International Journal of Bifurcation and Chaos, Vol. 16, No. 12 (2006) 3605–3616c© World Scientific Publishing Company

ANALYTICAL DYNAMICS OF NEURON PULSEPROPAGATION

PAUL E. PHILLIPSON∗ and PETER SCHUSTERInstitut fur Theoretische Chemie, Universitat Wien,

Wahringerstraße 17, A-1090 Wien, Austria∗Austrian Academy of Sciences, Dr. Ignaz Seipel-Platz 2,

A-1010 Wien, Austria

Received July 12, 2005; Revised January 11, 2006

The four-dimensional Hodgkin–Huxley equations describe the propagation in space and time ofthe action potential v(z) along a neural axon with z = x + ct and c being the pulse speed. Thepotential v(z), which is parameterized by the temperature, is driven by three gating functions,m(z), n(z) and h(z), each of which obeys formal first order kinetics with rate constants thatare represented as nonlinear functions of the potential v. It is shown that this system can beanalytically simplified (i) in the number of gating functions and (ii) in the form of associated ratefunctions while retaining to close approximation quantitative fidelity to computer solutions ofthe exact equations over the complete temperature range for which stable pulses exist. At a giventemperature we record two solutions (T < Tmax) corresponding to a high-speed and a low-speedbranch in speed-temperature plots, c(T ), or no solution (T > Tmax). The pulse is considered ascomposed of two contiguous parts: (i) a pulse front extending from v(0) = 0 to a pulse maximumv = Vmax, and (ii) a pulse back extending from Vmax through a pulse minimum Vmin to a finalregression back to v(z → ∞) = 0. An approximate analytic solution is derived for the pulsefront, which is predicted to propagate at a speed c(T ) = 1203 Θ

38 (T C) cm/sec, Θ = 3

T−6.310 in

close agreement with computer solution of the exact Hodgkin–Huxley equations for the entirepulse. These results provide the basis for a derivation of two-dimensional differential equationsystems for the pulse front and pulse back, which predict the pulse maximum and minimum overthe operational temperature range 0 ≤ T ≤ 25C, in close agreement with the exact equations.Most neuron dynamics studies have been based on voltage clamp experiments featuring externalcurrent injection in place of self-generating pulse propagation. Since the behaviors of the gatingfunctions are similar, it is suggested that the present approximations might be applicable tosuch situations as well as to the dynamics of myelinated fibers.

Keywords : Hodgkin–Huxley equations; action potentials; neuron models; nonlinear dynamics;neuron pulse propagation.

1. Neuron Pulse Propagation andthe Hodgkin–Huxley Equations

Conductance mechanisms for the propagation ofa pulse along an unmyelinated neural axon wereencapsulated within a predictive theory by theequations of Hodgkin and Huxley [1952]. These

equations became the prototype for description ofneural pulse propagation and provide the basis forall subsequent conduction models of neural behav-ior. The Hodgkin–Huxley equations relate the prop-agating action potential v to sodium, potassiumand leak conductances INa, IK, Ileak causing the

∗Permanent Address: Department of Physics, Box 390, University of Colorado, Boulder, CO 80309, USA E-mail:phillipe@colorado.edu

3605

3606 P. E. Phillipson & P. Schuster

propagation according to

Dd2v

dz2− c

dv

dz− 1

Cm[INa + IK + Ileak] = 0

with z = x + ct (1)

where the pulse is moving in the x-direction fromright to left with speed c at time t, Cm [1µ F/cm2]is the membrane capacitance, and the constant Dfunctions as a diffusion coefficient determined bythe axon radius, r [238µm], and the resistivity ofthe intracellular space R2 [35.4Ω · cm] according toD = r/2R2Cm [336.2 cm2/sec]. The dependencies ofthe conductances upon v were established by fit toexperimental data accumulated by clamped voltagestudies, and are given by

INa = gNam3h(v − ENa) IK = gKn4(v − EK)Ileak = gm(v − Vleak)

(2)

where [m,h, n] are three gating functions assumedto obey a formal first order kinetics according to

cdk

dz= Θ(T )[αk − (αk + βk)k],

Θ = 3ToC−6.3

10 , k ≡ [m,h, n]

