The same reach, a firmer hand

Miraia S. Chiou

Reading · Neuroscience

The same reach, a firmer hand

Hold the same position with different stiffness, then give both systems the same push.

2026-09-08 · 11 min read · 2 interactive demos

Hold an imaginary teacup loosely. Now imagine a bus taking a sharp turn, and hold the same cup more firmly. The cup can remain in exactly the same place while your response to a bump changes dramatically.

Position tells us where the hand is. Net force tells us the balance of its pushes and pulls. Neither completely describes how the hand will respond when the world pushes back.

Opposing muscles offer a vivid version of an Epistemic Physics theme: a balance can stay fixed while the amount behind it changes. Muscles that pull in opposite directions can become active together, a behavior called co-contraction. Their effects on posture and resistance to disturbance need separate coordinates.

Two pulls and a resting place

Let’s begin with a transparent mechanical model. Place a handle at position $x$ between two ideal linear spring-like elements. The left element pulls toward anchor $a$ with stiffness $k_{+}$ ; the right pulls toward anchor $b$ with stiffness $k_{-}$ . The subscripts label the two elements. They do not determine whether a force is positive: the anchor’s position does that.

Stiffness measures how much restoring force changes per unit displacement. In this one-dimensional model its units are newtons per metre, or N/m. Both stiffnesses are positive. The total force is

F (x) = k_{+} (a - x) + k_{-} (b - x) .

At an unloaded equilibrium $x_{*}$ , the opposing forces sum to zero. Solving $F (x_{*}) = 0$ gives

x_{*} = \frac{k_{+} a + k_{-} b}{k_{+} + k_{-}}, K = k_{+} + k_{-} .

Here $K$ is total stiffness. Rearrange the same force law:

F (x) = - K (x - x_{*}) .

We now have two useful coordinates. The ratio sets the resting place; the total sets resistance to displacement. Scale both stiffnesses by the same positive factor and the resting place stays fixed while resistance grows.

There is a small but revealing trap here. Adding the same amount to both stiffnesses preserves their ratio only when they were equal. Away from the center, preserving posture requires scaling both together or coordinating their changes in another ratio-preserving way.

If a steady external force $f$ also pushes the handle, equilibrium becomes

x_{f} = x_{*} + \frac{f}{K} .

An 8 N force displaces a 40 N/m system by 0.2 m. A 160 N/m system shifts by 0.05 m. The same resting place hides a fourfold difference in the response to the push.

Predict before pushing: move the target away from the center, increase total stiffness, and then press the push button. Will the unpushed target move? Will the displacement under load change?

Opposing pulls

Hold the pose, change the resistance

Two ideal spring-like muscles pull a handle toward opposite anchors. Drag the gold target, then strengthen both pulls together. Predict which handle will move farther under the same sustained push.

Total stiffness K 80 N/m Sustained external force 0 N

Left pullRight pullChosen stiffness40 N/m comparisonTarget

Left / right stiffness: 40.0 / 40.0 N/m
Unloaded equilibrium: 0.00 m
Chosen displacement: 0.0 cm
Comparison displacement: 0.0 cm

The same valley, steeper sides

These curves show elastic energy above the unloaded minimum. Increasing stiffness keeps the minimum in place and makes climbing away from it more costly.

Both unpushed handles settle at 0.00 m. The selected handle is 2.0 times as stiff, so the same force produces 0.50 times the displacement. The net force at unloaded equilibrium remains zero.

Chosen linear spring model, with anchors at -1 and +1 m. Stiffness is measured in newtons per metre. The handles show static equilibria after a sustained force, with no timing or overshoot simulation. Scaling both stiffnesses preserves their ratio; equal additive increases would preserve a centered target only.

The anchors and units are chosen to keep the arithmetic visible. Actual muscles have activation-dependent, length-dependent, and velocity-dependent behavior. The model captures one useful local relationship: opposing restorative contributions can preserve an equilibrium while changing its stiffness.

Three balances, three meanings for amount

You may have met the same weighted average in a neuron. A passive membrane is pulled toward channel reversal potentials $E_{1}, E_{2}$ by conductances $g_{1}, g_{2}$ . Its equilibrium voltage is $(g_{1} E_{1} + g_{2} E_{2}) / (g_{1} + g_{2})$ .

