Noisy Synaptic Conductance: Bug or a Feature?
Authors: Dmitri A. Rusakov, Leonid P. Savtchenko, Peter E. Latham
Journal: Trends in Neurosciences (2020)

Synaptic transmission is noisy, partially because neurotransmitter release is unreliable but also because synaptic receptor current varies stochastically.When synaptic input is below the cell spike threshold, adding noise may push it above the threshold, thus improving information transmission.Synaptic noise can widen the dynamic range of transmission.Synaptic noise allows synapses to communicate their degree of uncertainty.Noise is a ubiquitous feature of brain activity and is likely to be a critical element in any theory of how the brain works. More often than not, action potentials fail to trigger neurotransmitter release. And even when neurotransmitter is released, the resulting change in synaptic conductance is highly variable. Given the energetic cost of generating and propagating action potentials, and the importance of information transmission across synapses, this seems both wasteful and inefficient. However, synaptic noise arising from variable transmission can improve, in certain restricted conditions, information transmission. Under broader conditions, it can improve information transmission per release, a quantity that is relevant given the energetic constraints on computing in the brain. Here we discuss the role, both positive and negative, synaptic noise plays in information transmission and computation in the brain. More often than not, action potentials fail to trigger neurotransmitter release. And even when neurotransmitter is released, the resulting change in synaptic conductance is highly variable. Given the energetic cost of generating and propagating action potentials, and the importance of information transmission across synapses, this seems both wasteful and inefficient. However, synaptic noise arising from variable transmission can improve, in certain restricted conditions, information transmission. Under broader conditions, it can improve information transmission per release, a quantity that is relevant given the energetic constraints on computing in the brain. Here we discuss the role, both positive and negative, synaptic noise plays in information transmission and computation in the brain. Neurons communicate primarily through chemical synapses, and that communication is critical for proper brain function. However, chemical synaptic transmission appears unreliable: for most synapses, when an action potential arrives at an axon terminal, about half the time, no neurotransmitter is released and so no communication happens. Even strong and reliable synapses, such as the neuromuscular junction or the calyx of Held, operate by using multiple unreliable release sites [1.Tarr T.B. et al.Are unreliable release mechanisms conserved from NMJ to CNS?.Trends Neurosci. 2013; 36: 14-22Abstract Full Text Full Text PDF PubMed Scopus (41) Google Scholar]. Furthermore, when neurotransmitter is released at an individual synaptic release site, the size of the local postsynaptic membrane conductance change is also variable. Given the importance of synapses, the energetic cost of generating action potentials, and the evolutionary timescales over which the brain has been optimized, the high level of synaptic noise seems surprising. Here we ask: how surprised should we be? More to the point: is synaptic noise a bug, or is it a feature? Whilst the role of unreliable release on neural network performance has been explored in multiple studies (see [2.Stein R.B. et al.Neuronal variability: noise or part of the signal?.Nat. Rev. Neurosci. 2005; 6: 389-397Crossref PubMed Scopus (440) Google Scholar, 3.McDonnell M.D. Ward L.M. The benefits of noise in neural systems: bridging theory and experiment.Nat. Rev. Neurosci. 2011; 12: 415-426Crossref PubMed Scopus (0) Google Scholar, 4.Faisal A.A. et al.Noise in the nervous system.Nat. Rev. Neurosci. 2008; 9: 292-303Crossref PubMed Scopus (1693) Google Scholar] for informative reviews), here we expand on these views, in particular giving consideration to noisy synaptic conductance (see Glossary). When communicating information, noise would seem to be universally bad. Static on the radio, background noise during a conversation, grainy videos – they all make it harder to extract interesting information. However, for nonlinear systems – especially when a threshold is involved, as in spike generation – noise can be helpful [5.Edwards P.J. Murray A.F. Analog synaptic noise – implications and learning improvements.Int. J. Neural Syst. 1993; 4: 427-433Crossref PubMed Scopus (3) Google Scholar, 6.Murray A.F. Edwards P.J. Synaptic weight noise during multilayer perceptron training: fault tolerance and training improvements.IEEE Trans. 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This phenomenon is, very broadly, known as stochastic resonance [3.McDonnell M.D. Ward L.M. The benefits of noise in neural systems: bridging theory and experiment.Nat. Rev. Neurosci. 2011; 12: 415-426Crossref PubMed Scopus (0) Google Scholar,10.Stocks N.G. Suprathreshold stochastic resonance in multilevel threshold systems.Phys. Rev. Lett. 2000; 84: 2310-2313Crossref PubMed Scopus (378) Google Scholar,11.Mino H. Durand D.M. Enhancement of information transmission of sub-threshold signals applied to distal positions of dendritic trees in hippocampal CA1 neuron models with stochastic resonance.Biol. Cybern. 