Connecting neural mass models to functional imaging Olivier Faugeras, INRIA ● Basic neuroanatomy Basic neuroanatomy ● Neuronal circuits of the neocortex

Connecting neural mass models to

functional imagingOlivier Faugeras, INRIA

● Basic neuroanatomy

● Neuronal circuits of the neocortex

● Connectivity

● Mathematical framework

● Functional imaging

● Roadmap for future research

Olivier Faugeras Neuronal circuits of the neocortex, GDR 20/01/06

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Basic Neuroanatomy: the neocortex and the thalamusArea: 200,000 mm2.

Thickness: 2-3 mm, comprising 6 layers.

Neuron density: 100,000/ mm2.

Divided into specialized areas (100/hemisphere).

All input but the olfactory sense comes from the thalamus (divided into 50

nuclei).

Each part of the cortex is reciprocally connected to some nucleus in the

thalamus: as if it were an elaborate 7th layer.

The thalamus sends axons up to the cortex where they synapse in layers

III/IV

It receives axons originating in pyramidal cells in layers V/VI


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The neocortex: what kinds of cells

Two main types: pyramidal cells and interneurons

Pyramidal cells: excitatory, projecting intra- and

inter- area (60-80% of the population)

Interneurons: inhibitory, projecting intra-area

One minor type: spiny stellate, excitatory, projecting

intra-area.

The majority of cortical cells have inter-area projections


4

Cell populations of the six layersPyramidal cells:

deep: layers V and VI

superficial: layers II and III

Spiny stellate cells: mainly in layer IV

Inhibitory interneurons: all layers except I

Layer I (plexiform) has very few cell bodies; connections

between the interneurons and the apical dendrites of the

pyramidals

Olivier Faugeras Connectivity, GDR 20/01/06 5

Cortical connections

From Mumford 1991

Olivier Faugeras Connectivity, GDR 20/01/06 6

Cortico-cortical loops

Origin Destination

Deep pyramidal cells Layers I and VI(layer V)

Superficial pyramidal Layer IVCells

Superficial pyramidal Layers I and VICells

From Mumford 1991

Olivier Faugeras Mathematical framework, GDR 20/01/06 7

A neural mass model

Jensen and Rit 1995


The dynamical system


Bifurcation diagram

Branch of Hopf cycles Fold bifurcation of limit cycles

Saddle node

bifurcation with

homoclinic orbit


Alpha rhythms and spiking

Thesis work of François Grimbert: Grimbert and Faugeras 2005


Connecting point neural mass

models

1

2

3

Short-range afferences from excitatory

interneurons Short- and long-range afferences from

pyramidal cells

Afferences from the thalamus

Short- and long-range

efferences

Short-range efferences

Short-range efferences

Afferences from the thalamus

Afferences from the

thalamus

1: pyramidal cells

2: inhibitory interneurons

3: excitatory interneurons


Mathematical description

models the strength of the connections between pyramidal cells

For short-range connections, it is commonly assumed that

Modelling afferences to pyramidal cells

is some part of the neocortex

The axonal transmission delays and synaptic time constants can be included:

For long-range connections, information can be obtained from DTI data.


Integro-differential equationsThe resulting description:

is an integro-differential equation (see previous slide)

In order to analyse its well-posedness, think of it as a differential equation

on the infinite dimensional space where Y “lives”:

is typically a Banach space, e.g.,

(Local) existence and uniqueness can be obtained with reasonable assumptions

on the connection strengths, e.g., ,using for example Cauchy's theorem.

Numerical solutions can be computed using a fixed-point theorem

Olivier Faugeras Functional imaging, GDR 20/01/06 14

Functional Imaging: fMRI

● Deoxyhemoglobin is paramagnetic ● 40% of the oxygen delivered to the capillary bed is extracted● Substantial amount of dHb in the venous vessels ---> attenuation of MR signal● Brain activation:

● local flow ● oxygen metabolism

● Oxygen extraction reduced and venous blood more oxygenated: signal increases● Blood Oxygen Level Dependent (BOLD) effect


A model of the BOLD signal: the Balloon Model

Described by a nonlinear dynamic system of dimension 4 (Buxton et al. 97, 2004,

Deneux and Faugeras 2004)


Electroencephalography (EEG)

● Measures differences of potential on the scalp caused by cortical activity


Magneto-encephalography (MEG)

Measures the variations of magnetic field near the head caused by cortical activity


Models for EEG and MEG

● Not all cortical cells will induce measurable electromagnetic fields● Pyramidal cells (layers 3 and 6) are primarily responsible

● They create a primary current density at each point of the cortex● Which is related to the electrical potential by the Maxwell equations

● The magnetic field can then be computed from the Biot-Savart law


Identifying the parameters

The function

is proportional to the current density created by the post-synaptic potentials

of pyramidal cells:

where is equal to the average dendrite cross-section area multiplied by

the intracellular conductivity.

If we plug this into the MEEG direct problem we can predict the measurements

as a function of the unknowns.

If we plug this into the Balloon model, we can predict the fMRI measurements

as a function of the unknowns.


RoadmapExplore further the geometry and the physics of the human brain: xMRI, HARDI for

geometry

anatomical connectivity

conductivity tensor

Develop better physiological models of the relation between neural activity and the

BOLD, Optical Imaging signals.

Develop better neural mass models from neurophysiological data and first principles

(microscopic to mesoscopic)

Develop mathematical models of brain areas, explore their mathematical properties.

Use them in conjunction with functional imaging to identify their parameters, test their

validity.

Develop a computational interpretation of their behaviour, e.g. in visual perception.

Documents

Connecting neural mass models to functional imaging Olivier Faugeras, INRIA ● Basic neuroanatomy Basic neuroanatomy ● Neuronal circuits of the neocortex