• Linear / nonlinear time series analysis
• Uni- / Bivariate (Synchronization)
• Continuous / discrete time series
• Exemplary application to medical data
- EEG and neuronal recordings
- Epilepsy (“window to the brain”)
6 lectures of 2 hours: Thu, May 12 – Tue, May 31, 2016
Thomas Kreuz (ISC-CNR)([email protected]; http://www.fi.isc.cnr.it/users/thomas.kreuz/)
Time series analysis
• Lecture 1: Example (Epilepsy & spike train synchrony),
Data acquisition, Dynamical systems
• Lecture 2: Linear measures, Introduction to non-linear
dynamics, Non-linear measures I
• Lecture 3: Non-linear measures II
• Lecture 4: Measures of continuous synchronization
• Lecture 5: Measures of discrete synchronization
(spike trains)
• Lecture 6: Measure comparison & Application to epileptic
seizure prediction
Schedule
• Lecture 1: Example (Epilepsy & spike train synchrony),
Data acquisition, Dynamical systems
• Lecture 2: Linear measures, Introduction to non-linear
dynamics, Non-linear measures I
• Lecture 3: Non-linear measures II
• Lecture 4: Measures of continuous synchronization
• Lecture 5: Measures of discrete synchronization
(spike trains)
• Lecture 6: Measure comparison & Application to epileptic seizure prediction
Schedule
• General Introduction
• Example: Epileptic seizure prediction
• Data acquisition
• Introduction to dynamical systems
First lecture: Introduction
Non-linear model systems
Linear measures
Introduction to non-linear dynamics
Non-linear measures
- Introduction to phase space reconstruction
- Lyapunov exponent
Second lecture
Non-linear measures
- Dimension
[ Excursion: Fractals ]
- Entropies
- Relationships among non-linear measures
Third lecture
Motivation
Measures of synchronization for continuous data
• Linear measures: Cross correlation, coherence
• Mutual information
• Phase synchronization (Hilbert transform)
• Non-linear interdependences
Measure comparison on model systems
Measures of directionality
• Granger causality
• Transfer entropy
Fourth lecture
Motivation and examples
Measures of synchronization for discrete data (here: spike trains, but in principle can be any other kind of discrete data)
• Victor-Purpura distance
• Van Rossum distance
• ISI-distance
• SPIKE-distance (& Applications)
Fifth lecture
Spikes / Spike trains
Spike: Action potential (event in which the membrane potential of a neuron rapidly rises and falls.)
Spike train: Temporal sequence of spikes.
Basic assumptions:
All-or-non law: “There is no such thing as half a spike.”Either full response or no response at all(depending on whether firing threshold is crossed or not)
Spikes are stereotypical. Shape does not carry information.
Background activity carries minimal information. Only spike times matter.
Motivation: Spike train (dis)similarity
Three different scenarios:
1. Simultaneous recording of population
Neuronal correlations, pathology (e.g. epilepsy)
2. Repeated presentation of just one stimulus
Reliability
3. Repeated presentation of different stimuli
Stimulus discrimination, neural coding
• Monkey retina (functioning in vitro for ~ 15h)
• Multi-Electrode Array (MEA) recordings (512 electrodes)
• Complete populations of retinal ganglion cells (~ 100 RGCs)
1. Simultaneous recording: Example
0 1 2
60
0
Time [s]
# Tr
ial
One neuron, 60 repetitions: High reliability
2. Repeated stimulus presentation: Example
3. Different stimuli: Neural coding
Neural coding:Relationship between the stimulus and the individual or ensemble neuronal responses
Neural encoding: Map from stimulus to responseAim: Response prediction
Neural decoding: Map from response to stimulusAim: Stimulus reconstruction
Encoding Decoding
Stimulus
Response
Neural coding schemes I
Labelled line coding: Individual neurons code on their own.Identity of neuron that fires a spike matters.
Population coding: Joint activities of a number of neurons.Identity of the neuron is irrelevant. All that is important is that the spike is fired as part of the population response, not which neuron fired it.Advantages: Individual neurons are noisy, summed population is robust. Multi-coding possible. More spikes, thus faster.
See also: Sparseness vs. distributed representation in memory and recognition
Extreme sparseness: Grandmother cellJennifer Aniston neuron (concept cell)
Jennifer Aniston neuron
[Quian Quiroga et al. Nature (2005)]
Sensory-motor system: Cortical homunculus
[Wilder Penfield: Epilepsy and the Functional Anatomy of the Human Brain. 1954]
Primary somatosensory cortex Primary motor cortex
Neural coding schemes II
Rate coding: Most (if not all) information about the stimulusis contained in the firing rate of the neuron
Edgar Adrian 1929 (NP 1932): Firing rate of stretch receptor neurons in the muscles is related to the force applied to the muscle.
