Large scale models of the brain
Viktor Jirsa
Theoretical Neuroscience Group
Institut des Sciences du Mouvement
Anandamohan Ghosh
Rolf Kötter
Randy McIntosh
Young-Ah Rho
Michael Breakspear
Stuart Knock
Gustavo Deco
Information processing carried out by
large scale neural networks
Honey et al PNAS 2007, Ghosh et al
Plos CB 2008, Deco et al PNAS 2009
Izhikevich & Edelman 2008
Henry Markram – Blue Brain
Ananthanarayanan et al. IBM 2009
Motivation
Mean field models collapse the dynamic characteristics of a voxel into a
single neurocomputational unit of neurons with similar statistics
Deco et al. PLoS CB2009
Globally coupled network of Fitzhugh-Nagumo neurons
Network Dynamics
Coupled mean fields
• Define a continuous field parametrized by the dispersed
parameter
• Rewrite the network equations in terms of q(z,t)
Assisi, Jirsa, Kelso PRL2005
Stefanescu, Jirsa Plos CB 2009
•Express the dynamics of the network in z-space in terms of zspatial modes and the corresponding time dependent
amplitudes.
Assisi, Jirsa, Kelso PRL2005
Stefanescu, Jirsa Plos CB 2009
Mode Equations
The mode equations are given by,
Assisi, Jirsa, Kelso PRL2005
Stefanescu, Jirsa Plos CB 2009
Mode Dynamics
Network dynamics
Mode dynamics
Contour lines of equal mean field amplitude in
space
Assisi, Jirsa, Kelso PRL2005
Stefanescu, Jirsa Plos CB 2009
Neural field models
Full network
Reduced neural field
Jirsa & Stefanescu Bul. Math. Biol (in press)
Origin of ultraslow fluctuations: neural activity?
Simultaneous EEG and
fMRI study finds crosscorrelations between
BOLD signal and the
power fluctuations in
each frequency band.
Mantini et al. PNAS
2007
Generation of the rest state activity? Function?
• Product of chaotic processes involving the thalamocortical loop
(Lopes da Silva et al. 1997; Niedermeyer 1997)
• Distinct alpha generators (Nunez et al 2001)
• Large scale connectivity matrix and chaotic neural activity (Honey et
al PNAS 2007)
• Noise driven exploration of the high-dimensional phase space
defined by the network with time delays (Ghosh et al Plos CB2008)
• Stochastic Resonance in the network with time delays (Deco et al
PNAS 2009)
• « Rest state fluctuations reflect unconstrained but consciously
directed mental activity »
• Rest state network fluctuations observed in anaesthesized monkeys
(Vincent et al., Nature 2007)
Regional map of the primate brain (Kötter & Wanke, 2005)
Monkey
Human
Ghosh et al. PLoS CB 2008
Implementation of large scale model
N
ui (t )   [ g (ui , vi )  c  f ij u j (t  tij )]
j 1
tij 
d ij
v
vi (t )  (1 /  ) h(ui , vi )
g (u, v)   [v   u 
u
3
]
3
h(u, v)  (1 /  ) [u    bv]
Assisi, Jirsa Kelso PRL 2005
Stefanescu, Jirsa PLoS CB 2009
Jirsa, Stefanescu Bull.Math.Biol (in press)
Ghosh et al Plos CB 2008; Deco et al PNAS 2009
Linear stability analysis
x  g ( x(t ), x(t   ))
Let the solution be
g
x (t )
g
x(t   )
x  Ax(t )  Bx (t   )
linearization
x(t )  e t
characteristic equation in 
For N coupled FHN oscillators the characteristic equation is factorizable:
 A(1)
det 
 O
N

  tij
2

(


u

c
f
e
)

ij

j 1
A(i )  

1


O
0
A(2)


N
Characteristic equation:
b
N
 (  )(  (  u )  c fije
i 1

2
j 1
tij



 b  


) 1  0

Ghosh et al. PLoS CB 2008
Ghosh et al. PLoS CB 2008
Hemodynamic model: combining Balloon/Windkessel Model
with a model of how synaptic activity causes changes in regional flow
Linear coupling
term:
How evoked changes in blood flow
are transformed into a blood
oxygenation level dependent(BOLD)
Nonlinear coupling
term:
Balloon/Windkessel
model
Case 1
Case 3
Compare to Fox et al. PNAS 2005
Resting state network in BOLD signals
•
Task-negative regions:
MPF(medial prefrontal cortex),
PCC(posterior cingulate
precuneus), LP(Lateral parietal
cortex)
•
Task-positive regions:
IPS(intraparietal sulcus cortex),
FEF(the frontal eye field),
MT(middle temporal region)
Fox et al. PNAS (2005)
Cross correlations between six areas: Ghosh et al PLOS CB 2008
CCP
FEF
PCI
PCIP
PFCM
VACD
CCP
+
+
-
+
-
+
FEF
+
+
-
-
-
+
PCI
-
-
+
+
+
-
PCIP
+
-
+
+
-
-
PFCM
-
-
+
-
+
-
VACD
+
+
-
-
-
+
Compare to Fox et al. PNAS 2005
Forward EEG/MEG solution in realistic
head models
400fT
15uV
0fT
0 uV
-400fT
-15uV
Ghosh et al Plos CB 2008; Jirsa Phil. Trans. Royal Soc. A 2009
Qf = 1, Qs = 1
Honey et al PNAS 2007;
What is the dynamic mechanism leading to the
emergence of these coherent fluctuations?
Synchronization?
FitzHugh-Nagumo Neuron
g ( x, y )   [ y   x 
x
3
]
3
h( x, y )  (1 /  ) [ x    by ]
x1  g ( x1 , y1 )  c f12 x2 (t  t )  f13 x3 (t )  nx (t )
y1  h( x1 , y1 )  n y (t )
F
x2  g ( x2 , y2 )  c f 21x1 (t  t )  f 23 x3 (t )  nx (t )
y 2  h( x2 , y2 )  n y (t )
x3  g ( x3 , y3 )  c f 31x1 (t )  f 32 x2 (t )  nx (t )
y 3  h( x3 , y3 )  n y (t )
Rho, Jirsa & McIntosh (in preparation)
Rho, Jirsa & McIntosh (in preparation)
BOLD in CCA is correlated with coherence between PCI and CCP, and BOLD
time series are shifted with time lag(2.4sec).
Rho, Jirsa & McIntosh (in preparation)
Rho, Jirsa & McIntosh (in preparation)
Other working points, maybe self-sustained
oscillations?
Different working point:
What is the role of
synchronization?
Two clusters of synchronization
Deco, Jirsa, McIntosh et al. PNAS (2009)
Synchronization of clusters
Red – cluster 1
Black – cluster 2
Blue – difference
Power spectrum of ultraslow
oscillations with and without time
delay
Stochastic Resonance
Cross correlation as a function
of noise level
Maximal Power as a function of
noise level
Deco, Jirsa, McIntosh et al. PNAS (2009)
Summary of results
Rest state activity is interpreted as the « noise-driven exploration of
the equilibrium state of the brain network »
The space-time structure is crucial for the emergence of the rest
state networks.
Intermittent synchronization of subnetworks gives rise to ultra-slow
oscillations in BOLD signal.
Thank you
Codebox Research
ATIP (CNRS)
James S. McDonnell Foundation