Sleep, Criticality and Information Processing in the Brain

Transcription

Sleep, Criticality and Information
Processing in the Brain
Christian Meisel
National Institute of Mental Health
University Clinic Carl Gustav Carus Dresden
Eckehard Olbrich Peter Achermann
Sleep
Criticality
Information Processing in the Brain
Criticality
Sleep
Criticality
Sleep
Sleep
Criticality
Sleep
Criticality
A Theory for Sleep ?
‣observed in all species, from fruitflies to mammals
‣about one third of lifetime is spent asleep
‣„from the brain, for the brain“
‣role unknown
A Theory for Sleep ?
without sleep:
‣reduced responsiveness to stimuli
‣impaired information processing
‣reduced learning ...
Banks, J Clin Sleep Med, 2007
Mignot, PLoS Biol, 2008
... observations suggest that sleep may play an important role in
organizing or reorganizing neuronal networks in the brain
toward states where information processing is optimized
Two-process model of sleep
‣sleep
propensity increases during wake and decreases during sleep
- sleep homeostasis
(Achermann, Brain Res Bul,1992)
‣related to theta and slow-wave activity in the EEG
MIZ
]6 36 O4.'==4 (, C'3
O4/6 \6 3!!$,4-&4$. )'&;''. %4!' $0 123 4. !=''# -.( &+'&- -,&4<4&* 4. ;-@4./ 4=="!&%-&'( 0$% &;$ !")L',&!6 H'-. 123 #'% .$.>GBH !=''# '#4!$(' 4! #=$&&'( -&
&+' )'/4..4./ $0 '-,+ '#4!$('8 -.( 'E#%'!!'( %'=-&4<' &$ &+' )-!'=4.' <-="' $0 &+' T%!& .$.>GBH !=''# '#4!$(' N9JJiP6 BE#$.'.&4-= 0".,&4$.! ;'%' T&&'(
&+%$"/+ &+' (-&- #$4.&! N!$=4( ,"%<'!P6 F+' %'/%'!!4$. =4.' %'#%'!'.&! &+'&- #$;'% 4. ;-@4./ N4.&'%%"#&'( =4.'P6 O$% #%$,'("%' -.( !&-.(-%(4K-&4$.8 !'' O4/6 I6
!"##$%&'( )* &+' ,+-./'! $0 123 ("%4./ (-*&45' .-#!6 7897
:$;'<'%8 4& 4! #$!!4)=' &+-& &+' !-&"%-&4./ &45'>,$"%!' 4! .$&
-. 4.&%4.!4, #%$#'%&* $0 &+' #%$,'!!8 )"& - ,$.!'?"'.,' $0 ,$5#'.!-&$%* %'!#$.!' ("%4./ ;-@4./6 A. $&+'% ;$%(!8 &+'
%4!' $0 BBC &+'&- #$;'% ("%4./ #%$=$./'( ;-@4./ ,$"=( )'
.$& $.=* - !4/. $0 - %4!4./ !=''# #%$#'.!4&*8 )"& -=!$ -. 4.(4,->
&4$. $0 -. -.&-/$.4!&4, #%$,'!! ,$".&'%-,&4./ &+' %4!'6 3! %'!"=& $0 &+4! '00',&8 &+' %4!' %-&' $0 D%$,'!! 1 ;$"=( )'
#%$/%'!!4<'=* -&&'."-&'(8 %'!"=&4./ 4. - !-&"%-&4./ 'E#$.'.&4-=
&45'>,$"%!' $0 1236 F+' 0-,& &+-& &+'&- #$;'% 4. ;-@4./
!+$;'( - =4.'-% 4.,%'-!' -##'-%! &$ )' -& $((! ;4&+ &+4!
0$% )$&+8 &+' =-%/'!& <-="'! -%' ,'.&'%'( $<'% -.&'%4$% ,$%&4,-=
-%'-! NO4/6 MP6 3 0%$.&-= #%'($54.-.,' $0 123 ;-! ($,">
5'.&'( 0$% )-!'=4.' !=''#8 IQ8RJ #-%&4-= !=''# ('#%4<-&4$. R9
-.( &$&-= !=''# ('#%4<-&4$.6 989S 1#',&%-= /%-(4'.&! ,$5#"&'(
0%$5 -(L-,'.& -.&'%$#$!&'%4$% )4#$=-% %',$%(4./! !+$;'(
&+-& &+' 0%$.&-= #%'($54.-.,' $0 #$;'% 4. &+' I>:K )-.(
;-! =-%/'!& 4. &+' T%!& .$.>GBH !=''# '#4!$('8 (454.4!+'(
4. &+' !',$.( '#4!$(' -.( <-.4!+'( 4. &+' !")!'?"'.& &;$
'#4!$('!6 RJ F+' 4.&'%#%'&-&4$. &+-& &+'!' %'/4$.-= ,+-./'!
5-* %'U',& - +4/+'% V%',$<'%* .''(W $0 0%$.&-= +'&'%$5$(-=
-!!$,4-&4$. -%'-! $0 &+' ,$%&'E 4! !"##$%&'( )* #$!4&%$. '54!>
‣sleep
propensity increases during wake and decreases during sleep
- sleep homeostasis
(Achermann, Brain Res Bul,1992)
‣related to theta and slow-wave activity in the EEG
MIZ
]6 36 O4.'==4 (, C'3
O4/6 \6 3!!$,4-&4$. )'&;''. %4!' $0 123 4. !=''# -.( &+'&- -,&4<4&* 4. ;-@4./ 4=="!&%-&'( 0$% &;$ !")L',&!6 H'-. 123 #'% .$.>GBH !=''# '#4!$(' 4! #=$&&'( -&
&+' )'/4..4./ $0 '-,+ '#4!$('8 -.( 'E#%'!!'( %'=-&4<' &$ &+' )-!'=4.' <-="' $0 &+' T%!& .$.>GBH !=''# '#4!$(' N9JJiP6 BE#$.'.&4-= 0".,&4$.! ;'%' T&&'(
&+%$"/+ &+' (-&- #$4.&! N!$=4( ,"%<'!P6 F+' %'/%'!!4$. =4.' %'#%'!'.&! &+'&- #$;'% 4. ;-@4./ N4.&'%%"#&'( =4.'P6 O$% #%$,'("%' -.( !&-.(-%(4K-&4$.8 !'' O4/6 I6
‣Tononi: Synaptic
homeostasis is underlying
sleep
homeostasis
0$% )$&+8 &+' =-%/'!& <-="'!
-%' ,'.&'%'(
$<'% -.&'%4$% ,$%&4,-=
!"##$%&'( )* &+' ,+-./'! $0 123 ("%4./ (-*&45' .-#!6 7897
50
:$;'<'%8 4& 4! #$!!4)=' &+-& &+' !-&"%-&4./ &45'>,$"%!'
4! .$&
-. 4.&%4.!4, #%$#'%&* $0 &+' #%$,'!!8 )"& - ,$.!'?"'.,' $0 ,$5#'.!-&$%* %'!#$.!' ("%4./ ;-@4./6 A. $&+'% ;$%(!8 &+'
%4!' $0 BBC &+'&- #$;'% ("%4./ #%$=$./'( ;-@4./ ,$"=( )'
.$& $.=* - !4/. $0 - %4!4./ !=''# #%$#'.!4&*8 )"& -=!$ -. 4.(4,->
&4$. $0 -. -.&-/$.4!&4, #%$,'!! ,$".&'%-,&4./ &+' %4!'6 3! %'!"=& $0 &+4! '00',&8 &+' %4!' %-&' $0 D%$,'!! 1 ;$"=( )'
#%$/%'!!4<'=* -&&'."-&'(8 %'!"=&4./ 4. - !-&"%-&4./ 'E#$.'.&4-=
&45'>,$"%!' $0 1236 F+' 0-,& &+-& &+'&- #$;'% 4. ;-@4./
!+$;'( - =4.'-% 4.,%'-!' -##'-%! &$ )' -& $((! ;4&+ &+4!
Fig. 1
G. Tononi,
Cirelli
-%'-! NO4/6 MP6 3 0%$.&-= #%'($54.-.,'
$0C. 123
;-! ($,">
IQ8RJ
sleep ends.!=''#8
In addition to #-%&4-=
claiming a correspon!=''# ('#%4<-&4$. R9
5'.&'( 0$% )-!'=4.'
dence between the homeostatic
Process S and total
989S
-.( &$&-= !=''# synaptic
('#%4<-&4$.6
1#',&%-=
/%-(4'.&!
,$5#"&'(
strength, the hypothesis
proposes
specific
mechanisms,
whereby
synaptic
strength
would
0%$5 -(L-,'.& increase
-.&'%$#$!&'%4$%
)4#$=-% %',$%(4./! !+$;'(
during wakefulness and decrease during
and suggests why $0
the #$;'%
tight regulation
of
&+-& &+' 0%$.&-= sleep,
#%'($54.-.,'
4. &+'
I>:K )-.(
synaptic strength would be of great importance for
;-! =-%/'!& 4. &+'
T%!& .$.>GBH !=''# '#4!$('8 (454.4!+'(
the brain.
4. &+' !',$.( '#4!$(' -.( <-.4!+'( 4. &+' !")!'?"'.& &;$
'#4!$('!6 RJ F+' 4.&'%#%'&-&4$. &+-& &+'!' %'/4$.-= ,+-./'!
Synaptic homeostasis: a schematic
5-* %'U',& - +4/+'%
V%',$<'%* .''(W $0 0%$.&-= +'&'%$5$(-=
diagram
-!!$,4-&4$. -%'-! $0 &+' ,$%&'E 4! !"##$%&'( )* #$!4&%$. '54!>
The two-process model involving the circadian
The diagram in Fig. 2 presents a simplified version
Perspe
50
G. Tononi, C. Cirelli
sleep ends. In addition to claiming a correspondence between the homeostatic Process S and total
synaptic strength, the hypothesis proposes specific
mechanisms, whereby synaptic strength would
increase during wakefulness and decrease during
Figure 3. Evidence Supporting SH
sleep, and suggests why the tight regulation of
(A)
Experiments
in rats and mice show t
synaptic strength would be of great
importance
for
the brain.
phosphorylation levels of GluA1-AMP
Fig. 1 The two-process model involving the circadian
component (process C) and the homeostatic component
(process S).
Specifically, the curve in Fig. 1 can be interpreted
as reflecting how the total amount of synaptic
strength in the cerebral cortex (and possibly other
brain structures) changes as a function of wakefulness and sleep. Thus, the hypothesis claims that,
under normal conditions, total synaptic strength
increases during wakefulness and reaches a maximum just before going to sleep. Then, as soon as
sleep ensues, total synaptic strength begins to
decrease, and reaches a baseline level by the time
wake (data from rats are from Vyazovs
(B, B0 , and B00 ) Electrophysiological a
evoked responses using electrical stimu
diagram
Vyazovskiy et al., 2008) and TMS (in hu
et al., version
2013) shows increased slope
The diagram in Fig. 2 presents a simplified
of the main points of the hypothesis.
During
decreased slope after sleep. In (B), W
wakefulness (yellow background), we interact
onset and end of !4 hr of wake; S0 and
with the environment and acquire information
end of !4 hr of sleep, including at
about it. The EEG is activated, andand
the neuromodu0
latory milieu (for example, highsleep.
levels In
of (B
nor), pink and blue bars indica
adrenaline, NA) favors the storage of information,
deprivation and a night of recovery sleep
which occurs largely through long-term potenIn potentiation
vitro analysis of mEPSCs in rats
tiation of synaptic strength. This
occurs when the firing of a presynaptic
neuronfrequency
is
increased
and amplitude of m
followed by the depolarization or firing of a
and sleep deprivation (SD) relative to s
postsynaptic neuron, and the neuromodulatory
rats
are from Liu et al. (2010).
milieu signals the occurrence offrom
salient
events.
Strengthened synapses are indicated
in
red,
with
(C and C0 ) In flies, the number of sp
branches in the visual neuron VS1 incre
wake (ew) and decrease only if flies ar
(from Bushey et al., 2011). (C0 ) Str
adolescent mice show a net increas
density after wake and sleep deprivat
decrease after sleep (from Maret et al.,
16 Neuron 81, January 8, 2014 ª2014 Elsevier Inc.
Tononi and Cirelli, Sleep Med., 2006
Bushey et al., Science, 2011
Fig. 2
The synaptic homeostasis hypothesis.
Huber et al., Cereb. Cortex, 2012
Tononi and Cirelli, Nat. Neurosc., 2014
Cortical Network Dynamics
‣cortical
activity in superficial layers is
composed of cascades of activity
following a precise scaling relationship
!"#$%$&'(&)*+&&
,-)*)./"'0&$.&("'&"12
Fig. 2
Shew et al. • Dynamic Range Maximized during Neuronal Avalanches
15596 • J. Neurosci., December 9, 2009 • 29(49):15595–15600
‣first
processing in cortical networks, with peak
performance found under balanced conditions that generate neuronal avalanche
activity.
Fig. 2. Power law organization of neuronal avalanches identifies interacti
observed in organotypic cultures
cally synchronized neuronal groups. (A) Dorsal view of a macaque brain (l
showing the position of the microelectrode array (square) in pre-motor cor
Beggs and Plenz, J. Neurosci,
2003
scale). (B) Example period of the continuous local field potential (LFP) at
Materials and Methods
‣in
Monkey A Monkey B inB the LFP (nLFPs) that cross thresh
electrode. Peak negativeA deflections
line; −2.5SD) are indicative ofPMdlocal
PMd synchronized groups (red circles). (C
occurrence times for the 91-microelectrode array in A during ∼ 4 s of
Peterman et al., PNAS,nLFP
2009
M1
M1
M1
in an awake macaque monkey.
nLFPs
at different electrodes tend to cluste
single microelectrode on the array. Red box : region enlarged in D. (D) D
10 mm
tiotemporal cascade: nLPFs
in