Temporal coding: Precise spike timing carries information
Many studies: Temporal resolution on millisecond time scale
No absolute time reference in the nervous systemRelative timing to stimulus onset / other spikes, but also with
respect to ongoing brain oscillation
(Special cases: Latency code, Pattern code, Coincidence code)
Measures of spike train (dis)similarity
- Victor-Purpura distance (Victor & Purpura, 1996)
- van Rossum distance (van Rossum, 2001)
- Event synchronization (Quian Quiroga et al., 2002)
-
- Schreiber correlation measure (Schreiber et al., 2003)
- Hunter-Milton similarity (Hunter & Milton, 2003)
- ISI-distance (ISI = Inter-spike interval) (Kreuz et al., 2007)
- SPIKE-distance (Kreuz et al., 2013)
Overview and comparison: Kreuz et al. JNeurosci Methods, 2007; JNeurophysiol (2013)
Victor-Pupura: Sequence of elementary steps
0 1 2 3 4 5 6 7 8 9
-1
0
1
0
0.406
Output
Input
ISIs
Ratio
Time [s]
ISI-distance: DI=0.06
0 100 200 300 400 500 600 700 800
0
1
0
1
Time [ms]
Spike
trains
Ia
Sa
Motivation: SPIKE-distance
ISI-
Distance
SPIKE-
Distance
0 1 2 3 4 5 6 7 8 9 10 11
1
2
Spike
trains
Time [arbitrary unit]
t
t(1)
P (t) t
(1)
F (t)
t(2)
P (t) t
(2)
F (t)
x(1)
ISI (t)
x(2)
ISI (t)
x(1)
P (t) x
(1)
F (t)
x(2)
P (t) x
(2)
F (t)
tP
(1) (t)
tF
(1) (t)
tP
(2) (t) t
F
(2) (t)
SPIKE-distance
Visualization: Dissimilarity profile
0 200 400 600 800 1000 1200
0
0.4
2
1Spike
trains
S
Time [ms]
0 500 1000 1500 2000 2500 3000 3500 4000
0
0.5
50
25
Spike
trains
Sr
aS
r
a
Time [arbitrary units]
Causal (real-time) SPIKE-distance
Population averages
0 500 1000 1500 2000 2500 3000 3500 4000
40
30
20
10
Time [ms]
G1
G2
G3
G4
10 20 30 40
30
20
10
Spike trains
Sp
ike
tra
ins
S
Spike trains
10 20 30 40
30
20
10
Spike trains
Sr
Spike trains
G1 G2 G3 G4
G4
G3
G2
G1
Spike trains
< S >G
2 3 1 4
Spike train groups
G1 G2 G3 G4
G4
G3
G2
G1
Spike trains
< Sr >
G
0
0.2
0.4
0.6
0.8
1
2 3 1 4
Spike train groups
Advantages
• Perfect time resolution, no binning, no parameter
• Not invariant to shuffling of spikes among spike trains(in contrast to peri-stimulus time histogram, PSTH)
• Time-scale independence
• Computational efficiency
• Online monitoring (Real-time SPIKE-distance)
Applications: - Epilepsy
- Brain-machine interfacing
• Application to continuous data (e.g. EEG)
• Papers and Matlab source codes:
http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html
Leaders and followers
• Quantifying consistency in spatio-temporal propagation patterns
Comparison of continuous measure of synchronization
• Application to epileptic seizure prediction
• Predictive performance
• Statistical validation (Measure profile surrogates)
Today’s lecture
Examples of propagation patterns
• Avalanches• Tsunamis• Chemical waves and diffusion
processes• Epileptic seizures• Epidemic spread of diseases
Setup: Neuronal recordings, set of spike trains
Task: Sort spike trains from
Leader
to
Follower
(in terms of temporal sequence, not causality)
Finding propagation patterns in spike trains
Context
matters!
Coincidence detection
• Counts number of coincidences
• Maximum time lag adapted to local spike rate (scale-free):
2
},,,{min
1111
y
j
y
j
y
j
y
j
x
i
x
i
x
i
x
i
ij
tttttttt No parameter!
