A normative approach to neuromotor control

Abstract

While we can readily observe and model the dynamics of our limbs, analyzing the neurons that drive movement is not nearly as straightforward. As a result, their role in motor behavior (e.g., forward models, state estimators, controllers, etc.) remains elusive. Computational explanations of electrophysiological data often rely on firing rate models or deterministic spiking models. Yet neither can accurately describe the interactions of neurons that issue spikes, probabilistically. Here we take a normative approach by designing a probabilistic spiking network to implement LQR control for a limb model. We find typical results: cosine tuning curves, population vectors that correlate with reaching directions, low-dimensional oscillatory activity for reaches that have no oscillatory movement, and changes in neuron’s tuning curves after force field adaptation. Importantly, while the model is consistent with these empirically derived correlations, we can also analyze it in terms of the known causal mechanism: an LQR controller and the probability distributions of the neurons that encode it. Redesigning the system under a different set of assumptions (e.g. a different controller, or network architecture) would yield a new set of testable predictions. We suggest this normative approach can be a framework for examining the motor system, providing testable links between observed neural activity and motor behavior.

Appendix

The gradient for the cost function in Eq. 8 is as follows,

$$\begin{aligned}&L = \sum _k^N L_k + \lambda \sum _{i,j,p,q} \left[ (w^1_{ij})^2 + (w^2_{pq})^2\right] \end{aligned}$$

(10)

$$\begin{aligned}&L_k = \frac{1}{2n}(\mu (\Delta X^k)-\bar{O}^k)^T(\mu (\Delta X^k)-\bar{O}^k) \nonumber \\&\quad \quad = \frac{1}{2n}\sum _l^n (\mu _l(\Delta X^k)-\bar{o}^k_l)^2 \end{aligned}$$

(11)

$$\begin{aligned}&\text{ where } \nonumber \\&\bar{o}^k_l = p(o_l=1|\Delta X^k) \nonumber \\&\qquad = \sum ^M_v p(o_l=1|H^v) \prod ^m_i p(h_i|\Delta X^k) \end{aligned}$$

(12)

then we find the following,

$$\begin{aligned} \frac{\partial {}L_k}{\partial {}W^1_{pq}}= & {} \frac{-1}{n}\sum ^n_l[\mu _l(\Delta X_k)-\bar{o}_l]\left( \sum ^M_vp(o_l=1|h^v) ...\right. \nonumber \\&\left. \prod ^m_{j\ne {}p}p(h^v_j|x)dp(h^v_p|x)x_q\right) \end{aligned}$$

(13)

$$\begin{aligned} \frac{\partial {}L_k}{\partial {}B^1_{pq}}= & {} \frac{-1}{n}\sum ^n_l[\mu _l(\Delta X_k)-\bar{o}_l] \left( \sum ^M_vp(o_l=1|h^v)...\right. \nonumber \\&\left. \prod ^m_{j\ne {}p}p(h^v_j|x)dp(h^v_p|x)\right) \end{aligned}$$

(14)

$$\begin{aligned} \nonumber \frac{\partial {}L_k}{\partial {}W^2_{pq}}= & {} \frac{-1}{n}[\mu _p(\Delta X_k)-\bar{o}_p)]\left( \sum ^M_vdp(o_p=1|h^v)...\right. \nonumber \\&\left. \prod ^m_{j}p(h^v_j|x)h^v_q\right) \end{aligned}$$

(15)

$$\begin{aligned} \nonumber \frac{\partial {}L_k}{\partial {}B^2_{pq}}= & {} \frac{-1}{n}[\mu _p(\Delta X_k)-\bar{o}_p)]\left( \sum ^M_vdp(o_p=1|h^v)...\right. \nonumber \\&\left. \prod ^m_{j}p(h^v_j|x)\right) \end{aligned}$$

(16)