How do neurons compute? Two main theories compete: neurons could temporally integrate noisy inputs (rate-based theories) or they could detect coincident input spikes (spike timing-based theories). Correlations at fine timescales have been observed in many areas of the nervous system, but they might have a minor impact. To address this issue, we used a probabilistic approach to quantify the impact of coincidences on neuronal response in the presence of fluctuating synaptic activity. We found that when excitation and inhibition are balanced, as in the sensory cortex in vivo, synchrony in a very small proportion of inputs results in dramatic increases in output firing rate. Our theory was experimentally validated with in vitro recordings of cortical neurons of mice. We conclude that not only are noisy neurons well equipped to detect coincidences, but they are so sensitive to fine correlations that a rate-based description of neural computation is unlikely to be accurate in general.
What is the role of precise spike timing in neural computation? This fundamental issue remains highly controversial. The traditional view postulates that neurons temporally integrate noisy inputs, with precise spike timing playing little role in neural function (Adrian, 1934; Shadlen and Newsome, 1998; London et al., 2010): the output firing rate is essentially determined by the input firing rates. In the last decades, a number of authors have advocated radically different views, in which the temporal coordination of spikes plays a central role (Singer, 1999; VanRullen et al., 2005). A popular spike-timing theory proposes that neurons respond to precisely coincident spikes (Abeles, 1991; Softky and Koch, 1993; König et al., 1996; Kumar et al., 2010). This view is supported by experimental observations of synchrony in various areas of the nervous system (Usrey and Reid, 1999; Salinas and Sejnowski, 2001). For example, in the retina and the thalamus, neighboring cells are often synchronized at a fine timescale (Alonso et al., 1996; Brivanlou et al., 1998; Usrey et al., 1998; Meister and Berry, 1999); in olfaction, fine odor discrimination relies on transient synchronization between specific neurons (Stopfer et al., 1997).
However, the presence of synchrony does not by itself contradict the view that neural computation can be mainly described in terms of firing rates: it could be that synchrony is incidental and has no significant impact on neural function, if neurons are not very sensitive to it. Therefore, a crucial question to answer is whether neurons are sensitive to coincidences in their inputs, and to what extent. Independently of whether synchrony is functionally useful, this sensitivity determines whether the input-output function of neurons can be essentially described in terms of firing rates.
In a resting neuron, coincidence detection is a trivial property due to the all-or-none nature of neural firing: two coincident postsynaptic potentials (PSPs) superimpose and may reach spike threshold, when two temporally distant PSPs would remain subthreshold. In vivo, when the neuron is subjected to highly fluctuating synaptic activity, coincidence detection is more difficult to quantify. In vivo recordings in neurons of the visual cortex (Usrey et al., 2000) and somatosensory cortex (Roy and Alloway, 2001) with simultaneous recording of a pair of presynaptic neurons have shown that cortical neurons are more likely to fire when the two presynaptic neurons fire together rather than at distant times. However, it could be argued that coincidence between input spikes may not be the cause of increased firing but only correlates with it, since cortical neurons also received inputs from many other cells. In addition, it is not straightforward to understand the implications of these observations when thousands of synaptic inputs are considered.
Previous theoretical studies examined the role of input correlations in neuron models, but with specific input scenarios (Salinas and Sejnowski, 2000; Moreno et al., 2002; Rudolph and Destexhe, 2003b). Here we use a general probabilistic approach to understand and quantify coincidence sensitivity, which allows us to predict the impact of input coincidences on the output firing rate of a neuron, as a function of the statistics of background activity.
Materials and Methods
Slice preparation and solutions
Mice of either sex of the CBA (n = 3) or the C57BL/6 (n = 5) strain aged postnatal day 9–15 were decapitated under sodium-pentobarbital anesthesia in conformity with the rules set by the European Commission Council Directive (86/89/ECC) and approved by the local Swedish Animal Care and Use Committee (Permit N13/10). Transverse slices (250–300 μm) were collected when the corpus callosum (CC) joins the two hemispheres and the hippocampus covers the underlying subcortical structures (approximately at bregma −2.5 mm) using a vibratome (VT1200; Leica). Slices were incubated at 32°C in normal aCSF (see below) for 20–30 min, after which they were allowed to cool to room temperature. The cell-dense layer 2/3 region 100–300 μm below the pia was targeted in the primary auditory cortex, located ∼1.5 mm dorsal to the rhinal fissure (Fig. 1). Recordings were obtained within 4–5 h of the preparation.
The low-sodium, high-sucrose aCSF contained the following (in mm): 85 NaCl, 2.5 KCl, 1.25 NaH2PO4, 25 NaHCO3, 75 sucrose, 25 glucose, 0.5 CaCl2, 4 MgCl2, whereas the normal aCSF contained (in mm): 125 NaCl, 2.5 KCl, 1.25 NaH2PO4, 26 NaHCO3, 25 glucose, 2 CaCl2, and 1 MgCl2. These solutions were bubbled continuously with carbogen gas (95% O2–5% CO2), generating a pH of 7.4.