2010; 103: 227-236Crossref PubMed Scopus (20) Google Scholar], and it is observable in electrophysiological experiments: add a small amount of noise to subthreshold synaptic currents, and the probability of a spike can go way up [12.Stacey W.C. Durand D.M. Synaptic noise improves detection of subthreshold signals in hippocampal CA1 neurons.J. Neurophysiol. 2001; 86: 1104-1112Crossref PubMed Scopus (118) Google Scholar]. A simple multisynaptic neuron model (Figure 1A ) illustrates this point. With no noise in the synaptic conductance, there are very few spikes (Figure 1B), but adding a small amount of variability in the postsynaptic conductance can produce fairly reliable spikes (Figure 1C). Note, however, that decreasing threshold for the generation of an action potential can produce even more reliable spikes (Figure 1D), which has a higher somatic sodium conductance, making the cell more excitable compared with Figure 1B. Although in a wider context we cannot equate more spiking with more information, we will, in the interest of simplicity, assume that is the case here. Abstractly, the ability to communicate information depends on three things: the range of possible input signals, the noise around those signals, and the transformation from input to output (Box 1 provides a commonly accepted definition of information). For neurons, the transformation from input to output yields either a spike or no spike. In that case, communication is optimal when the integrated input to a neuron is either well above or well below threshold relative to the noise. In such a regime, a spike provides unambiguous information about the input: it tells downstream neurons whether the input was high or low. But typically, the signal and noise cannot be adjusted independently; in particular, the synaptic drive to a neuron is not bimodal – it is graded. So, the simple, optimal, regime is not realized. Consequently, the question of whether synaptic noise is a bug or a feature depends on how the signal and noise interact. For instance, if increasing noise made synaptic drive more bimodal, it could, at least in principle, increase information transmission. And we have already seen that adding noise can increase the reliability of output spikes (Figure 1C). But we have also seen that increasing cell excitability, and thus lowering the threshold for a spike, can do the same thing (Figure 1D). Which one is better, and when?Box 1Defining (Mutual) InformationSpike trains convey information about presynaptic input. Intuitively, this means that if you were to observe spikes from a neuron, you would know more about its synaptic drive than if you did not observe spikes. To quantify this intuition, we need a way to measure knowledge – or, as is much more common, uncertainty. There are many ways to do this, but an especially convenient one is entropy, which is a relatively direct measure of uncertainty; the higher the entropy of a distribution over some variable (i.e., the broader the distribution), the more the uncertainty about that variable (according to Equations (I), (II) as follows). With this choice, the difference in entropy is the Shannon mutual information [59.Shannon C.E. Weaver W. The Mathematical Theory of Communication. The University of Illinois Press, 1949Google Scholar], which indicates how much information about the input signal we can glean from the output signal (Figure I).The entropy of a continuous distribution P(x), often denoted H(x), isHx=−∫dxPxlog2Px.(I) If P(x) is discrete there is an analogous definition; the only difference is that the integral becomes a sum,Hx=−∑iPxilog2Pxi.(II) To compute mutual information, we need to compare the entropy before and after some observation (which for us is a spike train). Let's use s for the observation, and P(x | s) for the distribution of x (which for us is the synaptic drive) after observing a spike train. In that case, the conditional entropy, denoted H(x | s) – the average entropy after observing a spike train – isHxs=−∫dxdsPsPxslog2Pxs,(III) with integrals replaced by sums for discrete distributions. The mutual information, I(x,s), between x and s is the difference between the two entropies,Ixs=Hx−Hxs.(IV) This definition is sufficiently flexible that x could be continuous and s discrete, or vice versa. As is not hard to show, information is symmetric; it can also be writtenIxs=Hs−Hsx.(V) Symmetry is important because it tells us that the higher the entropy of a spike train (which typically translates to higher firing rate), the larger the potential for high information. Importantly, though, increasing the firing rate is not guaranteed to increase information, as that may also increase the second term. Spike trains convey information about presynaptic input. Intuitively, this means that if you were to observe spikes from a neuron, you would know more about its synaptic drive than if you did not observe spikes. To quantify this intuition, we need a way to measure knowledge – or, as is much more common, uncertainty. There are many ways to do this, but an especially convenient one is entropy, which is a relatively direct measure of uncertainty; the higher the entropy of a distribution over some variable (i.e., the broader the distribution), the more the uncertainty about that variable (according to Equations (I), (II) as follows). With this choice, the difference in entropy is the Shannon mutual information [59.Shannon C.E. Weaver W. The Mathematical Theory of Communication. The University of Illinois Press, 1949Google Scholar], which indicates how much information about the input signal we can glean from the output signal (Figure I). The entropy of a continuous distribution P(x), often denoted H(x), isHx=−∫dxPxlog2Px.