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. 1998 May 15;18(10):3870-96.
doi: 10.1523/JNEUROSCI.18-10-03870.1998.

The variable discharge of cortical neurons: implications for connectivity, computation, and information coding

M N Shadlen  1 , W T Newsome
Affiliations

The variable discharge of cortical neurons: implications for connectivity, computation, and information coding

M N Shadlen et al. J Neurosci. .

Abstract

Cortical neurons exhibit tremendous variability in the number and temporal distribution of spikes in their discharge patterns. Furthermore, this variability appears to be conserved over large regions of the cerebral cortex, suggesting that it is neither reduced nor expanded from stage to stage within a processing pathway. To investigate the principles underlying such statistical homogeneity, we have analyzed a model of synaptic integration incorporating a highly simplified integrate and fire mechanism with decay. We analyzed a "high-input regime" in which neurons receive hundreds of excitatory synaptic inputs during each interspike interval. To produce a graded response in this regime, the neuron must balance excitation with inhibition. We find that a simple integrate and fire mechanism with balanced excitation and inhibition produces a highly variable interspike interval, consistent with experimental data. Detailed information about the temporal pattern of synaptic inputs cannot be recovered from the pattern of output spikes, and we infer that cortical neurons are unlikely to transmit information in the temporal pattern of spike discharge. Rather, we suggest that quantities are represented as rate codes in ensembles of 50-100 neurons. These column-like ensembles tolerate large fractions of common synaptic input and yet covary only weakly in their spike discharge. We find that an ensemble of 100 neurons provides a reliable estimate of rate in just one interspike interval (10-50 msec). Finally, we derived an expression for the variance of the neural spike count that leads to a stable propagation of signal and noise in networks of neurons-that is, conditions that do not impose an accumulation or diminution of noise. The solution implies that single neurons perform simple algebra resembling averaging, and that more sophisticated computations arise by virtue of the anatomical convergence of novel combinations of inputs to the cortical column from external sources.