The internal pipette solution contained the following (in mm): 130 K-gluconate, 5 KCl, 10 HEPES, 1 EGTA, 2 Na2-ATP, 2 Mg-ATP, 0.3 Na3-GTP, 10 Na2-phosphocreatinine, adjusted to pH 7.3 with KOH, and the osmolarity was ∼290 mOsm. The recording solution also contained Neurobiotin (0.1%) to be able to reconstruct the recording site and the neuron morphology using a standard immunohistochemistry protocol with Streptavidin-Texas Red (Vector Labs).
Recording procedures and data acquisition
Slices were transferred to a recording chamber perfused (∼3 ml min−1) with oxygenated aCSF at room temperature (25 ± 2°C). The putative A1 pyramidal cells were viewed with an upright microscope (Zeiss Axioscope) using a 40× water-immersion objective (Achroplan, Zeiss) and infrared-differential interference optics equipped with a digital CCD camera (Orca 2, Hamamatsu). Whole-cell current-clamp recordings were done with a Multiclamp 700B amplifier (Molecular Devices) using borosilicate glass microelectrodes with a final tip resistance of Rp = 5–10 MΩ. The signals were filtered with a low-pass 4-pole Bessel filter at 10 kHz, sampled at 20 kHz and digitized using a Digidata 1422A interface (Molecular Devices). Capacitance neutralization was not fully applied, as it was not necessary for this study. The resting membrane potential of recorded cells varied between −75 mV and −63.5 mV (average −70 mV).
Because we injected highly fluctuating currents (Fig. 2a), the electrode produced artifacts that could not be well corrected with standard bridge compensation (Brette et al., 2008) (Fig. 2b): indeed, in addition to simulated synaptic noise, we injected large exponentially decaying EPSCs (instantaneous rise), and therefore currents were discontinuous. In addition, recordings in the same cell were long (up to a few hours) and electrode properties changed during the course of the experiment (typically, the series resistance increased; cell properties, on the other hand, were stable): in the experiments on coincidence sensitivity in cortical neurons (see Fig. 9b) (done after a couple of hours of recording in the same cell), the series resistance was 70–297 MΩ (median 120 MΩ—see Table 1). Thus, even though membrane resistance is high in these cells (several hundred MΩ), it was necessary to subtract the electrode response. Therefore, we used an offline electrode compensation procedure based on an electrode model. Traces are divided in 1 s slices, and we use a generic model fitting toolbox (Rossant et al., 2011) to fit a linear model of the neuron and electrode to the raw recorded trace: where τm and τe are the membrane and electrode time constants, R and Rs are the membrane and series resistance, and Vr is resting potential. These parameters are adjusted to minimize the Lp error between the model prediction Vmodel (Fig. 2c) and the raw trace Vraw, defined as: with p < 2. Using an Lp error rather than the more standard quadratic error reduces the impact of outliers, such as spikes. We detect spikes on the fully compensated trace, Vraw − Vmodel, which corresponds to what is not predicted by the linear model, in particular spikes (Fig. 2d). We use a manually selected threshold criterion (typically ∼20 mV) to identify spikes. The compensated membrane potential of the cell, which we use for subthreshold analyses, is Vraw − Ve (Fig. 2e).
The electrode compensation technique subtracts the voltage drop through the electrode that is produced by the injected current. However, it does not impact the filtering effect of the electrode, due to the non-zero response time (time constant τe). For this reason, spike height (difference between peak and spike threshold) varied systematically with the estimated electrode time constant τe (for cells C1–6: 64, 49, 33, 75, 79, and 57 mV; electrode time constant was 3 ms for C3 and 1.1 ms for C2, and <1 ms for all other cells; spike duration varied in the opposite direction).
Synaptic currents and data analysis
To measure the probability that a noisy neuron fires in response to a PSC of varying size (P(w); see Fig. 9), we inject a sum of a background noise and exponentially decaying PSCs. The background noise is an Ornstein-Uhlenbeck process (i.e., low-pass filtered white noise) with time constant τN = 10 ms (different mean and SD values were tested). After 9 s, we start injecting PSCs every 100 ms, with random size: PSC(t) = αwe−t/τs, where τs = 3 ms, α = 665 pA is a scaling factor (so that maximum depolarization is ∼25 mV), and w is a random number between 0.04 and 1. The total stimulation lasts ∼10 min.
To compute P(w), we divide the range of values for w (0.04–1) into 20 subintervals, and in each subinterval i, we calculate the proportion pi of trials in which the neuron spiked within 30 ms of the injected PSC. The error bar is the SD σi of this estimator: σi2 = pi(1− pi)/ni, where ni is the number of data points. Since we are in fact interested in the extra firing probability due to the injected PSC, we subtract the spontaneous firing probability pi0, which is estimated from the same trace, but using spikes produced in the 70 ms preceding each PSC. To compare with theoretical predictions, we measure the spike threshold, the membrane potential distribution and the size of PSPs. The spike threshold is estimated on each trace as the voltage at the maximum of the second derivative of the voltage trace before spikes (Henze and Buzsáki, 2001), and we use the median of all threshold values. The mean and SD of the membrane potential are measured in the first 9 s of the stimulation (which contains no PSC), after removing spikes, and we assume a Gaussian distribution. To measure the relationship between PSC size and PSP size, we injected a series of 10 identical EPSCs with peak α = 665 pA, and measured the average peak of PSPs.