(I) If P(x) is discrete there is an analogous definition; the only difference is that the integral becomes a sum,Hx=−∑iPxilog2Pxi.(II) To compute mutual information, we need to compare the entropy before and after some observation (which for us is a spike train). Let's use s for the observation, and P(x | s) for the distribution of x (which for us is the synaptic drive) after observing a spike train. In that case, the conditional entropy, denoted H(x | s) – the average entropy after observing a spike train – isHxs=−∫dxdsPsPxslog2Pxs,(III) with integrals replaced by sums for discrete distributions. The mutual information, I(x,s), between x and s is the difference between the two entropies,Ixs=Hx−Hxs.(IV) This definition is sufficiently flexible that x could be continuous and s discrete, or vice versa. As is not hard to show, information is symmetric; it can also be writtenIxs=Hs−Hsx.(V) Symmetry is important because it tells us that the higher the entropy of a spike train (which typically translates to higher firing rate), the larger the potential for high information. Importantly, though, increasing the firing rate is not guaranteed to increase information, as that may also increase the second term. To investigate how synaptic noise affects information transmission in realistic situations, we need to understand the sources of noise. There are two main sources. The first, and best studied, are release failures: when an action potential invades the presynaptic terminal, neurotransmitter may or may not be released [13.Redman S. Quantal analysis of synaptic potentials in neurons of the central nervous system.Physiol. Rev. 1990; 70: 165-198Crossref PubMed Scopus (443) Google Scholar,14.Jack J.J.B. et al.Quantal analysis of excitatory synaptic mechanisms in the mammalian central nervous system.Cold Spring Harb. Symp. Quant. Biol. 1990; 55: 57-67Crossref PubMed Scopus (12) Google Scholar]. The average release probability, Pr – calculated as one minus the probability of failure – ranges from 0.2 to 0.8 at central synapses [15.Oertner T.G. et al.Facilitation at single synapses probed with optical quantal analysis.Nat. 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This generates considerable noise: if ten action potentials arrive at a cell approximately simultaneously, typically somewhere between two and eight of them will cause a release of neurotransmitter, and on any one trial, anywhere from zero to ten will cause neurotransmitter release. The second source of noise is variability in the synaptic conductance (producing variability in the peak synaptic receptor current) in response to release of one neurotransmitter-filled synaptic vesicle. Fluctuations in the synaptic conductance depend on two main factors. First, synaptic vesicle content can vary depending on prior activity [20.Hanse E. Gustafsson B. Paired-pulse plasticity at the single release site level: an experimental and computational study.J. 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However, recent super-resolution studies have revealed a structural alignment of presynaptic release sites and postsynaptic receptor clusters [28.Tang A.H. et al.A trans-synaptic nanocolumn aligns neurotransmitter release to receptors.Nature. 2016; 536: 210-214Crossref PubMed Scopus (354) Google Scholar], which should minimise this source of noise, at least on time scale of minutes [29.Nair D. et al.Super-resolution imaging reveals that AMPA receptors inside synapses are dynamically organized in nanodomains regulated by PSD95.J. Neurosci. 2013; 33: 13204-13224Crossref PubMed Scopus (383) Google Scholar]. Nonetheless, excitatory postsynaptic currents recorded at individual synapses in response to synaptic vesicle release, in basal conditions, vary significantly. A recent study suggested that the architecture of common excitatory synapses is such that it maximises synaptic conductance variability [30.Savtchenko L.P. et al.Central synapses release a resource-efficient amount of glutamate.Nat. Neurosci. 