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Core Conductor Theory and Cable Properties of Neurons

Wilfrid Rall

10.1002/cphy.cp010103

Source: Supplement 1: Handbook of Physiology, The Nervous System, Cellular Biology of Neurons

Originally published: 1977

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Introduction
1.1 Core Conductor Concept
1.2 Perspective
1.3 Comment
1.4 Reviews and Monographs
2 Brief Historical Notes
2.1 Early Electrophysiology
2.2 Electrotonus
2.3 Passive Membrane Electrotonus
2.4 Passive Versus Active Membrane
2.5 Cable Theory
2.6 Core Conductor Concept
2.7 Core Conductor Theory
2.8 Estimation of Membrane Capacitance
2.9 Resting Membrane Resistivitiy
2.10 Passive Cable Parameters of Invertebrate Axons
2.11 Importance of Single Axon Preparations
2.12 Estimation of Parameters for Myelinated Axons
2.13 Space and Voltage Clamp
3 Dendritic Aspects of Neurons
3.1 Axon‐Dendrite Contrast
3.2 Microelectrodes in Motoneurons
3.3 Theoretical Neuron Models and Parameters
3.4 Class of Trees Equivalent to Cylinders
3.5 Motoneuron Membrane Resistivity and Dendritic Dominance
3.6 Dendritic Electrotonic Length
3.7 Membrane Potential Transients and Time Constants
3.8 Spatiotemporal Effects with Dendritic Synapses
3.9 Excitatory Postsynaptic Potential Shape Index Loci
3.10 Comments on Extracellular Potentials
3.11 Additional Comments and References
4 Cable Equations Defined
4.1 Usual Cable Equation
4.2 Steady‐state Cable Equations
4.3 Augmented Cable Equations
4.4 Comment: Cable Versus Wave Equation
4.5 Modified Cable Equation for Tapering Core
4.6 General Solution of Steady‐state Cable Equation
4.7 Basic Transient Solutions of Cable Equation
4.8 Solutions Using Separation of Variables
4.9 Fundamental Solution for Instantaneous Point Charge
5 List of Symbols
6 Assumptions and Derivation of Cable Theory
6.1 One Dimensional in Space
6.2 Intracellular Core Resistance
6.3 Ohm's Law for Core Current
6.4 Conservation of Current
6.5 Relation of Membrane Current to Vi
6.6 Effect of Assuming Extracellular Isopotentiality
6.7 Passive Membrane Model
6.8 Resulting Cable Equation for Simple Case
6.9 Physical Meaning of Cable Equation Terms
6.10 Physical Meaning of τ
6.11 Physical Meaning of λ
6.12 Electrotonic Distance, Length, and Decrement
6.13 Effect of Placing Axon in Oil
6.14 Effect of Applied Current
6.15 Comment on Sign Conventions
6.16 Effect of Synaptic Membrane Conductance
6.17 Effect of Active Membrane Properties
7 Input Resistance and Steady Decrement with Distance
7.1 Note on Correspondence with Experiment
7.2 Cable of Semi‐infinite Length
7.3 Comments about R∞, G∞, Core Current, and Input Current
7.4 Doubly Infinite Length
7.5 Case of Voltage Clamps at X1 and X2
7.6 Relations Between Axon Parameters
7.7 Finite Length: Effect of Boundary Condition at X= X1
7.8 Sealed End at X= X1: Case of B1 = 0
7.9 Voltage Clamp(V1 = 0) at X = X1: Case of B1 = ∞
7.10 Semi‐infinite Extension at X = X1: Case of B1 = 1
7.11 Input Conductance for Finite Length General Case
7.12 Branches at X = X1
7.13 Comment on Branching Equivalent to a Cylinder
7.14 Comment on Membrane Injury at X = X1
7.15 Comment on Steady Synaptic Input at X= X1
7.16 Case of Input to One Branch of Dendritic Neuron Model
8 Passive Membrane Potential Transients and Time Constants
8.1 Passive Decay Transients
8.2 Time Constant Ratios and Electrotonic Length
8.3 Effect of Large L and Infinite L
8.4 Transient Response to Applied Current Step, for Finite Length
8.5 Applied Current Step with L Large or Infinite
8.6 Voltage Clamp at X = 0, with Infinite L
8.7 Voltage Clamp with Finite Length
8.8 Transient Response to Current Injected at One Branch of Model
9 Relations Between Neuron Model Parameters
9.1 Input Resistance and Membrane Resistivity
9.2 Dendritic Tree Input Resistance and Membrane Resistivity
9.3 Results for Trees Equivalent to Cylinders
9.4 Result for Neuron Equivalent to Cylinder
9.5 Estimation of Motoneuron Parameters
Figure 1. Figure 1.

Cylindrical core conductor with the spreading distribution of electric current indicated (only roughly) by dashed lines, for 2 arrangements of electrodes; cylinders extend to (±) infinite length. A: both electrodes (cathode and anode) are extracellular; some current flows entirely extracellularly, directly from anode to cathode in the interpolar region; some extracellular current flows out into the extrapolar region before crossing the membrane. B: intracellular (micropipette) electrode provides source of current that spreads along the core before crossing the membrane to the extracellular volume that is isopotential with a ground electrode.
Figure 2. Figure 2.

