Firing Rate Neural Network Implementations of Model Predictive Control

Abstract

Human and animal brains perform planning to enable complex movements and behaviors. This process can be effectively described using model predictive control (MPC); that is, brains can be thought of as implementing some version of MPC. How is this done? In this work, we translate model predictive controllers into firing rate neural networks, offering insights into the nonlinear neural dynamics that underpin planning. This is done by first applying the projected gradient method to the dual problem, then generating alternative networks through factorization and contraction analysis. This allows us to explore many biologically plausible implementations of MPC. We present a series of numerical simulations to study different neural networks performing MPC to balance an inverted pendulum on a cart (i.e., balancing a stick on a hand). We illustrate that sparse neural networks can effectively implement MPC; this observation aligns with the sparse nature of the brain.

Figures (5)

• Figure 1: Block diagram of a neural network performing MPC. The controller is split into two parts; an offline state feedback law that corresponds to the unconstrained solution (inner loop) and an online neural network that ensures constraint satisfaction (outer loop).
• Figure 2: Structures of neural networks implementing MPC. $\Gamma$ corresponds to \ref{['eq: nn']}, $\Omega_1$ and $\Psi$ correspond to \ref{['sys: multilayer']}. $\Gamma + \Delta$ corresponds to \ref{['sys: perturbed']}, $\tilde{\Omega}_1$ and $\tilde{\Psi}$ correspond to a multilayer network where the layers only approximately satisfy \ref{['eq: factorization']}. Color of text corresponds to line color in Fig. \ref{['fig:comparison of trajectories']}.
• Figure 3: Control action ($u$) and trajectories of the cart position ($x_1$) and pendulum angle ($x_3$). MPC corresponds to the use of a traditional optimizer. The remaining lines correspond to the neural network implementations defined by the structures given in Fig. \ref{['fig:comparison of networks']}. Color of lines correspond to text color in Fig. \ref{['fig:comparison of networks']}.
• Figure 4: In-degree and out-degree distributions of the original network structure ($\Gamma$) and multilayer network ($\Omega_1$ & $\Psi$).
• Figure 5: The synaptic weight matrix corresponding to \ref{['eq: QP_slack']}. The core structure appears visually identical to the formulation without slack variables. Slack variable constraints are represented by the red nodes.

Theorems & Definitions (9)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof
• Remark 1
• Corollary 2
• proof