Neuronal arithmetic

Abstract

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 ina variety of different ways.

Figure 1. The rate-coded neuronal input–output relationship and possible arithmetic operations performed by modulatory inputs

a ∣ For rate-coded neuronal signalling, a driving input typically consists of asynchronous excitatory synaptic input from multiple presynaptic neurons firing in a sustained manner (shown in red). A neuron may also receive a modulatory input, such as inhibition (shown in green), that alters the way the neuron transforms its synaptic input into output firing rate (shown in blue). b ∣ The input–output (I–O) relationship between the total (or mean) driving input rate (d) and the response that is represented by the output firing rate (R). The arrow indicates the rheobase (minimum synaptic input that generates an action potential). c ∣ Rate-coded I–O relationships can be altered by changing the strength of the modulatory input (m), which may be mediated by a different inhibitory or excitatory input. If this shifts the I–O relationship along the x-axis to the right or left, changing the rheobase but not the shape of the curve, an additive operation has been performed on the input (shown by orange curves). This input modulation is often referred to as linear integration because the synaptic inputs are being summed. d ∣ An additive operation can also be performed on output firing. In this case a modulatory input shifts the I–O relationship up or down along the y-axis (shown by orange curves). e,f ∣ If the driving and modulatory inputs are multiplied together by the neuron, changing the strength of a modulatory input will change the slope, or gain, of the I–O relationship without changing the rheobase. A multiplicative operation can produce a scaling of the I–O relationship along either the x-axis (input modulation; e) or the y-axis (output modulation; f). Although both of these modulations change the gain of the I–O relationship, only output gain modulation scales the neuronal dynamic range (f).

Figure 2. Neuronal arithmetic during sparse coding

a ∣ Sparse coding relies on coincident synaptic input within a brief time-window (Δt) during which the inputs are integrated and potentially drive the cell to cross the action potential threshold. b ∣ Spike probability versus the number of coincident driving inputs (P_spike–N_exc) is the simplest way to quantify the neuron’s input–output (I–O) relationship under these conditions. c ∣ Background synaptic noise is important in determining the shape of the P_spike–N_exc relationship^, because it determines the width of the membrane voltage distribution. The simulation shows a simple integrate-and-fire cell with background voltage noise, driven with a synchronous synaptic input (top part). Shifting the voltage distribution without changing its shape by adding a hyperpolarizing current introduces a subtractive shift in the P_spike–N_exc relationship (bottom part; shift from black to orange traces). By contrast, decreasing the voltage noise increases the gain of the I–O relationship (shift from black to green traces). d ∣ The relationship between P_spike and driving inputs can also be defined in the temporal domain. In this case, the temporal correlation in the driving inputs is defined as the standard deviation of input spike times (σ_input)^,. Altering the level of noise changes the gain of the P_spike–σ_input relationship^,. e ∣ Changing the input resistance by altering a shunting conductance tends to shift the P_spike–σ_input relationship. This is due to scaling of the excitatory postsynaptic potential (EPSP) and voltage noise, together with altered summation of temporally dispersed inputs, as a result of changes in membrane time constant and thus EPSP shape (FIG. 3a). Parts d and e are modified, with permission, from REF. 48 © (2005) The American Physiological Society.

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

The effect of shunting inhibition on cerebellar granule cells (CGCs) was investigated in acute cerebellar slices using the dynamic clamp technique. a ∣ An excitatory postsynaptic potential (EPSP) evoked by an AMPAR (α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptor) synaptic conductance waveform (reversal potential 0 mV) that was injected during control conditions (shown in black) and during application of a 1 nanosiemens (nS) tonic shunting inhibitory conductance (reversal potential −75 mV). This inhibitory conductance scaled down the amplitude of the EPSP, consistent with Ohm’s law, and accelerated the decay of the EPSP as a result of a reduction in the membrane time constant (shown in green). b ∣ CGC firing responses to a noise-free excitatory conductance step (G_exc, shown in red). The upper traces show the voltage responses to G_exc under control conditions (shown in black) and during injection of an inhibitory conductance (shown in orange). c ∣ The relationship between mean firing rate of a CGC and step excitatory conductance in the presence and absence (control) of an inhibitory conductance. The inhibitory conductance produced a purely subtractive input modulation (x-axis shift; FIG. 1c) in the neuronal input–output relationship. d ∣ The relationship between current flow through the tonic inhibitory conductance and the driving excitatory conductance, for subthreshold voltages (+) and during spiking (Δ). The inhibitory current depends on the excitatory conductance as the subthreshold voltage depolarizes, as expected for a conductance. However, during sustained firing the shunting inhibitory conductance behaves like a constant current source. This explains why shunting inhibition has a multiplicative effect on subthreshold inputs but has an additive effect during sustained firing under low noise conditions. Figure is modified, with permission, from REF. 23 © (2003) Elsevier.