Quian Quiroga, Kreuz, Grassberger, PRE 2002
y
x
ti
x
ti
x
tj
y
Sp
ike
tra
ins
Time
t i
x = min{ti+1
x - ti
x, ti
x - ti-1
x }
2
Coincidence detection
Coincidence detection
t i
x = min{ti+1
x - ti
x, ti
x - ti-1
x }
2t i
y = min{t j+1
y - t j
y, t j
y - t j-1
y }
2t ij = min{t i
x,t j
y}
y
x
ti
x
ti
x
tj
y
tj
y
ti
x
ti
x
tj
y
tj
y
Time
Sp
ike
tra
ins
2
},,,{min
1111
y
j
y
j
y
j
y
j
x
i
x
i
x
i
x
i
ij
tttttttt
SPIKE-synchronization
0 100 200 300 400 500 600 700 800 900 1000
2
1
Spike trainsa
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
C=0.769
Time
Profileb
y
x
ti
x
ti
x
tj
y
tj
y
ti
x
ti
x
tj
y
tj
y
Time
Sp
ike
tra
ins
1 0
Ci
(1,2) =1 if min j (| ti
(1) - t j
(2) |) < t ij
(1,2)
0 else
ì
íï
îï
Mean value: Fraction
of coincident spikes
Kreuz et al. 2015; Mulansky et al. 2015
+1+1 0
0 200 400 600 800 1000
0
1
Reliable
Time
Spike
trains
C
1
SPIKE-synchronization
0 200 400 600 800 1000
0
1
Bursts
Time
Spike
trains
C
0
SPIKE-synchronization
0 200 400 600 800 1000
0
1
Random
Time
Spike
trains
C
0.3289
SPIKE-synchronization
Symmetric measure of coincidence: invariant to order of spikes and spike trains
0 100 200 300 400 500 600 700 800 900 1000
2
1
Spike trainsa
0 100 200 300 400 500 600 700 800 900 1000
0
0.5
1
C=0.769
Time
Profileb
SPIKE-order
• Asymmetric order indicator:
SPIKE-order : Is this spike leading or following?
Di
(n,m) = Ci
(n,m) ×sign(t j
(m) - ti
(n))
Dj
(m,n) = Cj
(m,n) ×sign(ti
(n) - t j
(m)) = -Di
(n,m)
)( ktD
SPIKE-order profile D
by construction D = 𝐷(𝑡) = 0
Very nice for color-coding of spikes, but not meaningful as a profile.
SPIKE-order and spike train order
• Asymmetric order indicator:
SPIKE-order : Is this spike leading or following?
Di
(n,m) = Ci
(n,m) ×sign(t j
(m) - ti
(n))
Dj
(m,n) = Cj
(m,n) ×sign(ti
(n) - t j
(m)) = -Di
(n,m)
)( ktD
SPIKE-order and spike train order
• Asymmetric order indicator:
SPIKE-order : Is this spike leading or following?
• Symmetric order indicator:
Spike train order : Are the two spike trains in the correct order?
Di
(n,m) = Ci
(n,m) ×sign(t j
(m) - ti
(n))
Dj
(m,n) = Cj
(m,n) ×sign(ti
(n) - t j
(m)) = -Di
(n,m)
Ei
(n,m) = Ci
(n,m) ×sign(t j
(m) - ti
(n))
Ej
(m,n) = Cj
(m,n) ×sign(t j
(m) - ti
(n)) = Ei
(n,m)
)( ktD
)( ktE
𝑛 < 𝑚
Spike train order profile E
0+1 -1 -1+1
Multivariate profiles
SPIKE-order
Spike train order-profile
SPIKE-order profile
Synfire pattern
Synfire indicator F
𝑭 =𝑫 𝒏<𝒎
ቀ𝑵 − 𝟏)σ𝒏𝑴𝒏
𝑫 𝒏,𝒎 =
𝒊
𝑫𝒊𝒏,𝒎
𝑫 𝒏<𝒎 =
𝒏<𝒎
𝑫 𝒏,𝒎
Finding the best spike train order: Maximal F
Simulated annealing
• Heuristic optimization algorithm
• Useful when the search space it too large for brute force (spike train permutations)
• Only tries small portion of possibilities: uses cost function and random variable to determine where to head next
• Search space for order of spike trains grows very fast when more spike trains are added
Number of spike trains N
Number of permutations N!
