Simulation of Biological Neural Networks

The list of success stories of the use of mathematics and mathematical simulation in physics, chemistry and engineering is endless. In the biological sciences the situation is different. The training of students in biology or medicine traditionally puts little emphasis on mathematics and physics, and skepticism towards any benefits of mathematics in describing living systems prevails in the biological and medical research communities. Neuroscience is maybe the biological subdiscipline where the use of mathematical techniques is most established and recognized. An important reason for this is the success of Hodgkin and Huxley [1] 50 years ago of describing signal transport in a single neuron (nerve cell) as a modified electrical circuit where the charge carriers are Na¢ , K¢ , Ca¢ £ ¢ , Cl¤ and other ions flowing through the neuron cell membrane. This mathematical formulation, known as Hodgkin-Huxley theory, could not only account for the results from experiments used to construct the model and fit the model parameters. From their model they could also predict the shape and velocity of the so called action potential which is a pulse-like electrical disturbance which travels down thin outgrowths, called axons, of neurons. From their model they calculated the propagation velocity of the action potential down their experimental system, the squid giant axon, to be 18.8 m/s (at 18.3¥ C) which was roughly 10% off the experimental value of 21.2 m/s. Such quantitatively accurate model predictions are rare in theoretical biology. (Thorough introductions to mathematical modeling of single neurons, including Hodgkin-Huxley theory, can be found in Refs. [2, 3, 4]). The success story of the Hodgkin-Huxley model has for two reasons made the life for theoretical neuroscientists easier than for modelers in other branches of biology: First, it has given the modelers a relatively firm starting point for mathematical explorations of both single neurons and neural networks. Secondly, experimental neuroscientists may have a more positive view on the potential benefit of mathematical modeling than their colleagues working in other fields of biology where such an example of a successful mathematical theory is still lacking. Moreover, due to its obvious success in describing action potentials, the Hodgkin-Huxley model has opened up for mathematical analysis of a variety of cell membrane phenomena. One example is the compartmental modeling of propagation of synaptic signals to the soma which is crucial for understanding the information processing properties of a single neuron [5, 6]. It has …