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Spike Train Correlations Induced
by Anatomical Microstructure
Stefan Rotter
Computational Neuroscience Lab
Bernstein Center Freiburg & Faculty of Biology
University of Freiburg, Germany
NeuroSeeker
MathStatNeuro workshop • Laboratoire J. A. Dieudonné, Nice • September 8 – 10, 2015
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Dynamic neuronal networks of the brain
Correlations and population signals
Disentangling multi-synaptic pathways
Inferring connectivity from correlations
Dynamic neuronal networks of the brain
Correlations and population signals
Disentangling multi-synaptic pathways
Inferring connectivity from correlations
s
e
ps
na
sy
dedrite
cell body
axon
Ramón y Cajal, 1900
Braitenberg, 1978
The leaky integrate-and-fire model
C U̇ +
1 U − Urest = I
R
dynamical variables:
X X
U (t)
membrane potential
I(t)
input current
fixed parameters:
C
membrane capacitance
R
membrane resistance
Urest
Uthresh
resting potential
threshold potential
Simplifications implied by the leaky integrate-and-fire model:
point neuron
linearity of integration
time-invariance
all parts of the neuron are iso-potential
linear differential equation / linear system
parameters do not change in time
Current injection into neurons
Chemical synapses
Source:
Mark Bear, Barry Connors, Michael Paradiso
Neuroscience: Exploring the Brain, Third Edition, 2006
Intracellular recording in vivo
20
−50
10
−52
−54
−56
−10
membrane potential (mV)
membrane potential (mV)
0
−20
−30
−40
−58
−60
−62
−64
−50
−66
−60
−70
−68
0
500
1000
time (ms)
1500
2000
2500
−70
0
50
100
150
time (ms)
200
250
300
courtesy of V. Bringuier and Y. Fregnac
The leaky integrate-and-fire neuron model
25
λE = 400 spikes/s
λI = 200 spikes/s
100
200
A = 2.7 mV
τm = 20 ms
Membrane potential (mV)
20
15
10
5
0
−5
−10
0
50
I(t) =
150
X
k
250
Time (ms)
Jk δ(t − tk )
300
350
400
(here: Jk = ±A)
450
500
The Mouse Cortex
2×87 mm3
Total volume
Total number of neurons
16 000 000
Number of sensory input fibres
< 1 000 000
Length of axonal tree
10–40 mm
Length of dendritic tree
4 mm
Range of axons
1/0.2 mm
Range of dendrites
0.2 mm
90 000 /mm3
Density of neurons
Density of axons
4 km/mm3
Density of dendrites
0.4 km/mm3
Density of synapses
700 000 000 /mm3
Synapses per neuron
8 000
Probability of synaptic contact
Relative density of axons
Relative density of dendrites
0.1
10
−5
/10−3
10−3
Valentino Braitenberg & Almut Schüz
Cortex: Statistics and Geometry of Neuronal Connectivity
Second Edition, Berlin: Springer, 1998
Biological neuronal networks (BNN)
excitatory neuron
inhibitory neuron
background neuron
High-Performance Neuro-Computing (HPNC)
compute cluster Hathor
# processors/cores
RAM
connectivity
24 × 2 × 2 = 96
24 × 8 GByte = 192 GByte
high-speed Infiniband
http://www.nest-initiative.org
Dynamic neuronal networks of the brain
Correlations and population signals
Disentangling multi-synaptic pathways
Inferring connectivity from correlations
Two different network topologies
Biological neurons
Hybrid neurons
Mean input to individual neurons is identical in both cases!
