Neural System Identification using Neural Differential Equations

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Neural system identification describes the task of constructing mathematical models that produce neural recordings. Finding a system that accurately predicts neural recordings could help with understanding the behavior of the brain, such as the connectivity between areas and how computations in the brain relate to behavior. However, system identification of the brain is difficult because the recordings can be noisy, the dynamics shows stochastic features and many parallel processes in the brain influence the recordings. A robust framework is necessary to tackle neural system identification. Neural observations can be modelled using Recurrent Neural Networks (RNN), since these networks are dynamic and have nested feedback loops. Because of these features, RNNs are capable of capturing the underlying non linear dynamics of neural observations. Neural Differential Equations (NDE) are differential equations with neural networks representing their vector fields. The NDE identifies continuous dynamics describing the unknown system instead of discrete dynamics of the RNN. Furthermore, training an NDE is more efficient than an RNN and adaptive step size solvers can be applied. In this thesis, an NDE is applied to neural data to identify the unknown system and learn latent dynamics that generate neural recordings of the motor and visual cortex. The state equation of the NDE is defined by concrete variables with respect to neural connectivity, which allows the integration of prior knowledge into these variables to evaluate hypotheses about the connectivity and size of neural structures. To improve the robustness of system identification, a novel method for online initialisation of the initial hidden state of the NDE is implemented. The NDE was validated on simulated data, after which it was applied to empirical data to gain useful insights about the computations in the brain. The NDE was able to predict the neural activity of individual trials and distinguish between different input conditions and areas. The main contribution is a powerful, continuous-time framework for neural system identification that enables hypothesis testing and online initial state estimation. This framework is beneficial for control problems where realtime system identification is required, but also for uncovering the brain’s computations through hypothesis testing.
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