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On the relationship between synaptic input and spike output jitter in individual neurons
What is the relationship between the temporal jitter in the arrival times of individual synaptic inputs to a neuron and the resultant jitter in its output spike? We report that the rise time of firing rates of cells in striate and extrastriate visual cortex in the macaque monkey remain equally sharp at different stages of processing. Furthermore, as observed by others, multiunit recordings from single units in the primate frontal lobe reveal a strong peak in their crosscorrelation in the 10–150 msec range with very small temporal jitter (on the order of 1 msec). We explain these results using numerical models to study the relationship between the temporal jitter in excitatory and inhibitory synaptic input and the variability in the spike output timing in integrateandfire units and in a biophysically and anatomically detailed model of a cortical pyramidal cell. We conclude that under physiological circumstances, the standard deviation in the output jitter is linearly related to the standard deviation in the input jitter, with a constant of less than one. Thus, the timing jitter in successive layers of such neurons will converge to a small value dictated by the jitter in axonal propagation times.
When a cortical neuron is presented with the appropriate stimulus it rapidly and reliably increases its probability of firing. Indeed, individual neurons from cortical area MT in the behaving monkey respond to a dynamic random dot motion stimulus with a highly reproducible temporal modulation of their firing rate, precise to a few milliseconds (1). Furthermore, the time that it takes for an MT cell to significantly modulate its firing rate—as determined by averaging over many presentations of an identical stimulus—is almost always less then 10 msec (2). This occurs in neurons that are at least six synapses removed from the periphery. This precision is surprising because the propagation of an input through a multilayer network with continuous mean rates causes rise times to become increasingly shallow (3).
Yet, this does not appear to be the case in cortex. The rise times of signals—as defined via the poststimulus time histogram—in extrastriate areas, such as V4 or MT, can be as rapid as those in primary visual cortex (V1). As the sensory triggered “wave” of activity propagates through many layers of neurons, such a signal does not appear to become appreciably rounded off, but is instead only delayed between consecutive stages. This can be assessed directly by comparing the rise time of neurons in a single cortical area to the same stimulus. The cell in Fig. 1A responded within about 30 msec to the onset of a black and white grating, whereas the neuron in Fig. 1B responded only after 100 msec. Yet the rise time in both neurons is equally fast, even though the signal must have traversed more processing stages in the second case.
Fig. 1C gives further testimony to this. The solid and broken lines show the average rise times for the visual responses of neurons in V1 and a higher order extrastriate cortex area, V4 (see Fig. 1 legend, and ref. 4). The average response latency (measured as the time when the response reaches onehalf its peak value) was 64 msec for the V1 neurons and 90 msec for the V4 neurons. This difference in latency presumably corresponds to many synaptic levels of processing interposed between areas V1 and V4. Yet, the average rise time of the firing rate in both sets of neurons is indistinguishable. How can neurons in later stage of processing retain a sharp onset when the signals are processed by neurons with passive membrane time constants of tens of milliseconds?
A similar problem arises in Abeles’s synfire model of cortical processing (5, 6). He proposes that sensory events trigger wavelike patterns that crisscross cerebral cortex. Underlying this model are small and coupled groups of neurons receiving synchronized input that excite all members of the group. This tightly coupled cell assembly excites a second group of neurons that triggers synchronized spiking activity in a third group of neurons, in a socalled synfire chain. Neurons might belong to more than one group, depending on their connectivity (7). Abeles formulated this model in response to his puzzling finding that the crosscorrelation function of firing of cortical cells frequently shows a narrow peak far removed from zero (up to 100 msec), raising the question of how such a wave of synchronized firing activity can propagate through many layers of intervening neurons, where the neurons possess passive time constants on the order of 10–20 msec (8, 9).