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Figures

Fig. 1.
Fig. 1.
Response variability of a neuron recorded from area MT of an alert monkey. A, Raster and peristimulus time histogram (PSTH) depicting response for 210 presentations of an identical random dot motion stimulus. The motion stimulus was shown for 2 sec. Raster points represent the occurrence of action potentials. The PSTH plots the spike rate, averaged in 2 msec bins, as a function of time from the onset of the visual stimulus. The response modulates between 15 and 220 impulses/sec. Vertical lines delineate a period in which spike rate was fairly constant. The gray region shows 50 trials from this epoch, which were used to construct B and C. B, Magnified view of the shaded region of the raster in A. The spike rate, computed in 5 msec bins, is fairly constant. Notice that the magnified raster reveals substantial variability in the timing of individual spikes. C, Frequency histogram depicting the spike intervals in B. The solid line is the best fitting exponential probability density function. D, Variance of the spike count is plotted against the mean number of spikes obtained from randomly chosen rectangular regions of the raster in A. Eachpoint represents the mean and variance of the spikes counted from 50 to 200 adjacent trials in an epoch from 100 to 500 msec long. The shaded region of A would be one such example. The best fitting power law is shown by the solid curve. Thedashed line is the expected relationship for a Poisson point process.
Fig. 2.
Fig. 2.
Three counting models for synaptic integration in the high-input regime. The diagrams (B, D, F) depict three strategies that would permit a neuron to count many input spikes and yet produce a reasonable spike output. For each of the strategies, model parameters were adjusted to produce an output spike count that is the same, on average, as any one input. The membrane state is represented by a particle that moves between a lower barrier and spike threshold (top bar). The height of the particle reflects the input count. Each EPSP drives the particle toward spike threshold, but the height decays to the ground state with time constant, τ (insets). When the particle reaches the top barrier, an action potential occurs, and the process begins again with the count reset to 0. A, Excitatory input to the model neurons. The 300 input spike trains are depicted as rows of a raster. Each input is modeled as a Poisson point process with a mean rate of 50 spikes/sec. The simulated epoch is 100 msec. C, E, G, Model response. The particle height is interpreted as a membrane voltage that is plotted as a function of time. These outputs were obtained using input spikes in A and the model illustrated in the middle column (B, D, F). B, C, Integrate-and-fire model with negligible inhibition and 20 msec time constant. To achieve an output of five spikes in the 100 msec interval, the spike threshold was set to 150 steps above the resting/reset state. Notice the regular interspike intervals in C. D, E, Coincidence detector. The spike threshold is only 16 steps above rest/reset, but the time constant must be 1 msec to achieve five spikes out. The coincidence detector fires if and only if there is sufficient synchronous excitation. F, G, Balanced excitation–inhibition. A second set of inputs, like the ones shown in A, provide inhibitory input. Each inhibitory event moves the particle toward thelower barrier. The spike threshold is 15 steps above rest/reset, and the time constant is 20 msec. The particle follows a random walk, constrained by the lower barrier and the absorption state at spike threshold. This model is most consistent with known properties of cortical neurons. A more realistic implementation is described in Appendix .
Fig. 3.
Fig. 3.
Conservation of response dynamic range. The spike rate of the model neuron is plotted as a function of the average input spike rate. A, Simulations with 300 excitatory inputs and 300 inhibitory inputs; parameters are the same as in Figure 2, F and G (barrier height, 15 steps; τ = 20 msec). The balanced excitation–inhibition model produces a response that is approximately the same as one of its many inputs. B, Simulations with 600 excitatory and inhibitory inputs. Open symbols and dashed curve show the response obtained using the same model parameters as in A. Solid symbolsand curve show the response when the barrier height is increased to 25 steps. These simulations suggest that a small hyperpolarization could be applied to enforce a unity gain input–output relationship when the number of active inputs is large.
Fig. 4.
Fig. 4.
Variability of the interspike interval. A, Frequency histogram of ISIs from one simulation using 300 inputs at 50 spikes/sec. Notice the substantial variability. The SD divided by the mean interval is known as the coefficient of variation of the interspike interval (CVISI). The value for this simulation is 0.9. The distribution is approximated by an exponential probability density (solid curve), which would predict CVISI = 1. B, Coefficient of variation of the interspike interval (CVISI) from 128 simulations using 300 and 600 inputs and a variety of spike rates. Each simulation generated 20 sec of spike discharge using parameters that led to a similar rate of discharge for input and output neurons (i.e., a common dynamic range). The average CVISI was 0.87.
Fig. 5.
Fig. 5.
Irregularity of the spike discharge is not merely a reflection of input spike irregularity. The graph compares the irregularity of the ISI produced by the balanced excitation–inhibition model with the irregularity of the intervals constituting the 300 excitatory and inhibitory input spike trains. The input spike trains were constructed by drawing intervals randomly from a gamma distribution. By varying the parameters of the gamma distribution, the input CVISI was adjusted from relatively regular to highly irregular (abscissa). Eachpoint represents the results of one simulation, using different parameters for the input interval distribution. Notice that the degree of input irregularity has only a weak effect on the distribution of output interspike intervals. Points abovethe main diagonal represent simulations in which the counting model produced a more irregular discharge than the input spike trains. Points below the main diagonal represent simulations in which the output is less irregular than the input spike trains. The dashed line is the least squares fit to the data. This line intersects the main diagonal at CVISI = 0.8. The best fitting line does not extrapolate to the origin, because the inputs are not necessarily synchronous.
Fig. 6.
Fig. 6.
Frequency histogram of the spike count variance-to-mean ratios obtained from the same simulations as in Figure 4B. For each of the simulations, the spikes were counted in 200 epochs of 100 msec duration. The variance in the number of spikes produced by the model in each of these epochs is proportional to the mean of the counts obtained for these epochs. Spike count variability is therefore conveniently summarized by the variance-to-mean ratio. The average ratio is 0.75 (arrow).
Fig. 7.
Fig. 7.
Redundancy necessitates shared connections. Three ensembles of neurons represent the quantities α, β, and γ. Each neuron that represents γ receives input from many neurons that represent β, and each neuron that represents β receives input from many neurons that represent α. A, There are no shared connections; each neuron receives a distinct set of inputs from its neighbor. The shaded neurons receive no common input, and the same can be said of any pair of neurons in the ensemble that represents β. The scheme would require an inordinately large number of neurons. B, Neurons share a fraction of their inputs. Theshaded neurons receive some of the same inputs from the ensemble that represents β. Likewise, any pair of neurons in the β ensemble receive some common input from the neurons that represent α. This architecture allows for redundancy without necessitating immense numbers of neurons. Neither the number of neurons nor the number of connections are drawn accurately. Simulations suggest that the pair ofshaded neurons might receive as much as 40% common input, and each needs about 100 inputs to compute with the quantity β.
