5. Noise in Spiking Neuron Models

In vivo recordings of neuronal activity are characterized by a high degree of irregularity. The spike train of individual neurons is far from being periodic and relations between the firing patterns of several neurons seem to be random. If the electrical activity picked up by an extra-cellular electrode is made audible by a loudspeaker then we basically hear - noise. The question whether this is indeed just noise or rather a highly efficient way of coding information cannot easily be answered. Listening to a computer modem or a fax machine might also leave the impression that this is just noise. Being able to decide whether we are witnessing the neuronal activity that is underlying the composition of a poem (or the electronic transmission of a love letter) and not just meaningless noise is one of the most burning problems in Neuroscience.

Several experiments have been undertaken to tackle this problem. It seems as if neurons can react in a very reliable and reproducible manner to fluctuating currents that are injected via intracellular electrodes. As long as the same time-course of the injected current is used the action potentials occur with precisely the same timing relative to the stimulation (Bryant and Segundo, 1976; Mainen and Sejnowski, 1995). A related phenomenon can be observed by using non-stationary sensory stimulation. Spatially uniform random flicker, for example, elicits more or less the same spike train in retinal ganglion cells if the same flicker sequence is presented again (Berry et al., 1997). A similar behavior has been reported for motion-sensitive neurons of the visual system in flies (de Ruyter van Steveninck et al., 1997) and monkey cortex (Bair and Koch, 1996). On the other hand, neurons produce irregular spike trains in the absence of any temporally structured stimuli. Irregular spontaneous activity, i.e., activity that is not related in any obvious way to external stimulation, and trial-to-trial variations in neuronal responses are often considered as noise (Shadlen and Newsome, 1994; Softky and Koch, 1993).

The origin of the irregularity in the electrical activity of cortical neurons in vivo is poorly understood. In spiking neuron models such as the integrate-and-fire or Spike Response Model (SRM), noise is therefore often added explicitly to neuronal dynamics so as to mimic the unpredictability of neuronal recordings. In this chapter we present three different ways to implement noise in models of neuronal networks, viz. escape noise (Section 5.3), slow noise in the parameters (Section 5.4), and diffusive noise (Section 5.5). In Section 5.6 we discuss the differences between subthreshold and superthreshold stimulation and explain its consequences for spike train variability. In the subthreshold regime, it is possible to relate the diffusive noise model to the escape noise model. Section 5.7 illustrates this relation. The noise models are finally applied to the phenomenon of stochastic resonance in Section 5.8 and compared with rate models in Section 5.9. Before we start with the discussion of the noise models, we review in Section 5.1 some experimental evidence for noise in neurons and introduce in Section 5.2 a statistical framework of spike train analysis.