αm =x

ex − 1, x =

25 − v

10, βm = 4exp

[− v

18

](3a)

αh = 0.07 exp[− v

20

], βh =

1ex + 1

, x =30 − v

10

(3b)

αn =x

10[ex − 1], x =

10 − v

10,

(3c)βn = 0.125 exp

[− v

80

]where the temperature dependence of thepulse enters by the multiplicative factor Θ(T )[Cooley & Dodge, 1966]. The conductances [gm,gk, gm] = [120, 36, 0.3mΩ−1/cm2] and the poten-tials [ENa, EK] = [115,−12mV] were determinedexperimentally, and Vleak = 10.5989mV fixed suchthat v(0) = 0. One of the successes of the Hodgkin–Huxley formulation is that the INa(v) and IK(v)functions, which fit with remarkable precision volt-age clamp data,1 succeeded also in predicting thespeed of a propagating pulse in accordance withexperiment. The complexity of these equations

stimulated attempts to simplify them in such away that on the one hand the qualitative behav-iors would be preserved while on the other theimportant features of the underlying dynamicalmechanisms could be better understood. The mostprominent formulation is the equations initiated byFitzHugh [1961], which approximates the Hodgkin–Huxley dynamics for the clamped case by postulat-ing a two-dimensional differential equation systemas an extension of the van der Pol equation for arelaxation oscillator [Jackson, 1991; Phillipson &Schuster, 2001, 2004]. Nagumo et al. [1962] subse-quently demonstrated that replacing the constantterm of current injection in FitzHugh’s equationby spatial variation of the action potential as inEq. (1) produces pulse solutions very similar tothose predicted by the Hodgkin–Huxley equations.As a result, the FitzHugh–Nagumo equations, asthey are known, are invariably invoked as a proto-typic encapsulation of Hodgkin–Huxley dynamics[Cronin, 1987; Murray, 1993; Koch, 1999; Gerstner& Kistler, 2002]. A comparative study [Phillipson &Schuster, 2005] of the Hodgkin–Huxley andFitzHugh–Nagumo equations have revealed bothqualitative and quantitative relationships betweenthese two systems. On the other hand, rather thanconsidering an analogical model, there also havebeen attempts to simplify the Hodgkin–Huxleyequations directly through various assumptionsabout the gating functions and separation of theneuronal dynamics into fast and slow processes[Cronin, 1987; Gerstner & Kistler, 2002] as well astransformation of the gating functions into equiv-alent potentials [Kepler et al., 1992]. The majorityof these studies has been concerned with reductionsof the voltage-clamped Hodgkin–Huxley equations.

The purpose of the present study is to considerapproximations of the Hodgkin–Huxley dynamicsfor pulse solutions which maintain the mathemat-ical structure of the gating functions. The nextsection presents the discussion of a computer con-firmed approximate conservation principle resultingin the formal elimination of the h-gate, combinedwith analytical modification of the αn rate function.Combination of the two approximations results ina substantially simpler ODE representation of thedynamics, which is quantitatively faithful to thepredictions of the Hodgkin–Huxley theory. A neuralpulse v(z) is characterized by a speed c and a pulse

1Equation (1) reduces to the voltage-clamped equations upon replacing the second derivative term by an external injectioncurrent and remapping the independent variable according to z → t with c = 1.

Analytical Dynamics of Neuron Pulse Propagation 3607

front which grows to a maximum Vmax followedby a pulse back which features a pulse minimumVmin. An analytic expression for the pulse speed andpulse front as a function of temperature is arrivedat. Combining the approximations with the pulsefront solution provides the basis for a subsequenttwo-dimensional dynamics which predicts the pulsemaximum as a function of the temperature. Herethe m- and n-gates function as concerted machineswhereby the former drives the formation of theaction potential pulse front while the latter guaran-tees that the front peaks at Vmax. This informationis then used to facilitate description of the evolutionof the pulse back within the framework of a paralleltwo-dimensional system. Now the m-gate is slavedby the action potential which is driven through itsminimum by the dynamics of the n-gate. Since theapproximate procedures proceeds from the partic-ular functional dependencies of the rates αk, βk ofEqs. (3) upon v defining the Hodgkin–Huxley equa-tions, the question naturally arises as to the suit-ability of the present assumptions when applied toother conductance models. This question and thediscussion of the limitations of the present approachwill be considered in the final section.

