Streaming Control of a Rotary Joint

Neuron model

Elevate is a phenomenological model with sodium, potassium, calcium, slow potassium and leak currents:

$$C\frac{dV}{dt}=I_0+g_e(E_e-V)+g_i(E_i-V) +\sum_{k\in\{\mathrm{Na},\mathrm{K},\mathrm{Ca},\mathrm{KCa},L\}}g_k(E_k-V).$$

$V,t,C,I_0$ denote voltage, time, capacitance and applied current. Excitatory and inhibitory conductances $g_e,g_i$ have reversals $E_e,E_i$; intrinsic conductance $g_k$ has reversal $E_k$. Units are mV, ms, nF, µS and nA; no cell-area calibration is assumed.

For gates $j\in\{m,h,n,c,q\}$,

$$\tau_j\frac{dU_j}{dt}=V-U_j,\qquad x_j=\sigma\!\left[a_j(U_j-b_j)\right],\qquad \sigma(z)=\frac{1}{1+\exp(-z)}.$$

$U_j$ is filtered voltage (mV), $\tau_j$ its timescale (ms), $x_j$ a dimensionless gate, $a_j$ the slope (mV$^{-1}$), $b_j$ the midpoint (mV), and $z$ the dimensionless sigmoid argument. In that order, slopes are $(0.2,-0.1,0.1,0.2,0.1)$ and midpoints $(-25,-40,-15,-45,-30)$ mV:

$$g_{\mathrm{Na}}=\bar g_{\mathrm{Na}}x_mx_h,\quad g_{\mathrm{K}}=\bar g_{\mathrm{K}}x_n,\quad g_{\mathrm{Ca}}=\bar g_{\mathrm{Ca}}x_c,\quad g_{\mathrm{KCa}}=\bar g_{\mathrm{KCa}}x_q,\quad g_L=\bar g_L.$$

$\bar g_k$ is maximum conductance. Both potassium currents use $E_{\mathrm K}$. KCa depends on filtered voltage, with no calcium-concentration state. All six states, $V$ and the five $U_j$, start at $-60$ mV.

Numerical update and spike output

Each 1 ms step uses 40 split substeps of $h=0.025$ ms with fixed input conductances. First update the filters using the old voltage:

$$U_j^+=V+(U_j-V)\exp(-h/\tau_j).$$

The superscript $+$ denotes an updated value. Evaluate gates and conductances from $U_j^+$, then update the membrane with conductances frozen:

$$G=g_e+g_i+\sum_k g_k,\qquad V_\infty=\frac{I_0+g_eE_e+g_iE_i+\sum_k g_kE_k}{G}, \qquad V^+=V_\infty+(V-V_\infty)\exp(-hG/C).$$

$G$ is total conductance (µS), $V_\infty$ the equilibrium voltage (mV), and $k$ indexes the five intrinsic currents. Each frozen sub-update is exact; their composition approximates the coupled dynamics. A crossing $V\leq0<V^+$ mV sets the window’s spike flag without resetting state.

Synaptic and local drive

Let $r\in\{f,e\}$ denote flexor or extensor, $\bar r$ its peer, and $n=0,1,\ldots$ the control window. Synapses decay, then receive spike increments:

$$\begin{aligned} g_{\mathrm{s},r}^{n} &=e^{-\Delta t/\tau_{\mathrm{s}}}g_{\mathrm{s},r}^{n-1} +w_{\mathrm{s}}S_r^{n-1},\\ g_{\mathrm{p},r}^{n} &=e^{-\Delta t/\tau_{\mathrm{p}}}g_{\mathrm{p},r}^{n-1} +w_{\mathrm{p}}M_{\bar r}^{n-1}. \end{aligned}$$

$g_{\mathrm{s},r}$ and $g_{\mathrm{p},r}$ are sensory AMPA and peer GABA conductances (µS). $S_r^{n-1}$ and $M_{\bar r}^{n-1}$ count the preceding window’s sensory and peer motor spikes. With $\Delta t=1$ ms, decay times are $(\tau_{\mathrm{s}},\tau_{\mathrm{p}})=(10,50)$ ms and per-spike weights are $(w_{\mathrm{s}},w_{\mathrm{p}})=(0.035,0.2)$ µS. Initial conductances and input counts are zero; increments are added in full, without timestep scaling.

Unfiltered local excitation switches from starting to tonic drive:

$$g_{\mathrm{local},r}(t)= \begin{cases} g_{\mathrm{start},r}, & 0\leq t<T_{\mathrm{start}},\\ g_{\mathrm{tonic}}, & t\geq T_{\mathrm{start}}. \end{cases}$$

$t$ is model time from startup, $T_{\mathrm{start}}=20$ ms, $(g_{\mathrm{start},f},g_{\mathrm{start},e})=(0.5,0)$ µS and $g_{\mathrm{tonic}}=0.04$ µS. At $t_n=n\Delta t$, the membrane inputs are

$$g_{e,r}^{n}=g_{\mathrm{local},r}(t_n)+g_{\mathrm{s},r}^{n}, \qquad g_{i,r}^{n}=g_{\mathrm{p},r}^{n}.$$

$g_{e,r}^{n}$ and $g_{i,r}^{n}$ are total excitatory and inhibitory conductances. Recorded settings appear in the parameter table.

