Information processing in neocortex can be very fast, indicating that neuronal ensembles faithfully transmit rapidly changing signals to each other. Apart from signal-to-noise issues, population codes are fundamentally constrained by the neuronal dynamics. In particular, the biophysical properties of individual neurons and collective phenomena may substantially limit the speed at which a graded signal can be represented by the activity of an ensemble. These implications of the neuronal dynamics are rarely studied experimentally. Here, we combine theoretical analysis and whole cell recordings to show that encoding signals in the variance of uncorrelated synaptic inputs to a neocortical ensemble enables faithful transmission of graded signals with high temporal resolution. In contrast, the encoding of signals in the mean current is subject to low-pass filtering.
The firing rate of many neurons in the cortex is known to depend on various aspects of stimuli in a smooth way. While this finding suggests that the graded rate of an individual neuron is used to distinguish between different stimuli, it can be estimated only after a sufficiently large number of spikes have occurred. For this reason, reliable rate signals are necessarily slow if obtained from single neurons and hence cannot account for the rapid information processing observed in the cortex (Thorpe et al. 1996). In contrast, at the level of populations, this problem of signal-to-noise can be overcome such that the population rate of an ensemble of neurons (i.e., the average number of spikes in the population per time interval) can be estimated on a time scale that is even smaller than the interspike intervals of the individual neurons. If the neuronal responses are statistically independent from each other, the achievable time resolution will in fact depend only on the total number of neurons in the ensemble: the larger the ensemble, the higher the temporal precision with which the population rate can be estimated. Although it is possible to overcome the limitations of temporal precision due to noise by using an increasing number of neurons, there is another constraint on the speed of signal transmission caused by the neuronal dynamics: intrinsic properties of individual neurons like the membrane time constant and population effects like synchronization can severely limit the ability of neuronal ensembles to realize rapid rate codes (Knight 1972). Therefore we here investigate to what extent rapid transmission of graded rate signals between populations of cortical neurons rely on the encoding strategy due to the neuronal population dynamics.
In contrast to the usual characterization of a signal by the temporally averaged mean and the variance components, we here consider the instantaneous distribution of input currents into the neurons of a functional ensemble at each moment in time. In this case, the synaptic inputs can also be divided into two components: one component is given by the input averaged over the ensemble, the other component is given by the deviations of individual inputs from the average: the population variance. Both components can, in general, fluctuate with time, contributing to the observable fluctuations in the synaptic currents of single neurons. The output of the ensemble can be described by an instantaneous population rate, estimated by the number of spikes emitted by the entire ensemble in small time intervals divided by the number of neurons. The population rate depends on the amplitudes of both components of the input. This reasoning allows one to conclude that signals delivered to a neuronal ensemble could in principle be carried by (encoded in) either the common, correlated part of the synaptic inputs to the neurons, or by the variance of the inputs across the population, or by both.
In this study, we examine the respective implications of these coding strategies for rapid and reliable transmission of information between neuronal ensembles. We show that correlated input currents cannot be used to transmit rapidly changing signals, whereas encoding the signal in the variance enables faithful signal transmission in the population rate on a millisecond time scale irrespective of the membrane time constants of the neurons.
integrate-and-fire model. The leaky integrate-and-fire neuron is characterized by its membrane potential, whose dynamics is described by the circuit equation (Laplicque 1907; Tuckwell 1988) (1) where τ denotes the membrane time constant, Vrest is the membrane resting potential, Isyn is the synaptic current, and Rin is the input resistance of a neuron. This equation is supplemented by the condition that each time the membrane potential hits the threshold potential, Vth, a spike is emitted and the membrane potential is instantaneously reset to a certain subthreshold level, Vreset.
To achieve an analytical understanding of the population dynamics, we consider an infinite population of identical integrate-and-fire neurons indexed by i, receiving input currents of the form (2) where μ(t) denotes the mean input across the population at time t, ηi(t) is Gaussian white noise with unit spectral density, and σ(t) is a scaling factor measuring how strongly the individual input currents deviate from the mean. Rigorously speaking, Gaussian white noise does not exist in nature, but it is understood as a reasonable idealization that together with Eq. 1 becomes a well-defined mathematical term in the sense of Langevin equations.