We seek to address this issue at the single cell level in the following manner. Suppose an instantaneous sensory event in the world triggers a volley of activity in n synaptic inputs. Let us assume that the arrival time of this input is centered around t = 0 and that its standard deviation in time, henceforth called input jitter, is σ_{in}. If we further suppose that these n synapses excite a pulse generating neuron, we can compute the standard deviation in time, termed the output jitter σ_{out}, of the spike triggered in response to this input. If σ_{out} > σ_{in}, then it is clear that each consecutive layer of spiking neurons will introduce more and more temporal jitter, compromising the ability of higher level neurons to sharply respond to this sensory input and rendering synfire assemblies difficult. Inversely, if σ_{out} < σ_{in} (as we will find) then, depending on other sources of temporal jitter, the temporal variability in spike times response to an abrupt input converges toward a fixed point. In the following, we investigate analytically the case of a perfect integrateandfire (I&F) model. Computer simulations using two different single cell models complement our results and confirm that under physiological circumstances, σ_{out} ≪ σ_{in}.
The I&F Model
We begin with the simplest model of a spiking cell (10). This I&F model consists of a capacitance, C, and a voltage threshold, V_{th} (Fig. 2A). Each synaptic input dumps positive or negative charge onto the capacitance, de or hyperpolarizing the membrane. Once V_{th} is reached, an output spike is generated and the membrane potential is reset to zero. The I&F unit is assumed to receive input from n excitatory synaptic inputs of equal weight, a_{E}, each of which can be activated independently of each other.
First we assume that n_{th} = n, where n_{th} = V_{th}/a_{E} is the number of inputs needed to reach threshold. Under these conditions, the time, t_{out}, for which the I&F unit generates an output pulse is the time that the nth input arrives, given by max(t_{1}, t_{2}, … t_{n}). The probability distribution of spike times for the I&F unit is where P{t_{1} < t} is the distribution for all of the synaptic inputs (the last step holds because all synaptic inputs are assumed to be independent of each other). The probability density of spike times, p_{out}(t), is the derivative of this distribution: Using Eq. 1, we can transform this into Here p_{input}(t) is the probability density of the synaptic input.
Let us assume that p_{input}(t) is a normal density with a temporal input jitter σ_{in}, as in Fig. 3 (for normal density). This allows us to make several observations. For n = 1, it trivially follows that the density of spike times is identical with the synaptic input density. For larger ns, the resulting density becomes narrower. Indeed, it can be shown that the output jitter, i.e., the standard deviation of p_{out}(t), for n + 1 inputs, is less than the jitter for n inputs. Thus, this rudimentary model achieves the goal of transmitting signals without temporal smear, although activity is significantly delayed. Indeed, jitter would be systematically reduced at each successive level in a multilayer network of such neurons because, as is evident upon inspection of Fig. 3, σ_{out} < σ_{in} (in the case of normally distributed synaptic input).
For an arbitrary density, p_{input}(t), p_{out}(t) scales with the input jitter. When changing the input jitter from σ_{in} to one can rescale time replacing t with tσ_{in}/. This leads to the same density function times a scaling factor σ_{in}/ (to account for the broader spread of the probability density, which has to sum to one). In other words, increasing the input jitter by α increases the output jitter by α.
Now let us deal with the more general case of n > n_{th} and where p_{input}(t) is the uniform density on the interval [0, 1]. (p_{input}(t) = 1 for t ∈ [0, 1]; p_{input}(t) = 0 otherwise.) This assures us that at t = 1 the voltage will be exactly na_{E} (in the absence of a threshold).
The probability that the voltage at time t has attained the value n_{th}a_{E} is given by the beta density (11–13): The standard deviation of this distribution is In the case of the uniform probability density assumed here, σ_{in} = 1/2 . This allows us to rescale Eq. 5, and we can write for the output jitter To arrive at some intuitive feeling for this result, let us assume that the number of inputs n is far above the number needed to reach threshold n_{th}. Making the approximations that n + 3 ≈ n + 2 ≈ n, that n + 1 − n_{th} ≈ n, and n_{th} + 1 ≈ n_{th} we arrive at (with n_{th} = V_{th}/a_{E}) Or, the output jitter is inversely proportional to the number of excitatory synaptic inputs. This makes sense, since the time it takes to reach threshold will be proportional to the drift that is dictated by n. The jitter in this time is inversely related to the drift. Of course, our derivation only holds under conditions when the membrane leak can be neglected.
