Cable Equations for Neurons

Generalized cable equations for neurons
In collaboration with Claude Bedard (postdoc in my laboratory, now CDI CNRS), we have investigated how to incorporate the effect of charge displacement into cable equations. When ion channels open or close, the flow of ions depends on the concentrations at either side of the membrane, and in all cable equations considered so far, this displacement of charges was considered as instantaneous. However, as seen above for modeling of LFPs (Section 1.3), charge displacement is not instantaneous, and may lead to important effects such as high-frequency filtering of LFPs. We incorporated this effect into cable equations, in order to determine its influence on several important properties of the neuron, such as the attenuation with distance and post-synaptic summation. We obtained a more accurate electrical description of the neuron for high frequencies, as well as predictions that can be tested experimentally. We could show that such modified cable equations are more accurate to model fast-frequency phenomena in neurons [1].

Although this approach could account for some measurements that the traditional cable equations cannot model, it was not entirely satisfactory. A more general approach was proposed later [2], where we considered the general problem of neurons embedded in media which are more complex than simple resistors. Unfortunately, the traditional Rall’s cable equation cannot be used in this case, because it was derived under the assumption that the medium is resistive or ohmic. We re-derived cable equations, from Maxwell equations of electromagnetism, and by making no such assumption. These generalized cable equations led to a few potentially important observations: first, the electric nature of the extracellular medium affects the attenuation of voltage along the dendrite, and second, it can induce resonances. Voltage attenuation is important for the neuron’s integrative properties, so these findings may have important consequences on the basic properties of neurons (see [2] for details).

This approach was later extended to magnetic fields generated by neurons [3]. Similar to electric signals, magnetic signals are also influenced by the nature of the extracellular medium. Because the properties of the medium influences neuronal properties, it also has an influence on electric and magnetic fields. This predicts specific frequency scaling properties of the power spectra of these signals, which should now be investigated experimentally.

[1] Bedard, C. and Destexhe, A. A modified cable formalism for modeling neuronal membranes at high frequencies. Biophys. J. 94:1133-1143, 2008 (see abstract) [2] Bedard, C. and Destexhe, A. Generalized cable theory for neurons in complex and heterogeneous media. Physical Review E 88:022709, 2013 (see abstract) [3] Bedard, C. and Destexhe, A. Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations. Phys. Rev. E 90: 042723, 2014 (see abstract)

Alain Destexhe