Living neuronal networks transform inputs through recurrent cellular and population dynamics, yet it is unknown which network architecture supports which computation. Neurons-on-a-chip turn this question into a design problem because microchannels guide axonal growth and set the network architecture. We introduce IC$^3$, an Integrated Characterisation of Communication-Driven Computation, which characterizes network state through neuronal dynamics, functional communication, and structural support. We implemented nine architectures \textit{in silico} as conductance-based spiking networks and tested each on frequency decoding, temporal-order discrimination, and fading memory. Predominantly feedforward circuits decoded best. Sequential Chain and Microchannel Diode had the lowest IC$^3$ and recruited a third of reachable neurons, yet achieved the two highest scores on both classification tasks. Across architectures, higher IC$^3$ went with lower classification scores. We term this new direction \emph{neurotopomorphic computing}, in which the physical organisation of neuronal connectivity is engineered as part of the computing substrate.

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