Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Diffusion Equation

  • Fidel SantamariaEmail author
  • Toma M. Marinov
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_186-3



The diffusion equation (DE) is a second-order parabolic partial differential equation describing mass transport phenomena due to thermal motion of particles (Crank 1975).

Detailed Description

Diffusion is an important transport mechanism in neurons (Bloodgood 2005; Ehlers et al 2007; Korkotian and Segal 2006; Makin and Malinow 2009; Qian and Sejnowki 1988; Renner et al. 2009; Sabatini et al. 2002; Santamaria 2006; Schmidt 2007a. Smith et al. 1993; Soler and Sabatini 2006). It takes place in the cytosol, the cell membrane and the endoplasmic reticulum. Subject to diffusion are both ions (Ca +, Na +, K +, Cl-, etc) and large complex molecules (lipids, proteins, DNA, RNA, etc.). The physical process of diffusion in absence of any chemical reactions is described by the diffusion equation:
$$ \frac{\partial C\left(\mathbf{x},t\right)}{\partial t}=\nabla \cdotp \left(D\left(\mathbf{x},t\right)\nabla C\left(\mathbf{x},t\right)\right) $$


Diffusion Equation Mean Square Displacement Anomalous Diffusion Differential Equation Model Parabolic Partial Differential Equation 
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This work was partially supported by NSF IOS-1209029 and NSF EF-1137897


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of BiologyUniversity of Texas at San AntonioSan AntonioUSA