Multiscale Modeling of Brain Dynamics

Peter RobinsonSchool of Physics, University of Sydney

Brain Dynamics Center, Westmead Hospital

& University of Sydney

Faculty of Medicine, University of SydneySupported by the ARC and NHMRC.

Kevin AquinoHomi Bahramali Matt Barton Lindsay Botha Paul Bourke Michael BreakspearParry Chen Po-Chia Chen Alan Chiang Jonathon Clearwater Nick CooperTim Cooper Peter DrysdaleBen FulcherCandy FungBiljana Germanoska Evian Gordon Stuart GrieveRon Grunstein

Alex Guinaudeau Rebecca Hamilton James Henderson Hal Henke Jackie HuberKim KaufmannCliff KerrJong-Won KimKrzysztof Kozak Anthony KrenselAndrew LaydenBelinda Liddle Peter LoxleyNeil Mahant Elie MatarSuzanne O’Connor Andrew Phillips Rebecca Powles

Collaborators: Chris Rennie Michelle RigozziPeter RileyJames Roberts Naomi RogersDonald RoweSacha van Albada Helena van der Merwe Rebecca WhitehouseLea Williams Keith Wong Jim WrightHui-Ying Wu

stimuli behavioral outputsmanipulations

observations

measurementsPROCESSES

The Big Picture

Measurement: what, why, how?

Integration

A First-Cut Model “Working Brain”

• Responds to stimuli, diurnal, circadian drives. Arousable.• Reproduces EEG, fMRI, etc.• Incorporates neuromodulation and simple behavioral feedbacks.• Starting point for further development.• Framework for integration & unification.

Modeling• We use a continuum model at scales of 0.1 mm to whole brain:

• Retains key anatomy and physiology at multiple scales.• Cortex approximated as 2D.• Include corticothalamic connections (plus others later).• Average over scales below about 0.1 mm (1000 neurons).• Seek partial differential equations for continuum fields.

• Such models date from 1950s on: Beurle, Nunez, Wilson, Cowan, Lopes da Silva, Freeman, Wright, Liley, Jirsa, Haken, Steyn-Ross, Sydney group, Coombes, others.

Neurons

• Excitatory (e) neurons excite others.• Inhibitory (i) neurons suppress others.• Inputs thru synapses on dendrites.• Firing triggered at axonal hillock.• Outputs via axon synaptic terminals.• e.g., Cortex contains:

• Long-range (several cm) excitatory neurons.• Mid-range (several mm) excitatory neurons.• Short-range (< 1 mm) excitatory neurons.• Short-range (< 1 mm) inhibitory neurons.

Kandel, Schwartz, & Jessell (2000)

Axonal hillock

Synapses and Dendrites

• Incident neurons transmit chemicalsignals to dendrites at synapses.• Chemical neurotransmitters are

released into the synaptic cleft, changing postsynaptic potential.

• Synaptic dynamics and dendriticpropagation smear signals over ~ 1-100 ms at the cell body.

Nolte (2002)

• Single cell response has a nonlinearthreshold firing rate behavior.

• Sigmoidal when averaged over apopulation:

Qa (Va) = Sa(Va).

• Cell body potentials Va approximately obey

• ab = mean activity from neural type b.• sab = mean strength of connections.• Nab= mean number of connections.

/)(max

1)(

Ve

QVS

Cell Body

. 1111

2

2

babababa sNV

dt

d

dt

d

• Activity spreads in a wavelike fashion with velocity vab and mean range rab.

• Approximate using a damped wave equation:• ab = vab / rab = damping rate.

• The propagator ab(0)(r,t) is the solution to

this equation for a -function input.

• Spatial part (effectively nonlocal):

. rr ),(),(121 22

2

2

2tQtr

tt babababab

Axonal Propagation

. rr rrr

arrabab etdt /)0(

0

)0( ~),(2)(2

Braitenberg & Shüz (1998)

The Model

• Our equations form a closed nonlinear set, parametrized physiologically:

Activity fields

ab

Va Cell-body potentials

Qa Firing rates

Propagation Synaptic/dendritic dynamics

Nonlinear thresholdresponse

t0Corticothalamic loop delay

Synaptodendritic response rates

GabGains

rabAxonal ranges

vabAxonal velocities

Qa,maxMaximum firing rate

SymbolQuantity

• Setting gives uniform nonlinearly determined steady states.• 2 stable steady states: low-e (normal) and high- e (seizure).• Only the seizure state survives at high stimulation levels.• Linear perturbations yield EEG spectra and ERPs.• Clarify links to physiology.

0,0

xt

Steady States, Response Properties

Coherence, Time Series, Stability

Theory DataEyes open

Eyes closed

Normalsleep

Deep sleep

Brain Resource International Database

• Brain Resource Ltd.• Spinoff 2001, ASX listed.• Approx. $40M market cap.• Database of circa 30 000 subjects, aged 6-80+. • Approx. 50 functional measures per subject +

MRI.• Excellent statistics.• Customers and labs in circa 10 countries.• 1st fully standardized international brain

function database. • Access via BRAINnet.