the same time bin or consecutive bins (∆t =
C
Local rate -> nLFP
cascade, whose size can be measured by its number of nLFPs (three cascades
Tagliazucchi et al., Front.
Phys., 2012
5 shown; gray areas). ∆t: discrete time window to detect simultaneous or su
Palva et al., PNAS, 2013
on the array. (E) Neuronal avalanche dynamics are identified when the size
Shriki et al., J. Neurosci,distribute
2013 according to a power law with exponent of −3/2. Calculated fr
LFP activity inD the awake
macaque monkey (cf. A, C). The cut-o
Priesemann et al., PLoSongoing
CB, 2013
Local synchrony -> nLFP
law reflects the finite size of the microelectrode array and changes with the
ray used for analysis. Four distributions from the same original data set (
over-plotted using different areas (inset), i.e. number of electrodes (N ). Gray
electrodes. The power law reflects interactions between neuronal groups at d
Bellay et al., in submission
the array and is not found
shuffled
data
lines). L: Spatial leng
nLFP for
peak amplitude
correlates with
local rate(broken
and synchrony of unit
human MEG, ECoG, fMRI
‣at
the level of individual neurons
nLFP (SD)
nLFP (SD)
‣in
awake monkeys
Organotypic cultures on microelectrode arrays.
Coronal slices from rat somatosensory cortex
(350 !m thick, postnatal day 0 –2; Sprague
Dawley) and the midbrain (VTA; 500 !m
thick) were cut and cultured on a poly-Dlysine-coated 8 ! 8 microelectrode array
(MEA) (Multi Channel Systems; 30 !m electrode diameter; 200 !m interelectrode distance). In this organotypic slice coculture, the
development of deep and superficial cortical
layers (Götz and Bolz, 1992; Plenz and Kitai,
1996) as well as neuronal avalanche activity
parallels that observed in vivo (Gireesh and
Plenz, 2008). In short, a sterile chamber attached to the MEA allowed for repeated recording from cultures for weeks. After plasma/
thrombin-based adhesion of the tissue to the
MEA, standard culture medium was added
1.4
(600 !l, 50% basal medium, 25% HBSS, 25%
horse serum; Sigma-Aldrich). MEAs were then
affixed to a slowly rocking tray inside a custombuilt incubator ("65° angle, 0.005 Hz frequency, 35.5 " 0.5°C) (Stewart and Plenz,
2008).
Spontaneous activity. Using a recording head
0
stage inside the incubator (MEA1060 w/blanking circuit; !1200 gain; bandwidth 1–3000 Hz;
12 bit A/D; range 0 – 4096 mV; Multi Channel
Systems), local field potential (LFP; 4 kHz sampling rate; reference electrode in bath) was ob1.4
tained from 1 h of continuous recordings of
extracellular activity (low-pass, 100 Hz, phaseneutral). To establish a correlation between
LFP and neuronal spiking activity, in n # 5
cultures, extracellular activity was recorded for
15 min at 25 kHz. In addition to extracting
LFP, the extracellular signal was filtered in the
0
frequency band 300 –3000 Hz and $78 single
units were identified per culture using threshold detection and PCA-based spike sorting
Fig. 1.
(Offline Sorter; Plexon).
activity.
right
left
left
right
left
10SD
200 ms
200
200
300
100
Monkey A
M1left
100
0.8
100
Monkey B
300
300
200
400
0
Local firing rate (Hz)
100
300
200
400
M1 left
M1right
PMd left
PMd right
Local firing rate (Hz)
0.8
1
2
3
1
2
3
1
2
3
units/10 ms (n)
0
1
2
3
units/10 ms (n)
(A) Position of microelectrode arrays in primary motor (M1) and
Dependence on E/I balance
hase Synchrony and Neuronal Avalanches
‣in vitro: systematic control of E/I balance
‣control GABAergic and Glutamatergic syn. transm.
organotypic
cortical cultures
J.‣
Neurosci.,
December 9, 2009 • 29(49):15595–15600
• 15597 J. Neurosci., January 18, 2012 • 32(3):1061–1072 • 1065
‣neuronal avalanches characteristic for E/I balance
ifiat occurred during the burst. We note that in previous work we
atple
spike count during a burst was proportional to the burst size
based on sum of nLFP (Shew et al., 2009). Thus, defining burst size
vity
spike count is appropriate for the computational model. Size
ns of these bursts were used to parameterize the model dynamics
c!or
as defined above. Moreover, each condition was modeled on 60
taonnection
matrices (10 different networks of each size) to repressible variability from one culture to another in the experiments.
rug
ars in the results represent variation among the dynamics of these
urtrenetworks.
el the pharmacological manipulations made in our experinist
simply multiplied either the positive or the negative entries of
tor matrix by a constant between zero and10681,• J.modeling
Neurosci., January 18,the
2012 • 32(3):1061–1072
Yang et al. • Phase Synchrony and Neuronal Avalanches
ction
oliither disfaciliation (AP5/DNQX) or disinhibition
(PTX) renear 1 respectively ( fAPL ! 0.52 " 0.01;
ANOVA, p # 0.01).
Although such antiSince the average wij value of the baseline connection
matrix
um
phase locking in general will reduce the
isfacilitation
or disinhibition resulted in a mean
w !1/Q or
20
magnitudeij of r, the high ! conditions still
pectively. As average wij is increased from !1/Q
resulted
the highest
average r due to the
toin"1/Q,
the
ng,
fact that anti-phase-locked groups were
of this computational model undergo an abrupt
change at the
always small compared with the phasened
nt specified by 1/Q. At the critical point the
computational
locked
groups.
xed
erates neuronal avalanches, i.e., the burst sizeMaximum
distribution
is a
entropy and onset of
red
with exponent near #1.5. Our computationalsynchrony
modelnear
is similar
! " 1 in
computational model
which have been used to study criticality in Toneural
networks
gain further insight into our experimental
dwe
Plenz, 2003; Haldeman and Beggs, 2005;
Kinouchi
and a similar investifindings,
we next performed
gation in a network-level computational
06; Larremore et al., 2011), but is different from
these because
model. Network-level phase synchrony has
ive
inhibitory neurons.
been investigated extensively in previous
FPs
theoretical and model studies using coupled
nalyzing the computational model data, aggregate
population
oscillators (Strogatz, 2001; Arenas et al.,
han
et al.,
J. Neurosci,
2008).
might interpret the sig!l#Naively,
b $t%,one
where
t) forShew
each model
site were
computed 2009
PN $t% "
L l from one electrode as one thenal recorded
ater
Figure
of neurons
siteal.,
N. To
more closely model
experimentally
oscillator. However, in reality,
each2. Burst area and duration have moderate mean and maximum entropy near ! & 1.
Yangatet
J. Neurosci,
2012anoretical
κ ... deviation from a power-law
‣moderate
mean and maximum
variability of synchronization
electrode records the collective activity of a
amplitudes, we used the response curve, R( S), to compute dynamic
range,
Optimization of certain information processing capabilites
i
. Neurosci., December 9, 2009 • 29(49):15595–15600
. To obtain response as a function of
us in the model, we simulated increasing
us amplitude S by increasing the number
ally activated neurons (S ! 1, 2, 4, 16,
128 initially active neurons). Finally, we
at our model is very similar to (N "
ensional directed percolation (Buice
wan, 2007). Therefore, at high dimen# 5) and weak coupling, it is expected
model behaves as a branching process,
! is the branching parameter and the
ower law is predicted at criticality.
est for statistical differences between
a one-way ANOVA followed by a
post hoc test was used.
ts
where 1[x] is the u
week ending
P H Y S I C A L R E V I E W L E T T E R S J. Neurosci., December 9, 2009 • 29(49):15595–15600
Shew et al.
• Dynamic Range Maximized during Neuronal Avalanches
• 15599 2005 ex
FEBRUARY
PRL 94, 058101
(2005)
range11of
network
where Smax
1.25/N
in steps o
and Sthe
are
stimulation
valueswhen
to 90%
and
10%clustersto
cluster
size. the
One
thousand
clusters were
simulated
at leading
each of 11allowing
min
several
different
of output
configurations
mimicked
doubly
peaked
distribution
produced
agreement
establishedan
theory,
model
levels
of !. Incultures
we bathed
cortical
inwith
picrotoxin,
agent
thatclusterto achieve significance [Fig. 2(b), center],
of the range
of R,therespectively.
meanthus
pij,maximizing
the mode
size PDFs near criticality (i.e., ! ! 1) fit a "3/2 power law very
the
number
of
metastable
states.
selectively blocked
inhibition
and
increased
!
[Fig.
1(b)]
closely (Fig. 2 B, right, black) (Harris, 1989; Zapperi et al., 1995).
Model.[7].
The
consisted
of weNcomputed
coupled,
binary-state neu#j!J(t)
ij to order
the p
branching
net- N
Themodel
hump
distribution
Sall-to-all
! 60 is
caused
" based
on by
CDFs of We also investigated dynamics in
Justinasthe
in the
experiment,near
! (Fig. 2 B).works.
simulated
spontaneous
activity
for different
values
ofarray
For all
Lyapunov
exponent
propagates
over the
the entire
electrode
rons (N $activity
250, that
500,
1000)
and
following
dynamical
rules:
If networks,
neuronthe
j Boolean
spikes
at time
t will,
We found that " and ! were almost linearly related (Fig. 2 D),
" did not settle at one value, but wandered itinerantly near
dyingwhich
out.supports
The model
alsointerpretation:
produced Insimilarity
the 1),
following
the experiments,
spiked at before
time
t
(i.e.,
s
which constitutes cr
(t)
$
then
postsynaptic
neuron
i will spike at
j
matrices and" metastable
states similar
to those
in livingregime,a mean (Fig. 4) [20]. In critical networks, this mean was
! 1 is close to criticality,
" # 1 identifies
the subcritical
indistinguishable
from zero (! and
! 1; "Copelli,
% 0), producing
" $ 1suggesting
is analog
the
supercritical
regime
of are
the model.
and 2),
(Fig.
that
the simple
time t ! 1networks
with probability
pij.toAs
such,
the
pbranching
N 2 numbers
represent2006)
ij
neutral dynamics on average. By neutral, we mean that
