Global Visualization Global Visualization of Neural Dynamicsof Neural Dynamics

Krzysztof Dobosz, Krzysztof Dobosz, Włodzisław DuchWłodzisław Duch

Department of InformaticsDepartment of InformaticsNicolaus Copernicus UniversityNicolaus Copernicus University, , Toruń, PolandToruń, Poland

Google: W. DuchGoogle: W. Duch

Neuromath, Jena, April 2008Neuromath, Jena, April 2008

Brain SpirographyBrain Spirography

Example of a pathological signal analysisExample of a pathological signal analysis

MotivationMotivation• Analysis of multi-channel, non-stationary, time series data.Analysis of multi-channel, non-stationary, time series data.• Neural respiratory rhythm generator (RRG): hundreds of Neural respiratory rhythm generator (RRG): hundreds of

neurons, what is the system doing?neurons, what is the system doing?• Information is in the trajectories, how to see them? Information is in the trajectories, how to see them?

• Component-based analysis.Component-based analysis.• Time-frequency analysis. Time-frequency analysis. • Recurrence plots.Recurrence plots.

Fuzzy Symbolic Dynamics (FSD), visualize + understand.Fuzzy Symbolic Dynamics (FSD), visualize + understand.

1.1. Understand FSD mappings using model data.Understand FSD mappings using model data.

2.2. First look at RRG data.First look at RRG data.

3.3. First look at real EEG data.First look at real EEG data.

Fuzzy Symbolic Dynamics (FSD)Fuzzy Symbolic Dynamics (FSD)Trajectory of dynamical Trajectory of dynamical systemsystem (neural activities, av. rates): (neural activities, av. rates):

1..1..{ ( )}t N

i i nx t

1. Standardize data.1. Standardize data.

2. Find cluster centers (e.g. by k-means algorithm): 2. Find cluster centers (e.g. by k-means algorithm): 11, , 2 2 ......

3. Use non-linear mapping to reduce dimensionality:3. Use non-linear mapping to reduce dimensionality:

T 1( ; , ) expkk k k k ky t x x

Localized probe function: Localized probe function:

sharp indicator functions => symbolic dynamics;sharp indicator functions => symbolic dynamics;

soft membership functions => fuzzy symbolic dynamics.soft membership functions => fuzzy symbolic dynamics.

Model, radial/linear sourcesModel, radial/linear sources

Sources generate waves on a gridSources generate waves on a grid

( ; ) coslij l l ijF t t p k p

( ; ) cos ,lij l l l i jR t t k r x y p

( ; ) , ,l lij ij ij

l l

A t F t p R t p p

Flat waveFlat wave

Radial wave

Relatively simple patterns arise, but slow sampling shows Relatively simple patterns arise, but slow sampling shows numerical artifacts.numerical artifacts.

Ex: one and two radial waves.Ex: one and two radial waves.

Respiratory Rhythm GeneratorRespiratory Rhythm Generator

3 layers, spiking neurons, output layer with 50 neurons3 layers, spiking neurons, output layer with 50 neurons

Sensitive differences?Sensitive differences?

FSD developmentFSD development

Optimization of parameters of probe functions to see more Optimization of parameters of probe functions to see more structure from the point of view of relevant task.structure from the point of view of relevant task.

Learning: supervised clustering, projection pursuit based on Learning: supervised clustering, projection pursuit based on quality of clusters => projection on interesting directions.quality of clusters => projection on interesting directions.

Measures to characterize dynamics: position and size of Measures to characterize dynamics: position and size of basins of attractors, transition probabilities, types of basins of attractors, transition probabilities, types of oscillations around each attractor.oscillations around each attractor.

Visualization in 3D and higher (lattice projections etc).Visualization in 3D and higher (lattice projections etc).

Tests on model data and on the real data. Tests on model data and on the real data.

Complex logicComplex logicComplex logicComplex logicWhat is needed to understand data with complex logic?What is needed to understand data with complex logic? cluster non-local areas in the cluster non-local areas in the XX space, use projections space, use projections WW..XX capture local clusters after transformation, use capture local clusters after transformation, use G(WG(W..XX) )

SVMs fail because the number of directions SVMs fail because the number of directions WW that should be that should be

considered grows exponentially with the size of the problem considered grows exponentially with the size of the problem nn..

What will solve it?What will solve it?

1.1. A class of constructive neural network solution with A class of constructive neural network solution with G(WG(W..XX)) functions with special training algorithmsfunctions with special training algorithms..

2.2. Maximize the leave-one-out error after projection: take localized Maximize the leave-one-out error after projection: take localized function function GG, count in a soft way cases from the same class as X., count in a soft way cases from the same class as X.

Projection may be done directly to 1D, 2D or higher.Projection may be done directly to 1D, 2D or higher.Examples: parity, monks. Examples: parity, monks.

X X''

' ,Q G C C X X

W W X X

Parity n=9Parity n=9Parity n=9Parity n=9Pursuite of the best “point of view” using simple gradient learning; cluster quality index shown below.No problem with large variance noise in 6 channels.

EEG exampleEEG example Data from two electrodes, BCI IIIaData from two electrodes, BCI IIIa

Alcoholics vs. controlsAlcoholics vs. controls

Colors: from blue at the beginning of the sequence, to red at the end.Colors: from blue at the beginning of the sequence, to red at the end.

Left: normal subject; right: alcoholic; task: two matched stimuli, Left: normal subject; right: alcoholic; task: two matched stimuli, 64 channels (3 after PP), 256 Hz sampling, 64 channels (3 after PP), 256 Hz sampling, 1 1 sec, 10 trialssec, 10 trials; single st alc; single st alc..

What can we learn?What can we learn?

FSD shows global mapping of the whole trajectory.FSD shows global mapping of the whole trajectory. Pairs of probe functions show different aspects.Pairs of probe functions show different aspects. Where is the trajectory most of the time? Where is the trajectory most of the time?

Low/high energy synchronization.Low/high energy synchronization.

Supervised clustering for characterization of the basins Supervised clustering for characterization of the basins of attractors, transition probabilities, types of oscillations of attractors, transition probabilities, types of oscillations around each attractor.around each attractor.

Clear differences between different conditions, perhaps Clear differences between different conditions, perhaps useful in classification and diagnosis, if standardized.useful in classification and diagnosis, if standardized.

More tests on real data needed. More tests on real data needed.

Future plansFuture plans

More complex models to understand More complex models to understand how to interpret the FSD plots. how to interpret the FSD plots.

Effects of various component-based transformations.Effects of various component-based transformations. Projection pursuit is important, raw signals quite messy.Projection pursuit is important, raw signals quite messy. Identifying interesting segments: projection pursuit in Identifying interesting segments: projection pursuit in

space and time.space and time. Learning of parameters of probe functions that show Learning of parameters of probe functions that show

interesting structures.interesting structures. Analysis of types of behavior using the models of spiking Analysis of types of behavior using the models of spiking

networks (RRG and other models).networks (RRG and other models). BCI applications? Many other things … BCI applications? Many other things …

Thank Thank youyoufor for

lending lending your your ears ears

......

Google: W. Duch => Papers & presentationsSee also http:www.e-nns.org