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