Or you may have met it in a Beta distribution. Positive parameters $α, β$ give mean $μ = α / (α + β)$ and total parameter amount $A = α + β$ . Under an evidence interpretation, prior parameters and observed trial amounts must be tracked consistently; neither parameter is automatically a raw count of independent observations.

Put their corresponding quantities together:

Question	Two restorative elements	Passive membrane	Beta belief
What are the positive weights?	$k_{+}, k_{-}$	$g_{1}, g_{2}$	$α, β$
What is their total?	$K = k_{+} + k_{-}$	$G = g_{1} + g_{2}$	$A = α + β$
Where is the balance?	$x_{*} = (k_{+} a + k_{-} b) / K$	$V_{*} = (g_{1} E_{1} + g_{2} E_{2}) / G$	$μ = (α \cdot 1 + β \cdot 0) / A$
What does a fixed input change?	$Δ x = f / K$ for a steady force	$Δ V = I / G$ for a steady current	$Δ μ = (y - μ) / (A + 1)$ for one outcome $y$

The final Beta expression follows by adding one binary outcome, $y \in {0, 1}$ :

μ_{new} = \frac{A μ + y}{A + 1}, μ_{new} - μ = \frac{y - μ}{A + 1} .

For the same mean and the same next outcome, a larger $A$ produces a smaller update. This is one precise sense of lower malleability, meaning reduced ease of changing a belief. In the spring system, larger $K$ produces a smaller displacement under a fixed force. In the passive membrane, larger $G$ produces a smaller voltage displacement under a fixed current.

These are aligned response laws with different units and mechanisms. Stiffness has units of N/m, conductance has units of electrical current per voltage, and Beta parameters are dimensionless. The shared division by a total does useful explanatory work without requiring those totals to be physically interchangeable.

A valley you can make steeper

A spring stores elastic energy. Relative to its minimum, the two-element system has energy

U (x) - U (x_{*}) = \frac{1}{2} K (x - x_{*})^{2} .

The demo draws this valley. The resting place is the bottom; stiffness is its curvature, the rate at which the slope changes. Strengthen both elements together and the valley steepens around the same bottom. A sustained external force tilts the energy landscape by adding $- f x$ , shifting the minimum by $f / K$ .

That picture suggests another connection. A sharply curved negative log probability also resists movement away from its preferred state in an optimization problem. For a Beta distribution with $α, β > 1$ , its negative log density, up to a constant, is

L (p) = - (α - 1) \log p - (β - 1) \log (1 - p) .

Its interior mode, the most probable value, is

p_{*} = \frac{α - 1}{α + β - 2}, L^{″} (p_{*}) = \frac{α - 1}{p_{*}^{2}} + \frac{β - 1}{(1 - p_{*})^{2}} .

This curvature is a mathematical measure of how steeply the density falls locally around its mode. It is distinct from mechanical stiffness and from inverse variance in general. To hold this mode fixed while steepening the loss, scale $α - 1$ and $β - 1$ together. Scaling $α$ and $β$ instead holds the mean fixed. Those two constructions are easy to confuse, and keeping them separate makes the analogy more useful.

Firmness is not a complete motion model

A motion has timing as well as a resting place. If we extend the spring model with a mass $m$ and damping coefficient $c$ , a displacement $δ = x - x_{*}$ follows

m \ddot{δ} + c \dot{δ} + K δ = f (t) .

A dot means one time derivative; two dots mean acceleration. Damping is a velocity-dependent force that removes mechanical energy. This chosen equation introduces new parameters that the static balance did not contain.

For positive $m, c, K$ , the undriven system settles toward equilibrium. Its damping ratio is $ζ = c / (2 \sqrt{m K})$ . Values below one permit decaying oscillations; at or above one the standard free response has no oscillatory ringing. Increasing $K$ while fixing $m$ and $c$ lowers $ζ$ . “Firmer” therefore does not automatically mean “no overshoot.” The stiffness, damping, inertia, and control policy work together.

The membrane comparison makes another useful distinction. For a passive membrane with capacitance $C$ , the relaxation time is $C / G$ . Increasing conductance both reduces the sustained displacement and shortens this first-order relaxation time. A second-order arm model can behave differently because it has inertia. The similar equilibrium formulas have not erased the dynamics.

