Recent experimental advances have made it possible to record up to several hundreds of neurons simultaneously in the cortex or in the retina. Analysing such data requires mathematical and numerical methods to describe the spatio-temporal correlations in population activity. This can be done thanks to Maximum Entropy method. Here, a crucial parameter is the product NxR where N is the number of neurons and R the memory depth of correlations (how far in the past does the spike activity affects the current state). Standard statistical mechanics methods are limited to spatial correlation structure with
R = 1 (e.g. Ising model) whereas methods based on transfer matrices, allowing the analysis of spatio-temporal correlations, are limited to NR = 20.
In the first part of the thesis we propose a modified version of the transfer matrix method, based on the parallel version of the Montecarlo algorithm, allowing us to go to NR = 100.
In the second part we present EnaS, a C++ library with a Graphical User Interface developed for neuroscientists. EnaS offers highly interactive tools that allow users to manage data, perform empirical statistics, modeling and visualizing results.
Finally, in a third part, we test our method on synthetic and real data sets. Real data set correspond to retina data provided by neuroscientists partners. Our non extensive analysis shows the advantages of considering spatio-temporal correlations for the analysis of retina spike trains, but it also outlines the limits of Maximum Entropy methods.
For more information about the software that I co-developed with my colleagues, please visit this page:
https://enas.inria.fr/
For more information about the publications, please visit this page:
https://scholar.google.fr/citations?user=L97ZODwAAAAJ
For the thesis, please visit this link:
https://www.theses.fr/178166669
Similar to Analysis of large scale spiking networks dynamics with spatio-temporal constraints: application to Multi-Electrodes acquisitions in the retina
DFA minimization algorithms in map reduceIraj Hedayati
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Similar to Analysis of large scale spiking networks dynamics with spatio-temporal constraints: application to Multi-Electrodes acquisitions in the retina (20)
6. Probabilistic
Models
Maximum
entropy
1 time-step memory
(Marre et al 09)
Generalized
Linear model
Point
process
General framework
(Vasquez et al 12)
Ising
(Schneidman
et al 06)
Triplets (Ganmor et al
09)
Spatio-
Temporal
Spatial
No memory
Limited to
1 time step
memory
Limited to
small scale
Neurons are considered
conditionally independent
given the past
Hawks
Linear Non
Linear model
# of neurons doubles
every 8 years !!
6
7. Goal
โข Definitions
โ Basic concepts
โ Maximum entropy principle (Spatial
& Spatio-temporal).
โข Montecarlo in the service of
large neural spike trains
โข Fitting parameters
โ Tests on synthetic data
โ Application on real data
โข The EnaS software
โข Discussion
Develop a
framework to fit
spatio temporal
maximum entropy
models on large
scale spike trains
7
8. Goal
โข Definitions
โ Basic concepts
โ Maximum entropy principle (Spatial
& Spatio-temporal).
โข Montecarlo in the service of
large neural spike trains
โข Fitting parameters
โ Tests on synthetic data
โ Application on real data
โข The EnaS software
โข Discussion
Develop a
framework to fit
spatio temporal
maximum entropy
models on large
scale spike trains
8
23. โ ๐คโฒ ๐ค = ๐โ ๐0
๐ท , if ๐คโฒ ๐ค is a legal transition
0 Otherwise
Transfer matrix
Non normalized
Perron-Frobenius
Theorem
Right eigenvector
Left eigenvector
๐ ๐
The biggest eigenvalue
๐ (. )
L(. )
Using Chapmanโ
Kolmogorov equation
๐ ๐0
๐
=
๐โ ๐0
๐
๐ ๐
๐โ๐ท+1 ๐ ๐ ๐โ๐ท
๐
๐ฟ(๐0
๐ท
)
๐ซ โ = log ๐ ๐
Direct computing of
the Kullback-Leibler
Divergence
๐ ๐พ๐ฟ ๐ ๐
๐
, ๐ ๐ = ๐ซ โ โ ๐ ๐
๐
โ โ ๐ฎ ๐ ๐
๐
2 ๐๐ท
Compute the
average of
monomials
๐ ๐๐ =
๐๐ซ โ
๐๐๐
23
Pressure Entropy
Empirical
probability
of the
potential
24. Setting the
constraints
Computing the
empirical
distribution
๐ ๐
๐
(๐๐)
Random set of
parameters
Computing the
predicted
distribution
๐ ๐(๐๐ก) ๐๐
Update the
parameters
Final set of
parameters
Predicted
distribution
Comparison
Transfer
Matrix
24
25. Limitation of the transfer matrix โ ๐ค๐ค
โฒ
2 ๐๐ท
2 ๐๐ท
โ ๐คโฒ ๐ค ๐, ๐ โ โ ๐ท๐๐ข๐๐๐ = 1 ๐ต๐ฆ๐ก๐
2 ๐๐ท ร 2 ๐๐ท = 22๐๐ท ๐ต๐ฆ๐ก๐๐ โ ๐คโฒ ๐ค
Memoryneed
Neuron number
Range: R = D+1 = 3
20 neurons
๏จ
1,099,511,627,776 ๐๐ต
25
26. ๐๐ = 20
Small scale Large scale
๐๐ > 20๐๐ โค 20
Transfer matrix Montecarlo
26
27. Computing the
predicted
distribution
๐ ๐(๐๐ก) ๐๐
Setting the
constraints
Computing the
empirical
distribution
๐ ๐
๐
(๐๐)
Random set of
parameters
Update the
parameters
Final set of
parameters
Predicted
distribution
Comparison
Transfer
Matrix
27
28. Goal
Develop a
framework to fit
spatio temporal
maximum entropy
models on large
scale spike trains
โข Definitions
โ Basic concepts
โ Maximum entropy principle (Spatial
& Spatio-temporal).
