Motor Systems

Derivation 1: Population Vector Algorithm

Each M1 neuron $i$ has a preferred direction $\mathbf{d}_i$ and fires according to a cosine tuning model. For a movement in direction $\mathbf{m}$:

$$r_i = b_i + k_i \cos(\theta_m - \theta_i) = b_i + k_i \, \hat{\mathbf{d}}_i \cdot \hat{\mathbf{m}}$$

The population vector is the weighted sum of preferred directions:

$$\mathbf{P} = \sum_{i=1}^{N} (r_i - b_i) \, \hat{\mathbf{d}}_i$$

Substituting the tuning model and using the identity for uniformly distributed preferred directions:

$$\mathbf{P} = \sum_{i=1}^{N} k_i (\hat{\mathbf{d}}_i \cdot \hat{\mathbf{m}}) \hat{\mathbf{d}}_i = \frac{N \bar{k}}{2} \hat{\mathbf{m}}$$

For uniformly distributed preferred directions and equal gains $k_i = k$, the population vector points in the true movement direction with magnitude proportional to $N$. This result relies on $\sum_i \hat{d}_{ix}\hat{d}_{iy} = 0$ and$\sum_i \hat{d}_{ix}^2 = N/2$ for 2D uniform distributions.

Derivation 2: Reinforcement Learning Model of Striatal Plasticity

Dopamine signals from the substantia nigra pars compacta (SNc) encode reward prediction errors (RPE), as described by the temporal difference (TD) learning rule. The RPE at time $t$ is:

$$\delta_t = r_t + \gamma V(s_{t+1}) - V(s_t)$$

where $r_t$ is reward, $\gamma$ is the discount factor, and $V(s)$ is the value function. Striatal synaptic weights update according to:

$$\Delta w_{ij} = \alpha \, \delta_t \, e_{ij}(t)$$

where $e_{ij}(t)$ is an eligibility trace capturing the history of pre-post coincidence:

$$e_{ij}(t+1) = \lambda \gamma \, e_{ij}(t) + x_i(t) \cdot y_j(t)$$

The direct pathway D1 neurons undergo LTP with positive $\delta$ (dopamine burst), strengthening rewarded actions. Indirect pathway D2 neurons undergo LTP with negative$\delta$ (dopamine dip), strengthening avoidance of punished actions. This actor-critic architecture implements a biologically plausible reinforcement learning algorithm.

Derivation 3: The Marr-Albus-Ito Model of Cerebellar Learning

The cerebellum implements a supervised learning algorithm where climbing fiber inputs from the inferior olive provide error signals. The Purkinje cell output is a weighted sum of granule cell (parallel fiber) inputs:

$$y(t) = \sum_{i=1}^{N_{GC}} w_i \, x_i(t) = \mathbf{w}^T \mathbf{x}(t)$$

The climbing fiber carries the error signal $e(t) = y^*(t) - y(t)$ where $y^*$ is the desired output. The parallel fiber-Purkinje cell synaptic weights update via long-term depression (LTD):

$$\Delta w_i = -\eta \, e(t) \, x_i(t)$$

This is equivalent to the Widrow-Hoff (LMS) learning rule. The granule cell layer performs a high-dimensional expansion: $N_{GC} \gg N_{\text{input}}$ (humans have ~50 billion granule cells). By the Cover theorem, this expansion makes the representation linearly separable with high probability. The convergence rate depends on the eigenvalues of the input correlation matrix:

$$\tau_{\text{learn}} \approx \frac{1}{\eta \lambda_{\min}(\mathbf{X}^T\mathbf{X})}$$

This model explains cerebellar involvement in vestibulo-ocular reflex adaptation, saccade calibration, and reaching error correction.

Derivation 4: Half-Center Oscillator Model

The half-center CPG can be modeled as two mutually inhibitory units with adaptation. Let $u_1, u_2$ represent the activities of flexor and extensor populations:

$$\tau \frac{du_1}{dt} = -u_1 + S\left(w_E u_1 - w_I u_2 - b \cdot a_1 + I_{\text{ext}}\right)$$

$$\tau \frac{du_2}{dt} = -u_2 + S\left(w_E u_2 - w_I u_1 - b \cdot a_2 + I_{\text{ext}}\right)$$

where $S(x) = 1/(1 + e^{-x})$ is the sigmoid activation, $w_I$ is the mutual inhibition strength, $w_E$ is self-excitation, and $a_i$ are adaptation variables with slow dynamics:

$$\tau_a \frac{da_i}{dt} = -a_i + u_i, \quad \tau_a \gg \tau$$

The oscillation period is approximately $T \approx 2\tau_a \ln\left(\frac{w_I + b}{w_I - b}\right)$when $w_I > b$. The duty cycle (fraction of time each half is active) can be modulated by asymmetric drive $I_{\text{ext}}$, explaining how descending signals from motor cortex control locomotion speed and gait.

Derivation 5: Optimal Feedback Control of Reaching

Todorov and Jordan (2002) proposed that the motor system implements optimal feedback control, minimizing a cost function that trades off accuracy and effort. For a reaching movement with state $\mathbf{x}$ (position, velocity) and control $\mathbf{u}$ (muscle forces):

$$J = \mathbf{x}(T)^T \mathbf{Q}_f \mathbf{x}(T) + \int_0^T \left[\mathbf{x}^T \mathbf{Q} \mathbf{x} + \mathbf{u}^T \mathbf{R} \mathbf{u}\right] dt$$

subject to linear dynamics $\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \text{noise}$. The optimal controller is:

$$\mathbf{u}^*(t) = -\mathbf{R}^{-1}\mathbf{B}^T \mathbf{P}(t) \hat{\mathbf{x}}(t)$$

where $\mathbf{P}(t)$ satisfies the Riccati equation and $\hat{\mathbf{x}}$ is the Kalman-filtered state estimate. A key prediction is the "minimum intervention principle": the controller only corrects deviations that affect task performance, allowing variability in task-irrelevant dimensions. This explains the observed structure of movement variability in reaching and grasping.