Connecting neural mass models to functional imaging Olivier Faugeras, INRIA ● Basic neuroanatomy Basic neuroanatomy ● Neuronal circuits of the neocortex Neuronal circuits of the neocortex ● Connectivity Connectivity ● Mathematical framework Mathematical framework ● Functional imaging Functional imaging ● Roadmap for future research Roadmap for future research

Olivier FaugerasNeuronal circuits of the neocortex, GDR 20/01/06 2 Basic Neuroanatomy: the neocortex and the thalamus Area: 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 7 th 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 FaugerasNeuronal circuits of the neocortex, GDR 20/01/06 3 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 FaugerasNeuronal circuits of the neocortex, GDR 20/01/06 4 Cell populations of the six layers Pyramidal 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 FaugerasConnectivity, GDR 20/01/065 Cortical connections From Mumford 1991

Olivier FaugerasConnectivity, GDR 20/01/066 Cortico-cortical loops From Mumford 1991

Olivier FaugerasMathematical framework, GDR 20/01/067 A neural mass model Jensen and Rit 1995

Olivier FaugerasMathematical framework, GDR 20/01/068 The dynamical system

Olivier FaugerasMathematical framework, GDR 20/01/069 Bifurcation diagram Branch of Hopf cycles Fold bifurcation of limit cycles Saddle node bifurcation with homoclinic orbit

Olivier FaugerasMathematical framework, GDR 20/01/0610 Alpha rhythms and spiking Thesis work of François Grimbert: Grimbert and Faugeras 2005

Olivier FaugerasMathematical framework, GDR 20/01/0611 Connecting point neural mass models 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 Afferences from the thalamus Afferences from the thalamus 1: pyramidal cells 2: inhibitory interneurons 3: excitatory interneurons

Olivier FaugerasMathematical framework, GDR 20/01/0612 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 FaugerasMathematical framework, GDR 20/01/0613 Integro-differential equations The 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 FaugerasFunctional imaging, GDR 20/01/0614 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 FaugerasFunctional imaging, GDR 20/01/0615 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 FaugerasFunctional imaging, GDR 20/01/0616 Electroencephalography (EEG) ● Measures differences of potential on the scalp caused by cortical activity

Olivier FaugerasFunctional imaging, GDR 20/01/0617 Magneto-encephalography (MEG) Measures the variations of magnetic field near the head caused by cortical activity

Olivier FaugerasFunctional imaging, GDR 20/01/0618 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 FaugerasMathematical framework, GDR 20/01/0619 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 FaugerasMathematical framework, GDR 20/01/0620 Roadmap Explore 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.