โข Montecarlo in the service of
large neural spike trains
โข Fitting parameters
โ Tests on synthetic data
โ Application on real data
โข The EnaS software
โข Discussion
28
34. Algorithm review Start: Random spike
train
Parameters ๐
๐ neurons.
Length = ๐
Choose a random
event and flip it
Compute ๐ฮโ ๐
๐ฮโ ๐ > ๐
๐ โ [0,1]
Choose between
[๐ท + 1, ๐ โ ๐ท โ 1]
No
Accept the
change
Reject the
change
Yes
Updated Montecarlo
spike train
๐๐๐๐๐ / Loop
34
Computed only
between [โ๐ท, +๐ท]
35. Hassan Nasser, Olivier Marre, and Bruno Cessac. Spike trains analysis using
Gibbs distributions and Montecarlo method. Journal of Statistical Mechanics:
Theory and experiments, 2013.
35
37. Goal
Develop a
framework to fit
spatio temporal
maximum entropy
models on large
scale spike trains
โข Definitions
โ Basic concepts
โ Maximum entropy principle (Spatial
& Spatio-temporal).
โข Montecarlo in the service of
large neural spike trains
โข Fitting parameters
โ Tests on synthetic data
โ Application on real data
โข The EnaS software
โข Discussion
37
38. Fitting parameters / concept
Maximizing entropy
(difficult because computing the exact entropy intractable)
โก
minimizing the divergence
๐ ๐พ๐ฟ ๐ ๐
๐
, ๐ ๐ = ๐ซ ๐ โ ๐ ๐
๐
โ โ ๐ฎ[๐ ๐
๐
]
Dudรญk, M., Phillips, S., and Schapire, R. (2004). Performance guarantees for
regularized maximum entropy density estimation. Proceedings of the 17th Annual
Conference on Computational Learning Theory.
Small scale: easy to compute
Large scale: hard to compute
- Bounding the negative log likelihood
Divergence
Iterations
Big ๐ ๐พ๐ฟ
Small ๐ ๐พ๐ฟ
38
- Relaxation
39. Fitting parameters / concept
Maximizing entropy
(difficult because computing the exact entropy intractable)
โก
minimizing the divergence
๐ ๐พ๐ฟ ๐ ๐
๐
, ๐ ๐ = ๐ซ ๐ โ ๐ ๐
๐
โ โ ๐ฎ[๐ ๐
๐
]
Dudรญk, M., Phillips, S., and Schapire, R. (2004). Performance guarantees for
regularized maximum entropy density estimation. Proceedings of the 17th Annual
Conference on Computational Learning Theory.
Small scale: easy to compute
Large scale: hard to compute
- Bounding the negative log likelihood
Divergence
Iterations
Big ๐ ๐พ๐ฟ
Small ๐ ๐พ๐ฟ
39
- Relaxation
40. Fitting parameters / concept
Hassan Nasser and Bruno Cessac. Parameters fitting for spatio-temporal
maximum entropy distributions: application to neural spike trains. Submitted to
Entropy.
- Bounding the Divergence
40
- With relaxation
42. Setting the
constraints
Computing the
empirical
distribution
๐ ๐
๐
(๐๐)
Random set of
parameters
Computing the
predicted
distribution
๐ ๐(๐๐ก) ๐๐
Update the
parameters
Final set of
parameters
Exact predicted
distribution
Comparison
Monte-
Carlo
Fitting
42
43. Updating the target distribution
๐2
๐ซ[๐ ]
๐๐๐ ๐๐ ๐
=
๐=โโ
โ
๐ถ๐๐ ๐
๐ ๐+๐น ๐๐ = ๐ ๐ ๐๐ +
๐
๐2
๐ซ[๐ ]
๐๐๐ ๐๐ ๐
๐ฟ ๐ +
1
2
๐,๐,๐
๐3
๐ซ[๐]
๐๐๐ ๐๐ ๐ ๐๐๐
๐ฟ๐ ๐ฟ ๐ ๐ฟ๐ + โฏ
New ๐ ๏จ New distribution.
