How does the size of a neural circuit influence its learning performance? Larger brains tend to be found in species with higher cognitive function and learning ability. Intuitively, we expect the learning capacity of a neural circuit to grow with the number of neurons and synapses. We show how adding apparently redundant neurons and connections to a network can make a task more learnable. Consequently, large neural circuits can either devote connectivity to generating complex behaviors or exploit this connectivity to achieve faster and more precise learning of simpler behaviors. However, we show that in a biologically relevant setting where synapses introduce an unavoidable amount of noise, there is an optimal size of network for a given task. Above the optimal network size, the addition of neurons and synaptic connections starts to impede learning performance. This suggests that the size of brain circuits may be constrained by the need to learn efficiently with unreliable synapses and provides a hypothesis for why some neurological learning deficits are associated with hyperconnectivity. Our analysis is independent of specific learning rules and uncovers fundamental relationships between learning rate, task performance, network size, and intrinsic noise in neural circuits.

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The cerebellum has a distinctive circuit architecture comprising the majority of neurons in the brain. Marr-Albus theory and more recent extensions demonstrate the utility of this architecture for particular types of learning tasks related to the separation of input patterns. However, it is unclear how the circuit architecture facilitates known functional roles of the cerebellum. In particular, the cerebellum is critically involved in refining motor plans even during the ongoing execution of the associated movement. Why would a cerebellar-like circuit architecture be effective at this type of 'online' learning problem? We build a mathematical theory, reinforced with computer simulations, that captures some of the particular difficulties associated with online learning tasks. For instance, synaptic plasticity responsible for learning during a movement only has access to a narrow time window of recent movement errors, whereas it ideally depends upon the entire trajectory of errors, from the movement's start to its finish. The theory then demonstrates how the distinctive input expansion in the cerebellum, where mossy fibre signals are recoded in a much larger number of granule cells, mitigates the impact of such difficulties. As such, the energy cost of this large, seemingly redundantly connected circuit might be an inevitable cost of precise, fast, motor learning.