network qualitatively captured salient features of cortical
ing the synaptic
coupling
strengths
between
system is supercriti
Stimulus-evoked
activity and dynamic
range each pair of neurons. The pij
slice culture.After obtaining " for a given experimental condition, we re-the distance between nearby trajectories remained nearly
over time.
Supercriticalparameter
networks hadof
chaotic
are asymmetric
the m
+model,
ptheji,response
positive,
time-independent,
uniformly
distribUsing p
this
we determine
the
branching
pa- S (forconstant
R as a function
of stimulus
amplitude
ijcorded
peristimulus
time
histograms
of
response
for
different
S,
see
sup#1 (! > 1; " # 0), where small initial distances
dynamics
rameter’s influence on the number of metastable states.
uted random
numbers
with
mean
and
SD
of
order
N
. If a set of
reflects the average
plemental
material,
available
at
www.jneurosci.org)
Typical
rebetween trajectories were amplified over time. Subcritical
We ran simulations in driven mode increasing ! from
sponse curves from experiments and simulations are shown in
neurons J(t)
spikes
time t, ofthen
probability
that
neuron
i fires at
networks
had attracting
dynamicsconsecutive
(! < 1; " " 0), time
where st
0.00 to
3.00Figure
inatincrements
0.02. the
Forfound
small
3, A and B, respectively.
We
thatnetworks
the shape of the
2
between trajectories shrank over time. Thus the
(N
the
number
of
metastable
states
peaked
at ). distances
response
curves
in$
the
model
closely
matched
the#
experimental
timeCapacity
t and
!Transmission
1 is" 64)
exactly
p
(t)
1
#
,
(1
p
To
the
balanced such that,
Shew et al. • Maximized Information
J. Neurosci.,
January
5, 2011 implement
• 31(1):55– 63 • 59
iJ
j!J(t)
ij
findings. When
excitatory
was reducedbranching parameter determined three dynamical regimes.
clearly supercritical
values
of !synaptic
[! # transmission
1:6; Fig. 3(a)],
1), thevariability
system was relatively
(required
a larger Bursting
(" #
probabilistic
nature
and
ofinsensitive
unitary
synaptic
efficacy,
neuron
nor
decays
with
tim
in cortical
slicei cultures
differs
from the
more
but for
larger
networks
this
peak
gradually
approached
! ! 1. This
possibility
in line with significant evinear
stimulus to evoke peak
a given
response).
When
inhibitoryis synaptic
Dynamic Range
* & 10 log10(Smax/Smin),
(2)
–VTA cocultures from rat (n ! 16),
closely parallel in vivo differentiand maturation of cortical superficial
2
continuous
activity
seen in awake, behaving animals [21].
the critical value
of ! was
! reduced
1:0
[for
Nongoing
!in!
1:00;
(Gireesh and Plenz, 2008), were
dence
that
activity
the
cortex is intimately
related
to
1),!
the2500,
system
was
hyperexcitable,
transmission
(" $
2
others
stimulus-evoked
activity
(Kenet
et!al.,
Ji and Wilson,
2007; have shown that when &0:6 cm slabs of
best least-squares
fit single
exponential
for
allsmall
data:
!2003;
with responses
that
saturate for relatively
stimuli.
In theHowever,
on 8 $ 8 integrated planar micro"!
1, Fig.
theLuczak
range
of
resulting
E/Is:d:,
condition
Han
et al.,
2008;
et stimuli
al.,While
2009).
Forcortical
instance, tissue
stimulusin vivo are isolated by cutting from incoming
1:03 $ 0:01, balanced
mean $
R2 with
!
0:98;
3(b)].
de arrays (Fig. 1A). LFPs were
in nonzero and nonsaturated
response
was recur
largest.
evoked activity
patterns
during ongoingconnections,
activity, at both the
they burst in ways remarkably similar to
most of these simulations used
few connections to reduce
ed after superficial layer differentipopulation level (Kenet et al., 2003; Han et al., 2008)
and
the
level
Figure 4.cultures
Network tuning
curve for
dynamic range
% near
criticality.
A, In experiments,
%
cortical
[22].
When
slabs
are
made
larger, burstcomputation Maximal
times (C
! 4),ofrange
increasing
C (Ji
to
161Wilson,
or 32 2007).
did Therefore,
#10 d in vitro, DIV) and analyzed
!and
!
dynamic
criticality,
spikeatsequences
aim
1 and drops rapidly with distance from criticality. Paired measurements
peaksour
closenext
to " !
becomes
more
suggesting
that
slabs
For each response
curve,
we
quantified
thefinding
range ofofstimuli
theing share
not change appreciably
ourwas
results
tract spatiotemporal clusters of
! 1symbol
also
to5,test
peak entropy
near
60 • J. Neurosci.,
January
2011[Fig.
• whether
31(1):55–3(b)].
63 our
Shew et(no-drug)
al. • Maximized
Information
and just
Transmission
the!same
shape;frequent,
normal
condition
wasCapacity
measured
before thethe
drug size
network
can
process,
i.e.,
the
dynamic
range
%
(see
Materials
and
stimulus-evoked
of the
entire
cortical
mantle
would% produce
condition.
Circles, Unpaired
measurement.
B, In simulations,
is also maximumcontinuous
for " ! 1.
The similarity matrices holds
the for
three
types of activity.
networks
(n ! 47 experiments). ExtracelluMethods). We found experimentally
that %
!10
5.0different
& 0.1 (mean
&
Symbol
indicates
network size (circles, N ! 250; squares, N ! 500; triangles, N ! 1000). Lines
Stimuli
consisted
of
amplitude
single
bipolar
t activity recorded simultaneously
produced offer
an intuitive
explanation
of&these
results.
SE) under
normal shocks
conditions,
! 2.4
in the
of order
representthrough
binned averages.
each%applied
400.1
times
inpresence
randomized
a
he LFP revealed that sizes of nLFP
Subcritical networks
units
weakly
coupled
so
their
PTX, and % have
! 3.4 &
0.3
for
AP5/DNQX
(Fig.
3C)
(F
!
11.3;
(2,44)
single electrode of the MEA within cortical layers II/III (a)
(see MaN = 40
p # 0.05 PTX and
AP5/DNQX
from
normal).
Importantly,
s correlated with the level of suactivity is uncorrelated,
producing
few
output
configuraterials
and Methods).
A binary
pattern wasthe
constructed to repre-7
ences
into distinguishable
network outputs, our result is also
dynamic
range
wassent
largest
unperturbed
in ms
which
eachin
response
duringnetworks,
the 20 2(b),
–500
after the
stimulus.
The6
eshold neuronal activity in the nettions that share
more
than
chance
similarities
[Fig.
N =separation,
16
closely
related
to
network-mediated
which has been
neuronal avalanches
are
most
likely
to
occur.
These
results
were
evokednetworks
entropy Hhave
was calculated
for the set of 400 stimulusFig. 1 B) (R ! 0.84 % 0.13, mean %
left]. In contrast,
supercritical
units
con5 at criticality, at the transition from
predicted
to
peak
ordered
to
robust for
differentevoked
networkactivation
sizes and different
maximal
stimulus
N=
8
patterns for each E/I. As found for ongoing
! 5 cultures). For each experimennected so strongly
that
activation
of
any
unit
usually
actidynamics
amplitudes, whether
or notthe
theevoked
response
curveswas
reached
satura1 for both
fine4(Bertschinger and Natschlager, 2004; Legenstein
activity,
entropy
highest
near ! !chaotic
and
Maass,
2007).
tion for
all conditions
(see
supplemental
material,
available
at
ndition, we first measured spontavates the whole
network.
The
one
resulting
metastable
and coarse spatial resolution (Fig. 3A, black shows 8 " 8, green3 In contrast, our results show that variability of
response to a given stimulus is highest at criticality. Further inwww.jneurosci.org).
Similar overall changes in % were also found
activity (Fig. 1C) and quantified
" 4) ( pstates
# 0.05).
state was highly
ordered,shows
butcoarse-grained
inhibited 4other
vestigation of reliability versus variability in cortical networks is
in our simulations (Fig. 3D) (F(2,195) ! 820; p # 0.05).
In Introduction,
we gave a simple
in which informa-2
viation of the observed spontaneous
[Fig. 2(b), right].
the branching
parameter
criti-example
WeWhen
then derived
a tuning curve
of % versus "was
by combining
all
warranted.
tion transmission from input to output was limited because of1
rk dynamics from neuronal avaexperimental
conditions
into one
scatter
plotprevailed,
(Fig. 4 A). These
Considering the simplicity of our model with all-to-all concal (! ! 1:0),
a mixture
of
variety
and
order
low entropy. With our measurements of network responses (i.e.,0
data&.demonstrate
that
%
is
maximized
and
its
variability
is
largest
nectivity,
absence of refractory period, and approximating inhidynamics by calculating " (Fig. Figure 3. Stimulus–response curves and dynamic range
A, Experimental
response
evoked (i.e.,
by current
stimulation
output)
toRstimuli
input),
we can of
directly test whether effi- 0.8 1.0 1.2 1.4 1.6 1.8
"
!
1.
These
findings
agree
well
with
our
model
including
near
bition
by
reducing !, the overall agreement in the %–"
ee Materials and Methods). In a sec- amplitude S for three example cultures with different " values. Orange arrows, Range
from
Smin to Smax; length
is proportional
to
cacy
of
information
transmission
is
optimized
when
entropy
is
changes in % as the system is pushed away from " ! 1, '10 dB
relationship between
experiment andσ
simulation is remarkable.
by
different
numbers
of
initially
activated
sites;
&
is
largest
for
ep, we measured the input/output &. Note that & is largest for " ! 1. B, Model response evoked
maximized.
This
idea
is
concisely
summarized
in
the
following
drop (10-fold reduction in Smax/Smin) for a 30% change in " (Fig.
The increase of variability in % as well as the drop in % due to
equation:
MI(S;R) that
!summary
H(
R)change
$
H(R!S).
Here,
is the
mu-" ! 1 was well matched between experiment and
Error bars,
1 SE.demonstrates
C, Experimental
statistics
"
mic range & of the cultured network ! ! 1. Like the experiment, each point is calculated from4 40
B).stimuli.
The tuning
curve
the
inforthe
dy- MI(S;R)
deviation
from
(b)
tual
information
of
stimulus
and
response,
which
quantifies
the
2.00 similarity supports the notion that universal
under
different
