Inferring Inference

Rajkumar Vasudeva Raju; Zhe Li; Scott Linderman; Xaq Pitkow

q-bio.NCOct 4, 2023

Inferring Inference

Authors: Rajkumar Vasudeva Raju, Zhe Li, Scott Linderman, Xaq Pitkow

Organizations: Department of ECE, Rice University · Department of Neuroscience, Baylor College of Medicine · Department of Statistics, Stanford University · Neuroscience Institute, Carnegie Mellon University ·

Abstract

Patterns of microcircuitry in the cerebral cortex suggest that the brain has an array of repeated canonical computational units. However, neural representations are distributed, so the relevant computations may only be related indirectly to single-neuron transformations. It thus remains an open challenge how to define canonical distributed computations. Here we integrate normative and algorithmic theories of neural computation to present a mathematical framework for inferring canonical distributed computations from large-scale neural activity patterns. At the normative level, we hypothesize that the brain learns a structured internal model of its environment and its latent causes. At the algorithmic level, we propose that this inference process is a nonlinear message-passing algorithm on a graph-structured model of the world. Given a time series of neural activity during a perceptual inference task, our analysis framework finds (i) the neural representation of the relevant latent variables, (ii) interactions between these latent variables that define the brain's internal model of the world, and (iii) message functions that specify the inference algorithm. With enough data, key computational properties are identifiable due to constraints from the symmetries inherent in any canonical computation, up to a joint global transformation of all interactions and the message-passing functions. As a demonstration of this framework, we analyze artificial neural recordings generated by a simulated brain that implicitly implements an approximate inference algorithm. Given simulated inputs and neural activity we successfully recover the latent variables, their neural representation and dynamics, and canonical message functions that govern the dynamics. Analysis of these models reveals features of experimental design required to extract canonical computations from neural data.

Figures & tables

Fig 1: Schematic of inferring inference . A : Mechanistic model has observations $\vo_{t}$ at time $t$ generated from an evolving world state $\vw_{t}$ . These observations affect the intrinsic dynamics of neural responses $\vr_{t}$ . B : We impose a low-dimensional normative interpretation upon these mechanistic dynamics, where the reduced dimensions encode approximate posterior probabilities $q$ over assumed task-relevant variables $\vs_{t}$ , which may differ from the true world state $\vw_{t}$ . C : The dynamics are structured according to a sparsely connected probabilistic graphical model, where signals propagate through task-relevant dimensions only along edges in the underlying graph. D : This propagation is assumed to be lawful, following a canonical nonlinear message function $\mathcal{M}$ that is shared by all edges, but is modulated by coupling strengths that are specific to each edge.

	Symbol	Meaning
World model	$\vw$	true world state
	$\vs$	brain’s assumed latent world state
	$\vo$	sensory observation
	$q(\cdot)$	approximate posterior over assumed world state
	$\vx$	parameter of posterior
	$t$	time

Table 1: Glossary of notation.

Fig 2: A model brain , implemented as a trained RNN ( A ), has neuronal dynamics ( B ) that are an approximate linear embedding of the TAP inference dynamics ( C ) on a binary probabilistic graphical model ( D ). In this illustration, the joint activity of the three gray neurons traces out a trajectory (time indicated as red to blue) in a 2D subspace corresponding to the inference dynamics of two interacting variables.

