bifurcations and multiscale cascades in cortical...
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OVERVIEW
Introduction
Criteria for a neuronal model
I: Neuronal dynamics
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley model
II: Ensembles of spiking neurons
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensembles
III: Mesoscopic neural mass models
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and induced
IV: A Neural field model of epilepsy
What is a neural field model?A neural field model of epilepsy
Computational Models of the
Brain
Michael Breakspear
Division of Mental Health Research,Queensland Institute of Medical Research
Brisbane, QLD
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Introduction: Why worry about large-scale brain activity?
151×MEG, 2×EMG
2×force sensors
Participants learned to perform a 3:5 paced bimanual tapping task:
Level 1: 1.11−0.67 Hz; Level 2: 1.67−1.0 Hz; Level 3: 2.22−1.33 Hz
Boonstra, et al. (2007) NeuroImage 36: 370-377
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Introduction: Why worry about large-scale brain activity?
Modulations of beta power were used to reconstruct sources in bilateral motor cortex using a dual-state beamformer technique
Boonstra, et al. (2007) NeuroImage 36: 370-377
force
EMG
Contralateral motor cortex
fast hand
slow hand
Results show first mode of PCA(all trials, all subjects)
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Introduction: Why worry about large-scale brain activity?
fast hand
slow hand
The (slow) movement cycle is associated with phase-amplitude coupling to a whole set of (rapid) oscillations in motor cortex
Data converges onto this pattern as the task is learned
MEG-EMG(“brain-body”)
MEG-MEG(“brain-brain”)
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Introduction: Criteria for a model of large-scale cortical activity
I: It should be derived, by first principles, from biophysical models of the “mass action” on neuronal populations
II: It should be self-organizating and support cognitive operations (language, inference, predictive modelling, computation)
III: It should describe the statistics of cortical activity. In particular, it should predict “the right class of statistics” evident in neuroscience data
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Introduction: Criteria for a model of large-scale cortical activity
Macroscopic
Microscopic
Mesoscopic
pictures downloaded from Google.com and Nunez (1997)
Whole BrainThe brain exhibits
structures at multiple scales– each with
distinct principles of organization and
distinct data set profiles
In this talk, we move from the
neuronal up to the whole brain scale
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Introduction: Criteria for a model of large-scale cortical activity
Macroscopic
Microscopic
Mesoscopic
Whole BrainThe brain exhibits
structures at multiple scales– each with
distinct principles of organization and
distinct data set profiles
In this talk, we move from the
neuronal up to the whole brain scale
Haider, McCormick (2009) Neuron
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Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley model
At the heart of the dynamical approach to neuroscience data are mathematical
formalisms of neuronal activity.
These are typically in the form of differential equations specifying how a set of neural variables V change in time, and
space depending on “state parameters”
( ) ( )),(,. tFtD xVxV a=
Differential operator containing temporal
and spatial derivatives
Neural States (firing rate, membrane
potential etc)
Equations which embody physiology
State parameters (Nernst potential, connectivity etc)
I: Neuronal dynamics
space time
Rather than seeking exact “closed form” solutions, dynamical system combines
analysis and geometry though the concept of “phase space analysis”
TIME SERIES PHASE SPACE
The function F defines a “vector field” on the “manifold” spanned by V.
Solution curves are tangent to the vector field
V
W
Z
VW
Z
time
I: Neuronal Dynamics
Geometry of spikes
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For example, a system with three variables V={V, W, Z}
An example of a chaotic flow …
V
V
time
W
time
Z
time
W
Z
TIME SERIES PHASE SPACE
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
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.. and an example of quasiperiodic burstingV
V
time
W
time
Z
time
W
Z
TIME SERIES PHASE SPACE
The system that these dynamics capture will be described later …
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
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Hartmann-Grobman theorem states that “all” nonlinear systems can be approximated (e.g. through truncation of Taylor expansion) by a linear system in the neighbourhood(s)
of their fixed point(s). Hence these are “canonical” fixed points
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
Simple linear system:
eigenvalues λ1, λ2 of A determine the stability of the single fixed point
has a single fixed point at the origin
λ1, λ2 same sign λ1>0, λ2<0
λ1, λ2 real
node saddleReal(λ) → stability
Imag(λ) → frequency
λ1, λ2 complex conjugates
spiral
FIXED POINTS
AZZ =
dt
d
Z t( )= Z 0( )etA .and time-dependent solutions:
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For example, the Van der Pol oscillator
Limit cycle attractor
Embedding manifold
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
,22
21 zzA += .arctan
1
2
= zzφ
… transformation of variables:
TIME SERIES PHASE SPACE
Z2 nullcline
Z1 nullcline
z2
NULLCLINES & MANIFOLDS
02 =z
( ) 01 122
1 =−− zzzµ
dz1
dt= z2 ,
dz2
dt= µ 1 − z1
2( )z2 − z1.