pharmacological
conditions
(*p
'
0.05
from
normal).
D,
Simulation
summary
statistics
for
"
comparing
different
namic range of a network due to a shift in E/I depends on both the
simulations. Such
on its response to a range of stimu4
information
transmission
(Rieke
et
al.,
1997;
Dayan
and
Abbott,
1. Measuring spontaneous and stimulus-evoked activity from
cortical
networks.
A,
Light-microscopic
image
of
a
ranges of ! (*p ' 0.05 from ! ! 1).
original, unperturbed state and the resulting change in ".
principles, independent of system details, are found
at criticality
plitudes
(Figs.
1
D,
3A).
These
mea16
1.75
osensory cortex and dopaminergic midbrain region (VTA) coronal slice cultured on a 60-channel microelectrode
2001). H( R) is the entropy of the full set of response
patterns
for
(Stanley,
1971;
Bak and Paczuski, 1995; Jensen,321998). The main
ents
were
performed
normal
Yellow dot,
Stimulation
site. Blackunder
dots, Recording
sites. B, Number of extracellular spikes correlates with the size of
all stimuli, whereas H(R!S) is the conditional entropy
(Riekedifference
et
Discussion
quantitative
was the lower % values for experiments
Metastable States
metastable states
Pattern Entropy
Haldeman and Beggs, PRL, 2005
Shew et al., J. Neurosci, 2009
Kinouchi and Copelli, Nat. Physics, 2006
What are the consequences of changes in synaptic
strength and consequently excitability on network
dynamics?
Does the sensitivity of neuronal avalanches and related
metrics to E/I conditions capture these effects?
Could these effects account for the observed
impairments to information processing in cortical
networks?
Study design
Fp1 Fpz Fp2
F7
F3
Fz
F4
F8
FC1 FC2
T3
C3
Cz
C4
T4
CP1 CP2
Pz P4
T5 P3
PO1 PO2
O1
Oz
T6
. . . . . . . . . . . .
Fp1 Fpz Fp2
F7
F3
C3
Cz
24h
Fp1 Fpz Fp2
F8
F7
F3
T4
CP1 CP2
Oz
T3
C3
F4
Cz
C4
F8
T4
CP1 CP2
T6
O2
36h
‣8 healthy subjects
‣sleep deprivation for a total of 40 hours
‣EEG every 3 hours, 27 channels
‣we used artefact free 20s segments (eyes
‣distribution of neuronal avalanches
‣mean and variability of synchronization
‣distribution of phase-lock intervals
Fz
FC1 FC2
C4
Pz P4
T5 P3
PO1 PO2
O1
12h
F4
FC1 FC2
T3
O2
0h
Fz
Pz P4
T5 P3
PO1 PO2
O1
Oz
T6
O2
p.s.
open condition)
Neuronal avalanches
Cascades of activity identified by two methods:
(A) Large positive or negative events on each channel
exceeding a certain threshold
(B) Events with high similarity - „coherence potentials“
Thiagarajan, PLoS Biol, 2010
Fp1 Fpz Fp2
F7
F3
Fz
F4
F8
FC1 FC2
T3
C3
Cz
C4
T4
CP1 CP2
Pz P4
T5 P3
PO1 PO2
O1
Oz
T6
O2
1s
5s
σ
p(S)
Neuronal avalanches
S
hours awake
∆D ... deviation from a power-law
σ ... branching parameter
n(2nd time bin)
σ=
n(1st time bin)
σ
p(S)
Neuronal avalanches
n
σ
hours awake
hours awake
σ ... branching parameter
th
=
th 3.8
=
R 4.2
=0
R .6
=0 5
R .75
=0
.8
Diff (σn, ∆Dn )
hours awake
∆D n
S
ve
i, j
j
synchronization. The calculation was performed on the samei, artifactSm
difference
estim
where
L ' 5000 ison
the the
length
of our
EEGL segments
in samples
and r(t) isabove
nchronization.
calculation
was
performed
same
artifactdifference
estimates
ti
VariabilityThe
and
mean
of
synchronization
free segments used for the derivation of coherence 1potentials. After film
Estimation
of
the
Kuramoto
order
parameter:
e segments used for
the derivation
of coherence
fil-obtained
of
the
goodness
% r !Hz),
t "& After
" first
r ! t " , Estimation
(3)
tering
the data in the
alpha band potentials.
(8 –12
we
a
phase
th
the
fit
of a powe
L
ti
ing the
data in synchronization
thetrace
alpha
band
–12in
Hz),
wealpha
obtained
aHz)
phase
the
fit
of
a
power-law
functio
t'1
‣phase
!i(t)
from(8each
EEG
trace
Ffirst
its Hilbert
transform
H[F
(t)]:
the
(8-16
and
theta
co
a
goodness-ofi(t) using
i
n
b
1
ce !i(t) from each EEG trace Fi(t) using its Hilbert transform H[Fi(t)]:i!j!t" a goodness-of-fit
test,
which
ve
bility
of
the
hyp
(4-8 Hz) frequency bands
r !H#F
t" "
e segments
, in samples and r(t)(4)
where L ' 5000 is the length
of
our
EEG
is Upon
i(t)$
ti
n
al.,
2009).
bility
of
the
hypothesis
that
! i ! t(t)$
" order
" arctan
. j'1
(2)
the Kuramoto
parameter:
H#F
Fi(t)
i
synthetic
data
w
Upon
fitting
the
e
al.,
2009).
‣phase ... ! i! t " " arctan
.
(2)
ti
fit
and ind
Fi(t)
data law
were
generate
which was
used as a time-dependent nmeasure synthetic
of phase synchrony
with
Next, we quantified the mean synchrony in each
EEG segment by
bu
1 channels
were
then
defin
n ' 27 being the numberr !oft "EEG
data.
i ! j !in
t " our
law
fit and individually
fit t
"
e
,
(4)
‣Kuramoto orderparameter ...
fo
Smirnov
statist
n of synchronization
xt, we quantified the mean synchrony
each EEG
segment
by
As a in
measure
for the
variability
we
derived
the
were then defined as the fr
j'1
L
b
model was larg
1 segment by
entropy of r(t) in each EEG
Smirnov
statistics for eachsos
%
r
!
t
"&
"
r
!
t
"
,
(3)
therefore
indic
‣mean synchronization
...
whichL was used as a time-dependent
measure model
of phasewas
synchrony
with
L
larger
than
theth
1
t'1
could
be
expla
B
data. indicate a higher
% r ! t "& "n ' 27 being
r ! t " ,the number of EEG channels
(3) in our
therefore
nv
fo
versely,
low
p
L
H
!
r
!
t
""
"
#
p
log
p
,
(5)
As
a
measure
for
the
variability
of
synchronization
we
derived
the
‣variabilitywhere
i
2 i
of synchronization
...our EEG segments in samples
L ' 5000 is the
and r(t)
isexplained
t'1 length of
could
be
by anbti
tion to describe
i'1
entropy
of
r(t)
in
each
EEG
segment
by
the Kuramoto order parameter:
w
so
versely, low p values
a
As a make
measure
here L ' 5000 is theAlength of ourwhere
EEGwe
segments
inB
anddistribution
r(t)
is
th
tion
ofempiric
some qu
tion
describe
the
B
estimated
asamples
probability
of r(t)toby
binning
values
d
e Kuramoto order parameter: into intervals. p is1 thenn the probability that r(t) falls
n
bution
with
exp
into
a
range
b
(
As
a
measure
quantifying
i
i
H ! r !it!""
"
#
p
log
p
,
(5)
i
2 i
j! t "
r
!
t
"
"
e
,
(4) to be robust
se
ti
r(t) $ bi)1. Similar
we found
results
tion of
some quantity X (in
n to Yang et al. (2012),
i'1
ti
w
n a broad range forj'1
over
the
number
of
bins
B
used.
We
applied
B
'
24
in
bution with exponent ), we
1
*
d
i
!
!
t
"
j estimated
where
we
a probability distribution
of r(t) by binning values
the
current
analysis.
r
!
t
"
"
e
,
(4)
which was used
as a time-dependent measure of phase synchrony with
d
n into
al
intervals.
p
is
then
the
probability
that
r(t)
falls
into
a
range
b
(
Derivation
of
the
distribution
of
PLIs.
PLIs
were
calculated
for
all
i
i
j'1
1 se
n ' 27 being the
number of EEG channels in ourp=0.003
data.
p=0.003
r(t)
$
b
.
Similar
to
Yang
et
al.
(2012),
we
found
results
to
be
robust
possible i)1
pairs of derivations of artifact-free EEG segments *D
of
for19.53
which
c
" s theti¥
As a measure
for
the
variability
of
synchronization
we
derived
the
NX i
overmeasure
a broad
range
for the
number
ofwith
bins as
B used.
Weanalysis
applied of
Bbution’s
'
24 in tail,
duration
(5000
samples,
same
segments
for the
coherhich was used as a time-dependent
of
phase
synchrony
entropy of r(t) the
in each
EEGanalysis.
segment by
d
current
ence
potentials
and
synchronization
measures).
The
analysis
was
perR
some
minimal
hours
awake
' 27 being the number of EEG channels in our
data.
alp
Derivation
of 4the
of band;
PLIs. PLIs
were
calculated
for
all
formed
for scale
(8 distribution
–16
Hz;
alpha
see
below)
and
scale
5
(4
–
8
for
which
the
cumulative
the minimumC
B we derived the
As a measure for the variabilitypossible
of synchronization
se
pairs
of
derivations
of
artifact-free
EEG
segments
of
19.53
s
Hz;
theta
band).
bution’s
tail,
i.e.,
the
N
n
number of Xcha
effect
in both
frequency
bands
(stronger
in
alpha)
th
tropy‣of
r(t) in observed
each EEG segment
by
H !wavelet
r(5000
! t "" "samples,
#
psame
p i , a scale-dependent
(5)estimate
i log
duration
segments
as for
the
analysis
of
coherHilbert
transforms.
To 2derive
ofofthePLI
tion
the
some
minimal
value
X
,
W
min
i'1
ence potentials and synchronization measures). The analysis was perR
!
"! "
!
"! "
!
"! " !
H(r(t)) n
<r(t)>n
!
"! "
!
Variability
and mean
of synchronization
PLI [sampling,
s]
B ‣
distribution
|!D|
of phase-lock intervals in the alpha (8-16
Hz) and theta (4-8 Hz) frequency bands
... continuous intervals with |∆Φi, j(t)| ≤ π/4
A
alpha band
!D n
n
p
P(PLI)
!-1
P. PLI
p=0.039
p=0.004
B
p=0.768
p<0.001
theta band
!D n
PLI
p
n
PLI
spectral power
p
C
spectral power
‣PLI
p=0.231
p ... likelihood for power-law
p=0.323
hours awake
Variability
and mean
of synchronization
PLI [sampling,
s]
B ‣
distribution
|!D|
of phase-lock intervals in the alpha (8-16
Hz) and theta (4-8 Hz) frequency bands
... continuous intervals with |∆Φi, j(t)| ≤ π/4
A
alpha band
!D n
n
p
P(PLI)
!-1
P. PLI
p=0.039
p=0.004
B
p=0.768
p<0.001
theta band
!D n
PLI
p
n
PLI
spectral power
p
C
spectral power
‣PLI
p=0.231
p=0.323
hours awake
‣Again: effect observed in both frequency bands
‣Cannot be explained by changes in power alone
Microelectrode recording under sleep deprivation
SITE MAPS
CM32 Package
Updated January 15, 2014
p.15
A8x4
Reference Site
3ch11 7
ch7
11
2ch10 6
ch6
10
4 ch8
8
ch4
12
1ch12 5
ch9
9
ch3
ch2
ch1
ch5
15
19 ch17 23 ch32 27ch28 31ch24
14
18ch20 22 ch29 26ch25 30 ch21
16
20ch19 24ch27 28ch23 32ch22
13
17ch18 21 ch31 25ch30 29ch26
ch14
ch15
ch16
ch13
Kappa (z-score)
0 h SD
2 h SD
4 h SD
6 h SD
Avalanche Size S
Sigma (z-score)
p(S)
NeuroNexus Technologies, Inc. ©2014 | 655 Fairfield Court, Suite 100, Ann Arbor, Michigan USA
Telephone: +1.734.913.8858 | Fax: +1.734.786.0069 | [email protected]
EEG
Fp1 Fpz Fp2
F7
F3
Fz
F4
F8
FC1 FC2
T3
C3
Cz
C4
T4
CP1 CP2
Pz P4
T5 P3
PO1 PO2
Oz
O2
p(S)
O1
T6
S
P(PLI)
∆Dn
