II Generalized Integrate-and-Fire Neurons

Chapter 7 Variability of Spike Trains and Neural Codes

The neuron models discussed in the previous chapters are deterministic and generate, for most choices of parameters, spike trains that look regular when driven by a constant stimulus. In vivo recordings of neuronal activity, however, are characterized by a high degree of irregularity. The spike train of an individual neuron is far from being periodic and correlations between the spike timings of neighboring neurons are weak. If the electrical activity picked up by an extra-cellular electrode is made audible by a loudspeaker then what we basically hear is: noise. The question whether this is indeed just noise or rather a highly efficient way of coding information cannot easily be answered. Indeed, 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 flicker is one of the most burning problems in neuroscience.

Several experiments have been undertaken to tackle this problem. It seems that a neuron in vitro, once it is isolated from the network, can react in a very reliable and reproducible manner to a fluctuating input current, and so can neurons in sensory cortex in vivo when driven by a strong time-dependent signal. 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.

The origin of this irregularity of neuronal dynamics in vivo is poorly understood. In integrate-and-fire models, noise is therefore often added explicitly to neuronal dynamics so as to mimic the unpredictability of neuronal recordings. How to add noise to neuron models is the topic of Chapters 8 and 9. The aim of the present chapter is a mere description and quantification of the variability of neuronal spike trains. We review in Section 7.1 some experimental evidence for noise in neurons and introduce in Sections 7.27.5 a statistical framework of spike train analysis. In particular, we present the definitions of firing rate, interval distribution, power spectrum, and renewal statistics. In Section 7.6 we ask whether the firing rate, which is such a useful measure for quantification of spike trains, can also be considered as the code used by neurons in the brain.