The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes. (English) Zbl 1311.92043

Summary: Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can, in principle, be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the full combinatorial data of a neural code.
Our main finding is that these objects can be expressed in a “canonical form” that directly translates to a minimal description of the receptive field structure intrinsic to the code. We also find connections to Stanley-Reisner rings (see e.g., [R. P. Stanley, Combinatorics and commutative algebra. 2nd ed. Basel: Birkhäuser (2005; Zbl 1157.13302)]) and use ideas similar to those in the theory of monomial ideals to obtain an algorithm for computing the primary decomposition of pseudo-monomial ideals. This allows us to algorithmically extract the canonical form associated to any neural code, providing the groundwork for inferring stimulus space features from neural activity alone.


92C20 Neural biology
94B60 Other types of codes


Zbl 1157.13302


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[1] Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Reading: Addison–Wesley · Zbl 0175.03601
[2] Averbeck, B. B., Latham, P. E., & Pouget, A. (2006). Neural correlations, population coding and computation. Nat. Rev. Neurosci., 7(5), 358–366.
[3] Ben-Yishai, R., Bar-Or, R. L., & Sompolinsky, H. (1995). Theory of orientation tuning in visual cortex. Proc. Natl. Acad. Sci. USA, 92(9), 3844–3848.
[4] Brown, E. N., Frank, L. M., Tang, D., Quirk, M. C., & Wilson, M. A. (1998). 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), 7411–7425.
[5] Cox, D., Little, J., & O’Shea, D. (1997). An introduction to computational algebraic geometry and commutative algebra. In Undergraduate texts in mathematics: Ideals, varieties, and algorithms (2nd ed.). New York: Springer.
[6] Curto, C., & Itskov, V. (2008). Cell groups reveal structure of stimulus space. PLoS Comput. Biol., 4(10).
[7] Danzer, L., Grünbaum, B., & Klee, V. (1963). Helly’s theorem and its relatives. In Proc. sympos. pure math. (Vol. VII, pp. 101–180). Providence: Am. Math. Soc. · Zbl 0132.17401
[8] Deneve, S., Latham, P. E., & Pouget, A. (1999). Reading population codes: a neural implementation of ideal observers. Nat. Neurosci., 2(8), 740–745.
[9] Eisenbud, D., Grayson, D. R., Stillman, M., & Sturmfels, B. (Eds.) (2002). Algorithms and computation in mathematics: Vol. 8. Computations in algebraic geometry with Macaulay 2. Berlin: Springer. · Zbl 0973.00017
[10] Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press. · Zbl 1044.55001
[11] Jarrah, A., Laubenbacher, R., Stigler, B., & Stillman, M. (2007). Reverse-engineering of polynomial dynamical systems. Adv. Appl. Math., 39, 477–489. · Zbl 1129.93026
[12] Kalai, G. (1984). Characterization of f-vectors of families of convex sets in R d . I. Necessity of Eckhoff’s conditions. Isr. J. Math., 48(2–3), 175–195. · Zbl 0572.52006
[13] Kalai, G. (1986). Characterization of f-vectors of families of convex sets in R d . II. Sufficiency of Eckhoff’s conditions. J. Comb. Theory, Ser. A, 41(2), 167–188. · Zbl 0599.52003
[14] Ma, W. J., Beck, J. M., Latham, P. E., & Pouget, A. (2006). Bayesian inference with probabilistic population codes. Nat. Neurosci., 9(11), 1432–1438.
[15] McNaughton, B. L., Battaglia, F. P., Jensen, O., Moser, E. I., & Moser, M. B. (2006). Path integration and the neural basis of the ’cognitive map’. Nat. Rev. Neurosci., 7(8), 663–678.
[16] Miller, E., & Sturmfels, B. (2005). Graduate texts in mathematics: Combinatorial commutative algebra. Berlin: Springer. · Zbl 1090.13001
[17] Nirenberg, S., & Latham, P. E. (2003). Decoding neuronal spike trains: how important are correlations? Proc. Natl. Acad. Sci. USA, 100(12), 7348–7353.
[18] O’Keefe, J., & Dostrovsky, J. (1971). The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res., 34(1), 171–175.
[19] Osborne, L., Palmer, S., Lisberger, S., & Bialek, W. (2008). The neural basis for combinatorial coding in a cortical population response. J. Neurosci., 28(50), 13522–13531.
[20] Pistone, G., Riccomagno, E., & Wynn, H. P. (2001). Computational commutative algebra in statistics. In Monographs on statistics and applied probability.: Vol. 89. Algebraic statistics, Boca Raton: Chapman & Hall/CRC Press. · Zbl 0960.62003
[21] Schneidman, E., Berry, M. II., Segev, R., & Bialek, W. (2006a). Weak pairwise correlations imply strongly correlated network states in a neural population. Nature, 440(20), 1007–1012.
[22] Schneidman, E., Puchalla, J., Segev, R., Harris, R., Bialek, W., & Berry II, M. (2006b). Synergy from silence in a combinatorial neural code. arXiv:q-bio.NC/0607017 .
[23] Shiu, A., & Sturmfels, B. (2010). Siphons in chemical reaction networks. Bull. Math. Biol., 72(6), 1448–1463. · Zbl 1198.92020
[24] Stanley, R. (2004). Progress in mathematics: Combinatorics and commutative algebra. Boston: Birkhäuser.
[25] Veliz-Cuba, A. (2012). An algebraic approach to reverse engineering finite dynamical systems arising from biology. SIAM J. Appl. Dyn. Syst., 11(1), 31–48. · Zbl 1235.92005
[26] Watkins, D. W., & Berkley, M. A. (1974). The orientation selectivity of single neurons in cat striate cortex. Exp. Brain Res., 19, 433–446.
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