Figure 4. Synaptic voltage noise and gain modulation of the rate-coded input–output relationship

a ∣ Schematic setup for intracellular voltage recordings from a neocortical pyramidal neuron in a cat under anaesthesia. The intracellular voltage is characterized by a high level of noise in each recording period. Blocking synaptic input with local injection of tetrodotoxin (TTX), eliminated the voltage noise and increased the input resistance of the cell (see the voltage response to a current step of 0.1 nanoamperes (nA)). b ∣ Blocking synaptic input with TTX reduces the variance of the intracellular voltage distribution in pyramidal neurons and demonstrates that some cortical cells operate under conditions of high levels of synaptically induced voltage noise (active) in vivo. c ∣ The effect of noise on the relationship between firing rate and current (F–I) for a cortical interneuron, recorded in an acute slice preparation. As the noise increases (σ indicates standard deviation of the voltage noise in mV), the foot of the F–I relationship becomes more pronounced, reducing the gain of the relationship and producing a more sigmoid shape. d ∣ The dynamic clamp configuration used to inject balanced excitatory (g_E) and inhibitory (g_I) synaptic conductance trains (resulting in zero net excitatory drive) into neocortical pyramidal cells, mimicking background synaptic input (noise) in the acute slice. Current steps were used as driving inputs to assay the neuronal responsiveness. I_syn is the synaptic current and E_E and E_I are the excitatory and inhibitory reversal potentials, respectively. e ∣ The relationship between firing rate and the amplitude of the driving current steps for different levels of synaptic conductance noise (multiples of X, a balance of excitatory (rate = 7,000 Hz) and inhibitory synaptic inputs (rate = 3,000 Hz)). As the level of noise increases the gain of the F–I relationship was modified in a divisive manner. Part a is modified, with permission, from Nature Reviews Neuroscience REF. 69 © (2003) Macmillan Publishers Ltd. All rights reserved. Part b is modified, with permission, from REF. 68 © (1999) The American Physiological Society. Part c is modified, with permission, from REF. 80 © (2006) Society for Neuroscience. Parts d and e are modified, with permission, from REF. 66 © (2002) Elsevier.

Figure 5. Inhibition-mediated gain modulation with noisy rate-coded synaptic input in cerebellar granule cells

The effect of shunting inhibition on the rate coded input-output (I–O) relationship of cerebellar granule cells (CGCs) driven with random trains of noisy, synaptic-like conductances in the acute cerebellar slice preparation using the dynamic clamp technique. a ∣ A CGC was excited by four summed Poisson trains of AMPAR (α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptor) synaptic conductances (G_exc) with a reversal potential of 0 mV under control conditions and during a 1 nanosiemens (nS) constant (tonic) inhibitory conductance. The resulting relationships between CGC firing rate and the mean excitation rate are shown in b. b ∣ Application of the tonic inhibitory conductance reduced the slope of the I–O relationship (divisive operation, shown in green) and introduced an additive offset (subtractive operation, shown in orange). c ∣ The relationship between the variance of random trains of synaptic conductances with constant amplitude and the total excitation rate (shown by grey circles). The variance increases with rate as predicted by Campbell’s theorem. σ² is the conductance variance, ν is the total synaptic rate, and G(t) is the time course of the synaptic conductance waveform. d ∣ The relationship between firing rate and mean voltage for control and during 1 nS tonic inhibition. In the presence of noise, this relationship is approximated by a power law, which when combined with conductance changes can perform a crude form of multiplication (shown by solid green and grey lines). In contrast to the noise-free case (FIG. 3), a shunting inhibitory conductance has a multiplicative effect on the rate-coded input–output relationship owing to synaptic inputs exhibiting an excitation-dependent variance. Parts a, b and d are reproduced, with permission, from REF. 23 © (2003) Elesvier.