To generate synaptic inputs with synchrony events (see Fig. 10), we first generate a set of Ne = 4000 (resp. Ni = 1000) independent excitatory (resp. inhibitory) spike trains with Poisson statistics and rate λe = 0.65 Hz (resp. λi = 1.3 Hz). Each spike triggers an exponentially decaying PSC with time constant τe = 3 ms (resp. τi = 10 ms) and peak value we = 13 pA (resp. wi = 5.7 pA). These values were chosen so that EPSPs and IPSPs are ∼±1 mV high (close to the average excitatory PSP size in the mouse auditory cortex in vitro; Oswald and Reyes, 2008). Synchrony events are generated according to a Poisson process with rate λc, and for each event we pick p excitatory synapses at random and make them simultaneously fire. To compensate, a random spike is suppressed for each new spike. For each cell, we test 4 values of p (5, 15, 25, 35) and 3 values of λc (5 Hz, 10 Hz, 20 Hz). The total stimulation lasts ∼14 min. We calculate the firing rate r for each pair of values (p, λc), and the error bar corresponds to the SD assuming Poisson statistics, that is, r/T, where T is the duration of the block.
We also measure the firing rate of neurons in response to spike trains with specified pairwise correlation c (Fig. 3c; see Fig. 13). The current is the same as in the previous paragraph, except that correlations are introduced differently (Kuhn et al., 2003; Brette, 2009): we generate a single reference Poisson spike train with rate λ/c, and for each target spike train, we copy every spike of the reference spike train with probability c. Correlations are only introduced in excitatory inputs. The stimulation lasts ∼5 min, during which we successively test 11 values of c regularly spaced between 0 and 0.01. The blocks are randomized, so that c does not increase monotonously during the stimulation.
Models were simulated with the Brian simulator (Goodman and Brette, 2009). We used integrate-and-fire models, where the membrane potential V(t) is governed by the following equation: where τm = 5 ms is the membrane time constant (taking into account the increase conductance in vivo; Destexhe et al., 2003), El = −65 mV is the resting potential and I(t) is the input (current times membrane resistance). The neuron fires when V(t) reaches the threshold θ = −55 mV, and is then reset to El and clamped at this value for a refractory period of 5 ms. Despite their simplicity, such simple models can predict the spiking responses of cortical neurons to time-varying currents injected at the soma with surprising accuracy (Gerstner and Naud, 2009; Rossant et al., 2011).
When calculating P(w) and coincidence sensitivity S (Figs. 4⇓⇓–7), a voltage noise is added as an Ornstein-Uhlenbeck process with SD σ and time constant τN. We estimate P(w), we calculate the probability that the model fires within 10 ms of an injected PSC, minus the probability that it fires spontaneously in the temporal window. PSCs are either instantaneous (dirac functions), giving exponential PSPs (time constant τm) or exponential (with time constant τe = 3 ms for excitatory synapses and τi = 10 ms for excitatory synapses), giving biexponential PSPs. The synaptic weight w corresponds to the peak value of the PSPs.
In the simulations with correlated inputs (see Figs. 10, 13, 14), we calculate the inhibitory weight to ensure that the mean total current is zero, which is given by the balance equation: where λe and λi are the excitatory and inhibitory rates. These PSP integrals can be analytically calculated.
In simulations with synaptic conductances instead of currents (see Fig. 12), the input current is: where Ee = 0 mV and Ei = −75 mV are the excitatory and inhibitory reversal potentials, and ge(t) and gi(t) are the excitatory and inhibitory conductances (in units of the leak conductance). These are sums of exponentially decaying conductances, with the same time constants as before. The membrane time constant was τm = 20 ms. Background activity consists of Ne = 4000 excitatory and Ni = 1000 inhibitory inputs at rate λe = λi = 1 Hz (Poisson processes). We set the individual peak conductances so that the mean total excitatory conductance is 〈ge〉 = 0.5 (in units of the leak conductance) and the mean total inhibitory conductance is 〈gi〉 = 3.25 (which ensures that the mean membrane potential is −65 mV).
The membrane potential distribution is denoted p(v) and the threshold θ. The probability that the voltage is above θ − w is: The coincidence sensitivity is then S = P(2w) − 2P(w) for two spikes, and Sp= P(pw) − pP(w) for p spikes. We then assume a Gaussian distribution for p(v) with SD σ (this is not a requirement of the theory, but it simplifies calculations), so that: where erf is the error function and the spike threshold θ is relative to the mean membrane potential.
To calculate the extra output rate due to synchrony events in the sparse synchrony scenario (see Fig. 10a), we first calculate the mean and variance of the membrane potential using Campbell's theorems, applied to the nonsynchronous inputs: Then the extra output rate is λcP(pw), where μ and σ are used in the definition of P. These formulae are in fact only valid in the subthreshold regime (for a nonspiking neuron), but the change induced by spikes is small when the time constant (5 ms in our simulations) is small compared with the typical interspike interval. More accurate expressions of the membrane potential distribution exist for a limited number of cases (Fourcaud and Brunel, 2002).
