What can topology tell us about the neural code?

Curto, Carina

doi:10.1090/bull/1554

Cited by 112 publications

(101 citation statements)

References 32 publications

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“…Algebraic topology is a mathematical tool that was borrowed to study hippocampal neuronal coding for spatial topology [60–62]. It is aimed to compute abstract topological properties from the derived topological object and use those to derive a group relationship within neurons.…”

Section: Figurementioning

confidence: 99%

Deciphering Neural Codes of Memory during Sleep

Chen

Wilson

2017

Trends in Neurosciences

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Memories of experiences are stored in the cerebral cortex. Sleep is critical for consolidating hippocampal memory of wake experiences into the neocortex. Understanding representations of neural codes of hippocampal-neocortical networks during sleep would reveal important circuit mechanisms on memory consolidation, and provide novel insights into memory and dreams. Although sleep-associated ensemble spike activity has been investigated, identifying the content of memory in sleep remains challenging. Here, we revisit important experimental findings on sleep-associated memory (i.e., neural activity patterns in sleep that reflect memory processing) and review computational approaches for analyzing sleep-associated neural codes (SANC). We focus on two analysis paradigms for sleep-associated memory, and propose a new unsupervised learning framework (“memory first, meaning later”) for unbiased assessment of SANC.

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Section: Figurementioning

confidence: 99%

Deciphering Neural Codes of Memory during Sleep

Chen

Wilson

2017

Trends in Neurosciences

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“…For instance, persistent homology [14] has been employed across fields, such as contagion maps [68] and materials science [69]. In neuroscience, it has also yielded quite impactful results [31,33,37,57,[70][71][72]. In this sense, the Blue Brain Project recently provided persuasive support based both on empirical data and theoretical insights for the hypothesis that the brain network comprises topological structures in up to eleven dimensions [51].…”

Section: Discussionmentioning

confidence: 99%

Topological phase transitions in functional brain networks

Santos

Raposo

Coutinho‐Filho

et al. 2018

Preprint

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Functional brain networks are often constructed by quantifying correlations among brain regions.Their topological structure includes nodes, edges, triangles and even higher-dimensional objects.Topological data analysis (TDA) is the emerging framework to process datasets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. Our results suggest that a major alteration in the pattern of brain correlations can modify the signature of such transitions, and may point to suboptimal brain functioning. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives towards establishing reliable topological and geometrical biomarkers of individual and group differences in functional brain network organization. 1 2 I. INTRODUCTIONTopology aims to describe global properties of a system that are preserved under continuous deformations and are independent of specific coordinates, while differential geometry is usually associated with the system's local features [1]. As they relate to the fundamental understanding of how the world around us is intrinsically structured, topology and differential geometry have had a great impact on physics [2, 3], materials science [4], biology [5], and complex systems [6], to name a few. Importantly, topology has provided [7-11] strong arguments towards associating phase transitions with major topological changes in the configuration space of some model systems in theoretical physics. More recently, topology has also started to play a relevant role in describing global properties of real world, data-driven systems [12]. This emergent research field is called topological data analysis (TDA) [13,14].In parallel to these conceptual and theoretical advances, the topology of complex networks and their dynamics have become an important field in their own right [15,16]. The diversity of such networks ranges from the internet to climate dynamics, genomic, brain and social networks [15,16]. Many of these networks are based upon intrinsic correlations or similarities relations among their constituent parts. For instance, functional brain networks are often constructed by quantifying correlations between time series of activity recorded from different brain regions in an atlas spanning the entire brain [16].Here we report the discovery of topological phase transitions in functional brain networks.We merge concepts of TDA, topology, geometry, physics, and high-dimensional network theory to describe the topological evolution of complex networks, such as the brain networks, as function of their intrinsic correlation level. We consider tha...

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“…We then consider a tripod with vertices c 0 (2/3), c 1 (2/3) and c 0 (0) = c 1 (0). Since by the shortest property and (14), all distances are ≤ 2a/3, the tripod cannot coincide with the configuration c 0 , c 1 . Note that he center of the tripod might lie on c 0 or c 1 , and one of the two geodesics forming the tripod might coincide with a portion of c 0 or c 1 , but this cannot happen for both of them simultaneously, as the center of the tripod cannot be c 0 (0) = c 1 (0), because the geodesic connecting c 0 (2/3) and c 1 (2/3) through that center has length at most 2a/3.…”

Section: Tripod Spacesmentioning

confidence: 99%

Topology and curvature of metric spaces

Joharinad

Jost

2019

Advances in Mathematics

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We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Busemann, our concept also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivialČech homology. This suggests a link of our geometrical method to the topological method of persistent homology employed in topological data analysis. We also investigate the geometry of general tripod spaces.

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What can topology tell us about the neural code?

Cited by 112 publications

References 32 publications

Deciphering Neural Codes of Memory during Sleep

Deciphering Neural Codes of Memory during Sleep

Topological phase transitions in functional brain networks

Topology and curvature of metric spaces

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