Correspondingly, the time evolution of the population density function P(V, t) then obeys the Fokker-Planck equation (3) where the probability flux J(V, t) is given by (4)
The flux consists of two components: the first term is the drift component, which is governed by the mean μ(t), and the second term is the diffusion component representing the effect of random fluctuations. The firing threshold and the reset potential impose boundary conditions on the flux, which imply P(Vth, t) ≡ 0 for all D = [σ(t)2]/2 > 0 (Risken 1984; Tsodyks and Sejnowski 1995). Therefore at threshold, the only contribution to the flux is given by the diffusion component.
intracellular recordings. Parasagittal slices 300-μm thick were obtained from Wistar rats (14–16 days old). Slices were incubated for 30 min at 32–34°C before being transferred to the recording chamber. The recordings were made at 32–34°C. Neurons were selected for recording according to the morphology of the soma and proximal dendrites, as visualized by IR-DIC optics using a Zeiss Axioscope and Hamamatsu CCD camera. The bathing solution consisted of (in mM) 125 NaCl, 25 NaHCO3, 25 glucose, 2.5 KCl, 2 CaCl2, 1.25 NaH2PO4, and 1 MgCl2. Whole cell recordings were made using patch pipettes (5–10 MΩ), containing (in mM) 110 K-gluconate, 10 KCl, 10 HEPES, 10 phosphocreatine(Na), 4 MgATP, 0.3 NaGTP, and 2% biocytin. Recorded voltage and current were amplified by axopatch 200/B amplifiers (Axon instruments). Acquisition was done using IGOR-Pro software (WaveMetrics). Injected current was calculated for each individual injection using random values drawn from a Gaussian distribution. The values for μ and σ2 were determined according to the discharge rate of the stimulated neuron. Current recordings used for stimulation were obtained in voltage-clamp mode and inverted before injection. Activity in the slice was induced by a gradual application of 4-aminopyridine (4-AP), a blocker of transient potassium current.
It is instructive to first analyze a simplified model, where the issue of population coding can be addressed in a mathematically rigorous way. We consider an (infinitely) large population of identical integrate-and-fire neurons indexed by i (see methods), each receiving fluctuating synaptic inputs of the following form (5)
Here stands for the instantaneous amplitude of the averaged, correlated part of the input (common to all the neurons), while are the deviations of individual inputs from the average (unique for every neuron). A global measure for the time-dependent input diversity is given by the instantaneous population variance of the input. In general, both of the variables, and , change in time, and thereby can serve as signals carrying information from presynaptic populations. The main goal of the analysis is to estimate how the activity of the population reflects the signals carried by either variable. In particular, we are interested in the conditions under which the instantaneous population rate will faithfully follow the analog value of the signal at any time (“perfect signaling”).
The analysis can be pursued to the ultimate solution, if Ĩi(t) are mutually independent and temporally uncorrelated random processes (i.e., Gaussian white noise; see methods). In that limiting case, diverges because of the vanishing correlation time. It is understood, however, that if the correlation time τc of a real current is sufficiently small (i.e., clearly smaller than the membrane time constant) essentially only the product is relevant. For the sake of convenience, we will therefore call σ2 somewhat incorrectly a population variance, too, in the following.
Because the momentary state of an integrate-and-fire neuron is defined only by its membrane potential V(t), the state of the whole population is completely characterized by a probability density function P(V; t), i.e., the fraction of neurons with membrane potential close to V. The time evolution of the density function is governed by both the average and the fluctuating components of the inputs via the so-called Fokker-Planck equation (Risken 1984) (see methods).
One can visualize this formulation by considering a collection of point-like particles moving independently along a one-dimensional axis under the combined influence of the deterministic force and a random, diffusing force which tends to equilibrate the particles along the axis. In this analogy, P(V) is just the density of particles on the axis of V.
The advantage of this approach is that one can derive an exact expression for the instantaneous firing rate of a population of neurons, which is given by the flux of particles to the firing threshold Vth (6)
The instantaneous population rate R(t) is defined here as the number of spikes emitted by the whole population in an (infinitely) small time bin around time t, normalized by the duration of the bin. Importantly, if the uncorrelated noise has a positive amplitude, only the diffusion component of the flux (see methods) is contributing to the firing rate, since in this case the density of particles at the threshold is zero. The preceding formula, together with the Fokker-Planck equation, represents a complete characterization of the population response in its dependence on the parameters of the input. In a stationary situation when μ and σ2 are constants, the density function adjusts its shape to the values of these parameters. The output rate, as stated in the preceding text, is therefore a function of both parameters (Roy and Smith 1969; Tuckwell 1988).
In the general situation, when the signals depend on time, the density function P(V; t) evolves according to the Fokker-Planck partial differential equation (see methods), thereby exhibiting a low-pass filtered response to changes in the signals μ(t) and σ2(t) with a filtering parametrized by the membrane time constant of the neurons in a population (Brunel and Hakim 1999). This means that the output rate R(t) given by Eq. 6 depends not only on the current values of the input parameters but also on their previous values, i.e., the instantaneous population rate does not faithfully reflect the instantaneous signals. However, if the modulations of the input signals around some baseline values are much faster than the membrane time constant, the shape of P(V; t) will be almost stationary, due to the above-mentioned filtering property of the Fokker-Planck equation.