Unstable spiral
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dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
Slow potassium channel
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
m∞(V ) =mmax
1+ e Vm −V( ) σ ,n∞(V ) =nmax
1+ e Vn −V( ) σ ,
Sigmoid shaped channel activation
functions
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
V
n
-8 -6 -4 -2 0 2 4 6 8 10
time (ms)
V1
n(V,t) n∞(V1)
n∞(V2)
V2V(t)
( ) ( )( )( )V
VnVn
dt
Vdn
nτ−= ∞)(
“instant” Na channel
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dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
dn
dt=
n∞ − n( )τ n
Slow potassium channel
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
n
V (mV)-80 -60 -40 -20 0 20
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
n
V (mV)n nullcline
V nullcline Heteroclinic orbit
Stable focus Saddle Unstable spiral
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
m∞(V ) =mmax
1+ e Vm −V( ) σ ,n∞ (V ) =nmax
1+ e Vn −V( ) σ ,
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2 4 6 8 10-80
-70
-60
-50
-40
-30
-20
-10
0
10
dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
“Low threshold” parameters (Izhikevich 2005)
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
time (ms)
V (mV)
V (mV)
n
suprathreshold
subthreshold
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
m∞(V ) =mmax
1+ e Vm −V( ) σ ,n∞ (V ) =nmax
1+ e Vn −V( ) σ ,dn
dt=
n∞ − n( )τ n
Slow potassium channel
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dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
dn
dt=
n∞ − n( )τ n
Slow potassium channel
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
0 10 20 30 40 50-80
-70
-60
-50
-40
-30
-20
-10
0
10
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50-80
-70
-60
-50
-40
-30
-20
-10
0
10
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
n
V (mV)
V
time (ms)
(d)(c)
n V
(b)(a)
m∞(V ) =mmax
1+ e Vm −V( ) σ ,n∞ (V ) =nmax
1+ e Vn −V( ) σ ,
Low threshold parameters
(Izhikevich 2005)
“saddle node” bifurcation
→ current “integrators”
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Individual Neurons
Depending on the strength of synaptic currents, these neurons exhibit either by stochastic (left) or periodic/deterministic (right) firing rates
dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
,tan stochastictcons III += Noisy (Poisson) modulation of constant background rate
subthreshold + noise suprathreshold + noise
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dV
dt= gNam∞ V( )× V − VNa( )+ gKn V( )× V − VK( )+ gL × V − VL( )+ I ,
Conductance based model of transmembrane voltage due
to ion channel currents
dn
dt=
n∞ − n( )τ n
Slow potassium channel
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
m∞(V ) =mmax
1+ e Vm −V( ) σ ,n∞ (V ) =nmax
1+ e Vn −V( ) σ ,
0 20 40 60 80 100 120-80
-70
-60
-50
-40
-30
-20
-10
0
n
V
time (ms)
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120-80
-70
-60
-50
-40
-30
-20
-10
0
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-80 -60 -40 -20 0 20-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120-80
-70
-60
-50
-40
-30
-20
-10
0
n
V (mV)
V
time (ms)
n
V (mV)
V
time (ms)
V (mV)
High threshold parameters (Izhikevich
2005)
“Hopf” bifurcation
→ input “resonators”
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Conductance based model of transmembrane voltage due
to ion channel currents
Slow potassium channel
Dynamical systems frameworkSimple neuronal model: Integrators and resonators
Hodgkin-Huxley modelI: Neuronal dynamics
CdV t( )
dt= g
Naf
NaV t( )( )× V t( )− V
Na( )+ gK
fK
V t( )( )× V t( )− VK( )
+gL
× V t( )− VL( )+ I ,
fNa (V ) = m(V )M h(V )H
fK (V ) = n(V )N
( ) ( )( )( )V
VmVm
dt
Vdm
mτ−= ∞)(
( ) ( )( )( )V
VhVh
dt
Vdh
mτ−= ∞)(
( ) ( )( )( )V
VnVn
dt
Vdn
nτ−= ∞)(
Fast sodium channel
sodium inactivation channel
-1 -0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
h
V (mV)
m
n
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How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
1. Make a two-compartment neuron, with synaptic and dendritic compartments
HOW TO CONNECT NEURONS
( ) ( ) ,somaion
ionsomaionsoma IVVVf
dt
dVC +−×= ∑ Conductance based model of
cell soma compartment
( ) ..2 tett ααη −=
Post-synaptic currents
Synaptic and dendritic filtering 0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
time (ms)
nu(t
)
1250 −= msα
1400 −= msα
( ) ( ) ( ).∑ −=m
msynapse
msoma TITttI η
Passive dendritic compartment
II: Ensembles of spiking neurons
Clouds and moments
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( )( ) ,,H∑ ++−=j
sensorynoisejjjcsynapse IItxI τδ
Microscopic Ensemble
In this example, stimulus-induced currents induce a change from stochastic (time-space) activity to coherent periodic dynamics
Z2. Couple neurons together through synaptic currents
Synaptic currents in each individual
neuron Interneuron connectivity (H) and
efficacy (c)
Stochastic inputs (continuous Poisson
noise)
Afferent input from sensory neurons (on
or off)
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
Cell j firing event
Saddle node bifurcation
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( )( ) ,,H∑ ++−=j
sensorynoisejjjcsynapse IItxI τδZ2. Couple neurons together through synaptic currents
Synaptic currents in each individual
neuron Interneuron connectivity (H) and
efficacy (c)
Stochastic inputs (continuous Poisson
noise)
Afferent input from sensory neurons (on
or off)
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
Cell j firing event
Microscopic Ensemble
In this example, stimulus-induced currents induce a change from stochastic (time-space) activity to coherent periodic dynamics
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Microscopic Ensemble
When the stimulus is present, the deterministic dynamics cause a contraction in the ensemble density “cloud”.