2013; 16: 10-12Crossref PubMed Scopus (42) Google Scholar]. If so, it would appear that individual excitatory synaptic connections 'prefer' to be noisy. An additional source of noise affecting synaptic circuit signalling is stochastic fluctuations of the cell membrane potential. Its multiple origins and its effect on performance have recently been reviewed in detail [31.Yarom Y. Hounsgaard J. Voltage fluctions in neurons: signal or noise?.Physiol. Rev. 2011; 91: 917-929Crossref PubMed Scopus (39) Google Scholar] and are outside the scope of this present opinion article. Figure 1 shows that if the spike threshold is set too high, noise from variability in the conductance can improve information transmission. This is not true of release failures: lowering the probability of release decreases the drive to a postsynaptic cell. However, if we conserve transmitter expenditure, by increasing the conductance as we lower the release probability so that the product of conductance and Pr stays constant, the story changes. In this regime, the mean is independent of the release probability and the variance scales as (1– Pr)/Pr. The variance thus increases as Pr decreases – exactly what we need to drive a cell when the threshold is set too high. Consequently, both failures and conductance variability could be beneficial for information transfer. This is illustrated in Figure 2. Figure 2A shows that reliable synapses (Pr = 1) with constant conductance are perfectly efficient, but only when the overall input is above the threshold (compare left and right). In comparison, unreliable synapses (Pr < 1) with conductance scaled to conserve transmitter expenditure (as described earlier) can transfer more information when below threshold, but their performance is worse above threshold (Figure 2B). Not surprisingly, reliable synapses with variability in conductance will behave in a similar way to unreliable synapses with constant conductance (Figure 2C). The picture that emerges from this analysis is that when the threshold is set too high, noise increases information transmission; otherwise, noise decreases information transmission (see Figure I in Box 2). This suggests that the optimal strategy is to reduce noise as much as possible, then set the threshold for spike generation to maximize information transmission. As we have seen, increasing release probability and reducing conductance variability both reduce noise. In addition, the nervous system has another option: increase the redundancy of synaptic connections – by allowing individual axons to make multiple synaptic connections on a postsynaptic cell [32.Bloss E.B. et al.Single excitatory axons form clustered synapses onto CA1 pyramidal cell dendrites.Nat. Neurosci. 2018; 21: 353-363Crossref PubMed Scopus (59) Google Scholar]. This averages the noise and so partially compensates for synaptic unreliability [33.Zador A. Impact of synaptic unreliability on the information transmitted by spiking neurons.J. Neurophysiol. 1998; 79: 1219-1229Crossref PubMed Scopus (162) Google Scholar,34.Manwani A. Koch C. Detecting and estimating signals over noisy and unreliable synapses: information-theoretic analysis.Neural Comput. 2001; 13: 1-33Crossref PubMed Scopus (76) Google Scholar]. This is, in fact, the strategy taken by reliable synapses, such as the neuromuscular junction or the calyx of Held, which are equipped with multiple unreliable release sites [1.Tarr T.B. et al.Are unreliable release mechanisms conserved from NMJ to CNS?.Trends Neurosci. 2013; 36: 14-22Abstract Full Text Full Text PDF PubMed Scopus (41) Google Scholar].Box 2Computing (Mutual) InformationTo gain an understanding of how mutual information depends on parameters, consider a simplified scenario in which there is a single presynaptic neuron that either does or does not fire, the output spikes tell us whether or not it fired, time is divided into small bins, say 10 ms, and there are either zero spikes or one spike in each bin.In this scenario, it is not hard to show that the mutual information (Box 1) is given byIxs=hqp−phq.(I) where p is the probability that the presynaptic neuron fires, q denotes the probability that the postsynaptic neuron emits a spike given that the presynaptic neuron fired, and h(z) is the entropy of a Bernoulli random variable with probability z,hz=−zlog2z−1−zlog21−z.(II) Assume the postsynaptic neuron fires when its input exceeds a threshold, θ, and that the input drive is Gaussian with mean μ and standard deviation σ. In that case, q is a cumulative normal function of (μ – θ)/σ. A plot of mutual information versus μ is shown in Figure I for various values of the noise (the thicker the line, the larger the noise). If the mean synaptic drive is above threshold, noise hurts, because it decreases the probability that a presynaptic spike will cause a postsynaptic one; if it is below threshold, noise helps, because it increases the probability that a presynaptic spike will cause a postsynaptic one. In addition, the higher the firing rate of the presynaptic neuron, the higher the mutual information (compare red and blue curves).These observations suggest that mutual information is maximized when the input is bimodal [either well above or well below threshold, so that q = 1 according to Equation II, which means h(q) is zero], the noise is small, and the output firing rate is large. However, it is unlikely that these are achievable. For instance, presynaptic spikes are spread out over time. Consequently, the synaptic drive is a continuous function of time, and neurons spike as soon as the drive crosses threshold. Thus, synaptic drive is rarely bimodal. In addition, there are energetic constraints on firing rate, so it can't be arbitrarily large. Note, though, that in the brain, the input information is much larger than the output information – about 1000 times larger, since each neuron receives about 1000 inputs. Consequently, the information lost due to noise added by failures and variability in the size of postsynaptic currents is likely to be small compared with the information that is thrown away by the finite capacity of neurons, an argument made by Levy and Baxter [37.Levy W.B. Baxter R.A. Energy-efficient neuronal computation via quantal synaptic failures.J. Neurosci. 