Nonlinear Model Predictive Control of a Conductance-Based Neuron Model via Data-Driven Forecasting

TL;DR

This paper addresses how to achieve anticipatory control of a conductance-based neuron when only membrane voltage is observable and intrinsic currents are unknown. It combines nonlinear model predictive control (MPC) with a data-driven forecasting (DDF) model built from a Radial Basis Function Network to predict voltage dynamics, enabling MPC to compute current injections that steer spiking behavior. The approach is validated on Connor-Stevens Hodgkin-Huxley-type neurons, demonstrating successful homogeneous and heterogeneous control as well as precise spike-timing control, despite limited state information and noise. The work highlights the potential of data-driven MPC for neural control, suggesting applicability to larger networks and real-world experimental and clinical scenarios, while noting the need for estimators and robust formulations in more complex settings.

Abstract

Objective. Precise control of neural systems is essential to experimental investigations of how the brain controls behavior and holds the potential for therapeutic manipulations to correct aberrant network states. Model predictive control, which employs a dynamical model of the system to find optimal control inputs, has promise for dealing with the nonlinear dynamics, high levels of exogenous noise, and limited information about unmeasured states and parameters that are common in a wide range of neural systems. However, the challenge still remains of selecting the right model, constraining its parameters, and synchronizing to the neural system. Approach. As a proof of principle, we used recent advances in data-driven forecasting to construct a nonlinear machine-learning model of a Hodgkin-Huxley type neuron when only the membrane voltage is observable and there are an unknown number of intrinsic currents. Main Results. We show that this approach is able to learn the dynamics of different neuron types and can be used with MPC to force the neuron to engage in arbitrary, researcher-defined spiking behaviors. Significance. To the best of our knowledge, this is the first application of nonlinear MPC of a conductance-based model where there is only realistically limited information about unobservable states and parameters.

Figures (7)

• Figure 1: Open- vs Closed-Loop Control. (a) (Above) Block diagram of open-loop control. A command signal is applied to the system irrespective of the system output. (Below) Diagram of typical optogenetic stimulation experiment. A light source is applied to a cell expressing the appropriate light-sensitive opsin. For open-loop stimulation, the intensity of the light is determined before recording and may or may not cause the cell to fire. (b) (Above) Block diagram of closed-loop control. A command signal is calculated by the controller based on the state error - the difference between the system state trajectory and reference trajectory. (Below) Diagram of a voltage-clamp experiment. The cell is held at a specified voltage by injecting current determined by an online comparison of the membrane voltage with the desired reference voltage.
• Figure 2: Receding Horizon of MPC. Starting at the top row, (left) the system state trajectory (red) is being controlled to follow the reference trajectory (black). At the current time step (vertical dotted line) the controller finds the optimal set of inputs that minimize the loss function for a specified time horizon (shown with a gray box). In this case, the controller looks ahead 5 time steps (black dashed curve). Given the state at the current time ($t_0$), the controller uses a model to predict where the system will be across the future time horizon (red dashed curve). The inputs into the system (right) are optimized in discrete-time, and the input into the system is held constant between model time steps (solid blue curve). The predicted optimal future inputs (dashed blue curve) are calculated across the future time horizon. However, only the first of these values (circled in black) is used as input in the next time step before the optimization procedure begins again. From the top to bottom rows, we see how the controller may pick new optimal inputs given updates in the model predictions and by having access to new reference trajectory values (black dotted curve).
• Figure 3: CS Model Behaviors. (a) The spiking pattern of a Type-I (above) and Type-II (below) CS model in response to a 300 ms 9 $\mu$A step current. (b) The firing rate of the CS model as a function of step current amplitude. Notice that the Type-I model's firing rate increases approximately linearly after the input passes the firing threshold while the Type-II model abruptly jumps in firing rate. (c) The effect of the noise current on the CS models when stimulated with the same step current from panel A.
• Figure 4: DDF Model Forecasts. Each CS model was stimulated with 5 seconds of a known $I_{inj}(t)$ and unknown $I_{noise}(t)$. Each of the DDF models were fit using only the observations of the CS model voltages and the $I_{inj}(t)$. To evaluate the fit of the DDF models, their forecasted membrane voltages and spike trains were compared to CS models stimulated by a validation set of injected and noise currents. The forecasts were completely open-loop, where the DDF model was not corrected based on errors in predictions. (Left) CS model state trajectories (colored) and DDF model forecasts (black) on 2000 ms of validation data. (Middle) The $I_{inj}(t)$ (black) and $I_{noise}(t)$ (orange) currents for the section of validation data. The injected currents used to train and validate the model were obtained using the chaotic Lorenz 63 system. Poisson neurons were used to produce the noise current and had balanced excitation and inhibition. The amplitudes of the noise currents were chosen to result in an SNR of 5 compared to the injected current. This can be seen in the differing scales of the injected and noise currents between the two models. (Right) In black are the DDF forecasted rasters and directly below are the CS model spikes when only stimulated by $I_{inj}(t)$. We see a strong similarity between the two spike trains indicating that the DDF model learned much of the CS model dynamics. Although the DDF model did not accurately forecast every spike in the validation data (a), this was largely due to the unknown sources of noise. Repeated simulations of the CS model with the same injected current but different noise currents (yellow) show that the CS model without $I_{noise}(t)$ is deficient in predicting the noisy spike trains.
• Figure 5: MPC via DDF Diagram. (Red) The CS neuron receives an input $I_{inj}(t)$ from the controller at time step n which is held constant across a time interval of $\Delta t$ seconds. (Blue) The DDF model gets an update of the CS model membrane voltage every $\Delta t$ seconds. Given the membrane voltage $V_n$, time-embedded state history $S_n$, and discrete-time input $I_n$, the controller finds an optimal input for the next time step $I_{n+1}$. Given this optimized input, the DDF model makes a prediction of the CS model membrane voltage at the next $\Delta t$ time step, $V_{n+1}$. The controller uses the DDF model to simulate 5 $\Delta t$ time steps into the future (the time horizon) to find a sequence of optimized inputs by minimizing the loss function. (Purple) The first of these optimized inputs $I_{n+1}$ is used as the next injected current into the CS neuron.