Multi-level Models

The brain is a prototype of a hierarchical system, as Fig. 1 shows. More precisely, it is hierarchical dynamical system. To specify a dynamical system, characteristic state variables and evolution equations governing the change of state should be defined. At the molecular level, the dynamic laws can be identified with chemical kinetics, at the channel level with biophysically detailed equations for the membrane potential, and at the synaptic and network levels with learning rules to describe the dynamics of synaptic modifiability (see Table 1). Neurons are considered the classical building blocks of the brain. Small-scale models are focused on from the dynamics of subneuronal structures via single neuron dynamics to small networks. The fundamental method is based on the Hodgkin– Huxley equations. Not a whole neuron, but a part of it, namely the giant axon of the squid, was studied by Hodgkin and Huxley (1952), who quantitatively described the electrogenesis of the action potential. Two channels, the fast sodium channel and the delayed rectifier potassium channel, were included. The total membrane current is the sum of the individual currents transported through individual channels. Channels were assumed to be either in an “open” or “closed” state, and the probability of the transition between them was described by first-order kinetics, with voltagedependent rate “constants.” Three elementary processes namely sodium activation, sodium inactivation, and potassium activation were found, and therefore three (“gating”) variables, m, h, and n, respectively, were defined. The experiments were done under the “space clamp” method to eliminate the spatial variation of the membrane potential V . The mathematical consequence is that the partial differential equation is reduced to a four-dimensional system of ordinary differential equations, where the variables are the membrane potential and the three gating variables.