Quantifying brain state transition cost via Schrödinger Bridge

Abstract

Quantifying brain state transition cost is a fundamental problem in systems neuroscience. Previous studies utilized network control theory to measure the cost by considering a neural system as a deterministic dynamical system. However, this approach does not capture the stochasticity of neural systems, which is important for accurately quantifying brain state transition cost. Here, we propose a novel framework based on optimal control in stochastic systems. In our framework, we quantify the transition cost as the Kullback-Leibler divergence from an uncontrolled transition path to the optimally controlled path, which is known as Schrödinger Bridge. To test its utility, we applied this framework to functional magnetic resonance imaging data from the Human Connectome Project and computed the brain state transition cost in cognitive tasks. We demonstrate correspondence between brain state transition cost and the difficulty of tasks. The results suggest that our framework provides a general theoretical tool for investigating cognitive functions from the viewpoint of transition cost.

Optimal Control Costs of Brain State Transitions in Linear Stochastic Systems

“…Third, for the control cost, we adopt the KL divergence between the uncontrolled and controlled processes ( Dai Pra, 1991 ; Léonard and Modal, 2014 ; Chen et al, 2016a ; Kawakita et al, 2022 ). A KL divergence is a metric that measures the closeness between two probability distributions.…”

“…The Schrödinger bridge problem was subsequently found to be equivalent to an optimal control problem, and has been studied in the field of control theory ( Dai Pra, 1991 ; Léonard and Modal, 2014 ; Chen et al, 2016b ). To our knowledge, our recent study ( Kawakita et al, 2022 ) is the first work in neuroscience to use the KL minimization problem in examining control cost in brain dynamics.…”

“…In other words, controlling transitions to some states incurs greater “costs” than controlling transitions to others. Providing a theoretical framework for quantifying transition costs, or control costs, is important for evaluating the difficulty of the shifts between various brain states, and possibly in explaining cognitive loads ( Braun et al, 2021 ; Kawakita et al, 2022 ), sleep-awake differences ( Deco et al, 2019 ), habituation of cognitive tasks ( Szymula et al, 2020 ), and psychiatric disorders ( Braun et al, 2021 ) with a quantifiable measure. Therefore, the development of such a framework for quantifying control cost in the brain is a vital topic in neuroscience.…”

“…In this study, we propose a novel framework to quantify control costs in linear stochastic neural systems. This framework takes advantage of both the linear control theoretic framework ( Gu et al, 2015 ; Kim et al, 2018 ; Szymula et al, 2020 ) and the control cost proposed in our previous work ( Kawakita et al, 2022 ). That is, we modeled brain dynamics using linear stochastic differential equations, and defined the control cost as the Kullback-Leibler (KL) divergence between the uncontrolled and controlled processes ( Fig.…”

“…That is, we modeled brain dynamics using linear stochastic differential equations, and defined the control cost as the Kullback-Leibler (KL) divergence between the uncontrolled and controlled processes ( Fig. 1 ) as we did in Kawakita et al (2022) . Thanks to the linearity, we can obtain an analytical expression of the control cost in the stochastic system.…”