Population fluctuations depend on correlations
20
20
10
20
Biological
pre neurons
D
100
1000
exc
500
50
0
100
200
2000
Neuron ID
1500
20
inh
1500
Counts
Neuron ID
2000
Counts
C
10
Hybridpre
neurons
100
1000
500
50
0
0
100
Time (ms)
Kriener et al., Neural Computation, 2008
200
0
Time (ms)
Population fluctuations depend on correlations
20
20
10
20
Biological
pre neurons
D
100
1000
exc
500
50
0
100
200
2000
Neuron ID
1500
20
inh
1500
Counts
Neuron ID
2000
Counts
C
10
Hybridpre
neurons
100
1000
500
50
0
0
100
200
0
Time (ms)
Time (ms)
X X
X
Var
Xi =
Var Xi +
Cov Xi , Xj
i
Kriener et al., Neural Computation, 2008
i
i6=j
Shared input structure differs
A
C
B
Dale:
−1
10
D
Hybrid:
k
k
l
l
0
10
E
−2
10
up http://www.nature.com/natureneuroscience
Recording from localized neuronal populations
Buzsáki, Nature Neuroscience, 2004
Figure 1 Unit isolation quality varies as a function of distance from the electrode. Multisite electrodes
(a wire tetrode, for example) can estimate the position of the recorded neurons by triangulation.
trode/o
lighted
Because
invasive
neuron
Further
ative ac
gives r
require
Because
majorit
comput
the neo
to hypo
mini- a
blobs),
of prin
types. T
the sim
neuron
to the h
The
extrace
matical
rons7,17
neuron
Tuning of the LFP during arm movements
3
4
2
5
1
6
8
7
Contralateral movement
LFP (µV)
8
7
6
5
4
3
2
1
750
100
50
0
−500
0
Time (ms)
750
Mehring et al., Nature Neuroscience, 2003
20
20
...
Decoding of arm movements
0.5
0.8
P
0.4
0.6
0.4
0
Dist.−prox. (mm)
d 50
0.2
0
0.6
LFP
LFP
0.3
SUA
MUA
0.2
LFP&SUA
e
LFP&MUAcc=(0.74,0.83)
0.1
Chance level
0
10
20
30
40
0
Number of electrodes
Mehring et al., Nature Neuroscience,
2003
−50
−50
0
Lateral (mm)
50
0
LFP
cc=(0.74,0.83)
e
0
−50
−50
50
0
50
Dist.−prox. (mm)
b
1
Dist.−prox. (mm)
a
Dist.−prox. (mm)
d 50
SUA
0
Lateral (mm)
c
0
−50
−50
Sim.rec.
Diff.days
f
1
cc=(0.93,0.84)
Chance level
50
0.
0.
0
0.8
0.6
cc
0.4
0.2
0
Lateral (mm)
0
0
50
2
4
6
8
Number of electrodes
−50
−50
SUA
0
Po
0.
0
Dynamic neuronal networks of the brain
Correlations and population signals
Disentangling multi-synaptic pathways
Inferring connectivity from correlations
Neuronal microcircuits
Convergent synaptic input onto inhibitory interneurons.
dimensional rendering of axonal contacts onto a postsynaptic neuron.
s at the top represent cell bodies of neurons within the functionally
ane. Axons of a horizontally tuned neuron (cell 4; green) and a
tuned neuron (cell 10; red) descend and make synapses (small yellow
o dendrites of an inhibitory interneuron (cyan). The axonal and
segments leading to the convergence were independently traced by a
180 270
90
Direction (degrees)
360
synaptic potentials (EPSPs), in current–clamp mode (Fig. 1 C
and D). In 270 experiments, we took measurements from 1,345
neurons and 3,446 pairs of neurons. To ensure statistical ro-
second person, blind to the original segmentation (thick tracing). Cell bodies
and axons coloured by orientation preference, as in Fig. 1b. Scale bar, 50 mm.
b, c, Electron micrographs showing the synapses onto the inhibitory neuron
from cell 4 (b) and cell 10 (c) with corresponding colours overlaid. Scale bar,
1 mm. d, e, Orientation tuning curves derived from in vivo calcium imaging of
the cell bodies of cell 4 (d) and cell 10 (e). Coloured bars and arrows, stimulus
orientation and direction. DF/F, change in fluorescence. Error bars, 6s.e.m.
