that the bridge between these two extremes is the canonical cortical circuit consisting of recurrent excitation and lateral inhibition. We and others have shown that such a simple circuit can explain the neural firing dynamics in diverse tasks such as working memory and binocular rivalry (e.g. see here). The hope is that we can connect the genetic and molecular perturbations to the circuit dynamics and then connect the circuit dynamics to behavior. In this sense, we can circumvent the really hard problem of how the canonical circuits are connected to each other. This may not lead to a complete understanding of the brain or the ability to treat all disorders but it may give insights into how genes and medication act on cognitive function.
SPAUN (Semantic Pointer Architecture Unified Network) is a model of a functioning brain out of Chris Eliasmith’s group at the University of Waterloo. I first met Chris almost 15 years ago when I visited Charlie Anderson at Washington University, where Chris was a graduate student. He was actually in the philosophy department (and still is) with a decidedly mathematical inclination. SPAUN is described in Chris’s paper in Science (obtain here) and in a forthcoming book. SPAUN can perform 8 fairly diverse and challenging cognitive tasks using 2.5 million neurons with an architecture inspired by the brain. It takes input through visual images and responds by “drawing” with a simulated arm. It decodes images, extracts features and compresses them, stores them in memory, computes with them, and then translates the output into a motor action. It can count, copy, memorize, and do a Raven’s Progressive Matrices task. While it can’t learn novel tasks, it is pretty impressive.
However, what is most impressive to me about SPAUN is not how well it works but that it mostly implements known concepts from neuroscience and machine learning. The main newness was putting it all together. This harkens back to what I called the Hopfield Hypothesis, which is that we already know all the elementary pieces for neural functioning. What we don’t know is how they fit and work together. I think one of the problems in computational neuroscience is that we’re too timid. I first realized this many years ago when I saw a talk by roboticist Rodney Brooks. He showed us robots with very impressive capabilities (this was when he was still at MIT) that were just implementing well-known machine learning rules like back-propagation. I recall thinking that robotics was way ahead of us and that reverse engineering may be harder than engineering. I also think that we will likely construct a fully functioning brain before we understand it. It could be that if you connect enough neurons together that incorporate a set of necessary mechanisms and then expose it to the world, it would start to develop and learn cognitive capabilities. However, it would be as difficult to reverse engineer exactly what this constructed brain was doing as it is to reverse engineer a real brain. It may also be computationally undecidable or intractable to a priori determine the essential set of necessary mechanisms or the number of neurons you need. You might just have to cobble something together and try it out. A saving grace may be that these elements may not be unique. There could be a large family of mechanisms that you could draw from to create a thinking brain.
I’m giving a computational neuroscience lunch seminar today at Johns Hopkins. I will be talking about my work with Michael Buice, now at the Allen Institute, on how to go beyond mean field theory in neural networks. Technically, I will present our recent work on computing correlations in a network of coupled neurons systematically with a controlled perturbation expansion around the inverse network size. The method uses ideas from kinetic theory with a path integral construction borrowed and adapted by Michael from nonequilibrium statistical mechanics. The talk is similar to the one I gave at MBI in October. Our paper on this topic will appear soon in PLoS Computational Biology. The slides can be found here.
I’m currently at the Mathematical Biosciences Institute at The Ohio State University at a workshop on Mathematical Challenges in Neural Network Dynamics. My slides are here.
A new paper by Steve Gotts, myself, and Alex Martin has officially been published in the journal Cognitive Neuroscience:
Stephen J. Gotts, Carson C. Chow & Alex Martin (2012): Repetition priming and repetition suppression: Multiple mechanisms in need of testing, Cognitive Neuroscience, 3:3-4, 250-259 [PDF]
This paper is a review of the topic but is partially based on the PhD thesis work of Steve Gotts when we were both in Pittsburgh over a decade ago. Steve was a CNBC graduate student at Carnegie Mellon University and came to visit me one day to tell me about his research project to reconcile the psychological phenomenon of repetition priming with a neurophysiological phenomenon called repetition suppression. It is well known that performance improves when you repeat a task. For example, you will respond faster to words on a random list if you have seen the word before. This is called repetition priming. The priming effect can occur over time scales as short as a few seconds to your life time. Steve was focused on the short time effect. A naive explanation for why you would respond faster to priming is that the pool of neurons that code for the word become slightly more active so when the word reappears they fire more readily. This hypothesis could only be tested when electrophysiological recordings of cells in awake behaving monkeys and functional magnetic resonance imaging data in humans finally became available in the mid-nineties. As is often the case in science, the opposite was observed. Neural responses actually decreased and this was called repetition suppression. So an interesting question arose: How do you get priming with suppression? Steve had a hypothesis and it involved work I had done so he came to see if I wanted to collaborate.
