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  • Perspective
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Reconstructing computational system dynamics from neural data with recurrent neural networks

Nature Reviews Neuroscience volume 24, pages 693–710 (2023)Cite this article

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Abstract

Computational models in neuroscience usually take the form of systems of differential equations. The behaviour of such systems is the subject of dynamical systems theory. Dynamical systems theory provides a powerful mathematical toolbox for analysing neurobiological processes and has been a mainstay of computational neuroscience for decades. Recently, recurrent neural networks (RNNs) have become a popular machine learning tool for studying the non-linear dynamics of neural and behavioural processes by emulating an underlying system of differential equations. RNNs have been routinely trained on similar behavioural tasks to those used for animal subjects to generate hypotheses about the underlying computational mechanisms. By contrast, RNNs can also be trained on the measured physiological and behavioural data, thereby directly inheriting their temporal and geometrical properties. In this way they become a formal surrogate for the experimentally probed system that can be further analysed, perturbed and simulated. This powerful approach is called dynamical system reconstruction. In this Perspective, we focus on recent trends in artificial intelligence and machine learning in this exciting and rapidly expanding field, which may be less well known in neuroscience. We discuss formal prerequisites, different model architectures and training approaches for RNN-based dynamical system reconstructions, ways to evaluate and validate model performance, how to interpret trained models in a neuroscience context, and current challenges.

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Fig. 1: State spaces, vector fields and trajectories.
Fig. 2: Dynamical system reconstruction via recurrent neural networks.
Fig. 3: Dynamical system reconstruction of simulated and real physiological data by recurrent neural networks.
Fig. 4: Architectures used for dynamical system reconstruction.
Fig. 5: Interpreting the relationship of a data-inferred recurrent neural network to the biological substrate.

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Data availability

All data used to create the RNN reconstructions in Fig. 3 are publicly available. See Supplementary Methods for details.

Code availability

All codes used to create the RNN reconstructions in Figs. 2 and 3 are publicly available. The code for the models used in Fig. 1b,d is publicly available. See Supplementary Methods for details.

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Acknowledgements

D.D. discloses support for this work from the German Research Foundation (DFG) through individual grants (Du 354/10–1; Du 354/15–1), within research cluster FOR-5159 (“Resolving prefrontal flexibility”; Du 354/14–1) and through Germany’s Excellence Strategy EXC 2181/1–390900948 (STRUCTURES). The authors thank A. Draguhn, C. Lapish, J. Mikhaeil, K. Mitchell, A. Meyer-Lindenberg, Z. Monfared and R. Traub for providing detailed feedback and suggestions on this article, L. Judith for providing the EEG reconstructions in Fig. 3d, J. Hyman for providing the multiple single-unit data used in Fig. 3e, F. Hess for generating the DS reconstruction used in Supplementary Fig. 4 and M. Brenner for providing the code for the RNN animation.

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Authors and Affiliations

  1. Dept. of Theoretical Neuroscience, Central Institute of Mental Health, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany

    Daniel Durstewitz, Georgia Koppe & Max Ingo Thurm

  2. Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany

    Daniel Durstewitz

  3. Faculty of Physics and Astronomy, Heidelberg University, Heidelberg, Germany

    Daniel Durstewitz

  4. Dept. of Psychiatry and Psychotherapy, Central Institute of Mental Health, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany

    Georgia Koppe

  5. Hector Institute for Artificial Intelligence in Psychiatry, Central Institute of Mental Health, Medical Faculty Mannheim, Heidelberg University, Mannheim, Germany

    Georgia Koppe

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  1. Daniel Durstewitz
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All authors reviewed and/or edited the manuscript before submission and researched data for the article. D.D. wrote the article and contributed substantially to the discussion of the content.

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Correspondence to Daniel Durstewitz.

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Supplementary information

Glossary

Activation function

The non-linear function in a neural network computed on the inputs to a unit (node) of the network.

Attractor

A subset of the state space of a DS towards which the DS evolves over time from the basin of attraction of the attractor; it can, for example, be a single point (point attractor), a closed orbit (limit cycle) or a complex, fractal geometrical structure (chaotic attractor).

Autoencoder

A type of (usually non-linear) neural network architecture used to learn a compressed (lower-dimensional) representation of the data in an unsupervised manner, consisting of an encoder that maps input data to the lower-dimensional latent representation and a decoder that reconstructs the input data from the encoded representation.

