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Dynamics of Population Rate Codes in Ensembles of Neocortical Neurons

¹ Department of Neurobiology, The Weizmann Institute of Science, Rehovot, 76100, Israel; ² Institute of Theoretical Physics, University of Bremen, D-28334 Bremen, Germany

Submitted 29 April 2003; accepted in final form 29 September 2003

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
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.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
General methods

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 denotes the membrane time constant, V_rest is the membrane resting potential, I_syn is the synaptic current, and R_in is the input resistance of a neuron. This equation is supplemented by the condition that each time the membrane potential hits the threshold potential, V_th, a spike is emitted and the membrane potential is instantaneously reset to a certain subthreshold level, V_reset.

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, _i(t) is Gaussian white noise with unit spectral density, and (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(V_th, t) 0 for all D = [(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.

Data collection

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 M), 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 ² 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.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
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 _i(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 _c 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 ² 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 V_th

(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 ² 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 ²(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).

View larger version (22K): [in this window] [in a new window]   FIG. 1. Simulations of the probability density function evolution in response to fast changes in input mean. A population of 20,000 integrate- and fire-neurons with = 10 ms, Vth = 1, Vrest = Vreset = 0 are simulated (see METHODS), with constant input variance = 0.3 and µ(t) oscillating between the values of 0.4 and 0.9, applied for 200 ms. A: overlayed probability density functions computed for subsequent 1-ms time bins, where µ(t) oscillates with a frequency of 50 Hz. B: same as A, but now µ(t) oscillates with a frequency of 1,000 Hz. C: the SD of the density function averaged over positive values of voltage plotted as a function of the frequency.

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 ²(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.

View larger version (34K): [in this window] [in a new window]   FIG. 2. Response of neocortical neurons to abrupt changes in input parameters µ and 2. In each case, 4,000 different virtually white noise current traces (sampling interval in all of the experiments is sample = 0.25 ms) were injected into a pyramidal neuron sequentially. A, bottom: an example of a single current trace injected. , the moment at which the amplitude of the mean current was increased. Middle: histogram of the "population" response with a time bin of 1 ms. At the transition point, the mean current was increased from 120 to 200 pA. Top: raster plot of spike trains for 10 randomly chosen trials. Solid red lines show the stationary levels of the response before and after the transition. B: same as in A, but the set of current time courses featured a change in the population variance from 22.5 to 90 (pA)2 sec. Histogram binning is 1 ms.

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 ²) 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 ² 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 ²(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 ²(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,² is identical in both cases, µ = µ_min, ² = ²_max and µ = µ_max, ² = ²_min).

View larger version (48K): [in this window] [in a new window]   FIG. 3. Fast signaling by current variance. The response of a pyramidal neuron to virtually white noise current injection (sample = 0.25 ms) in which 2 temporally uncorrelated signals are encoded. A: instantaneous values of mean current. B: instantaneous values of population variance. C: instantaneous response [peristimulus time histogram (PSTH)] in time bins of 1 ms. For every time bin, the PSTH was computed from the spiking responses of a neuron to repeated injections of different current traces as explained in the text. D and E: instantaneous response plotted vs. the instantaneous value of µ and 2, respectively, after compensating for a 1-ms delay. Every point on the graph represents a pair [µ(t), PSTH(t)] or [2(t), PSTH(t)] for consecutive time bins. D: correlation coefficient R = 0.15. E: the line is a linear regression. The sample correlation coefficient is = 0.79, and the sample correlation ratio is 0.65, which is very close to 2 and hence confirms the linear relationship (Stuart and Ord 1994). Overall, 5,600 current traces were injected (i.e., the observed correlation reflects the precision that would be achieved in a population at a time scale of 1 ms in a population of about 5,000 neurons.) F: the cross-correlation values between the population response and the signals carried by the mean and the variance of the input currents.

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.

View larger version (29K): [in this window] [in a new window]   FIG. 4. Response of neocortical neurons to currents obtained from voltage-clamp experiments. A, bottom: an example of a segment of a current obtained in a voltage-clamp recording from a slice with low activity level (see METHODS). At the transition time, a constant value was added to the current. Top: histogram of 4,000 responses to different current segments. B: same as in A but with a step change in the variance of the current. This was achieved by switching to current traces recorded at higher activity level.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
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.

	ACKNOWLEDGMENTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
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.

GRANTS

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.

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

^* G. Silberberg and M. Bethge contributed equally to this work.

Address for reprint requests and other correspondence: M. Bethge, Redwood Neuroscience Institute, 1010 El Camino Real #380, Menlo Park, CA 94025 (E-mail: mbethge{at}rni.org).

	REFERENCES

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Brunel N, Chance FS, Fourcaud N, and Abbott LF. Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86: 2186–2189, 2001.[CrossRef][ISI][Medline]

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Laplicque L. Recherches quantiatives sur l'excitation electrique des nerfs traitee comme une polarization. J Physiol Pathol Gen 9: 620–635, 1907.

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