I joined the math department at Pitt in the fall of 1998 (the webpage has a nice picture of Bard Ermentrout, Rodica Curtu and Pranay Goel standing at a white board). I had just come from doing a post doc with Nancy Kopell at BU. At that time, the computational neuroscience community was interested in how a population of spiking neurons would become synchronous. The history of synchrony and coupled oscillators is long with many threads but I got into the game because of the weekly meetings Nancy organized at BU, which we dubbed “N-group”. People from all over the Boston area would participate. It was quite exciting at that time. One day Xiao Jing Wang, who was at Brandeis at the time, came to give a seminar on his joint work with Gyorgy Buzsaki on gamma oscillations in the hippocampus, which resulted in this highly cited paper. What the paper was really about was how inhibition could induce synchrony in a network with heterogeneous connections. It had already been shown by a number of people that a network with inhibitory synapses could synchronize a network of spiking neurons. This was somewhat counter intuitive because the conventional wisdom was that inhibition would lead to anti-synchrony. The key ingredient was that the inhibition had to be slow. Xiao Jing argued from his simulations that the hippocampus had a sweet spot for synchronization for the gamma band (i.e. frequencies around 40Hz). I was highly intrigued by his result and spent the next two years trying to understand the simulations mathematically. This resulted in four papers:
C.C. Chow, J.A. White, J. Ritt, and N. Kopell, `Frequency control in synchronized networks of inhibitory neurons’, J. Comp. Neurosci. 5, 407-420 (1998). [PDF]
J.A. White, C.C. Chow, J. Ritt, C. Soto-Trevino, and N. Kopell, `Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons’, J. Comp. Neurosci. 5, 5-16 (1998). [PDF]
C.C. Chow, `Phase-locking in weakly heterogeneous neuronal networks’, Physica D 118, 343-370 (1998). [PDF]
C.C. Chow and N. Kopell, `Dynamics of spiking neurons with electrical coupling’, Neural Comp. 12, 1643-1678 (2000). [PDF]
In a nutshell, these papers showed that in a heterogeneous network, neurons will tend to synchronize around the time scale of the synaptic inhibition, which in the case of the inhibitory neurotransmitter receptor GABA_A is around 25 ms or 40 Hz. When the firing frequency is too high the neurons tend to fire asynchronously and when the frequency is too slow, neurons tend to stop firing all together.
Steve read my papers (and practically everything else) and thought that this might be the resolution of his question. Now, it had also been known for a while that when neurons fire they tend to slow down. This is due to both spike-frequency adaptation and synaptic depression, so repetition suppression is not entirely surprising since when neurons are stimulated they will tend to fire slower. What is surprising is that slowing down makes you respond faster. Steve thought that maybe suppression synchronized neurons and made them more effective in getting downstream neurons to fire. In essence, what he needed to find was a mechanism that increases the gain of a neuron for a decrease in input and synchrony was a solution. I helped him work out some technical details and he wrote a very nice thesis showing how this could work and match the data. He then went on to work with Bob Desimone and Alex Martin at NIH. However, we never wrote the theoretical paper from his thesis because of a critique that we never got around to answering. The issue was that if a lowering of network frequency can elicit priming then why does a reduction in contrast in the primed stimulus, which also reduces network frequency, not do the same? This came up after Steve had left and I turned my attention to other things. The answer is probably because not all frequency reductions are equal. A reduction in contrast lowers the total input to the early part of the visual system while synaptic depression will have the largest effect on the most active neurons. The ensuing dynamics will likely be different but we never had the time to fully flesh this out. Although, I always wanted to get back to this, the project sat idle for me for about eight years until Steve sent me an email one day saying that he’s writing a review with Alex on the topic and wanted to know if I wanted to be included. I was delighted. The paper covers all the current theories for priming and suppression and is accompanied by commentaries from many of the key players in the field. I’ve just covered a small part of the many interesting issues brought up in the review.