Montecarlo
Taylor
Expansion
Previous
distribution
Exponential decay of correlation
๏จ
In practice n is finite
(If ๐น ๐๐๐๐๐)๐ = ๐๐ข๐๐๐ก๐๐๐(๐)
43
โ๐๐๐ฃ๐ฆ ๐ก๐ ๐๐๐๐๐ข๐ก๐
50. Data Courtesy: Michael J. Berry II (Princeton University)
and Olivier Marre (Institut de la vision, Paris).
Purely spatial pairwise
Pairwise with 1 time-step memory
Binned at 20 ms
Application on retinal data
50
Schneidman et al 2006
51. Real data: 20 neurons
Spatial Pairwise Spatio-temporal Pairwise
51
52. Real data: 40 neurons
Pairwise Spatial Pairwise Spatio-temporal
52
53. Goal
Develop a
framework to fit
spatio temporal
maximum entropy
models on large
scale spike trains
โข Definitions
โ Basic concepts
โ Maximum entropy principle (Spatial
& Spatio-temporal).
โข Montecarlo in the service of
large neural spike trains
โข Fitting parameters
โ Tests on synthetic data
โ Application on real data
โข The EnaS software
โข Discussion
53
54. Event neural assembly Simulation
(EnaS)
V1
2007
V2
2010
V3
2014
Thierry Viรฉville
Bruno Cessac
Juan-Carlos Vasquez / Horacio-Rostro Gonzalez/Hassan Nasser
Selim Kraria
+ Graphical user interface
Goal:
Analyzing spike trains
Share research advances with the community
C++ & Qt (interface Java, Matlab, PyThon)
54
55. Architecture
EnaS
RasterBlock Gibbs Potential
Graphical User
interface
- Data management
- Formats
- Empirical statistics
- Grammar
- Defining models
- Generating artificial
spike trains
- Fitting
- Montecarlo process
(Parallelization)
- Interactive environment
- Visualization of
stimulus and response
simultaneously.
- Demo
Contributions Contributions
55
61. Map: C++ data container Sorting in a chosen order
โฆ
61
62. Map: C++ data container Sorting in a chosen order
โฆ
It appeared two times!
62
63. Architecture
EnaS
RasterBlock Gibbs Potential
Graphical User
interface
- Data management
- Grammar
- Empirical statistics
- โฆ
- Defining models
- Generating artificial
spike trains
- Fitting
- Montecarlo process
(Parallelization)
- Interactive environment
- Visualization of
stimulus and response
simultaneously.
- Demo
63
64. Parallelization of Montecarlo process
๐๐ก๐๐๐๐
xx
x
x x
x
x x
x
x x
x x
x
x x
x
x
x x
Personal Multi-processors computer: 2-8 processors
Cluster (64 processors machines at INRIA)
OpenMp
64
MPI More processors / More time consuming in our case
72. Synthetic data Vs Real data
Synthetic data
Potential shape is known
(monomials are known)
Real data
72
Potential shape is
unknown
(monomials are
unknown)Fitting only
Guessing the shape
+
Fitting
73. Monomials
Model
Canonical
Ising, pairwise with
delay, triplets, โฆ
Small scaleLarge scale
- Big computation time
- Non Observed
monomials
- Estimation errors.
Pre-Selection
73
Rodrigo Cofre & Bruno Cessac
40 neurons
74. Making sense of parameters
Model
parameters
Evaluate the importance
of particular type of
correlations
Possibility of generalize
the model prediction on
new stimulus
74
76. EnaS
Retina Spike sorting Spike trainStimulus
Visualization
Visualization
+
Empirical analysis
+
Maximum Entropy
modelling
NowFuture
- More empirical
observation
packages
- More neural
coding
functionalities
Spike sorting
- Receptive
field
- Neurons
selection
Type
identification
- Stimulus
design
- Features
extraction
76
Retina
models
VirtualRetina
77. Next โฆ
Starting a company in IT/Data Analytics:
โ First prize in innovative project competition (UNICE Foundation).
โ Current project: Orientation in education using real surveys.
โ EnaS is in perspective in collaboration with INRIA.
Caty Conraux & Vincent Tricard
77
80. Appendix
โข Tuning Ntimes.
โข Tuning Nflip.
โข Validating montecarlo algorithm.
โข Tunnig delta.
โข MPI Vs OpenMP, memory.
โข Why MPI is not better than OpenMP?
โข Computational complexity of the Montecarlo algorithm.
โข Review of Montecarlo / Nflip.
โข Number of Iterations for fitting.
โข Fluctuations on parameters / Non existing monomials.
