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Workshop “Advances in the collective
behaviour of complex systems”
Potsdam, September 2016
Temporal correlations in neuronal spikes
induced by noise and periodic forcing
J. M. Aparicio Reinoso, M. C. Torrent, Cristina Masoller
Physics Department, Universitat Politecnica de Catalunya
[email protected]
www.fisica.edu.uy/~cris
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Over the years, Arkady’ work has
been a great source of inspiration
2 HAPPY BIRTHDAY!
Wittenberg 2004
The University of Ohio,
April 2016
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Introduction
• Motivation: spiking lasers
that mimic neuronal
behavior
• Symbolic method of time-
series analysis
Results:
• Response to a weak
periodic input:
comparison of optical and
neuronal spikes
• Analysis of ISI sequences
generated by single-
neuron models
Summary
Outline
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Laser
dynamics
Time-series
analysis
Neuron
dynamics
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In our lab: experiments with
semiconductor lasers
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Laser Mirror
WHAT DO LASERS
HAVE TO DO WITH NEURONS?
Similar statistics
of inter-spike
intervals?
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MOTIVATION
5
“a computer that is
inspired by the brain.”
Neuro-synaptic architecture
allows to do things like image
classification at a very low
power consumption.
Science 345, 668 (2014)
• Spiking lasers: photonic neurons?
• potential building blocks of brain-
inspired computers.
• Ultra fast ! (micro-sec vs. mili-sec)
HOW SIMILAR NEURONAL AND OPTICAL
SPIKES ARE?
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6
FHN model
Increasing
the noise
strength
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7
Increasing
the noise
strength
SCL with
feedback
FHN model
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A. Longtin et al, PRL 67 (1991) 656
Optical ISI distribution, data
collected in our lab
Neuron inter-spike interval (ISI)
distribution
when modulation is applied to the
laser current
Similarity of neuronal &
optical spikes?
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A. Longtin IJBC 3 (1993) 651
Optical ISIs Neuronal ISIs
A. Aragoneses et al, Opt. Exp. (2014)
M. Giudici et al, PRE 55, 6414 (1997)
D. Sukow and D. Gautheir, JQE (2000)
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In the spike rate?
Is the timing of the spikes relevant?
• Rate-based information encoding is slow.
• Temporal codes transmit more information.
How neurons encode
information?
HOW TEMPORAL CORRELATIONS CAN
BE IDENTIFIED AND QUANTIFIED?
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Inter-spike-intervals
serial correlation coefficients
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HOW TO INDENTIFY TEMPORAL STRUCTURES?
RECURRENT / INFREQUENT PATTERNS?
iii ttI 1
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Symbolic method of analysis
of ISI sequences
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Ordinal analysis
─ Advantage: the probabilities uncover temporal correlations.
The OP probabilities allow to identify frequent
patterns in the ordering of the data points
Brandt & Pompe, PRL 88, 174102 (2002)
Random data
OPs are
equally probable
─ Drawback: we lose information about the actual values.
1 2 3 4 5 6
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550 555 560 565 570 575 580 585 590 595 6000
0.2
0.4
0.6
0.8
1
To fix ideas: the logistic map
x(i+1)=4x(i)[1-x(i)]
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
0 100 200 300 400 500 6000
0.2
0.4
0.6
0.8
1
Time series Detail
Histogram ordinal patterns D=3 Histogram x(i)
1 2 3 4 5 60
50
100
150
200
Forbidden
pattern
• Ordinal analysis provides
complementary
information.
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1 1.2 1.4 1.6 1.8 2-0.2
-0.1
0
0.1
0.2
0.3
Map parameter
P-1
/6
3.5 3.6 3.7 3.8 3.9 4-0.2
-0.1
0
0.1
0.2
0.3
Map parameter
P-1
/6
012
021
102
120
201
210
Logistic map Tent map
Ordinal bifurcation diagrams
3.5 3.55 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.950
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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With D=3 we can study correlations among 4 spikes.
210
0321
With D=4
The number of patterns
grows with the length of
the pattern as D!
012
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D=5: 5!=120 patterns
How to quantify the information?
‒ Permutation entropy
How to select optimal D?
depends on:
─ The length of the data.
─ The length of the correlations.
iip pps log
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Contrasting empirical optical spikes
with synthetic neuronal spikes
- do they have similar ordinal statistics?
- are there more/less frequent patterns?
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Ordinal analysis of ISI correlations
Close to threshold Higher pump current
A. Aragoneses, S. Perrone, T. Sorrentino, M. C.
Torrent and C. Masoller, Sci. Rep. 4, 4696 (2014)
Grey region
computed with
binomial test:
probabilities
consistent with
uniform
distribution with
95% confidence
level.