When synaptic inputs are modeled as conductances (see Fig. 12), we calculate the membrane potential distribution and EPSP size using the effective time constant approximation (Richardson and Gerstner, 2005), that is, we define the average total conductance as gtot = 1 + 〈ge〉 + 〈gi〉 (in units of the leak conductance) and the effective time constant as τeff = τm/gtot, and we use a linear approximation of the membrane equation using these effective parameters: where V0 is the average membrane potential: With this approximation, which takes into account the change in effective time constant and resistance, we can use exactly the same analytical methods as before (calculate the EPSPs analytically, and use Campbell's theorems to calculate the membrane potential distribution). It is possible to calculate the membrane distribution more precisely with more sophisticated methods (Rudolph and Destexhe, 2003a; Richardson and Gerstner, 2006; Richardson and Swarbrick, 2010), but this simple method was sufficient in our case.
To motivate this study, Figure 3 shows the sensitivity of a cortical neuron (layer 2/3 of primary auditory cortex, Fig. 1) to fine correlations in its inputs. We injected a fluctuating current in vitro composed of a sum of 4000 excitatory and 1000 inhibitory random spike trains (Fig. 3a, each presynaptic spike triggers a postsynaptic current), with Poisson statistics. Excitation and inhibition were balanced on average, as in the sensory cortex in vivo (Destexhe et al., 2003), and this cell did not fire when the inputs were uncorrelated (Fig. 3a). We introduced synchrony events where p randomly selected presynaptic excitatory spikes occurred at the same time (Fig. 3b), without changing individual spike train statistics. The neuron fired at 2 Hz when synchrony events involved only p = 15 synapses, that is, <0.4% of all synapses, and it fired at 5.5 Hz with p = 25. This exquisite sensitivity was also seen when homogeneous pairwise correlations with smaller higher-order correlations were introduced in the inputs (Fig. 3c): the cell fired at 3 Hz with input correlation c = 0.002 (0.2%) and 6 Hz with c = 0.004. Thus, pairwise correlations that are so small that they could probably not even be observed in a paired recording (a pair of coincident spikes between two given presynaptic neurons occurs every 150 s with c = 0.004 and firing rate F = 0.6 Hz) still have a dramatic impact on postsynaptic firing rate. We observed the same phenomenon in simple integrate-and-fire neuron models, which suggests that it does not rely on specific cellular mechanisms.
Coincidence sensitivity: a simple probabilistic model
How general is this property? To quantify coincidence sensitivity, we compare the impact of two coincident versus two noncoincident spikes on a neuron with noisy background synaptic activity (Fig. 4). If two input spikes are added on top of the background activity, the neuron will fire more spikes on average. This average extra number of spikes can be measured by repeating the same protocol over many trials and computing the poststimulus time histogram (PSTH): the extra number of spikes is the integral of the PSTH above the baseline (Fig. 4a, yellow line). This corresponds to the “spike efficacy” defined by Usrey et al. (2000). In Figure 4, the neuron model fired on average 0.15 extra spikes in response to two noncoincident spikes and 0.26 extra spikes in response to two coincident spikes. Therefore, the average extra number of spikes due to input coincidence was S = 0.26 − 0.15 = 0.11 spikes. We define this quantity as the coincidence sensitivity, S: S > 0 means that the neuron fires more when its inputs are coincident. It depends on neuronal and synaptic properties, and on the statistics of background activity.
Coincidence sensitivity can be quantified with a simple probabilistic approach (Fig. 5). When excitation and inhibition are balanced on average, the membrane potential distribution peaks well below threshold (Fig. 5a). The neuron fires in response to an input spike if its membrane potential Vm is close enough to the spike threshold θ. More precisely, if w is the size of the postsynaptic potential (PSP), then the neuron fires if Vm + w > θ. Graphically, the probability P(w) that the neuron fires (Fig. 5a, inset) is then the integral of the membrane potential distribution between θ and θ − w. If two temporally distant spikes are received, the probability that the neuron fires is just 2P(w), corresponding to the orange area in Figure 5a. If two coincident spikes are received, the probability that the neuron fires is P(2w), that is, the integral of the membrane potential distribution between θ and θ − 2w. Thus, the coincidence sensitivity is the difference between these two probabilities: S = P(2w) − 2P(w), corresponding to the red area in Figure 5a. This approach extends to the coincidence sensitivity with p spikes Sp, defined as the difference in the average extra number of spikes with p coincident versus noncoincident input spikes (Fig. 3b).
This simplified description is not entirely accurate, because an input spike may bring the membrane potential slightly below threshold, which would not immediately trigger an output spike but still increase the probability of firing at a later time, because of background fluctuations. Therefore, our description is an approximation that is valid when the duration of a PSP is short compared with the time constant of background fluctuations, and otherwise underestimates the true probability P(w). This is shown in Figure 5c, where P(w) was numerically estimated in a neuron model with background noise, as a function of the time constant τN of that noise (dashed line). The theoretical prediction with the probabilistic model corresponds to the asymptotic value for large τN. For fast fluctuations (small τN), the true value of P(w) is significantly larger than our prediction. However, the error we make has the same magnitude and sign for P(2w) (two coincident spikes) and for 2P(w) (two noncoincident spikes), so that the coincidence sensitivity S, which is the difference, is in fact well approximated by our simple probabilistic model (Fig. 5d).