2002; 22: 4746-4755Crossref PubMed Google Scholar]. To gain an understanding of how mutual information depends on parameters, consider a simplified scenario in which there is a single presynaptic neuron that either does or does not fire, the output spikes tell us whether or not it fired, time is divided into small bins, say 10 ms, and there are either zero spikes or one spike in each bin. In this scenario, it is not hard to show that the mutual information (Box 1) is given byIxs=hqp−phq.(I) where p is the probability that the presynaptic neuron fires, q denotes the probability that the postsynaptic neuron emits a spike given that the presynaptic neuron fired, and h(z) is the entropy of a Bernoulli random variable with probability z,hz=−zlog2z−1−zlog21−z.(II) Assume the postsynaptic neuron fires when its input exceeds a threshold, θ, and that the input drive is Gaussian with mean μ and standard deviation σ. In that case, q is a cumulative normal function of (μ – θ)/σ. A plot of mutual information versus μ is shown in Figure I for various values of the noise (the thicker the line, the larger the noise). If the mean synaptic drive is above threshold, noise hurts, because it decreases the probability that a presynaptic spike will cause a postsynaptic one; if it is below threshold, noise helps, because it increases the probability that a presynaptic spike will cause a postsynaptic one. In addition, the higher the firing rate of the presynaptic neuron, the higher the mutual information (compare red and blue curves). These observations suggest that mutual information is maximized when the input is bimodal [either well above or well below threshold, so that q = 1 according to Equation II, which means h(q) is zero], the noise is small, and the output firing rate is large. However, it is unlikely that these are achievable. For instance, presynaptic spikes are spread out over time. Consequently, the synaptic drive is a continuous function of time, and neurons spike as soon as the drive crosses threshold. Thus, synaptic drive is rarely bimodal. In addition, there are energetic constraints on firing rate, so it can't be arbitrarily large. Note, though, that in the brain, the input information is much larger than the output information – about 1000 times larger, since each neuron receives about 1000 inputs. Consequently, the information lost due to noise added by failures and variability in the size of postsynaptic currents is likely to be small compared with the information that is thrown away by the finite capacity of neurons, an argument made by Levy and Baxter [37.Levy W.B. Baxter R.A. Energy-efficient neuronal computation via quantal synaptic failures.J. Neurosci. 2002; 22: 4746-4755Crossref PubMed Google Scholar]. Taken to its extreme, if neurons can optimise their threshold, zero noise should always be optimal. However, the continual learning that goes on in the brain involves constant adjustment of synaptic strengths and cell excitability. This should make threshold optimisation a continuously evolving process – one that can never be perfectly realised. Indeed, the large range of timescales over which changes in synaptic strength and in cell excitability occur [35.Ma Z. et al.Cortical circuit dynamics are homeostatically tuned to criticality in vivo.Neuron. 2019; 104: 655-664Abstract Full Text Full Text PDF PubMed Scopus (78) Google Scholar,36.Turrigiano G.G. et al.Activity-dependent scaling of quantal amplitude in neocortical neurons.Nature. 1998; 391: 892-896Crossref PubMed Scopus (1644) Google Scholar] suggest that brain circuits cannot rely on permanently optimised thresholds. Thus, it may be advantageous to use synaptic noise to ensure that some spikes are transmitted rather than relying on an optimal threshold. In this context, release probability is not strictly noise: its use-dependent changes may represent a certain 'informative' operation that contributes to the transmitted message. So far, we have defined optimality purely with respect to information transmission. However, even if a circuit is optimal from an information transmission standpoint, it is unlikely to be optimal once the cost of neurotransmitter release is considered. There is a relatively simple reason for that: spikes have limited bandwidth (sustained firing rates are almost always well below 100 Hz, with only occasional bursts of a few spikes at 1 kHz), so substantial input information, which occurs when release probability is high and/or input activity is intense, is largely wasted. As shown previously [37.Levy W.B. Baxter R.A. Energy-efficient neuronal computation via quantal synaptic failures.J. Neurosci. 2002; 22: 4746-4755Crossref PubMed Google Scholar], for reasonable output firing rates, release probabilities that maximize information per release are in the range 0.2–0.8; exactly what is seen in vivo. Subsequent studies taking into account short-term plasticity [38.Salmasi M. et al.Short-term synaptic depression can increase the rate of