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For patients with medically intractable epilepsy, there have been few effective alternatives to resective surgery, a destructive, irreversible treatment. A strategy receiving increased attention is using interictal spike patterns and continuous EEG measurements from epileptic patients to predict and ultimately control seizure activity via chemical or… (More)

Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset… (More)

It has long been known that the method of time-delay embedding can be used to reconstruct non-linear dynamics from time series data. A less-appreciated fact is that the induced geometry of time-delay coordinates increasingly biases the reconstruction toward the stable directions as delays are added. This bias can be exploited, using the diffusion maps… (More)

- Tyrus Berry, Timothy Sauer
- 2014

a r t i c l e i n f o a b s t r a c t We introduce a theory of local kernels, which generalize the kernels used in the standard diffusion maps construction of nonparametric modeling. We prove that evaluating a local kernel on a data set gives a discrete representation of the generator of a continuous Markov process, which converges in the limit of large… (More)

We develop a method from semiparametric statistics (Cox, 1972) for the purpose of tracking links and connection strengths over time in a neuronal network from spike train data. We consider application of the method as implemented in Masud and Borisyuk (2011), and evaluate its use on data generated independently of the Cox model hypothesis, in particular… (More)

Long time series of monosynaptic Ia-afferent to alpha-motoneuron reflexes were recorded in the L7 or S1 ventral roots in the cat. Time series were collected before and after spinalization at T13 during constant amplitude stimulations of group Ia muscle afferents in the triceps surae muscle nerves. Using autocorrelation to analyze the linear correlation in… (More)

Long time series of Schaffer collateral to CA1 pyramidal cell presynaptic volleys (stratum radiatum) and population spikes (stratum pyramidale) were evoked (driven) in rat hippocampal slices. From the driven CA1 region in normal [K+] perfusate, both population spike amplitude and an input-output function consisting of population spike amplitude divided by… (More)

Many signals measured from the nervous system exhibit apparently random variability that is usually considered to be noise. The development of chaos theory has revealed that such random appearing variability may not, in fact, be random, but rather may be deterministic behavior that can reveal important information about the system's underlying mechanisms.… (More)

An approach to discriminating deterministic versus stochastic dynamics from neuronal data is presented. Direct tests for determinism are emphasized, as well as using time series with clear physical correlates measured from small ensembles of neurons. Surrogate data are used to provide null hypotheses that the dynamics in our data could be accounted for by… (More)

- Timothy Sauer
- 2013

Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior. This article is an overview of numerical solution methods for SDEs. The solutions are stochastic processes that represent diffusive dynamics, a common modeling assumption in many application areas. We… (More)