Every network has a price list for computation

What the work claims

This is a theory-and-analysis paper: a mathematical framework plus applications to one model circuit, human connectomes, and artificial recurrent networks. No new experimental data. The central claim is that computation can be defined as a goal-directed transition of a network's activity from one state to another, that the cost of such a transition has a closed form in terms of the network's wiring through the controllability Gramian, and that the resulting cost distribution, which the authors call the computational affordance landscape, predicts what real circuits are for.¹

Three findings carry the claim. In a model of the fly head-direction circuit, the single cheapest mode of the landscape is precisely the derivative of the activity bump with respect to heading angle, meaning the network's least expensive computation is updating the heading representation, its established biological function, and the optimal input pattern that performs it matches the known anatomy of the circuit's shift neurons. In 100 human connectomes, the heterogeneity of the landscape, summarized as the interquartile range of its eigenvalues, correlates with position on the sensorimotor-association axis (Spearman r = -0.64, p = 0.006): sensory networks have heterogeneous, specialist landscapes while association networks have homogeneous, generalist ones. And in artificial recurrent networks trained on decision-making tasks, landscape heterogeneity increases monotonically over training: learning sculpts the price list toward a few cheap computations.

How it works

The setup is classical network control theory. A network of N nodes has activity vector x obeying dynamics with Jacobian W about an operating point and input structure B. To drive the network from an initial state to a target state over a time horizon, the inputs must compensate for the gap between the target and where the network would drift on its own; the minimum input energy has the closed form D-transpose Wc-inverse D, where D is that gap and Wc is the controllability Gramian, the integral of the dynamics' propagator weighted by where inputs land. The inverse Gramian therefore assigns a cost to every direction in activity space: the computational affordance landscape. States reachable cheaply form an ellipsoid whose axes are the Gramian's eigenvectors, elongated along cheap modes and pinched along expensive ones.¹

Two structural facts drop out. The cost of exciting a mode falls with its dynamical timescale: slow modes are cheap, fast modes expensive, with the closed form 2 lambda over e to the 2 lambda T minus 1 for symmetric networks. And the input structure B matters independently of wiring: the same target pattern can be cheap under one placement of inputs and expensive under another, because B reshapes the landscape itself. Throughout, the authors set the horizon to 1 time unit and assume every node receives input independently.

The applications are chosen to stress-test the framework against known biology. The fly circuit is a ring attractor of 25 neurons with cosine-tuned local excitation and broad inhibition, which sustains a localized bump tracking heading. Linearizing about a stable bump, the landscape has one mode far cheaper than all others, and it is the spatial derivative of the bump, the pattern that rotates it. The optimal input increases drive to neurons ahead of the bump and decreases drive behind it, matching how the fly's noduli neurons unbalance the two shift-neuron populations during turns. In humans, structural connectivity from 100 Human Connectome Project subjects is parcellated into the canonical Yeo networks, and the interquartile range of each landscape's eigenvalues, with network size regressed out, is compared against the sensorimotor-association axis. In artificial networks, 64-unit recurrent networks trained on five decision-making task variants are tracked step by step, and the landscape's spread grows monotonically in every task, over 10 seeds each.

Where a skeptic should push

The most load-bearing assumption is linearity. Everything is computed from a linearization about a fixed point, in a rate-based model with no spikes, no plasticity, and no noise except a small weight perturbation in the fly circuit. The authors are straightforward about this: the landscape describes the neighborhood of an operating point, and extension to nonlinear regimes requires estimating the Jacobian, which they note can be done from data. But a cost measure that is exact for linear dynamics and local for everything else can mislead where it matters most, namely far from fixed points, during the large, fast transitions that stimulation experiments actually induce. The fly result partly defends the framework because the rotational symmetry of the ring makes one linearization representative of the whole circuit; most networks enjoy no such symmetry.

Second, the cost functional is input energy, a stand-in for metabolic or hardware effort that has no direct calibration against anything biological. That the fly's cheapest mode matches its function is genuinely striking, but one circuit is one circuit, and the risk of selection bias is real: the authors applied the framework where structure, dynamics, and function were already jointly understood. Third, the human result is a correlation across 17 networks in 100 subjects; impressive, but the landscapes there are computed from diffusion-derived structural connectivity with B set to identity, so the numbers inherit every simplification of that pipeline. None of this voids the framework, but it calibrates the weight the findings can bear: a powerful lens, not a verdict.

Engineering affordance into organoid tissue

The non-obvious implication is that this framework hands organoid intelligence its missing assay. The field's central question is not whether a culture is active but what it can compute, and this paper turns that into a property of measurable structure. The Jacobian of an organoid's dynamics can be estimated from multi-electrode recordings, the authors explicitly note, which means the affordance landscape could in principle be computed for a dish of living neural tissue: a quantitative map of which activity transitions the substrate supports at low input cost and which it effectively forbids. That converts substrate characterization from vibes to numbers, and it sets an honest burden of proof. A claim that an organoid implements some computation should be preceded by the demonstration that the target transition is affordable on that tissue's landscape, not merely that the tissue produces rich-looking dynamics.

Just as important is the B matrix result. Where inputs land changes the landscape independently of the wiring, so electrode placement and stimulation targeting in an organoid system are not merely measurement choices: they are ways of rewriting the substrate's price list. An activity pattern that is prohibitively expensive under one electrode configuration may be cheap under another, which reframes hardware design as computational co-design. For a field that treats the interface as plumbing, this is a substantive promotion: the interface is part of the computation's cost structure.

The genuine threat is what the training result implies about generality. If learning a task sculpts the landscape toward heterogeneity, carving a few cheap modes and taxing everything else, then a substrate trained hard on one task is being deliberately specialized, and its price list tilts against other computations. A general-purpose biological computer may then be a structural contradiction: every hour of training that makes an organoid better at your benchmark makes it, by this measure, less general. Combined with the fly result, the lesson is sobering. Evolution aligns a circuit's cheapest computation with its function; random self-assembly does no such aligning, and training undoes generality. An organoid whose landscape has not been characterized and deliberately shaped is a substrate with an unknown price list, and the framework says that is not a detail but the main fact. The counterweight is that this is a linear, energy-based account: biological tissue may exploit regimes the Gramian cannot see, so a high computed cost should be read as a hypothesis to test with closed-loop stimulation, not a ceiling.

The bottom line

Established: a control-theoretic cost for activity transitions, computable from network structure, defines a computational affordance landscape; in the fly head-direction model the cheapest mode is the heading-update computation with anatomy-consistent optimal inputs; in 100 human connectomes landscape heterogeneity tracks the sensorimotor-association axis (r = -0.64, p = 0.006); and training recurrent networks monotonically increases landscape heterogeneity. Not established: that the linear, energy-based cost predicts behavior far from fixed points, that it calibrates against metabolic reality, or that any of this survives the move to spiking, plastic tissue. What would confirm it: Jacobian-from-data estimates of organoid or cortical landscapes whose cheap modes predict which computations the tissue can be driven to perform, validated in closed loop. What would weaken it: nonlinear stimulation experiments where expensive directions turn out to be easy. For organoid intelligence the actionable residue is twofold: adopt the landscape as a substrate assay, and treat electrode and stimulation design as computational co-design, because where inputs land rewrites what the tissue can afford.