Structural and Functional Connectomics

These are preparatory thoughts and draft slides for a lecture on connectomics and functional circuit analysis in Bruno Olshausen’s Neural Computation (VS265) course at the University of California Berkeley on October 16, 2014. The early portion is dedicated to motivating students to devote a considerable part of their university education to learning the necessary foundational tools required to navigate in the rapidly evolving field of neuroscience. The latter part draws upon our collaboration with the Allen Institute for Brain Science focusing first on how the data is acquired and why it might be important to adapt existing or develop new tools to assist in data acquisition, and, second, on the challenges involved in learning models of the visual cortex from the mouse data collected by Clay Reid’s team at the Allen Institute.

Introduction: Some unsolicited recommendations

I’m going to begin with what at first blush may seem like an odd question in this context: What should you study to be prepared for a career in neuroscience in the 21^st Century? I maintain that, with rare exception, you probably shouldn’t rely on simply copying your adviser’s curriculum—unless your adviser is Larry Abbott, Bill Bialek, Christof Koch, Bruno Olshausen, Tommy Poggio, Clay Reid or any of a number of other scientists with similar pedigree, but rather consider the trajectory of several famous scientists who started out in physics before turning their attention to neurobiology, e.g., Francis Crick, Leon Cooper, Alan Hodgkin, John von Neumann, Erwin Schrö̈dinger and Leó Szilárd—all of whom just happen to have been awarded a Nobel Prize or should have been in my humble opinion.

Here are some subjects I would recommend that you master early in your education: differential, integral and multivariate calculus, probability theory including stochastic processes, statistics, ordinary and partial differential equations, linear algebra, graph theory and combinatorics. They’re all math you say? Whether you’re an experimentalist building tools to study the brain or a theorist trying to explain the data produced by such tools so as to construct models, math is essential to understanding the brain.

Mathematics provides a wide range of explanatory and predictive models, but you have to know how to apply those models to physical phenomena. How does a budding neuroscientist acquire such know-how if the field of neuroscience doesn’t offer a rich portfolio of mathematical models? The answer is that you don’t have to rely on neuroscience as your sole source of models. Physics is the best source of readily applicable models and there plenty of books featuring examples of how they are derived and applied [12].

And so I would add to the list of subjects worthy of study early in your education, physics and in particular classical physics including mechanics, optics, thermodynamics and electromagnetic theory. Mechanics may seem irrelevant given the scale of interactions we observe in the brain, but, mechanics is the subject in which the scientific method and the role of mathematical models are first explored, and so I suggest you bear with it and identify with Archimedes, Galileo and Newton.

Here’s a test of your basic understanding of electromagnetic theory, and, in particular, its role in reactions similar to those responsible for electrochemical signalling in the brain: Explain how a rechargeable Lithium battery works and why they can cause fires. Start with the simpler case of a copper-cathode and a zinc-anode and your choice of electrolyte, and then move on to the Lithium case. Make sure you can precisely state what it means for a current to flow in a given direction¹. Give it a shot and then check your understanding here and here.

Quantum theory and optics play a role in such neurobiologically relevant technologies as optogenetics that makes it possible to excite and inhibit neurons with light and two-photon microscopy² which when combined with genetically encoded calcium indicators (GECIs) such as CCaMP is the basis for calcium imaging. Here’s an example of a relatively simple question that an engineer or scientist might need to answer in the process of adapting or applying these technologies: What is the minimum frequency required for a photon to eject an electron from a magnesium surface?³

As shining examples of how far one can go in the field of neurobiology equipped with a physics background, Ed Boyden—the lead author on the original 2005 optogenetics paper [4] published in Nature—and Winfried Denk—the lead author on the 1990 paper [8] appearing in Science first describing two-photon fluorescence microscopy—were both trained as physicists. In addition to their considerable scientific achievements, both men have patented numerous inventions that have significantly advanced the field of neurobiology⁴.

Continuing with my recommendations for advancing your scientific careers, I suggest you sample from organic chemistry⁵, cellular and molecular biology, and the cognitive and social sciences to better understand how the scientific method is applied in the practice of different fields and to study additional examples of how to model physical and psychological phenomena. Taking a course on the history of science or reading a biography of Charles Darwin or William James will help you appreciate the challenges faced in confronting a subject when starting from a meager or misleading basis of understanding.