E
B
A
0.2
0.1
λ: -5 mm
L: 2 mm
a
Excitatory target
Inhibitory target
c
F
C
30 um
2
1
11
3
9
4
8
7
d
0.3
0
100
200
300
0
100
200
300
6
0.1
G
10
100 µm
0.0
0.2
12
b
0.3
Reciprocal
Non-Reciprocal
Overall (p)
Connection Probability Connection Probability Connection Probability
0
5
0.0
0.3
0.2
0.1
0.0
0
100
200
300
Intersomatic Distance [um]
12
13
5
11
12
13
11
an
ab
th
ne
tie
bi
ex
ne
re
a
fir
fo
(P
iso
nu
ta
te
pr
bu
D
5
9
9
te
re
th
3A
sy
ce
ev
te
10
2
8
2
10
8
6
6
3
4
14
3
1
7
7
Characterized pyramidal neuron
Synapse
4
Dendritic fragment
Cell body in EM volume
From anatomy to connectivity graphs. a, Three-dimensional
(dendritic fragments) are drawn as squares. (From top to bottom Fig.
and left
1. toPair-wise connectivity. (A) Morphological staining of a cluster of 12
of the dendrites, axons and cell bodies of 14 neurons in the
right: functionally characterized cells 5, 2, 7; 13, 6, 14; 1; 10; 11, 3; 9; 12,
4; and
8.)
cells
recorded
simultaneously. (B) Region of the somatosensory cortex where
lly imaged plane (coloured according to their orientation preference, c, Three-dimensional rendering of the arbors and cell bodies of functionally
recordings were carried out. (C) Connectivity diagram of neurons in D. (D)
as in Fig. 1b), and the dendrites and cell bodies of all their
characterized neurons, along with postsynaptic targets that either receive
Example
ptic targets traced in the EM volume (magenta, excitatory targets;
convergent input from multiple functionally characterized neurons,
or were of recorded traces in an experimental session. A different neuron is
Mouse
cortex
Rat
somatosensory
remaining
neurons werecortex
recorded
bitory targets;
spines on visual
postsynaptic
targets not shown;
themselves functionally characterized (Supplementary Movie 5). d,stimulated
A subset of and the responses of the
ntary MovieCalcium
5). Scale bar, 100
mm. b, Directed
of the the
network graph showing
only the connections in c, all independently
verified in columns). [Scale bars:
(displayed
horizontal,
100 recording
ms; vertical, 1 in
mVvitro
(15 mV
imaging
innetwork
vivo diagram
+ electron
microscopy
in vitro
12-fold
patch
lly characterized cells and their targets, derived from a. Postsynaptic (from top to bottom and left to right: functionally characterized cells
5, action
2, 7; 13, potentials)]. (E–G) Connection probability profiles as a function of
for
et al.,
Perin
et
al.,
PNAS,
2011
(magenta) Bock
and inhibitory
(cyan)Nature,
targets with 2011
cell bodies contained
6; 10; 11, 3; 12, 9, 8 and 4).
distance. Error bars represent SEM.
e EM volume are drawn as circles. Other postsynaptic targets
AT U R E | VO L 4 7 1 | 1 0 M A R C H 2 0 1 1
©2011 Macmillan Publishers Limited. All rights reserved
2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1016051108
ro
se
20
nu
cr
so
Sm
in
by
ap
m
sh
nu
Sy
How does connectivity induce correlations. . .
Pernice, Staude, Cardanobile, Rotter, PLoS Computational Biology 7(5): e1002059, 2011
Pernice, Staude, Cardanobile, Rotter, Phys Rev E 85: 031916, 2012
Pernice, Rotter, Journal of Statistical Mechanics P03008, 2013
Rangan, PRL 102(15): 158101, 2009; PRE 80(3): 036101, 2009
Trousdale, Hu, Shea-Brown, Josić, PLoS Computational Biology 8(3): e1002408, 2012
. . . and to which degree can
connectivity be inferred from correlations?