To verify that the density function is indeed stable to very fast modulations of the input signals, we performed numerical simulations of the activity of a large population of integrate- and-fire neurons receiving noisy input currents with an instantaneous mean value oscillating in time with increasing frequency. The time evolution of the density function was then computed from the results of the simulations and its stability was assessed. In Fig. 1 A and B, we show a sample of probability density functions across the population computed over subsequent 1-ms bins for two different values of the frequency. Obviously, the variations in the density function are smaller for higher frequency. To quantify this result, we plot on Fig. 1C the SD of the density function averaged over all positive values of the voltage as a function of the frequency of the signal. For very high frequency the SD reduces to a low residual value that is explained by the finite size of the neuronal population. Qualitatively similar results were obtained for oscillating instantaneous variance in the input current (results not shown).
This result, together with Eq. 6, implies that the output population rate R(t) will be proportional to the quickly changing instantaneous values of the population variance of the uncorrelated component of the inputs, σ2(t), and ignores the modulation in the mean. In other words, the population can faithfully transmit rapid signals, if they are encoded in the amplitude of the uncorrelated “noise.”
Intuitively, this effect can be understood most easily in a population of neurons simultaneously receiving excitatory and inhibitory inputs. An excess of excitation at a given time will drive the neurons toward threshold. Some neurons will be caused to fire and immediately afterward will be synchronously refractory. To achieve a rapid response the excitatory pulse would have to be huge, which in turn would temporarily saturate the population activity. Therefore the analog signal cannot be transmitted this way because the population activity will not be able to remain constantly at the intermediate values dictated by the signal. If, in contrast, the inhibition is increased simultaneously together with the excitation, this implies an increased population variance of the inputs. In this case, only a fraction of the neurons will receive strong excitatory pulses and thereby respond rapidly, whereas a large proportion will be less affected or driven away from threshold. In this way, the balance of excitation and inhibition randomly selects changing subsets of neurons that are driven across threshold. This mechanism avoids population saturation and the number of firing neurons at each moment in time (population rate) faithfully reflects the graded signal encoded in the population variance of the input.
Because this theoretical prediction could be of fundamental importance for the issue of the neural code in neocortex, we undertook a series of experiments to test its validity (methods). Experimental testing is clearly warranted due to the many simplifications in the mathematical model, most notably the neglect of the kinetics of ionic channels.
We prepared an ensemble of virtually white noise current traces (see caption Fig. 2) characterized by particular μ(t) and σ2(t) and injected them into neocortical neurons while monitoring their spiking response. Ideally we would then proceed to estimate the instantaneous firing rate of a large neuronal population with every neuron receiving a different current trace from the prepared ensemble. Because intracellular recordings from so many neurons would be extremely time consuming, we instead repeatedly injected into single neurons, but every time choosing a different current trace from the same ensemble. The “population” activity can then be estimated by computing the average number of spikes emitted by the neuron in subsequent time bins of 1 ms, i.e., a so called peri-stimulus-time-histogram (PSTH). Substituting the PSTH for the population firing rate can be justified only if every neuron exhibits the same time course of the PSTH response (up to a possible scaling factor) because, in this case, subsequent responses obtained in single neurons would accurately represent different neurons in a population.
In our first series of experiments, we compared the response to two simple forms of input signals, for which either μ(t) or σ2(t) increases abruptly at a particular time (Fig. 2, A and B). Equation 6 predicts that there should be a gradual response to the jump in μ and an instantaneous initial response to a change in σ2 because the latter enters as a multiplying factor in the expression for the response. The amplitudes of the signals were calibrated such that the steady-state level of a neuron's firing was identical in both cases because this implies that the average firing rates are almost identical in both cases too. However, the time profile of the PSTH in response to abrupt increase in μ and σ2 was very different: whereas in the first case there was a gradual change in the PSTH, in the second case we observed an instantaneous initial response with subsequent decay to the new stationary level. These results are in agreement with the theoretical prediction (Eq. 6).
Similar results were obtained in all 18 of a wide variety of neocortical neurons, which included pyramidal neurons and different types of interneurons. We emphasize that in both cases, the output rate continued to change after the transition, when the input signal (the values of either μ or σ2) stayed constant at a different level. This means that the instantaneous population rate did not follow the step in the signal amplitude when this was modulating the mean current to the population. On the other hand, the rate response faithfully followed the step in the population variance σ2 but was then also slowly changed, thereby decreasing to the steady-state discharge level. Note that in both cases, the responses did NOT faithfully reflect the overall shape of the single step and showed temporal modulations, which are also a prediction of the theory presented in the preceding text.
