Beyond the Euclidean brain: inferring non-Euclidean latent trajectories from spike trains

Abstract

Neuroscience faces a growing need for scalable data analysis methods that reduce the dimensionality of population recordings yet retain key aspects of the computation or behaviour. To extract interpretable latent trajectories from neural data, it is critical to embrace the inherent topology of the features of interest: head direction evolves on a ring or torus, 3D body rotations on the special orthogonal group, and navigation is best described in the intrinsic coordinates of the environment. Accordingly, we recently proposed the manifold Gaussian process latent variable model (mGPLVM) to simultaneously infer latent representations on non-Euclidean manifolds and how neurons are tuned to these representations. This probabilistic method generalizes previous Euclidean models and allows principled selection between candidate latent topologies. While powerful, mGPLVM makes two unjustified approximations that limit its practical applicability to neural datasets. First, consecutive latent states are assumed independent a priori, whereas behaviour is continuous in time. Second, its Gaussian noise model is inappropriate for positive integer spike counts. Previous work in Euclidean LVMs such as GPFA has shown significant improvements in performance when modeling such features appropriately (Jensen et al., 2021). Here, we extend mGPLVM by incorporating temporally continuous priors over latent states and flexible count-based noise models. This improves inference on synthetic data, avoiding negative spike count predictions and discontinuous jumps in latent trajectories. On real data, we also mitigate these pathologies while improving model fit compared to the original mGPLVM formulation. In summary, our extended mGPLVM provides a widely applicable tool for inferring (non-)Euclidean neural representations from large-scale, heterogeneous population recordings. We provide an efficient implementation in python, relying on recent advances in approximate inference to e.g. fit 10,000 time bins of recording for 100 neurons in five minutes on a single GPU.

Linking Neural Manifolds to Circuit Structure in Recurrent Networks

“…Each column r :,k ∈ R N of this matrix represents population activtiy at time t k . If sufficiently many time points are recorded, these vectors will accumulate in R N , allowing to extract the neural manifold and the collective dynamics from the activity [31,32,35,38,43,44], as illustrated in Fig. 1F.…”

“…Each column r :,k ∈ R N of this matrix represents the population activity at time t k as a N -dimensional vector. If sufficiently many time points are recorded, dimensionality reduction can be robustly performed on the resulting N -dimensional point-cloud, enabling the decoding of the collective dynamics from the population activity [26][27][28][29][30][31].…”

“…These latent variables can be read out from the population activity (using either a linear or non-linear decoder), revealing the low-dimensional dynamics driving their trajectory (Fig. 1E) [21,[29][30][31]. Hence, in this view, neural computation is described by a dynamical flow in the lowdimensional neural manifold ( Fig.…”

“…For example, ring models of head-direction [18] or orientation tuning of visual receptive fields [17] have d = 1; whereas a model of rodent place cells where each neuron is characterised by its place cell center in the two-dimensional environment [40] has field dimension d = 2. The dimension of the neural manifold is denoted by (capital) D and refers to the minimal number of variables necessary to describe the population activity [41], determined by linear [33,35,42] or nonlinear [31,32,43,44] dimensionality reduction methods. Moreover, the collective dynamics in the neural manifold might generate a set of attractor states with dimension D A ≤ D. For example, a classic attractor state in the Hopfield model of associative memory [45] is a point with D A = 0 whereas a line attractor in the ring model has D A = 1.…”

“…(iii) Can we extract knowledge about the circuit structure from neuronal activity? While many studies have focused on extracting the neural manifold from experimental data [26,27,29,30], an established method for extracting the similarity space, without explicitly referring to tuning curves or selectivity, is missing.…”