An Introduction to Core-conductor Theory

Transcription

1 An Introduction to Core-conductor Theory I often say when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge of it is of a meagre and unsatisfactory kind. Lord Kelvin 1 Introduction It would not be inaccurate to suggest that the manner in which electric current flows along a semi-insulated conductor forms a central conceptual core in theoretical neuroscience. The passive properties of dendrites and the propagating action potential are only explicable within the context of the cable equation. The cable equation is a second order in space, first order in time, partial differential equation 1. A form of this equation was probably first derived by William Thomson ( ), later Lord Kelvin, during his involvement with the laying, design and analysis of the first trans-atlantic telegraph cables beginning in The cable equation is based on the simple notion of transverse current leakage between an inner and outer conductor due to an imperfect insulator as a consequence of the longitudinal flow of current within the inner conductor. 2 Assumptions the cell membrane is conceived of as cylindrical boundary of finite thickness separating the intra-cellular fluid (ICF) and the extra-cellular fluid (ECF). the ECF and ICF are homogeneous, isotropic and Ohmic. cylindrical symmetry is assumed for all variables subsequently defined i.e f(r, x, θ, t) f(r, x, t) f(x, t) (r will obviously constant at the level of the membrane). an electrostatic ( quasi-static ) approximation will be adequate. 1 mathematicians would call this a parabolic partial differential equation, physicists would be more inclined to call it a diffusion or heat equation 1

2 currents in the inner and outer conductors will flow in the longitudinal (x) direction only. trans-membrane current flows in the radial (r) direction only inner conductor insulator (membrane) a outer conductor Figure 1: Cross-section of the model cable illustrating the geometry of the inner conductor, insulator and outer conductor. 3 Definitions I o (x, t) = the total longitudinal current flowing in the +x direction in the outer conductor. (A) I i (x, t) = the total longitudinal current flowing in the +x direction in the inner conductor. (A) J m (x, t) is the membrane current density flowing from inner conductor to outer conductor. (A m 2 ) K m (x, t) is the membrane current per unit length flowing from inner conductor to outer conductor. (A m 1 ) 2 K eo (x, t) is the current per unit length due to external sources applied in a cylindrically symmetric manner to the outer conductor. (A m 1 ) K ei (x, t) is the current per unit length applied to the inner conductor. (A m 1 ) 2 this is the neurophysiological or physiologists convention and is often referred to as the inward negative definition in that the trans-membrane current is defined as negative when it flows from outer conductor to inner conductor. We will return to the issue of conventions at the end of this chapter. 2

3 V i (x, t) is the potential of the inner conductor. (V ) V o (x, t) is the potential of the outer conductor. (V ) V m (x, t) = V i (x, t) V o (x, t) is the trans-insulator (i.e trans-membrane) potential. (V ) r i is the resistance per unit length of the inner, cylindrically symmetric conductor. (Ω m 1 ) r o is the resistance per unit length of the outer, cylindrically symmetric conductor. (Ω m 1 ) a is the internal radius of the cylindrical shell. (m) r i x PSfrag replacements a x Figure 2: Geometry of the inner conductor Based on these definitions and assumptions we are now able to represent concisely in diagrammatic form the structure of the cable. This is illustrated in Figure 3. Note that we have collapsed the cylindrical symmetry of the cable. 4 Derivation of the core-conductor equations While it is quite easy to derive the cable equation directly without noting any intermediate equations it is more profitable to derive a limited form of the cable equation, called the core-conductor equations, which require no assumptions be made about the structure of the cell membrane or insulator. The core-conductor equations are easily derived by the use of Kirchoff s electrical circuit laws and a spatially discrete cable. 3

4 I i (x,t) K ei(x,t) x inner conductor I i K ei (x+ x,t) x (x+ x,t) r i x K m (x,t) x (a) V i (x,t) r i x K m (x+ x,t) x (b) V i (x+ x,t) r i x insulator (membrane) I o I o (x,t) (x+ x,t) (c) (d) r x o V o (x,t) r x o V o (x+ x,t) r x o K eo (x,t) x K eo (x+ x,t) x outer conductor x x+ x Figure 3: Schematic diagram of the model cable based on the definitions of the proceeding sections 4.1 Kirchoff s First Law (conservation of current) By referring to nodes (a) and (c) in Figure 3 and by noting that charge (current) must be conserved we obtain I i (x, t) + K ei (x, t) x = I i (x + x, t) + K m (x, t) x node (a) (1) I o (x, t) + K m (x, t) x = I o (x + x, t) + K eo (x, t) x node (c) (2) rearranging, dividing both equations through by x and taking the limit as x 0 we obtain I i (x, t) x I o (x, t) x = K ei (x, t) K m (x, t) (3) = K m (x, t) K eo (x, t) (4) 4