Undirected and directed graphs
3
3
2
2
4
4
1
5
1
5
7
6
7
6
A graph consists of vertices (nodes) and edges (links). Each
edge connects one pair of vertices. Connections can be either
undirected (left) or directed (right).
The adjacency matrix of an undirected graph
3
2
4
1
5
0
0
0
1
1
0
0
0
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
0
0
0
0
1
1
1
0
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
1
1
0
7
6
The adjacency matrix A = (aij ) fully describes the graph, provided that each vertex has been assigned a unique label. We
have aij = 1 if vertex i and vertex j are connected, and aij = 0
otherwise. Because edges are undirected, the matrix is symmetric, i.e. aij = aji for all i and j, and A = AT .
The adjacency matrix of a directed graph
3
2
4
1
5
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
1
1
1
0
7
6
Again, the adjacency matrix A = (aij ) fully describes a graph
with labeled vertices. We set aij = 1 if there is a link from j to i,
and aij = 0 otherwise. Because here edges are directed, the
matrix is asymmetric, i.e. aij 6= aji and A 6= AT . The transposed
matrix AT corresponds to a graph with all arrows reversed.
The in-degree and out-degree of a directed graph
3
2
4
1
5
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
1
1
1
0
7
6
The in-degree of a vertex is the number of its incoming edges,
the number of its outgoing edges is called out-degree. In the
adjacency matrix, the in-degree is the sum of all entries in the
corresponding row. The out-degree is the sum of all entries in
the corresponding column.
Graphs with weighted or multiple edges
excitatory neuron
inhibitory neuron
background neuron
Paths in a graph
3
2
3
2
4
4
1
1
5
5
7
7
6
6
Starting from a given node, one can follow the links (if there are
any) through the graph, respecting their orientation. If the outdegree of a vertex is larger than one, the path bifurcates. The
graph that indicates which nodes can be reached after two hops
can have multiple edges between any two vertices (not shown).
Powers of the adjacency matrix
3
2
0
0
0
0
0
1
0
0
1
0
1
1
2
0
0
0
0
0
1
0
1
0
1
0
1
2
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
1
1
2
4
1
5
7
6
The graph that indicates which vertex can be reached after two
hops corresponds to the square of the adjacency matrix A2 . Because this derived graph can have multiple edges between any
two vertices, A2 can have entries greater than one. Accordingly,
the matrix power Ak corresponds to paths of length k.
The Hawkes process
A network of N nodes (neurons) is described in terms of
spike train
firing rate
external input
interaction kernel
si (t) =
X
Hawkesmo
mo
Hawkes
δ(t − tik )
k yi (t) = si (t)
Spi
Spik
wh
wher
Rat
Rate
y0 ≥ 0
G(t) = gij (t)
yiy(t)
i (t
Hawkes,Journal
Journalofofthe
theRoyal
RoyalStatistical
StatisticalSoc
So
A.A.Hawkes,
Its dynamics is defined in terms of the linear integral equation
Z ∞
y(t) = y0 +
G(τ )s(t − τ ) dτ = y0 + (G ∗ s)(t).
−∞
Hawkes, 1971
Stationary firing rates
Assuming stationarity y(t) = y, one has
Z ∞
y = y0 +
G(τ )y dτ = y0 + Gy
−∞
with
G=
Z
∞
G(τ ) dτ.
−∞
If the matrix 1 − G is invertible we have
y = [1 − G]−1 y0 .
If, in addition, |λ| < 1 for all eigenvalues λ of G, the usual
geometric series expansion suggests a decomposition into
contributions of recurrent pathways of all orders
y = [1 − G]−1 y0 =
∞
hX
n=0
i
Gn y0 = y0 + Gy0 + G2 y0 + . . .
Stationary correlations
Assuming joint stationarity, the pulse-coded interactions are
conveniently quantified by the covariance functions
cij (τ ) = Cov si (t + τ ), sj (t) = si (t)sj (t + τ ) − si (t) sj (t) .
