Petr Lánský1 and Jean-Pierre Rospars2
1 Institute of Physiology, Academy of Sciences of the Czech Republic, Vídeňská 1083, 142 20 Prague 4,
2Département de Zoologie and Laboratoire de Biométrie, Institut National de la Recherche Agronomique, 78026 Versailles Cedex, France
Intensity of environmental stimulation is encoded in the nervous system by the frequency of action potentials , and the olfactory system is no exception to this rule. Electrophysiological investigations have shown that the frequency of action potentials increases with odor concentration in neuroreceptors ([6, 20, 23, 51] in insects; [11, 15, 41] in Amphibians). In most of these experimental recordings neurons do not fire perfectly regularly, which indicates that stochastic mechanisms are also present. Olfactory coding begins with transduction processes that take place in the dendritic cilia of specialized receptors (neuroreceptors) in the vertebrate olfactory mucosa and insect antennal sensilla. Odorant molecules bind to receptor proteins borne by the dendritic membrane, and a cascade of events follows that ultimately evokes a generator potential (see e.g., [1, 7, 8, 28]). When the generator potential is high enough, action potentials are generated and propagated to the second-order neurons. Each second-order neuron receives the excitatory terminals of a larger number (of the order of 103) of neuroreceptors and of inhibitory local neurons (see e.g., [46, 52]). These local neurons are excited by neuroreceptors, second-order neurons and neurons of higher centers. Thus, a second-order neuron can be inhibited as a result of its own activity (feedback inhibition) or from the activity of its neighbors (lateral inhibition). All this network activity is likely involved in intensity coding.
In parallel with the experimental studies, mathematical models were devised in an attempt to formalize the results of the experiments. Models of the neuroreceptor were based on physical properties of neurons [21, 22, 38] and global models relating directly the electrical (or behavioral) response to the concentration of odorant were proposed (see review in [12, 27]). In addition to these more or less specialized models, a whole range of mathematical models of neurons exists that were devised to take into account only the most basic and generic properties of neurons . Most of these models describing the dynamics of interspike intervals (ISIs) are one dimensional, i.e. they model the time evolution of the neuronal membrane potential at only one point, the spike trigger zone. We term the membrane potential at the trigger zone the axonal potential. However, the stimulating actions impinging on the neuron take place on the dendrites (and soma in vertebrates), at points not located at the trigger zone. We term the sum of all dendritic contributions the dendritic potential.
The action potential (spike) is produced when the axonal potential exceeds a voltage threshold S. Formally, it corresponds to the first passage time (FPT) of a stochastic process X across the threshold S. The one-dimensional models are generally based on two assumptions. The first is that after spike generation, the axonal potential is reset to a constant X(0) = x0 (reviewed in ) or to a random variable X0 [32, 36]. The second assumption, implicitly contained in the fact that the models are unidimensional, is that the dendritic potential is also reset at the moment of spike generation. Consequently, the ISIs are independent random variables (for a stationary input they form a renewal process), whose probability density function is the only feature that differentiates the models that