Flow of electric current from a microelectrode whose tip penetrates the cell body (soma) of a neuron. Full extent of dendrites is not shown. External electrode to which the current flows is at a distance far beyond the limits of this diagram.

From Rall 139
Figure 3. Figure 3.

Arbitrary dendritic branching to illustrate the subscript notation used to treat this problem. Originally L0 represented actual trunk length; subsequently, I have preferred to use ��0 for actual length and L0 = ��0/λ0 as the dimensionless electrotonic length of a dendritic trunk.

From Rall 139
Figure 4. Figure 4.

Symmetrical dendritic tree, illustrative of a class of trees that can be mathematically transformed into an equivalent cylinder 141,142,143,144 and into approximately equivalent chains of equal compartments. Dashed lines divide both the tree and the cylinder into 5 equal increments of electrotonic distance, having equal membrane surface area. In one specific example 141,143, the dendritic trunk diameter was 10 μm, and successive branch diameters were 6.3, 4.0, 2.5, 1.6, and 1.0 μm, which satisfy both symmetry and the constraint on d3/2 values. Chain of 5 compartments corresponds to the 5 increments of electrotonic distance above. Chain of 10 compartments, used for Fig. 7, 8, and 10, represents the soma as compartment 1, and progressive electrotonic distance out into the dendrites is represented by compartments 2–10.
Figure 5. Figure 5.

Distributions of membrane depolarization for an excitatory conductance step in the peripheral half of a dendritic tree, for several values of T = t/τ. In the central half of the dendritic tree (i.e., X = 0 to X = 0.5) the conductances are assumed to remain at their resting values. Over the peripheral half of the dendritic tree (i.e., X = 0.5 to X = 1.0) the conductance (Ge) is assumed to step from zero to a value of 2Gr at T = 0.

Calculations are based on equations in 141
Figure 6. Figure 6.

Transients of membrane depolarization at the central end, the peripheral end, and at the middle of a dendritic tree. A: 3 curves at left illustrate the same problem as in Fig. 5; uppermost curve corresponds to the peripheral end (X = 1); intermediate curve corresponds to X = 0.5; lowest curve corresponds to the central end (X = 0). B: 3 curves at right illustrate the response to a square conductance pulse. On step is the same as in A; off step occurs 0.2 units of T later; the duration of the square conductance pulse is indicated by vertical dashed lines.

For equations used to calculate the off transients, see 141
Figure 7. Figure 7.

Computed transients at the soma (compartment 1) for 4 synaptic inputs that differ only in their input locations. In each case, the synaptic input consisted of Ge = Gr in 2 compartments for a time interval, T = 0 to T = 0.25, indicated by heavy black bar. In A, the input was applied to compartments 2 and 3, in B to 4 and 5, in C to 6 and 7, and in D to 8 and 9

See Fig. 4 for the relation of such compartments to the dendritic branches; see 143 for equations and discussion of compartmental modeling
Figure 8. Figure 8.

Effect of two spatiotemporal sequences on transient soma‐membrane depolarization. Two input sequences, A→B→C→D and D→C→B→A, are indicated at upper left and upper right, respectively; component input locations are the same as in Fig. 7. Time sequence is indicated by means of the 4 successive time intervals, Δt1, Δt2, Δt3, and Δt4, each equal to 0.25 τ. Dotted curve shows the computed effect of Ge = 0.25 Gr in compartments 2–9 for the period τ = 0 to l = τ

For further details, see 143
Figure 9. Figure 9.

Right: computed EPSP's (solid traces) generated by synaptic currents of different time courses (dotted traces). Synaptic currents are assumed to be uniformly distributed over entire soma‐dendritic surface. Arrows on the lowermost EPSP indicate its half‐width. Left: plot of paired shape index values for EPSP's generated by uniformly distributed synaptic input with coordinates as labeled. The •, ▴, and ▪ on the plot represent the shapes of corresponding EPSP's on the right side of the figure

See 149
Figure 10. Figure 10.

Right: diagrammatic representation of the transformation of soma‐dendritic receptive surface of a neuron into a chain of 10 equal compartments. Below are graphs of computed EPSP's occurring in compartment 1, obtained with the compartmental model for the medium synaptic current time course (upper graph, dotted line). Synaptic current introduced equally to all compartments gave the upper computed EPSP (▴). Synaptic current localized to a single compartment gave the lower 3 computed EPSP's: compartment 1(), compartment 4 (), or compartment 8 (). Left: EPSP shape index values for computed EPSP's shown at right. Dashed line, locus of EPSP shapes generated when synaptic input is limited to the numbered compartment. Solid line, locus for spatially uniform depolarization of the cell. Note that scales are in units of dimensionless ratio t/τ

See 149
Figure 11. Figure 11.

