Biological systems are often complex, nonlinear and time-varying. The modeling of biological systems, therefore, presents significant challenges that are not overcome by the classical linear methods. In recent decades, intensive research has begun to produce methods for analyzing and modeling isolated classes of nonlinear systems. However, this vast class of models still presents many challenges, especially in complex biological systems. In this research, two novel methods are introduced for analyzing time series resulting from nonlinear systems. In the first approach, we study a class of dynamical systems that are nonlinear, discrete and with a latent state-space. We solve the probabilistic inference problem in these latent models using a variational autoencoder (VAE). Compared to continuous latent random variables, the inference of discrete latent variables is more difficult to solve. However, stochastic variational inference provides us with a general framework that tackles the inference problem for this class of model. We focused on an important neuroscience application – inferring pre- and post-synaptic activities from dendritic calcium imaging data. For it, we developed families of generative models, a deep convolutional neural network recognition model, and methods of inference using stochastic gradient ascent VAE. We benchmarked our model with both synthetic data, which resembles real data, and real experimental data. The framework can flexibly support rapid model prototyping. Both the generative model and recognition model can be changed without perturbing the inference. This is especially beneficial for testing different biological hypotheses. As a second approach, we treat a subclass of nonlinear autoregressive models: linear-time-invariant-in-parameters models. This class of models is useful and easy to work with. We propose an identification algorithm that simultaneously selects the model and does parameter estimation. The algorithm integrates two strategies: set-based parameter identification, and evolutionary algorithms that optimize fitness measures derived from these solutions. The algorithm can identify nonlinear models in novel noise scenarios. We show the performance of the algorithm in various simulated systems and practical datasets. We demonstrate its application to identify causal connectivity in a graph. This problem is often posed in recovering functional connectivity in the brain. The main contribution of this thesis is that we provide two framework for identifying nonlinear, biological systems from time series data. These two classes of nonlinear models and their applications are significant as each class is broad enough for modeling many complicated biological systems. We develop general, fast algorithms for learning these systems from data for these two model classes.
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