Ernest Rutherford, then Lord Rutherford, once said that ‘‘all science is either physics or stamp collecting’’, a statement that did not win him any friends in the many disciplines besides physics in which individuals who think of themselves as scientists ply their trade. However, the statement is misleading in its representation of science. In particular, it assumes a ‘‘one size fits all’’ characterization of what constitutes a scientific theory and how best to proceed in doing science.

Physics has a relatively simple ontology when compared with the biological sciences in which there are a lot of names used to categorize organisms, their constituent parts and the complex chemical reactions that govern their behavior. This state of affairs will likely get worse before it gets better, but be thankful that you have Google and Wikipedia to augment your memory, and become facile in using these and other analytical and information retrieval tools in your studies.

I’m overstating the much touted purity and elegant simplicity of physics. As it is often taught in college, there tend to be fewer terms to memorize and a deeper mathematical basis to fall back on when intuition fails us, but things are messier than they might seem from their presentation in textbooks. For example, many molecules of interest are formed by covalent bonds that depend on one sort of electron sharing, but you soon learn that there are other common molecules, like table salt and sulphuric acid, that consist of atoms held together by ionic bonds.

Moreover, unless you read your textbook carefully, you might miss out on the fact that a lot of molecules are held together by a combination of covalent and ionic bonds, and there other types of bonds, for example metallic and hydrogen bonds, that might not even appear in your introductory text. For a geologist or metallurgist, these simple basic types of bonds combine to give rise to an astounding diversity of materials with interesting and useful properties.

If you really want to commit some raw facts to memory, I suggest that you master the top few rows of the periodic table. You’ll probably never have to venture much beyond Osmium (Os) (#76) and you’ll be in pretty good shape if you only master Hydrogen (H) (#1) through Calcium (Ca) (#20). As an exercise, think about why the periodic table is different from, say, the modern version of the taxonomy developed by Carl Linnaeus, which is different from a list of the names of all the organs or bones in the human body.

Finally, despite how computer science is often perceived by other disciplines—primarily by those whose training predates the present era of ubiquitous computing, open-source software and the Internet, I maintain that an understanding of algorithms, machine learning, signal processing, computational complexity and programming is essential to the education of a modern neuroscientist. Relative to the other disciplines mentioned above, the importance of computer science will increase as the technologies for extracting information from the brain improve.

There are two reasons for the importance of computer science that we’ll explore in this lecture: (a) the amount of raw data required to understand many of the relevant phenomena in brain science defies direct human understanding unaided by computer analyses and the means for searching, sifting, factoring and fitting models, and (b) the complexity of the computations that give rise to cognitive phenomena is such that often the clearest explanation is an algorithm coupled with an implementation of the algorithm that simulates the phenomena of interest and thereby makes predictions that can be verified by experiment.

Explanations: Different principles dominate at different scales

Richard Feynman described how you might build half-scale versions of the sort of machine tools you would find in a well equipped machine shop, then use those machines to manufacture tiny half-scale versions of those, and so on down to the tiniest machines that could manipulate individual atoms and assemble complex molecules. It did not escape him that nature had already succeeded in this endeavor, albeit starting at the smallest and working its way up to the scale we are most familiar with.

Feynman provided insights into the challenges faced by natural and man-made tiny machines, describing how, as you descend to smaller and smaller scales, different physical laws dominate. For example E. Coli use corkscrew-shaped flagella and molecule-sized motors to propel themselves through a watery fluid which is for them a viscous medium as thick as molasses. For organisms operating at millimeter scales, surface tension is an important consideration; at sub nanometer scales, Van der Waals force—the sum of the attractive / repulsive forces between molecules other than those due to covalent bonds—starts to play a key role.

Let’s follow the steps involved in how a mosquito causes West Nile Virus or Equine Encephalitis in a human host. You’re familiar with the mostly Newtonian forces that dominate at the scale or your everyday perception of the natural world—meters or kilometers serve to measure size or distance on a gross scale and centimeters work fine for most of the things we handle. The mosquito landing on your arm in the tropics is typically less than a centimeter and its larger features can be measured in millimeters. Smaller features like its hypodermic-like proboscis are best measured in microns.