2 2
4 24
6 720
8 40320
10 3628800
𝑭 =𝑫 𝒏<𝒎
ቀ𝑵 − 𝟏)σ𝒏𝑴𝒏
Optimal order and significance
Poisson test sets
Neuronal data
• Giant depolarizing potentials
• Calcium imaging
• Rat hippocampus
Global event detection
Neuronal data
Climate science data: El Nino
• Sea Surface Temperatures (SST)• Deviations from long term mean in degrees• Setting threshold to gaussian filtered data gives spikes
Results of the method
• Synfire indicator Fu for the unsorted spike trains
• Synfire indicator Fs for the sorted spike trains
• Significance of Fs for the sorted spike trains
• Order of the sorted spike trains (leaders – followers)
Conclusions
• Leader to follower dynamics and the significance of the result
• Parameter free and time scale adaptive
• Conseptually intuitive interpretation
• Low computational cost
• Can be applied to all kinds of discrete events
Outreach
• Available online: • ISI-distance• SPIKE-distance• SPIKE-synchronization
SPIKY (Matlab GUI) and
PySpike (Python) packages
cSPIKE (Matlab cmd, MEX) coming soon
Download-page • Papers and Matlab source codes:
http://www.fi.isc.cnr.it/users/thomas.kreuz/sourcecode.html• Python source codes:
http://mariomulansky.github.io/PySpike/
Py Spike
Introduction and motivation
Comparitive investigation:
Predictive performance of measures of synchronization
Statistical validation of seizure predictions:
The method of measure profile surrogates
Summary and outlook
Predictability of epileptic seizures
- Content -
~ 1 % of world population suffers from epilepsy
~ 22 % cannot be treated sufficiently
~ 70 % can be treated with antiepileptic drugs
~ 8 % might profit from epilepsy surgery
Exact localization of seizure generating area
Delineation from functionally relevant areas
Aim: Tailored resection of epileptic focus
Predictability of epileptic seizures
- Introduction: Epilepsy -
Intracranially implanted electrodes
TBARTBPR TBAL
TBPL
TL TR
FLRFPR
FPLFLL
TLL
TLR
RL
RL
RL
EEG containing onset of a seizure (preictal and ictal)
L
R
EEG in the seizure-free period (interictal)
L
R
Predictability of epileptic seizures
- Motivation I -
Open questions:
Does a preictal state exist?
Do characterizing measures allow a reliable detection of this
state?
Goals / Perspectives:
Increasing the patient‘s quality of life
Therapy on demand (Medication, Prevention)
Understanding seizure generating processes
Predictability of epileptic seizures
- Motivation II -
State of the art:
Reports on the existence of a preictal state, mainly based on
univariate measures
Gradual shift towards the application of bivariate measures
Little experience with continuous multi-day recordings
No comparison of different characterizing measures
Mostly no statistical validation of results
Predictability of epileptic seizures
- Motivation III -
Why bivariate measures?
Synchronization phenomena key feature for establishing the
communication between different regions of the brain
Epileptic seizure: Abnormal synchronization of neuronal ensembles
First promising results on short datasets:
“Drop of synchronization” before epileptic seizures *
* Mormann, Kreuz, Andrzejak et al., Epilepsy Research, 2003; Mormann, Andrzejak, Kreuz et al., Phys. Rev. E, 2003
I. Continuous EEG – multichannel recordings
II. Calculation of a characterizing measure
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
- Sensitivity
Performance
- Specificity
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
1 2 3 4 5 6 7 80
0.5
1M
-3
0
3
x (
t)
20 40 60 80 100 120 140 160
-303
y (
t)
t [s]
Predictability of epileptic seizures
- Moving window analysis -
Window
Chan. 1
Chan. 2
1 2 3 4 5 6 7 80
0.5
1M
-3
0
3
x (
t)
20 40 60 80 100 120 140 160
-303
y (
t)
t [s]
Predictability of epileptic seizures
- Moving window analysis -
Window
Chan. 1
Chan. 2
1 2 3 4 5 6 7 80
0.5
1M
-3
0
3
x (
t)
20 40 60 80 100 120 140 160
-303
y (
t)
t [s]
Predictability of epileptic seizures
- Moving window analysis -
Window
Chan. 1
Chan. 2
1 2 3 4 5 6 7 80
0.5
1M
-3
0
3
x (
t)
20 40 60 80 100 120 140 160
-303
y (
t)
t [s]
Predictability of epileptic seizures
- Moving window analysis -
Window
…
Chan. 1
Chan. 2
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Zeit [Tage]
a)R
H
sensitivenot
sensitivenot
specific specific
For this channel combination:
Reliable seperation preictal interictal impossible !
Predictability of epileptic seizures
- Example: Drop of synchronization as a predictor -
Time [Days]
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Zeit [Tage]
a)R
H
Predictability of epileptic seizures
- Example: Drop of synchronization as a predictor -
Selection of best channel combination :
Clearly improved seperation preictal interictal
Significant ? Seizure times surrogates
Time [Days]
Introduction and motivation
Comparitive investigation:
Predictive performance of measures of synchronization
Statistical validation of seizure predictions:
The method of measure profile surrogates
Summary and outlook
Predictability of epileptic seizures
- Content -
I. Continuous EEG – multichannel recordings
II. Calculation of a characterizing measure
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
- Sensitivity
Performance
- Specificity
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
10
15
3
6
17
1
6
5
3
Anfälle
Zeit [Std.]