Using Fourier transforms
Z ∞
ˆ
f (ω) =
f (t)e−iωt dt
and
−∞
fˆ(0) =
Z
∞
f (t) dt
−∞
Using Wiener-Hopf theory, Hawkes (1971) obtained for linearly
interacting point processes
−1 −1
ĉij (ω) = Ĉ(ω) = 1 − Ĝ(ω) Y 1 − Ĝ(ω)∗
where Y = diag(y). For the integrated covariances, one gets
C = Ĉ(0) =
1−G
−1 −1
Y 1 − GT
.
Expanding correlations
Influence of direct and indirect connections
Assuming again |λ| < 1 for all eigenvalues λ of G, the expansion
If absolute
value of eigenvalues
of Gthe
< 1,covariance
Y =1
of [1 − G]−1 can
be exploited
to re-write
(Y = 1)
C = (1 − G )−1 (1 − G T )−1 = (1 + G + G 2 ...)(1 + G T + (G T )2 ...)
∞
hX
∞
ih X
m i
∞
T�
C = [1 − G]= 1 +[1G−+ G
=
G
T ]
T
2 G T 2
G + GG + G + (G ) + ... =
G m (G T )n
−1
T −1
n
n=0
=
1 + G + GT + G 2 + G
T
=G +
G
2
+
G
+
T 2
GT2
m=0
m,n=0
T
+ GG T + . . .
+
GG
+
. . . +1
with matrix elements
e.g.
�
�
X
TX
[G 2 ]ij =
Gik G(1,1)
G G
(2,0)
kj
2
T [GG ]ij =
gij ≡ G ij =
gik gkj
, gij ≡ GG ij =
gikk gjkik , jk. . .
k
(Pernice,
k Staude, Cardanobile, Rotter 2011, compare Rangan
k PRE 2009)
corresponding to shared input contributed by different types of
multi-synaptic pathways (motifs).
Pernice et al., 2011
Shared input motifs contributing to correlations
Fluctuations of population activity
Fluctuations of stationary population activity
X
S(t) =
si (t)
i
can be expanded into contributions from all auto- and
cross-covariances
X
Cov si (t + τ ), sj (t) .
Cov S(t + τ ), S(t) =
ij
For a Hawkes process, we can exploit the power series
expansion for the matrix of integrated covariances
X
X X (n,m) X X (n,m) X
Cij =
gij
=
gij
=
N 2 g (n,m) .
ij
ij
nm
nm ij
nm
5
Negative feedback decorrelates activity
g(m,n)
g(m,n)
B
0.010
0.005
0.000
0.005
0.08
0.04
0.00
0.04
theoretical mean
random output mean
fixed output mean
p = 0.1
p = 0.25
1.0
Average correlations in (almost) regular networks do not depend
on fine-scale structure. For dominant recurrent inhibition, motifs
of uneven order m + n contribute negatively.
Non-regular networks: cliques of excitatory hubs
Patchy and/or cell-type specific connectivity
dence on self-inhibition
from
linear theory, Fig. 3.
Correlations
in networks of
LIFthe
neurons
An increase of Vθ causes weaker impulse responses and
ĉij (ω = 0)
cross-correlation
FIG. 2. (color
(a) Scatter plot of covariances between
Re[ĉijonline)
(ω = 310 Hz)]
Im[ĉ
310
Hz)]
ij (ω = C
neuron pairs
(red:
kj (ω = 0), green: Im[Ckj ](ω = 0.31kHz),
Pernice et al., Phys Rev E 2012
black: Re[Ckj ](ω = 0.31kHz)). Inset: Standard deviation
Higher-order cumulants
A
D
A
D
A
D
A
D
Time
Multivariate Hawkes process
B
B
C
B
B
Hawkes, Oakes, J Appl Prob, 1974
Saichev, Sornette, Phys Rev E, 2014
Jovanović, Hertz, Rotter, Phys Rev E, 2015
C
C
C
A
B
C
D
e.g. the classic text by Felsenstein [27]), the numbe
terms grows very quickly with increasing n (see Figur
- source [27]) and thus computing k i (t) quickly becom
impractical.