Inversion• Fitting predictions to data yields best estimates of parameters for individuals

• Can map parameters and combine consistently with other measures:

2040

60

2040

60800

10

20

30

40

50

Absence Seizures

• Linear instability at 3 Hz.• Ramping se up and down yields

start and end of ‘spike and wave’ oscillations via supercritical Hopf bifurcation.

υse

e (s

-1)

1

2

Time (s)

e (s

-1)

Time (s)

Fre

quen

cy (

Hz)

Time (s)

e (s

-1)

Time (s)

e (s

-1)

(t)

(t-

τ)

(t-2τ)

Time (s)

Fz

(µV

)

Time (s)

Fre

quen

cy (

Hz)

Time (s)

Time (s)

Fz

(µV

)F

z (µ

V)

(t)

(t-

τ)

(t-2τ)

Ocular Dominance and Orientation Preference• Orientation preference (OP) varies with position in each OD band.• Singularities, or pinwheels, occur mostly near OD band centers.• V1 is tessellated into hypercolumns; boundaries nonunique.• Each hypercolumn corresponds to a visual field (VF).

Kandel, Schwartz, & Jessell (1995)

Gamma Oscillations, Binding

• Scenes are analyzed via several feature-sensitive paths.• How are these aspects bound into one percept?• Firing of simultaneously stimulated cells in the visual

cortex is highly correlated over many mm.• Correlation functions (CFs) usually peak at T=0, even

when large conduction delays exist.• CFs are highest for nearby cells with similar feature preference. • Do gamma oscillations reflect or mediate binding, or are they epiphenomena?

Engel, Konig, Kreiter, Schillen, & Singer (1992)

• Use of patchy propagators yields new transfer functions and spectra.

• Waves obey Schroedinger equation.

• Resonances at and gamma

frequencies.

Gamma Resonances from Patchy Propagators

P(k,ω)

Kk

• Peak at T=0. Spatial and temporal extents

consistent with data.

• 1 long bar crossing different VFs produces a

stronger correlation than 2 separate short bars.

• Consistent with summation over stimuli and

infill of missing contours:

Engel, Konig, Kreiter, Schillen, & Singer (1992)

Gamma Correlations

Dworetzky (1994)

• Conflicting stimuli presented to 4 sites:

• 1 and 2 have vertical OP.

• 3 and 4 have horizontal OP.

• Correlations segment the scene into objects.

• Correlations between groups destroyed.

• Theory explains this effect via superposition:

Scene Segmentation

Engel, Konig, & Singer (2002)

S1

S2

S3

S1+S2

• How does the brain move between arousal states?• Develop and apply a quantitative, physiologically-based model of arousal

dynamics, with parameters from experiment.• Brainstem ascending arousal system must be integrated, plus circadian oscillations.• Physiological Modeling and Parameter Constraints

Arousal Dynamics

• Diffusely projecting brainstem nuclei control sleep-wake cycle:

• MA (monoaminergic)• ACh (cholinergic)

• Circadian (C) and Homeostatic (H) drives integrated in VLPO• Mutual MA-VLPO inhibition gives flip-flop behavior• Mean ACh and ORX inputs included

Model Dynamics• Neuronal population modeling predicts mean voltages Vi and firing rates Qi.

• Physiology & dynamics constrain parameters via a few experiments.• Dynamics accords with experiment:

Orexin, Narcolepsy, and Modafinil• Orexin group has input to the MA group.• Reducing this results in smaller hysteresis loop: age, narcolepsy.

• Stability of wake and sleep states reduced.• Modafinil pharmacokinetics imply stronger MA input• This restores hysteresis loop: antinarcoleptic.

A First-Cut Model “Working Brain”

• Responds to stimuli, diurnal, circadian drives. Arousable.• Reproduces the range of results discussed + others.• Incorporates neuromodulation, simple behavioral feedback.• Starting point for further development, detailed analysis of subsystems.• Framework for integration & unification.• Basal ganglia being incorporated.

macro

micro

fast slow

imaging

intracellular

basic features fine detail

Summary• Our continuum model tractably includes many features of neurophysiology, anatomy, measurement, and behavior from the microscale up.• Unifies many phenomena across scales.• Provides an approximate framework for interrelating observations.• Parameters lie in physiological ranges.• Many successful predictions including:

– Steady states, stability, spectra, coherence, correlations, seizures– EEGs, ERPs, SSEPs, ECoGs, fMRI connections.– Gamma phenomena in perception.– Arousal Dynamics: normal, abnormal, drugs.– Parameter space structure of states, parameter mapping.

• Ongoing: basal ganglia, parkinson’s, gamma-theta correlations, development, network connections, pharmacology, …• Future: attention, learning, plasticity, memory, pharmacology, cerebellum, …