Pa
tie
nt
30 60 90 120 150 180
A
B
C
D
E
F
G
H
I
I. Database
Seizures
Time [h]
I. Continuous EEG – multichannel recordings
II. Calculation of a characterizing measure
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
- Sensitivity
Performance
- Specificity
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
• Cross Correlation Cmax
• Mutual Information I
• Indices of phase synchronization
based on
and using
• Nonlinear interdependencies Ss and Hs
• Event synchronization Q
Synchronization Directionality
• Nonlinear interdependencies Sa and Ha
• Delay asymmetry q
- Shannon entropy (se)
- Conditional probabilty (cp)
- Circular variance (cv)
- Hilbert phase (H)
- Wavelet phase (W)
W
cv
H
cv
W
cp
H
cp
W
se
H
se , , , , ,
II. Bivariate measures
- Overview -
II. Bivariate measures
- Cross correlation and mutual information -
1.0
0.5
0.0
Cmax I
* *
1.0
0.5
0.0
Cmax I
* *1.0
0.5
0.0
Cmax I
**
0 2
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
0
/2
3/2
0 2
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
0
/2
3/2
0 2
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
0
/2
3/2
II. Bivariate measures
- Phase synchronization -
II. Bivariate measures
- Nonlinear interdependencies -
No coupling:
X
II. Bivariate measures
- Nonlinear interdependencies -
Strong coupling:
1 2
-4
0
4
Kanal 2
Kanal 1
0
25
Q*
q*
Zeit [s]
II. Bivariate measures
- Event synchronization and Delay asymmetry I -
Time [s]
Chan. 1
Chan. 2
I. Continuous EEG – multichannel recordings
II. Calculation of a characterizing measure
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
- Sensitivity
Performance
- Specificity
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
III. Seizure prediction statistics
- Steps of analysis -
Measure profiles of all neighboring channel combinations
Statistical approach:
Comparison of preictal and interictal
amplitude distributions
Measure of discrimination: Area below the
Receiver-Operating-Characteristics (ROC) - Curve
Mormann, Kreuz, Rieke et al., Clin Neurophysiol 2005
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Sensitiv
ität
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Sensitiv
ität
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Sensitiv
ität
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
Se
nsitiv
itä
t
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
-4 -2 0 2 40
0.01
0.02
0.03
0.04
0 0.5 10
0.5
1
ROC-Fläche: 0.86
Sensitiv
ität
1-Spezifizität
III. Seizure prediction statistics: ROC
Sen
siti
vit
y
1 - Specificity
ROC-Area
a)
0
0.5
1
ROC-Fläche: 0Se
nsitiv
itä
t
b)
0
0.5
1
ROC-Fläche: 1Se
nsitiv
itä
t
c)
0
0.5
1
ROC-Fläche: -1Se
nsitiv
itä
t
d)
0 0.5 10
0.5
1
ROC-Fläche: -0.003Se
nsitiv
itä
t1-Spezifizität
III. Seizure prediction statistics: ROC
ROC-Area
ROC-Area
ROC-Area
ROC-Area
1 - SpecificityS
ensi
tiv
ity
Sen
siti
vit
yS
ensi
tiv
ity
Sen
siti
vit
y
1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
Zeit [Tage]
cv
H
Time Profile : BON , TR08-TR09
0.5 0.6 0.7 0.8 0.90
1
2
3
4
cv
H
%
InterPrä
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Se
nsitiv
itä
t
1 - Spezifizität
ROC-Fläche: 0.67
III. Seizure prediction statistics: Example
Sen
siti
vit
y
1 - Specificity
ROC-Area
Time [days]
e
For each channel combination 2 * 4 * 2 = 16 combinations
III. Seizure prediction statistics
- Parameter of analysis -
• Smoothing of measure profiles (s = 0; 5 min)
• Length of the preictal interval (d = 5; 30; 120; 240 min)
• ROC hypothesis H
- Preictal drop (ROC-Area > 0, )
- Preictal peak (ROC-Area < 0, )
Optimization criterion for each measure: Best mean over patients
Mormann, Kreuz, Rieke et al., Clin Neurophysiol 2005
I. Continuous EEG – multichannel recordings
II. Calculation of a characterizing measure
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
- Sensitivity
Performance
- Specificity
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
IV. Statistical Validation
- Problem: Over-optimization -
Given performance: Significant or statistical fluctuation?