We then proceeded to test our main theoretical prediction by studying the population response to signals that undergo rapid ongoing changes in time. To this end, we injected the same type of current forms as in the first experiment, whereas now both μ(t) and σ2(t) are rapidly fluctuating all the time. In other words, the two signals were present simultaneously in the input. To separate the effectiveness of these two signals, we constructed μ(t) and σ2(t) by randomly assigning new values to each of them independently at every millisecond drawn from uniform distributions (see Fig. 3, A and B). The ranges for both kinds of input signals were calibrated for each neuron individually, such that the corresponding output ranges of the firing rates were of the same size (i.e., the calibration makes sure that the stationary firing rate for constant μ,σ2 is identical in both cases, μ = μmin, σ2 = σ2max and μ = μmax, σ2 = σ2min).
The spiking PSTH response to these currents was obtained (Fig. 3C). Strikingly, the instantaneous population rate reliably followed the signal carried by the variance of the input currents and there was no observable correlation with the signal contained in the mean (Fig. 3F). To quantitatively test the prediction contained in Eq. 6 about the dependency of the response on the components of the input, we plotted the instantaneous values of the PSTH response versus the instantaneous variance of the currents (Fig. 3E). In agreement with the prediction, the points in this graph scatter around the straight line passing through the origin. The same graph, but with the signal contained in the average current, μ does not result in any significant dependency (Fig. 3D), again following the theoretical prediction. Very similar results were obtained in all four pyramidal neurons that were tested. We therefore conclude that substituting the PSTH responses obtained in single neurons for instantaneous population rate of large neuronal populations is justified and that the instantaneous population rate reflects the signal carried by the rapidly changing instantaneous variance of the input.
The preceding experiments confirm that signaling with variance applies to neurons in the cortex and does not rely on the particular choice of the model neuron used in the theoretical analysis. However, because in these experiments the injected currents were artificial and in particular did not correctly reflect the temporal correlations of real input currents, we in addition obtained realistic synaptic currents from whole cell voltageclamp recordings in cortical slices with different levels of excitation (methods). With these currents, we performed a control experiment by injecting them into a neuron and recording the discharge responses for both, step changes in mean current and step changes of the variance.
As in the first experiments, the discharge response increased gradually when we injected recorded currents containing abrupt changes in the mean current (Fig. 4A), whereas the change in discharge was much faster when the variance of the current increased abruptly (Fig. 4B). These results demonstrate that the difference between signaling by variance and signaling by mean persists with real synaptic noise currents (Brunel et al. 2001). In particular, signaling by variance does not require white noise currents but is possible with the synaptic currents generated in neocortex.
Finally, in this study, we did not consider the effects of changing neuronal conductances, which would result from the barrage of excitatory and inhibitory synaptic inputs. Experimentally, this issue could be addressed by employing the dynamic clamp technique, which remains a challenge for future studies. In a series of simulations (results not shown) with populations of integrate-and-fire neurons receiving inputs in the form of noisy conductance changes, we found qualitatively the same effects as presented in this paper.
Cortical neurons in vivo fire irregularly (Softky and Koch 1993) and in an apparently irreproducible manner (Schiller et al. 1976; Vogels et al. 1989). As a possible explanation, this phenomenon has been suggested to originate from a balance of excitatory and inhibitory synaptic inputs (Gerstein and Mandelbrot 1964; Shadlen and Newsome 1995), in which case the mean becomes small and signaling by variance is particularly relevant. The resulting variability, however, if considered noise, clearly impairs the precision of rate estimates. This raises the question whether there is a functional reason that may justify this loss. To this puzzle, our study contributes a novel, independent argument demonstrating that in case of neocortical neurons, signaling by variance, in contrast to the mean, allows for rapid population rate codes. This result does not rely on synaptic connectivity within an ensemble but reflects the basic biophysical properties of populations of neocortical neurons. It could be generated either by the dynamics of a network in which the neurons are embedded or by the activity of presynaptic ensembles. Indeed, network models have been developed the activity of whic can rapidly follow the input (Abbott and vanVreeswijk 1993; Battaglia and Treves 1998; Tsodyks and Sejnowski 1995; vanVreeswijk and Sompolinsky 1996). In conclusion, our work suggests that signaling by variance may be important as being a simple mechanism for desynchronizing neuronal populations, something that is necessary for the realization of rapid population rate codes. Future experiments are required to clarify whether this is indeed the case.
M. Tsodyks and K. Pawelzik acknowledge the very productive atmosphere during their stay in the Institute for Theoretical Physics at the University of California, Santa Barbara. H. Markram is the Helen and Stanley Diller Professor in Neuroscience.
Present addresses: G. Silberberg and H. Markram, Brain and Mind Institute, EPFL, Lausanne 1015, Switzerland.
This study was supported by the Office of Naval Research, McDonnel-Pew program, Human Frontier Science Program, Deutsche Forschungsgemein-schaft (via SFB 517), and the Israel Academy of Science. In part it was supported by the National Science Foundation Grant PHY99-07947.
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↵* G. Silberberg and M. Bethge contributed equally to this work.
- Copyright © 2004 by the American Physiological Society