The Leaky I&F Model
Although this I&F model has desirable properties, it is not physiologically plausible and leaves many questions open. What happens if p_{input}(t) is not uniform? Furthermore, neurons are not perfect integrators. Rather, they have a passive membrane time constant that is on the order of 10–20 msec in pyramidal cells in vivo (for review, see ref. 8). If only a small number of synapses are active, the membrane potential can decay significantly between inputs. These concerns are addressed using the leaky or forgetful I&F model. It is obtained from the I&F model by addition of a resistance R, endowing the unit with a passive time constant τ = RC (Fig. 2B). Although quite simple, leaky I&F units have been successfully used to mimic many aspects of the temporal dynamics of cortical pyramidal cells (5, 14–17).
In the subthreshold region, the voltage is governed by where the current I(t) is given by the superposition of random excitatory inputs. The input comes in the form of n short, depolarizing current pulses of amplitude a_{E}, and m hyperpolarizing current pulses of amplitude a_{I}. Each input has an associated probability density function p_{input}(t), with a mean of t_{mean} and input jitter σ_{in}.
Each excitatory input pulse depolarizes the membrane, pushing V_{m} closer to the threshold V_{th}, whereas the inhibitory input displaces V_{m} away from V_{th}. Unlike the nonleaky I&F model, V_{m} decays exponentially to zero in the absence of input. What can we say about the relationship between σ_{in} and σ_{out}, that is the standard deviation in time in which V_{m} reaches threshold? We have not managed to place firm boundaries on the jitter for a leaky I&F unit, forcing us to rely on computer simulations.
We numerically solve Eq. 8 via Matlab. We assume V_{th} = 16 mV, τ = 10 msec and a_{E} = a_{I} = 0.23 mV, such that n_{th} = 70 simultaneous excitatory synaptic inputs that are required for the unit to fire. Fig. 4A illustrates one sample path for V_{m}(t), for excitatory, rectangular current pulses (using α functions did not lead to any significant difference) of 1 msec width, 0.23 nA amplitude, and with a Gaussian input probability density p_{input}. For n = 250 excitatory inputs in the absence of any inhibition, approximation gives σ_{out} ≈ 0.116 σ_{in} (Fig. 5A). The simulated curve is always below the estimate, confirming the intuition that for large number of synaptic inputs the decay can be neglected.
What happens in the presence of inhibition? On general grounds, one expects the overall jitter to increase, due to the introduction of additional degrees of freedom. Indeed, if 62 inhibitory synapses are added, the output jitter is larger than in the noinhibition case, yet still substantially below the identity function (Fig. 5B). We conclude that a leaky I&F unit can reduce the jitter in its input in the presence of massive synaptic input.
A Cortical Pyramidal Cell
It could be argued that an I&F unit does not represent a reasonable model of a cortical cell. Although substantial evidence has accumulated in favor of neurons having an abrupt voltage threshold (for a detailed discussion see refs. 17 and 18), voltagedependent channels inactivate and can change the effective threshold depending on the history of V_{m}. We address this point by carrying out simulations in a detailed model of a regularfiring, neocortical pyramidal cell. The arcanae of these simulations are described in detail in ref. 18, and we will only summarize the salient points here.
The morphology of this cell (Fig. 2C) was derived from a layer 5 pyramidal cell in primary visual cortex filled with horseradish peroxidase during in vivo experiments in the anesthetized, adult cat (9). Fig. 2D illustrates part of the compartmental model that was simulated using the single cell simulator NEURON (19, 20). In the complete model the cell has 163 branches, consisting from 1 to 10 compartments, with the most distal tip 1387 μm from the soma. The soma comprises only 2% of the total membrane area. The basal dendrites, including the apical obliques that are located in layer 5, account for approximately 60% of the membrane area, with the apical trunk and apical tuft taking the remainder.
We assume here that the dendritic tree is passive, with R_{m} = 100 kΩcm^{2}, R_{i} = 200 Ωcm, C_{m} = 1 μF/cm^{2}, and E_{leak} = −66 mV. We mimicked the synaptic background activity by incorporating the steadystate conductance (and driving potential) effect of 4000 excitatory inputs of the voltageindependent AMPA (αamino3hydroxy5methyl4isoxazoleproprionic acid) type 500 GABA_{A} and 500 GABA_{B} inhibitory synapses, all activated independently of each other on the basis of a Poisson process with a rate of 0.5 Hz (21). Synapses are not distributed uniformly along the membrane. Inhibition is more prevalent on and near the soma, whereas excitation is more distal (22, 23). This distribution was modeled by making the surface density of synapses a function of distance from the soma. This lowers the effective R_{m} to about 11 kΩcm^{2} at the soma (where the density of inhibitory synapses is large) and to 50 kΩcm^{2} for distal locations.
