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What can topology tell us about the neural code?
Carina Curto
TL;DR
Neural codes raise the question of how collective activity represents the outside world, and topology is a natural tool for studying their properties. This survey connects neural activity to topological structures and reviews applications including data analysis and neural-code constraints. It highlights place-cell topology and structural results about convex codes, while noting that a complete characterization of convex codes remains open.
Problem
Method
The talk surveys topological data analysis and neural-code topology, relating activity patterns to simplicial complexes and examining local obstructions to convexity.
Source: §3 · p. 6, §4.1 · p. 8, §6.1 · p. 13Results
Place-cell co-firing can reveal represented-space homology, and convex realizability requires no local obstructions.
Source: §3 · p. 6, §6.1 · p. 13Takeaways & Limitations
Place-cell codes can reflect the topology of represented space, while local-obstruction tests provide a necessary condition for convex realization.
Source: §3 · p. 6, §6.1 · p. 13A complete characterization of convex codes is lacking, and the minimum dimension needed for a convex realization remains open.
Source: §6.2 · p. 14
Abstract
from arXiv · showhide
Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze emerging data sets. Consistent with this trend, the last decade has seen an uptick in the use of topological ideas and methods in neuroscience. In this talk I will survey recent applications of topology in neuroscience, and explain why topology is an especially natural tool for understanding neural codes. Note: This is a write-up of my talk for the Current Events Bulletin, held at the 2016 Joint Math Meetings in Seattle, WA.
1. Introduction
Topology is increasingly used to analyze neuroscience’s expanding data and network-focused research, but neural codes are the talk’s central motivation: their questions often contain topology in disguise. The talk surveys these applications and argues topology may play a fundamental role in theories of how the brain works.
1. Introduction
- Neuroscience’s growing data sets and focus on networks make topological methods useful for analyzing emerging research.The talk identifies data analysis and network study as two reasons topology is increasingly applied.Source: §1 · p. 1
- Neural codes ask how collective neuron activity represents information about the outside world, a question the talk treats as a topological problem in disguise.The talk devotes most of its attention to this third reason for topology’s relevance.Source: §1 · p. 1
- The talk surveys neuroscience applications of topology and suggests it may have a fundamental role in theories of how the brain works.It moves from neural-code examples to other applications, then returns to intrinsic neural-code properties.Source: §1 · p. 1
2. Neurons: nodes in a network or autonomous sensors?
Neurons can be understood both as network components whose activity emerges from network properties and as sensors whose activity relates to external stimuli. Examples including orientation-tuned, place, and grid cells motivate neural coding as a complementary perspective and topology as a natural tool for its open questions.
2. Neurons: nodes in a network or autonomous sensors?
- Hubel and Wiesel’s orientation-tuned neurons collectively form a neural code for angles in the visual field, with individual neurons responding rapidly to bars at particular angles.Source: §2 · p. 2
- Hippocampal place cells increase firing as an animal passes through a localized place field, acting as position sensors in space.The account notes that place cells were mysterious because the hippocampus has no obvious sensory pathway.Source: §2 · pp. 2–3
- Entorhinal grid cells encode position through periodic place fields arranged in a hexagonal lattice.The paper says these cells were discovered by Edvard and May-Britt Moser, former postdocs of O’Keefe.Source: §2 · p. 3
- Neural network theory explains activity through network properties, while neural coding theory focuses on the relation between activity and external stimuli.The paper presents both perspectives as valid but leading to different models of neural behavior.Source: §2 · p. 3
- Because coding properties of single neurons are often better known than their networks, open questions about neural codes remain, and topology is a natural tool for studying them.Neural codes concern how activity represents sensory inputs such as sight, smell, hearing, touch, and taste.Source: §2 · p. 4
3. Topology of hippocampal place cell codes
Place fields can be treated as a good cover of an environment, allowing co-firing patterns to define a nerve whose homology reflects the represented space. The section connects that inference to place-cell recordings and contrasts it with the fundamental-domain construction needed to interpret grid-cell fields.
3. Topology of hippocampal place cell codes
- Simultaneous recordings and statistical inference showed that an animal’s position could be inferred from population place-cell activity.The population code was understood to encode position collectively, despite earlier limits on recording one neuron at a time.Source: §3 · p. 4
- Place fields approximately cover the environment and are approximately convex, motivating the good-cover assumption needed to apply the Nerve Lemma.Under that assumption, the nerve and covered space have the same homology groups.Source: §3 · pp. 4–5
- The Nerve Lemma can fail for a non-good cover: a disconnected intersection produces a nerve that misrepresents the topology of the annulus.In the illustrated bad cover, the nerve is homotopy-equivalent to a point rather than the covered annulus.Source: §3 · p. 5, Figure 4 · p. 6
- Co-firing subsets of place cells can estimate the nerve of their fields, whose homology matches the environment when the fields form a good cover.This lets activity reveal represented-space topology without knowing the place fields themselves.Source: §3 · pp. 5–6
- Place-cell activity in an environment with a hole can detect nontrivial first homology and distinguish it from a square box or two-arm maze.The paper says simultaneous place-cell activity can be used to infer the nerve even when the fields are unknown.Source: §3 · p. 6
- In a flexible-track experiment, place fields moved with the track while preserving relative location sequences despite changes in angles and movement direction.The authors interpret the fields as respecting topological aspects more than metric features.Source: §3 · p. 7
- Grid fields have multiple disconnected components in the full environment, violating the good-cover assumption; restricting to a fundamental domain gives each field one convex component.The Nerve Lemma can then be used to infer the fundamental domain’s topology from grid-cell spiking.Source: §3 · p. 7, Figure 6 · p. 7
- Identifying the labeled edges of the hexagonal fundamental domain yields a torus, the space represented by grid cells.The construction identifies one pair of edges to form a cylinder, then matches the remaining edges after a twist.Source: §3 · pp. 7–8
4. Topology in neuroscience: a bird’s-eye view
The survey groups neuroscience applications of topology into TDA on data, upgrades from graphs to simplicial complexes, and analysis of neural codes. Examples include persistent-homology analyses of neural activity and topological analyses of brain networks.
