# SYNAPTIC TRANSMISSION AND FOKKER-PLANCK EQUATION

### Elso Drigo Filho, Marcelo Araujo

#### Abstract

An important neurologic process consists in a time dependent transmission of the electric signal between neurons. The description of such signal is the objective of this work. In this way, the Fokker-Planck equation with a term of control which depends on time is used. The applied force is characterized by the existence of a barrier that increases with time and reduces the diffusion of particles. The solution of the equation is obtained by an ansatz that satisfies the initial conditions of the problem. Numerical examples of the time evolution of the found solutions are analyzed by calculating the escape rate and the time necessary to across the barrier for different values of diffusion constant.

#### References

- Hille, B., 1992. Ionic Channels of Excitable Membranes, Sinauer Associates INC., Sunderland, 2nd edition.
- Ramakrishnan, S., Hasler, P. E., Gordon, C., 2011. Floating Gate Synapses With Spike-Time-Dependent Plasticity, IEEE Trans. Biomed. Circuits Syst., v. 5 244-252.
- Fassio, A., et. al., 2011. Synapsins: From synapse to network hyper excitability and epilepsy, Semin. Cell Dev. Biol., 8p.
- Joshi, J., et. al., 2011. A Biomimetic Fabricated Carbon Nanotube Synapse for Prosthetic Applications, Life Science Systems and Applications Workshop, 139- 142.
- England, P. M., 2010. Bridging the Gaps between Synapses, Circuits, and Behavior, Chem. & Biol. 17, 607-615.
- Guo, D., Li, C., 2010. Signal propagation in feed forward neuronal networks with unreliable synapses, J. Comput. Neurosci., v. 30, 567-587.
- Fallon, J. R., 2011. Calcium channels put synapses in their place, Nat. Neurosci., v. 14, 536 - 538.
- Coffey, W., 2004. The Langevin equation: with applications to stochastic problems in physics, chemistry, and electrical engineering, River Edge: World Scientific, 2nd edition.
- Curtiss, C. F., Bird, R. B., 1997. Fokker-Planck equation for the one-molecule distribution function in polymer mixtures and its solution, Jour. Chem. Phys., v. 106, n. 23.
- Lee, K., Sung, W., 2002. Ion transport and channel transition in biomembranes, Phys. A, v. 315 79 - 97.
- Risken, H., 1989. The Fokker-Planck Equation, Springer, 2nd edition.
- Reichl, L. E., 1988. A Modern Course in Statistical Physics, Wiley-Interscience, 2nd edition.
- Ames, W. F., 1992. Numerical methods for partial differential equations, Academic Press INC., San Diego.
- Tatari, M., Dehghan, M., Razzaghi, M., 2007. Application of the Adomian decomposition method for the FokkerPlanck equation, Math. and Comp. Model., v. 45, 639-650.
- Hänggir, P., Talkner, P., Borkovec, M., 1990. Reactionrate theory: fifty years later Kramers, Rev. Mod. Phys., v. 62.
- Lenzi, E. K., Anteneodo, C., Borland, L., 2001. Escape time in anomalous diffusive media, Phys. Rev. E, v. 63, 051109 (1-5).

#### Paper Citation

#### in Harvard Style

Drigo Filho E. and Araujo M. (2012). **SYNAPTIC TRANSMISSION AND FOKKER-PLANCK EQUATION** . In *Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-8425-97-3, pages 59-63. DOI: 10.5220/0003757100590063

#### in Bibtex Style

@conference{icores12,

author={Elso Drigo Filho and Marcelo Araujo},

title={SYNAPTIC TRANSMISSION AND FOKKER-PLANCK EQUATION},

booktitle={Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2012},

pages={59-63},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003757100590063},

isbn={978-989-8425-97-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 1st International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - SYNAPTIC TRANSMISSION AND FOKKER-PLANCK EQUATION

SN - 978-989-8425-97-3

AU - Drigo Filho E.

AU - Araujo M.

PY - 2012

SP - 59

EP - 63

DO - 10.5220/0003757100590063