Subsections

• Literature

7.5 Summary

In this chapter we have focused on the signal transmission properties of a population of identical and independent spiking neurons that are firing asynchronously. The state of asynchronous firing may be particularly interesting for information transmission, since the system can respond rapidly to changes in the input current. For slow noise such as noisy reset or correlated diffusive noise, the signal gain defined as the amplitude of the population activity divided by that of the input current shows no cut-off at high frequencies (Brunel et al., 2001; Gerstner, 2000b; Knight, 1972a). The effective cut-off frequency of the system is therefore given by the input current. For real neurons, changes in the input current are of course limited by the opening and closing times of synaptic channels. The conclusion is that the response time of the system is determined by the time-course of the synaptic currents (Treves, 1993) and not by the membrane time constant.

These insights may have important implications for modeling as well as for interpretations of experiments. It is often thought, that the response time of neurons is directly related to the membrane time constant $\tau_{m}^{}$. In neural network modeling, a description of the population activity by a differential equation of the form

$$\tau_{m}^{} {{\text{d}}A \over {\text{d}}t}$$ (7.80)

is common practice. The results presented in this chapter suggest that, in some cases, the population activity A can respond more rapidly than the input potential h. In particular, the response is faster than the time course of the membrane if either (i) noise is slow or (ii) the amplitude of the signal is larger than the noise amplitude.

We have used escape noise models to illustrate the differences between high and low levels of noise. For neurons with high noise, the kernel $\mathcal {L}$ describes a low-pass filter and the activity follows during a transient the input potential h, i.e.,

A(t) = g[h(t)] . (7.81)

The transient is therefore slow. On the other hand, if the noise level is low so that the interval distribution has a narrow peak, then the activity follows the derivative h' and the transient is fast.

If neurons with either `low' noise or `slow' noise are in a state of asynchronous firing, the population activity responds immediately to an abrupt change in the input without integration delay. The reason is that there are always some neurons close to threshold. This property suggests that a population of neurons may transmit information fast and reliably. Fast information processing is a characteristic feature of biological nervous systems as shown by reaction time experiments.

The theoretical results on signal transmission properties of neuronal populations can be related to single-neuron experiments. Instead of observing a population of identical and independent neurons in a single trial, the spiking activity of a single neuron is measured in repeated trials. The experimental PSTH response to an input current pulse exhibits qualitatively the same type of noise-dependence as predicted by the population theory. Furthermore, reverse correlation experiments can be related to the linear signal transmission function $\hat{{G}}$($\omega$) that can be calculated from population theories.

Literature

The signal transmission properties of single neurons and populations of neurons have been studied by numerous authors. We refer the interested reader to the early papers of Knight (1972a), Knight (1972b), Knox (1974), and Fetz and Gustafsson (1983) as well as the more recent discussions in Abeles (1991), Poliakov et al. (1997), Gerstner (2000b), and Brunel et al. (2001).