Fig. 8.
Fig. 8.
Cross-correlation response histograms from a pair of simulated neurons. The correlograms represent the relative change in response from one neuron when a spike has occurred in the other neuron at a time lag indicated along the abscissa. The spike train for each neuron was simulated using the random walk counting model with 300 excitatory and 300 inhibitory inputs. Plots A–F differ in the amount of common input that is shared by the simulated pair. A small central peak in the correlogram is apparent when the pair of neurons share 20–50% of their inputs.
Fig. 9.
Fig. 9.
Effect of common input on response covariance. The correlation coefficient is plotted as a function of the fraction of shared excitatory and shared inhibitory input to a pair of model neurons. Each point was obtained from 20 sec of simulated spike discharge using a variety of model parameters (input spike rate, number of inputs, and barrier height). In each simulation, the output spike rate was approximately the same as the average of any one input (within a factor of ±0.25). The best fitting plane through the origin is shown. A substantial degree of shared input is required to achieve even modest correlation.
Fig. 10.
Fig. 10.
Weak correlation limits the fidelity of a neural code. The plot shows the variability in the number of spikes that arrive in an average ISI from a pool of input neurons modeled as Poisson point processes. Pool size is varied along theabscissa. In one ISI, the expected number of input spikes equals the number of neurons. Uncertainty is the SD of the input spike count divided by the mean. For one input neuron, the uncertainty is 100%. The diagonal gray line shows the expected relationship for independent Poisson inputs; uncertainty is reduced by the square root of the number of neurons. If the input neurons are weakly correlated, then uncertainty approaches an asymptote of r¯ (see Appendix ). For an average correlation of 0.2, the uncertainty from a pool of 100 neurons (arrow) is approximately the same as for five independent neurons or, equivalently, the count from one neuron in an epoch of five average ISIs.
Fig. 11.
Fig. 11.
Homogeneity of synchrony among input and output ensembles of neurons. A, Normalized cross-correlogram from a pair of neurons receiving 300 excitatory and inhibitory inputs, the typical pairwise cross-correlogram of which is shown in B. The pair share 40% common excitatory and inhibitory input. The CCG was computed from 80 1 sec epochs. The simulation produced a correlation coefficient of 0.29. B, The average correlogram for pairs of neurons serving as input to the pair of neurons, whose CCG is shown in A. The correlogram was obtained from 80 1 sec epochs using randomly selected pairs of input neurons. The mean correlation coefficient, r¯, was 0.3. Vertical scale reflects percent change in the odds of a spike, relation to background. C, Spike interval histogram for the output neurons. Synchrony among input neurons does not lead to detectable structure in the output spike trains (CVISI = 0.94).
Fig. 12.
Fig. 12.
Propagation of noise among ensembles of neurons. Each of the neurons depicted in this figure emits a spike train idealized as a Poisson point process. The three neurons at thetop compute the average spike rate among their inputs and emit the answer as a Poisson spike train. During a 100 msec epoch, the input neurons (bottom) discharge at 100 spikes/sec. Each input neuron is therefore expected to generate 10 spikes, but in any one epoch the count may vary. The bottom graph shows the Poisson distribution of spike counts from one input neuron. Themiddle row of graphs shows the probability density of the quantity that the output neuron has computed: the number of spikes per input neuron. The neuron obtains an estimate of the input spike count by calculating with one (A) or more neurons (B, C). The value is represented as an expectation, 〈λT〉, which can be thought of as a desired rate times the epoch duration. In any one epoch, the neuron emits a Poisson spike train at a rate, λ, resulting in N(T) spikes. The distribution of N(T) from many 100 msec repetitions is shown at the top. A, If the output neuron receives input from just one input neuron, the variance of the input count, var [〈λT〉], is 10. The output neuron emits an average of 10 spikes, but the variance is 20, reflecting the sum of input and (its own) Poisson variance. B, If there are many independent inputs, then the variance of the mean input count is negligible (delta function; middle plot). The output neuron emits an average of 10 spikes, and the variance is 10, the amount of variance expected for a Poisson spike train. C, If there are many weakly correlated input neurons, then the variance of the mean input count is approximately 10 times the average correlation coefficient among the input neurons. If  r¯ = 0.2, then the variance is 2. The output neuron emits an average of 10 spikes, but the total variance is 12. Notice that in A and C, the variance of the output spike count exceeds the variance of the inputs.
Fig. 13.
Fig. 13.
Stable solution for the variance-to-mean spike count in networks of neurons when the computation involves three independent quantities. A, Numerical approximation to the stable solution. We simulated the response by estimating three values using 100 weakly correlated neurons per value ( r¯ = 0.2). The three means were combined as a sum and difference (a + b − c) to generate the expected spike count from the output neuron, the spike train of which was modeled as a renewal process with  CVISI = 0.8. The variance-to-mean ratio for the input neurons is shown along theabscissa. The output variance-to-mean ratio is plotted along the ordinate. The least squares fit line crosses the main diagonal at a ratio of 1.6. This variance-to-mean ratio would be common to input and output neurons that compute similarly complex quantities. B, Effect of correlation on the stable variance-to-mean ratio. Each point represents a numerical approximation like the one obtained in A. The procedure was repeated for a range of average correlation coefficients. The solid line is the theoretical result (Eq. 18). Simulations led to larger estimates of variance, because the number of neurons is finite.
Fig. 14.
Fig. 14.
A single-compartment model using brief synaptic conductance changes instead of voltage steps. The model behaves like the simpler random walk counting model. The resting potential is −70 mV, τm = 20 msec, and spike threshold is −55 mV. A, Response to 0.75 nA current step. The trace lasts 250 msec. B, Fifteen synchronous excitatory synaptic inputs depolarize the neuron sufficiently to trigger a spike. C, Response to simulated current steps of varying size. The f–I plot depicts the steady-state firing rate as a function of injected current. D, Response to 300 excitatory and 300 inhibitory inputs. Each input neuron spikes at an average of 80 spikes/sec. The size of the inhibitory conductance was adjusted to produce an average spike rate similar to the input rate. Notice that the membrane voltage hovers near spike threshold. E, ISI distribution from the simulated response, which includes the trace in C. Solid line is an exponential fit to the distribution. F, Average synaptic conductance preceding a spike. The spike-triggered average excitatory conductance is shown by the solid curve (left ordinate scale); the inhibitory conductance is shown by the dashed curve(right ordinate scale). Although the effective membrane time constant is ∼0.5 msec, the synaptic activity affects the neuron for 5–10 msec preceding a spike.
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