5 4.2 Kirchoff s Second Law (conservation of energy) From Figure 3 and a simple application of Ohm s Law we obtain V i (x, t) V i (x + x, t) = r i xi i (x + x, t) nodes (a) and (b) (5) V o (x, t) V o (x + x, t) = r o xi o (x + x, t) nodes (c) and (d) (6) dividing through by x and taking the limit as x 0 we obtain V i (x, t) x V o (x, t) x = r i I i (x, t) (7) = r o I o (x, t) (8) However V m = V i V o and thus by subtracting equation (8) from (7) we obtain V m (x, t) x = r o I o (x, t) r i I i (x, t) (9) 4.3 The core-conductor equations The key to understanding the derivation of the core-conductor equations is in the combination of equations (3), (4), and (9). Therefore differentiating equation (9) with respect to x we obtain 2 V m (x, t) I o (x, t) I i (x, t) = r x 2 o r i x x (10) and substituting in equations (3) and (4) we obtain 2 V m (x, t) x 2 = (r o + r i )K m (x, t) r o K eo (x, t) r i K ei (x, t) (11) This result is known as the core-conductor equation, and forms the skeleton about which the full cable equation can be developed. 5

6 5 Derivation of the Cable Equation In the derivation of the core-conductor equation no assumptions were made regarding the form of the total trans-membrane current per unit length, K m (x, t), in that the electrical properties of the insulator (membrane) were ignored. However early electro-physiological studies had established that, to good approximation, the electrical properties of a cell membrane could be represented by an equivalent circuit consisting of a capacitor and resistor in parallel. Still assuming cylindrical symmetry then a small section of our model dendrite or axon (i.e cable) of length x can be represented diagrammatically as in Figure 4. ĉ m ( x) is the capacitance of our small section of cable (which has a total surface area of 2πa x). Similarly ˆr m ( x) is the trans-membrane resistance of this small section of cable. inner conductor V i (x, t) K m (x, t) x ĉ m ( x) ˆr m ( x) = 1/ĝ m ( x) PSfrag replacements outer conductor V o (x, t) Figure 4: Equivalent electrical circuit for a cylindrically symmetric segment of cell membrane. ĉ m ( x) is the total capacitance and ˆr m ( x) is the total (transmembrane) resistance of this small segment of cable. By referring to Figure 4 the membrane current per unit length is then easily seen to be given by the sum of a resistive and capacitive component K m (x, t) x = ĉ m ( x) t [V i(x, t) V o (x, t)] + [V i (x, t) V o (x, t)]/ˆr m ( x) capacitive resistive = ĉ m ( x) t [V i(x, t) V o (x, t)] + [V i (x, t) V o (x, t)]ĝ m ( x) = ĉ m ( x) V m(x, t) t + ĝ m ( x)v m (x, t) (12) 6

7 As defined the membrane capacitance and membrane resistance are functions of x. In order to take the limit as x 0 a more explicit relationship between these quantities and x is required. Therefore assuming that ĉ m ( x) and ˆr m ( x) are not functions of x we can define the the following relationships ĉ m ( x) = c m x = 2πaC m x (13) ĝ m ( x) = g m x = 2πaG m x (14) where c m and g m are the capacitance and conductance per unit length respectively (F m 1, S m 1 ) and C m and G m are the capacitance and conductance per unit area respectively (F m 2, S m 2 ). By inserting these definitions into equation (12) and dividing both sides through by x we obtain V m (x, t) K m (x, t) = c m t V m (x, t) = c m t + V m g m (15) + V m /r m (16) Be careful to note the meaning and units of r m in the last equation! By substituting equation (16) into (11) and rearranging we obtain the following second-order partial differential equation r m c m V m t = r { } m 2 V m r o + r i x + r ok 2 eo (x, t) + r i K ei (x, t) V m (17) This result is commonly referred to as the cable equation. Ignoring any external current input (i.e K ei = K eo = 0) it is often convenient to rewrite the above equation as τ m V m t = λ 2 2 V m c x V 2 m (18) where λ c = r m /(r o + r i ) and τ m = r m c m 3. For reasons that will soon become apparent λ c is called the cable space constant and τ m is called the membrane time constant. Further by defining the dimensionless quantities X = x/λ c and T = 3 the student should note that r m c m = R m C m, where R m = 1/G m 7