Studying an entire ensemble of neurons allows one to visualise the evolution of the entire distribution of activity
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
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Microscopic Ensemble
When the stimulus is present, the deterministic dynamics cause a contraction in the ensemble density “cloud”.
Studying an entire ensemble of neurons allows one to visualise the evolution of the entire distribution of activity
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
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e.g. Harrison et al. (2005), Deco et al. (2007) PLoS CB,
Breakspear et al. (2008) Brain Imaging and Behaviour
Ensemble density models
where p describes a neural ensemble density of states v(t), evolving according to deterministic f and stochastic D processes
( ) .. pDfp ∇−−∇=
f=“drift”
D=“diffusion”
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
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Two ensemble network
First ensemble receives “sensory input”
Inter-network coupling modulated by an “attentional modulation”
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
Buchel, C., Friston, K.J., (1997) Cereb. Cortex. 7: 768–778.Friston KJ, Büchel C (2000) Proc. Natl. Acad. Sci. USA. 97, 7591-7596.
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Mean synaptic current
Inter-spike variance
Inter-spike kurtosis
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
ENSEMBLE 1Stimulus onset Stimulus offset
Inter-spike interval (t)
Inte
r-sp
ike
inte
rval
(t-
1)
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Inter-spike variance
How to connect neuronsEnsemble dynamics: Kinetics and moments
Networks of ensemblesII: Ensembles of spiking neurons
Mean synaptic current
ENSEMBLE 1
Mean synaptic current
ENSEMBLE 2
Inter-spike variance
Inter-spike covariance
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V i
Q i
( ) sensorynoiseiieeae IIf
dt
d ++= ςµςµµ,,,
VT0
1
µ
ς
Mean ensemble membrane potential ,
µ=<V>
Mean ensemble firing rate
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
.26.0
,0.0
,015.0
3
2
1
−=Λ=Λ=Λ
Lyapunov spectra (tendency of orbits to diverge/converge)
III: Mesoscopic neural mass models
The centre of mass
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
( ) ( ),)1(.,,, iV
jV
iiiV
ii
QCQCGWZQVfdt
dV −++=
V j
Q jV i
Q i
C
Interactions between two neural subsystems modifying the evolution of the local pyramidal cells to,
Coupling parameter
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V
Z
W
V2
V1
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
C=0
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V
Z
W
V2
V1
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
C=0.16
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
λ
Λ1
Λ2
λ1
λ2
C
‘blowout’ bifurcation (Ott, Sommerer 1994)
-0.6 -0.4 -0.2 0 0.2 0.4-0.6
-0.4
-0.2
0
0.2
0.4
V1
V2
Λ1
λ1
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V
Z
W
V2
V1
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
C=0.12
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
V
Z
W
Vi-Vj
W1-W2 time
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
P
G(Q)
Q
G-1(P)
G-1(P)
N
xi
xj
A
λ
Λ1
Λ2
λ1
λ2
C
λp
‘bubbling’ (Ashwin et al. 1994)
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Monoaminergic neurons
V i
Q i
V i
Q iV i
Q i
V i
Q iV i
Q i
CC
A cortical array is constructed by coupling
an ensemble of subsystems
-30 -20 -10 0 10 20 300
0.2
0.4
0.6
0.8
1
.),( h
ji
KjiH
xx −=
h=1
h=0.1
h=0.01
h=4
The strength of the coupling drops inversely with
subsystem separation.
