RESEARCH STATEMENT. Nora Youngs, University of Nebraska - Lincoln

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1 RESEARCH STATEMENT Nora Youngs, University of Nebraska - Lincoln 1. Introduction Understanding how the brain encodes information is a major part of neuroscience research. In the field of neural coding, the relationship between stimulus and response, and the creation of a dictionary to relate the two, is one area of research which has been very frequently investigated [1, 2, 3]. Though this question has been studied heavily, it is not the only useful approach. Much can also be learned from investigating the structure of the neural code on its own. This question is important from a coding theory perspective, as it addresses error-correcting properties and decoding efficiency [4]. Also, it has been shown that the structure of the neural code can reflect the topological structure of the stimulus space [5]. My research has thus far focused on this second question - what can be learned from the intrinsic structure of a neural code? Given a set of neurons {1,..., n}, we define a neural code C {0, 1} n as a set of binary patterns of neural activity. Each codeword c = (c 1,..., c n ) corresponds to a subset of neurons supp(c) def = {i c i = 1} which were recorded to be on simultaneously at some point. This type of code (without specific firing rates or times) is often referred to as a combinatorial neural code [6, 7]. One common way to obtain a neural code is with receptive fields. A receptive field is a map f i : X R 0 from a space of stimuli, X, to the average firing rate of a neuron i in response to each stimulus. Commonly, the set U i X where f i takes on positive values is called a receptive field for the ith neuron. We obtain a neural code from a receptive field as follows: Definition. Let X be a stimulus space and U = {U 1,..., U n } a collection of receptive fields with U i X. The receptive field (RF) code of U is C(U) def = {c {0, 1} n ( ) ( ) U i \ U j }. i supp c j / supp c If X R d, and U i is convex for all i, then we say C(U) is a convex receptive field code. The particular example of receptive fields that has inspired much of my research is hippocampal place cells [8, 9]. Place cells are neurons whose receptive field is a 2-dimensional spatial region, the place field. These 2D regions overlap to create a set of smaller regions, each of which activates a particular set of neurons and thus has the same corresponding codeword. Receptive field codes are found in many other situations, such as orientation-selective neurons in visual cortex [10, 11], or those neurons which code for head direction. In such situations, how can we extract the structure of the receptive fields from the neural code? What can we learn about the dimension of the underlying space? To make progress on these questions and others like them, we turn to algebraic geometry, which provides a natural framework for extracting topological and geometric structures from a space. 1

2 2. The Neural Ring Although any code may be realized as a receptive field code in R, there are codes which cannot be realized as convex receptive field codes in R d, for any d. Examples of this occur on as few as 3 neurons. The questions we hope to answer are the following: given a code C, is C a convex receptive field code? If so, what is the minimal dimension d for which we can realize this code as a receptive field code in R d? To work towards answers for these questions, we define an algebraic object called the neural ring. This, and much of the following section, is laid out in detail in our recent paper [12] The neural ideal for general codes. Definition. Given a code C {0, 1} n, we define the corresponding neural ideal I C F 2 [x 1,..., x n ] as def I C = {f F 2 [x 1,..., x n ] f(c) = 0 for all c C}. That is, we define I C to be the set of polynomials which evaluate to 0 on all codewords: C is exactly the variety V (I C ). We define the neural ring to be F 2 [x 1,..., x n ]/I C, which is exactly the ring of functions C {0, 1}. This abstract definition is a natural way to store the code, but in order to extract the combinatorial information that defines the code in a simpler way, we must look further. By finding a set of generators for I C, we can understand the structure more concretely. To this end, define the ideal generated by the so-called Boolean relations B = {x i (1 x i ) i = 1,..., n}. Then, for each v {0, 1} n, define the polynomial ρ v = v i =1 x i v i =0 (1 x i); note ρ v is essentially an indicator polynomial for v, i.e., ρ v (c) = 1 c = v. Define J C = ρ v v / C. Then, I have proven that we can decompose I C as as follows: Theorem. With B and J C defined as above, I C = B + J C. Here B, as the ideal generated by the Boolean relations, does not depend on C at all. That is, because the code is binary, the polynomial x i (1 x i ) will evaluate to 0 on any v {0, 1} n. Thus, all the specific information in the code is contained within J C. Note that we can define this ideal for any binary code C, not just a receptive field code. The generators of J C belong to a class of polynomials we have called pseudo-monomials. This list of pseudo-monomial generators for J C are often no simpler to deal with than the code itself; indeed, if the code has fewer than 2 n 1 words, there are more generators for J C than there were codewords in C. Therefore, we would like to find a different set of generators of the same ideal. The most efficient choice we have found so far is a set of generators we refer to as the canonical form of J C (written CF (J C )) made up of the minimal pseudo-monomials of J C. The combinatorial information that defines the code is still contained within these generators, and for receptive field codes they have a nice interpretation The neural ideal for receptive field codes. Given a set U of receptive fields, we can encode their intersection information in an ideal: Definition. Given a set of receptive fields, U = {U 1... U n }, define the corresponding ideal I U as follows: def I U = x i (1 x i ) ( ) ( ) U i U j. i σ j τ i σ j τ 2