A
hours awake
PLI [sampling,
B s]
A
H(r(t)) n
|!D|
<r(t)>n
B
p=0.003
p=0.003
hours awake
hours awake
P(PLI)
!-1
P. PLI
C
p awake
hours
PLI
PLI
disconnected the recording amplifiers during stimulation, significantly reducingFp1stimulus
Fpz Fp2 artifacts (Multi Channel Systems). Sample
rate and filtering
was
identical
to that used for spontaneous activity
F7
F8
recordings. F3 Fz F4
Pharmacology. FC1
BathFC2
application of antagonists of fast glutamatergic or
T3
C3
Cz
C4
T4
GABAergic synaptic
transmission
was used to change ratios of excitaCP1 CP2
tion to inhibition
(E/I
).
The
normal
(no-drug) followed by a drug
P3
Pz
P4
T5
T6
PO1during
PO2within
Shew et al. • Dynamic
Range Maximized
Neuronal
condition
was studied
3 hAvalanches
to minimize nonstationarities durO1
O2
Oz
ing development.
Stock
solutions were prepared for the GABAA receptor antagonist picrotoxin (PTX), the NMDA receptor antagonist
disconnected(2 R)-amino-5-phosphonovaleric
the recording amplifiers during
stimulation,
signifiacid (AP5),
and the AMPA
receptor
cantly reducing
stimulus
artifacts (Multi Channel Systems).
Sample
antagonist
6,7-dinitroquinoxaline-2,3-dione
(DNQX). Six
microlimedium
ters of was
theseidentical
stock solutions
were
added
600 !l of culture
rate and filtering
to that
used
fortospontaneous
activity
recordings. to reach the following working concentrations (in !M): 5 PTX, 20
AP5, Bath
10 AP5
! 0.5 DNQX,
and 20 AP5 !
DNQX.
After recording,
Pharmacology.
application
of antagonists
of 1fast
glutamatergic
or
the drug medium was replaced with 300 !l of drug-free conditioned
GABAergic synaptic
transmission was used to change ratios of excitamedium (collected from the same culture the previous day) mixed
tion to inhibition
Theunconditioned
normal (no-drug)
by arecovered
drug
!l of).fresh,
medium.followed
Most cultures
with 300(E/I
condition was
studied
within
3
h
to
minimize
nonstationarities
durto criticality within "24 h.
ing development.
Stock solutions
were
prepared
for For
theeach
GABA
Spontaneous
cluster size and
response
to stimulus.
electrode,
A re- we
identified
negative peaks
in the
(nLFPs)receptor
that wereantagonist
more negative
ceptor antagonist
picrotoxin
(PTX),
theLFP
NMDA
than #4 SDs of the electrode
noise.
We then
a cluster
of nLFPs
(2 R)-amino-5-phosphonovaleric
acid
(AP5),
andidentified
the AMPA
receptor
on the array as a group of consecutive nLFPs each separated by less than
antagonist 6,7-dinitroquinoxaline-2,3-dione
(DNQX). Six microliS
a time " (Beggs and Plenz, 2003). The threshold " was chosen to be greater
!l ofwithin
culture
medium
ters of these than
stock
wereofadded
to 600
thesolutions
short timescale
interpeak
intervals
a cluster,
but less
!
M
):
5
to reach thethan
following
working
concentrations
(in
" $ 8620
% 71
the longer timescale of intercluster quiescent periods (PTX,
1068
Neurosci.,
January
• 32(3):1061–1072
AP5, 10 AP5ms
! for
0.5allDNQX,
AP5
!18,12012
DNQX.
After
recording,
cultures;and
see• J.20
also
supplemental
material,
available
at www.
jneurosci.org).
Results were
large range in
the choice of "
l ofa drug-free
conditioned
the drug medium
was replaced
withrobust
300 !for
near
1
respectively
(
f
!
0.52
"
0.01;
APL
(data
not
shown).
The
size
s
of
a
cluster
was
quantified
as the
absolute
medium (collected from the same culture the previous day)
mixed
ANOVA, pwithin
# 0.01).
AlthoughSimilarly,
such anti- the size R of an
sum
of
all
nLFP
amplitudes
a
cluster.
medium.
Most
cultures
recovered
with 300 !l of fresh, unconditioned
phase locking
in general
will reduce
evoked response was
quantified as
the absolute
sum ofthe
nLFPs within 500
to criticality within "24 h. magnitude of r, the high ! conditions still
ms following a stimulus.
resulted
in the
highest
average
r due
to the
SpontaneousDefinition
cluster size
response
toavalanches,
stimulus.
For
each
electrode, we
#. For
the
probability
of and
factneuronal
that anti-phase-locked
groups
were density funcidentified negative
peaks
in
the
LFP
(nLFPs)
that
were
more
negative
$ $ #3/2
tion (PDF) of cluster
sizesmall
s follows
a power
always
compared
with law
the with
phase-slope
than #4 SDs(Beggs
of theand
electrode
noise.
We
then
a cluster of nLFPs
locked(see
groups.
Plenz, 2003)
Fig.
2 A).identified
Thus, the corresponding
cumulahours
awake
tiveadensity
function
(CDF)
for
clustereach
sizes,separated
FNA(%), which
specifies
on the array as
group
of
consecutive
nLFPs
by
less
thanthe
PLI [sampling,
s] sizes
Maximum
entropy
and
onset
of power-law function,
B
fraction
of
measured
cluster
s
&
%
,
is
a
#1/2
The threshold
a time " (BeggsNAand Plenz, 2003).
1"
inwas chosen to be greater
synchrony
near
#1
'l/L)
'l/%!)"for
F timescale
(%) $ (1 #of
(1 # intervals
l & s & L. Here we define a novel
than the short
interpeak
computational
model within a cluster, but less
nonparametric measure,
#
,
to
quantify
the
difference
between an experTo gain further insight
into our
experimental
$ 86 % 71
than the longer
timescale of intercluster
quiescent
periods
(" reference
%
),
and
the
theoretical
CDF,
imental cluster size
CDF,
F(
findings, we next performed a similar investiNA
ms for all cultures;
see
also
supplemental
material,
available
F (%),
gation in a network-level computational at www.
in vitro
J. Neurosci., December 9, 2009 • 29(49):15595–15600 • 15597
‣observations
p(S)
EEG
A
P(PLI)
∆Dn
Yang et al. • Phase Synchrony and Neuronal Avalanches
in EEG
during sleep deprivation
are in agreement with a
shift towards increased
excitability where larger
events dominate
dynamics
A
H(r(t)) n
|!D|
jneurosci.org). Results were robust
for a largephase
range
in the
model. Network-level
synchrony
haschoice of "
1 mwas
extensively
in
previous
(data not shown). The size s#been
of&a1investigated
cluster
quantified
as
the absolute(1)
' and model
(FNAstudies
(% k) (
F(%coupled
k)),
theoretical
using
m
sum of all nLFP amplitudes within
a cluster.
Similarly, the size R of an
k$1
oscillators (Strogatz, p=0.003
2001; Arenas et al.,
p=0.003
evoked response
was quantified
as Naively,
the sizes
absolute
sum
of nLFPs
500
onelogarithmically
might
interpret
thespaced
sig- within
where %k are m $2008).
10 burst
between the
nal recorded from one electrode as one thems followingminimum
a stimulus.
and maximum observed burst size. Using CDFs rather than
oretical oscillator.
However,awake
in reality, each
hours
awake
hours
neuronal
avalanches,
the probability
density
funcDefinition
of #. For
PDFs
to calculate
#electrode
avoids
the sensitivity
to binning
records
the collective
activity ofin
a constructing a
p
hours
awake
Compared
other
nonparametric
comparisons
of
CDFs,
C (PDF) ofPDF.
$
$
#3/2e.g.,
tion
cluster
size s to
follows
a
power
law
with
slope
population of E and I neurons. Thus, our
Kolmogorov–Smirnov
and
Kuiper’s
test,
as
well
as
other
methods,
experimental
pharmacological
E–I
manipu(Beggs and Plenz, 2003) (see Fig. 2 A). Thus, the corresponding cumula- #
lations effect
not only the
coupling
betweenavalanches (see
more accurately
deviation
from
tive density function
(CDF) measures
for
cluster
sizes, FNA
(%),neuronal
which
specifies the
electrodes,
but alsoatthe
coupling among the
supplemental material,
available
www.jneurosci.org).
fraction of measured cluster local
sizes s & %,measured
is a #1/2
function,
by apower-law
single
Dynamic range. Afterneurons
measuring responses
to aelecrange of stimulus
'l/%Since
trode.
the
intrinsic
properties
of
a
FNA(%) $ (1 amplitudes,
# 'l/L) #1we
(1 #
)
for
l
&
s
&
L.
Here
we
define
a novel
used the response curve, R( S), to compute
dynamic
theoretical
oscillator
do
not
depend
on
counonparametric
measure,
#
,
to
quantify
the
difference
between
an
experrange,
pling, the interpretation of one electrode as
%),
and isthe
theoretical
reference
CDF,
imental cluster size CDF, F(
one
oscillator
problematic.
Therefore,
we
NA
*
&
10
log
(
S
/S
)
,
(2)
10
max
min
developed a computational model with the
F (%),
!
!-1
P. PLI
P(PLI)
<r(t)>n
B
Figure 2. Change in the ratio of excitation/inhibition moves cortical networks away from
criticality. A, Left, PDFs of spontaneous cluster sizes for normal (no-drug, black), disinhibited
(PTX, red), and hypoexcitable (AP5/DNQX, blue) cultures. Broken line, #3/2 power law. Cluster
size s is the sum of nLFP peak amplitudes within the cluster; P(s) is the probability of observing
a cluster of size s. Right, Corresponding CDFs and quantification of the network state using #,
which measures deviation from a #1/2 power law CDF (broken line). Vertical gray lines, The 10
distances summed to compute #, shown for one example PTX condition (red). B, Simulated
cluster size PDFs (left) and corresponding CDFs (right) for different values of the model control
parameter *. C, Summary statistics of average # values for normal, hypoexcitable, and disinhibited conditions (*p & 0.05 from normal). D, In simulations, # accurately estimates *.
Broken line, # $ *. Colored dots, Examples shown in B.
actually fires at time t ! 1 only if piJ(t) - )(t), where )(t) is a random
number from a uniform distribution on [0,1],
Figure 2. Change
in the ratio of excitation/inhibition moves cortical networks away from
s i ( t ' 1 ) & . / p iJ ( t ) ( ) ( t )0
criticality. A, Left, PDFs of spontaneous cluster sizes for normal (no-drug,
black),(3)disinhibited
& ./1 (
( 1 ( p ij ) ( ) ( t )0 ,
(PTX, red), and hypoexcitable (AP5/DNQX, blue)
j!J ( t ) cultures. Broken line, #3/2 power law. Cluster
sizewhere
s is the1[x]
sum of
nLFP
peak
amplitudes
within
cluster;
P(s) is thewe
probability
is the unit step function. Likethe
our
experiments,
exploreofa observing
a cluster
s. Right,
Corresponding
CDFs and
of pthe