It may be 1-D or 2-D
Cortical clustering:
Relatively sparse excitatory connectivity between local modules (cortical columns), each consisting of densely interconnected excitatory
and inhibitory neurons
Image from Nunez 1997 (Neocortical Rhythms)
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
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2-D Isotropic (symmetric) 1-D anisotropic
II: Ensembles of spiking neurons From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
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Narrow footprint: Local synaptic inputsBroad synaptic
footprint: No input
From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
Real brain tractography
centrality
Anatomical connectivity matrix
( ) Ifdt
diieea
e += ςµςµµ,,,
Spontaneous cortical activity
Functional connectivity matrix
Honey, Kotter, Breakspear, Sporns (2007) PNAS
( ) ( ) ττµτ
dttHRFtS e∫ −= )(
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From ensembles to mean fieldsA local mean field oscillator
Mean field oscillations: Spontaneous and inducedIII: Mesoscopic neural mass models
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IV: A neural field model of epilepsy
Primary generalized seizures are those which arise almost simultaneously across the entire cortex.
IV: A “neural field” model of epilepsy
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IV: A neural field model of epilepsy
Absence (“Petit Mal”) seizures Tonic-clonic (“Grand Mal”) seizures
We sought a unifying explanation of the two major forms of generalized seizures seen in clinical medicine.
3 Hz “spike and wave”Temporally symmetric and ~stationary
10 Hz irregular slowing to spike and wavesHighly asymmetric and nonstationary
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IV: A neural field model of epilepsy
Va membrane potentials
Φa field potentials
Qa firing rates
a=e,i cortical populations
a=s specific thalamic n.
a=r reticular thalamic n.
To do so, we employed a neural “mean field” approach,
where D is a (spatiotemporal) differential operator acting on V, a vector of neural states
which vary continuously across a smooth cortical
sheet.
( ) ( )),(,. tFtD xVxV a=
Each of the entries of V, represents the mean value of a neural state in a
local populations (~cm2).
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IV: A neural field model of epilepsy
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
+∂∂
++∂∂=
ttD
βααβα
),,(),(),()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
∇−+
∂∂+
∂∂= aaa
aa v
ttD γγ
γ
( ) [ ],),(, 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
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IV: A neural field model of epilepsy
μse μse
Φe
(s-1)
Φe
(s-1)
3 Hz instability
μse
Φe
(s-1)
10 Hz instability
μse
Φe
(s-1)
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
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Discussion
Deco et al. (2007) PLoS CB, Breakspear et al. (2008) Brain Imaging and Behaviour
I : Cortical field models
where
describes propagating cortical field potentials in cortical, thalamic and reticular populations
.21 222
2
2
2
∇−+
∂∂+
∂∂= aaa
aa v
ttD γγ
γ
( ) [ ],),(, tVStD aaa rr =φ
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
II: Ensemble density models
where p describes a neural ensemble density of states v(t), evolving according to deterministic f and stochastic D processes
( ) ( )..
∂∂
∂∂+
∂∂−=∇−−∇=
v
pD
vv
fptrpDfp
III: Neural mass models
describes local cortical ensembles (j) interacting through induced synaptic currents, with distant ensembles (k) in the presence of
noise.
( ) ( ) .,,, , σςςµςµµIgHf
dt
d
k
keckj
je
je
je
jea
je ++= ∑
Breakspear, Stam, Williams (2003) JCNS
Honey, Kotter, Breakspear, Sporns (2007) PNAS
Jirsa & Haken (1996) PRL, Robinson (2002) PRE, Breakspear et al. (2006) Cerebral Cortex
Probabilistic inference
Self organization
and complexity
Detailed biophysical
validity
?Cortical Statistics?
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Discussion
Hence we require a model that allows multistability and non-Gaussian statistics at all scales
Freyer, Aquino, Robinson, Ritter, Breakspear (2009). Journal of Neuroscience
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Acknowledgements
Local Team Members
Stuart Knock, Kevin Aquino, Angela Langdon, Mika Rubinov, Mark Schira, Stewart Heitmann, Tjeerd Boonstra, Muhsin Karim, Norman Ferns,
Tamara Powell, Nicky Kochan
Funding Sources
James S. McDonnell Foundation (Brain NRG)ARC “Thinking Systems” Special Initiative
Further Reading
Izhikevich E (2005) Dynamical systems in neuroscience: The geometry of excitability and bursting. MIT Press.Jirsa, V., McIntosh, A.R. (2007) Handbook of Brain Connectivity. Springer.
Deco, G., Jirsa, V.K., Robinson, P.A., Breakspear, M., Friston, K.J. (2007) PLoS CB. (in press).Breakspear, M., Knock, S. (2008) Kinetic models of brain activity. Brain Imaging and Behaviour (in press).