8 t/τ m equation (18) can be written as the following parabolic partial differential equation V m T = 2 V m X 2 V m (19) Solutions of the partial differential equations so defined depend, in addition to the stated electrical properties, on the initial conditions and the boundary conditions (i.e the value of V m at either ends of the cable). Further, under certain assumptions, equations (17-19) are able to describe the passive properties of multiply branched and tapered cables. The ability to incorporate multiply branched cables is of particular interest in describing the passive electrical properties of dendrites and axons. The interested reader should consult Rall (1989) for further details. 5.1 The cable space constant Let a constant trans-membrane current be applied at some point along the model cable. As the membrane capacitance becomes polarised with the passage of time the trans-membrane current density will become a function of axial distance only. Thus for sufficiently long times Vm 0. This corresponds to a steady state. t For increasing distances away from this site of current injection the membrane potential would be expected to decrease due to the continual trans-membrane leakage of current. Thus at steady state the partial differential cable equation is reduced to an ordinary differential equation (ODE) λ 2 d 2 V m c = V dx 2 m (20) As can be verified by direct substitution, a general solution to this ODE for constant current applied at x = 0 is 4 V m (x) = A exp[x/λ c ] + B exp[ x/λ c ] (21) where A and B will depend on the boundary conditions. For boundary-value 4 This is easily derived by substituting exp[ λx] into the differential equation. The solution of the resulting characteristic equation (which is a simple quadratic equation) defines the solution basis set. The student should refer to any textbook on the solution of ordinary differential equations to be reminded of the procedure. 8

9 solutions it is often convenient to use another pair of solutions to the second-order ordinary differential equation which are (as can be verified by direct substitution) V m (x) = A cosh(x/λ c ) + B sinh(x/λ c ) (22) where cosh(x) = (exp[x] + exp[ x])/2 and sinh(x) = (exp[x] exp[ x])/2. The space constant is useful as a parameter for estimating how far an axon or dendrite can propagate activity passively. Typical values for large myelinated axons are 1 2 mm whereas for small unmyelinated axons or dendrites the values range upwards from 30 µm. We will now consider the determination of the coefficients for the following three cases infinite cylinder a finite cylinder of length l with a sealed ends at x = 0 and x = l a finite cylinder of length l with a sealed end at x = 0 and a killed end at x = l (i.e the end at x = l is clamped to the resting membrane potential V m = 0) infinite cylinder For the case in which the cable extends from to + we require V ( ) = V ( ) = 0 and thus for x 0 A = 0 and B = V 0 in equation (21), where V 0 is the membrane potential at the site of the current passing electrode. Similarly for x 0 A = V 0 and B = 0 and thus V m (x) = V 0 exp[ x /λ c ] (23) for a constant current applied at x = 0. 9

10 5.1.2 finite cylinder, sealed end at x = l The idea of the sealed end is that axial current is prevented from flowing across a boundary. For the case now considered this boundary will be the end of the cable at x = l. Further we will assume that a current, I app, is injected at x = 0 at the inner conductor and V m has attained its steady-state value. By referring to equation (9) it is seen that the following conditions must be satisfied dv m dx = r i I app x=0 dv m dx = 0 (24) x=l These conditions are known as Neumann conditions 5. These boundary conditions are used to solve for the coefficients A and B in our general solution (22) by observing that dv m dx = A λ c sinh(x/λ c ) + B λ c cosh(x/λ c ) (25) Thus at x = 0 r i I app = B/λ c i.e B = r i I app λ c, whereas at x = l which gives 0 = A sinh(l/λ c ) + B cosh(l/λ c ) A = B coth(l/λ c ) = r i I app λ c coth(l/λ c ) where coth(x) = cosh(x)/ sinh(x). Thus the solution is V m (x) = r i I app λ c [coth(l/λ c ) cosh(x/λ c ) sinh(x/λ c )] (26) by rearranging the above equation we get 5 More specifically Neumann conditions specify normal gradients on the boundary. 10

11 V m (x) = r ii app λ c cosh(l/λ c ) cosh(l/λ c ) sinh(l/λ c ) [cosh(l/λ c) cosh(x/λ c ) sinh(l/λ c ) sinh(x/λ c )] which can be rewritten as V m (x) = V m(0) cosh[(l x)/λ c ] cosh(l/λ c ) 0 x l sealed end at x = l (27) with V m (0) = r i I app λ c coth(l/λ c ) and where we have used the identity cosh(a B) = cosh(a) cosh(b) sinh(a) sinh(b) V/V(0) killed end sealed end infinite cable x (in units of the space constant) Figure 5: Attenuation of the membrane potential as a function of distance in a cable with differing boundary conditions. The middle curve is a plot of equation (23) for x 0. Distances are dimensionless and are in units of λ c. Note that the attenuation of voltage with distance for the infinite cable lies in between that for the sealed end (upper curve) and the killed end (lower curve) cables of length λ c. Both finite cables have an electrotonic length, L = l/λ c, of unity. 11