network state using #,
rangeof
ofsize
network
excitability
by tuning
thequantification
mean value of
ij from 0.75/N
aim of obtaining results which were more
which
measures
deviation
from
a
#1/2
power
law
CDF
(broken
line).
Vertical
lines, The 10
where Smax and Smin
to
1.25/N
in
steps
of
0.05/N
by
scaling
all
p
are
the
stimulation
values
leading
to
90%
and
10%
by
a
constant.
For
suchgray
small
ij
m comparable to our experiments,
1directly
of the range
of
R,
respectively.
mean
p
,
the
model
reduces
to
probabilistic
integrate-and-fire,
i.e.,
p
2
NA
distances summed
to compute #, shown for one example PTX condition (red).iJ B, Simulated
ij
# & 1 ' while keeping
(F (%kthe
) (model
F(% as
)),simple as
(1)
Model. The model
consisted
ofPLI
N all-to-allk coupled, binary-state neu- cluster
#j!J(t)
to order
N #2
accuracy. IfCDFs
the(right)
meanfor
pijdifferent
is exactly
N #1of, then
n control
PLI
mpossible.
k$1
sizepijPDFs
(left) and
corresponding
values
the model
!
The computational model was com-
"
Beggs and Plenz, J. Neurosci, 2003
Yang et al., J. Neurosci, 2012
Interpretation: Criticality ?
(1) All those seemingly different findings are precisely captured by
a critical branching process.
261
non-trivial, active steady state, in which [I] < 1. The
stability of the trivial steady state is determined by the
spectrum of the Jacobian matrix J|x0 ∈ R10×10 , where
Jij = ∂ ẋi /∂xj . The steady state is asymptotically stable
if all eigenvalues of J|x0 have a negative real part [31].
If the variables xi are ordered as in Eqns. (1), the nonvanishing entries of J|x0 are
J1,1
J1,4
J2,1
J2,2
J3,3
J4,4
J5,5
J6,4
J7,10
J9,9
J10,10
= J7,3 = J7,4 = J7,5 = J7,6 = J7,8 = J7,9 = −i ,
= J3,4 = p ,
= J5,3 = J8,3 = J9,4 = J10,5 = J10,8 = i ,
= J4,5 = J6,7 = J6,9 = J9,10 = r ,
= −2i ,
= −i + (k − 1)p ,
= J7,7 = J8,8 = −i − r ,
= −2kp ,
= −i + r ,
= −r ,
= −2r .
3
0.1
inactive phase
0.0
2
4
6
8
k
The characteristic polynomial can be factored into 7
linear factors with negative real roots and a remaining
third order polynomial
P (λ) = (−r−λ)3 (−i−λ)2 (−i−r−λ)(−2r−λ)f (λ) , (5)
where
!
"
f (λ) = i i2 + 2i ((1 − k)p + r) + (1 − 2k)pr
!
"
+ 5i2 + (1 − k)pr + 3i ((1 − kp) + r) λ
active phase
[F ]
ate: From Black Swans to Dragon-Kings
Droste et al., JRS Interface, 2013
FIG. 1. Bifurcation diagram for the static network. Plot2005
ted is theHaldeman
steady state and
densityBeggs,
of firingPRL,
neurons
[F ] over
the network’s mean degree k. The solid line marks stable
Shew
etstatic
al., J.system,
Neurosci,
2009
steady states
of the
the dashed
line unstable
ones. At k = kc ≈ 5.6, the inactive steady state loses stability in a transcritical bifurcation. The respective transition from an inactive to an active phase is already observed
in individual-based simulations with N = 106 neurons (circles). Note that the critical k is nicely predicted by the
link-level approximation Eqns. (1), but underestimated by
a MCA at mean-field level (dotted lines). Parameters used
were p = 0.2, i = 0.95, r = 0.4. Nontrivial steady states were
calculated using AUTO[33].
(6)
+ (4i + (1 − k)p + r) λ2 + λ3 .
In other to assess the stability of x0 without explicitly
calculating the roots of f (λ), we use the Routh-Hurwitz
theorem [32]. It states that the roots of a polynomial
The role of the transcritical bifurcation is illustrated
in a representative bifurcation diagram in Fig. 1. The diagram shows that the analytically predicted bifurcation
point is in very good agreement with numerical results
261
non-trivial, active steady state, in which [I] < 1. The
stability of the trivial steady state is determined by the
spectrum of the Jacobian matrix J|x0 ∈ R10×10 , where
Jij = ∂ ẋi /∂xj . The steady state is asymptotically stable
if all eigenvalues of J|x0 have a negative real part [31].
If the variables xi are ordered as in Eqns. (1), the nonvanishing entries of J|x0 are
J1,1 = J7,3 = J7,4 = J7,5 = J7,6 = J7,8 = J7,9 = −i ,
J1,4 = J3,4 = p ,
J2,1 = J5,3 = J8,3 = J9,4 = J10,5 = J10,8 = i ,
J2,2 = J4,5 = J6,7 = J6,9 = J9,10 = r ,
J3,3 = −2i ,
J4,4 = −i + (k − 1)p ,
J5,5 = J7,7 = J8,8 = −i − r ,
J6,4 = −2kp ,
Yang et al. • Phase Synchrony and
Neuronal
J7,10
= −iAvalanches
+r ,
J9,9 = −r ,
J10,10 = the
−2rburst.
.
of spikes that occurred during
We note that in previous work we
active phase
3
0.1
inactive phase
[F ]
ate: From Black Swans to Dragon-Kings
0.0
2
4
6
8
k
J. Neurosci.,2013
January 18, 2012 • 32(3):1061–1072
Droste et al., JRS Interface,
FIG. 1. Bifurcation diagram for the static network. Plot2005
ted is theHaldeman
steady state and
densityBeggs,
of firingPRL,
neurons
[F ] over
the network’s mean degree k. The solid line marks stable
Shew
etstatic
al., J.system,
Neurosci,
2009
steady states
of the
the dashed
line unstable
showed that spike count
a burstpolynomial
was proportional
the burst
Theduring
characteristic
can be to
factored
intosize
7
ones. At k = kc ≈ 5.6, the inactive steady state loses stalinear
factors(Shew
with et
negative
realThus,
roots defining
and a remaining
definition based on sum
of nLFP
al., 2009).
burst size bility in a transcritical bifurcation. The respective transithirdisorder
polynomial
observed
in terms of spike count
appropriate
for the computational model. Size tion from an inactive to an active phase is already
in individual-based simulations with N = 106 neurons (cir3 to parameterize
distributions of these bursts
used
the model dynamics
P (λ) =were
(−r−λ)
(−i−λ)2 (−i−r−λ)(−2r−λ)f
(λ) , (5)
cles). Note that the critical k is nicely predicted by the
in terms of ! as defined above. Moreover, each condition was modeled on 60 link-level approximation Eqns. (1), but underestimated by
where
a MCA at mean-field level (dotted lines). Parameters used
different connection matrices (10
! 2different networks of each size)
" to repre- were p = 0.2, i = 0.95, r = 0.4. Nontrivial steady states were
f (λ)from
= i ione
+ 2i
((1 − k)p
+ r) + (1in−the
2k)pr
sent the possible variability
culture
to another
experiments.
calculated using AUTO[33].
! 2
"
(6)
+ 5i +
(1 − k)pr
+ 3i ((1
− dynamics
kp) + r) λof these
The error bars in the results represent
variation
among
the
60 different networks.
+ (4i + (1 − k)p + r) λ2 + λ3 .
The role of the transcritical bifurcation is illustrated
To model the pharmacological manipulations made
in
our
experiin a representative bifurcation diagram in Fig. 1. The diIn other to assess the stability of x0 without explicitly
ments, we simply multiplied
either the positive or the negative entries of agram shows that the analytically predicted bifurcation
calculating the roots of f (λ), we use the Routh-Hurwitz
the connection matrix
by a [32].
constant
between
and of
1, modeling
the point is in very good agreement with numerical results
theorem
It states
that zero
the roots
a polynomial
disconnected the recording amplifiers during
stimulation,
signifi- amplifiers during stimulation, signifidisconnected
the recording
cantly reducingFp1stimulus
artifacts
(Multi
Channel
Systems).
Sample
cantly reducing stimulus artifacts (Multi Channel Systems). Sample
Fpz Fp2
rate and filtering
was identical
to that used
spontaneous
activity
ratefor
and
filtering was
identical to that used for spontaneous activity
F7
F8
recordings. F3 Fz F4
recordings.
Pharmacology. FC1
BathFC2
application of antagonists
of fast glutamatergic
or
Pharmacology.
Bath application
of antagonists of fast glutamatergic or
T3
C3
Cz
C4
T4
GABAergic synaptic
transmission
was
used
to
change
ratios
of
excitaGABAergic
synaptic
transmission
was used to change ratios of excitaCP1 CP2
tion to inhibition
(E/I
).
The
normal
(no-drug)
followed
by
a
drug
P3
Pz
P4
tion
to
inhibition
(E/I
).
The
normal
(no-drug) followed by a drug
T5
T6
PO1during
PO2within
Shew et al. • Dynamic
Range Maximized
Neuronal
J. Neurosci.,
December 9, 2009 • 29(49):15595–15600 • 15597
condition
was studied
3 hAvalanches
to minimize
nonstationarities
condition
was studied durwithin 3 h to minimize nonstationarities
durO1
O2
Oz
ing development.
Stock
solutions were prepared
for the GABA
ing development.
Stock
solutions were prepared for the GABAA reA receptor antagonist picrotoxin (PTX), the NMDA
receptor antagonist
ceptor antagonist
picrotoxin (PTX), the NMDA receptor antagonist
disconnected(2 R)-amino-5-phosphonovaleric
the recording amplifiers during
stimulation,
signifiacid (AP5),
and the AMPA
receptor
(2
R)-amino-5-phosphonovaleric
acid (AP5), and the AMPA receptor
cantly reducing
stimulus
artifacts (Multi Channel
Systems).
Sample
antagonist
(DNQX).
Six
microliantagonist
(DNQX). Six microlil of culture
medium
ters of was
theseidentical
stock solutions
were
added
600of!these
rate and filtering
to that
used
fortoters
spontaneous
activity
stock solutions were added to 600 !l of culture medium
1070 • J. Neurosci., January 18, 2012 • 32(3):1061–1072
!Mfollowing
): 5 PTX,working
20
to reach(inthe
concentrations (in !M): 5 PTX, 20
recordings. to reach the following working concentrations
AP5,
10
AP5
!
0.5
DNQX,
and
20
AP5
!
1
DNQX.
After
recording,
AP5 ! 0.5 DNQX,
Pharmacology. Bath application of antagonists ofAP5,
fast10
glutamatergic
or and 20 AP5 ! 1 DNQX. After recording,
l ofdrug
drug-free
conditioned
the drug medium was replaced with 300 !the
medium
was
replaced with 300 !l of drug-free conditioned
GABAergic synaptic
transmission was used to change
ratios ofday)
excitamedium (collected from the same culture
the previous
mixed
medium
(collected from
the same culture the previous day) mixed
tion to inhibition
Theunconditioned
normal (no-drug)
followed
by arecovered
drug
!l of).fresh,
medium.
cultures
with 300(E/I
l of fresh,
unconditioned medium. Most cultures recovered