Good measure: „Correspondence“ seizure times - measure profile
To test against null hypothesis:
Correspondence has to be destroyed
I. Seizure times surrogates II. Measure profile surrogates
Randomization
of measure profiles
Randomization
of seizure times
IV. Statistical Validation
- Seizure times surrogates -
Random permutation of the time intervals between actual
seizures: Seizure times surrogates
Calculation of the seizure prediction statistics for the original as well
as for 19 surrogate seizure times ( p=0.05)
Andrzejak, Mormann, Kreuz et al., Phys Rev E, 2003
1 2 3 4 5
TL01-TL02
TL02-TL03
TL03-TL04
TL04-TL05
TL05-TL06
TL06-TL07
TL07-TL08
TL08-TL09
TL09-TL10
TR01-TR02
TR02-TR03
TR03-TR04
TR04-TR05
TR05-TR06
TR06-TR07
TR07-TR08
TR08-TR09
TR09-TR10
Zeit [Tage]
Ka
na
lko
mb
ina
tio
n
- Results: Measure profiles of phase synchronization -
Time [days]
Ch
ann
el c
om
bin
atio
n
Discrimination of amplitude distributions Interictal Preictal
1. Global effect:
All Interictal All Preictal (1)
2. Local effect:
Interictal per channel comb Preictcal per channel comb (#comb)
Results
- Evaluation schemes -
Mormann, Kreuz, Rieke et al., Clin Neurophysiol 2005
1 2 3 4 5
TL01-TL02
TL02-TL03
TL03-TL04
TL04-TL05
TL05-TL06
TL06-TL07
TL07-TL08
TL08-TL09
TL09-TL10
TR01-TR02
TR02-TR03
TR03-TR04
TR04-TR05
TR05-TR06
TR06-TR07
TR07-TR08
TR08-TR09
TR09-TR10
Zeit [Tage]
Ka
na
lko
mb
ina
tio
n
- First evaluation scheme -
Time [days]
Ch
ann
el c
om
bin
atio
n
0
0.2
0.4
0.6
0.8
1
Cmax
Ise
Hcp
H
cv
Hse
Wcp
Wcv
W Ss
Hs
QS
aH
a
q
Maße
| R
OC
-Flä
che |
n.s.p = n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s. n.s.
Results: First evaluation scheme
| RO
C-A
rea
|
Measures
Discrimination of amplitude distributions Interictal Preictal
1. Global effect:
All Interictal All Preictal (1)
2. Local effect:
Interictal per channel comb Preictcal per channel comb (#comb)
Results
- Evaluation schemes -
Mormann, Kreuz, Rieke et al., Clin Neurophysiol 2005
1 2 3 4 5
TL01-TL02
TL02-TL03
TL03-TL04
TL04-TL05
TL05-TL06
TL06-TL07
TL07-TL08
TL08-TL09
TL09-TL10
TR01-TR02
TR02-TR03
TR03-TR04
TR04-TR05
TR05-TR06
TR06-TR07
TR07-TR08
TR08-TR09
TR09-TR10
Zeit [Tage]
Ka
na
lko
mb
ina
tio
n
- Second evaluation scheme -
Time [days]
Ch
ann
el c
om
bin
atio
n
1 2 3 4 5
TL01-TL02
TL02-TL03
TL03-TL04
TL04-TL05
TL05-TL06
TL06-TL07
TL07-TL08
TL08-TL09
TL09-TL10
TR01-TR02
TR02-TR03
TR03-TR04
TR04-TR05
TR05-TR06
TR06-TR07
TR07-TR08
TR08-TR09
TR09-TR10
Zeit [Tage]
Ka
na
lko
mb
ina
tio
n
- Second evaluation scheme -
Time [days]
Ch
ann
el c
om
bin
atio
n
1 2 3 4 5
TL01-TL02
TL02-TL03
TL03-TL04
TL04-TL05
TL05-TL06
TL06-TL07
TL07-TL08
TL08-TL09
TL09-TL10
TR01-TR02
TR02-TR03
TR03-TR04
TR04-TR05
TR05-TR06
TR06-TR07
TR07-TR08
TR08-TR09
TR09-TR10
Zeit [Tage]
Ka
na
lko
mb
ina
tio
n
- Second evaluation scheme -
Time [days]
Ch
ann
el c
om
bin
atio
n
0 10
0.01
0.02
0.03 TL01-TL02
0 10
0.02
0.04
0.06
0.08
TL02-TL03
0 10
0.01
0.02
0.03 TL03-TL04
0 10
0.02
0.04 TL04-TL05
0 10
0.02
0.04
0.06 TL05-TL06
0 10
0.02
0.04
TL06-TL07
0 10
0.02
0.04
0.06
0.08 TL07-TL08
0 10
0.02
0.04
TL08-TL09
0 10
0.02
0.04
0.06 TL09-TL10
0 10
0.01
0.02
0.03 TR01-TR02
0 10
0.02
0.04
TR02-TR03
0 10
0.05
0.1
TR03-TR04
0 10
0.05
0.1
TR04-TR05
0 10
0.02
0.04
TR05-TR06
0 10
0.02
0.04
0.06
0.08 TR06-TR07
0 10
0.02
0.04
TR07-TR08
0 10
0.02
0.04
TR08-TR09
0 10
0.02
0.04
0.06
0.08
TR09-TR10
1 1.5 2-1
0
1
Inter
Prä
Results: Preictal and interictal distributions
e
0
0.2
0.4
0.6
0.8
1
| R
OC
-Flä
che |
Cmax
Ise
H
cp
Hcv
Hse
Wcp
Wcv
W Ss
Hs
QS
aH
a
q
Maße
.05p = .05 .05 .05 .05 .05 .05 .05 n.s. n.s. n.s. .05 n.s. .05
Results: Second evaluation scheme
| RO
C-A
rea
|
Measures
Predictability of epileptic seizures
- Summary I: Comparison of measures -
General tendency regarding predictive performance:
- Phase synchronization based on Hilbert Transform
- Mutual Information, cross correlation
- …
- Nonlinear interdependencies
Measures of directionality among measures of synchronization
No global effect, but significant local effects
Introduction and motivation
Comparitive investigation:
Predictive performance of measures of synchronization
Statistical validation of seizure predictions:
The method of measure profile surrogates
Summary and outlook
Predictability of epileptic seizures
- Content -
* Kreuz, Andrzejak, Mormann et al., Phys. Rev. E (2004)
Mostly not sufficient data for „Out of sample“ – study (Separation in
training- and test sample)
„In sample“ – Optimization (Selection)
(Best parameter, best measure, best channel, best patient, …)
Statistical fluctuations difficult to estimate
Seizure prediction
- Problem : Statistical validation -
I. Continuous EEG multi channel recordings
II. Calculation of characterizing measures
III. Investigation of suitability for prediction by means of a
seizure prediction statistics
IV. Estimation of statistical significance
Predictability of epileptic seizures
- Procedure -
- Patient A (18 channel combinations)
- Phase synchronization and event synchronization Q
- ROC, same optimization, for every channel combination
- Method of measure profile surrogates
H
cv
IV. Statistical Validation
- Problem: Over-optimization -
Given performance: Significant or statistical fluctuation?
Good measure: „Correspondence“ seizure times - measure profile
To test against null hypothesis:
Correspondence has to be destroyed
I. Seizure times surrogates II. Measure profile surrogates
Randomization
of measure profiles
Randomization
of seizure times
0
0.5
1Ori)
0
0.5
1S1)
0
0.5
1S2)
0
0.5
1S3)
1 2 3 4 50
0.5
1
Zeit [Tage]
S4)
Measure profile surrogates
Zeit [Tage]
Time [days]
Time [days]
• Formulation of constraints in cost function E
• Minimization among all permutations of the original measure profile
• Iterative scheme: Exchange of randomly chosen pairs
Measure profile surrogates
- Simulated Annealing I -
Schreiber, Phys. Rev. Lett., 1998
• Cooling scheme (Temp. T→0), abort at desired precision
Probability of acceptance:
104
105
106
107
10-6
10-5
Te
mp
era
tur
104
105
106
107
10-5
10-4
10-3
10-2
Iterationsschritte
Ko
ste
nfu
nktio
n
Measure profile surrogates
- Simulated Annealing II -
18 channel combinations
(Phase synchronization)
Co
st f
un
ctio
nT
emp
erat
ure
Iteration steps
Measure profile surrogates
- Simulated Annealing III -
Properties to maintain:
Recording gaps are not permuted
Ictal and postictal intervals are not permuted
Amplitude distribution Permutation
Autocorrelation Cost function
1
0
0 1
)(
N
n
nn xxN
C
max
1
)]()([
OriSurr CCE
1 2 3 4-1
-0.5
0
0.5
1
Zeit [Tage]
C ()
Measure profile surrogates
- Original autocorrelation functions (Phase sync.) -
Time [days]
1 2 3 4-1
-0.5
0
0.5
1
Zeit [Tage]
C ()
Measure profile surrogates
- Original autocorrelation functions (Phase sync.) -
Time [days]
0
0.5
1Ori)
0
0.5
1S1)
0
0.5
1S2)
0
0.5
1S3)
1 2 3 4 50
0.5
1
Zeit [Tage]
S4)
Measure profile surrogates
Time [days]
0
0.5
1Ori)
0
0.5
1S1)
0
0.5
1S2)
0
0.5
1S3)
1 2 3 4 50
0.5
1
Zeit [Tage]
S4)
Measure profile surrogates
Time [days]
Measure profile surrogates
- Two evaluation schemes -
• Each channel combination separately
• Selection of best channel combination
0
1 TL01-TL02
0
1 TL02-TL03
0
1 TL03-TL04
0
1 TL04-TL05
0
1 TL05-TL06
0
1 TL06-TL07
0
1 TL07-TL08
0
1 TL08-TL09
0
1 TL09-TL10
0
1 TR01-TR02
0
1 TR02-TR03
0
1 TR03-TR04
0
1 TR04-TR05
0
1 TR05-TR06
0
1 TR06-TR07
0
1 TR07-TR08
0
1 TR08-TR09
0
1 TR09-TR10
Results: Phase synchronization
|ROC|
0
1 TL01-TL02
0
1 TL02-TL03
0
1 TL03-TL04
0
1 TL04-TL05
0
1 TL05-TL06
0
1 TL06-TL07