Fig 5: Importance of experiment design for inferring latent structure. Each row corresponds to a different range of amplitude scaling factor of the inputs $\tilde{g}_{o}$ relative to the coupling strengths. The top, middle, and bottom rows correspond to low, medium, and high input gain ranges, respectively. The first column from the left shows a scatter plot of two components of the ground truth latent activity $\lpr x_{1},x_{2}\rpr$ of the example TAP brain in Fig 3 . The second column shows the histogram of one component of the ground truth latent activity $\lpr x_{2}\rpr$ . The third column from the left shows scatter plots of the initial ICA estimate of the neural embedding matrix vs. the ground truth. Similarly, the fourth column corresponds to the EM estimate of the neural embedding. The rightmost column shows a scatter plot of the inferred vs. the ground truth latent activity. When low-gain inputs (top row/red) are used, the latent dynamics are in a localized regime and the resulting ICA estimate $\hat{R}$ is poor. This leads to degenerate solutions with Particle EM. On the other hand, when the input gain is too high (bottom row/blue), the latent activity is more biased towards maximal and minimal values. The resulting ICA estimate of the neural embedding $\hat{R}$ has large deviations from the ground truth along multiple dimensions, again leading to suboptimal inference of the latent dynamics. However, when the input gain is the range $[20,25]$ (middle row/green), the latent dynamics exhibit a fairly uniform distribution of the latent states. In this regime, ICA is able to obtain a good initial estimate $\hat{R}$ , with which Particle EM correctly estimates the neural encoding dimensions and the latent activity.

Fig 6: The importance of using the correct inductive bias. The TAP brain, from Fig 3 (with dense, recurrent connectivity shown in A , visualized using a subset of 50 out of 500 total neurons) was trained to possess implicit low-dimensional nonlinear graph-structured interactions (with interactions shown in B ). Panels C , D and E show results of our inference methods under different assumptions about the nonlinear dynamics: a polynomial message function (top, ‘Polynomial messages’: a representation that includes the true latent dynamics), an MLP message function operating over the latent graph (middle, ‘MultiLayer Perceptron messages’: closely related to Graph Neural Networks [ 115 ] ), and MLP latent dynamics, a generic all-to-all nonlinear dynamical system without graph structure (bottom, ‘Generic Recurrence’). Fig 7 shows all four model variants, including MLP message and update functions. Given enough data, all of these methods can accurately reproduce the targeted neural activity and the underlying latents (not shown). However, with limited data (and neuroscience data is inevitably severely limited), the two structured dynamics models substantially outperform the unstructured MLP latent dynamics model at both reproducing neural activity ( C ) and latent variables ( D ) — even when these inputs and trajectories have never been seen before (as shown here). Here we used only 100 sequences with 15 time steps for this inference. Moreover, the structured models provide interpretability through their connectivity ( E ), whereas a generic dynamical system does not have explicit structure.

Fig 7: Inference performance of all four latent dynamics models in the data-limited regime. Columns: polynomial message function, MLP message function, MLP message and update functions, and MLP latent dynamics (see Materials and methods). Rows 1–3: estimated versus true neural activity, latent variables, and couplings $J_{ij}$ ; couplings are not defined for the MLP latent dynamics model. Bottom row: correlation and mean squared error (MSE) between estimated and true neural activity ( $\vr$ ) and latent variables ( $\vx$ ).

Fig 8: Degeneracies in our parameterization. A : Latent states decoded by a particle filter (blue) versus true latent states (black). This visualization shows that the decoded trajectory is a linear transformation of the true trajectory. B,C : If we fix the embedding to the correct value, and relearn the remaining parameters twice, then the two sets of parameters (red, blue) both provide reasonable matches to the message-passing parameters and couplings of the true model (black). Couplings $\hat{J}_{ij}$ (estimates) and $J_{ij}$ (true) are related by a linear transformation for each of the two solutions, shown in panel D . $\hat{J}_{ij}$ in panel C and the corresponding message-passing parameters $\hat{G}$ in panel B result in an inference algorithm that is equivalent to the true model. E . Log-eigenvalues of the curvature (Hessian) of the mean squared prediction error for the latent dynamics at the true parameter values. Small values indicate degeneracies. F . The $G$ and $J$ components (top and bottom rows, respectively) of the eigenvectors corresponding to the three smallest eigenvalues. Parameter perturbations along eigenvector 0 correspond to the linear degeneracy between the interactions and the message passing parameters seen in B–D.

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Abstract

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