2. Approximations ofthe Hodgkin–Huxley Equations

Figure 1 shows computer solution of the Hodgkin–Huxley equations for an oft-quoted temperature ofT = 18.5C. Since the pulse peak at any tem-perature is limited by the magnitude of sodiumpotential ENa [Phillipson & Schuster, 2005] thescaled dimensionless quantity v/ENa [solid blackline] is plotted on the unit scale. On the samescale are the dimensionless gating functions whichare similarly restricted [0 ≤ m,h, n ≤ 1]. Recall-ing that the pulse is traveling from right to left, itcan be conceptually divided into two parts (i) thepulse front extending from v(0) = 0 to the pulsemaximum, (ii) the pulse back extending from themaximum past the pulse minimum and finally ter-minating at v(z → ∞) = 0. The pulse front isdominated by the relatively fast variable m duringwhich the sodium channel is open while the gateh which tends to shut down the sodium channeland the gate n which opens the potassium chan-nel hardly come into play [FitzHugh, 1969]. Thegating functions have rest solutions k0(v) for which(dk/dz) = 0, k0 = αk(v)/αk(v) + βk(v), k = m,n, h.

Fig. 1. Computer solution of the Hodgkin–Huxley equations (1)–(3) for T = 18.5C. From r = 238 µ, R2 = 35.4 Ω · cm,Cm = 1 µF/cm2 reported by Hodgkin and Huxley for the squid giant axon [Hodgkin & Huxley, 1952] then D in Eq. (1)is D = 336.2 cm2/sec. The initial conditions are v(0) = 0, k(0) = αk(0)/(αk(0) + βk(0)) = [m(0), h(0), n(0)] =[0.0529324, 0.596121, 0.317677]. Black line: v(z)/ENa, ENa = 115 mV, dashed black line: αn(v) of Eq. (3c), green, red andblue lines: gating functions m, h and n from Eqs. (3) for k = m, h, n, respectively, green dashed line: stationary solu-tion m0 = αm/(αm + βm) for dm/dz = 0, αm, βm from Eq. (3a), purple line: βn(v) of Eq. (3c), and dashed purple line:βn(0) = 0.125.

3608 P. E. Phillipson & P. Schuster

The initial rest state is fixed by v(0) = 0 and[m0(0), h0(0), n0(0)] where the specific values aregiven in the caption of Fig. 1. Progressing towardsthe maximum the m-gate (green line) almost par-allels v/ENa, saturating in the vicinity of the maxi-mum at approximately unity. At the same time, theh and n gates evolve slowly with increasing z withh diminishing and n increasing from their rest val-ues to cross each other at roughly a midway point(h(0) + n(0))/2 prior to the pulse maximum. Alsoshown is a plot of the rate function αn (dashedblack line). This rate function is seen to almostduplicate the scaled action potential v/ENa itself.On the basis of these considerations, the Hodgkin–Huxley model will be subjected to the following twoapproximations

h(0) + n(0) = n(v) + h(v) . . . conservation and(4a)

αn =v

ENa+ α0, α0 = 0.058198

. . . global linearization. (4b)

Reducing the dimensionality of the system byexpressing h in terms of n according to Eq. (4a),computer solution of Eq. (1) using these approxima-tions is shown in Fig. 2. The values obtained at T =18.5C for [c , Vmax, Vmin] are [1875 (1873) cm/sec,91.1 (90.6)mV, −9.8(−9.67)mV] where the figures

Fig. 2. Computer solution of the Hodgkin–Huxley equa-tions (1)–(3) for T = 18.5C with approximations. The fol-lowing approximations were applied: (1) h eliminated accord-ing to h(z) = h(0) + n(0) − n(z), h(0) = 0.596121, n(0) =0.317677, and (2) αn = (v/ENa) + αn(0), αn(0) = 0.058198.The line of symmetry indicates the crossover point (h(0) +n(0))/2 at which h = n.

