1. Department of Neuroscience, Physiology and Pharmacology, University College London, Gower Street, London WC1E 6BT, UK
2. These authors contributed equally to this work.

Correspondence to: R. Angus Silver¹ Correspondence and requests for materials should be addressed to R.A.S. (Email: a.silver@ucl.ac.uk).

To act as computational devices, neurons must perform mathematical operations as they transform synaptic and modulatory input into output firing rate¹. Experiments and theory indicate that neuronal firing typically represents the sum of synaptic inputs^{1, 2, 3}, an additive operation, but multiplication of inputs is essential for many computations¹. Multiplication by a constant produces a change in the slope, or gain, of the input–output relationship, amplifying or scaling down the sensitivity of the neuron to changes in its input. Such gain modulation occurs in vivo, during contrast invariance of orientation tuning⁴, attentional scaling⁵, translation-invariant object recognition⁶, auditory processing⁷ and coordinate transformations^{8, 9}. Moreover, theoretical studies highlight the necessity of gain modulation in several of these tasks^{9, 10, 11}. Although potential cellular mechanisms for gain modulation have been identified, they often rely on membrane noise and require restrictive conditions to work^{3, 12, 13, 14, 15, 16, 17, 18}. Because nonlinear components are used to scale signals in electronics, we examined whether synaptic nonlinearities are involved in neuronal gain modulation. We used synaptic stimulation and the dynamic-clamp technique to investigate gain modulation in granule cells in acute slices of rat cerebellum. Here we show that when excitation is mediated by synapses with short-term depression (STD), neuronal gain is controlled by an inhibitory conductance in a noise-independent manner, allowing driving and modulatory inputs to be multiplied together. The nonlinearity introduced by STD transforms inhibition-mediated additive shifts in the input–output relationship into multiplicative gain changes. When granule cells were driven with bursts of high-frequency mossy fibre input, as observed in vivo ^{19, 20}, larger inhibition-mediated gain changes were observed, as expected with greater STD. Simulations of synaptic integration in more complex neocortical neurons suggest that STD-based gain modulation can also operate in neurons with large dendritic trees. Our results establish that neurons receiving depressing excitatory inputs can act as powerful multiplicative devices even when integration of postsynaptic conductances is linear.

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