Abstract
This paper discusses ergodic properties and circular statistical characteristics in neuronal spike trains. Ergodicity means that the average taken over a long time period and over smaller population should equal the average in less time and larger population. The objectives are to show simple examples of design and validation of a neuronal model, where the ergodicity assumption helps find correspondence between variables and parameters. The methods used are analytical and numerical computations, numerical models of phenomenological spiking neurons and neuronal circuits. Results obtained using these methods are the following. They are: a formula to calculate vector strength of neural spike timing dependent on the spike train parameters, description of parameters of spike train variability and model of output spiking density based on assumption of the computation realized by sound localization neural circuit. Theoretical results are illustrated by references to experimental data. Examples of neurons where spike trains have and do not have the ergodic property are then discussed.
Highlights
The ergodicity concept was introduced among others by Ludwig Eduard Boltzmann (1844–1906; after 1870; Boltzmann1898) in statistical physics in his kinetic theory of gases
The highest firing rate neurons in humans are in the auditory pathway, and they typically do not exceed a rate of R = 500 per second
Vector strength r of a spike train with the beta distribution (5) of interspike intervals (ISIs) can be parameterized by a, b and scaled by s
Summary
The ergodicity concept was introduced among others by Ludwig Eduard Boltzmann (1844–1906; after 1870; Boltzmann1898) in statistical physics in his kinetic theory of gases. The ergodic hypothesis states that an average taken over a smaller set of particles, or just one particle, and a longer time period should be equal to the average over a larger set of particles and shorter period of time. Systems fulfilling this hypothesis are called ergodic. The elementary neural signals are recorded as all-ornone events, action potentials, or spikes, which occur in volleys called spike trains.
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