The somatic membrane includes eight voltagedependent conductances, two sodium conductances (fast and persistent), a delayed rectified potassium conductance, a calcium and a calciumdependent potassium current, an A and an M type of potassium conductance, and an anomalous or inward rectified conductance (for details, see ref. 18). The final somatic input resistance was 12.5 MΩ, the time constant 9 msec, and the somatic resting potential −65.6 mV. For this cell, the voltage threshold to initiate an action potential V_{th} ≈ −49 mV.
To mimic the synaptic input, we randomly generated 500 locations for excitatory synapses and 125 locations for inhibitory synapses, taking into account their anatomical distributions (see above). For each block of trials onehalf (n = 250 and m = 62, respectively) of these synapses were randomly triggered. Excitation is of the fast, voltageindependent AMPA type and is modeled by an α function with t_{peak} = 0.5 msec, g_{peak}=1.5 nS, and E_{syn} = 0 mV. Inhibitory synaptic input is of the GABA_{A} type, with t_{peak} = 10 msec, g_{peak} = 1 nS, and E_{syn} = −70 mV. We simulated input jitter times, σ_{in} of 0.5–3.5 msec. For larger values of σ_{in}, frequently trials would not lead to the generation of a spike (at our level of excitation). Fig. 4B shows sample traces of V_{m} at the soma, while Fig. 5 plots the resultant jitter. It is obvious that the variance of the output jitter of this pyramidal cell is approximately linear in the input jitter and always falls below the identity line.
Spiking Jitter Following a Step Increase in Firing Rate
So far we have treated the case of a wave of elevated firing probability arriving at a neuron and triggering a spike. How does this situation relate to a step increase in activity (similar to that generated by an appropriate visual stimuli when recording from a cortical cell)?
Let us assume that the input to a nonleaky I&F unit abruptly switches at t = 0 from a low spontaneous value to a much higher rate. Since the rise time in the average firing rate (Fig. 1C) occurs within a fraction of a time constant, we will neglect the membrane decay during this portion of the input. We will approximate the voltage trace by Wiener process with drift. We will express the new, elevated value for the drift as μ_{w} = a_{E}λ_{E} − a_{I}λ_{I} and the variance parameter σ_{w}^{2} =a_{E}^{2} λ_{E} + a_{I}^{2}λ_{I}, where λ_{E} and λ_{I} denote the new excitatory and inhibitory input rate. Assuming that the nonleaky I&F unit had the voltage V_{0} at this time, it is straightforward to compute (24–26) that the probability density of the first spike following the input is given by the inverse Gaussian density Assuming that V_{0} is uniformly distributed within the interval [0, V_{th}], we can average over all possible values of V_{0}: From this average we can compute the mean and standard deviation of the first spike to occur following the step increase in input. The mean latency is and the jitter around this latency is Assuming that the nonleaky I&F unit discharges at a rate of 50 Hz following the input increase, by back substitution (and assuming, as before, a_{E} = a_{I} = 0.23 mV, λ_{E} = 4λ_{I}) we have λ_{E} = 4.63 msec^{−1} and a latency of first spike of 10 ± 1.55 msec. Thus, we have transformed a problem of jitter following a step increase in the input to the problem treated above, e.g., a probability density of the time of first spike p_{spike}(t) in output. Instead of a Gaussian density as we have assumed in our simulations, the spikes would be distributed according to Eq. 9. Yet for our parameter values, the difference between these two distributions is negligible. Moreover, the value of standard deviation lies within the range of assumed jitter levels.