4.1. “Traditional” TDA applied to neuroscience data sets.
- TDA estimates point-cloud shape by building nested simplicial complexes from intersecting balls and tracking homology cycles across scales.The complexes may be Cech or Vietoris-Rips; persistent classes can indicate structure such as a 3-sphere.Source: §4.1 · p. 8
- Persistent-homology analysis found similar topological structure in spontaneous and evoked primary-visual-cortex activity, consistent with a two-sphere.The results were interpreted in the context of neural coding.Source: §4.1 · pp. 8–9
4.2. An upgrade to network science.
- Neuroscience network analyses upgrade graphs to simplicial complexes, using higher-order interactions or the graph’s clique complex.For weighted graphs, edge-weight thresholds generate a sequence of clique complexes for persistent-homology analysis; other sequences can capture network evolution.Source: §4.2 · pp. 9–10
- Clique-complex homology detects graph-determined topology: higher-order simplices fill some graph cycles, while others may remain and higher-dimensional cycles may emerge.The network model can therefore represent more than pairwise edges alone.Source: §4.2 · pp. 9–10
- Topological methods have been applied to brain-network data, including fMRI, PET, morphology, metabolism, EEG, and neuronal functional connectivity.Applications include studies of clinical groups and drug-induced or learning-related states.Source: §4.2 · p. 10
5. The code of an open cover
The section connects combinatorial neural codes to open covers and distinguishes the information in a cover’s code from that in its nerve. It also shows that some codes cannot come from good or convex covers.
5. The code of an open cover
- A combinatorial neural code is a collection of binary patterns, with each digit indicating whether a neuron is active or silent.The simplicial complex of a code is the smallest abstract simplicial complex containing its codewords.Source: §5 · p. 11
- The code of an open cover assigns codewords to regions defined by intersections of its open sets.The code records set-containment relationships that the nerve cannot recover; the nerve is the smallest simplicial complex containing the code.Source: §5 · pp. 11–12
- The code’s combinatorial data can also be encoded algebraically using the neural ideal.The passage compares this encoding with Stanley-Reisner ideals for simplicial complexes.Source: §5 · p. 12
- Every binary code can be realized as an open-cover code, but not every code can arise from a good or convex cover.The lemma supplies a three-neuron example of this restriction.Source: §5 · p. 12
- A code containing 110 and 101 but neither 100 nor 111 cannot be realized by a good or convex cover.The code forces one covering set to be disconnected, violating the good-cover condition.Source: §5 · p. 12
6. Using topology to study intrinsic properties of neural codes
The section asks whether a neural code can be realized by convex sets, and shows how topological obstructions can rule out some realizations. It also describes classes that do have convex realizations while noting that the general characterization remains open.
6.1. What can go wrong.
- A local obstruction arises when code-imposed containment conflicts with the contractibility required by the Nerve Lemma for convex intersections.The resulting nerve is noncontractible, although convexity would make it contractible.Source: §6.1 · p. 13
- A neural code has no local obstructions exactly when every link of a simplex missing from the code is contractible [8].The links are taken for simplices in the code's simplicial complex that are not themselves codewords.Source: §6.1 · p. 13
- A code can lack local obstructions yet still fail to have a convex realization, so the obstruction test is not sufficient [24].The cited example nevertheless has a good cover; its nonconvexity relies on arguments not obviously topological.Source: §6.1 · p. 13
6.2. What can go right.
- Intersection-complete codes and some generalizations have constructive convex realizations, and additional realizable classes are known [8].Source: §6.2 · pp. 13–14
- A complete characterization of convex codes and the minimum dimension for a convex realization remain open.Source: §6.2 · p. 14
7. Codes from networks
Simple feedforward networks constrain neural codes through the geometry of hyperplane arrangements. Their codes have convex realizations, but not every convex code comes from such a network, and a full characterization remains unknown.
7. Codes from networks
- A simple feedforward network has one input and one output layer; its output neurons define hyperplanes whose chambers correspond to codewords.The hyperplanes lie in the positive orthant of R^k, where k is the number of input neurons.Source: §7 · p. 14, Figure 9 · p. 14
- Every feedforward code is realizable by a convex cover, but not every convex code is a feedforward code.Every simplicial complex can arise as the simplicial complex of a feedforward code, so that feature alone does not account for the difference.Source: §7 · pp. 14–15
- A complete characterization of feedforward codes is still unknown, and the interplay between network theory and neural coding remains early in development.Source: §7 · pp. 14–15