Cable theory

Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by modeling dendrites and axons as cylinders composed of segments with capacitances c m {displaystyle c_{m}} and resistances r m {displaystyle r_{m}} combined in parallel (see Fig. 1). The capacitance of a neuronal fiber comes about because electrostatic forces are acting through the very thin lipid bilayer (see Figure 2). The resistance in series along the fiber r l {displaystyle r_{l}} is due to the axoplasm's significant resistance to movement of electric charge. r m = R m 2 π a   {displaystyle r_{m}={frac {R_{m}}{2pi a }}}     (1) c m = C m 2 π a   {displaystyle c_{m}=C_{m}2pi a }     (2) r l = ρ l π a 2   {displaystyle r_{l}={frac { ho _{l}}{pi a^{2} }}}     (3) Δ V = − i l r l Δ x   {displaystyle Delta V=-i_{l}r_{l}Delta x }     (4) ∂ V ∂ x = − i l r l   {displaystyle {frac {partial V}{partial x}}=-i_{l}r_{l} }     (5) 1 r l ∂ V ∂ x = − i l   {displaystyle {frac {1}{r_{l}}}{frac {partial V}{partial x}}=-i_{l} }     (6) Δ i l = − i m Δ x   {displaystyle Delta i_{l}=-i_{m}Delta x }     (7) ∂ i l ∂ x = − i m   {displaystyle {frac {partial i_{l}}{partial x}}=-i_{m} }     (8) i c = c m ∂ V ∂ t   {displaystyle i_{c}=c_{m}{frac {partial V}{partial t}} }     (9) i r = V r m {displaystyle i_{r}={frac {V}{r_{m}}}}     (10) ∂ i l ∂ x = − i m = − ( V r m + c m ∂ V ∂ t ) {displaystyle {frac {partial i_{l}}{partial x}}=-i_{m}=-left({frac {V}{r_{m}}}+c_{m}{frac {partial V}{partial t}} ight)}     (11) 1 r l ∂ 2 V ∂ x 2 = c m ∂ V ∂ t + V r m {displaystyle {frac {1}{r_{l}}}{frac {partial ^{2}V}{partial x^{2}}}=c_{m}{frac {partial V}{partial t}}+{frac {V}{r_{m}}}}     (12) λ = r m r l {displaystyle lambda ={sqrt {frac {r_{m}}{r_{l}}}}}     (13) V x = V 0 e − x λ {displaystyle V_{x}=V_{0}e^{-{frac {x}{lambda }}}}     (14) x λ = 1 {displaystyle {frac {x}{lambda }}=1}     (15) V x = V 0 e − 1 {displaystyle V_{x}=V_{0}e^{-1}}     (16) V λ = V 0 e = 0.368 V 0 {displaystyle V_{lambda }={frac {V_{0}}{e}}=0.368V_{0}}     (17) τ = r m c m .   {displaystyle au =r_{m}c_{m}. }     (18) r m r l ∂ 2 V ∂ x 2 = c m r m ∂ V ∂ t + V {displaystyle {frac {r_{m}}{r_{l}}}{frac {partial ^{2}V}{partial x^{2}}}=c_{m}r_{m}{frac {partial V}{partial t}}+V}     (19) λ 2 ∂ 2 V ∂ x 2 = τ ∂ V ∂ t + V {displaystyle lambda ^{2}{frac {partial ^{2}V}{partial x^{2}}}= au {frac {partial V}{partial t}}+V}     (20) Classical cable theory uses mathematical models to calculate the electric current (and accompanying voltage) along passive neurites, particularly the dendrites that receive synaptic inputs at different sites and times. Estimates are made by modeling dendrites and axons as cylinders composed of segments with capacitances c m {displaystyle c_{m}} and resistances r m {displaystyle r_{m}} combined in parallel (see Fig. 1). The capacitance of a neuronal fiber comes about because electrostatic forces are acting through the very thin lipid bilayer (see Figure 2). The resistance in series along the fiber r l {displaystyle r_{l}} is due to the axoplasm's significant resistance to movement of electric charge. Cable theory in computational neuroscience has roots leading back to the 1850s, when Professor William Thomson (later known as Lord Kelvin) began developing mathematical models of signal decay in submarine (underwater) telegraphic cables. The models resembled the partial differential equations used by Fourier to describe heat conduction in a wire. The 1870s saw the first attempts by Hermann to model neuronal electrotonic potentials also by focusing on analogies with heat conduction. However, it was Hoorweg who first discovered the analogies with Kelvin's undersea cables in 1898 and then Hermann and Cremer who independently developed the cable theory for neuronal fibers in the early 20th century. Further mathematical theories of nerve fiber conduction based on cable theory were developed by Cole and Hodgkin (1920s–1930s), Offner et al. (1940), and Rushton (1951). Experimental evidence for the importance of cable theory in modelling the behavior of axons began surfacing in the 1930s from work done by Cole, Curtis, Hodgkin, Sir Bernard Katz, Rushton, Tasaki and others. Two key papers from this era are those of Davis and Lorente de Nó (1947) and Hodgkin and Rushton (1946). The 1950s saw improvements in techniques for measuring the electric activity of individual neurons. Thus cable theory became important for analyzing data collected from intracellular microelectrode recordings and for analyzing the electrical properties of neuronal dendrites. Scientists like Coombs, Eccles, Fatt, Frank, Fuortes and others now relied heavily on cable theory to obtain functional insights of neurons and for guiding them in the design of new experiments. Later, cable theory with its mathematical derivatives allowed ever more sophisticated neuron models to be explored by workers such as Jack, Rall, Redman, Rinzel, Idan Segev, Tuckwell, Bell, and Iannella. Note, various conventions of rm exist.Here rm and cm, as introduced above, are measured per membrane-length unit (per meter (m)). Thus rm is measured in ohm·meters (Ω·m) and cm in farads per meter (F/m). This is in contrast to Rm (in Ω·m²) and Cm (in F/m²), which represent the specific resistance and capacitance respectively of one unit area of membrane (in m2). Thus, if the radius, a, of the axon is known, then its circumference is 2πa, and its rm, and its cm values can be calculated as: These relationships make sense intuitively, because the greater the circumference of the axon, the greater the area for charge to escape through its membrane, and therefore the lower the membrane resistance (dividing Rm by 2πa); and the more membrane available to store charge (multiplying Cm by 2πa).The specific electrical resistance, ρl, of the axoplasm allows one to calculate the longitudinal intracellular resistance per unit length, rl, (in Ω·m−1) by the equation: The greater the cross sectional area of the axon, πa², the greater the number of paths for the charge to flow through its axoplasm, and the lower the axoplasmic resistance.