Optimal Control of Models of Neural Population Dynamics

• Applicant: Professor Dr. Klaus Obermayer
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics ; Experimental and Theoretical Network Neuroscience
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 523380209

Neural systems are described mathematically as controlled dynamical systems. They are affected by external stimulating inputs ("control"), which can be natural - via synaptic connections - or artificial - via external electric or magnetic stimulation. The mathematical approach can help to understand the underlying mechanisms responsible for the reaction of neural systems to stimulation. Here we will study the design of efficient control signals using methods from nonlinear optimal control theory (OCT) and investigate their properties, with applications being twofold: In "synthetic" scenarios, one can study how to manipulate neural systems efficiently (e.g. modulate brain activity). In "analytic" scenarios, one can study the natural design of neural dynamical systems in terms of optimization principles (e.g. enforced by evolution). The efficiency of a control signal is measured in terms of a cost functional that trades the strength of the input against the closeness to a target state. In this project, we use Pontryagin’s Principle to numerically compute a cost gradient. We then approach the optimal control iteratively. The conventional optimal control approach is limited to settings where an exact time-dependent target state can be defined. By contrast, any scenario where one would try to enforce oscillations, synchrony of the network dynamics, or locking, irrespective of the phases at control onset or of the exact shape of the oscillations, requires an extension of the standard approach. Here we want to study how to adapt the OCT approach to such settings. To this end, we suggest novel cost functionals and study their applicability and usefulness in models of single neural populations, network motifs, and large whole-brain models. We will focus on two neural mass models of different levels of complexity: the comparatively simple Wilson-Cowan model, which describes the time-evolution of the activation of recurrently coupled excitatory and inhibitory populations of neurons, and a mean-field model that is derived from a network of randomly connected excitatory and inhibitory adaptive exponential integrate-and-fire neurons (“mean-field AdEx model”). The computation of optimal control signals requires numerical simulations of the state evolution of these models, for which we use Neurolib, an open-source Python software framework for modeling neural dynamics. We will provide additional algorithms as an open-source extension to Neurolib, which will enable us to compute optimal control signals or the aforementioned (and other) models, for deterministic and noisy systems, and for models of single neural populations and networks thereof. The software will be implemented in a modular fashion, such that it can easily be adjusted to the requirements of specific scenarios to facilitate future studies of stimulation of neural systems.

• DFG Programme: Research Grants