It is not at all clear what technology will attain “human-level” intelligence first. Robin Hanson proposes brain emulation (e.g see here). I’ve been skeptical of emulation and am leaning towards machine learning (e.g. see here). However, given the recent technological advances of connectomics and 3D printing, brain emulation or rather replication might not be as distant as I thought. 3D printing is a technology to manufacture any 3 dimensional object by sequentially depositing 2 dimensional layers. You can find out more about it here including building your own 3D printer . People now regularly use open source software to take any object they may want, slice it into 2 dimensional layers then print it. The technology has reached the point where you can print with any material that can be squirted including biological material (see video here). People in the field are currently gearing up to print complete organs like kidneys and the liver. It is not overly far-fetched that they could print out an entire brain in the future. Recent progress in connectomics can be tracked here. The current state of the art involves taking electron microscope images of thin slices of neural tissue. The hard part is to reassemble these 2D slices back into a 3D brian, the reverse of 3D printing. However, perhaps what we can do is to 3D print the images first to obtain a faithful 3D reconstruction of the brain and then use the model to assist in the software reconstruction. If you had molecular level image resolution, you could even try to print out a functioning brain, complete with docked synaptic vesicles ready to be released!
Andrew Huxley, died last week at the age of 94. Huxley, with his research advisor Alan Hodgkin, proved that the mechanism for action potential propagation in nerve cells were due to the passage of ions through voltage gated ion channels in the cell membrane. In the course of their work, they developed the Hodgkin-Huxley model of the neuron and launched the field of computational neuroscience. While their work was a monumental achievement and deserved a Nobel prize, it still built upon the work of many others. I recommend reading the Nobel address of both Huxley (see here) and Hodgkin (see here) for the story of their discovery.
I’m currently at the New Jersey Institute of Technology for the ninth annual Frontiers in Applied and Computational Mathematics conference. Here are the slides for my talk. It’s on computational neuroscience and has nothing to do with obesity. Also, it only seems like lots of slides because of the animations.
A new paper in Physical Review E is now available on line here. In this paper Michael Buice and I show how you can derive an effective stochastic differential (Langevin) equation for a single element (e.g. neuron) embedded in a network by averaging over the unknown dynamics of the other elements. This then implies that given measurements from a single neuron, one might be able to infer properties of the network that it lives in. We hope to show this in the future. In this paper, we perform the calculation explicitly for the Kuramoto model of coupled oscillators (e.g. see here) but it can be generalized to any network of coupled elements. The calculation relies on the path or functional integral formalism Michael developed in his thesis and generalized at the NIH. It is a nice application of what is called “effective field theory”, where new dynamics (i.e. action) are obtained by marginalizing or integrating out unwanted degrees of freedom. The path integral formalism gives a nice platform to perform this averaging. The resulting Langevin equation has a noise term that is nonwhite, non-Gaussian and multiplicative. It is probably not something you would have guessed a priori.
| Michael A. Buice1,2 and Carson C. Chow11Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, Maryland 20892, USA
2Center for Learning and Memory, University of Texas at Austin, Austin, Texas, USA
Received 25 July 2011; revised 12 September 2011; published 17 November 2011
Complex systems are generally analytically intractable and difficult to simulate. We introduce a method for deriving an effective stochastic equation for a high-dimensional deterministic dynamical system for which some portion of the configuration is not precisely specified. We use a response function path integral to construct an equivalent distribution for the stochastic dynamics from the distribution of the incomplete information. We apply this method to the Kuramoto model of coupled oscillators to derive an effective stochastic equation for a single oscillator interacting with a bath of oscillators and also outline the procedure for other systems.
Published by the American Physical Society
I’m in beautiful Marseille again for a workshop on spike-timing dependent plasticity (STDP). My slides are here. The paper in which this talk is based can be obtained here. This paper greatly shaped how I think about neuroscience. I’ll give a summary of the paper and STDP for the uninitiated later.
