Every network has a price list of computations it finds easy

What the work claims

This is a theory-and-application paper: a framework, three demonstrations at different scales, and an explicit account of its own limits. The claim is that network structure, together with where inputs can enter, defines a computational affordance landscape: a complete price list over all possible activity states, in which cost is the minimum input energy required to drive the network to a given state. Because the price list derives from the controllability Gramian, a classical object from linear control theory, structure and computational capability are linked mechanically rather than by correlation.¹

The three demonstrations are specific. In a model of the fly head-direction circuit, the cheapest computation derived from the landscape is shifting the activity bump around the ring, which is precisely the circuit's known biological function, and the optimal input pattern (increase drive ahead of the bump, decrease it behind) matches the known anatomy of shift inputs. In structural connectomes of 100 human subjects, the unevenness of the price list varies systematically along the sensorimotor-association axis, with sensory networks more heterogeneous than association networks. And in recurrent neural networks trained on decision-making tasks, training monotonically increases that unevenness: learning sculpts the landscape of affordable computations.

How it works

Define a computation as a goal-directed transition of network activity from an initial state to a target state, and price it by the energy of the input that implements it. Linearizing the dynamics about an operating point gives a closed form: the minimum cost is a quadratic function defined by the inverse of the controllability Gramian, and the set of states reachable within unit cost forms an ellipsoid whose axes are the Gramian's eigenvectors and whose radii are the square roots of its eigenvalues. The inverse Gramian is the affordance landscape: eigenmodes with small inverse eigenvalues are cheap to excite, those with large ones are expensive. For symmetric connectivity the mode costs take a closed form in the eigenvalues of the connectivity matrix, with slow modes cheap and fast modes dear, and the authors use a time horizon of one and independent input to every node throughout.

For the fly circuit they linearize a ring-attractor model of 25 neurons, with connectivity shaped by local excitation and global inhibition, about a stable bump of activity representing heading. The landscape's single cheapest mode corresponds to the derivative of the bump with respect to heading angle, meaning that rotating the compass is the computation the structure most readily supports; the second-cheapest resembles sharpening the bump and the most expensive is changing its overall amplitude, which makes functional sense for a circuit whose job is to hold a steady internal heading signal. The required optimal input matches what shift neurons and the noduli are known to provide when self-motion updates the fly's heading.

For the human brain they build structural connectivity matrices from diffusion MRI of 100 unrelated Human Connectome Project subjects, mean age 29.1 plus or minus 3.7 years, using constrained spherical deconvolution, anatomically constrained tractography, and SIFT2 weighting in MRtrix3, parcellated into 400 regions and assigned to the Yeo networks. They summarize each network's landscape by the interquartile range of its eigenvalue distribution, after regressing out network size, and find landscape heterogeneity correlates with position on the sensorimotor-association axis (Spearman rank correlation of minus 0.64, p = 0.006): sensory networks support a narrow set of computations cheaply, association networks support many computations nearly uniformly. A bilateral-symmetry analysis adds that modes spread evenly across hemispheres are consistently cheaper than lateralized ones, across all seven networks (median per-subject Spearman correlations from minus 0.23 to minus 0.46; Wilcoxon signed-rank p below 10 to the minus 16, 100 subjects per network). In the artificial networks, 64-unit recurrent networks trained on five decision-making task variants show the interquartile range of landscape eigenvalues increasing monotonically over training, averaged over ten seeds.

Where a skeptic should push

The most load-bearing assumption is linearization. All price lists are computed about a fixed point, so the landscape describes the neighborhood of an operating point, not the global dynamics. The authors are upfront: the framework extends to arbitrary operating points only if you can estimate the Jacobian there, analytically or from data, which is precisely where a biological preparation becomes hard. The fly ring is the favorable case, since rotational symmetry makes one linearization representative of the whole circuit; most real circuits do not have that symmetry.

