Alexandre René

Neuronal activity is found to lie on low-dimensional manifolds embedded within the high-dimensional neuron space. Variants of principal component analysis are frequently employed to assess these manifolds. These methods are, however, limited by assuming a Gaussian data distribution and a flat manifold. In this study, we introduce a method designed to satisfy three core objectives: (1) extract coordinated activity across neurons, described either statistically as correlations or geometrically as manifolds; (2) identify a small number of latent variables capturing these structures; and (3) offer an analytical and interpretable framework characterizing statistical properties by a characteristic function and describing manifold geometry through a collection of charts. To this end, we employ normalizing flows (NFs), which learn an underlying probability distribution of data by an invertible mapping between data and latent space. Their simplicity and ability to compute exact likelihoods distinguish them from other generative networks. We adjust the NF's training objective to distinguish between relevant (in manifold) and noise dimensions (out of manifold). Additionally, we find that different behavioral states align with the components of the latent Gaussian mixture model, enabling their treatment as distinct curved manifolds. Subsequently, we approximate the network for each mixture component with a quadratic mapping, allowing us to characterize both neural manifold curvature and non-Gaussian correlations among recording channels. Applying the method to recordings in macaque visual cortex, we demonstrate that state-dependent manifolds are curved and exhibit complex statistical dependencies. Our approach thus enables an expressive description of neural population activity, uncovering nonlinear interactions among groups of neurons.

Neural circuit models are essential for integrating observations of the nervous system into a consistent whole. Public sharing of well-documented codes for such models facilitates further development. Nevertheless, scientific practice in computational neuroscience suffers from replication problems and little re-use of circuit models. One exception is a data-driven model of early sensory cortex by Potjans and Diesmann that has advanced computational neuroscience as a building block for more complex models. As a widely accepted benchmark for correctness and performance, the model has driven the development of CPU-based, GPU-based, and neuromorphic simulators. On the 10th anniversary of the publication of this model, experts convened at the Käte Hamburger Kolleg Cultures of Research at RWTH Aachen University to reflect on the reasons for the model’s success, its effect on computational neuroscience and technology development, and the perspectives this offers for the future of computational neuroscience. This report summarizes the observations by the workshop participants.

Fitting models to data is an important part of the practice of science. Advances in machine learning have made it possible to fit more—and more complex—models, but have also exacerbated a problem: when multiple models fit the data equally well, which one(s) should we pick? The answer depends entirely on the modelling goal. In the scientific context, the essential goal is replicability: if a model works well to describe one experiment, it should continue to do so when that experiment is replicated tomorrow, or in another laboratory. The selection criterion must therefore be robust to the variations inherent to the replication process. In this work we develop a nonparametric method for estimating uncertainty on a model’s empirical risk when replications are non-stationary, thus ensuring that a model is only rejected when another is reproducibly better. We illustrate the method with two examples: one a more classical setting, where the models are structurally distinct, and a machine learning-inspired setting, where they differ only in the value of their parameters. We show how, in this context of replicability or “epistemic uncertainty”, it compares favourably to existing model selection criteria, and has more satisfactory behaviour with large experimental datasets.

One challenge of physics is to explain how collective properties arise from microscopic interactions. Indeed, interactions form the building blocks of almost all physical theories and are described by polynomial terms in the action. The traditional approach is to derive these terms from elementary processes and then use the resulting model to make predictions for the entire system. But what if the underlying processes are unknown? Can we reverse the approach and learn the microscopic action by observing the entire system? We use invertible neural networks to first learn the observed data distribution. By the choice of a suitable nonlinearity for the neuronal activation function, we are then able to compute the action from the weights of the trained model; a diagrammatic language expresses the change of the action from layer to layer. This process uncovers how the network hierarchically constructs interactions via nonlinear transformations of pairwise relations. We test this approach on simulated datasets of interacting theories and on an established image dataset (MNIST). The network consistently reproduces a broad class of unimodal distributions; outside this class, it finds effective theories that approximate the data statistics up to the third cumulant. We explicitly show how network depth and data quantity jointly improve the agreement between the learned and the true model. This work shows how to leverage the power of machine learning to transparently extract microscopic models from data.

