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
Olivier Faugeras Neuronal circuits of the neocortex, GDR 20/01/06
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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
Olivier Faugeras Neuronal circuits of the neocortex, GDR 20/01/06
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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
Olivier Faugeras Mathematical framework, GDR 20/01/06 8
The dynamical system
Olivier Faugeras Mathematical framework, GDR 20/01/06 9
Bifurcation diagram
Branch of Hopf cycles Fold bifurcation of limit cycles
Saddle node
bifurcation with
homoclinic orbit
Olivier Faugeras Mathematical framework, GDR 20/01/06 10
Alpha rhythms and spiking
Thesis work of François Grimbert: Grimbert and Faugeras 2005
Olivier Faugeras Mathematical framework, GDR 20/01/06 11
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
Olivier Faugeras Mathematical framework, GDR 20/01/06 12
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.
Olivier Faugeras Mathematical framework, GDR 20/01/06 13
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
Olivier Faugeras Functional imaging, GDR 20/01/06 15
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)
Olivier Faugeras Functional imaging, GDR 20/01/06 16
Electroencephalography (EEG)
● Measures differences of potential on the scalp caused by cortical activity
Olivier Faugeras Functional imaging, GDR 20/01/06 17
Magneto-encephalography (MEG)
Measures the variations of magnetic field near the head caused by cortical activity
Olivier Faugeras Functional imaging, GDR 20/01/06 18
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
Olivier Faugeras Mathematical framework, GDR 20/01/06 19
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.
Olivier Faugeras Mathematical framework, GDR 20/01/06 20
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.
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