INTERNSHIP : ACTUATOR AND SENSOR PLACEMENT FOR NEURAL FIELDS

INTERNSHIP
• 2027
• POSSIBLE PH.
D. CONTINUATION Actuator and sensor placement for neural fields Laboratory Laboratoire des signaux et systèmes (L2S), CNRS, CentraleSupélec, Université Paris-Saclay, 91190 Gif-sur-Yvette, France Advisors Jean AURIOL, Lucas BRIVADIS, Antoine CHAILLET1 Duration 5 to 6 months, spring 2027 (flexible starting date) Keywords infinite-dimensional systems integro-differential equations spectral theory observability & controllability optimal placement mathematical neuroscience I. Context and scientific objectives Neural fields are nonlinear integro-differential equations that model the activity of neuronal populations [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 applications 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 resolution. 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 disorders 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 measured or actuated throughout the entire spatial domain. In practical applications, however, only a limited number of local actuators and sensors are available. This limitation raises important 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. L2S Mathematical setting On a spatial domain , the neuronal activity z(t, x) follows the Amari-type neural field equation, controlled by m electrodes with spatial profiles bi and observed at p locations x1, . . . , xp 2 where w is the synaptic kernel, S a sigmoidal activation function and I an external input. Around a stationary pattern z?, the linearized dynamics read ˙ = A + Bu, y = C, with A = −1(−Id +K) and K the integral operator with kernel w(x, y)S0(z?(y)). Since K is compact, A is a compact perturbation of −−1Id. Exact controllability with finitely many actuators is therefore out of reach, but the unstable part of the spectrum is finite-dimensional. The relevant notions are then approximate observability, detectability and stabilizability, and the question is how they depend on the positions xk and on the supports of the bi. Note also that pointwise measurements are unbounded on L2( ), which calls for a careful choice of the state space. An example: the ring model Consider a single population on a ring ( = S1), a classical model for orientation selectivity in the visual cortex, with a translation-invariant kernel w(x, y) = W(x − y). Around a homogeneous equilibrium, the linearized dynamics decouple into Fourier modes, and an unstable mode n corresponds to the emergence of an n-periodic activity pattern [1]. A single point sensor is never enough to reconstruct such a pattern, since the pattern can always be rotated so that the sensor lies on one of its nodes. Two sensors are enough, provided their distance is not a multiple of /n. For heterogeneous kernels, several populations or spatially extended electrodes, such explicit placement rules are no longer available. 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. More precisely, we will address the following questions.
• Where should a finite number of electrodes be placed to ensure the observability (or detectability) and controllability (or stabilizability) of a linearized neural field?
• How can electrode placement be optimized to enhance brain stimulation effects or to gather the most informative data on brain structures?
• Do placement rules obtained on a finite-dimensional approximation remain valid for the original infinite-dimensional system?
• (Longer term) How can effective output-feedback control laws be designed to achieve selective disruption of pathological neural oscillations? III. Scientific approach To address these questions, we will first consider a single neural population and simplify the dynamics through appropriate linearization. We will then project the kernel appearing in the integro-differential equation onto a functional basis (Fourier modes, eigenfunctions of K, . . . ) to obtain a finite-dimensional system, and adjust appropriate techniques developed for finite-dimensional systems to assess the controllability and observability of this reduced model. In particular, it 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]. Depending on the interests of the student, a second step will consist in working directly on the infinite-dimensional system (late lumping), for instance to design observers based on a few pointwise measurements using backstepping or Lyapunov techniques. The theoretical results will be illustrated by numerical simulations. Mathematical tools. The work will rely on spectral theory of compact operators, semigroups and infinite-dimensional linear systems, Hautus-type tests, Galerkin approximations, as well as optimal control and the optimization of Gramian-based criteria. IV. Profile and environment Profile. Master 2 student (or equivalent) in applied mathematics, PDE analysis or control theory, with a solid background in functional analysis and differential equations. Some familiarity with control theory and scientific computing (Python, Matlab or Julia) is appreciated. No prior knowledge of neuroscience is required. Environment. The intern will be hosted at L2S (Gif-sur-Yvette) and supervised by the three advisors, whose expertise covers infinite-dimensional systems, nonlinear control, neural fields and brain stimulation. This internship can lead to a Ph.
D. position. V. Application To apply, send an email with your CV and a transcript to Jean Auriol, Lucas Brivadis and Antoine Chaillet (firstname.lastname@centralesupelec.fr). VI. References [1] P.
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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.