in bold are the values for the exact solutions of theHodgkin–Huxley equations. The general trend ofdecreasing pulse maximum with increasing temper-ature [Huxley, 1959a] is shown in Table 1. Agree-ment between the exact computer solution withinapproximately 1% extends over the realistic temper-ature range for the axon. At a given temperature,the Hodgkin–Huxley equations also predict the exis-tence of lower speed pulse train solutions [Huxley,1959b; FitzHugh, 1969] which are similarly repro-duced with comparable fidelity with the approxi-mations of Eqs. (4). For example, at temperatures6.3C and 18.5C the present approximations resultin pulse trains at speeds of 188 (214) and 537 (560)cm/sec, respectively. The complimentary nature ofh and n has been noted before based upon the sim-ilarity of their time constants [αh,n + βh,n]−1, sug-gesting thereby that n and h sum approximately tosome constant [Gerstner & Kistler, 2002]. The iden-tification of the constant with the sum itself at restmakes the stronger statement that the interdepen-dent dynamics of these two gates are constrained byconservation of this sum whose value is fixed at rest.This is not to suggest that the underlying chemicalmechanisms for these two gates rigidly lock themby a common push-pull mechanism. What is sug-gested is that the chemical mechanisms are suffi-ciently complementary as to be approximated bysuch a mechanism. The further approximation ofαn as simply linear in the action potential is con-sidered global which holds best over the significantregions of the pulse, as can be seen from Fig. 1. Itfails only when v is small near the beginning andend of the pulse.

The approximations introduced above could bejustifiable to a different degree of reliability at dif-ferent temperatures. In addition, the full Hodgkin–Huxley equations sustain two pulse solutions at dif-ferent speeds as a function of temperature. Thereis a maximum temperature above which no sta-ble pulse solution exists [Phillipson & Schuster,2005] and there is a maximum speed c at a tem-perature a few degrees below the maximum tem-perature. Whether or not our approximations arevalid only for one or a few temperatures or rep-resentative in general for the whole temperaturerange is worth being investigated. In Fig. 3 theprediction of the speed as a function of temper-ature, c(T ) and the existence of two solutions orno solution of the Hodgkin–Huxley scenario (redline) is compared therefore with the predictions ofthe approximate equations (4a, 4b) (green line).

Analytical Dynamics of Neuron Pulse Propagation 3609

Table 1. Comparison of approximations for pulse speed, pulse maximum and pulse minimum.

T Hodgkin–Huxleya Eqs. (4)b Pulse Front Theoryc Pulse Back Theoryd

[C] [cm/sec] [mV] [cm/sec] [mV] [cm/sec] [mV] vf (0) [mV] vf (0)

0 949.51 (105.7) 936 (105) 928 (107) [0.4204]

5 1170.01 (103.7) 1157 (103) 1140 (105) [0.2976]

6.3 1231.47 (103.0,−10.9) 1219 (102) 1203 (104) [0.2727] −10.9 [0.5436]

10 1415.26 (100.5,−10.7) 1405 (100) 1401 (101) [0.2141] −10.8 [0.5686]

15 1681.10 (96.5,−10.3) 1676 (95) 1722 (96) [0.1577] −10.3 [0.6015]

18.5 1873.33 (90.6,−9.7) 1875 (91) 1989 (91) [0.1297] −9.8 [0.6294]

20 1955.05 (88.0,−9.3) 1961 (89) 2115 (88) [0.1199] −9.4 [0.6448]

25 2205.45 (77.2,−7.5) 2237 (80) 2599 (78) [0.0946] −7.8 [0.6880]

aComputer solutions of Hodgkin–Huxley equations (1)–(3) [Phillipson & Schuster, 2005]. The numbers in parentheses for eachtemperature are (Vmax, Vmin).bComputer solutions of Hodgkin–Huxley equations with the approximations of Eqs. (4a) and (4b): h(v) eliminated by condi-tion Eq. (4a): h(v) = h(0) + n(0)− n(v) and the exact rate function αn of Eq. (3c) replaced by the approximate rate functionof Eq. (4b). The number in parentheses for each temperature is pulse maximum.cPulse speed c(T ) = 1203 Θ(T )