โข Epsilon on fitting parameters.
โข Binning.
โข Tests with several stimulus.
โข Granot-Atedgi et al 2013
โข Granot-Atedgi et al 2013
80
85. โข Multiprocessors computers:
โ Personal computer (2-8 processors).
โ Cluster (64 processors machines at INRIA).
โข Parallel programming frameworks:
โ OpenMp: The processors of the same computer divide
the tasks (live memory (RAM) is shared).
โ MPI: several processors on each computer share the
task (Memory in not shared).
4 processors
๏จ Time/4.
Parallelization
64 processors ๏จ Time/64.
85
86. MPI
โข OpenMP is limited to the number of processors
on a single machine.
โข With MPI, 64 processors x 10 machine ๏จ 640
processors.
โข Although we though it would take less time
with MPI, but โฆ! Master
computer1 cluster of 64 proc
Another cluster of 64 proc
Another cluster of 64 proc
Another cluster of 64 proc
The whole
Montecarlo
Spike train
At each change of the memory, there will be a communication between the clusters and the master
๏จAt each flip ๏จ loss of time in communication more than computing 86
88. Computational complexity
Taken for running this algorithm:
๐ถ๐๐๐๐ข๐ก๐๐๐ ๐ก๐๐๐ = ๐. ๐. ๐๐ก๐๐๐๐ . ๐กฮโ ๐
๐กฮโ ๐
= ๐๐ก(๐ฟ)
Start: Random
spike train
Choose a random
event and flip it
Compute ๐ฮโ ๐
๐ฮโ ๐ > ๐
๐ โ [0,1]
No
Accept the
change
Reject the
change
Yes
Updated Montecarlo
spike train
Loop : ๐. ๐. ๐๐ก๐๐๐๐ 2- In each loop, computing ๐ฮโ ๐ needs to
perform a loop over the monomials.
1- We have a loop over ๐. ๐. ๐๐ก๐๐๐๐ .
On a cluster of 64 processors:
- 40 Neurons Ising: 10 min
- 40 Neurons Pairwise: 20 min
88
89. Start: Random spike
train
Parameters ๐
๐ neurons.
Length = ๐๐ก๐๐๐๐
Choose a random
event and flip it
Compute ๐ฮโ ๐
๐ฮโ ๐ > ๐
๐ โ [0,1]
No
Accept the
change
Reject the
change
Yes
Updated Montecarlo
spike train
Tuning
Loop : ๐๐๐๐๐=
Algorithm
review
๐ ร ๐ ร ๐๐ก๐๐๐๐
Choose between
[๐ท, ๐๐ก๐๐๐๐ โ ๐ท]
89
90. How many iterations do we need?
โข ๐๐ < 20:
โ 50 parallel + 100 sequential
โข ๐๐ < 150:
โ 100 parallel + 100
sequential
90
91. ๐ on parameters fitting
โข Dudik et al does not allow that:
โข ๐ฝ๐ > ๐๐ || ๐ฝ๐ > 1 โ ๐๐. In this case โ ๐ฝ๐ = 0.9 ๐๐
โข We avoid dividing by 0 (
โฆ
๐ ๐
) โฆ by replacing
putting ๐๐ = โโ
91
92. Problem of non- observed monomials
Central limit theorem (Fluctuations on monomials averages)
๏จ๐ ๐ ๐๐ =
๐๐ซ
๐๐
; ๐ ๐
โ
๐๐ + ๐ผ =
๐๐ซ
๐๐
+
๐๐2 ๐ซ
๐๐2 + โฆ =
๐๐ซ
๐๐
+ ๐๐ณ + โฏ
๏จ๐ = ๐โ + ๐ : Fluctuations on parameters
๏จ๐ = ๐ณโ1 ๐ผ
Covariance matrix ๐ณ๐๐ =
๐2 ๐ซ
๐๐ ๐ ๐๐ ๐
Convex
Computing ๐ณ over 1000 potential
shows that a big percentage of ๐ณ is
zero ๏จ ๐ณโ1
will have big value ๏จ
flucutations on ๐ are big.
๐ณ
92
93. Binning
โข Change completely
the statistics.
โข 700% of more new
patterns appear when
we bin at 20.
โข Should be studied
rigorously.
93
95. 95
- Loss of information
- Loosing biological scale
- More dense spike train
- Less non-observed monomials
Why spike trains have been binned in the literature?
- No clear answer.
- Relation between taking binning as a substitute for
memory is not convincing.
- Might be because it allows having more monomials ๏จ
Less dangerous for convexity ๏จ convergence is more
guaranteed.
96. Making sense of parameters
Stimulus 1
Stimulus 2
Stimulus 4
= ๐ ๐ ๐1
= ๐ ๐ ๐2
= ๐ ๐ ๐3
= ๐ ๐ ๐4
Stimulus 3
96