Using the SVD to estimate receptive fields

Spatio-temporal receptive fields can be hard to visualize. They can also be quite noisy. Thus, it’s desirable to find a low-dimensional approximation to the RF that is both easier to visualize and less noisy. The SVD is frequently used in neurophysiology for this purpose. Reading the Wikipedia page on the SVD, you might have trouble understanding how the SVD is relevant to RF estimation. Here’s a quick explanation of why you would want to use the SVD to estimate RFs.

The singular value decomposition (SVD) of a real rectangular matrix M factorizes the matrix into components \mathbf{USV'}, where the S matrix is diagonal, and composed of real, non-negative numbers. The numbers on the diagonal of S are called the singular values of M. Here I will use the convention that the singular values are sorted in descending order.

Both U and V are orthonormal matrices, meaning that \mathbf{U'U = I} and \mathbf{V'V = I}. Each column of U and V is a (left or right, respectively) singular vector of M. Writing out the result of the factorization explicitly, you get:

\mathbf{M_{i,j} = (USV')_{i,j} =} S_{1,1}U_{1,i}V_{1,j} + S_{2,2}U_{2,i}V_{1,j} + \ldots

Thus, the SVD factorizes an arbitrary matrix into a sum of outer products. Furthermore, each term in this series of outer products has decreasing influence. Thus, by truncating the SVD after a given number of singular values, you obtain an low-rank approximation of the original matrix.

So how is this relevant to RF estimation? Well, suppose that your receptive field is arranged as a matrix R, where the first dimension represents time, and the second dimension represents a single dimension of space. Now take the SVD of R. Then S_{1,1}U_{1,i}V_{1,j} is an approximation of \mathbf{R_{ij}} as an outer product of two vectors. So you’re saying that the receptive field can be approximated as the product of a time filter \mathbf{U_{1,i}} and a space filter \mathbf{V_{1,j}}; in other words, you’re saying the RF is separable in space-time. Furthermore, this is the best such approximation possible, in a certain mathematical sense.

Separable receptive fields are easier to visualize than non-separable RFs, they’re of lower-dimensionality, and typically have higher SNRs. Is it reasonable to assume that a RF is separable, though? You can gauge this by looking at the sequence of singular values. If a receptive field is truly separable, then only its first singular value should be non-zero.

In reality, noise will mean that a measured RF will never be truly separable. The ratio of the square of the first singular value to the sum of the squares of all singular values can be used as an index of separability (here for example). Furthermore, if you plot the singular values, you should see that they drop off rapidly after the first few (as with an eigenvalue decomposition), and this can be used as a means of finding a good point to truncate the decomposition. More formal criteria can be derived through cross-validation or bootstrapping.

Now, it’s possible to approximate a non-separable matrix using the SVD as well; you simply truncate after n singular values rather than the first. If you did this on the spatiotemporal RF of an LGN cell, you might find the first singular vectors to correspond to the early center response with the second corresponding to the late surround response (see above, from Wolfe and Palmer 1998).

It’s important to keep in mind, however, that the first and second left singular vectors will be orthogonal, while the traditional decomposition of an LGN RF might involve non-orthogonal vectors. It’s possible to recover a more “natural” non-orthogonal decomposition lying in the subspace of the truncated SVD based on other criteria like sparseness (see Construction of direction selectivity poster here).

If you have 3 dimensions instead of 2, you can simply bunch two dimensions together (in Matlab, using reshape) before performing the SVD.

The SVD is useful in other contexts in neurophysiology as well. It can usually directly on a design matrix X where you would use the eigenvalue decomposition on X’X; whitening images is one application of this. It’s an important tool to add to your belt.

Hexagonal orientation maps in V1

Interesting paper from Se-Bum Paik and Dario Ringach in this month’s issue of Nature Neuroscience on the origins of the orientation map in V1. Dr. Ringach has been developing a model of V1 orientation selectivity for a number of years now, the statistical connectivity hypothesis, based on the idea that the retinotopic map in V1 is at the source of orientation selectivity. In an article in the Journal of Neurophysiology, he showed that by simply pooling from a small number of LGN afferents which are physically close (and thus have close receptive field centers) together on the retinotopic map in V1, you end up with orientation-selective subunits, as shown in the image below. Note that the resulting receptive fields are not perfect matches for V1 simple cells; in particular their aspect ratio is off by a factor 2. Rather, the hypothesis is that the orientation bias stemming from the retinotopic map seeds the development of orientation selectivity in V1 which is refined by Hebbian mechanisms.

In the latest paper, the authors contend that this statistical connectivity mechanism can explain the existence of orientation selectivity maps in V1. Orientation columns form striking patterns visible through optical imaging; an example of such an orientation map is shown at the top, taken from Kandel and Schwartz. Continue reading

The far-reaching influence of sparse coding in V1

Introduction

Olshausen and Field (1996) made a big splash in visual neurophysiology and machine learning by offering an answer to a provocative question:

Why are simple cell receptive fields (RFs) organized the way they are?

After all, they could just as well be shaped like elongated sine waves, as in Fourier analysis, or they could be like bigger LGN receptive fields. Yet, evolution has selected local, oriented, band-pass receptive fields that resemble Gabor functions. Why?

Leibniz offered that we live in the best of all possible worlds, and while this idea was ridiculed by many, the co-inventor of calculus may have been on to something. Sensory systems are under a wide array of conflicting constraints: they should minimize energy consumption and wiring length without sacrificing acuity, for example. Perhaps the visual system is the way it is because it satisfies an optimality principle.

One desirable property for a neural code is sparseness. A vector is sparse if most of its elements are close to zero, and a few of its elements are large in magnitude. In a neural context, that means that a minority of neurons are needed to encode a typical stimulus.

Continue reading