withMost
300 !
condition was
studied within
3 h h.
to minimize nonstationarities
to criticality
within "24
to criticality withindur"24 h.
ing development.
Stock solutions
were
prepared
for
theeach
GABA
Spontaneous
cluster size and
response
to stimulus.
For
electrode,
Spontaneous
cluster
size we
and response to stimulus. For each electrode, we
A reidentified
negative peaks
in the
(nLFPs)
that were
more peaks
negative
identified
negative
in the LFP (nLFPs) that were more negative
ceptor antagonist
picrotoxin
(PTX),
theLFP
NMDA
receptor
antagonist
than
#4
SDs
of
the
electrode
noise.
We
then
identified
a
cluster
of
nLFPs
than
#4
SDs
of
the
electrode
(2 R)-amino-5-phosphonovaleric acid (AP5), and the AMPA receptor noise. We then identified a cluster of nLFPs
on the array as a group of consecutive nLFPsoneach
by less of
than
the separated
array
as amicroligroup
consecutive nLFPs each separated by less than
antagonist 6,7-dinitroquinoxaline-2,3-dione
(DNQX).
Six
S
" was
chosenand
to be
greater
a time " (Beggs and Plenz, 2003). The threshold
Figure 7. Phase synchrony and neuronal avalanches in a network-level computational model of E–I neurons. A, A recurrently
" (Beggs
Plenz,
2003). The threshold " was chosen to be greater
a time
!l ofwithin
culture
medium
ters of these than
stock
wereofadded
to 600
connected population of 80% excitatory and 20% inhibitory neurons were divided into 59 groups with Q neurons in each group
thesolutions
short timescale
interpeak
intervals
a cluster,
but less
than the short
timescale
of interpeak intervals within a cluster, but less
!periods
M): 5 (PTX,
to reach thethan
following
(in the
(Q % 30, 40, 50, 60, 70, 80 were tested). Each group models the neurons nearby one recording site in the experiment. B, Spike
" $ 8620
% of
71intercluster quiescent periods (" $ 86 % 71
the longerworking
timescale concentrations
of intercluster quiescent
than
longer timescale
rasters
two Avalanches
groups. (blue, E; red, I ). C, Population signals corresponding to the two group rasters shown in B. The spikes
1068
Neurosci.,
January
• 32(3):1061–1072
• Phase Synchrony
andfrom
Neuronal
AP5, 10 AP5ms
! for
0.5allDNQX,
AP5
!18,12012
DNQX.
recording,
cultures;and
see• J.20
also
supplemental
material,
atseewww.
ms
forAfter
all available
cultures;
also supplemental material, availableYangatet al.www.
generated by a group at each time step were summed (light colored traces) and bandpass filtered (dark colored traces) to generate
jneurosci.org).
Results were
large
range in
theResults
choicewere
of " robust for a large range in the choice of "
l ofa drug-free
conditioned
the drug medium
was replaced
withrobust
300 !for
jneurosci.org).
population signals. D, Phase of these filtered population signals was obtained using the Hilbert transform. E, An example dynamic
near
1
respectively
(
f
!
0.52
"
0.01;
phase histogram and synchrony trace. Phase synchrony among the 59 groups was analyzed exactly as in the experiments. F, Bursts
APL
(data
not
shown).
The
size
s
of
a
cluster
was
quantified
as
the
absolute
(data not shown).
The size s of a cluster was quantified as the absolute
medium (collected from the same culture the previous
day) mixed
ANOVA, pwithin
# 0.01).
AlthoughSimilarly,
such anti- the size R of an
of activity were modeled one at a time. Burst size distributions were used to compute !. Neuronal avalanches (! " 1) were
sum
of
all
nLFP
amplitudes
a
cluster.
sumcultures
of all nLFP
amplitudes within a cluster. Similarly, the size R of an
medium.
Most
with 300 !l of fresh, unconditioned
phase locking
in general
will reduce
the recovered
observed only when E and I were balanced such that, on average, the spike count of the population did not grow or decay over
evoked response was quantified as the absolute
sumresponse
of nLFPswas
within
500 as the absolute sum of nLFPs within 500
evoked
quantified
consecutive time steps. Bimodal size distributions (! # 1) occurred when inhibition was suppressed. And a lack of large-sized
still
to criticality within "24 h. magnitude of r, the high ! conditions
ms following a stimulus.
msr following
a stimulus.
bursts marked the disfacilitated distribution (! $ 1).
resulted
in
the
highest
average
due
to
the
SpontaneousDefinition
cluster size
response toavalanches,
stimulus. For
each electrode,
we
#. For
probability
func- avalanches, the probability density funcof and
#. For neuronal
Definition
factneuronal
that anti-phase-lockedthe
groups
were of density
identified negative
peaks
in
the
LFP
(nLFPs)
that
were
more
negative
$ $ size
#3/2
tion (PDF) of cluster
sizesmall
s follows
a power
law
with
always
compared
with
the(PDF)
phase-slope
tion
of cluster
s follows a power law with slope $ $ #3/2
respectively. The baseline, normal condition of the computational
than #4 SDs(Beggs
of theand
electrode
noise.
We
then
a cluster
of nLFPs
locked(see
groups.
Plenz, 2003)
Fig.
2 A).identified
Thus,(Beggs
the corresponding
cumulaand Plenz,
2003)
(see Fig. 2 A). Thus, the corresponding cumulaDiscussion
model was chosen such that activity in the network propagated in a
hours
awake
tiveadensity
function
(CDF)
for
clustereach
sizes,tive
FNAdensity
(%), which
specifies
thefor cluster sizes, FNA(%), which specifies the
on the array as
group
of
consecutive
nLFPs
separated
by
less
than
function
(CDF)
balanced
manner.
More
specifically,
we
established
the
magnitude
of
The ability of a network of cortical n
PLI [sampling,
s] sizes
entropy
of power-law function,
BMaximum
fraction
of measured
cluster
s &and
%", onset
is fraction
a #1/2
of to
measured
cluster sizesFigure
s & %2., is Change
a #1/2
power-law
function,
connections
between
neurons
such
that a single active
excitatory
dynamics
in
the
ratio
of
excitation/inhibition
moves
cortical
networks
away
from
and
Plenz,
2003).
The
threshold
was
chosen
be
greater
a time " (Beggs
Figure
2.
Change
in
the
ratio
of
moves
cortical
networks
away fromdepends on recurrent inter
1 in NA
synchrony
near
NA
#1
#1
'l/L)
'l/%!)"for
F
(
%
)
$
(1
#
(1
#
l
&
s
&
L.
Here
we
define
a
novel
'
'
neuron
was
expected,
on
average,
to
excite
exactly
one
more
neuron
and
inhibitory components of the n
F
(
%
)
$
(1
#
l/L)
(1
#
l/
%
)
for
l
&
s
&
L.
Here
we
define
a
novel
criticality. A, Left, PDFs of spontaneous cluster sizes for normal
(no-drug,
criticality.
A, Left,black),
PDFs ofdisinhibited
spontaneous cluster sizes for normal (no-drug, black), disinhibited
than the short timescale of interpeak
intervals
computational
model within a cluster, but less
in
the
following
time
step.
Disinhibition
and
disfacilitation
disspontaneously
nonparametric measure,
#
,
to
quantify
the
difference
between
an
expernonparametric
#, to quantify
the
between
anblue)
exper(PTX, red),
anddifference
hypoexcitable
(AP5/DNQX,
cultures. Broken
line,and
#3/2
power law.(AP5/DNQX,
Cluster
To gain further insight
into our
experimental
(PTX, red),
hypoexcitable
blue) cultures. Broken line, #3/2 power law. Cluster emerging network-leve
$ measure,
86 % 71
than the longer
timescale of intercluster
quiescent
periods
(" reference
%
),
and
the
theoretical
CDF,
imental cluster size
CDF,
F(
the
referencewithin
CDF,
cluster size CDF, F(%),sizeand
findings, we next performed aimental
similar investis is the
sumtheoretical
of nLFP peak amplitudes
the cluster;
thesum
probability
of observing
sizeP(s)
s isisthe
of nLFP peak
amplitudes within the cluster; P(s) is the probability of observing
ms for all cultures;
supplemental
material,
NA available at www.
FNA(%), see also gation
in a network-level
computational
F
(%),
a cluster of size s. Right, Corresponding CDFs and quantification
network
using #, CDFs and quantification of the network state using #,
a clusterofofthesize
s. Right,state
Corresponding
jneurosci.org). Results were robust
for a largephase
range
in the
model. Network-level
synchrony
haschoice of "
which measures deviation from a #1/2 power law CDF (broken
Vertical
gray lines,
10 power law CDF (broken line). Vertical gray lines, The 10
whichline).
measures
deviation
fromThe
a #1/2
1 mwas
extensively
in
previous
1 mdistances
(data not shown). The size s#been
of&a1investigated
cluster
quantified
as
the
absolute
NA
#
,
shown
for
one
example
PTX
condition
(red).
B,
Simulated
summed
to
compute
NA
#
,
shown for one example PTX condition (red). B, Simulated
distances
summed
to
compute
' and model
(F studies
(% k) (
F(%coupled
# & 1 (1)
'
(F (%k) ( F(%k)),
(1)
k)),
theoretical
using
mcluster.
m
sum of all nLFP amplitudes within
a
Similarly,
the
size
R
of
an
k$1
cluster
size
PDFs
(left)
and
corresponding
CDFs
(right)
for
different
values
of
the
model
control
k$1
cluster size PDFs (left) and corresponding CDFs (right) for different values of the model control
oscillators (Strogatz, p=0.003
2001; Arenas et al.,
p=0.003
normal, *
hypoexcitable,
disin-of average # values for normal, hypoexcitable, and disinparameter *. C, Summary statistics of average # values for
evoked
response
was
quantified
as
the
absolute
sum
of
nLFPs
within
500
. C, Summary and
statistics
parameter
2008).
Naively,
one
might
interpret
the
sigwhere %k are m $ 10 burst sizes logarithmically
the sizes logarithmically spaced between the
where %spaced
are mbetween
$ 10 burst
k
#
accurately
estimates
*. normal). D, In simulations, # accurately estimates *.
hibited
conditions
(*p
&
0.05
from
normal).
D,
In
simulations,
nal recorded from one electrode as one thehibited conditions (*p & 0.05 from
ms followingminimum
a stimulus.
and maximum observed burst size.
Using CDFs
rather than
minimum
and maximum
observed
burst
size.
Using
CDFs
rather shown
than in B.
oretical
oscillator.
However,
in
reality,
each
#
$
*
.