Discussion
The aim of this study was to explain some puzzling observations regarding the preservation of highly accurate spike timing in cortical networks. Naive intuition suggests that in a layered network of meanrate “units” (such as those at the heart of most neural networks), precise temporal information will become smeared out. Experimentally, as we have shown in Fig. 1, this is not true for neurons in different cortical areas in the macaque monkey. We can show analytically that for an uniformly distributed synaptic input to a nonleaky I&F model, a linear relationship exists between σ_{out} and σ_{in}, with the constant of proportionality depending inversely on n. Using numerical methods to deal with temporal dispersion and nonlinear processing existing in biophysical realistic models of individual neurons, we can show that under physiological conditions σ_{out} ≈ α σ_{in} with α <1. α depends on various parameters, in particular on n_{th} and n, and is larger if both excitatory as well as inhibitory synaptic input is included. Lastly, we show that the situation of an abrupt increase of the excitatory and inhibitory synaptic input to a nonleaky I&F unit gives rise to a probability distribution of spikes with a certain jitter that now propagates to the following layer of processing.
For a cascade of such neurons the output jitter will converge to zero. However, in real networks this is unlikely to be the case since we have neglected several additional sources of timing variability. The two most dominant sources are likely to be (i) the inhomogeneous spike propagation times between consecutive layers of neurons due to variations in the diameter and length of the associated axons and axonal termination (wiring jitter) and (ii) jitter in the delay between presynaptic spike and the opening of the postsynaptic synaptic channels (synaptic jitter). Thus, even if all neurons in one layer spike at precisely the same time, the synaptic induced conductance change in their postsynaptic target cells will not be perfectly synchronized. Manor and others (27) simulated spike propagation in a highly branched layer 5 pyramidal cell axonal tree from somatosensory cortex and found that the jitter in the arrival times at the different synaptic boutons due to purely geometrical factors was on the order of ±0.5 msec.Thus, the output jitter for many layers of such a network will converge to a fixpoint that is different from zero but bounded by the “jitter” in the anatomical connections and in synaptic jitter, which would presumably be in the order of several milliseconds.
The effect of strong dendritic nonlinearities is unclear. However, if such voltagedependent membrane components would act to more rapidly depolarize the membrane, such as dendritic spikes, the output jitter should be smaller than estimated here. σ_{out} can be maximized by combining excitatory inputs with the largest amount of inhibition compatible with the cell still firing (since inhibition increases the variance of the membrane potential). This is precisely the situation we simulated in Fig. 5B: any more inhibition and the cell would have failed to generate spikes in a growing fraction of all trials.
Several groups have previously investigated the preservation of spike timing using the dynamic equations describing network activity (in particular refs. 28 and 29). Not taking into account any processing at the single cell level and assuming a meanfield approach, both studies conclude that the jitter between two consecutive stages decreases as 1/; that is, for relatively large number of afferent excitatory synapses, σ_{out} will remain small. We investigate the specific case of a nonleaky I&F unit, and conclude that the dependency on n could be even stronger (see Eq. 7) and show that this approximation is also valid for a biophysical quite detailed model of a layer 5 pyramidal cell. Thus, our simulations are in qualitative agreement with those of Abeles (29) and Herrmann (28).
We conclude that layers of pulsegenerating neurons can preserve the temporal jitter of spike times and that this jitter will converge to a small fixpoint. This property of neurons provides one of the biophysical substrates necessary for exploiting the detailed timing information inherent in spike trains as is frequently asserted (30–32).
Acknowledgments
We thank G. Holt for his invaluable advice at all stages of this project, in particular in his formulation of Eqs. 1, 2, 3 and his help with NEURON. We thank Carrie J. McAdams for providing some of the neurophysiological data in Fig. 1 and for assisting in their analysis. We also thank J. Sýkora and M. Stemmler for helpful comments. This research was supported by the National Institute of Mental Health through the Center for Cell and Molecular Signaling, by a National Institute of Mental Health grant to C.K., and by National Institutes of Health Grant EY05911 to J.M.
Note
. We recently learned that Diesmann et al. (33) have independently studied the relationship between input and output jitter in a different type of neuron model and have arrived at qualitatively similar conclusions.
Footnotes

↵ To whom reprint requests should be addressed. email: koch{at}klab.caltech.edu.

David Mumford, Brown University, Providence, RI

Abbreviations: I&F, integrateandfire; AMPA, αamino3hydroxy5methyl4isoxazoleproprionic acid.
 Received May 21, 1996.
 Accepted November 18, 1996.
 Copyright © 1997, The National Academy of Sciences of the USA
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