Tracking the topology of neural manifolds across populations

TL;DR

The method of analogous cycles is introduced, which allows for robust and falsifiable inference of matchings between features of neural manifolds that code, at the population level, for related nonlinear topological features.

Abstract

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and \emph{in vivo} experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.

Figures (4)

• Figure 1: Topological comparison of neural manifolds across populations. A. Tuning curves assemble to represent linear (left) and non-linear(right) neural manifolds. B. Detection of circular features from a dissimilarity matrix via persistent homology. Given a dissimilarity matrix (left), a filtration of simplicial complexes (middle) encodes the system at varying thresholds. The birth and death thresholds for topological features (middle) are summarized in a persistence diagram (right). C. The method of analogous cycles matches features. Given systems $P, Q$, with dissimilarity matrices $D_P, D_Q$, compute corresponding persistence diagrams $\text{\sc pd}(P), \text{\sc pd}(Q)$ (top, bottom). For a cross-system dissimilarity matrix $D_{P,Q}$ (middle row, left), create a witness filtration (middle row, center), with features summarized in the witness persistence diagram (middle row, right). The point (star) indicates one shared feature between the two systems. The analogous cycles method matches any representations of this feature in $\text{\sc pd}(P)$ (pink circle) and $\text{\sc pd}(Q)$ (pink diamond) consistent with $D_{P,Q}.$ The null model matching matrix (far right) gives the probability that the method returns a match between each pair of points in $\text{\sc pd}(P)$ and $\text{\sc pd}(Q)$ under the geometric null model (Methods and Materials). Shading at the $(i,j)$ entry indicates that the corresponding pair is matched through the computed $\text{\sc wpd}(P,Q).$
• Figure 2: Analogous cycles correctly match features across a simulated visual system. A. Architecture of the simulation. Stimulus videos are presented to a population of simulated spiking simple cells. Spikes from these cells are presented to populations tuned to orientation and direction features of the stimulus. B. Stimuli are 40,000-frame videos of a circular grating with fixed orientation, with the center moving in a fixed direction with fixed speed, looping to the opposite edge at boundaries. Orientations, starting locations, and movement directions are sampled uniformly. C. Simulation of spiking simple cells. Firing rate is computed as the rectified dot product between a modified Gabor filter and frames of the stimulus video. Spikes are sampled from the resulting inhomogeneous Poisson process. D. Firing rates of orientation sensitive cells are given by $f_{\text{orientation}},$ which is described by a feed-forward neural network trained to approximate a unimodal tuning curve on orientations using the simple cell spike trains as input (see SI Section 2A3.) Output is obtained by sampling from the resulting firing rates using an inhomogeneous Poisson process. Direction cells are simulated similarly. E. Analogous cycles match cycles across systems. (left) All three significant circular features of the stimulus are matched to those of the simple cells. Matches are indicated by color. All matches are statistically significant. (top right) One circular feature (teal diamond) in the simple cell neural manifold is matched to the unique significant feature (teal cross) of orientation sensitive population, implying a projection of encoded information from the simple cells. (bottom right) No matches are found between the simple cells and direction cells, indicating that the direction cells encode a novel feature.
• Figure 3: Analogous cycles assign semantics to simulated navigational system. A. Firing rate simulation. Given a rat trajectory, the grid cells are simulated using a continuous attractor networks model. The head-direction(HD) cells are simulated by tuning curves. The conjunctive cells firing rates are computed by the minimum firing rates of the grid cells and HD cells. B. (Top) Analogous cycles between the grid and conjunctive cells indicated by the colors. The teal and yellow points in the conjunctive PD encode the torus arising from the grid cell organization. (Bottom) Analogous cycles between the HD and conjunctive cells. The single significant feature in $\text{\sc pd}(\text{head})$ is analogous to a combination of the significant features in $\text{\sc pd}(\text{conj})$. Since the diamond and square points of the conjunctive PD are analogous to the two points in the grid PD, we conclude that the cross point of the conjunctive PD must encode the cyclicity of HD cells.
• Figure 4: Analogous cycles for neural coding propagation on experimental data. A. Schematic (top) of dual region two-photon calcium imaging in primary (V1) and anterolateral (AL) visual areas (left). Spike trains were inferred from the calcium sensor dynamics (right). B. The mouse was presented with a video consisting of drifting gratings of four orientations. Each stimulus was repeated 20 times. For each cell, the trial-aggregated spike train was computed. C. The analogous cycles method identifies a pair of points in the V1 and AL persistence diagram that encodes the same circular information.