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Neuronal arithmetic

Key Points

  • A neuron can rapidly combine and transform the information it receives through its synaptic inputs before the information is converted into neuronal output. This transformation can be defined by the neuronal input–output (I–O) relationship. Changes in the I–O relationship can correspond to distinct arithmetic operations.

  • During sustained rate-coded signalling, the neuron operates as a signal integrator (over a given time-window) and the neuronal I–O relationship is defined as the dependence of sustained output firing rate on the input rate. During sparse temporally correlated signalling, the neuron acts as a coincidence detector and the I–O relationship can be defined in terms of the dependence of spike probability on the number of coincident inputs or their temporal correlation.

  • Additive and subtractive operations performed on driving inputs by distinct modulatory synaptic input correspond to shifts in the I–O relationship, without a change of shape. Multiplicative and divisive operations correspond to increases or decreases in the slope, or gain, of the I–O relationship. Both of these operations have been observed in vivo during different tasks.

  • Additive operations are essential for linearly combining signals and for controlling the number of inputs required for signalling. In the temporal domain, additive operations control the width of the temporal correlation window that the neuron can respond to and thus the temporal properties of signals that can propagate through the network.

  • Multiplicative operations, or gain changes, are important for signal amplification, normalization and preventing saturation of firing, thereby allowing efficient information transmission. Gain changes are essential for coordinate transforms and have been proposed to control the functional connectivity of networks. In the temporal domain, neural gain also controls the 'roll-off' of the temporal correlation window for synaptic integration.

  • Both morphologically simple neurons and those with extensive dendritic trees possess a number of biophysical mechanisms, including inhibition, short-term synaptic plasticity, synaptic noise and somatic and dendritic conductances that enable them to perform additive and multiplicative operations on their synaptic inputs.

  • Some biophysical mechanisms, such as voltage noise, are general in that they perform the same arithmetic operation (multiplication) on sustained rate-coded and sparse temporally coded signals. However, others seem to be tuned for either sustained rate-coding or sparse coding regimes.

  • Widespread presynaptic and dendritic mechanisms enable spatially segregated inputs to be multiplied together, although other local dendritic conductances seem to be tuned to detect spatio-temporally correlated input onto a specific dendritic branch.

  • An extensive tool kit of nonlinear mechanisms confers considerable computational power on individual neurons, enabling them to perform a range of arithmetic operations on signals encoded in a variety of different ways.


The vast computational power of the brain has traditionally been viewed as arising from the complex connectivity of neural networks, in which an individual neuron acts as a simple linear summation and thresholding device. However, recent studies show that individual neurons utilize a wealth of nonlinear mechanisms to transform synaptic input into output firing. These mechanisms can arise from synaptic plasticity, synaptic noise, and somatic and dendritic conductances. This tool kit of nonlinear mechanisms confers considerable computational power on both morphologically simple and more complex neurons, enabling them to perform a range of arithmetic operations on signals encoded in a variety of different ways.

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Figure 1: The rate-coded neuronal input–output relationship and possible arithmetic operations performed by modulatory inputs.
Figure 2: Neuronal arithmetic during sparse coding.
Figure 3: Subthreshold effects of shunting conductance and its effect on the input–output relationship of a cerebellar granule cell in the absence of noise.
Figure 4: Synaptic voltage noise and gain modulation of the rate-coded input–output relationship.
Figure 5: Inhibition-mediated gain modulation with noisy rate-coded synaptic input in cerebellar granule cells.
Figure 6: Short-term synaptic depression converts inhibition-mediated additive shifts in the rate-coded input–output relationship into multiplicative gain changes in morphologically simple and complex cells.
Figure 7: Clustered synaptic input activates local dendritic nonlinearities which could form the basis of branch specific computation.
Figure 8: Summary of biophysical mechanisms underlying rapid neuronal arithmetic.


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I would like to thank D. Attwell, G. Billings, T. Branco, M. Carandini, E. Chaigneau, T. Fernandez-Alfonso, M. Farrant, T. Margrie, A. Roth, J. Rothman, D. Ruedt, J. Sjöström, V. Steuber, K. Vervaeke and D. Ward for comments on the manuscript and J. Rothman and M. Farinella for help with preparing figures. This work was funded by the Medical Research Council, the Biotechnology and Biological Sciences Research Council, and the Wellcome Trust. R.A.S. holds a Wellcome Trust senior research fellowship in basic biomedical science.

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Supplementary information

Supplementary information S1 (box)

Determining arithmetic operations from rate-coded input-output relationships and the confounding effects of nonlinearity (PDF 385 kb)

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Neuromatic (electrophysiology acquisition and analysis software)

NeuroML (simulator independent language for defining biologically detailed neuronal and network models)

NEURON simulation environment


Neuronal gain

The slope of the neuronal input–output relationship. Changing the neuronal gain alters the sensitivity of a neuron to changes in driving inputs.

Coordinate transform

The conversion of one coordinate system to another — for example, the brain converts visual information from retina-centric to body-centric coordinates, taking account of gaze angle during visually guided reaching.

Driving force

The difference between the membrane potential and the reversal potential of an ionic conductance.

Membrane time constant

The product of the capacitance and resistance of the cell membrane, characterizing how rapidly a current changes the membrane potential. The smaller the time constant the faster the membrane potential can change.

Space constant

The distance over which a voltage imposed at a point will have decreased to 1/e (Euler's number) (37%) of its original value; a measure of how far a subtheshold potential will spread along an axon or dendrite. The thinner the dendritic branch, the shorter its space constant.

Reversal potential

The potential at which the electrochemical gradient is zero and there is no net current flow though a conductance.

Shunting inhibitory conductance

(Also known as silent inhibitory conductance.) An inhibitory conductance that reduces the depolarization produced by an excitatory current by increasing membrane conductance.

Ohm's law

The current across a conductor is linearly proportional to the potential difference (voltage).

AND–NOT operation

A logical operation in which an output only occurs when one input is on and the other is off.

Drifting gratings

A visual stimulus that consists of alternating black and white bars that drift in a particular direction.


The minimum current or conductance that produces an action potential in a neuron.

Dynamic clamp

An electrode-based electrophysiological device that uses an electrical circuit or computer simulation to inject artificial conductances into a neuron.

Campbell's theorem

A mathematical theorem that describes the relationship between the frequency of brief, randomly occurring, exponentially decaying events and the time-averaged value of their mean and the variance.

Power law

A mathematical relationship between two quantities (for example, x and y) of the form y(x)=kxn, in which k is a constant and n is the exponent. Such relationships have the special property that they are scale invariant.

Low-pass filter

A device that attenuates high frequency signals but allows low frequency signals to pass unhindered.

Cable theory

This refers to the mathematical equations that describe how electrical signals propagate in space and time in spatially extended neurons.

Two-photon glutamate uncaging

The release of glutamate from a 'caged' glutamate compound using two-photon excitation.

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Silver, R. Neuronal arithmetic. Nat Rev Neurosci 11, 474–489 (2010).

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