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Topological analysis of population activity in visual cortex - PubMed

  • ️Tue Jan 01 2008

Topological analysis of population activity in visual cortex

Gurjeet Singh et al. J Vis. 2008.

Abstract

Information in the cortex is thought to be represented by the joint activity of neurons. Here we describe how fundamental questions about neural representation can be cast in terms of the topological structure of population activity. A new method, based on the concept of persistent homology, is introduced and applied to the study of population activity in primary visual cortex (V1). We found that the topological structure of activity patterns when the cortex is spontaneously active is similar to those evoked by natural image stimulation and consistent with the topology of a two sphere. We discuss how this structure could emerge from the functional organization of orientation and spatial frequency maps and their mutual relationship. Our findings extend prior results on the relationship between spontaneous and evoked activity in V1 and illustrates how computational topology can help tackle elementary questions about the representation of information in the nervous system.

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Figures

Figure 1
Figure 1

Topological equivalence in rubber-world. The figure illustrates the notion of equivalence by showing several objects (topological spaces) connected by the symbols ~ when they are equivalent or by ≁ when they are not. The reader should think that all the objects shown are made of an elastic material and visualize the equivalence of two spaces by imagining a deformation between to objects.

Figure 2
Figure 2

Betti numbers provide a signature of the underlying topology. Illustrated in the figure are five simple objects (topological spaces) together with their Betti number signatures: (a) a point, (b) a circle, (c) a hollow torus, (d) a Klein bottle, and (e) a hollow sphere. For the case of the torus (c), the figure shows three loops on its surface. The red loops are “essential” in that they cannot be shrunk to a point, nor can they be deformed one into the other without tearing the loop. The green loop, on the other hand, can be deformed to a point without any obstruction. For the torus, therefore, we have b1 = 2. For the case of the sphere, the loops shown (and actually all loops on the sphere) can be contracted to points, which is reflected by the fact that b1 = 0. Both the sphere and the torus have b2 = 1, this is due to the fact both surfaces enclose a part of space (a void).

Figure 3
Figure 3

Barcodes and Rips complexes. The figure illustrates the construction of the Rips complex and the generation of barcodes (only the first three Betti numbers are displayed) for 50 points randomly sampled from the surface of a torus. Panels a to d show the barcode “sliced” at different values of ε (the horizontal axis) with the corresponding Rips complexes shown to the right. The corresponding Betti numbers for each level of ε can be obtained by counting the number of horizontal lines crossed by the vertical red line in each graph.

Figure 4
Figure 4

Animations of barcodes and Rips complexes for data points consistent with a circle (top), sphere (center), and torus (bottom). The animation proceeds from low values of ε to high values (that is, from fine to coarse spatial scales). Clicking on each of the figures will link to a movie.

Figure 5
Figure 5

Testing the statistical significance of barcodes. (a) We assume an initial population of Poisson-spiking neurons tuned for orientation. (b) The simulated response of this population to the presentation of different orientations is collected into a data matrix (or point cloud). (c) Analysis of the simulated data shows a long interval with a signature (b0, b1) = (1, 1), which correctly identifies the underlying object as a circle. (d) We also compute the barcodes by shifting the relative positions of the columns (data shuffling). In this case, the statistical distributions of spike counts for each axis remain unchanged, but their relationship is destroyed. By computing the distribution of maximal b1 lengths under this null hypothesis, we can evaluate the likelihood that our data was generated by the null hypothesis that there the coordinates of the points are independent.

Figure 6
Figure 6

The ability of the method in recovering the underlying structure depends on the mean firing rate and number of cells. Small p-values are regions where the algorithm correctly identified the circle (a) and the torus (b), as the likelihood of obtaining such barcodes by chance is very low. There is a trade-off between number of neurons and maximal spike rates. The more neurons the smaller the firing rates can be to be able to detect the structures at the same level of statistical significance.

Figure 7
Figure 7

Robustness of the method to changes in count statistics and in the homogeneity of firing rates in the population. Top, simulation of Poisson firing and non-uniform rates. Middle, non-Poisson statistics and uniform rates. Bottom, non-Poisson statistics and non-uniform rates. It can be seen that there is little impact of these in the ability of the method to recover the structure of the embedded circle. Compare with Figure 6a.

Figure 8
Figure 8

Experimental recordings in primary visual cortex. (a) Insertion sequence of a multi-electrode array into primary visual cortex (Nauhaus & Ringach, 2007). (b) Natural image sequences, sampled from commercial movies, were used to stimulate all receptive fields of neurons isolated by the array. In the spontaneous condition, activity was recorded while both eyes were occluded.

Figure 9
Figure 9

Estimation of topological structure in driven and spontaneous conditions. (a) Ordering of topological signatures observed in our experiments. Each triplet (b0, b1, b2) is shown along an illustration of objects consistent with these signature. (b) Distribution of topological signatures in the spontaneous and natural image stimulation conditions pooled across the three experiments performed. Each row correspond to signatures with a minimum interval length (denoted as the threshold) expressed as a fraction of the covering radius of the data cloud (see Appendix A for the definition of the covering radius).

Figure 10
Figure 10

A possible origin of the spherical topological structure in visual cortex. (a) Cartoon organization of orientation and spatial frequency maps in V1. An orientation map tiling V1 is shown along the locations of extremes spatial frequency tuning. Extreme spatial frequency selectivity (solid circles indicates high spatial frequency preference; dashed circles represents low spatial frequency preference) tends to overlap with orientation pinwheels. The white transparent disk indicates a localized region of activity in this hypercolumn. (b) A the activity profile shifts to different locations on the cortex the resulting topological structure of the population response is equivalent to that of a sphere, where extreme spatial frequencies are mapped to the poles and orientation is coded by the azimuth (Bressloff & Cowan, 2003).

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