12 5.1.3 finite cylinder, killed end at x = l The idea of the killed end is that one of the boundaries is clamped to the resting membrane potential, which in the absence of ionic batteries will be V m = 0. Such a boundary condition is known as a Dirichlet condition. Again we assume that a current, I app, is applied to the inner conductor at x = 0 and that V m has achieved its steady state. Therefore our solution (equation (22) ) must satisfy the following conditions dv m dx x=0 = r i I app V m (l) = 0 (28) In a manner similar to that of the last section our solution is V m (x) = V m(0) sinh[(l x)/λ c ] sinh(l/λ c ) 0 x l killed end at x = l (29) with V m (0) = r i I app λ c tanh(l/λ c ) and where we have made used of the identity sinh(a B) = sinh(a) cosh(b) cosh(a) sinh(b). 5.2 The membrane time constant The other extreme case of the cable equation is when Vm = 0. The cable is then x collapsed to a single isopotential element and the cable equation then becomes the following ordinary differential equation τ m dv m dt = V m (30) which has the general solution V m (t) = A exp[ t/τ m ] (31) where A is a constant that will depend on the initial conditions. Typical values for τ m range from 5 50 ms. 12

13 5.3 The addition of ionic batteries PSfrag replacements Trans-membrane current is mediated by the flux of ions, through protein pores, between the intra-cellular and extra-cellular fluids. The flux of each type of ion will depend on its electro-chemical gradient, which in turn depends on the trans-membrane concentration gradients and the cell membrane potential, and the selective permeability of the membrane to any given ionic species. Further, the membrane permeability for any given ion may be regulated by one or more conductances. All of these conductances may be regulated by endogenous ligands (e.g neurotransmitters), membrane potential and the concentrations of other ions in the extracellular and intracellular fluids. The total trans-membrane current density will depend on the contributions of the individual ionic currents. Figure 6 displays a circuit diagram describing the electrical properties of nerve membrane which incorporates the individual ionic currents. intracellular J m (x, t) G i (x, t) G j (x, t) G k (x, t) C m J c (x, t) E i E j E k J i (x, t) J j (x, t) J k (x, t) extracellular Figure 6: An equivalent electrical circuit of a small section of neuron membrane. See text for definitions. From Figure 6 it is easily ascertained that J m = J c + k J k (32) where J m is the total trans-membrane current flowing per unit area (A m 2 ) which V is composed of a capacitive current (J c = C m m t ) and k ionic currents (J k ). For each ionic conductance we have 13

14 V i 2πaJ k(x) x 2πaG k (x) x E k = V o and thus J k = (V m E k )G k (33) where E k is the equilibrium or reversal potential for the ionic species k and where the explicit dependence of J k and G k on x has been removed. E k represents the membrane potential at which there is no net trans-membrane current for ions of type k. E k is determined for a given internal and external ion concentrations by the Nernst equation E k = RT F ln Co k C i k (34) where C o k and Ci k are the external and internal ion concentrations respectively. By noting that K m (x, t) x = 2πaJ m (x, t) x the cable equation (equation 17) can be re-written as (see also equation 12) 1 2 V m 2πar i x 2 = C m V m t + k (V m E k )G k (x, t, V m ) J ei (x, t) (35) where r o r i and J ei (x, t) is an internally applied (positive outwards) current density (A m 2 ). Note that the dependence of trans-membrane ionic conductances on the membrane potential, time and space is made explicit. There may be more than one conductance associated with any given ion. The essence of the Hodgkin-Huxley equations, which describe mathematically the form of the action potential, is in the removal of any explicit time dependence of the ionic conductances by the use of empirically determined auxiliary differential equations. 14

15 5.3.1 A note on conventions By lumping all trans-membrane currents into a single term, J op ion, and assuming spatial homogeneity equation (35) becomes C m dv m dt + J op ion = J ei which implies that an externally applied current ( outward positive or inward negative ) will depolarise the cell membrane whereas transmembrane ion currents ( outward positive ) will hyperpolarise the cell membrane. The superscript, op, indicates the outward positive convention. By noting that J op ip ion = Jion we can rewrite the above equation so that all currents have a consistent effect of the membrane potential i.e C m dv m dt = J ip ion + J ei where J ip ion = k (E k V m )G k. It is important to appreciate that such a convention is of no conceptual significance rather it allows all currents to be treated consistently without making a special case for ionic currents. 6 References Johnston D and Maio-sin Wu S. (1995). Foundations of Cellular Neurophysiology, MIT Press, Cambridge,MA. Rall, W. (1989). Cable theory for dendritic neurons, in C. Koch and I. Segev (eds), Methods in Neuronal Modeling, MIT Press, Cambridge, MA, chapter 2, pp Weiss J L (1996). Cellular Biophysics, Volume 2: Electrical Properties, MIT Press, Cambridge, MA. 15