The parasites that cause malaria are around 10 microns at the stage in their life cycle when they are transferred to a human host by a mosquito bite—a red blood cell is about 8 microns. For these one-celled organisms, a droplet of water is an ocean and diffusion is a fast and efficient means of transport. Viruses like those that cause West Nile Fever or Equine Encephalitis are on the order of 40 nanometers and their structure can be seen clearly using transmission electron microscopy with samples at cryogenic temperatures—see here.

Viruses are essentially capsules of RNA or DNA that encode their genetic information. They don’t have the machinery required to replicate and so they rely on injecting their genetic information into unwilling hosts, and then hijacking the machinery in the cell nucleus to reproduce so many copies of themselves that the host membrane eventually ruptures spilling even more virus into the extracellular fluid.

At this scale, the molecular machinery includes enzymes that enable reactions and complex molecules that assemble proteins and convert DNA into RNA (called RNA polymerase) and RNA into DNA (called DNA polymerase), one sort of which is called reverse transcriptase and is necessary in order for a virus to insert its genetic code into that of its host, thus making the host create many copies of the virus as a side effect of its normal cellular maintenance.

Our cells also perform quantum machinations on a routine basis. Rhodopsins found in our retinas and throughout nature convert photons into changes in membrane potential in a process called transduction that involves a cascade of electrical, chemical and even mechanical signals. Rhodopsins are the molecular basis for optogenetics. In this case, we are talking about interactions occurring at the scale of femtometers, and bioengineers and nanotechnologists building to tools for biologists to use at this scale have to think in terms of processes that play out in time measured as femtoseconds⁶.

The Experimentalists: Plus a couple of token theoreticians

Michael Faraday 1791-1867
Faraday’s inventions including the electric motor (1821).

James Clerk Maxwell 1831-1879
Oliver Heaviside 1850-1950
Carl Friedrich Gauss 1777–1855
Maxwell’s equations as simplified by Heaviside (1861).

Ernest Rutherford 1871-1937
Rutherford’s lab where he developed his atomic model (1911).

Ernst Ruska 1906–1988
Max Knoll 1897-1969
Early transmission electron microscope developed in (1931).

Alan Lloyd Hodgkin 1914–1998
Andrew Huxley 1917-2012
Equivalent circuit model for the Hogkin-Huxley model.
Equipment used in developing action-potential model (1952).

David Hubel 1925-2013
Torsten Wiesel 1924-
Experimental apparatus to study the cat visual cortex (1959).

Herman Carr 1924-2008
Raymond Damadian 1936-
Early magnetic resonance device used in first human scan (1971).

Scientists and engineers working on the Large Hadron Collider.
View down the 27km circular tunnel enclosing the LHC accelerator.

Ed Boyden 1979-
Karl Deisseroth 1971-
Rhodopsin in rod outer segment discs of the retina.
Channelrhodopsin optogenetic excitation and inhibition (2005).

Breaking the ‘‘Laws’’ of Physics: ‘‘No’’ is not an acceptable answer

I’ve yet to see an experimentalist who was satisfied with the technologies he or she could purchase off the shelf. They’ll often buy one off the shelf, e.g., an electron microscope from Zeiss or an fMRI machine from Siemens or GE, but it isn’t long before they have the cover off and are tinkering with the innards. Most experimentalists are wary of ‘‘laws’’ and ‘‘fundamental limits’’ as they often as not turn out to be misleading or flat out wrong. A good example is the diffraction limit for the resolution of visible light microscopy.⁷.

Conventional light microscopy resolution is inversely proportional to the wavelength of light used to illuminate the target and so light microscopy resolution is limited to a few hundred nanometers, e.g., visible-light microscope with high numerical aperture can achieve resolution of 250 nm. Super-resolution imaging beats the diffraction limit by either exploiting other physical principles, e.g., capturing the information in evanescent waves, or by using multiple images and computation to achieve sub-pixel resolution. For example, STORM — provide 20 nm in the lateral dimensions and 50 nm in the axial dimension, whereas LIMON — in biological applications offer 25 nm resolution in all three dimensions.

Mark Schnitzer and researchers in his lab at Stanford have built a miniature fluorescent microscope that can be surgically attached to the head of a mouse allowing the animal to move about freely while simultaneously recording from hundreds of neurons using calcium imaging⁸ (VIDEO). The 1.9 g camera offers a wide field of view compared with other technologies and introduces fewer motion artifacts than two-photon microscopy in awake, head-fixed mice responding to experimental stimuli.