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

a ∣ A morphologically simple cerebellar granule cell (CGC) with AMPAR (α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid receptor)-mediated synaptic conductance trains showing short-term depression (+STD; shown by blue traces) measured by stimulating mossy fibres at two frequencies. Artificial synaptic conductance trains without STD were also constructed (−STD; shown by red traces). Dashed lines show the time averaged conductance (G_exc) with and without STD. b ∣ The relationship between CGC firing rate and G_exc obtained with the dynamic clamp technique. Depressing and non-depressing synaptic conductance trains produced similar CGC firing rates in control conditions and during a 0.5 nanosiemens (nS) constant (tonic) inhibitory conductance, indicating that they are integrated in a similar manner. Lines are fits to Hill-type functions (Supplementary information S1 (box)) and error bars show the standard error of the mean. c ∣ The relationship between G_exc and mean input rate across four inputs with and without STD. Without STD the relationship is linear, as expected for summing identical waveforms. STD introduces a nonlinear saturating exponential function between input rate and G_exc. d ∣ CGC input–output (I–O) relationships with and without STD and their modulation by a 0.5 nS tonic inhibitory conductance. STD in the driving input (c) amplifies the gain change observed with tonic inhibition by transforming the additive component into a multiplicative scaling. e ∣ A morphologically complex layer 5 pyramidal neuron model. The model shows random locations of excitatory synapses (shown by red circles) and inhibitory synapses (shown by green circles) distributed over the basolateral dendritic tree, and spiking is shown below. This simulation was built with neuroConstruct and run on the NEURON simulator. f ∣ A conductance train and the I–O relationship for non-depressing excitatory synaptic input (−STD) for control conditions (shown by red squares) and for various rates of synaptic inhibition (shown by yellow squares). Synaptic inhibition introduced an additive shift along the driving axis consistent with linear integration. g ∣ Same as f but including STD in the excitatory synaptic inputs (+STD alone, shown by dark blue circles; +STD and inhibition, shown by grey circles). STD changed the effects of inhibition from largely additive (f) to a largely multiplicative operation (g). Scaling down of the I–O relationship indicates an output gain modulation (FIG. 1f). Synaptic conductance waveforms and STD used in the simulation were matched to experimental data for layer 5 synaptic connections. Part a is modified, with permission, from REF. 90 © (2003) Society for Neuroscience. Parts b–g are modified, with permission, from Nature REF. 110 © (2009) Macmillan Publishers Ltd. All rights reserved.

Figure 7. Clustered synaptic input activates local dendritic nonlinearities which could form the basis of branch specific computation

a ∣ Cartoons showing layer 5 cortical pyramidal cells with the location of activated synaptic inputs and the recording electrode. ‘Within branch’ refers to two inputs (A and B) that synapse onto the same dendritic branch (left), and ‘between branches’ corresponds to two inputs that synapse onto different branches (right). Traces show the somatic responses to a paired-pulse synaptic stimulation protocol. Black traces show the two synaptic inputs stimulated individually, blue traces show the predicted response for simultaneous activation (assuming linear summation) and red traces show the measured response for intermediate strength stimulation. The within-branch response was supralinear, whereas the response between branches was linear. This effect was blocked by a selective NMDAR (N-methyl-d-aspartate receptor) antagonist. b ∣ Responses for the two input scenarios; the between-branches configuration (shown in green; the dashed line indicates linearity) and the within-branch configuration (shown in red). The nonlinear effects of NMDARs are seen when excitatory postsynaptic potentials (EPSPs) are a few millivolts in amplitude. However, these experiments were carried out in the presence of a GABA_A (α-aminobutyric acid type A) receptor antagonist, which may favour the occurrence of NMDA spikes. c ∣ The difference between the arithmetic sum of individual inputs (shown by blue squares) and paired-pulse protocols with different intervals between the pulses. The supralinear response of the within-branch configuration occurs when the two synaptic inputs occur within approximately 40 ms indicates that synaptic NMDAR activation has a ‘memory’ of prior input onto the branch over this timescale. d ∣ A cartoon of a pyramidal cell with n_i clustered synaptic inputs on each of the dendritic branches. A nonlinear dendritic mechanism, such as NMDAR spikes, introduces a nonlinear sigmoidal input–output function (S(n_i)) to each of the subcompartments. The weights of the subcompartments (α_i) are then combined with the somatic spike threshold nonlinearity (g) to produce output (y). It has been proposed that the presence of multiple dendritic subcompartments each with a nonlinear thresholding element could enable an individual pyramidal cell to act like a two-layer network of neurons, thereby enhancing its computational power. Parts a–c modified, with permission, from Nature Neuroscience REF. 156 © (2004) Macmillan Publishers Ltd. All rights reserved. Part d modified, with permission, from REF. 157 © (2003) Elsevier.

Figure 8. Summary of biophysical mechanisms underlying rapid neuronal arithmetic

Mechanisms that may underlie rapid addition (shown in red) and subtraction (shown in orange) or multiplication (shown in blue) and division (shown in green) of synaptic input in cells with simple and complex morphologies. The upper two rows correspond to sustained rate-coded signalling while the bottom two correspond to sparse temporally coded signalling. As indicated by the graphs on the left, rows one and three correspond to additive operations, whereas rows two and four correspond to multiplicative operations. Overlap of boxes between additive and multiplicative regions indicates mixed operations. DAP, depolarizing after potential; NMDA, N-methyl-daspartate; STD, short term depression.