Many Neurons, General Case, Connection with Integrate and Fire Model

Abstract

The lighthouse model treated in Chaps. 5 and 6 had the advantage of simplicity that enabled us to study in particular the effects of delay and noise. In the present chapter we want to treat the more realistic model of Chap. 7 in detail. It connects the phase of the axonal pulses with the action potential U of the corresponding neuron and takes the damping of U into account. Furthermore, in accordance with other neuronal models, the response of the dendritic currents to the axonal pulses is determined by a second-order differential equation rather than by a first-order differential equation (as in the lighthouse model). The corresponding solution, i.e. the dendritic response, increases smoothly after the arrival of a pulse from another neuron. When the dendritic currents are eliminated from the coupled equations, we may make contact with the integrate and fire model. Our approach includes the effect of delays and noise. We will treat the first-order and second-order differential equations for the dendritic currents using the same formalism so that we can compare the commonalities of and differences between the results of the two approaches. The network connections of excitatory or inhibitory nature may be general, though the occurrence of the phase-locked state requires a somewhat more restricted assumption.