Scatter diagram of normalized shape indices of the EPSP's from the motoneurons for which a time constant was obtained. •, EPSP's recorded from knee flexor motoneurons; ○, EPSP's recorded from ankle extensor motoneurons. Two dashed lines show theoretical boundaries for the shape indices with the assumed values of parameters (α = 12–100; ρα = 4–25; L = 0.75–1.5). The set of areas, each bounded by a continuous line, show the theoretical boundaries for particular distances

See 90
Figure 12. Figure 12.

Shape index plot for evoked EPSP's for which impedance measurements were available. Time to peak as abscissa; half‐width as ordinate; scales in milliseconds. •, EPSP's accompanied by a measurable impedance change; ○, EPSP's not accompanied by a detectable impedance change. Dotted outline, scatter of shape index values for observed miniature EPSP (for reference)

See 149,174
Figure 13. Figure 13.

Theoretically calculated relation between intracellularly and extracellularly recorded action potentials. Uppermost curve, experimental “AB spike” followed by an “A spike” as recorded intracellularly from a cat motoneuron by Nelson & Frank 128. The other intracellular curve represents the theoretically calculated passive electrotonic spread into a dendritic cylinder of infinite length; it corresponds to a radial distance, R = 18 [i.e., an electrotonic distance, x/λ, of (R − 1/40 = 0.425; this would correspond to about 600 μm in the examples considered]. Extracellular curves were calculated on the assumption of radial symmetry. Curve for R = 18 has been multiplied by 10 to aid the comparison of shape. Curve at R = 3 has a shape extremely similar to that at R = 1, except that the peak at R = 1 has an amplitude about 5 times that at R = 3

See 128,142
Figure 14. Figure 14.

Computed isopotential contours for a spherical soma with 7 cylindrical dendrites, of which only 3 can be seen in the plane shown here. Relative to 1 dendrite shown on the vertical (polar) axis, 3 dendrites were equally spaced at 60° from the polar axis, and 3 more dendrites were equally spaced at the equator (see inset). The soma was the sink for extracellular current; the dendritic cylinders were sources of extracellular current corresponding to passive electrotonic spread at the time of the peak of an antidromic action potential. For this calculation, dendritic λ was set equal to 40 times the somatic radius. Numbers labeling the contours correspond to the quantity Ve/(INRe/4πb), whereIN is the total current flowing from dendrites to soma, Re represents extracellular volume resistivity, and b represents the soma radius. For the particular case of the peak somatic action potential in a cat motoneuron, this numerical quantity expresses the value of Ve approximately in millivolts. This is because of the following order of magnitude consideration: IN is of the order 10−7 A, because the peak intracellular action potential is of the order 10−1 V, and the whole neuron instantaneous conductance is of the order 10−6 Ω−1;Re/4πb is of the order 104 Ω, because the soma radius, b, lies between 25 and 50 μm, and the effective value of Re probably lies between 250 and 500 Ω cm

See 142
Figure 15. Figure 15.

Cortical symmetry and synchronous activation of the mitral cell population. A: schematic diagram of experimental recording situation. Microelectrode (ME) penetrates the olfactory bulb; reference electrode (RE) is distant. Mitral cells are arranged in an almost spherical cortical shell; their axons all project into the lateral olfactory tract. Single‐shock stimulation to the lateral olfactory tract results in synchronous antidromic activation of the mitral cell population. B: complete spherical symmetry of a cortical arrangement of mitral cells. Cone indicates a volume element associated with one mitral cell; arrows indicate extracellular current generated by this mitral cell; current is as though confined within its cone when activation is synchronous for the population. C: punctured spherical symmetry. Arrows inside the cone represent the primary extracellular current generated per mitral cell; dashed line (with arrows) represents the secondary extracellular current per mitral cell. Location of the reference electrode along the resistance of this secondary pathway serves as a potential divider. D: the potential divider aspect (C) combined with a compartmental model. Relations of both the microelectrode (ME) and the reference electrode (RE) to the primary extracellular current (PEC) and the secondary extracellular current (SEC) are shown. Generator of extracellular current (GEC) is a compartmental model representing the synchronously active mitral cell population. Solid arrows adjacent to the compartmental model represent the direction of membrane current flow at the moment of active, inward, somamembrane current (heavy black arrow); dendritic membrane current is outward. Open arrows represent the direction of extracellular current flow (both PEC and SEC) at this same moment

See 147,151
Figure 16. Figure 16.

Derivation of cable equation. Relation between cylindrical core conductor length increments and the lumped elements of the electric equivalent circuit are shown. A and B: relation of core current to the increment in voltage and in length (see Eq. 24–26). C and D: relation between membrane current and change in core current (see Eq. 27). E: membrane current divided into 2 parallel components, one capacitive and one resistive (see Eq. 33). F: lumped circuit approximation to a continuous cable, sometimes called a ladder network.
Figure 17. Figure 17.

Relation of applied current (at both intracellular and extracellular points) to membrane current and to longitudinal current (both intracellular core current and extracellular longitudinal current). A: 2 microelectrodes with a core conductor diagram somewhat similar to that used by Taylor 182. Intracel., intracellular; extracel., extracellular. B: lumped parameter circuit diagram used for an application of Kirchoff's law for conservation of current (see Eq. 51–56).
Figure 18. Figure 18.

