Ken Miller, Ph.D.

Research Interests

Theoretical Neuroscience

My lab's interests focus on understanding the cerebral cortex. We use theoretical and computational methods to unravel the circuitry of the cerebral cortex, the rules by which this circuitry develops or "self-organizes", and the computational functions of this circuitry. Our guiding hypothesis -- motivated by the stereotypical nature of cortical circuitry across sensory modalities and, with somewhat more variability, across motor and "higher-order" cortical areas as well -- is that there are fundamental computations done by the cortical circuit that are invariant across highly varying input signals. In some way that does not strongly depend on the specific content of the input, cortex extracts invariant structures from its input and learns to represent these structures in an associative, relational manner. We (and many others) believe the atomic element underlying these computations is likely to be found in the computations done by a roughly 1mm-square chunk of the cortical circuit. To understand this element, we have focused on one of the best-studied cortical systems, primary visual cortex, and also have interest in any cortical system in which the data gives us a foothold (including rodent whisker barrel cortex, studied here at Columbia by Randy Bruno; monkey area LIP, studied here by Mickey GoldbergJackie Gottlieb and Mike Shadlen; and the primate ventral visual stream, studied here by Elias Issa and  Niko Kriegeskorte).

The function of this element depends both on its mature pattern of circuitry and on the developmental and learning rules by which this circuitry is shaped by the very inputs that it processes. Thus we focus both on understanding how the mature circuitry creates cortical response properties (see lab publications on Models of Neuronal Integration and Circuitry) and on how this circuitry is shaped by input activity during development and learning (see lab publications on Models of Neural Development).

While I was at UCSF, I also had an experimental component to my lab, focused on the study of neuronal responses in cat visual cortex and LGN (the nucleus providing visual input to cortex); see lab publications on Experimental Results.

Courses Taught

Introduction to Theoretical Neuroscience


Advanced Topics in Theoretical Neuroscience


Student Journal Club: Neural Circuits


Responsible Conduct of Research



See my old home page (click 'Publications') to view PDFs of articles. Also see my google scholar page and the Center for Theoretical Neuroscience Publications.

Reverse Chronological Order, since 2006

Links of Interest

Links of Interest

Linear Algebra for Theoretical Neuroscience 

These are some notes I've written to try to teach linear algebra and related aspects of linear differential equations to students of theoretical neuroscience. I've also included a nice set of notes written by Philip (Flip) Sabes of UCSF when we were co-teaching a course; these are best read after Part 3. Most neuroscience students seem to find they never make it through Part 4, which is on the Fourier transform -- too much detail and too little motivation -- but the rest seems to work reasonably well in getting the conceptual ideas across to motivated biologists who don't have much background. Part 4 stands alone, and can be omitted, but read it if you'd like to better understand the Fourier transform and in particular understand that it is just another coordinate transformation (a particular one that diagonalizes a particular set of matrices or linear operators and hence is used particularly often).

Although these notes work reasonably well -- particularly if they're used in or followed by a course in which the ideas are used in the context of real biological problems -- they also leave a lot to be desired. They need many more figures, many more neuroscience examples, and more and better problems. I'd also like to include an introductory chapter reminding people of basics of 1-dimensional linear differential equations and the exponential function, before heading into multiple dimensions (Jan. 2019: this has now been added, it is part 0). I'd like to include some chapters on probability, and in particular on Poisson and Gaussian distributions (and the linear algebra leads naturally to understanding multi-dimensional Gaussian distributions). At that point, it would probably become "Mathematics for Theoretical Neuroscience" rather than "Linear Algebra for Theoretical Neuroscience". Part 3, which deals with non-normal matrices -- matrices that do not have a complete orthonormal basis of eigenvectors -- needs to be completely rewritten: since it was written, I've learned that non-normal matrices have many features not predicted by the eigenvalues that are of great relevance in neurobiology and in biology more generally, and the notes don't deal with this (this is discussed in our paper Balanced amplification: A new mechanism of selective amplification of neural activity patterns, Neuron 61:635-648, and also in a paper by Mark Goldman in the same issue of Neuron; a beautiful book on the mathematical aspects is L.N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, 2005). And Part 4 runs out of steam where I start talking about the connections between vectors and functions, matrices and linear operators, Kronecker deltas and Dirac deltas, and even more where it talks about multi-dimensional Fourier transforms. This all needs more work. Just haven't had the time. If you are interested in taking on any of these projects, particularly (but not limited to) figures, examples, or problems, or in adding other useful pieces of mathematics, let me know. Perhaps we can collaboratively build a useful resource.

All feedback on making these notes better will be appreciated. I'm not certain when I'll have time to implement them, but hope to.

