Lecture series: Data analysis Lectures: Each Tuesday at 16:00 (First lecture: May 21, last lecture: June 25) Thomas Kreuz, ISC, CNR

Lecture 1: Example (Epilepsy & spike train synchrony), Data acquisition, Dynamical systems Lecture 2: Linear measures, Introduction to non-linear dynamics Lecture 3: Non-linear measures Lecture 4: Measures of continuous synchronization Lecture 5: Measures of discrete synchronization (spike trains) Lecture 6: Measure comparison & Application to epileptic seizure prediction Schedule

Introduction to data / time series analysis Univariate: Measures for individual time series - Linear time series analysis: Autocorrelation, Fourier spectrum - Non-linear time series analysis: Entropy, Dimension, Lyapunov exponent Bivariate: Measures for two time series - Measures of synchronization for continuous data (e.g., EEG) cross correlation, coherence, mutual information, phase synchronization, non-linear interdependence - Measures of directionality: Granger causality, transfer entropy - Measures of synchronization for discrete data (e.g., spike trains): Victor-Purpura distance, van Rossum distance, event synchronization, ISI-distance, SPIKE-distance Applications to electrophysiological signals (in particular single-unit data and EEG from epilepsy patients) Epilepsy – “window to the brain” Overview of lecture series

Example: Epileptic seizure prediction Data acquisition Introduction to dynamical systems First lecture

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 Schreiber correlation measure 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

# Trial 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 response Aim: Response prediction Neural decoding: Map from response to stimulus Aim: Stimulus reconstruction EncodingDecoding Stimulus Response

Neural coding schemes 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. Faster. See also: Sparseness vs. distributed representation in memory and recognition Extreme sparseness: Grandmother cell Jennifer 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 cortexPrimary motor cortex

Neural coding schemes Rate coding: Most (if not all) information about the stimulus is 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 system  Relative 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 T, Haas J, Morelli A, Abarbanel HDI, Politi A: Measuring spike train synchrony. JNeurosci Methods 165, 151 (2007) Kreuz T, Chicharro D, Houghton C, Andrzejak RG, Mormann F: Monitoring spike train synchrony. JNeurophysiol 109, 1457 (2013)

Victor-Pupura: Sequence of elementary steps

Van Rossum: D R (τ R =0.1)=1.61

ISI-distance: D I =0.06

Motivation: SPIKE-distance ISI- Distance SPIKE- Distance

SPIKE-distance

Visualization: Dissimilarity profile

Causal (real-time) SPIKE-distance

Instantaneous clustering

Selected averaging

Population averages

Internally triggered averaging

Application to continuous data

Representations Dissimilarity matrix of size N^2 * #(t): Full representation (as seen in movie) Instantaneous dissimilarity (one frame of movie) Temporal averaging (selective, triggered) Spatial averaging - Synchronization among spike train groups (or full population  Measure profile) Temporal and spatial averaging: Overall synchrony

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:

Comparison of spike train distances Capability to reproduce known clustering Comparison of continuous measure of synchronization Application to epileptic seizure prediction Predictive performance Statistical validation Secondary time series analysis / Analysis of measure profiles The method of measure profile surrogates Today’s lecture

Measure comparison

- Associate neuronal network („Black box“) - Time series from 29 neurons (each points) - Two synaptically coupled clusters of 13 neurons (1 and 2), remaining 3 neurons are coupled to all other (shared, S) Validation: Hindemarsh-Rose simulations

HR spike trains from cluster 1: D I =0.019

HR spike trains from clusters 1 and 2: D I =0.032

Minimum cost D V of transforming one train into the other Only three possible transformations: - Adding a spike (cost 1) - Deleting a spike (cost 1) - Shifting a spike (Parameter: Cost c V ) Low c V : D V ~ Difference in spike count (rate code distance) High c V : D V ~ # non-aligned spikes (coincidence distance) Reminder: Victor & Purpura distance D V [Victor & Purpura, J Neurophysiol 76, 1310 (1996)]

Time scale dependence: D V

Distance matrix (pairwise similarities): D I

Hierarchical cluster tree (dendrogram): D I C2C2 C1C1 CSCS

Single linkage algorithm First, the closest pair of spike trains is identified and thereby linked by a П-shaped line, where the height of the connection measures the mutual distance. These two spike trains are merged into a single element, and the next closest pair of elements is then identified and connected. The procedure is repeated iteratively until a single cluster remains. Distance between a pair of clusters:

Confusion matrix : # spike trains from cluster classified as belonging to cluster Correct clustering: diagonal Quantification: Normalized confusion entropy For H=1: Cluster separation Assessing cluster quality

Quantifying clustering performance

Time scale dependence: D V

Clustering performance: Parameter dependence

Performance comparison: Clustering

Correlation among spike train distances

Clustering of spike train distances

Epileptic seizure prediction

 Lecture 6b

Introduction to data / time series analysis Univariate: Measures for individual time series - Linear time series analysis: Autocorrelation, Fourier spectrum - Non-linear time series analysis: Entropy, Dimension, Lyapunov exponent Bivariate: Measures for two time series - Measures of synchronization for continuous data (e.g., EEG) cross correlation, coherence, mutual information, phase synchronization, non-linear interdependence - Measures of directionality: Granger causality, transfer entropy - Measures of synchronization for discrete data (e.g., spike trains): Victor-Purpura distance, van Rossum distance, event synchronization, ISI-distance, SPIKE-distance Applications to electrophysiological signals (in particular single-unit data and EEG from epilepsy patients) Epilepsy – “window to the brain” Overview of lecture series

Thanks a lot for your patience and attention!