Electric equivalent circuit model of synaptic membrane. Per unit area, Cm is the membrane capacity, Gr is the resting membrane conductance in series with battery Er representing the resting emf, Gε is the synaptic excitatory conductance in series with battery Eε representing the synaptic excitatory emf, and Gε is the synaptic inhibitory conductance in series with battery Eε representing the synaptic inhibitory emf (see Eq. 59 and 60)

This model was based on those of Fatt & Katz 47,48 and of Coombs, Eccles, and Fatt 29; see also Hodgkin & Katz 79
Figure 19. Figure 19.

Steady states for infinite and semi‐infinite lengths. A and B: semi‐infinite length extending from a sealed end (at X = 0) out toward X = ∞ an intracellular electrode applies I0 at X = 0; placement of the extracellular electrode is not critical because extracellular isopotentiality is assumed. C and D: doubly infinite length, with symmetry about intracellular electrode that introduces I0 at X = 0. E and F: intracellular electrodes at X = X1 and X = X2; I1 and I2 are those currents needed to clamp V to V1 at X = X1, and V to V2 at X = X2; the core conductor is sealed at X = 0.
Figure 20. Figure 20.

Finite length of core conductor, from X = 0 to X = X1. Core conductor is sealed at X = 0, but the boundary condition at X = X1 is adjustable. Core current, ii1 at X = X1 depends on the conditions there: whether a sealed or leaky termination, or whether a branch point. In B, the parent branch (trunk) has a diameter d0; one daughter branch has a diameter d11 and extends from X =X1 toX = X21; the other daughter branch has a diameter d12 and extends to X22
Figure 21. Figure 21.

Decrement of V with distance for different boundary conditions at the far end of a cylinder of finite length. Curves A, B, and C correspond to a sealed‐end boundary condition (dV/dX = 0) at X = 0.5, 1.0, and 2.0, respectively (see Eq. 104 for B1 = 0 in Eq. 101). Curves E, F. and G correspond to a voltage‐clamped boundary condition (V = 0, meaning Vm = Er) at X = 0.5, 1.0, and 2.0, respectively (see Eq. 107 for B1 = ∞ in Eq. 101). Curve D is a simple exponential (Eq. 110) corresponding to B1 = 1 in Eq. 101.

From Rall 139
Figure 22. Figure 22.

Branching diagram (upper left) and graph (below) showing steady‐state values of V as a function of X in all branches and trees of the neuron model, for steady current injected into the terminal of one branch. BI and BS designate the input branch and its sister branch, respectively; P and GP designate their parent and grandparent branch points, respectively; BC‐1 and BC‐2 designate first‐ and second‐cousin branches, respectively, with respect to the input branch; OT designates the other trees of the neuron model. Model parameters are N = 6, L = 1, M = 3, with equal electrotonic length increments ΔX = 0.25 assumed for all branches. Ordinates of graph express V/IRτ∞ values, as defined by Eq. 133

See 150
Figure 23. Figure 23.

Two examples of peeling the membrane time constant away from the first equalizing time constant, τ1; data are slopes (dV/dt) of response at a motoneuron soma to an applied current step. A: top, start of the membrane potential response to hyperpolarizing current steps with 5 traces superimposed. Tangents on the slope are drawn at the 1st, 3rd, 5th, and 7th ms for dV/dt determinations. Plot of dV/dt on logarithmic scale vs. time shows an almost linear late part of its slope but considerable deviations from linearity in the initial part (bottom). Replots of these deviations during the initial 4 ms on logarithmic scale (▴) reveal the second‐order time constant τ1, (dashed line). [From Lux et al. 119.] B: semilogarithmic plot of the slope (dV/dt) vs. t, of the response to a constant current step applied to a motoneuron soma. •, the dV/dt values plotted on a log scale; straight line through the tail has a slope implying ∞m, = 5.3 ms. ○, difference between the dV/dt values of the smooth curve and those of the dashed line extrapolated back from the straight tail. Heavier dashed line through ○ has a slope implying τ1 = 1.0 ms

From Burke & ten Bruggincate 18
Figure 24. Figure 24.

Two different graphs based on Eq. 161, as plotted by Davis & Lorente de No 33. A: curves plotted to the same amplitude scale show both slowing and reduced amplitude with distance. B: curves replotted relative to the steady‐state value at each location [i.e., V (X, T)/V (X,∞)] to highlight the changing shape of the delayed rise.
Figure 25. Figure 25.

Response function at the input branch terminal (solid curve), compared with 2 asymptotic cases (dashed curves); ordinates are plotted on a logarithmic scale. Inset, neuron model with input injected at one branch terminal. Solid curve represents the response function as T → ∞ (Eq. 182); its left intercept represents 1.0, and N = 6. Lower dashed curve is a straight line representing uniform decay and representing the asymptotic behavior of the response function as T → ∞ (Eq. 182); its left intercept represents a value of 1/6 because NL = 6. Upper dashed curve represents the asymptotic behavior as T → 0; this also represents the response for a semi‐infinite length of terminal branch (Eq. 183). For any combination that makes Q0 R∞/∞ = 1 mV. the values of the functions plotted here would correspond to V in millivolts

See 161
Figure 26. Figure 26.