3 Since the sets σ, τ need not be disjoint, this seems different from the similarly-generated I C. However, I proved the following useful theorem: Theorem. If C = C(U) for U = {U 1,... U n } a set of receptive fields, then I C = I U. By pulling out the trivial combinatorial information (e.g. that U 1 U 1 ) into B, I then showed that we can interpret the canonical form to give minimal combinatorial information about the receptive fields: Theorem. Let C {0, 1} n be a neural code, and let U = {U 1,..., U n } be any collection of open sets (not necessarily convex) in a nonempty stimulus space X such that C = C(U). The canonical form of J C is: J C = { x σ σ is minimal w.r.t. U σ = }, { xσ { i τ (1 x i ) σ, τ, σ τ =, U σ, i τ U i X, and σ, τ are each minimal w.r.t. U σ i τ U i }, i τ (1 x i ) τ is minimal w.r.t. X i τ U i }. We call the above three (disjoint) sets of relations comprising CF (J C ) the minimal Type 1 relations, the minimal Type 2 relations, and the minimal Type 3 relations, respectively. The minimal pseudo-monomial generators that give the canonical form (and thus the minimal combinatorial information) can be found algorithmically from the code itself. An algorithm for this process is published in our paper. I have written Matlab code (since many neuroscience researchers use Matlab for their own data) which can take as input a neural code and give the canonical form for the ideal. The workflow runs as follows: neural code C {0, 1} n neural ideal J C = {ρ v v / C} primary decomposition of J C canonical form CF (J C ) minimal RF structure of C As an example of the power of this algorithm, consider the following code on 5 neurons. After applying the algorithm as indicated in the workflow above, we get minimal information to help us draw the picture shown at right. C = {00000, 10000, 01000, 00100, 00001, 11000, 10001, 01100, 00110, 00101, 00011, 11100, 00111}. 3. Ongoing Work and Future Projects 3.1. Maps Between Codes. Currently, I am taking this research in two different directions that stem from the approach outlined above. Firstly, I am investigating the relationship between natural maps between the codes and the corresponding maps between the algebraic objects corresponding to each code. Several natural maps on codes exist, namely permutation of the indices, inclusion of one code into another with more codewords, and truncation to a shorter code. However, considering the induced maps on 3