In a related application of computational photography, a team led by Alipasha Vaziri and Ed Boyden [28] have demonstrated how to image the 302 neurons in an awake, behaving nematode—C. Elegans, the only organism for which we have a complete connectome [30]—using light-field microscopy in combination with 3-D deconvolution.

Researchers at HHMI Janelia showed (VIDEO) it is possible to record the behavior of large populations of neurons in a zebrafish by exploiting the fact that this organism is almost entirely transparent in its larval stage. To accomplish this they used light-sheet microscopy in which the illumination laser is focused with a cylindrical lens to create a cross-sectional light sheet which is then scanned over the entire volume. Variants that involve alternative illumination strategies coordinated with microscope-objective focus provide precise depth-of-field control and minimize fluorophore bleaching thereby lengthening recording time⁹.

For an excellent overview of the current state of the art in two-photon imaging and its application in studying synaptic plasticity, you might check out Karel Svoboda from HHMI Janelia describing the recent advances in optical imaging of individual synapses (VIDEO) and the application of these techniques to studying plasticity and signaling in single synapses (VIDEO).

In these lectures, you’ll also learn about a method of activating molecules such as glutamate that stimulate learning and signalling with light which is called photo activation or more commonly the shorter but less intuitive term uncaging referring to the technique of first inactivating (caging) the molecule you want to control by attaching an additional molecular group, and then subsequently activating (uncaging) the molecule with light using techniques similar to those used in optogenetics.

There are several types of electron microscope neuroscientsts can choose from for imaging biological targets. Here’s a quick summary of the different types of electron microscopes and their resolving power:

• SEM — area square millimeters, depth of field millimeters, resolution of nanometers;

• AFM — area 150 × 150 micrometers, depth of 10-20 micrometers, sub-nanometer resolution¹⁰;

• STM — scanning tunneling — 0.1 nm lateral resolution and 0.01 nm depth resolution;

• TEM — transmission electron — resolution of a few nanometers — area similar to SEM;

• FIB — focused ion beam — resolution of 5 nm with the added benefit that the 3-D volume of the specimen can be examined without using a conventional microtome by precisely ablating top surface of the specimen thereby achieving higher resolution in the z-axis;

Scientists at HHMI Janelia Farm Campus have been experimenting with the FIB technologies. Clay Reid and his colleagues at Harvard modified their TEM to increase the field of view so it could be used for connectomics. As I describe in the lecture, their modifications involved a significant engineering effort. Here is a list of some of the scientists working at the forefront of micro-scale connectomics:

Davi Bock, HHMI Janelia Farm Campus
Winfried Denk, Max Planck, Tübingen
Josh Lichtman, Harvard University
Clay Reid, Allen Institute for Brain Science
Sebastian Seung, Princeton University

There is more to getting good electron micrographs than picking the right microscope. The tissue has to be carefully prepared in a process that involves several time-consuming steps and a number of chemicals many of which are very toxic. First the tissue is fixed using formaldehyde or other chemical fixatives to preserve the tissue from degradation, and maintain the structure of the cell and its sub-cellular components. Next it is dehydrated and the water replaced with a supporting matrix like paraffin or epoxy resins. Finally the tissue is stained, embedded in a block often of the same material used for the supporting matrix and mounted in the device used for thin sectioning. Here’s a manual describing protocols for preparing tissue samples for electron microscopy.

Osmium-based stains are common and have been used for many decades to stain the lipids that make up the membranes of cell bodies, neuropil and their respective organelles. There is some recent work in using fluorescent probes to image lipids, but there remains a lot of room for developing better stains. It was big news the Kwanghun Chung et al [6] paper appeared in Nature describing CLARITY a tissue preparation technology that renders the tissue transparent to visible light. CLARITY isn’t the only option for clarifying biological samples, e.g., the Clear^T technology described in Kuwajima et al [19] was published almost simultaneously with the CLARITY paper albeit in a much less prestigious venue.. Here is a VIDEO demonstrating CLARITY combined with immunofluorescent labeling to visualize the neurons in a whole mouse brain.