Computed voltage time course at the input‐receiving branch terminal (solid curve) and at the soma (lower dashed curve) for a particular time course [I (T)] of injected current (upper dashed curve). Neuron model is shown at upper right, and parameters used were N = 6, M = 3, X1 − 0.25, X2 = 0.5, X3 = 0.75, and L = 1. Ordinate values for the solid curve using scale at left represent dimensionless values of V (L, T)/(2MRT∞ Ip e) [see 161]. Soma response (lower dashed curve) has been amplified 200 times; the ordinate values, using scale at right, represent dimensionless values of V (0, T)/(2MRT∞ Ip e) [(see 161]. Factor 2MRT∞ Ip e equals 8 × (4.56 RN) × (Ip e) which is approximately equal to 100 times the product RN and Ip. For example, if RN = 106Ω and Ip = 10−8 A, the above factor is approximately 1 V; then the left‐hand scale can be read in volts for V (L, T) and the right‐hand scale can be read in volts for V(0,T).
Figure 27. Figure 27.

Semi‐log plots of transient membrane potential vs. T at successive sites along the mainline in the neuron model for transient current injected into the terminal of one branch. BI designates the input branch terminal, while P, GP, and GGP designate the parent, grandparent, and great grandparent nodes, respectively, along the mainline from BI to the soma. Response at the terminals of the trees not receiving input directly is labeled OT. Model parameters, neuron branching diagram, and current time course are the same as in Fig. 26. Ordinate values represent dimensionless values of V (X, T)/(2MRT∞ Ip e) where V (X, T) was obtained as the convolution of I (T) with K (X, T; L) defined in reference 161.
Figure 28. Figure 28.

Semi‐log plots of voltage vs. T at all the branch terminals in the neuron model for transient current injected into the terminal of one branch. Refer to Fig. 26 for model parameters and input current time course. BI and BS designate the input branch terminal and the sister branch terminal, respectively. BC‐1 and BC‐2 designate the terminals of the 2 first‐cousin and the 4 second‐cousin branches, respectively, in the input tree, while OT designates the branch terminals of the other 5 trees. Transients are computed and scaled as indicated in Fig. 27

See 161


Figure 1.

Cylindrical core conductor with the spreading distribution of electric current indicated (only roughly) by dashed lines, for 2 arrangements of electrodes; cylinders extend to (±) infinite length. A: both electrodes (cathode and anode) are extracellular; some current flows entirely extracellularly, directly from anode to cathode in the interpolar region; some extracellular current flows out into the extrapolar region before crossing the membrane. B: intracellular (micropipette) electrode provides source of current that spreads along the core before crossing the membrane to the extracellular volume that is isopotential with a ground electrode.



Figure 2.

Flow of electric current from a microelectrode whose tip penetrates the cell body (soma) of a neuron. Full extent of dendrites is not shown. External electrode to which the current flows is at a distance far beyond the limits of this diagram.

From Rall 139


Figure 3.

Arbitrary dendritic branching to illustrate the subscript notation used to treat this problem. Originally L0 represented actual trunk length; subsequently, I have preferred to use ��0 for actual length and L0 = ��0/λ0 as the dimensionless electrotonic length of a dendritic trunk.

From Rall 139


Figure 4.

Symmetrical dendritic tree, illustrative of a class of trees that can be mathematically transformed into an equivalent cylinder 141,142,143,144 and into approximately equivalent chains of equal compartments. Dashed lines divide both the tree and the cylinder into 5 equal increments of electrotonic distance, having equal membrane surface area. In one specific example 141,143, the dendritic trunk diameter was 10 μm, and successive branch diameters were 6.3, 4.0, 2.5, 1.6, and 1.0 μm, which satisfy both symmetry and the constraint on d3/2 values. Chain of 5 compartments corresponds to the 5 increments of electrotonic distance above. Chain of 10 compartments, used for Fig. 7, 8, and 10, represents the soma as compartment 1, and progressive electrotonic distance out into the dendrites is represented by compartments 2–10.



Figure 5.

Distributions of membrane depolarization for an excitatory conductance step in the peripheral half of a dendritic tree, for several values of T = t/τ. In the central half of the dendritic tree (i.e., X = 0 to X = 0.5) the conductances are assumed to remain at their resting values. Over the peripheral half of the dendritic tree (i.e., X = 0.5 to X = 1.0) the conductance (Ge) is assumed to step from zero to a value of 2Gr at T = 0.

Calculations are based on equations in 141


Figure 6.

Transients of membrane depolarization at the central end, the peripheral end, and at the middle of a dendritic tree. A: 3 curves at left illustrate the same problem as in Fig. 5; uppermost curve corresponds to the peripheral end (X = 1); intermediate curve corresponds to X = 0.5; lowest curve corresponds to the central end (X = 0). B: 3 curves at right illustrate the response to a square conductance pulse. On step is the same as in A; off step occurs 0.2 units of T later; the duration of the square conductance pulse is indicated by vertical dashed lines.