A combinatorial challenge
n=2
u1
t1
n=3
t2
u1
u1
u2
t1
t2
n=4
t3
t1
t2
t3
u1
u1
u1
u2
u2
u2
u3
u3
t1
t2
t3
t4
t1
t2
u1
t3
t4
t3
t4
u1
u2
t1
t2
t3
t4
t1
t2
t1
t2
t3
t4
n
Terms in nth order density
2
1
3
4
4
26
5
236
6
2, 752
7
39, 208
8
660, 302
9
12, 818, 912
10
282, 137, 824
11
6, 939, 897, 856
12
188, 666, 182, 784
13
5, 617, 349, 020, 544
14
181, 790, 703, 209, 728
15
6, 353, 726, 042, 486, 272
16
238, 513, 970, 965, 257, 728
17
9, 571, 020, 586, 419, 012, 608
18
408, 837, 905, 660, 444, 010, 496
19 18, 522, 305, 410, 364, 986, 906, 624
20 887, 094, 711, 304, 119, 347, 388, 416
FIG. 3.Jovanović,
NumberHertz,
of terms
in Phys
ki (t)Rev
forE,a 2015
given n
Rotter,
Dynamic neuronal networks of the brain
Correlations and population signals
Disentangling multi-synaptic pathways
Inferring connectivity from correlations
Inferring directions from a symmetric matrix?
Pernice & Rotter, JSTAT, 2013
Fundamental degeneracy of the problem
Given the covariance matrix Ĉ(ω), can one solve the equation
Ĉ(ω) =
1 − Ĝ(ω)
−1 −1
Y 1 − Ĝ(ω)∗
for the connectivity matrix Ĝ(ω)? The Idea followed here is to
determine “some” square root of the inverse covariance matrix
Ĉ −1 (ω) = B̂(ω)∗ B̂(ω)
and extract Ĝ(ω) from the relation (assuming Y is known)
B̂(ω) = Y −1/2 1 − Ĝ(ω) .
However, for any unitary matrix U satisfying U ∗ U = U U ∗ = 1
the matrix  = U B̂ provides an equivalent solution
Â∗ Â = B̂ ∗ U ∗ U B̂ = B̂ ∗ B̂.
Searching for sparse networks
(a)
0
0.05
(b)
0
(c)
0.05
0
5
5
10
0.00 10
0.00 10
neuron
5
15
(d)
0
0
5
10 15
0.05
15
(e)
0.05
0
10
0.00 10
15
15
neuron
5
5
10 15
neuron
0.05
10 15
0
5
0
5
0.05
FN
FP
0
5
10 15
neuron
TN
0.00
15
(f)
TP
0.05
0.05
0
5
10 15
0
10
20
30
40
0 10 20 30 40
neuron
TP
FN
FP
TN
Method: stochastic minimization of the L1 -norm of the coupling
matrix, using the Cholesky decomposition to initiate the search
Inference of connectivity from noise-free Ĉ(0)
1.2
/ [G(0)]
1.0
n=50
(b)
n=300
gE =0.005
gE =0.01
gE =0.02
gE =0.05
0.9
0.8
0.6
0.4
0.2
0.00.0 0.1 0.2 0.3
p
(c)
1.0
true positive rate
(a)
0.8
0.7
0.6
0.5
0.4
0.3
0.0 0.1 0.2 0.3
p
0.20.0
p=0.28
p=0.22
p=0.1
p=0.04
0.2 0.4 0.6 0.8
false positive rate
Inference of connectivity from estimated Ĉ(ω)
n=100
(b)
1.0
0.6
0.4
0.2
0.9
0.8
20 ×103 s
5 ×103 s
2 ×103 s
20 ×103 s
5 ×103 s
2 ×103 s
true positive rate
true positive rate
0.8
(c)
1.0
n=300
0.6
0.4
area under ROC
(a)
1.0
0.2
0.00.0 0.2 0.4 0.6 0.8 1.0 0.00.0 0.2 0.4 0.6 0.8 1.0
false positive rate
false positive rate
0.8
0.7
0.6
0.5
0.2
n=100, all
n=300, all
n=100, =0
n=300, =0
tsim [ms]
2.0
1e4
tate xt , consisting of hand x-/y-position and velocity, how
evolved
to influence that activity by visual
ccording
to
the
following
linear
stochastic
difference
equation:
stimulation.
