1. Global Visualization of Neural Dynamics
Krzysztof Dobosz, Włodzisław Duch
Department of Informatics
Nicolaus Copernicus University, Toruń, Poland
Google: W. Duch
Neuromath, Jena, April 2008

2. Example of a pathological signal analysis
Brain Spirography
Example of a pathological signal analysis

3. Motivation
Analysis of multi-channel, non-stationary, time series data.
Neural respiratory rhythm generator (RRG): hundreds of neurons, what is the system doing?
Information is in the trajectories, how to see them?
Component-based analysis.
Time-frequency analysis.
Recurrence plots.
Fuzzy Symbolic Dynamics (FSD), visualize + understand.
Understand FSD mappings using model data.
First look at RRG data.
First look at real EEG data.

4. Fuzzy Symbolic Dynamics (FSD)
Trajectory of dynamical system (neural activities, av. rates):
1. Standardize data.
2. Find cluster centers (e.g. by k-means algorithm): m1, m2 ...
3. Use non-linear mapping to reduce dimensionality:
Localized probe function:
sharp indicator functions => symbolic dynamics;
soft membership functions => fuzzy symbolic dynamics.

5. Model, radial/linear sources
Sources generate waves on a grid
Flat wave
Radial wave
Relatively simple patterns arise, but slow sampling shows numerical artifacts.
Ex: one and two radial waves.

6. Respiratory Rhythm Generator
3 layers, spiking neurons, output layer with 50 neurons

7. Sensitive differences?

8. FSD development
Optimization of parameters of probe functions to see more structure from the point of view of relevant task.
Learning: supervised clustering, projection pursuit based on quality of clusters => projection on interesting directions.
Measures to characterize dynamics: position and size of basins of attractors, transition probabilities, types of oscillations around each attractor.
Visualization in 3D and higher (lattice projections etc).
Tests on model data and on the real data.

9. Complex logic What is needed to understand data with complex logic?
cluster non-local areas in the X space, use projections W.X
capture local clusters after transformation, use G(W.X-q)
SVMs fail because the number of directions W that should be
considered grows exponentially with the size of the problem n.
What will solve it?
A class of constructive neural network solution with G(W.X-q) functions with special training algorithms.
Maximize the leave-one-out error after projection: take localized function G, count in a soft way cases from the same class as X.
Projection may be done directly to 1D, 2D or higher. Examples: parity, monks.

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

11. EEG example
Data from two electrodes, BCI IIIa

12. Alcoholics vs. controls
Colors: from blue at the beginning of the sequence, to red at the end.
Left: normal subject; right: alcoholic; task: two matched stimuli, 64 channels (3 after PP), 256 Hz sampling, 1 sec, 10 trials; single st alc.

13. What can we learn? FSD shows global mapping of the whole trajectory.
Pairs of probe functions show different aspects.
Where is the trajectory most of the time? Low/high energy synchronization.
Supervised clustering for characterization of the basins of attractors, transition probabilities, types of oscillations around each attractor.
Clear differences between different conditions, perhaps useful in classification and diagnosis, if standardized.
More tests on real data needed.

14. Future plans
More complex models to understand how to interpret the FSD plots.
Effects of various component-based transformations.
Projection pursuit is important, raw signals quite messy.
Identifying interesting segments: projection pursuit in space and time.
Learning of parameters of probe functions that show interesting structures.
Analysis of types of behavior using the models of spiking networks (RRG and other models).
BCI applications? Many other things …

15. Thank you for lending your ears ...
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