For equations used to calculate the off transients, see 141


Figure 7.

Computed transients at the soma (compartment 1) for 4 synaptic inputs that differ only in their input locations. In each case, the synaptic input consisted of Ge = Gr in 2 compartments for a time interval, T = 0 to T = 0.25, indicated by heavy black bar. In A, the input was applied to compartments 2 and 3, in B to 4 and 5, in C to 6 and 7, and in D to 8 and 9

See Fig. 4 for the relation of such compartments to the dendritic branches; see 143 for equations and discussion of compartmental modeling


Figure 8.

Effect of two spatiotemporal sequences on transient soma‐membrane depolarization. Two input sequences, A→B→C→D and D→C→B→A, are indicated at upper left and upper right, respectively; component input locations are the same as in Fig. 7. Time sequence is indicated by means of the 4 successive time intervals, Δt1, Δt2, Δt3, and Δt4, each equal to 0.25 τ. Dotted curve shows the computed effect of Ge = 0.25 Gr in compartments 2–9 for the period τ = 0 to l = τ

For further details, see 143


Figure 9.

Right: computed EPSP's (solid traces) generated by synaptic currents of different time courses (dotted traces). Synaptic currents are assumed to be uniformly distributed over entire soma‐dendritic surface. Arrows on the lowermost EPSP indicate its half‐width. Left: plot of paired shape index values for EPSP's generated by uniformly distributed synaptic input with coordinates as labeled. The •, ▴, and ▪ on the plot represent the shapes of corresponding EPSP's on the right side of the figure

See 149


Figure 10.

Right: diagrammatic representation of the transformation of soma‐dendritic receptive surface of a neuron into a chain of 10 equal compartments. Below are graphs of computed EPSP's occurring in compartment 1, obtained with the compartmental model for the medium synaptic current time course (upper graph, dotted line). Synaptic current introduced equally to all compartments gave the upper computed EPSP (▴). Synaptic current localized to a single compartment gave the lower 3 computed EPSP's: compartment 1(), compartment 4 (), or compartment 8 (). Left: EPSP shape index values for computed EPSP's shown at right. Dashed line, locus of EPSP shapes generated when synaptic input is limited to the numbered compartment. Solid line, locus for spatially uniform depolarization of the cell. Note that scales are in units of dimensionless ratio t/τ

See 149


Figure 11.

Scatter diagram of normalized shape indices of the EPSP's from the motoneurons for which a time constant was obtained. •, EPSP's recorded from knee flexor motoneurons; ○, EPSP's recorded from ankle extensor motoneurons. Two dashed lines show theoretical boundaries for the shape indices with the assumed values of parameters (α = 12–100; ρα = 4–25; L = 0.75–1.5). The set of areas, each bounded by a continuous line, show the theoretical boundaries for particular distances

See 90


Figure 12.

Shape index plot for evoked EPSP's for which impedance measurements were available. Time to peak as abscissa; half‐width as ordinate; scales in milliseconds. •, EPSP's accompanied by a measurable impedance change; ○, EPSP's not accompanied by a detectable impedance change. Dotted outline, scatter of shape index values for observed miniature EPSP (for reference)

See 149,174


Figure 13.

Theoretically calculated relation between intracellularly and extracellularly recorded action potentials. Uppermost curve, experimental “AB spike” followed by an “A spike” as recorded intracellularly from a cat motoneuron by Nelson & Frank 128. The other intracellular curve represents the theoretically calculated passive electrotonic spread into a dendritic cylinder of infinite length; it corresponds to a radial distance, R = 18 [i.e., an electrotonic distance, x/λ, of (R − 1/40 = 0.425; this would correspond to about 600 μm in the examples considered]. Extracellular curves were calculated on the assumption of radial symmetry. Curve for R = 18 has been multiplied by 10 to aid the comparison of shape. Curve at R = 3 has a shape extremely similar to that at R = 1, except that the peak at R = 1 has an amplitude about 5 times that at R = 3

See 128,142


Figure 14.

Computed isopotential contours for a spherical soma with 7 cylindrical dendrites, of which only 3 can be seen in the plane shown here. Relative to 1 dendrite shown on the vertical (polar) axis, 3 dendrites were equally spaced at 60° from the polar axis, and 3 more dendrites were equally spaced at the equator (see inset). The soma was the sink for extracellular current; the dendritic cylinders were sources of extracellular current corresponding to passive electrotonic spread at the time of the peak of an antidromic action potential. For this calculation, dendritic λ was set equal to 40 times the somatic radius. Numbers labeling the contours correspond to the quantity Ve/(INRe/4πb), whereIN is the total current flowing from dendrites to soma, Re represents extracellular volume resistivity, and b represents the soma radius. For the particular case of the peak somatic action potential in a cat motoneuron, this numerical quantity expresses the value of Ve approximately in millivolts. This is because of the following order of magnitude consideration: IN is of the order 10−7 A, because the peak intracellular action potential is of the order 10−1 V, and the whole neuron instantaneous conductance is of the order 10−6 Ω−1;Re/4πb is of the order 104 Ω, because the soma radius, b, lies between 25 and 50 μm, and the effective value of Re probably lies between 250 and 500 Ω cm

See 142


Figure 15.

