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The computational properties of neural systems are often thought to beimplemented in terms of their network dynamics. Hence, recovering the systemdynamics from experimentally observed neuronal time series, like multiplesingle-unit (MSU) recordings or neuroimaging data, is an important step towardunderstanding its computations. Ideally, one would not only seek a state spacerepresentation of the dynamics, but would wish to have access to its governingequations for in-depth analysis. Recurrent neural networks (RNNs) are acomputationally powerful and dynamically universal formal framework which hasbeen extensively studied from both the computational and the dynamical systemsperspective. Here we develop a semi-analytical maximum-likelihood estimationscheme for piecewise-linear RNNs (PLRNNs) within the statistical framework ofstate space models, which accounts for noise in both the underlying latentdynamics and the observation process. The Expectation-Maximization algorithm isused to infer the latent state distribution, through a global Laplaceapproximation, and the PLRNN parameters iteratively. After validating theprocedure on toy examples, the approach is applied to MSU recordings from therodent anterior cingulate cortex obtained during performance of a classicalworking memory task, delayed alternation. A model with 5 states turned out tobe sufficient to capture the essential computational dynamics underlying taskperformance, including stimulus-selective delay activity. The estimated modelswere rarely multi-stable, but rather were tuned to exhibit slow dynamics in thevicinity of a bifurcation point. In summary, the present work advances asemi-analytical (thus reasonably fast) maximum-likelihood estimation frameworkfor PLRNNs that may enable to recover the relevant dynamics underlying observedneuronal time series, and directly link them to computational properties.

A State Space Approach for Piecewise-Linear Recurrent Neural Networks for Reconstructing Nonlinear Dynamics from Neural Measurements