Fluctuation-driven versus mean-driven regime
In Figure 5, the neuron responds more to coincident than to noncoincident spikes (S > 0) because the membrane potential is more likely to be near average than near threshold, i.e., the membrane potential distribution p(Vm) is decreasing. This reflects the fact that the neuron was in a “fluctuation-driven” regime, because of the balance between excitation and inhibition. If the average synaptic current is suprathreshold (more excitation than inhibition), then the situation is reversed (Fig. 6). Indeed, in this case the neuron fires regularly at a rate defined by the average current (“mean-driven” regime) and spends more time near threshold than far from it (Fig. 6a,b and membrane potential distribution in Fig. 6c, see voltage traces). It follows that two coincident spikes have a smaller impact on output firing than two noncoincident spikes, that is, S < 0 (Fig. 6c, area with red contour). Figure 6d shows the measured coincidence sensitivity of a neuron model with background noise where the mean was varied, for two different noise variances. Coincidence sensitivity is positive when the average membrane potential (measured without threshold) is far below threshold, and it changes sign when it approaches threshold (more precisely, S is expected to change sign when the average membrane potential is within 2 PSPs of the threshold—PSP size was 1 mV in this figure). Thus, neurons are sensitive to coincidences in the fluctuation-driven (or balanced) regime, not in the mean-driven regime.
Which one of these two situations is a better description of membrane potential dynamics in vivo, fluctuation-driven or mean-driven? In Figure 7, we show membrane potential distributions in in vivo intracellular recordings in various areas, extracted from a number of previous studies. In anesthetized preparations, these distributions are often bimodal, which reflects “up” and “down” states (Constantinople and Bruno, 2011). Since down states are quiescent, only the depolarized mode (up state) is relevant to this study. In all cases, the membrane potential distribution decreases toward threshold, suggesting that neurons are in a fluctuation-driven rather than mean-driven regime. Therefore, we now focus on the fluctuation-driven regime.
Influence of background noise statistics
Quantitatively, coincidence sensitivity depends on PSP size and background noise statistics (e.g., mean and variance of the membrane potential). In our probabilistic model, S increases monotonically with PSP size w (Fig. 8a, left), until 2w is the difference between average membrane potential and threshold (10 mV in Fig. 8a), which corresponds to the inflection point of the sigmoid P(w) (Fig. 5a, inset). The relationship with the SD σ of the membrane potential is more surprising: it appears that S is maximal for an intermediate value of σ, for example ∼2 mV when w ≈ 5 mV. Intuitively, it can be explained as a trade-off: if there is little background noise (small σ), then two spikes are unlikely to make the neuron fire, whether they are coincident or not, and therefore S is small; if there is too much noise (large σ), then the membrane potential distribution is flat (all voltages are equally likely) and two coincident spikes have the same effect as two noncoincident spikes. These theoretical predictions accurately matched numerical results obtained in neuron models with simulated input currents (Fig. 8a, right).
It could be argued that 5 mV is unreasonably large for a PSP. However, the same analysis applies when considering p small PSPs instead of 2 large PSPs (Fig. 8b, p = 10 and w is varied between 0 and 1 mV): 10 coincident PSPs of 1 mV produce on average 0.5 more output spikes than if they were not coincident. In fact, when the number of input spikes p is large, the coincidence sensitivity Sp is mainly determined by the total depolarization pw, because small isolated PSPs have little effect on output firing (Fig. 5b: the orange area is small). Figure 8c shows the relationship between Sp and pw for different values of p (between 10 and 30) and with w varying between 0 and 10 mV/p (shaded blue), in a simulated neuron model. The dashed line shows the theoretical prediction with p = ∞ (i.e., many small PSPs).
Coincidence sensitivity in vitro
We then verified our theoretical predictions in auditory cortical neurons in vitro (Fig. 9). We generated background noisy currents with specified mean and SD, and injected them into the soma with additional exponentially decaying currents of various sizes, representing EPSCs, PSCs (Fig. 9a) (we address the more realistic case of synaptic conductances below). We recorded the spikes produced by the cell in response to this stimulation and calculated the probability P(w) that the cell fires as a function of PSP size w (Fig. 9b). To compare with our theoretical predictions, we measured the membrane potential distribution in the cell and the spike threshold (see Materials and Methods). Our simple probabilistic model could predict P(w) with good accuracy over the tested range of background statistics (Fig. 9b). The theoretical prediction tends to slightly underestimate P(w), as we already discussed previously (Fig. 5). From the measured P(w), we could calculate the coincidence sensitivity S = P(2w) − 2P(w), which was also well predicted by our theory (Fig. 9c). Consistently with our theoretical analysis, all cells (n = 6) were very sensitive to coincidences.
