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Čech complex the metric-ball version of a more general object. Given a collection $\mathcal{U} = \{U_\alpha\}$ of (say, compact) subsets of a topological space $X$ one builds the nerve $\mathcal{N}(\mathcal{U})$ as follows. The k-simplices of $\mathcal{N}(\mathcal{U})$ correspond to nonempty intersections of $k+1$ distinct elements of $\mathcal{U}$. For example, vertices of the nerve correspond to elements of $\mathcal{U}$, edges correspond to pairs in $\mathcal{U}$ which intersect nontrivially, and so on. This definition preserves faces: the faces of a k-simplex are obtained by removing corresponding elements of $\mathcal{U}$, leaving the resulting intersection still non-empty.
Nerve Lemma
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Theorem. If $\mathcal{U}$ is a finite collection of open contractible subsets of $X$ with all non-empty intersections of subcollections of $\mathcal{U}$ contractible, then nerve $\mathcal{N}\big(\mathcal{U}\big)$ is homotopic to the union $\bigcup_\alpha U_\alpha$. |
Applications
Neuroscience
The work of Curto and ltskov considers how neural activity can represent external environments. In particular, the authors consider the impacts of external stimuli on certain place cells in the dorsal hippocampus of rats. These cells experience dynamic electrochemical activity which is known to strongly correlate to the rat's location.
Each such cell group is assumed to determine a specific location in space, and the collection of such location patches, or place fields, forms a collection $\mathcal{U}$ satisfying the hypotheses of the Nerve Lemma, assuming that place fields exist and are stable, omni-directional, and have firing fields that are convex. Curto and ltskov argue that these assumptions are generally satisfied for place fields of dorsal hippocampal place cells recorded from a freely foraging rat in a familiar open field environment.
In an experiment, the place cells are monitored, with activity recorded as a time series. These time series show occasional bursts of activity, such spike trains being common data forms in neuroscience. Cell groups are identified by correlating spike train firings that occur at the same time (within a window of time based on some parameter chosen appropriately). The correlation of spiking activity provides the intersection data of the place fields.
This, in turn, allows for the computation of the nerve of the place fields based on spike train correlation. These investigations suggest that rats may build a structured representation of their external environment within an abstract space of stimuli.
Of note to topological data analysis is the lack of metric data: the construction of the physical environment is purely topological and can be effected without reference to coordinates.
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