Ordinal bifurcation diagram
P =1 /6; N > 10,000 ISIs
EEL
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Empirical laser data
DK
iciii )4sin()2sin(2
1
Circle map data
Minimal model of ISI correlations:
modified circle map
=0.23, K=0.04, D=0.002
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• Same “clusters” & same hierarchical structure.
• Modified circle map: minimal model for ordinal correlations.
• Same qualitative behavior found with other lasers & feedback conditions.
= natural frequency
forcing frequency
K = forcing amplitude
D = noise strength
Lang-Kobayashi
time-delay model
Model equations and parameters: A. Aragoneses et al, Sci. Rep. 4, 4696 (2014)
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The circle map describes many excitable systems.
The modified circle map has been used to describe
spike correlations in biological neurons.
A. B. Neiman and D. F. Russell, Models of stochastic
biperiodic oscillations and extended serial correlations in
electroreceptors of paddlefish, PRE 71, 061915 (2005)
Connection with neurons
0 2 4 6 8 10-0.06
-0.04
-0.02
0
0.02
0.04
Modulation amplitude (arb. units)
P-1
/6
Empirical laser data
Modulation amplitude applied
to the laser current (%)
)4sin()2sin(2
1 iiii
K
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FHN model
Gaussian white noise and subthreshold
(weak) modulation: a0 and T such that
spikes are only noise-induced.
Time series with 100,000 ISIs simulated.
T=20
D=0.015
FHN model
0 2 4 6 8 10-0.06
-0.04
-0.02
0
0.02
0.04
Modulation amplitude (arb. units)
P-1
/6
Empirical laser data
Modulation amplitude applied
to the laser current (%)
Good
qualitative
agreement
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Analysis of synthetic ISI sequences
generated by single-neuron models
- more/less frequent patterns encode
information?
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FHN model: role of the noise
strength
a0=0
a0=0.02
T=10
a0=0.02
T=20
• No noise-induced temporal ordering.
• External periodic input induces temporal ordering.
• Preferred ordinal patterns depend on the noise strength.
• Resonant-like behavior.
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Role of the modulation
amplitude
T=20
D=0.035 T=20
D=0.015
• The amplitude of the (weak)
modulation does not modify the
preferred and the infrequent
patterns.
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Role of the modulation period
a0=0.02
D=0.015 a0=0.02
D=0.035
Which is the underlying mechanism? A change of the spike rate?
No direct
relation.
• More probable patterns depend on the period of the
external input and on the noise strength.
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Relation between ordinal
probabilities and C1 , C2
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Varying noise strength, modulation amplitude
and period: all datasets collapsed
clear trend with C2, no trend with C1
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Permutation entropy: computed from ordinal
probabilities & normalized to maximum value.
Are there longer temporal correlations?
28
!log
log
L
ppS
ii
PE
• Modulation period T > I induces long temporal correlations
• Sharp transition seen in Spe at T=10 not detected by C1 or C2
C1 & C2
I4.7
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Statistically significant results?
Influence of the length of the data
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a00 a00 a0=0
• Long datasets are need for a robust estimation of the ordinal
probabilities.
01 & 10
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Take home message: • ordinal analysis is useful for understanding data, uncovering patterns,
• for model comparison, parameter estimation, classifying events, etc.
• robust to noise and artifacts in the data.
Main conclusions • Correlations in optical & neuronal spike sequences compared: good
qualitative agreement.
• Minimal model for optical spikes identified: a modified circle map.
• FHN model with subthreshold modulation and Gaussian white noise o There are preferred ordinal patterns which depend on the noise strength and on
the period of the input signal, but not on (weak) amplitude of the signal.
o resonance-like behavior: certain periods and noise levels maximize the
probabilities of the preferred patterns, enhancing temporal order.
Open issues (ongoing and future work): • Hierarchical & clustered structure: universal feature of excitable systems?
• Mathematical insight: can we calculate the probabilities analytically?
• Role of coupling? induce preferred/infrequent patterns?
• Compare with empirical data (single-neuron ISI sequences)
What did we learn?
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<[email protected] >
Papers at: http://www.fisica.edu.uy/~cris/
THANK YOU FOR YOUR
ATTENTION !
Unveiling the complex organization of recurrent patterns in spiking dynamical
systems
A. Aragoneses, S. Perrone, T. Sorrentino, M. C. Torrent and C. Masoller,
Sci. Rep. 4, 4696 (2014).
Emergence of spike correlations in periodically forced excitable systems
J. A. Reinoso, M. C. Torrent, C. Masoller
PRE in press (2016) http://arxiv.org/abs/1510.09035
Analysis of noise-induced temporal correlations in neuronal spike sequences
J. A. Reinoso, M. C. Torrent, C. Masoller
EPJST in press (2016).