Many of the new technologies for probing the secrets of neural tissue don’t involve any hardware at all, but rather they exploit existing biological molecules, e.g., channelrhodopsins used in optogenetics, and whole organisms, e.g., retroviral gene delivery, or synthetic biological agents—see here for the announcement of a new PLoS (Public Library of Science) journal devoted to Neuroscience and Synthetic Biology. With the success of optogenetics and numerous examples from research on gene sequencing, e.g., the use of a thermostable polymerase found in T. aquaticus and applied in PCR (Polymerase Chain Reaction) to generate many copies of a gene, the search for biomolecules that can be used to encode, decode and transfer information.

Kording et al [18, 32] propose the idea of using a polymerase that has misincorporation rate—the probability that the polymerase will incorporate the wrong nucleotide at the location of a given base pair in a DNA sequence— is sensitive to the local concentration-density of calcium cations—positive calcium ions, Ca²⁺—to create a DNA ticker tape that could encode information about spiking neurons. Basically this polymerase would be expressed in neurons and spontaneously produce DNA records of neuronal activity that could be subsequently read off by gene-sequencing technology to recover spike chains.

In another proposal exploiting high-throughput gene sequencing technology, Zador et al [20, 31] propose a method of using a retrovirus to infer the connectome of living brains. The basic approach relies on the ability to genetically engineer cells capable of producing a DNA barcode in the form of a DNA sequence that would uniquely identify each cell. A retrovirus is then employed to transfer the cell’s barcode to each of its synapse-connected neighbors. All of the component technologies are currently possible and Zador’s lab at Cold Spring Harbor is working to assemble them into a reliable technology.

Terry Sejnowski, Tom Bartol and Justin Kinney from the Salk Institute
Kristen Harris from the University of Texas Austin
Daniel Keller from the É́cole Polytechnique Fédérale de Lausanne

This VIDEO from which the above image is taken shows a biologically-accurate model consisting of 450 synapses, 69 axons, and 77 dendritic spines. The model includes a complete—to the extent that we understand the relevant biology—account of the biophysics allowing researchers to set the initial conditions and the run a Monte-Carlo simulation of the neural circuitry. The simulation software¹¹, called MCell, is able to reproduce the behavior of all the biologically active elements in the model including neurons and their associated neuropil, astrocytes that surround the neurons, and changes in the local field potentials in the extracellular matrix. Sejonowski and the rest of the team created the model to test a hypothesis about the role of astrocytes in neural computation. See the article on the HHMI website describing this research.

Functional Connectomics: What does that circuit compute?

Neuroscience has no shortage of models, but it’s a good bet that most of them aren’t very good at explaining what’s actually happening down at the level of circuits. I take that back; the Hodgkin-Huxley model does a good job of predicting how an isolated giant squid axon will respond to a change in its membrane potential. I’m being facetious; I think the H-H model is a wonderful achievement and it has had a significant impact on the field of neurobiology [1, 13]. But it doesn’t tell us very much about the behaviour of circuits composed of interconnected neurons, though there are plenty of scientists that would argue against that view.

Structural connectomics will tell us what those circuits look like, but alone it is like someone gave us the wiring diagram of a computer but didn’t tell us how resistors, capacitors, inductors, transistors and conducting and insulating materials work. Actually, we sort of know how an individual transistor—a neuron in our computer analogy—works if you believe H-H. Unfortunately, the analogy breaks down when you consider that the mammalian brain has hundreds if not thousands of different types of neurons that traffic in an equally daunting number of synaptic neurotransmitters and diffuse neuromodulators¹² that propagate signals by a half-dozen or so pathways—chemical, electrical, genetic, diffusion, hormonal, mechanical.

We don’t really understand the role of astrocytes and other glia, but increasingly it appears that they play an important role. We’re not completely sure how circuit-spanning neural oscillations—alpha, beta, gamma, delta, theta, mu waves—influence the local behaviour of individual circuits, but it’s clear they do so. Eve Marder showed us that you could easily spend a lifetime studying such a ‘‘simple’’ circuit as the stomatogastric ganglion of lobsters or crabs and still come up with new revelations about the function of real neural circuits performing what are arguably pretty simple computations, at least when taken against the backdrop of all the other functions we don’t have a clue about [21, 3, 11]. A complete computational model will likely have to account for some of the more exotic signalling pathways that have been proposed including ephaptic coupling¹³ [2] and ectopic signal transmission¹⁴ [7]

Will having more data really make a difference given the current state of knowledge? How much do we understand about, say, the neural circuitry responsible for bird song? Answer: Less than you might imagine, but we have made headway [9]. If birdbrains are hard to crack, why not work on something ‘‘really simple’’ like C. Elegans for which we already have the connectome for all 302 neurons? The OpenConnectome Project is a collection of scientists—including amateurs and professionals if that distinction even makes sense here—working to do just that. Unfortunately, the 302 neurons that comprise the C. Elegans nervous system consist of roughly 151 distinct types of neurons—the worm is bilaterally symmetric—that are spiking but not in the same way that mammalian neurons are spiking. By the way, someone asked if nematodes have neuromodulators. Apparently they do [10].