Colored
dots,
Examples
Broken
line,
Broken line, # $ *. Colored dots, Examples shown in B.
hours
awake
hours
awake
neuronal
avalanches,
the probability
density
Definition
of #. For
PDFs
to calculate
avoids
the sensitivity
to
binning
PDFs
to calculate
#funcavoidsathe sensitivity to binning in constructing a
electrode
records
the collective
activity
ofin
a constructing
p #awake
hours
PDF.
Compared
other nonparametric
comparisons
CDFs,
e.g.,
tion (PDF) of
cluster
size s to
follows
a ofpower
with
slope
PDF.
Compared
to#3/2
other
nonparametric comparisons of CDFs, e.g.,
population
E and Ilaw
neurons.
Thus,
our $of$
#
and
Kuiper’s
test,
as
well
as
other
methods,
experimental
pharmacological
E–I
manipu# - )(t), where )(t) is a random
test,
as well
as other
(Beggs and Plenz, 2003) (see Fig. 2 A). Thus, the corresponding
cumula-and Kuiper’s
actually
fires
at time
t ! 1 methods,
only if piJ(t)
actually fires at time t ! 1 only if piJ(t) - )(t), where )(t) is a random
NA
lations effect
not only the
coupling
between
more accurately
deviation
from
neuronal
avalanches
(see deviation from neuronal avalanches (see
more
accurately
measures
tive density function
(CDF) measures
for
cluster
sizes,
F
(
%
),
which
specifies
the
number
from
a
uniform
distribution
on
[0,1],
number from a uniform distribution on [0,1],
electrodes,
but alsoatthe
coupling
among the material, available at www.jneurosci.org).
supplemental material,
available
supplemental
fraction of measured
cluster local
sizesneurons
s & %,measured
is awww.jneurosci.org).
#1/2
power-law
function,
by
a
single
elecFigure
2.
Change
in
the
ratio
of
moves cortical networks away from
Dynamic range. After measuring responses
to a range
of stimulus
Dynamic
range.
After measuring responsess ito
a range
(t '
1 ) & of
. /stimulus
p iJ ( t ) ( ) ( t )0
s i ( t ' 1 ) & . / p iJ ( t ) ( ) ( t )0
'l/%Since
trode.
the
intrinsic
properties
of
a
FNA(%) $ (1 amplitudes,
# 'l/L) #1we
(1 #
)
for
l
&
s
&
L.
Here
we
define
a
novel
criticality.
A, Left,
PDFstoofcompute
spontaneous
cluster sizes for normal (no-drug,
black),(3)disinhibited
used the response curve, amplitudes,
R( S), to compute
dynamic
,
we
used
the
response
curve,
R(
S),
dynamic
(3)
& ./1 (
( 1 ( p ij ) ( ) ( t )0
theoretical
oscillator
do not depend
on cou- an exper& ./1 (
( 1 ( p ij ) ( ) ( t )0 ,
nonparametric
measure,
#
,
to
quantify
the
difference
between
range,
(PTX, red), and hypoexcitable (AP5/DNQX, blue)
j!J ( t ) cultures. Broken line, #3/2 power law. Cluster
range,
j!J ( t )
pling, the interpretation of one
electrode as
in vitro
Model
p(S)
EEG
P(PLI)
∆Dn
A
A
1, in good agr
ments (Fig. 7F )
We found t
model was tune
characteristics o
served in our e
tantly, average
network synchro
near the critical
by ! % 1 (Fig. 8
these quantities r
above ! % 1, as o
(Fig. 8A–C, righ
synchrony (Fig
burst synchron
matched the ex
there were quan
the computatio
ments. First, the
and peak entrop
cally shifted to s
those observed
the range of obt
in the model—
pletely suppress
! % 1.6 as fou
experiments. Fi
anti-phase lock
differences coul
simplifications m
model, includin
large-scale netw
layers, and binar
emphasize that
for both the exp
tional model; n
moderate averag
der E–I conditio
avalanches.
H(r(t)) n
|!D|
!
!
!-1
P. PLI
C
P(PLI)
<r(t)>n
B
"
"
PLI
PLI
%),
and isthe
theoretical
reference
CDF,
imental cluster size CDF, F(
sizewhere
s is the1[x]
sum of
nLFPunit
peakstep
amplitudes
within
cluster;
P(s) isisthe
probability
ofafunction.
observingLike
one
oscillator
problematic.
Therefore,
we
is )the
function.
Likethe
our
experiments,
we
explore
where
1[x]
the
unit
step
our
experiments,
we
explore
a
NA
*
&
10
log
(
S
/S
)
,
(2)
*
&
10
log
(
S
/S
,
(2)
10
max
min
10 max
min
developed a computational model with the
F (%),
#, the mean value of p from 0.75/N
a cluster
s. Right,
Corresponding
CDFs and
of pthe
network
stateby
using
rangeof
ofsize
network
excitability
by tuning
thequantification
mean
of
0.75/N
rangevalue
of network
excitability
tuning
ij from
ij
aim of obtaining results which were more
Figure 4. Distribution of PLI in a model exhibiting
self-organized criticality. A Through an ad
which
measures
deviation
from
a
#1/2
power
law
CDF
(broken
line).
Vertical
gray
lines,
The
10
where Smax and Smin
to
1.25/N
in
steps
of
0.05/N
by
scaling
all
p
are
the
stimulation
values
leading
to
90%
and
10%
by
a
constant.
For
such
small
topology,
Bornholdt
toward
connectivity independent of initial co
where
Smax and Smin are the stimulation values leading to 90% and 10% ij to 1.25/N in steps of 0.05/N
bythe
scaling
allmodel
pij byself-organizes
a constant.
Fora characteristic
such small
directly
comparable
to
our
experiments,
m
1
characteristic connectivity of approximately K~2:55 in a network of 1024 nodes for three different in
of the range
of
R,
respectively.
mean
p
,
the
model
reduces
to
probabilistic
integrate-and-fire,
i.e.,
p
2
NA
#
,
shown
for
one
example
PTX
condition
(red).
B,
Simulated
distances
summed
to
compute
of
the
range
of
R,
respectively.
mean
p
,
the
model
reduces
to
probabilistic
integrate-and-fire,
i.e.,
p
2
while
keeping
the
model
as
simple
as
ij
iJ
~6.
B
At
this
self-organized
connectivity
the
network
exhibits
a phase transition between order
K
ij
iJ
ini
#&1'
(F (%k) ( F(%k)),
(1)
#1not change their state along the attractor as a fu
component
C(K)
as the fraction
nodes that
Model. The model
consisted
ofPLI
N all-to-all coupled,
binary-state
neu- cluster
to order
N #2
accuracy. Ifneuthe(right)
mean
pijdifferent
ispijexactly
N #1
then
n control
PLI
mpossible.
Model. The
model consisted
of#Nj!J(t)
all-to-all
coupled,
binary-state
#j!J(t)
to order
Nof,#2
accuracy.
Ifdefined
the mean
pij is ofexactly
Ndo
, then n
k$1
sizepijPDFs
(left)
and
corresponding
CDFs
for
values
the
model
K for a network of 1024 nodes. The data were measured along the dynamical attractor reached by the syst
!
The computational model was com-
Beggs and Plenz, J. Neurosci, 2003
Meisel et al., PLoS CB, 2012
(2) Tuning of one parameter is sufficient to account for all the
observations during sleep deprivation:
the balance between excitation and inhibition
(3) A change in the E/I balance towards higher excitation is known
to occur during sleep deprivation.
50
the brain.
diagram
(process S).
of the main points of the hypothesis. During
about it. The EEG is activated, and the neuromodulatory milieu (for example, high levels of noradrenaline, NA) favors the storage of information,
which occurs largely through long-term potentiation of synaptic strength. This potentiation
occurs when the firing of a presynaptic neuron is
milieu signals the occurrence of salient events.
Strengthened synapses are indicated in red, with
(3) A change in the E/I balance towards higher excitation is known
to occur during sleep deprivation.
50
the brain.
diagram
(process S).
of the main points of the hypothesis. During
about it. The EEG is activated, and the neuromodulatory milieu (for example, high levels of noradrenaline, NA) favors the storage of information,
which occurs largely through long-term potentiation of synaptic strength. This potentiation
occurs when the firing of a presynaptic neuron is
milieu signals the occurrence of salient events.
Strengthened synapses are indicated in red, with
(4) A critical branching process captures other experimental
observations: maximal dynamic range, maximal pattern entropy, powerlaw scaling of avalanche durations, relations between scaling exponents,
optimal information transmission (mutual information between stimulus
and response), ...
Hypothesis:
Sleep reorganizes cortical network dynamics to a critical state
and thereby assures optimal computational capabilities for the
time awake.
The Journal of Neuroscience, October 30, 2013 • 33(44):17363–17372 • 17363
Systems/Circuits
Fading Signatures of Critical Brain Dynamics during
Sustained Wakefulness in Humans
Christian Meisel,1,2,3 Eckehard Olbrich,4 Oren Shriki,3 and Peter Achermann5,6
Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany, 2Department of Neurology, University Clinic Carl Gustav Carus, 01307
Dresden, Germany, 3Section on Critical Brain Dynamics, National Institute of Mental Health, Bethesda, Maryland, 4Max Planck Institute for Mathematics in
the Sciences, 04103 Leipzig, Germany, 5Institute of Pharmacology and Toxicology and Zurich Center for Integrative Human Psychology, University of
Zurich, 8057 Zurich, Switzerland, and 6Neuroscience Center Zurich, University and ETH Zurich, Zurich, Switzerland
1
Sleep encompasses approximately a third of our lifetime, yet its purpose and biological function are not well understood. Without sleep
optimal brain functioning such as responsiveness to stimuli, information processing, or learning may be impaired. Such observations
suggest that sleep plays a crucial role in organizing or reorganizing neuronal networks of the brain toward states where information
processing is optimized.
Increasing evidence suggests that cortical neuronal networks operate near a critical state characterized by balanced activity patterns,
which supports optimal information processing. However, it remains unknown whether critical dynamics is affected in the course of

Sleep, Criticality and Information Processing in the Brain

Transcription

Similar documents

NASAL-AIRE® II Dare to Dream

In 2011 - Atlanpole

1. go iceblocking 2. eat mulberries straight from the tree

Internationalisation – the Strategic Way

Be Good to Yourself ? You gotta be darn near a saint, but

key figure s

for a great night`s sleep!

How to Tie a Woggle

Vertafore SalesTrack Infographic

CC6022 SS6020 Convoluted Foam Mattress Cushion Insert