Cortical symmetry and synchronous activation of the mitral cell population. A: schematic diagram of experimental recording situation. Microelectrode (ME) penetrates the olfactory bulb; reference electrode (RE) is distant. Mitral cells are arranged in an almost spherical cortical shell; their axons all project into the lateral olfactory tract. Single‐shock stimulation to the lateral olfactory tract results in synchronous antidromic activation of the mitral cell population. B: complete spherical symmetry of a cortical arrangement of mitral cells. Cone indicates a volume element associated with one mitral cell; arrows indicate extracellular current generated by this mitral cell; current is as though confined within its cone when activation is synchronous for the population. C: punctured spherical symmetry. Arrows inside the cone represent the primary extracellular current generated per mitral cell; dashed line (with arrows) represents the secondary extracellular current per mitral cell. Location of the reference electrode along the resistance of this secondary pathway serves as a potential divider. D: the potential divider aspect (C) combined with a compartmental model. Relations of both the microelectrode (ME) and the reference electrode (RE) to the primary extracellular current (PEC) and the secondary extracellular current (SEC) are shown. Generator of extracellular current (GEC) is a compartmental model representing the synchronously active mitral cell population. Solid arrows adjacent to the compartmental model represent the direction of membrane current flow at the moment of active, inward, somamembrane current (heavy black arrow); dendritic membrane current is outward. Open arrows represent the direction of extracellular current flow (both PEC and SEC) at this same moment

See 147,151


Figure 16.

Derivation of cable equation. Relation between cylindrical core conductor length increments and the lumped elements of the electric equivalent circuit are shown. A and B: relation of core current to the increment in voltage and in length (see Eq. 24–26). C and D: relation between membrane current and change in core current (see Eq. 27). E: membrane current divided into 2 parallel components, one capacitive and one resistive (see Eq. 33). F: lumped circuit approximation to a continuous cable, sometimes called a ladder network.



Figure 17.

Relation of applied current (at both intracellular and extracellular points) to membrane current and to longitudinal current (both intracellular core current and extracellular longitudinal current). A: 2 microelectrodes with a core conductor diagram somewhat similar to that used by Taylor 182. Intracel., intracellular; extracel., extracellular. B: lumped parameter circuit diagram used for an application of Kirchoff's law for conservation of current (see Eq. 51–56).



Figure 18.

Electric equivalent circuit model of synaptic membrane. Per unit area, Cm is the membrane capacity, Gr is the resting membrane conductance in series with battery Er representing the resting emf, Gε is the synaptic excitatory conductance in series with battery Eε representing the synaptic excitatory emf, and Gε is the synaptic inhibitory conductance in series with battery Eε representing the synaptic inhibitory emf (see Eq. 59 and 60)

This model was based on those of Fatt & Katz 47,48 and of Coombs, Eccles, and Fatt 29; see also Hodgkin & Katz 79


Figure 19.

Steady states for infinite and semi‐infinite lengths. A and B: semi‐infinite length extending from a sealed end (at X = 0) out toward X = ∞ an intracellular electrode applies I0 at X = 0; placement of the extracellular electrode is not critical because extracellular isopotentiality is assumed. C and D: doubly infinite length, with symmetry about intracellular electrode that introduces I0 at X = 0. E and F: intracellular electrodes at X = X1 and X = X2; I1 and I2 are those currents needed to clamp V to V1 at X = X1, and V to V2 at X = X2; the core conductor is sealed at X = 0.



Figure 20.

Finite length of core conductor, from X = 0 to X = X1. Core conductor is sealed at X = 0, but the boundary condition at X = X1 is adjustable. Core current, ii1 at X = X1 depends on the conditions there: whether a sealed or leaky termination, or whether a branch point. In B, the parent branch (trunk) has a diameter d0; one daughter branch has a diameter d11 and extends from X =X1 toX = X21; the other daughter branch has a diameter d12 and extends to X22



Figure 21.

Decrement of V with distance for different boundary conditions at the far end of a cylinder of finite length. Curves A, B, and C correspond to a sealed‐end boundary condition (dV/dX = 0) at X = 0.5, 1.0, and 2.0, respectively (see Eq. 104 for B1 = 0 in Eq. 101). Curves E, F. and G correspond to a voltage‐clamped boundary condition (V = 0, meaning Vm = Er) at X = 0.5, 1.0, and 2.0, respectively (see Eq. 107 for B1 = ∞ in Eq. 101). Curve D is a simple exponential (Eq. 110) corresponding to B1 = 1 in Eq. 101.

From Rall 139


Figure 22.