4 the neural rings does not lead to natural ring maps. In particular, two neural rings are isomorphic as rings exactly when the two corresponding codes have the same number of codewords, regardless of the number of neurons or any other structure. Therefore, I have been considering the neural rings as F 2 [x 1,..., x n ]/B-modules instead, which preserves the structure from the code and the inherent correspondence between neuron i and the variable x i. This has led to a much more natural notion of neural ring homomorphism as a module homomorphism Data analysis. I am also working (along with Aubrey Thompson, an undergraduate student in our research group) to optimize the existing Matlab code and improve our algorithm for finding the canonical form. For example, the algorithm currently requires finding the code s complement, which is computationally expensive, but we are confident this step can be avoided. Once the code is optimized, it is our intention to apply these methods to real neural data from place cells to see how it performs and what kinds of structures are obtained from actual place cells. The natural maps which motivate my most recent work are especially important in data analysis. When analyzing an actual neural code from data, it is vital to consider possible sources of error such as missing neurons or activity patterns, and how the structures would look if the errors were corrected Future directions. We know that some codes can be written as convex receptive field codes in R d for some d, while others cannot. For those which can, there is a minimal d which makes this possible. Though some broad classes of convex receptive field codes are known (e.g. simplicial complex codes), it is not yet certain how to detect for any code whether or not it is convex. For those codes C that are convex, it is also unknown how to detect the minimum dimension d for which C is convex in R d. Some lower bounds are given by classical results such as Helly s theorem [13] but I am interested in improving these results. For those codes which are known to be convex in R 2, particularly place cell codes, we know that the canonical form of the neural ideal gives us all the information necessary to describe the accompanying layout of place fields. However, we do not yet know an algorithmic process by which this picture can be generated, and developing one is an interesting problem Problems for Undergraduate Students. The mathematical problems motivated by neuroscience research can be attacked from many different angles, so the work is very approachable for undergraduate students with many types of mathematical background. A student with more interest in programming or algorithms could work on the (probably difficult) problem of how to take a convex code and draw the corresponding receptive fields in some automated way. For students who are learning to code, work on these algorithms or similar ones, or on particular classes of examples, could be done in Matlab, Macaulay2, or Sage. For those who want to try working in algebra or even topology, projects are possible researching the algebraic properties of specific classes of examples, and their related simplicial complexes. 4

5 References [1] E. N. Brown, L. M. Frank, D. Tang, M. C. Quirk, and M. A. Wilson. A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells. J Neurosci, 18(18): , [2] S. Deneve, P. E. Latham, and A. Pouget. Reading population codes: a neural implementation of ideal observers. Nat Neurosci, 2(8):740 5, [3] W. J. Ma, J. M. Beck, P. E. Latham, and A. Pouget. Bayesian inference with probabilistic population codes. Nat Neurosci, 9(11):1432 8, [4] C Curto, V Itskov, K Morrison, Z Roth, and J.L. Walker. Combinatorial neural codes from a mathematical coding theory perspective. Neural Computation, 25(25), [5] C. Curto and V. Itskov. Cell groups reveal structure of stimulus space. PLoS Computational Biology, 4(10), [6] E. Schneidman, J. Puchalla, R. Segev, R. Harris, W. Bialek, and M. Berry II. Synergy from silence in a combinatorial neural code. arxiv:q-bio.nc/ , [7] L. Osborne, S. Palmer, S. Lisberger, and W. Bialek. The neural basis for combinatorial coding in a cortical population response. Journal of Neuroscience, 28(50): , [8] J. O Keefe and J. Dostrovsky. The hippocampus as a spatial map. preliminary evidence from unit activity in the freely-moving rat. Brain Research, 34(1): , [9] B. L. McNaughton, F. P. Battaglia, O. Jensen, E. I. Moser, and M. B. Moser. Path integration and the neural basis of the cognitive map. Nat Rev Neurosci, 7(8):663 78, [10] D.W. Watkins and M.A. Berkley. The orientation selectivity of single neurons in cat striate cortex. Experimental Brain Research, 19: , [11] R. Ben-Yishai, R. L. Bar-Or, and H. Sompolinsky. Theory of orientation tuning in visual cortex. Proc Natl Acad Sci U S A, 92(9):3844 8, [12] C. Curto, V. Itskov, A. Veliz-Cuba, and N. Youngs. The neural ring : an algebraic tool for analyzing the intrinsic structure of neural codes. Bulletin of Mathematical Biology, 75(9), [13] Ludwig Danzer, Branko Grünbaum, and Victor Klee. Helly s theorem and its relatives. In Proc. Sympos. Pure Math., Vol. VII, pages Amer. Math. Soc., Providence, R.I.,