To understand the behavior of a neural circuit it is a presupposition that we have a model of the dynamical system describing this circuit. This model is determined by several parameters, including not only the synaptic weights, but also the parameters of each neuron. Existing works mainly concentrate on either the synaptic weights or the neural parameters. In this paper we present an algorithm to reconstruct all parameters including the synaptic weights of a spiking neuron model. The model based on works of Eugene M. Izhikevich (Izhikevich 2007) consists of two differential equations and covers different types of cortical neurons. It combines the dynamical properties of Hodgkin-Huxley-type dynamics with a high computational efficiency. The presented algorithm uses the recordings of the corresponding membrane potentials of the model for the reconstruction and consists of two main components. The first component is a rank based Genetic Algorithm (GA) which is used to find the neural parameters of the model. The second one is a Least Mean Squares approach which computes the synaptic weights of all interconnected neurons by minimizing the squared error between the calculated and the measured membrane potentials for each time step. In preparation for the reconstruction of the neural parameters and of the synaptic weights from real measured membrane potentials, promising results based on simulated data generated with a randomly parametrized Izhikevich model are presented. The reconstruction does not only converge to a global minimum of neural parameters, but also approximates the synaptic weights with high precision.