Back propagation through time

(BPTT). A gradient-based training algorithm for training RNNs; BPTT computes the gradients (partial derivatives) of the loss function between RNN-generated outputs and target values and propagates these backwards through time to update the RNN weights.

Basin of attraction

The set of initial conditions from which the trajectory of a DS will eventually converge into the attractor (in the limit t → ∞).

Bifurcation

A sudden qualitative (topological) change in the state space and behaviour of a DS as one or more of its parameters cross a certain threshold, usually involving the creation or destruction of attractors.

Decoder

A component of a neural network model that maps the latent state of a model back into observation space (in other words, the space of the observed data).

Delay coordinate map

A map that embeds an observed time series into a space in which the resulting trajectory will be diffeomorphic to the true trajectory of the observed system.

Diffeomorphism

A bijective (1:1 and onto) function that maps one differentiable manifold onto another such that both the function and its inverse are continuously differentiable (implying a 1:1 relation also between gradients).

Dynamical system

A system that evolves in time (and possibly along other dimensions such as space) according to a set of rules or equations in a state space, which is the space spanned by all its dynamical variables.

Encoder

A component of a neural network model that maps input data (observations) into a latent space, in which an RNN may operate.

Equilibrium point (state)

Steady state of a DS described by differential equations, in which — when exactly placed at this point — the state of a DS would not change anymore (the same type of object is called a fixed point in a discrete-time DS).

Exploding or vanishing gradient problem

The problem that in RNNs or deep neural networks, the gradients of the loss function will eventually diverge (‘explode’) or vanish during the training process, if not controlled in some way.

Feedforward neural network

A neural network in which connections between nodes exclusively point in one direction, leading from input to final output.

Flow

A function that maps states of a DS to future or past states, given by the solution to the system of differential equations describing the DS.

Fractal dimensionality

The dimensionality of a geometrical object is commonly thought to be an integer number, but chaotic sets often have a self-similar geometrical structure that is more accurately captured by a non-integer (such as a transcendental real) number.

Gradient descent

A class of optimization techniques that aim to find a (local) minimum of a differentiable objective function (such as a loss function) by iteratively adjusting model parameters such that they are pushed into directions of descending slope (gradients).

Initial condition

The state in state space of a DS from which a trajectory originates (starts).

Invariant sets

Sets of states in state space of a DS, in which the state of the DS remains for all time under the action of the flow (the dynamical rules of the system).

Latent model

A statistical or ML model that contains unobserved (latent) variables that need to be inferred in order to account for the data observed.

Limit set

A set of states into which a DS converges as time goes to infinity.

Loss function

A function (also known as cost or objective function) that quantifies the mismatch between outputs predicted by a model and the target or desired outputs (it could be a negative likelihood, for instance).

Manifold

Any topological space that locally resembles Euclidean space (that is, for which there exists a continuous (bijective) function, with continuous inverse, that maps any neighbourhood of any point in that space to an open ball of Euclidean space).

Recurrent neural network

A type of neural network in which connections also recurrently couple different network units, in other words can run both forwards and backwards, unlike in feedforward neural networks.

State

A (vector) point in state space.

State (or phase) space

The space of all possible states a DS may be in, which is spanned by all dynamical variables of the DS.

Teacher forcing

A technique used in training algorithms for sequence generation and DS reconstruction tasks, in which during training (but not during model deployment), the latent states of an RNN are pushed to agree with the observations (in DS reconstruction models, specific, recently developed amendments of these techniques are used).

Temporal delay embedding

The vector space produced by the delay coordinate map.

Training algorithm

An algorithmic procedure by which the parameters of an ML model are obtained given a specified loss function and a set of training data as targets.

Training data

The set of sampled data points used for training an ML model (part of the acquired empirical data are usually held back as validation and test sets and not used for training).

Trajectory or orbit

The sequence or continuous series of states a DS moves through, starting from some initial condition, as time progresses (for a continuous-time DS, it is formally the solution curve from a specific initial condition).

Turing complete

A system that can emulate the operations of any Turing machine, a general model of computation.

Variational autoencoder

A specific type of autoencoder in which the latent states are probabilistic (treated as random variables), such that the encoder and decoder operate on probability distributions rather than on single data points.

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Durstewitz, D., Koppe, G. & Thurm, M.I. Reconstructing computational system dynamics from neural data with recurrent neural networks. Nat. Rev. Neurosci. 24, 693–710 (2023). https://doi.org/10.1038/s41583-023-00740-7

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