Joint dynamics model

Two muscles drive a spring-damped joint. Zero points downward; positive angles point right. Only the tip mass contributes gravity:

$$(J_0+mL^2)\ddot{\theta}+b\dot{\theta}+k(\theta-\theta_r)+mgL\sin\theta =\tau_{\max}(a_f-a_e).$$

$\theta$ is angle (rad); dots denote derivatives in seconds. $J_0$ is base inertia (kg m$^2$), $m$ tip mass (kg), $L$ arm length (m), $g$ gravity (m/s$^2$), $\theta_r$ spring rest angle (rad), $b$ damping (N m s/rad), and $k$ stiffness (N m/rad). $a_f,a_e\in[0,1]$ are flexor/extensor activations; $\tau_{\max}$ is maximum muscle torque (N m).

At mechanical step $n$, muscle $i\in\{f,e\}$ updates as

$$a_{i,n+1}=\min\!\left(1, a_{i,n}\exp[-\Delta t/\tau_a]+q\,s_{i,n}\right).$$

Here $\Delta t=0.001$ s, $\tau_a$ is activation decay time (s), $q$ the dimensionless spike increment, and $s_{i,n}$ the preceding window’s motor spike count per neuron. Updated activations set torque. Semi-implicit Euler updates velocity, then angle; stops at $\theta_r\pm1$ rad clamp angle and zero velocity.

The joint starts at $\theta_r=\pi/2$ (90°), with zero velocity and activation. Total inertia is $0.00508$ kg m$^2$; initial gravity torque is $-0.003924$ N m. Flexor raises the mass against gravity; extensor lowers it.

Before each update, sensory spikes excite flexor for $\theta<\theta_r-\theta_s$ and extensor for $\theta>\theta_r+\theta_s$. The sensory centre is $\theta_r$, threshold $\theta_s=0.25$ rad, and minimum interspike interval 10 ms. Each channel repeats while beyond threshold.

Learning rule

The separate native adaptation protocol uses bounded coordinate search to adjust tonic excitation $g_{\mathrm{tonic}}$ and sensory synaptic strength $w_{\mathrm s}$, symmetrically across flexor and extensor. Reciprocal inhibition and mechanics remain fixed. This is a model-specific application of derivative-free direct search, using global movement errors:

$$\mathcal L= \left(\frac{f-f_\star}{f_\star}\right)^2+ \left(\frac{A-A_\star}{A_\star}\right)^2.$$

$\mathcal L$ is dimensionless loss. Frequency $f$ (Hz) is the reciprocal of mean cycle period; excursion $A$ (rad) is the mean peak-to-peak angle across measured cycles. Positive targets $f_\star,A_\star$ default to 0.65 Hz and $\pi/3$ rad (60° peak-to-peak). The two relative errors receive equal weight.

Alternate between the two conductances. For the selected coordinate $j$, test both proposals

$$p'_j=\operatorname{clip}\!\left((1+d\rho_j)p_j,\;0.01,\;0.10\right), \qquad d\in\{-1,+1\}.$$

$p_j$ is the current conductance and $p'_j$ a proposal, both in µS; $d$ chooses decrease or increase, $\rho_j$ is that coordinate’s fractional probe size, and $\operatorname{clip}$ bounds conductance to 0.01–0.10 µS. Keep the baseline unless an eligible proposal lowers $\mathcal L$ by more than $10^{-6}$; choose the better eligible proposal. Probe sizes start at 10% and halve after an unsuccessful comparison, down to 1.25%.

Each baseline or probe lasts at least 26 s of model time. Discard the first 15 s and measure the final five complete cycles, delimited by upward crossings of $\theta_r$. Eligibility requires no stop hits and all cycle-repeatability errors below 1%: maximum relative period and turning-point errors, and worst-cycle normalized root-mean-square errors for angle and velocity. Period errors use the median period; turning-point and angle errors use median excursion, and velocity errors use the median waveform’s velocity range. Unsettled segments extend in 5 s increments up to 60 s. Unsettled probes are excluded; an unsettled baseline skips its update without shrinking the probe size.

Neural, synaptic, muscle and pending spike states continue between probes. After 16 coordinate attempts, freeze the parameters and validate for 60 s both continuously and from a fresh reset, assessing the final ten complete cycles after settling. Local repeatability alone does not guarantee target acquisition or repeatability after reset.

Architecture Pseudocode

Each node retains state. These Python-like helpers summarize execution.

Setup

dt = 1 * ms
controllers = paired_controllers(neuron="Elevate")
joint = loaded_joint(initial_angle=90 * degrees)

warm_up_then_reset(controllers)
start_together(*controllers, joint)

Warm-up resets controller states, inputs and model clocks before execution.

Each control window

Each node waits for its release and both preceding input windows, servicing messages and enforcing its lag limit. Startup inputs are empty; silence is published explicitly.

def run(node):
    for window in fixed_schedule(period=dt):
        inputs = receive_previous_window(window)
        outputs = node.update(inputs)
        publish_window(window, outputs)  # Include windows with no spikes.
        record_timing(deadline=window.release + dt)

Late nodes catch up without dropping windows or shifting deadlines.

Controller and joint updates

Controllers share this update but receive different starting excitation.

def update_controller(controller, sensory_spikes, peer_spikes, time):
    drive = starting_drive(controller) if time < 20 * ms else tonic_drive
    controller.excite(local_drive=drive, sensory_spikes=sensory_spikes)
    controller.inhibit(peer_spikes)
    return controller.advance(dt).spikes


def update_joint(joint, motor_spikes, time):
    sensory_spikes = sense_angle(joint, time)
    activation = decay_then_activate(joint.muscles, motor_spikes)
    torque = opposing_muscle_torque(activation)
    advance_mechanics(joint, torque, dt)
    return sensory_spikes

Network diagram

Parameters

Values come from the selected recording’s native snapshot and configuration.