Linking fast and slow: The case for generative models

Abstract

A pervasive challenge in neuroscience is testing whether neuronal connectivity changes over time due to specific causes, such as stimuli, events, or clinical interventions. Recent hardware innovations and falling data storage costs enable longer, more naturalistic neuronal recordings. The implicit opportunity for understanding the self-organised brain calls for new analysis methods that link temporal scales: from the order of milliseconds over which neuronal dynamics evolve, to the order of minutes, days, or even years over which experimental observations unfold. This review article demonstrates how hierarchical generative models and Bayesian inference help to characterise neuronal activity across different time scales. Crucially, these methods go beyond describing statistical associations among observations and enable inference about underlying mechanisms. We offer an overview of fundamental concepts in state-space modeling and suggest a taxonomy for these methods. Additionally, we introduce key mathematical principles that underscore a separation of temporal scales, such as the slaving principle, and review Bayesian methods that are being used to test hypotheses about the brain with multiscale data. We hope that this review will serve as a useful primer for experimental and computational neuroscientists on the state of the art and current directions of travel in the complex systems modelling literature.

Keywords: Bayesian statistics; Dynamical systems; Generative models; Hidden Markov models; Hierarchical modelling; Temporal scales.

Figure 1.

Important concepts of the theory of dynamical systems. (A) An example of flow in state-space (grey arrows), governing the evolution of trajectories (coloured curves) from different initial states (coloured circles). (B) The corresponding trajectories in the time domain for both x₁ and x₂ axes. (C) An example of bifurcation of the flow: trajectories converge towards a fixed point of state space when the bifurcation parameter α is below a critical value α_c, and towards a limit cycle when the bifurcation parameter is above the critical value (Andronov-Hopf bifurcation). (D) An example of a multistable system: the attractor to which the trajectory evolves depends on the initial state, as indicated by the colours of the trajectories.

Figure 2.

Taxonomy of the different modelling frameworks discussed here. The key factors guiding the selection of a particular framework are the nature of dynamics (discrete or continuous) and the nature of state space (stochastic or deterministic). In addition, the nature of the inputs (stochastic or deterministic) has relevance for continuous deterministic systems. In the particular case of continuous deterministic systems with stochastic inputs, one can use a linear response function, which is the first-order term of the Volterra kernel representation of the system, to directly approximate the outputs from the inputs without reference to the states. Effectively, this implies that the dynamics do not need to be integrated over time, which greatly simplifies model inversion. In all other cases, model inversion entails tracking the states or their distribution through time.

Figure 3.

Multiscale dynamics of brain signals: mapping slow and fast variables. (A) Slow quantities in the brain, such as synaptic efficacy between regions, exhibit large time constants and evolve slowly over time. The evolution of two slow variables are illustrated here, as yellow and purple lines. (B) The evolution of slow variables can also be represented by dynamics in a slow state space. (C) Importantly, for every location in the slow state space (horizontal axes), there is a corresponding mode of fast dynamics (vertical axis). Three modes are depicted here, numbered 1–3. These fast dynamics give rise to rapid brain signals, such as field potentials in pyramidal neurons (D). The mathematical relationship between the slow and the fast timescale is given by dt = εdT (ε ≪ 1); in other words, the dynamics at faster scale t unfolds over a fraction (ε) of the slower scale T. In summary, the brain is understood to navigate slowly (A) through a repertoire of fast stable dynamics (D). Crucially, the slow variables are directly linked to the dynamics of the fast variables (C). Similarly, changes in the fast variables’ dynamics can be attributed to changes in the slow variables. Therefore, modelling the complex dynamics of multiscale dynamical systems can be simplified by focusing on the dynamics of the slow variables and the mapping from slow to fast variables.

Figure 4.

Illustration of the centre manifold theorem with a three-dimensional dynamical system. (A) The three-dimensional state-space of the system. The blue surface is the centre manifold, and gives a height x = h(θ₁, θ₂) to each point of the (θ₁, θ₂) plane. Trajectories initialized away from the manifold converge rapidly towards the surface (black curves). This is due to the presence of a strong flow orthogonal to the centre manifold (B) for a section of state space. The strong flow (green arrows) converges towards the centre manifold (blue curve). The flow parallel to the manifold (blue arrows) is weaker by orders of magnitude. Hence, trajectories quickly collapse to the centre manifold before evolving alongside it. This is reflected in the exponential decay of the distance to the manifold (C). Hence, the x-component of the trajectory is well approximated by a static function from the location on the (θ₁, θ₂) plane (D). This can motivate an adiabatic approximation: as the rapidly changing x component of the trajectory can be approximated by the mapping h(θ₁, θ₂), we may consider x as a spurious dimension of the system and restrict our description to the evolution on the (θ₁, θ₂) plane; in other words, we can approximate the fast vanishing states by a fixed mapping from the slow states.

Figure 5.

Hierarchical modelling approach used to link slow effects to fast observations. First, the authors extracted sliding window data from their LFP time series (A). Then, they estimated the power spectral density for each window, to produce a time-frequency representation of the data (B). Then they fitted a state-space model, called a canonical microcircuit (CMC) dynamic causal model (DCM), for each window of power spectral densities (C). This resulted in a time course of posterior densities for the parameters of the DCM models. Finally, the authors added a second level to the model to test for between-window effects, enabling them to evaluate hypotheses of interest, that is, the interaction between interventions (PTZ concentration, presence or absence of NMDAr-Ab) and the parameters of the CMC model (D). Adapted with permission from R. E. Rosch et al. (2018).