What the nervous system actually adjusts

Mechanical impedance describes how a system responds to imposed movement, including elastic and velocity-dependent effects. In multiple dimensions, resistance can also depend on the direction of the disturbance. Think of a valley that is narrow across a ridge but gentle along it.

Burdet and colleagues used a robotic interface to place human reaching movements in unstable dynamics. Participants learned to stabilize their movements, and the study examined selective changes in impedance geometry. This makes the “same task, different resistance” idea experimentally concrete. Burdet et al., 2001.

Franklin and colleagues then examined reaching in unstable force fields of differing strength. Their results connected adaptation of arm impedance with the cost of muscle activation. The interesting strategy is therefore more selective than making every direction maximally stiff: protection against disturbance has costs, and the task gives those costs a context. Franklin et al., 2004.

Our two-element picture gives a first coordinate system for these ideas. A human arm adds multiple joints, nonlinear muscles, reflexes, sensory delays, and learning. Each addition gives us another measurable way to ask how the same successful reach can be produced.

From a hand to an intended cursor movement

A neural interface adds a new bridge: recorded neural activity must be translated into an action. A decoder is the algorithm that performs that translation. It may estimate an intended velocity from noisy observations, combine that estimate with an internal prediction, and send a command to a cursor or device.

The ReFIT Kalman filter work by Gilja and colleagues studied neural cursor control in two monkeys. Its training procedure incorporated assumptions about intended movement during feedback control, and improved performance relative to the comparison decoder. The closed loop matters: the animal sees the cursor, responds to it, and thereby changes the neural signals that the algorithm receives. Gilja et al., 2012.

A Kalman filter combines a predicted state with a new measurement according to their modeled uncertainty. The following laboratory explores that generic estimation principle. It is a tracking illustration, rather than a reproduction of ReFIT or a biological model of intention.

Prediction opens the estimate; a reading pulls it together

Keep track when the sensor goes quiet.

Follow the same moving signal with a Kalman filter and a fixed-gain memory. Scrub through the readings. During a sensor outage, watch the Kalman band widen; then see how strongly the returning measurement pulls.

Expected motion per step · SD 0.15 Sensor noise · SD 0.60 Fixed memory's gain 0.20

WorldReadingsKalmanFixed gain

Drag the gold time handle to walk forward or back through the readings.

Walk through the experiment Step 46 / 80

Predicted mean / SD: -0.549 / 0.663
Updated mean / SD: -0.881 / 0.445
This reading: -1.153
Kalman gain: 0.550

The reading pulls the prediction 55.0% of the way toward it. The fixed memory always moves 20% of the distance from its own estimate to the reading.

SD means standard deviation; the shaded band is the Kalman estimate ±2 model SDs. The world takes independent steps with SD 0.15, with an added jump in the jump scenario. The sensor's SD is known to the filter; the expected motion is your model choice. The same random sequence is reused as you adjust the settings. Both filters use only readings up to the selected step.

Notice how the questions stack. Where does the user intend to go? How uncertain is our estimate? What command should we issue? How resistant should the device be to unexpected contact? A useful controller keeps these questions connected while giving each its own quantity. Decoder uncertainty does not by itself specify muscle stiffness or a safe physical response.

Three experiments to take with you

Begin with the simplest perturbation experiment: hold the same equilibrium, apply the same small force, and measure displacement. You have exposed a hidden total that an unperturbed position could not reveal.

Then move into two dimensions. Replace the stiffness number by a matrix, a table describing how displacement in each direction changes force in each direction. Draw its equal-energy ellipses beside an estimator’s uncertainty ellipses. Do the directions needing stronger mechanical protection coincide with those that are poorly estimated? A task can make that relation worth learning rather than assuming.

Finally, imagine a neural interface that learns both a movement decoder and a policy for compliance, the willingness of a device to yield under force. It could adapt its preferred movement and its resistance to disturbance separately, using feedback from the person and the environment. Evidence geometry suggests a useful design question here: what is the current balance, what supports it, and which perturbation would reveal the difference?

Follow this intoComputer Science ↗Engineering ↗