Optimal Control Costs of Brain State Transitions in Linear Stochastic Systems

Abstract

The brain is a system that performs numerous functions by controlling its states. Quantifying the cost of this control is essential as it reveals how the brain can be controlled based on the minimization of the control cost, and which brain regions are most important to the optimal control of transitions. Despite its great potential, the current control paradigm in neuroscience uses a deterministic framework and is therefore unable to consider stochasticity, severely limiting its application to neural data. Here, to resolve this limitation, we propose a novel framework for the evaluation of control costs based on a linear stochastic model. Following our previous work, we quantified the optimal control cost as the minimal Kullback-Leibler divergence between the uncontrolled and controlled processes. In the linear model, we established an analytical expression for minimal cost and showed that we can decompose it into the cost for controlling the mean and covariance of brain activity. To evaluate the utility of our novel framework, we examined the significant brain regions in the optimal control of transitions from the resting state to seven cognitive task states in whole-brain imaging data. We found that, in realizing the different transitions, the lower visual areas played a commonly significant role in controlling the means, while the posterior cingulate cortex played a commonly significant role in controlling the covariances.

Optimal Control Costs of Brain State Transitions in Linear Stochastic Systems

“…For proof of this transformation ( Eq. 25 ), see Kamiya et al (2022) . Thus, Equation 23 can be seen as the cost for controlling the mean from μ 0 to μ T .…”

“…In the equation, , and . For the definitions and detailed derivations, see Kamiya et al (2022) . Π( t ), G ( t ), and Ψ( t , s ) depend on A , C , Σ 0 , and Σ T .…”

“…Furthermore, the formulation given in Equation 25 indicates a correspondence between the stochastic control cost and the deterministic control cost ( Kamiya et al, 2022 ). We can easily see that, if the input matrix B is identical to the diffusion matrix C , the mean control cost ( Eq.…”

“…25 ) is that one can obtain the directions in which controlling the mean is easy or difficult. In the deterministic control system, it is classically known that the eigenvectors of the control Gramian matrix (see Kamiya et al, 2022 ) can be interpreted as the direction in which the control is easy/difficult ( Yan et al, 2015 ; Brunton and Kutz, 2022 ). The same logic can be applied to the mean control cost that shares the similar formulation to the Gramian in the deterministic system: the eigenvectors of Equation 26 specify the directions in which the system is good at in controlling its mean.…”

“…3 ). This minimum is known to exist, and the optimal value is readily solved (see Kamiya et al, 2022 ). This metric has been used as the control cost in previous studies in neuroscience ( Gu et al, 2017 ; Kim et al, 2018 ; Stiso et al, 2019 ; Cornblath et al, 2020 ; Deng and Gu, 2020 ; Braun et al, 2021 ).…”