New applications: “networks of networks”
State model:
t+1
(1)
activity
= MUA
F · xt + omulti-unit
+ wt
LFP local field potentials
where wt is a zero-mean Gaussian noise process with covariance
matrixECoG
Q and o aelectro-corticogram
constant offset. In addition, we assumed that
MREG
MR-encephalography
he observed neuronal
activity depended linearly on the hand
tate according to(“gradient-less
the following linear
equation:
imaging”)
NIRS Near-infrared spectroscopy
Pistohl et al., 2008
10 mm
10 µm
Hubel, 1995
The visual cortex in a monkey, stained by the Go
tate xt , consisting of hand x-/y-position and velocity, how
evolved
to influence that activity by visual
ccording
to
the
following
linear
stochastic
difference
equation:
stimulation.
New applications: “networks of networks”
State model:
t+1
(1)
= F · xt + o + w t
where wt is a zero-mean Gaussian noise process with covariance
matrix Q and o a constant offset. In addition, we assumed that
he observed neuronal activity depended linearly on the hand
tate according to the following linear equation:
Pistohl et al., 2008
10 mm
10 µm
Hubel, 1995
The visual cortex in a monkey, stained by the Go
Continuous variable systems
Consider a multi-component system, with state variables y(t),
driven by fluctuating input x(t). Assume that the system is
characterized by a coupling matrix of response kernels G(t)
and satisfies the linear consistency equation
y(t) = x(t) + (G ∗ y)(t).
In the Fourier domain, after taking expectations, we have
and
−1 ŷ = 1 − Ĝ
x̂
∗ −1 ∗ −1
ŷ ŷ = 1 − Ĝ
x̂x̂ 1 − Ĝ∗ .
In the case of the Hawkes process, the source term x̂x̂∗ is
replaced by Ŷ . In other words, the spikes of the neurons in the
network generate their own “driving noise”.
Application to ECoG data
Ball et al., 2009; Derix et al., 2012
Efficient inference of connectivity
Niederbühl, Diploma thesis, 2014
Different frequency channels in ECoG
Niederbühl, Schiefer, work in progress
Application to MREG data
Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013
Pernice, Niederbühl, LeVan, work in progress
Application to MREG data
Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013
Pernice, Niederbühl, LeVan, work in progress
Application to MREG data
Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013
Pernice, Niederbühl, LeVan, work in progress
Conclusions
I Linear Hawkes processes are useful models for networks of irregularly
spiking neurons. Recurrent networks of LIF neurons can be matched to
an equivalent Hawkes process. We expect that this also applies to real
nerve cells in neocortical networks.
I Dynamic properties of spiking networks of finite size with arbitrary
topology can be inferred analytically, using matrix algebra, provided
couplings are weak and spike trains are irregular.
I Pairwise correlations in large networks have strong impact on the
amplitude of population signals (LFP, ECoG, EEG, MREG, fMRI). The
contribution of specific multi-synaptic pathways (motifs) to pairwise and
higher-order correlations can be computed for any given network.
I The micro-topology of sparse networks can be approximately recovered
from the covariance matrix, employing compressed sensing methods.
Directed links can be inferred from non-directed zero-lag covariances,
but accounting for temporal information improves the inference.
Stefano
Alexander
Volker
Jonathan
John
Stojan
Marcel
Benjamin
Jürgen
Pierre
Tonio
Markus
Sina
Cardanobile
Niederbühl
Pernice
Schiefer
Hertz
Jovanović
Sauerbier
Staude
Hennig
LeVan
Ball
Kern
Bert
Thanks!