I'd be very happy if you linked to this page. I'd prefer you link rather than posting the material yourself, both because (1) that way the creative commons license stays with the material and (2) that way people are always pointed to the latest versions, in case I should find the time to update. Here are the notes:

And here's Flip Sabes' notes on linear algebraic equations, SVD, and the pseudo-inverse:

Linear Algebra for Theoretical Neuroscience by Kenneth D. Miller is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.
Linear Algebraic Equations, SVD, and the Pseudo-Inverse by Phillip N. Sabes is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.

See here.

See here.

  • Marek Edelman's stunning account of the Warsaw Ghetto Uprising.
    And another short but incredibly moving account of the Warsaw Ghetto.
  • Richard Feynmann on Cargo Cult Science: "... a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty--a kind of leaning over show how you're maybe wrong, that you ought to have when acting as a scientist ..."
  • [July 2011: The following references to the Bush administration are obviously old and getting older; alternatively, think "Fox News" or "Republicans". The main ideas below are ageless.]
    Just a gentle reminder that the problems we face in the age of Bush are as old as the hills. The struggle for what is good and decent in the face of these forces is a never-ending one, perhaps nothing less than the human condition:
    • "Woe to those who make unjust laws, to those who issue oppressive decrees, to deprive the poor of their rights and withhold justice from the oppressed of my people, making widows their prey and robbing the fatherless."
      Isaiah 10:1-2, The Bible, written around 700 B.C.
    • "To think of the future and wait was merely another way of saying one was a coward; any idea of moderation was just another attempt to disguise one's unmanly character; ability to understand the question from all sides meant that one was totally unfitted for action; fanatical enthusiasm was the mark of a real man... Anyone who held violent opinions could always be trusted, and anyone who objected to them became a suspect."
      -- Thucydides, the Father of History, writing about the Greek Civil Wars of 427 B.C. (full source)
    • "Naturally, the common people don't want war; neither in Russia nor in England nor in America, nor for that matter in Germany. That is understood. But, after all, it is the leaders of the country who determine the policy and it is always a simple matter to drag the people along, whether it is a democracy or a fascist dictatorship or a Parliament or a Communist dictatorship ... the people can always be brought to the bidding of the leaders. That is easy. All you have to do is tell them they are being attacked and denounce the pacifists for lack of patriotism and exposing the country to danger. It works the same way in any country."
      -- Herman Goering, Nazi leader, while being held in Nuremberg jail during the war crimes trials. (full source)
    • "There was no point in seeking to convert the intellectuals. For intellectuals would never be converted and would anyway always yield to the stronger, and this will always be `the man in the street.' Arguments must therefore be crude, clear and forcible, and appeal to emotions and instincts, not the intellect. Truth was unimportant and entirely subordinate to tactics and psychology."
      -- Josef Goebbels, Nazi leader (quoted in this article on the process by which the US gov't made the decision to launch the war on Iraq and persuaded the country to follow; see also this excellent article on the same subject).
    • "`[Bush] was thinking about invading Iraq in 1999,' said author and journalist Mickey Herskowitz [KM: Herskowitz was working in 1999 as ghost-writer of Bush's autobiography and had around 20 meetings with him; he was later replaced]. `It was on his mind. He said to me: 'One of the keys to being seen as a great leader is to be seen as a commander-in-chief.' And he said, 'My father had all this political capital built up when he drove the Iraqis out of Kuwait and he wasted it.' He said, 'If I have a chance to invade....if I had that much capital, I'm not going to waste it. I'm going to get everything passed that I want to get passed and I'm going to have a successful presidency.''
      "Bush and his advisers were sold on the idea that it was difficult for a president to accomplish an electoral agenda without the record-high approval numbers that accompany successful if modest wars."
      "First, we simply do not defeat an incumbent president in wartime. After wars surely, but never in their midst. Republicans have been spinning this fact for months, and they are correct."
      -- Mark Mellman, Kerry pollster, in an analysis written two days before the Nov. 2004 election that accurately predicted Bush's vote to 0.1%.
      (KM adds: and of course it's not just the war in Iraq. The key part of the strategy is to preside over an eternal and never-ending "War on Terror".)

See her website.

See my Op-Ed here.

Lab Members & Co-Conspirators

Lab Members & Co-Conspirators

All current Center for Theoretical Neuroscience members can be found on the People page.

All former Center for Theoretical Neuroscience members can be found on the Alumni page.

Page Last Updated: 10/20/2023

Contact Ken

Ken Miller

[email protected]




Mailing Address:
Dept. of Neuroscience 
3227 Broadway, L6-070
Mail Code 9864
New York, NY 10027

Office Address
Room 70, 6th Floor
Jerome Green Science Building
3227 Broadway
(between 129th and 130th on W. Side of Broadway; near 125th St. stop of 1 train.)