Second, minimum input energy is a proxy, not a metabolic measurement. The cost functional integrates squared input norm, which is a control-theoretic convenience; the brain pays in ATP, transmitter release, and conduction, and the mapping between the two is assumed rather than derived. Third, the human result rests on diffusion tractography, whose edge weights are known to carry reconstruction biases that SIFT2 mitigates but does not eliminate; a landscape computed from a weighted connectome is a landscape computed from an estimate. Fourth, the human statistics, while significant, are group-level summaries of a young healthy cohort, and the sensorimotor-association axis correlation explains a fraction of the variance. None of these caveats overturn the fly result, which is the strongest because function and anatomy are independently known, but they calibrate how far the framework should be stretched before it is validated against measured energetics.

Affordance landscapes for training organoids

The non-obvious implication for organoid intelligence is that this framework turns a vague worry into a measurable quantity. The field's standard anxiety is that organoids lack developmental patterning, so their connectivity is not cortex, so why expect cortical computation? Here is a way to make that precise without a connectome: the affordance landscape depends only on an effective connectivity estimate and an input matrix, and effective connectivity in a dish is exactly what multi-electrode stimulation and recording can estimate today, via the Jacobian-from-data route the authors name as future work. A lab can perturb a culture through its electrodes, fit the local linear dynamics, compute the Gramian, and ask empirical questions: how heterogeneous is this tissue's price list, how does it compare to the homogeneous-versus-heterogeneous signature of association versus sensory cortex, and how does it change with maturation, patterning protocols, or training.

The training result carries a sharper message. Learning progressively increases landscape heterogeneity: training makes a network cheap at its task and expensive at everything else. Applied to organoids, this reframes the reservoir-versus-trained-substrate debate. If goal-directed adaptation in living tissue follows the same law, then training an organoid for one task will steepen its landscape, specializing it and narrowing what it can subsequently do cheaply. Untrained tissue with a broad, flat landscape is a generalist substrate; trained tissue is a specialist. Neither is categorically better, but they are different products, and a field that sells organoids as reconfigurable general-purpose computers needs to reckon with the possibility that every training run quietly taxes that generality. Measuring the landscape before and after closed-loop training would turn this from philosophy into data.

The threats deserve equal weight. One is a measurement trap: the Gramian's eigendecomposition is cubic in node count, so a naive application to a dense full-culture effective-connectivity matrix is a computational wall, and low-rank approximations will become an arena where results are sensitive to choices a reader cannot easily audit. Another is more delicate and cuts the other way: this is also a steering manual. A framework that prices the minimum inputs required to drive a network into any desired activity pattern is, transparently, a framework for optimal control of neural tissue, and it applies as readily to a living preparation as to a model. Dual-use review for biological computing should treat minimum-energy control analysis of cultures as a capability, not merely a metric. Finally, the familiar hype correction: the brain's energy-efficiency narrative implicitly assumes its structure is well matched to its computations, and this paper is among the first to make that matching quantifiable, which also means it can fail, and in a dish with arbitrary wiring it very likely does, at first.

The bottom line

Established: a controllability-Gramian framework prices all activity transitions of a linearized network, and in the fly compass circuit the cheapest computed transition coincides with known function and known input anatomy; in human connectomes landscape heterogeneity tracks the sensory-association distinction, and in trained recurrent networks it rises monotonically with training. Hypothesis, not established: that the linearized price list predicts metabolic cost or causal necessity in real tissue, and that organoid cultures exhibit landscapes distinguishable from random controls. What would confirm it: direct estimation of affordance landscapes from stimulation-recorded effective connectivity in vitro and in vivo, validated against measured energy use. What would weaken it: evidence that nonlinear dynamics away from the fixed point dominate real computation, making the local landscape unrepresentative. For organoid intelligence the residue is a metric with a mechanism behind it: perturb, estimate the local dynamics, compute the price list, and let the landscape, not the narrative, tell you what the tissue can do.