Impact of sparse synchrony events on output firing
To understand the impact of this property when many synaptic inputs are considered, we analyzed the sparse synchrony scenario presented in Figure 3b with our theoretical approach (Fig. 10a). In that scenario, the neuron receives 4000 excitatory and 1000 inhibitory spike trains with Poisson statistics. Excitation and inhibition are balanced (mean total current is zero), so that the neuron fires irregularly at low rate (3 Hz). Synchrony events are introduced at random times (also with Poisson statistics), by shifting p random excitatory spikes to the event time. This protocol leaves individual spike train statistics unchanged but modifies the correlations. We can use our probabilistic model to predict the extra firing rate due to these synchrony events (see Materials and Methods) (Fig. 10a, left), and the prediction agrees very well with numerical simulations (Fig. 10a, right). The impact of synchrony on output firing rate is dramatic: by introducing synchrony between <1% of all synapses without changing the input firing rates, the output firing rate increases from 2 Hz to 28 Hz in Figure 10a. On the other hand, if we increase the firing rate of p excitatory inputs without changing the correlations while maintaining the excitatory-inhibitory balance (by increasing the inhibitory rate), then the output firing rate hardly changes (Fig. 10b). This is not so surprising since the total rate of excitatory input spikes is hardly modified when the rates of only 1% of synapses are increased and therefore the variance of the membrane potential changes very little (precisely, the relative change is 1 + (pλc)/(Neλe), where λc is the rate of the p inputs, λe is the initial excitatory rate and Ne is the number of excitatory inputs).
We then tested the impact of sparse synchrony in cortical cells, by injecting synthesized currents made of sums of excitatory and inhibitory PSCs with synchrony events (Fig. 10c). In all tested cells (n = 4), we observed that inserting synchrony events without changing input rates had a dramatic impact on the cell's firing rate, with only a few synapses involved in each synchrony event (<1% in all cases). Firing rates tended to be lower than in model simulations (Fig. 10a) because the spike threshold was high in these cells (>30 mV above resting potential), and perhaps because of adaptive properties, which were not included in the models.
Effect of temporal jitter and synaptic unreliability
In our analysis, we compared precisely coincident spikes with temporally distant spikes. How does coincidence detection depend on the delay between input spikes, that is, on the temporal precision of coincidences? Consider a sum of coincident PSPs, and introduce a temporal jitter in spike times (Fig. 11a): the peak voltage decreases with the amount of jitter σj (defined as the SD of spike times). It is possible to calculate the average peak voltage as a function of σj (Fig. 11b): it decreases with a characteristic time close to the decay time constant of PSPs, that is, close to the membrane time constant τm (dashed line). In our theoretical model, coincidence sensitivity is determined by the peak size of combined PSPs, therefore we expect neurons to be sensitive to coincidences when the temporal jitter is smaller than τm. This is confirmed by numerical simulations (Fig. 11c): coincidence sensitivity Sp quickly increases with the number of input spikes when σj < τm (left; τm = 5 ms here), and introducing a 3 ms temporal jitter does not significantly change the impact of sparsely synchronous inputs on output firing rate (Fig. 11, right).
An additional source of variability in vivo is the probabilistic nature of synaptic transmission: presynaptic spikes are transmitted with some probability α < 1. However, it does not degrade the temporal precision of coincidences, and therefore this fact does not qualitatively impact coincidence sensitivity (Fig. 11d). For example, when synapses transmit spikes with probability 0.5, sparse synchrony still has a dramatic impact on output firing rate (left). For a synchrony event consisting of p presynaptic spikes, on average αp synchronous spikes are seen on the postsynaptic side, and therefore the output firing rate is essentially determined by the effective number of synchronous synapses αp, with little dependence on transmission probability α (Fig. 11d, right). Therefore, the impact of stochastic synaptic transmission on coincidence detection is simply to increase the number of synchronous presynaptic spikes for a given postsynaptic effect by a factor 1/α.
Effect of synaptic conductances
In all the results we have shown, the synaptic inputs were modeled as currents rather than conductances. Considering the more realistic case of synaptic conductances has two main consequences: 1) the total conductance is increased (by a factor of ∼5; Destexhe et al., 2003) and therefore EPSP size is decreased by the same amount, 2) the effective membrane time constant is reduced in the same proportion. In our models, we considered a short membrane time constant (∼5 ms) to take this effect into account. For the problem we are considering, the sensitivity to excitatory coincidences, this is likely to be sufficient. First, the excitatory reversal potential is high compared with the spike threshold (typically Ee ≈ 0 mV), and therefore the driving force is not very variable. Second, our measurements of coincidence sensitivity only involve a few input spikes (2 in Fig. 5a, a few tens in Fig. 10), and therefore these additional spikes should have virtually no effect on the total conductance (always <1%).
To demonstrate this point, we measured the coincidence sensitivity of a neuron model with a background of random excitatory and inhibitory input spikes (Fig. 12a), with the same statistics as in Figure 10 (with no correlations), except the inputs were modeled as synaptic conductances: the excitatory (resp. inhibitory) current is Ie(t) = ge(t)(Ee − V) (resp. Ii(t) = gi(t)(Ei − V)), where Ee = 0 mV (resp. Ei = −75 mV) is the excitatory (resp. inhibitory) reversal potential. The mean excitatory conductance was half the leak conductance, while the mean inhibitory conductance was 3.25 times the leak conductance. Thus, the total conductance was ∼5 times the leak conductance, so that the effective membrane time constant was ∼5 times smaller than the membrane time constant (τeff = τm/(1 + 0.5 + 3.25)). We then measured the coincidence sensitivity by injecting excitatory input spikes with variable amplitude, as in Figure 9 (again, the inputs were modeled as conductances) (Fig. 12b). We made theoretical predictions with the same formulae as before, but we used the effective time constant and resistance to (approximately) calculate the membrane distribution and EPSP size. In this effective time constant approximation, the driving forces are replaced by their average (Ee − V0 and Ei − V0) and the leak conductance is replaced by the mean total conductance (Richardson and Gerstner, 2005). We then used exactly the same methods as before. It is possible to calculate the membrane distribution more precisely (Richardson and Gerstner, 2006; Richardson and Swarbrick, 2010), but this simple current-based method was already accurate enough in our case. Figure 12b shows that our theoretical prediction remains reasonably accurate in this complex situation, which is far from the ideal setting (synaptic conductances rather than currents, shot noise rather than diffusion).
