Vladimir Itskov

Neural representations in the olfactory bulb (OB) remain poorly understood compared to well-established coding principles in other sensory systems. This difficulty arises because a natural coordinate system for olfactory stimulus space has not yet been identified, complicating the characterization of neural receptive fields analogous to those in other sensory modalities. Moreover, the extent to which these tuning properties are spatially organized within the OB remains unclear. To advance our understanding of the olfactory coding, we introduce an interpretable method of Shapley fields, as an olfactory analog of retinotopic receptive fields. Shapley fields are spatial distributions of chemical feature importance for the tuning of OB glomeruli. We used this tool to investigate chemotopy in the OB with diverse sets of chemical features using widefield epifluorescence recordings of the mouse dorsal OB in response to stimuli across a wide range of the chemical space. We found that Shapley fields reveal a weak chemotopic organization of the chemical feature sensitivity of dorsal OB glomeruli. This organization was consistent across animals and mostly agreed across very different chemical feature sets: (i) the expert-curated PubChem database features and (ii) features derived from a Graph Neural Network trained on human olfactory perceptual tasks. This chemotopy involved at least three chemical feature dimensions, rather than one or two dominant properties. Moreover, we found that the principal components of the Shapley fields corresponded to single commonly accepted chemical classification groups. This implies that these chemical groups could have been identified purely from the mouse olfactory responses. Our findings suggest that Shapley fields may serve as a chemical feature-agnostic method for investigating olfactory perception.

Representations of chemical features in the neural activity of the olfactory bulb (OB) are not well-understood, unlike the neural code for stimuli of the other sensory modalities. This is because the space of olfactory stimuli lacks a natural coordinate system, and this significantly complicates characterizing neural receptive fields (tuning curves), analogous to those in the other sensory modalities. The degree to which olfactory tuning is spatially organized across the OB, often referred to as chemotopy, is also not well-understood. To advance our understanding of these aspects of olfactory coding, we introduce an interpretable method of Shapley fields, as an olfactory analog of retinotopic receptive fields. Shapley fields are spatial distributions of chemical feature importance for the tuning of OB glomeruli. We used this tool to investigate chemotopy in the OB with diverse sets of chemical features using widefield epifluorescence recordings of the mouse dorsal OB in response to stimuli across a wide range of the chemical space. We found that Shapley fields reveal a weak chemotopic organization of the chemical feature sensitivity of dorsal OB glomeruli. This organization was consistent across animals and mostly agreed across very different chemical feature sets: (i) the expert-curated PubChem database features and (ii) features derived from a Graph Neural Network trained on human olfactory perceptual tasks. Moreover, we found that the principal components of the Shapley fields often corresponded to single commonly accepted chemical classification groups, that therefore could be “recovered” from the neural activity in the mouse OB. Our findings suggest that Shapley fields may serve as a chemical feature-agnostic method for investigating olfactory perception.

Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasi-periodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics.

Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. Here we introduce a Hodge-theoretic approach to spectral theory and dimensionality reduction for probability distributions on sequences and simplicial complexes. We demonstrate that this Hodge theory has desirable properties with respect to natural null-models, where the underlying vertices are independent. We prove that for the case of independent vertices in simplicial complexes, the appropriate Laplacians are multiples of the identity and thus have no meaningful Fourier modes. For the null model of independent vertices in sequences, we prove that the appropriate Hodge Laplacian has an integer spectrum, and describe its eigenspaces. We also prove that the underlying cell complex of sequences has trivial reduced homology. Our results establish a foundation for developing Fourier analyses of probabilistic models, which are common in theoretical neuroscience and machine-learning.

We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We construct a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural assumptions. We also illustrate the usability of this method in simulations.

Neural codes serve as a language for neurons in the brain. Open (or closed) convex codes, which arise from the pattern of intersections of collections of open (or closed) convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly understood. Here we find that a code that can be realized by a collection of open convex sets may or may not be realizable by closed convex sets, and vice versa, establishing that open convex and closed convex codes are distinct classes. We establish a non-degeneracy condition that guarantees that the corresponding code is both open convex and closed convex. We also prove that max intersection-complete codes (i.e., codes that contain all intersections of maximal codewords) are both open convex and closed convex, and provide an upper bound for their minimal embedding dimension. Finally, we show that the addition of non-maximal codewords to an open convex code preserves convexity.

Neural codes serve as a language for neurons in the brain. Convex codes, which arise from the pattern of intersections of convex sets in Euclidean space, are of particular relevance to neuroscience. Not every code is convex, however, and the combinatorial properties of a code that determine its convexity are still poorly understood. Here we find that a code that can be realized by a collection of open convex sets may or may not be realizable by closed convex sets, and vice versa, establishing that open convex and closed convex codes are distinct classes. We also prove that max intersection-complete codes (i.e. codes that contain all intersections of maximal codewords) are both open convex and closed convex, and provide an upper bound for their minimal embedding dimension. Finally, we show that the addition of non-maximal codewords to an open convex code preserves convexity.

Significance: Detecting structure in neural activity is critical for understanding the function of neural circuits. The coding properties of neurons are typically investigated by correlating their responses to external stimuli. It is not clear, however, if the structure of neural activity can be inferred intrinsically, without a priori knowledge of the relevant stimuli. We introduce a novel method, called clique topology, that detects intrinsic structure in neural activity that is invariant under nonlinear monotone transformations. Using pairwise correlations of neurons in the hippocampus, we demonstrate that our method is capable of detecting geometric structure from neural activity alone, without appealing to external stimuli or receptive fields.

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a two-dimensional environment, and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations. Remarkably, we found similar results during non-spatial behaviors such as wheel running and REM sleep. This suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits, and is not merely a consequence of position coding. We propose that clique topology is a powerful new tool for matrix analysis in biological settings, where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown.

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one-layer feedforward networks and identified a large class of combinatorial codes that indeed cannot be shaped by the feedforward architecture alone. However, these codes are difficult to distinguish from codes that share the same sets of maximal activity patterns in the presence of subtractive noise. When we coarsened the notion of combinatorial neural code to keep track of only maximal patterns, we found the surprising result that all such codes can in fact be realized by one-layer feedforward networks. This suggests that recurrent or many-layer feedforward architectures are not necessary for shaping the (coarse) combinatorial features of neural codes. In particular, it is not possible to infer a computational role for recurrent connections from the combinatorics of neural response patterns alone. Our proofs use mathematical tools from classical combinatorial topology, such as the nerve lemma and the existence of an inverse nerve. An unexpected corollary of our main result is that any prescribed (finite) homotopy type can be realized by a subset of the form [ ], where [ ] is a polyhedron.