How is having a complete connectome and recordings of neural activity across circuits spanning thousands if not millions of neurons really going to help things? Let’s take a sample one-millimeter-square section through the mouse visual cortex which is mostly between 0.6 and 1.2 mm in thickness. First thing to consider is that we would get a lot of EM data—roughly a petabyte¹⁵ at the current approximately 40 nm resolution—if we used the same sort of technologies that Allen will be using.

Let’s suppose we can identify the inputs and outputs by some means. We can easily create a neural network (NN) with a few billion tunable weights and assign the input neurites to NN input units and the output neurites to NN output units. Divide the data into training and testing, and then try this a few million times—sampling from the class of network topologies and weight distributions—and see how well the models can predict the behaviour of the circuit on the test set.

Suppose we’re lucky and we do a good job on the test data. What have we proved? How do you think such a result would be received by the neuroscience community. I was assuming that all of the data—training and test—was from a single organism, say a mouse. Suppose the data includes recordings from multiple mice—more than one and fewer than 100, and suppose for the sake of argument that we can do a pretty good job of extracting roughly the same circuit from each animal for some instantiation of ‘‘same.’’

Once again, suppose we’re lucky and our model generalizes across different mice. Now how do you think neuroscientists would react? Consider the recent run of papers claiming—at least according to the hyperbolic accounts in the popular press—to be able to ‘‘read minds’’ by analyzing fMRI scans [25, 26, 24, 16]. What do you think of the results reported in these papers? They do seem to show some generalization between individuals.

I’d like you to think about these questions and prepare a few of your own for class. Do you have a better way of tackling the problem of making sense of all the data we’re expecting from the technologies discussed above? Do you think the rest of the field has a good handle on how to proceed? If you were going to try to fit a model to the data, what family of models would you choose? In the few last paragraphs of this annotated version of the lecture, I’ll lay out a straw-man reductionist research program for you to critique and improve upon.

There is a relatively straightforward approach to modeling neural tissue that attempts to leverage what we know about the brain at the level of individual neurons in order to infer the function of larger circuits. Christof Koch and Henry Markram have proposed variants of this approach [17, 22]. Not everyone agrees we even need such models; Winfried Denk believes an accurately annotated serial-scan reconstruction will suffice for most scientific purposes, perhaps with immunofluorescent labeling as a post process to identify specific cell and synapse classes [27, 23]. In a recent discussiona at Google, he countered my examples of how dynamics might help to resolve disputes about specific hypotheses with suggestions for using pharmacological interventions to collect the relevant evidence.

I’m not convinced by Winfried’s argument, but, in any case, for now let’s continue with the modeling discussion. The basic idea is to build accurate models for each cell type using a detailed molecular modeling tool such as MCell or Neuron. To do so, we collect a subset of the calcium-imaging recordings containing representative instances of each cell type and fit a multi-compartmental Hodgkin-Huxley model to each subset. The resulting cell-type-specific models are likely to be our best bet in terms of accurately modeling the dynamics, but they will be very expensive computationally to simulate.

However, we can use the H-H model to train a simpler but computationally-tractable model such as a leaky-integrate-and-fire model of the sort that Izhikevich and Edelman [14] used in their large-scale simulations. Substituting the simpler but substantially faster models for the H-H models, we can then build a variant of the Izhikevich and Edelman model, but with a significantly more accurate model of the network than Izhikevich and Edelman in terms of cell-type-specific models for individual neurons, their connections and estimates of their synaptic strength.

The resulting complete model of the target tissue will have a number of additional global parameters that can be tuned using the calcium-imaging training data. This is obviously a very sketchy description, but you can probably get a pretty good idea of how things would play out. Suppose that this model accurately predicts the test data. Do you find it any more compelling as an explanatory model for cortical computations? What computational experiments would you suggest as additional tests of its adequacy?

References

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