Approximation of the Solution of the Dendritic Cable Equation by a Small Series of Coupled Differential Equations

In most literature on artificial neural network models the node (neuron) is represented as a point, learning rules are based on activity of other nodes and the activity of the node itself. Models incorporating local learning rules allows increase and decrease of weights on bases of activities of other nodes if the synapses are placed close together on the same dendrite. These models model the biological dendrite by solving an equivalent electrical circuit, consisting of several compartments. Current models solve the electrical circuit model by the numerical calculation of a recurrence relation in which the current of a compartment is expressed in the value of the current in a next compartment. In this way it is always necessary to model the whole dendrite, even if we are only interested in local effects. In this paper an alternative mathematical description is proposed on bases of which local activities can be calculated by only simulating a part of the dendrite model.