Erratum: In my talk I said that I had reduced the models to disjunctive normal form. Actually, I had it backwards. I reduced it to conjunctive normal form. I’ll attribute this mixup to jet lag and lack of sleep.
I just returned from an excellent meeting in Marseille. I was quite impressed by the quality of talks, both in content and exposition. My talk may have been the least effective in that it provoked no questions. Although I don’t think it was a bad talk per se, I did fail to connect with the audience. I kind of made the classic mistake of not knowing my audience. My talk was about how to extend a previous formalism that much of the audience was unfamiliar with. Hence, they had no idea why it was interesting or useful. The workshop was on mean field methods in neuroscience and my talk was on how to make finite size corrections to classical mean field results. The problem is that many of the participants of the workshop don’t use or know these methods. The field has basically moved on.
In the classical view, the mean field limit is one where the discreteness of the system has been averaged away and thus there are no fluctuations or correlations. I have been struggling over the past decade trying to figure out how to estimate finite system size corrections to mean field. This led to my work on the Kuramoto model with Eric Hildebrand and particularly Michael Buice. Michael and I have now extended the method to synaptically coupled neuron models. However, to this audience, mean field pertains more to what is known as the “balanced state”. This is the idea put forth by Carl van Vreeswijk and Haim Sompolinsky to explain why the brain seems so noisy. In classical mean field theory, the interactions are scaled by the number of neurons N so in the limit of N going to infinity the effect of any single neuron on the population is zero. Thus, there are no fluctuations or correlations. However in the balanced state the interactions are scaled by the square root of the number of neurons so in the mean field limit the fluctuations do not disappear. The brilliant stroke of insight by Carl and Haim was that a self consistent solution to such a situation is where the excitatory and inhibitory neurons balance exactly so the net mean activity in the network is zero but the fluctuations are not. In some sense, this is the inverse of the classical notion. Maybe it should have been called “variance field theory”. The nice thing about the balanced state is that it is a stable fixed point and no further tuning of parameters is required. Of course the scaling choice is still a form of tuning but it is not detailed tuning.
Hence, to the younger generation of theorists in the audience, mean field theory already has fluctuations. Finite size corrections don’t seem that important. It may actually indicate the success of the field because in the past most computational neuroscientists were trained in either physics or mathematics and mean field theory would have the meaning it has in statistical mechanics. The current generation has been completely trained in computational neuroscience with it’s own canon of common knowledge. I should say that my talk wasn’t a complete failure. It did seem to stir up interest in learning the field theory methods we have developed as people did recognize it provides a very useful tool to solve the problems they are interested in.
Here are some links to previous posts that pertain to the comments above.
J Neurophysiol. 2011 Jul 20. [Epub ahead of print] The role of mutual inhibition in binocular rivalry. Seely J, Chow CC. Binocular rivalry is a phenomenon that occurs when ambiguous images are presented to each of the eyes. The observer generally perceives just one image at a time, with perceptual switches occurring every few seconds. A natural assumption is that this perceptual mutual exclusivity is achieved via mutual inhibition between populations of neurons that encode for either percept. Theoretical models that incorporate mutual inhibition have been largely successful at capturing experimental features of rivalry, including Levelt's propositions, which characterize perceptual dominance durations as a function of image contrasts. However, basic mutual inhibition models do not fully comply with Levelt's fourth proposition, which states that percepts alternate faster as the stimulus contrasts to both eyes are increased simultaneously. This theory-experiment discrepancy has been taken as evidence against the role of mutual inhibition for binocular rivalry. Here, we show how various biophysically plausible modifications to mutual inhibition models can resolve this problem. PMID: 21775721 [PubMed - as supplied by publisher] Paper can be downloaded here.
Each second you’re taking in perhaps a million bits of information. See here for the estimate. Most of those bits don’t affect your brain or your life at all. The information you take in just walking down the street is immense yet you probably ignore most of it. However, a single bit of information can change your entire life. A blood test that indicates that your LDL concentration is too high can cause you to change what you eat and how much you exercise. A single yes or no answer can change your mood for a day or the rest of your life. You could imagine how such sensitivity would arise if the bit were anticipated as when you wait for an answer. The really interesting problem is how we are able to react to unanticipated bits of great importance such as when you feel your chair suddenly shaking. Our brains are shaped to find those needles in the haystack.