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

The computational properties of neural systems are often thought to be implemented in terms of their network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single-unit (MSU) recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a state space representation of the dynamics, but would wish to have access to its governing equations for in-depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi-analytical maximum-likelihood estimation scheme for piecewise-linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation-Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, the approach is applied to MSU recordings from the rodent anterior cingulate cortex obtained during performance of a classical working memory task, delayed alternation. A model with 5 states turned out to be sufficient to capture the essential computational dynamics underlying task performance, including stimulus-selective delay activity. The estimated models were rarely multi-stable, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi-analytical (thus reasonably fast) maximum-likelihood estimation framework for PLRNNs that may enable to recover the relevant dynamics underlying observed neuronal time series, and directly link them to computational properties

Near Saddle-Node Bifurcation Behavior as Dynamics in Working Memory for Goal-Directed Behavior

In consideration of working memory as a means for goal-directed behavior in nonstationary environments, we argue that the dynamics of working memory should satisfy 3 Current correspondence: RIKEN, Frontier Research Program, Lab. For Information Representation, Hirosawa, 2-1, Wako-shi, Saitama, 351-01, Japan. email: [email protected] two opposing demands: long-term maintenance and quick transition. These two characteristics are contradictory within the linear domain. We propose the near saddle-node bifurcation behavior of a sigmoidal unit with a self-connection as a candidate of the dynamical mechanism that satisfies both of these demands. It is shown in evolutionary programming experiments that the near saddle-node bifurcation behavior can be found in recurrent networks optimized for a task that requires efficient use of working memory. The result suggests that the near saddle-node bifurcation behavior may be a functional necessity for survival in non-stationary environments..