Impact of input correlations without synchrony events
One may argue that the sparse synchrony scenario we described in Figure 10 is rather specific in that we introduce strong depolarizations, which also increases higher order correlations (although this is consistent with intracellular recordings in the auditory cortex in vivo; DeWeese and Zador, 2006). However, the results still hold if correlations are introduced in a way that minimizes these higher orders, as in Figure 3c: a fixed pairwise correlation c is introduced between all excitatory spike trains with firing rate r, by copying every spike from a common reference spike train with rate r/c to any given target spike train with probability c (Kuhn et al., 2003; Brette, 2009). Introducing very small correlations (<1%) in this way also has a dramatic impact on the output firing rate of cortical cells (Fig. 13a,b—note the very small scale of the horizontal axis). The same phenomenon is seen in neuron models (Fig. 13c, left). These correlations are so small that they could probably not be seen in a paired recording: indeed, a correlation of 1% between two spike trains with rate 1 Hz results in one coincidence every 100 s on average.
This surprising sensitivity can be explained by the remark that the variance of the total synaptic input increases with input correlation, but a small pairwise correlation is still significant if there are many input pairs (approximately N2 for N synapses). More precisely, in a fluctuation-driven regime, the output rate critically depends on the input variance because the mean synaptic input is (by definition) below threshold. Suppose the total input consists of N synaptic currents: X = ΣIk, and these currents have variance σ2 and pairwise correlation c (c = covar(Ik,Ij)/σ2). If these currents were uncorrelated (c = 0), then the input variance would simply be Nσ2. When they are correlated, the input variance is the sum of the covariances all pairs of currents: var(X) = Σcovar(Ik,Ij). A simple calculation gives: var(X) = Nσ2 + cN(N − 1)σ2. Thus, because there are N(N − 1) pairs of currents, the variance of the total input X is mainly determined by the input correlations, unless c is negligible compared with 1/N (which is already very small). This is shown in Figure 13c (right), where the SD of the total current increases very quickly with correlation (note that the largest correlation considered is c = 1%). Thus, very small pairwise correlations produce strong effects simply because there are many pairs, as was previously observed at network level in the retina (Schneidman et al., 2006).
Transmission of correlations
Precise coincidences increase the firing rate of a postsynaptic neuron, but is precise spike timing preserved in the operation? Figure 14a shows a model example where the postsynaptic neuron fires at 2 Hz when the inputs are uncorrelated, and 8 Hz when synchrony is introduced (p = 10). Additional spikes caused by synchrony events are precisely time-locked to them, with each event triggering on average 0.16 extra spike (Fig. 14b). Therefore if several neurons receive independent inputs except for common synchrony events (involving 0.25% of synapses), then their firing is strongly correlated at a fine timescale (Fig. 14c). These correlations may then impact target postsynaptic neurons.
In Figure 14, output correlation is an order of magnitude larger than spike input correlation (between any two input spike trains), that is, spike correlation increases in the process. This may seem to contradict recent studies showing that the output correlation of a pair of neurons is always smaller than input correlation (de la Rocha et al., 2007; Shea-Brown et al., 2008), but it should be stressed that these studies compare the spike output correlation (correlation between two output spike trains) with the correlation between the two total synaptic currents. Total current correlation is essentially the sum of pairwise correlations (for all pairs of synaptic spike trains) and is therefore one order of magnitude larger (Renart et al., 2010; Rosenbaum et al., 2011). We illustrate this point in Figure 15, where we simulated two neurons with correlated input spike trains, with pairwise correlation c (Fig. 15a). The correlation between the two total inputs (ccurrent) is an order of magnitude larger than the spike input correlation c (Fig. 15a, inset). This is a simple effect of pooling (Rosenbaum et al., 2011): the covariance between the two total inputs is the sum of the covariances of all pairs of synaptic currents, that is, cN2σ2, and therefore the correlation is ccurrent = cN/(1 + c(N − 1)). For a large number of synapses N, this is close to 1, unless correlations are very small. As a result, the spike output correlation is much larger than the spike input correlation (Fig. 15b), even though it is smaller than the total current input correlation (Fig. 15c), in agreement with previous studies. In fact, the amplification of correlations is so strong that network stability requires that inhibition be correlated with excitation (Renart et al., 2010).