Branching diagram (upper left) and graph (below) showing steady‐state values of V as a function of X in all branches and trees of the neuron model, for steady current injected into the terminal of one branch. BI and BS designate the input branch and its sister branch, respectively; P and GP designate their parent and grandparent branch points, respectively; BC‐1 and BC‐2 designate first‐ and second‐cousin branches, respectively, with respect to the input branch; OT designates the other trees of the neuron model. Model parameters are N = 6, L = 1, M = 3, with equal electrotonic length increments ΔX = 0.25 assumed for all branches. Ordinates of graph express V/IRτ∞ values, as defined by Eq. 133

See 150


Figure 23.

Two examples of peeling the membrane time constant away from the first equalizing time constant, τ1; data are slopes (dV/dt) of response at a motoneuron soma to an applied current step. A: top, start of the membrane potential response to hyperpolarizing current steps with 5 traces superimposed. Tangents on the slope are drawn at the 1st, 3rd, 5th, and 7th ms for dV/dt determinations. Plot of dV/dt on logarithmic scale vs. time shows an almost linear late part of its slope but considerable deviations from linearity in the initial part (bottom). Replots of these deviations during the initial 4 ms on logarithmic scale (▴) reveal the second‐order time constant τ1, (dashed line). [From Lux et al. 119.] B: semilogarithmic plot of the slope (dV/dt) vs. t, of the response to a constant current step applied to a motoneuron soma. •, the dV/dt values plotted on a log scale; straight line through the tail has a slope implying ∞m, = 5.3 ms. ○, difference between the dV/dt values of the smooth curve and those of the dashed line extrapolated back from the straight tail. Heavier dashed line through ○ has a slope implying τ1 = 1.0 ms

From Burke & ten Bruggincate 18


Figure 24.

Two different graphs based on Eq. 161, as plotted by Davis & Lorente de No 33. A: curves plotted to the same amplitude scale show both slowing and reduced amplitude with distance. B: curves replotted relative to the steady‐state value at each location [i.e., V (X, T)/V (X,∞)] to highlight the changing shape of the delayed rise.



Figure 25.

Response function at the input branch terminal (solid curve), compared with 2 asymptotic cases (dashed curves); ordinates are plotted on a logarithmic scale. Inset, neuron model with input injected at one branch terminal. Solid curve represents the response function as T → ∞ (Eq. 182); its left intercept represents 1.0, and N = 6. Lower dashed curve is a straight line representing uniform decay and representing the asymptotic behavior of the response function as T → ∞ (Eq. 182); its left intercept represents a value of 1/6 because NL = 6. Upper dashed curve represents the asymptotic behavior as T → 0; this also represents the response for a semi‐infinite length of terminal branch (Eq. 183). For any combination that makes Q0 R∞/∞ = 1 mV. the values of the functions plotted here would correspond to V in millivolts

See 161


Figure 26.

Computed voltage time course at the input‐receiving branch terminal (solid curve) and at the soma (lower dashed curve) for a particular time course [I (T)] of injected current (upper dashed curve). Neuron model is shown at upper right, and parameters used were N = 6, M = 3, X1 − 0.25, X2 = 0.5, X3 = 0.75, and L = 1. Ordinate values for the solid curve using scale at left represent dimensionless values of V (L, T)/(2MRT∞ Ip e) [see 161]. Soma response (lower dashed curve) has been amplified 200 times; the ordinate values, using scale at right, represent dimensionless values of V (0, T)/(2MRT∞ Ip e) [(see 161]. Factor 2MRT∞ Ip e equals 8 × (4.56 RN) × (Ip e) which is approximately equal to 100 times the product RN and Ip. For example, if RN = 106Ω and Ip = 10−8 A, the above factor is approximately 1 V; then the left‐hand scale can be read in volts for V (L, T) and the right‐hand scale can be read in volts for V(0,T).



Figure 27.

Semi‐log plots of transient membrane potential vs. T at successive sites along the mainline in the neuron model for transient current injected into the terminal of one branch. BI designates the input branch terminal, while P, GP, and GGP designate the parent, grandparent, and great grandparent nodes, respectively, along the mainline from BI to the soma. Response at the terminals of the trees not receiving input directly is labeled OT. Model parameters, neuron branching diagram, and current time course are the same as in Fig. 26. Ordinate values represent dimensionless values of V (X, T)/(2MRT∞ Ip e) where V (X, T) was obtained as the convolution of I (T) with K (X, T; L) defined in reference 161.



Figure 28.

Semi‐log plots of voltage vs. T at all the branch terminals in the neuron model for transient current injected into the terminal of one branch. Refer to Fig. 26 for model parameters and input current time course. BI and BS designate the input branch terminal and the sister branch terminal, respectively. BC‐1 and BC‐2 designate the terminals of the 2 first‐cousin and the 4 second‐cousin branches, respectively, in the input tree, while OT designates the branch terminals of the other 5 trees. Transients are computed and scaled as indicated in Fig. 27

See 161
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Article Title: Core Conductor Theory and Cable Properties of Neurons
 

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Wilfrid Rall. Core Conductor Theory and Cable Properties of Neurons. Compr Physiol 2011, Supplement 1: Handbook of Physiology, The Nervous System, Cellular Biology of Neurons: 39-97. First published in print 1977. doi: 10.1002/cphy.cp010103