[Chaos in the Brain] Nonlinear dynamical analysis for neural signals Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST

Detecting chaos in experimental data Bottom-up approach We can apply nonlinear dynamical system methods to the dynamical equations, if we know the set of equations governing the basic systems variables. Top-down approach However, the starting point of any investigation in experiments is usually not a set of differential equations, but rather a set of observations. The way to get from the observations of a system with unknown properties to a better understanding of the dynamics of the underlying system is nonlinear time series analysis. Starting with the output of the system, and working back to the state space, attractors and their properties.

General strategy of nonlinear dynamical analysis Nonlinear time series analysis is a procedure that consists of three main steps: (i) reconstruction of the system’s dynamics in the state space using delay coordinates and embedding procedure. (ii) characterization of the reconstructed attractor using various nonlinear measures (iii) checking the validity (at least to a certain extent) of the procedure using the surrogate data methods.

Reconstruction of system dynamics [problem] our measurements usually do not have a one to one correspondence with the system variables we are interested in. For instance, the actual state space may be determined by ten variables of interest, while we have only two time series of measurements; each of these time series might then be due to some unknown mixing of the true system variables.

Delay coordinate and Embedding procedure With embedding, one time series are converted to a series or sequence of vectors in an m-dimensional embedding space. If the system from which the measurements were taken has an attractor, and if the embedding dimension m is sufficiently high, the series of reconstructed vectors constitute an ‘equivalent attractor’ (Whitney, 1936). Takens has proven that this equivalent attractor has the same dynamical properties (dimension, Lyapunov spectrum, entropy etc.) as the true attractor (Takens, 1981). We can obtain valuable information about the dynamics of the system, even if we don't have direct access to all the systems variables.

Takens has shown that, if we measure any single variable with sufficient accuracy for a long period of time, it is possible to reconstruct the underlying dynamic structure of the entire system from the behavior of that single variable using delay coordinates and the embedding procedure. Takens’ Embedding theorem (1981)

Time-delay embedding We start with a single time series of observations. From this we reconstruct the m-dimensional vectors by taking m consecutive values of the time series as the values for the m coordinates of the vector. By repeating this procedure for the next m values of the time series we obtain the series of vectors in the state space of the system. The connection between successive vectors defines the trajectory of the system. In practice, we do not use values of the time series of consecutive digitizing steps, but use values separated by a small ‘time delay’ d.

Stam, 2005

Parameter choice Time delay d: a pragmatic approach is to choose l equal to the time interval after which the autocorrelation function (or the mutual information) of the time series has dropped to 1/e of its initial value. Embedding dimension m: repeat the analysis (for instance, computation of the correlation dimension) for increasing values of m until the results no longer change; one assumes that is the point where m>2d (with d the true dimension of the attractor).

Spatial Embedding The m coordinates of the vectors are taken as the values of the m time series at a particular time; by repeating this for consecutive time points a series of vectors is obtained. The embedding dimension m is equal to the number of channels used to reconstruct the vectors. The spatial equivalent of the time delay d is the inter electrode distance. The advantage of spatial embedding is that it achieves a considerable data reduction, since the dynamics of the whole system is represented in a single state space. The disadvantage is that the spatial ‘delay’ cannot be chosen in an optimal way. Some groups advocated spatial embedding (Lachaux et al., 1997), whereas others suggested it may not be a valid embedding procedure (Pritchard et al., 1996, 1999; Pezard et al., 1999).

Nonlinear dynamical analysis attractor Jeong, 2002

How to quantify dynamical states of physiological systems Physiological system States Physiological Time series Embedding procedure (delay coordinates) 1-dimensional time series  multi-dimensional dynamical systems Attractor in phase space Dynamical measures (L1, D2) A deterministic (chaotic) system Topologically equivalent

C(r)  r D2 D2 algorithm Nonlinear measure: correlation dimension (D2)

Correlation integral attractor Scaling region

Nonlinear measures: The first positive Lyapunov exponent

Why determinism is important? Whether a time series is deterministic or not decides our approach to investigate the time series. Surrogate data method This method detects nonlinear determinism. Surrogate data are linear stochastic time series that have the same power spectra as the raw time series. They are randomized to destroy any deterministic nonlinear structure that may be present. Statistical differences of nonlinear measures between the raw data and their surrogate data imply the presence of nonlinear determinism in the original data.

Stam, 2005

Bursting as an information carrier of temporal spiking patterns of nigral dopamine neurons (a) Dopamine neurons in substantia nigra Substantia nigra, a region of the basal ganglia that is rich in dopamine-containing neurons, is thought to be etiologies of Parkinson’s disease, Schizophrenia, Tourette's syndrome etc.

Electrophysiology of DA neurons in substantia nigra Irregular and complex single spiking and bursting states in vivo The presence of nonlinear deterministic structure in ISI firing patterns (Hoffman et al. Biophysical J, 1995) Deterministic structure of ISI data produced by nigral DA neurons reflects interactions with forebrain structures (Hoffman et al. Synapse 2000)

No determinism of non-bursting DA neurons Histogram D2s of ISI data of DA neurons D2s of surrogate ISI data Embedding dim. vs. D2

Nonlinear determinism of bursting DA neurons D2s of ISI data of DA neuronsD2s of ISI surrogate data HistogramEmbedding dim. vs. D2

The source of nonlinear determinism in ISI firing patterns of DA neurons Materials (a)Estimation of correlation dimension (b)Surrogate data method (c)Burst separation method Methods (a) Non-bursting neurons (3/7) (b) Bursting neurons (4/7) (ISI <80ms, 160ms) 7 Male Sprague-Dawley rats anesthetized with chloral hydrate Original ISI Burst time series Single spike time series

Nonlinear determinism of burst time series D2s of ISI burst time seriesD2s of its surrogate ISI time series

No determinism of single spike time series D2s of ISI single spike time seriesD2s of its surrogate ISI time series

Nonlinear determinism of inter-burst interval data D2s of IBI data D2s of surrogate IBI data

Suprachiasmatic nucleus(SCN) Computational modeling of single neurons and small neuronal circuits

SCN neurons exhibit the circadian rhythm in their mean firing rates.

Spontaneous Spiking activity of SCN neurons SCN neurons exhibit irregular spontaneous firing patterns, accompanied by intermittent bursts, and thus generate complex ISI patterns, although the average SFR seems to maintain a circadian rhythm.

Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. I. Nonlinear Dynamical Analysis (Jeong et al., 2005) Among 173 neurons, 16 neurons were found to exhibit deterministic ISI patterns of spikes.

Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. II. Fractal stochastic Analysis. (Kim et al., 2003) 1/f A

Temporal Dynamics Underlying Spiking Patterns of the Rat Suprachiasmatic Nucleus in vitro. II. Fractal stochastic Modeling A B A CD