Sensitivity to fine correlations
Our results show that neurons are highly sensitive to input correlations in the balanced (or fluctuation-driven) regime, when their timescale is smaller than the integration time constant. The required number of synchronous inputs for a strong effect is ∼p ≈ 10–20 (assuming PSP sizes ∼1 mV, close to the average excitatory PSP size in the mouse auditory cortex in vitro (Oswald and Reyes, 2008); more generally: p = (threshold − average Vm)/PSP size). To have an idea of how strong this requirement is, let us consider the average number of excitatory input spikes within an integration window of 5 ms: with N excitatory inputs at rate F, we obtain F × N × 5 ms. In Figure 10, for example (F = 1 Hz and N = 4000), we obtain on average 20 spikes in one integration window. Therefore, a relatively small excursion above this average number has a very strong impact on postsynaptic firing. These results are consistent with recent findings that a small number (20–40) of synchronous thalamic inputs can reliably drive cortical neurons (Bruno and Sakmann, 2006; Wang et al., 2010). In the latter study, authors found that spike efficacy was maximal for ∼p = 30 synchronous inputs (that is, the ratio P(pw)/p, called reliability per synchrony magnitude, is maximal for p = 30). Using parameter values from that study, our theoretical formula predicts a very similar value, p = 27.
In this study, we defined the coincidence sensitivity as the difference in the average number of postsynaptic spikes (spike efficacy described by Usrey et al., 2000) produced by two synchronous input spikes versus two asynchronous spikes. Similar ideas were introduced by Abeles, who defined the “coincidence advantage” as the number of asynchronous spikes required to produce 0.5 average extra spike, divided by the number of synchronous spikes required to produce the same number of spikes (Abeles, 1982, 1991). The coincidence advantage corresponds to the ratio between the total colored area (blue + red) and the blue area in Figure 5b, that is, P(pw)/(pP(w)), where p is the number of synchronous EPSPs required to produce 0.5 extra spike on average (in our terms, this is approximately p = (θ − μ)/w). Indeed, each asynchronous spike produces a postsynaptic spike with probability P(w), and therefore 0.5/P(w) asynchronous spikes produce on average 0.5 postsynaptic spike. Thus the coincidence advantage is 0.5/(pP(w)) = P(pw)/(pP(w)) (colored area divided by blue area in Fig. 5b), for this particular value of p. This definition is related to the coincidence sensitivity Sp for p spikes (Fig. 8c), which is Sp = P(pw) − pP(w), except our definition is a difference while the coincidence advantage is a ratio. As we previously explained, it turned out that theoretical predictions worked better with a difference than with a ratio, because the errors in the estimation of P(2w) and 2P(w) appear to approximately cancel (Fig. 5c).
In a recent theoretical study, it was found that a precise arrangement of excitation and inhibition could result in partial cancellation of correlations in a network, that is, pairwise correlations scale as 1/N as the network size increases (Renart et al., 2010). In the light of our results, it should be stressed that this does not mean that correlations become negligible, in the sense that correlations of order 1/N still modulate the output rate (correlations should be small compared with 1/N, e.g., 1/N2, to be negligible). Indeed, the linear decrease in correlation is compensated by a linear increase in the number of synaptic inputs. From this point of view, describing the resulting network state as “asynchronous” may be misleading.
Fluctuation-driven versus mean-driven
The main condition for neurons to be sensitive to coincidences is that the average input current is subthreshold, that is, that neurons are in a “fluctuation-driven” regime, as opposed to the “mean-driven” regime, where neurons fire regularly at a rate determined mainly by the mean current. This happens in particular when excitation balances inhibition on average, which occurs in vivo in the high-conductance regime induced by intense synaptic activity (Destexhe et al., 2003). Another signature of fluctuation-driven regimes is the temporal irregularity of spike trains. Recently, it was found that neurons in sensory cortices fire irregularly, while neurons in the motor cortex fire more regularly, but at a variable rate (Shinomoto et al., 2009). This would suggest that neurons in the sensory but not in the motor cortex are sensitive to coincidences. However, spike train regularity may also indicate strong oscillatory activity rather than a mean-driven regime, and therefore we cannot draw a firm conclusion from this observation. In vivo intracellular recordings in many areas (Fig. 8), including the primary motor cortex (Brecht et al., 2004), all indicate that the membrane potential distribution decreases toward spike threshold, which supports the notion that neurons are generally in a fluctuation-driven regime.
Spike-timing versus rate in noisy neurons
Our results support the view that synchrony plays an important role in neural computation. A central point in the debate between spike-timing and rate-based theories of neural computation is whether background activity should be treated as “noise” or “signal” (Stein et al., 2005; London et al., 2010)—that is, intertrial variability could reflect differences in network state rather than intrinsic noise. While our approach is agnostic about this point, our results show that, even if background activity reflects intrinsic or irreducible noise (e.g., chaos) (London et al., 2010), neurons are still extremely sensitive to the relative spike timing of their inputs. In other words, the presence of high voltage fluctuations implies that computation and coding are stochastic, but not that they are based on rate only. On the contrary, the fact that even very tiny correlations (of order 1/N) have tremendous postsynaptic impact suggests that neural computation is generally not satisfactorily described in terms of rates—whether synchrony is functionally useful or not.
This work was supported by the European Research Council (ERC StG 240132) and by the Swedish Research Council (Grant 80326601).
- Correspondence should be addressed to Romain Brette, Equipe Audition, DEC, Ecole Normale Supérieure, 29, rue d'Ulm, 75005 Paris, France.