Thermal noise due to surface-charge effects within the Debye layer of endogenous structures in dendrites

Roman R. Poznanski

Phys. Rev. E 81, 021902 – Published 2 February 2010

Abstract

An assumption commonly used in cable theory is revised by taking into account electrical amplification due to intracellular capacitive effects in passive dendritic cables. A generalized cable equation for a cylindrical volume representation of a dendritic segment is derived from Maxwell’s equations under assumptions: (i) the electric-field polarization is restricted longitudinally along the cable length; (ii) extracellular isopotentiality; (iii) quasielectrostatic conditions; and (iv) homogeneous medium with constant conductivity and permittivity. The generalized cable equation is identical to Barenblatt’s equation arising in the theory of infiltration in fissured strata with a known analytical solution expressed in terms of a definite integral involving a modified Bessel function and the solution to a linear one-dimensional classical cable equation. Its solution is used to determine the impact of thermal noise on voltage attenuation with distance at any particular time. A regular perturbation expansion for the membrane potential about the linear one-dimensional classical cable equation solution is derived in terms of a Green’s function in order to describe the dynamics of free charge within the Debye layer of endogenous structures in passive dendritic cables. The asymptotic value of the first perturbative term is explicitly evaluated for small values of time to predict how the slowly fluctuating (in submillisecond range) electric field attributed to intracellular capacitive effects alters the amplitude of the membrane potential. It was found that capacitive effects are almost negligible for cables with electrotonic lengths $L > 0.5$ , contributes up to 10% of the signal for cables with electrotonic lengths in the range between $0.25 < L < 0.5$ , and dominates the membrane potential for electrotonically short cables $(L < 0.2)$ . These results show that electrotonically short dendritic cables with both ends sealed are prone to significant neurobiological thermal noise due to intracellular capacitive effects. The presence of significant thermal noise weakens the assumption of intracellular isopotentiality when approximating dendrites with compartments.

Received 23 September 2009

DOI:https://doi.org/10.1103/PhysRevE.81.021902

Authors & Affiliations

Roman R. Poznanski^*

Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia

^*roman.poznanski@um.edu.my

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Vol. 81, Iss. 2 — February 2010

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Figure 1
A schematic cable of a dendritic segment as volume element $(ν)$ with length increment $Δ x$ and electrotonic length $L$ . The arrow indicates the convention that positive charge is in the direction of increasing $x$ , which is the physical distance along the cable (centimeters).Reuse & Permissions
Figure 2
A circuit representing a patch of passive membrane. The intracellular medium containing the endogenous core is represented by longitudinal capacitance $(C_{i})$ of the cable (F/cm) in parallel with the intracellular resistivity $(R_{i})$ of the cable $(Ω cm)$ .Reuse & Permissions
Figure 3
Intrinsic electrical noise (measured as a percentage of the subthreshold response) that is thermally generated and which produces a $0.8 μ V$ transmembrane voltage shift at physiological temperatures for a 10 mV subthreshold response in passive cables with electrotonic length of $L = 1$ (i). The thermal noise is considered negligible and at the lower limit of noise. For $L = 0.5$ (ii) there is a $14 μ V$ transmembrane voltage shift which is still considered negligible and within the expected values of a signal-to-noise ratio. When $L = 0.25$ (iii) there is a 0.2 mV transmembrane voltage shift which will distort the signal by 2%. This contributes to the signal-to-noise ratio. When $L = 0.2$ (iv) there is a 0.5 mV transmembrane voltage shift which will distort the signal by up to 5%. When $L = 0.15$ (v) there is a 1.6 mV transmembrane voltage shift which will distort the signal by up to 16%. When $L = 0.135$ (vi) there is a 2.5 mV transmembrane voltage shift which will distort the signal by up to 25%.Reuse & Permissions
Figure 4
The influence of thermal noise on the transient voltage decay. The voltage response is compared to the response in a passive cable $V_{0}$ for several values of electrotonic length (i) $L = 0.2$ , (ii) $L = 0.25$ , (iii) $L = 0.5$ , and (iv) $L = 1.0$ . The solution with thermal noise $(γ = 0.001)$ has effect on the voltage response for several values of times $(T = t / τ_{m})$ in the submillisecond range. The solution with no thermal noise $(γ = 0)$ was calculated with the Poisson transformed Green’s function expression with ten terms in the series. The responses were measured near the origin in the submillisecond interval $0 < T < 0.01$ for $γ = 0.001$ and $r_{i} λ I_{o} = 1 mV$ .Reuse & Permissions
Figure 5
The influence of thermal noise on voltage attenuation with distance along a cable of electrotonic length $(L)$ . The voltage response is compared to the response in a passive cable $V_{0}$ for several values of electrotonic length (i) $L = 0.2$ , (ii) $L = 0.25$ , (iii) $L = 0.5$ , and (iv) $L = 1.0$ . The solution with no thermal noise $(γ = 0)$ was calculated with the Poisson transformed Green’s function expression with ten terms in the series. The responses were measured at $T = 0.001$ being in the submillisecond interval $0 < T < 0.01$ for $γ = 0.001$ and $r_{i} λ I_{o} = 1 mV$ .Reuse & Permissions

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