An Algorithm for Pattern Discovery in Time Series

We present a new algorithm for discovering patterns in time series and other sequential data. We exhibit a reliable procedure for building the minimal set of hidden, Markovian states that is statistically capable of producing the behavior exhibited in the data -- the underlying process's causal states. Unlike conventional methods for fitting hidden Markov models (HMMs) to data, our algorithm makes no assumptions about the process's causal architecture (the number of hidden states and their transition structure), but rather infers it from the data. It starts with assumptions of minimal structure and introduces complexity only when the data demand it. Moreover, the causal states it infers have important predictive optimality properties that conventional HMM states lack. We introduce the algorithm, review the theory behind it, prove its asymptotic reliability, use large deviation theory to estimate its rate of convergence, and compare it to other algorithms which also construct HMMs from data. We also illustrate its behavior on an example process, and report selected numerical results from an implementation.Comment: 26 pages, 5 figures; 5 tables; http://www.santafe.edu/projects/CompMech Added discussion of algorithm parameters; improved treatment of convergence and time complexity; added comparison to older method

The Computational Structure of Spike Trains

Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its complexity. Starting from spike trains, our approach finds their causal state models (CSMs), the minimal hidden Markov models or stochastic automata capable of generating statistically identical time series. We then use these CSMs to objectively quantify both the generalizable structure and the idiosyncratic randomness of the spike train. Specifically, we show that the expected algorithmic information content (the information needed to describe the spike train exactly) can be split into three parts describing (1) the time-invariant structure (complexity) of the minimal spike-generating process, which describes the spike train statistically; (2) the randomness (internal entropy rate) of the minimal spike-generating process; and (3) a residual pure noise term not described by the minimal spike-generating process. We use CSMs to approximate each of these quantities. The CSMs are inferred nonparametrically from the data, making only mild regularity assumptions, via the causal state splitting reconstruction algorithm. The methods presented here complement more traditional spike train analyses by describing not only spiking probability and spike train entropy, but also the complexity of a spike train's structure. We demonstrate our approach using both simulated spike trains and experimental data recorded in rat barrel cortex during vibrissa stimulation.Comment: Somewhat different format from journal version but same conten

Measuring information-transfer delays

In complex networks such as gene networks, traffic systems or brain circuits it is important to understand how long it takes for the different parts of the network to effectively influence one another. In the brain, for example, axonal delays between brain areas can amount to several tens of milliseconds, adding an intrinsic component to any timing-based processing of information. Inferring neural interaction delays is thus needed to interpret the information transfer revealed by any analysis of directed interactions across brain structures. However, a robust estimation of interaction delays from neural activity faces several challenges if modeling assumptions on interaction mechanisms are wrong or cannot be made. Here, we propose a robust estimator for neuronal interaction delays rooted in an information-theoretic framework, which allows a model-free exploration of interactions. In particular, we extend transfer entropy to account for delayed source-target interactions, while crucially retaining the conditioning on the embedded target state at the immediately previous time step. We prove that this particular extension is indeed guaranteed to identify interaction delays between two coupled systems and is the only relevant option in keeping with Wiener’s principle of causality. We demonstrate the performance of our approach in detecting interaction delays on finite data by numerical simulations of stochastic and deterministic processes, as well as on local field potential recordings. We also show the ability of the extended transfer entropy to detect the presence of multiple delays, as well as feedback loops. While evaluated on neuroscience data, we expect the estimator to be useful in other fields dealing with network dynamics

Model-free reconstruction of neuronal network connectivity from calcium imaging signals

A systematic assessment of global neural network connectivity through direct electrophysiological assays has remained technically unfeasible even in dissociated neuronal cultures. We introduce an improved algorithmic approach based on Transfer Entropy to reconstruct approximations to network structural connectivities from network activity monitored through calcium fluorescence imaging. Based on information theory, our method requires no prior assumptions on the statistics of neuronal firing and neuronal connections. The performance of our algorithm is benchmarked on surrogate time-series of calcium fluorescence generated by the simulated dynamics of a network with known ground-truth topology. We find that the effective network topology revealed by Transfer Entropy depends qualitatively on the time-dependent dynamic state of the network (e.g., bursting or non-bursting). We thus demonstrate how conditioning with respect to the global mean activity improves the performance of our method. [...] Compared to other reconstruction strategies such as cross-correlation or Granger Causality methods, our method based on improved Transfer Entropy is remarkably more accurate. In particular, it provides a good reconstruction of the network clustering coefficient, allowing to discriminate between weakly or strongly clustered topologies, whereas on the other hand an approach based on cross-correlations would invariantly detect artificially high levels of clustering. Finally, we present the applicability of our method to real recordings of in vitro cortical cultures. We demonstrate that these networks are characterized by an elevated level of clustering compared to a random graph (although not extreme) and by a markedly non-local connectivity.Comment: 54 pages, 8 figures (+9 supplementary figures), 1 table; submitted for publicatio

Markov processes follow from the principle of Maximum Caliber

Markov models are widely used to describe processes of stochastic dynamics. Here, we show that Markov models are a natural consequence of the dynamical principle of Maximum Caliber. First, we show that when there are different possible dynamical trajectories in a time-homogeneous process, then the only type of process that maximizes the path entropy, for any given singlet statistics, is a sequence of identical, independently distributed (i.i.d.) random variables, which is the simplest Markov process. If the data is in the form of sequentially pairwise statistics, then maximizing the caliber dictates that the process is Markovian with a uniform initial distribution. Furthermore, if an initial non-uniform dynamical distribution is known, or multiple trajectories are conditioned on an initial state, then the Markov process is still the only one that maximizes the caliber. Second, given a model, MaxCal can be used to compute the parameters of that model. We show that this procedure is equivalent to the maximum-likelihood method of inference in the theory of statistics.Comment: 4 page