Bifurcations and multiscale cascades in cortical dynamics Michael Breakspear, Stuart Knock, UNSW, Australia Kevin Aquino, Peter Robinson, James Roberts, University of Sydney John Terry, Serafim Rodriguez, University of Bristol, UK
Jan 02, 2016
Bifurcations and multiscale cascades in cortical dynamics
Michael Breakspear, Stuart Knock, UNSW, Australia
Kevin Aquino, Peter Robinson, James Roberts, University of Sydney
John Terry, Serafim Rodriguez, University of Bristol, UK
Bifurcations and multiscale cascades in cortical dynamics
1: Mean Field Brain Modelling
Robinson et al. (2002) Phys Rev. E 65:041924.
Corticothalamic connectivity
where
describes dendritic filtering of synaptic currents
),2,(),(),(),( 0ttvtvtvtVD sasiaieaea −++= rrrr φφφα
1111
2
2
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛++
∂∂
=tt
Dβααβα
),,(),(),()2,(),( 0 tvtvtvttvtVD nanrarsaseaea rrrrr φφφφα +++−=
8 9 10 11 12 13
11.76
11.78
11.8
11.82
11.84
11.86
model
time (sec)
Φe
79 80 81 82 83
-100
-50
0
50
100
EEG
time (sec)
Pz
Va membrane potentials
Φa field potentials
Qa firing rates
Cortico-cortical connectivty
where
describes propagating cortical field potentials
Jirsa & Haken (1996) PRL
.21 222
2
2
2 ⎥⎦
⎤⎢⎣
⎡∇−+
∂∂
+∂∂
= aaaa
a vtt
D γγγ
( ) [ ],),(, tVStD aaa rr =φ
a=e,i cortical populations
a=s specific thalamic n.
a=r reticular thalamic n.
0 2 4 6 8 10 126
8
10
12
14
time (sec)
Φe
strongly damped
μse μse
Φe
(s-1)
Φe
(s-1)
3 Hz instability
μse
Φe
(s-1)
10 Hz instability
μse
Φe
(s-1)
Bifurcations and multiscale cascades in cortical dynamics
2: Unifying explanation of seizures
Breakspear et al. (2006) Cerebral Cortex doi:10.1093/cercor/bhj072
Global mode instabilities
1. Ignore spatial derivatives ( ) & 2. explore nonlinear bifurcations by changing μse 02 =∇
model Cortex-reticular–specific loop
data
model Cortex-specific loop
data
Fp1
Time:174 (s)
Fp2
F3 F4
C3 C4
P3 P4
O1 O2
F7 F8
T5 T6
Fz
Cz
Pz
T3 T4Frequency (Hz)
2
2.5
3
3.5
4
4.5
5
1. Nonlinear frequency coupling
Data
172 173 174 175 176 177 1781.5
2
2.5
3
3.5
4
4.5
Time (s)
Frequency (Hz)
Petitmal Seizure Data: Frequency Locking
Fp1Fp2F3F4C3C4P3P4O1O2F7F8T5T6FzCzPzT3T4
2. Amplitude modulation
Data
time (s)
Frequency (Hz)
P3
5 10 15 20 25 30 35
2
4
6
8
10
12
160 165 170 175 180 185 190 1950
0.5
1
1.5
2
2.5
3
3.5x 104 P3
Time (s)
Power (
μV4Hz-
1s-1)
time (s)
Frequency (Hz)
Fp1
5 10 15 20 25 30 35
2
4
6
8
10
12
160 165 170 175 180 185 190 1950
0.5
1
1.5
2
2.5
3x 105 Fp1
Time (s)
Power (
μV4Hz-
1s-1)
time (s)
Frequency (Hz)
Fz
5 10 15 20 25 30 35
2
4
6
8
10
12
160 165 170 175 180 185 190 1950
0.5
1
1.5
2
2.5x 105 Fz
Time (s)
Power (
μV4Hz-
1s-1)
Bifurcations and multiscale cascades in cortical dynamics
3: Impact of global coupling on local mean fields
Global corticocortical coupling
Reintroduce spatial dependency in Absence seizure (3Hz spike and wave) regimen
Model
7 8 9 10 11 12 130
2
4
6
8
10x 104 node(2)
Time (s)
Power (
μV4Hz-
1s-1)
time (s)
Frequency (Hz)
node(2)
7 8 9 10 11 12 13
2
4
6
8
10
12
7 8 9 10 11 12 130
1
2
3
4
5x 105 node(3)
Time (s)
Power (
μV4Hz-
1s-1)
time (s)
Frequency (Hz)
node(3)
7 8 9 10 11 12 13
2
4
6
8
10
12
7 8 9 10 11 12 130
0.5
1
1.5
2
2.5
3x 106 node(1)
Time (s)
Power (
μV4Hz-
1s-1)
time (s)
Frequency (Hz)
node(1)
7 8 9 10 11 12 13
2
4
6
8
10
12
Model
7 8 9 10 11 12 131.5
2
2.5
3
3.5
4
4.5
Time (s)
Frequency (Hz)
Model Petitmal Seizure: Frequency Locking
node(1)node(2)node(3)
Hindmarsh Rose: Between scales
Hindmarsh Rose: Between systems
Bifurcations and multiscale cascades in cortical dynamics
4: Multiscale temporal effects in neural systems
( ) ,,, 321 Ixxxfdt
d+= a
X
Fast timescales, x1,2
Slow timescales, x3( ) ( ),, 11ji
ci
i
xxHIfdt
d++= X
Xa
Coupled nodes:
Single node:
Burst (slow) synchrony: c~0.35
Spike (slow+fast) synchrony: c>0.45
Dhamala et al. (2004) PRL
0 100 200 300 400 500 600 700 800 900 1000-2
0
2
0 100 200 300 400 500 600 700 800 900 1000-0.1
0
0.1
0 100 200 300 400 500 600 700 800 900 1000-0.5
0
0.5
0 100 200 300 400 500 600 700 800 900 1000-5
0
5
0 100 200 300 400 500 600 700 800 900 1000-5
0
5
0 100 200 300 400 500 600 700 800 900 1000-10
0
10
c
sca
leco
arse
fin
e
0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
100
0.2
0.4
0.6
0.8
scalefine coarse
…and their anti-symmetric component,
0 50 100 150 200-0.2
0
0.2
0.4
0.6
0.8
1
1.2bgrckmy
'.MSMSaMS −=
t
aMS
-200 -150 -100 -50 0 50 100 150 200-0.2
0
0.2
0.4
0.6
0.8
1
1.2bgrckmy
Calculate inter-scale effects as,
( ) ( ) ( ) .)(,)(,, 211121 ss txtxMIssMS ττ +=
Nakao et al. (2001) IJCB
tM
S
Healthy cortical Activity
Seizures
Bifurcations and multiscale cascades in cortical dynamics
5: Coupling of spatiotemporal scales in cortical activity
μse
Φe
(s-1)
10 Hz instability
10 10.1 10.2 10.30
0.1
0.2
0.3
0.4
0.5
10 10.1 10.2 10.30
0.2
0.4
0.6
Symmetric Antisymmetric
0 0.5 1 1.5 2 2.5 3 3.5 4
-50
0
50
100
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
0
2
x 10-3M
od
el
Da
ta
sca
leco
arse
f
ine