### Computational Neuroscience

#### Simulating neurons

Understanding the brain is the next frontier in understanding all animals including humans. Much is known about various regions of the brain and their role in animal behavior, function, and cognition. However, understanding how the various brain function is tied to the electrical activity in individual neurons is still largely unknown. Considering that the human brain contains approximately 100 billion neurons, the computational challenge of simulating even a small functional part of the brain is massive. Clearly, algorithms for speeding up the simulations are essential to achieving that goal.

The Chopp group has developed novel algorithms for solving the Hodgkin-Huxley cable equations that focus computational effort on the parts of neurons where action potentials and other non-equilibrium activity is taking place. The key is a combination of spatial and temporal adaptivity that permits different components of the same system to use vastly different time step sizes without losing fidelity of the overall solution. The figure at the left illustrates this adaptivity where an action potential is generated in the soma of the right neuron, which then triggers the left neuron to fire. The colors indicate the voltage, while the level of transparency indicates the local time step size. We see that large portions of the two cells require little attention while the regions with non-equilibrium voltage get full attention.

### Computational Neuroscience Papers

## Neuroscience Articles

Author(s) | Title/Journal | Year |
---|---|---|

R. Kublik and D. L. Chopp | A Locally Adaptive Time-Stepping Algorithm for the Solution to Reaction-Diffusion Equations on Branched Structures, Advances in Computational Mathematics, 42(3):621-649 | 2016 |

M. J. Rempe, N. Spruston, W. L. Kath, and D. L. Chopp | Compartmental neural simulations with spatial adaptivity. Journal of Computational Neuroscience, 25(3):465-480 | 2008 |

M. J. Rempe and D. L. Chopp | A predictor-corrector algorithm for reaction-diffusion equations associated with neural activity on branched structures. SIAM Journal of Scientific Computing, 28(6):2139-2161 | 2006 |