Every network has a price list, and learning rewrites it

What the work claims

This is a methods paper with three demonstrations. The core move is to define computation as a controlled transition: the network sits in an initial activity state, an external input drives it to a target state, and the cost is the control energy that input must supply, computed from linearized dynamics about a given state. The affordance landscape is the full distribution of those costs across all target patterns; its eigenmodes sort computations from cheap to expensive, and the claim is that this ordering is biologically meaningful. In a 25-neuron ring-attractor model of the fly compass, the single cheapest mode turns out to be the spatial derivative of the activity bump with respect to heading: exactly the update-the-heading computation the circuit is known to perform, with the optimal input profile, more drive to neurons ahead of the bump, less behind, matching the anatomy of the fly's shift-neuron and noduli circuitry.¹

The second claim is macroscale. Structural connectomes from 100 Human Connectome Project subjects show that each cortical network's landscape heterogeneity, summarized by the interquartile range of its eigenvalues after regressing out network size, tracks the network's position on the sensorimotor-association axis: sensory networks have heterogeneous landscapes, a few cheap modes among many expensive ones, consistent with specialized processing, while association networks such as the default mode network have homogeneous landscapes, uniformly affordable access to many states, consistent with domain-general processing. The Spearman correlation between heterogeneity and axis rank is -0.64 with p = 0.006. The third claim is about learning: 64-unit recurrent networks trained on each of five perceptual decision-making tasks show monotonically increasing landscape heterogeneity over training across all tasks and ten seeds per task. Training does not just fit a function; it sculpts the price list, making the task's modes cheap and everything else expensive.

How it works

The framework is classical optimal control applied locally. Write the network's dynamics, linearize about an operating state, and the minimum-energy input that steers the system between two states is governed by the controllability structure of the linearized matrix; each eigenmode of the associated operator carries an energy cost, and the spectrum of those costs is the affordance landscape. Cheap modes are activity patterns the wiring practically invites; expensive modes are patterns the structure resists. Because everything is linear-algebraic, the landscape is computable for any network where you can specify connectivity and a linearization point: an insect circuit model, a human structural connectome, or a trained artificial network. The cost is input energy, not spikes or ATP, a distinction that matters and that the paper is honest about: this is a control-theoretic proxy, not a metabolic measurement.

Where a skeptic should push

The load-bearing assumption is linearization. The landscape is computed about a fixed operating state, so it characterizes the network's affordances only locally; brains and organoids operate far from fixed points, moving through regimes where the Jacobian changes constantly. The fly result is reassuring because the cheapest mode matches known function, but it is also the case where the attractor story is strongest and the linearization most defensible. At the human scale, the analysis rests on diffusion-MRI tractography, a structural estimate whose edge weights are model-dependent and whose relationship to effective, causal coupling is indirect; the reported correlation of -0.64 across networks, while significant, pools across very different systems and its mechanistic reading is suggestive rather than demonstrated.

Third, the training result risks the opposite overreach: heterogeneity rose monotonically in 64-unit rate networks on five tasks, but those networks are trained with backpropagation to criterion, and the result says nothing yet about whether biological plasticity, operating under Dale's law and local learning rules, sculpts landscapes the same way. Finally, note what cheapness does not tell you: a mode can be cheap because the network computes it well or because the linearization makes it trivially reachable. Affordance is capacity, not usefulness; the framework prices computations but cannot rank their value.

A spec sheet for what living tissue affords

For organoid intelligence this is close to a missing metrology. The field's core anxiety is that every dish is a different computer: same protocol, different wiring, different competence. Task benchmarks score one computation; the affordance landscape is a task-agnostic spec sheet that prices all of them at once, computed from measurable quantities, effective connectivity and the local dynamics around an operating state, both accessible in organoids through perturbation and recording. Two operational uses follow. Quality control becomes comparative rather than pass-fail: instead of asking whether a culture clears a threshold on one task, ask whether its landscape matches a reference, and whether its cheap modes are the ones your application needs. And substrate selection becomes rational: given a fixed application requiring certain activity transitions, pick the dish whose landscape makes those transitions cheap, the way the fly's wiring makes heading updates cheap.¹

The training result carries the sharpest implication, and it cuts both ways. If closed-loop training of living tissue sculpts its landscape the way backprop sculpts the RNNs here, then training for task A does not leave the rest of the substrate neutral: other computations get pricier as A gets cheap. That is a concrete, testable warning for the emerging practice of repeatedly training the same organoid across successive tasks, competence in the new task may be purchased with the erosion of the old one. The optimistic mirror image is that a measured landscape gives you a training audit: recompute it after a training epoch and you can see, in numbers, what the tissue just became good at and what it gave up. Neither capability currently exists in OI practice, where training success is judged on output error alone.

The threat to watch is over-interpretation: because the metric is cheap to compute and produces pretty spectra, it will be tempting to publish landscapes as if they were capabilities. A cheap mode is an invitation, not a computation; until someone demonstrates that externally supplied control inputs can actually drive an organoid along its predicted cheap modes, the landscape is a hypothesis about the substrate, not a property of it.

The bottom line

Established: a control-theoretic cost framework can be computed across scales, and in three settings it tracks biological intuition sharply, the fly compass's cheapest mode is heading update with anatomy-consistent inputs, human sensory networks have more heterogeneous affordance landscapes than association networks, and task training progressively increases landscape heterogeneity in artificial recurrent networks. Open: whether linearized affordances predict what nonlinear, spiking, plastic tissue can actually be driven to do. For biological computing, the near-term value is as a comparative substrate metric and a training audit: price what the tissue affords before and after training instead of trusting task scores alone. What would confirm it: perturbation experiments in organoids showing that externally driven transitions along predicted cheap modes require less energy than expensive ones. What would break it: evidence that effective, dynamic coupling, not structural connectivity, dominates real transition costs, which would make landscapes computed from connectomes or effective matrices miss the operative variable.