Title: Actuator and sensor placement for neural fields
Laboratory: Laboratoire des signaux et syst`emes, CNRS, CentraleSupélec, Université Paris
Saclay, 91190, Gif-sur-Yvette, France
Internship Advisors: Jean AURIOL, Lucas BRIVADIS, Antoine CHAILLET1.
Description of the Internship
I. Context and scientific objectives
Neural fields are nonlinear integro-differential equations that model the activity of neuronal pop-
ulations [1, 2]. They provide a continuum approximation of brain structures, inspired by
the high density of neurons and synapses in the brain. The infinite-dimensional nature of neural
fields allows for accounting for the spatial heterogeneity of neuronal activity and the complex synaptic
connections between them. Additionally, their delayed versions can consider the non-instantaneous
communication between neurons. Unlike numerical models that represent each individual neuron with
a set of differential equations, neural fields are conducive to mathematical analysis. A wide range of
mathematical tools is now available to predict, analyze, and control their behavior. These
tools can assess the existence of stationary patterns [3, 4], perform stability analysis [5], and facilitate
feedback stabilization [6].
This balance between biological significance and abstraction accounts for the diverse applica-
tions of neural fields, which span across various areas such as the primary visual cortex [7, 8], the
auditory system [9], working memory [10], sensory cortex [11], and deep brain structures related to
Parkinson’s disease [6]. The advancement of modern technologies, such as multi-electrode arrays and
calcium imaging, enables the measurement of neuronal activity with increasingly high spatial resolu-
tion. By utilizing these measurements to estimate the synaptic distribution among neurons, we can
better understand the internal organization of specific brain structures. In our most recent research,
we developed adaptive reconstruction techniques [12] to estimate the evolution of the system in
real-time. This estimation of synaptic distribution could be instrumental in enhancing the feedback
control of neuronal populations. A particularly relevant example of this is deep brain stimulation
(DBS), which involves electrically stimulating deep brain structures associated with neurological disor-
ders like Parkinson’s disease [13]. Research has shown that stimulation proportional to the activity of
a brain structure known as the subthalamic nucleus is sufficient to disrupt brain oscillations associated
with Parkinson’s disease [6].
Although these estimation and control strategies represent significant theoretical breakthroughs,
they often rely on stringent assumptions, such as requiring that the neuronal population be mea-
sured or actuated throughout the entire spatial domain. In practical applications, however, only a
limited number of local actuators and sensors are available. This limitation can raise impor-
tant questions regarding observability and controllability. For example, depending on the kernel that
governs neuronal interconnections, one might wonder what information is necessary to reconstruct the
entire spatiotemporal state of a brain structure, or at the very least, to obtain a significant spatial
average of that state. Furthermore, in systems composed of different neuronal populations, only
certain populations can be effectively actuated or measured. In the context of Parkinson’s disease,
one may wonder how this limited information can be used to successfully compensate for pathological
neural oscillations.
Research goal: The general objective of this internship is to analyze the impact of actuator
and sensor placement on the control and estimation of neural fields comprising multiple neural
populations. Specifically, we aim to analyze how electrodes be placed to ensure the observability
and controllability of the neural field?
II. Scientific approach
To address the challenging questions of controllability and observability for neural fields, we will
consider projecting the kernel appearing in the integro-differential equations onto a functional basis
to obtain a finite-dimensional system. Then, we will adjust appropriate techniques developed for finite-
dimensional systems to assess controllabilty and observability of this reduced model. In particular, it
1The advisors are with Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des signaux et syst`emes, 91190,
Gif-sur-Yvette, France. Emails: {jean.auriol,lucas.brivadis,antoine.chaillet}@centralesupelec.fr.
will be possible to use optimal control methods to effectively position the available actuators and
sensors in the spatial domain. The main challenge here is to ensure that the terms neglected during
this projection do not undermine the observability/controllability of the original infinite-dimensional
system, which is known as the spill-over effect [14]. During the internship, we will consider a single
neural population and simplify the dynamics through appropriate linearization. This will allow
for the development of suitable actuator and sensor placement methods for this simplified infinite-
dimensional system.
III. Application
To apply, write an email with your CV and a transcript to Jean Auriol, Lucas Brivadis and Antoine
Chaillet: firstname.lastname@centralesupelec.fr. This internship can lead to a possible Ph.D.
position.
IV. References
[1] P.C. Bressloff. Spatiotemporal dynamics of continuum neural fields. Journal of Physics A: Mathematical
and Theoretical, 45(3), 2012.
[2] S. Coombes, P. beim Graben, R. Potthast, and J. Wright. Neural Fields: Theory and Applications. Springer,
2014.
[3] L. Brivadis, C. Tamekue, A. Chaillet, and J. Auriol. Existence of an equilibrium for delayed neural fields
under output proportional feedback. Automatica, 151:110909, 2023.
[4] O. Faugeras, R. Veltz, and F. Grimbert. Persistent neural states: stationary localized activity patterns
in nonlinear continuous n-population, q-dimensional neural networks. Neural Computation, 21(1):147–187,
January 2009.
[5] G. Faye and O. Faugeras. Some theoretical and numerical results for delayed neural field equations. Physica
D: Nonlinear Phenomena, 239(9):561–578, 2010. Mathematical Neuroscience.
[6] A. Chaillet, G. Detorakis, S. Palfi, and S. Senova. Robust stabilization of delayed neural fields with partial
measurement and actuation. Automatica, 83:262–274, Sep. 2017.
[7] M. Bertalm´ıo, L. Calatroni, V. Franceschi, B. Franceschiello, and D. Prandi. Cortical-Inspired
Wilson–Cowan-Type Equations for Orientation-Dependent Contrast Perception Modelling. Journal of
Mathematical Imaging and Vision, 63(2):263–281, 2021.
[8] D.A. Pinotsis, N. Brunet, A. Bastos, C.A. Bosman, V. Litvak, P. Fries, and K.J. Friston. Contrast gain
control and horizontal interactions in v1: A dcm study. NeuroImage, 92:143–155, 2014.
[9] U. Boscain, D. Prandi, L. Sacchelli, and G. Turco. A bio-inspired geometric model for sound reconstruction.
The Journal of Mathematical Neuroscience, 11(1):2, 2021.
[10] C.R. Laing, W.C. Troy, B. Gutkin, and G.B. Ermentrout. Multiple bumps in a neuronal model of working
memory. SIAM Journal on Applied Mathematics, 63(1):62–97, 2002.
[11] G. Detorakis and N. Rougier. Structure of receptive fields in a computational model of area 3b of primary
sensory cortex. Frontiers in computational neuroscience, 8, 2014.
[12] L. Brivadis, A. Chaillet, and J. Auriol. Adaptive observer and control of spatiotemporal delayed neural
fields. Systems & Control Letters, 186:105777, 2024.
[13] P. Limousin, P. Krack, P. Pollak, C. Benazzouz, A.and Ardouin, D. Hoffmann, and A. Benabid. Electrical
stimulation of the subthalamic nucleus in advanced Parkinson’s disease. N Engl J Med, 339(16):1105–1111,
January 1998.
[14] K. Morris. Controller Design for Distributed Parameter Systems. Springer, 2020