0
1 TL07-TL08
0
1 TL08-TL09
0
1 TL09-TL10
0
1 TR01-TR02
0
1 TR02-TR03
0
1 TR03-TR04
0
1 TR04-TR05
0
1 TR05-TR06
0
1 TR06-TR07
0
1 TR07-TR08
0
1 TR08-TR09
0
1 TR09-TR10
Results: Event synchronization
|ROC|
0
1 TL01-TL02
0
1 TL02-TL03
0
1 TL03-TL04
0
1 TL04-TL05
0
1 TL05-TL06
0
1 TL06-TL07
0
1 TL07-TL08
0
1 TL08-TL09
0
1 TL09-TL10
0
1 TR01-TR02
0
1 TR02-TR03
0
1 TR03-TR04
0
1 TR04-TR05
0
1 TR05-TR06
0
1 TR06-TR07
0
1 TR07-TR08
0
1 TR08-TR09
0
1 TR09-TR10
Results: Phase synchronization
|ROC|
0
1 TL01-TL02
0
1 TL02-TL03
0
1 TL03-TL04
0
1 TL04-TL05
0
1 TL05-TL06
0
1 TL06-TL07
0
1 TL07-TL08
0
1 TL08-TL09
0
1 TL09-TL10
0
1 TR01-TR02
0
1 TR02-TR03
0
1 TR03-TR04
0
1 TR04-TR05
0
1 TR05-TR06
0
1 TR06-TR07
0
1 TR07-TR08
0
1 TR08-TR09
0
1 TR09-TR10
Results: Event synchronization
|ROC|
Results
- Each channel combination separately -
Phase synchronization:
Event synchronization:
Nominal size: p = 0.05 (One-sided test with 19 surrogates)
Independent tests: q = 18 (18 channel combinations)
At least r rejections:
Significant,
Null hypothesis rejected !
kqkq
rk
ppk
qP
)1(
0000011.0)8( rP
0015.0)5( rP
-1
0
1 Phasensynchronisationa)
RO
C-A
rea
-1
0
1 Event Synchronisationb)
RO
C-A
rea
Results
- ES II: Selection of best channel combination -
Event synchronization
Phase synchronization
Measure profile surrogates
- Two Evaluation schemes -
• Each channel combination separately
Null hypothesis H0 I :
Measure not suitable to find significant number of local effects
predictive of epileptic seizures.
Null hypothesis H0 II :
Measure not suitable to find maximum local effects
predictive of epileptic seizures.
• Selection of best channel combination
Measure profile surrogates
- Two Evaluation schemes -
• Each channel combination separately
Null hypothesis H0 I :
Measure not suitable to find significant number of local effects
predictive of epileptic seizures.
Null hypothesis H0 II :
Measure not suitable to find maximum local effects
predictive of epileptic seizures.
• Selection of best channel combination
-1
0
1 Phasensynchronisationa)
RO
C-A
rea
-1
0
1 Event Synchronisationb)
RO
C-A
rea
Results
- ES II: Selection of best channel combination -
Event synchronization
Phase synchronization
0
1 PhasensynchronisationR
OC
-Flä
ch
e
0
1 Event Synchronisation
RO
C-F
läch
e
Results
- Selection of best channel combination -
Significant!
Null hypothesis H0 II rejected
Not significant!
Null hypothesis H0 II accepted
Event synchronization
Phase synchronization| R
OC
-Are
a |
| RO
C-A
rea
|
Measure profile surrogates
- Summary II: Measure profiles surrogates -
Method for statistical validation of seizure predictions
Test against null hypothesis Level of significance
Estimating the effect of „In sample“ – optimization
Phase synchronization more significant than event synchronization.
Given example:
Discrimination of pre- and interictal intervals:
Introduction and motivation
Comparitive investigation:
Predictive performance of measures of synchronization
Statistical validation of seizure predictions:
The method of measure profile surrogates
Summary and outlook
Predictability of epileptic seizures
- Content -
Predictability of epileptic seizures
- Summary and outlook -
Retrospective investigation:
Evidence of significant changes before seizures
Measures good enough for prospective application ???
• Lecture 1: Example (Epilepsy & spike train synchrony),
Data acquisition, Dynamical systems
• Lecture 2: Linear measures, Introduction to non-linear
dynamics, Non-linear measures I
• Lecture 3: Non-linear measures II
• Lecture 4: Measures of continuous synchronization
• Lecture 5: Measures of discrete synchronization
(spike trains)
• Lecture 6: Measure comparison & Application to epileptic
seizure prediction
Schedule