A generalized neuronal activation function derived from ion-channel characteristics

Abstract

An abstract model neuron, simple enough to be used in analytical studies of neural networks, is defined. It incorporates neuronal adaptation, i.e. the coupling between neuronal activity and excitability, which can control the complexity of neural network dynamics. A generalized neuronal activation function is defined as the output activity, i.e. The firing rate of action potentials, as a function of an input and an excitability variable. For a biological neuron, the intracellular concentration of Ca ions is shown to be a useful excitability variable. The principal features of the generalized activation function are derived by means of a multicompartment model neuron with ionic currents described by Hodgkin-Huxley-type equations. As a relevant neuron type for investigations of cortical associative memory, a neocortical pyramidal cell of the regularly-spiking type is selected. An approximate analytical consideration suggests a simple form of independent variable of the generalized activation function. The complete numerical treatment verifies this form and displays a threshold character of the activation function. The derivation thereby pmvides an interpretation of dynamic threshold models of neuronal adaptation in terms of ionic mechanism. The threshold with respect to the input has well-defined dynamics directly given by the intracellular Ca dynamics. A comparison of the response of the Hodgkin-Huxley-type and reduced-model neumns to a varying input variable indicates that the reduced model is a usefui approximation even for the detailed behaviour of the firing rate. The generalized activation function also accounts for the dependence on neuromodnlation via an adaptability parameter. An absmt model neural network of units thus defined has the capacity to describe complex network dynamics and the simplicity to allow elucidation of mechanisms and applications to large-scale systems.