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus nonrandom weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the nonlinearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.

Understanding how rich dynamics emerge in neural populations requires models exhibiting a wide range of behaviors while remaining interpretable in terms of connectivity and single-neuron dynamics. However, it has been challenging to fit such mechanistic spiking networks at the single-neuron scale to empirical population data. To close this gap, we propose to fit such data at a mesoscale, using a mechanistic but low-dimensional and, hence, statistically tractable model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous pools of neurons and modeling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to optimize parameters by gradient ascent on the log likelihood or perform Bayesian inference using Markov chain Monte Carlo (MCMC) sampling. We illustrate this approach using a model of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived and show that both single-neuron and connectivity parameters can be recovered from simulated data. In particular, our inference method extracts posterior correlations between model parameters, which define parameter subsets able to reproduce the data. We compute the Bayesian posterior for combinations of parameters using MCMC sampling and investigate how the approximations inherent in a mesoscopic population model affect the accuracy of the inferred single-neuron parameters. Understanding how rich dynamics emerge in neural populations requires models exhibiting a wide range of behaviors while remaining interpretable in terms of connectivity and single-neuron dynamics. However, it has been challenging to fit such mechanistic spiking networks at the single-neuron scale to empirical population data. To close this gap, we propose to fit such data at a mesoscale, using a mechanistic but low-dimensional and, hence, statistically tractable model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous pools of neurons and modeling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to optimize parameters by gradient ascent on the log likelihood or perform Bayesian inference using Markov chain Monte Carlo (MCMC) sampling. We illustrate this approach using a model of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived and show that both single-neuron and connectivity parameters can be recovered from simulated data. In particular, our inference method extracts posterior correlations between model parameters, which define parameter subsets able to reproduce the data. We compute the Bayesian posterior for combinations of parameters using MCMC sampling and investigate how the approximations inherent in a mesoscopic population model affect the accuracy of the inferred single-neuron parameters. Understanding how rich dynamics emerge in neural populations requires models exhibiting a wide range of behaviors while remaining interpretable in terms of connectivity and single-neuron dynamics. However, it has been challenging to fit such mechanistic spiking networks at the single-neuron scale to empirical population data. To close this gap, we propose to fit such data at a mesoscale, using a mechanistic but low-dimensional and, hence, statistically tractable model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous pools of neurons and modeling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to optimize parameters by gradient ascent on the log likelihood or perform Bayesian inference using Markov chain Monte Carlo (MCMC) sampling. We illustrate this approach using a model of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived and show that both single-neuron and connectivity parameters can be recovered from simulated data. In particular, our inference method extracts posterior correlations between model parameters, which define parameter subsets able to reproduce the data. We compute the Bayesian posterior for combinations of parameters using MCMC sampling and investigate how the approximations inherent in a mesoscopic population model affect the accuracy of the inferred single-neuron parameters.

Dynamical models are often required to incorporate both delays and noise. However, the inherently infinite-dimensional nature of delay equations makes formal solutions to stochastic delay differential equations (SDDEs) challenging. Here, we present an approach, similar in spirit to the analysis of functional differential equations, but based on finite-dimensional matrix operators. This results in a method for obtaining both transient and stationary solutions that is directly amenable to computation, and applicable to first order differential systems with either discrete or distributed delays. With fewer assumptions on the system's parameters than other current solution methods and no need to be near a bifurcation, we decompose the solution to a linear SDDE with arbitrary distributed delays into natural modes, in effect the eigenfunctions of the differential operator, and show that relatively few modes can suffice to approximate the probability density of solutions. Thus, we are led to conclude that noise makes these SDDEs effectively low dimensional, which opens the possibility of practical definitions of probability densities over their solution space.