38 from Eq. (8). The number in parentheses is pulse maximum from computer solution of

Eqs. (9), (10a) and (10b), the number in square brackets is the value of vf (0) parameterizing the driving front solution Eq. (7)at which the pulse is stable. Only four figures are reported although computations were done through 16 figures to achievethe reported maximum.dPulse minimum from computer solution of Eqs. (9), (11a) and (11b). The number in square bracket is the value of n(Vmax)parameterizing the solution at which the pulse is stable. Four figures are reported although computations were done through16 figures to achieve the reported minimum. Initial condition is (dv/dz)|0 = 0 and v(0) = Vmax as input from the values inthe adjacent pulse front theory column.

Fig. 3. The speed as a function of temperature, c(T ). Com-putations were performed for the exact Hodgkin–Huxleyequations (1)–(3) [red curve; Phillipson & Schuster, 2005],for the approximations shown in Eqs. (4) [green curve] and,in addition, for one further simplification, which replacesthe exponential function in βn(v) by the linear approxi-mation: βn(v) = 0.125 · exp(−v/80) ≈ 0.125 · (1 − v/80)[blue curve]. The solutions on the upper branch from lowtemperatures up to a temperature between 25C and 28Cappear as single pulses, whereas pulse trains are obtainedfor rest of the curve c(T ), i.e. on the lower branch andfrom maximum speed to maximum temperature on the upperbranch.

In addition, we show also the results of a furtherapproximation consisting of a linearization of βn(z),βn(z) = 0.125 exp(−v(z)/80) ≈ 0.125(1 − v(z)/80)(blue line). The qualitative and quantitative agree-ment between all three curves is remarkably good:Apart from the region around maximum speedand maximum temperature, T > 27C, the solu-tion of the exact Hodgkin–Huxley equations andboth approximations agree within the width of thelines. It is to be noted that all three curves — theexact Hodgkin–Huxley solution and the two approx-imations lie closest together on the upper branchbetween the temperatures 6.3C and 18.5C, therange for which the Hodgkin–Huxley theory hasbeen calibrated. Thus, this is also the region wherethe solutions are least sensitive to approximations.(In detail, the two approximations are compen-satory when considered over the entire temperaturerange since the the approximations (4a, 4b) devi-ate more strongly from the exact solution than thecurve for (4a, 4b) and linearization of βn(z).) Inaddition, the lower branch of the exact Hodgkin–Huxley solutions including the upper branch aboveT ≈ 25C (see [Phillipson & Schuster, 2005]) gives

3610 P. E. Phillipson & P. Schuster

rise to pulse trains whereas only single pulses arerecorded as solutions at the upper branch at tem-peratures T < 25C.2 This complex scenario iscorrectly reproduced by both approximations.3 Assaid in the previous paper [Phillipson & Schuster,2005] the extension of the plot in Fig. 3 to lowertemperatures shows an exponential temperaturedependence of the speed that simply reflects theexpression for Θ(T ) in Eq. (3a).

3. Analytic Determination ofthe Pulse Speed

Since the steep rise of the pulse front is precededand accompanied by a steep rise of the m gate,one possible simplification reducing the complex-ity of the dynamics is a quasi steady-state approx-imation [Gerstner & Kistler, 2002], in which m isreplaced by the resting function m0(v). The timedependence of this resting gate m0(v) is shownby the green dashed line in Fig. 1. It differs fromthe exact gate m in two respects: (i) The formerpeaks at the pulse maximum while the latter peaksforward of that point, and (ii) m0 roughly paral-lels the rise of v/ENa but m more closely followsthe course of v/ENa itself. Because of these dif-ferences, replacing m0(v) in Eq. (1) yields pulsespeeds of the order of 5900 cm/sec independent oftemperature while the exact pulse speeds are ofthe order of 1000–2300 cm/sec over the temperaturerange of 0–30C. The quasi steady-state approxi-mation, which, for example, played a role in theMorris–Lecar model interpreting clamped voltagedata [Morris & Lecar, 1981], is not valid for a prop-agating pulse front. This reinforces, on the otherhand, the notion that the pulse speed is predomi-nantly controlled by the dynamics of the pulse front.A first approximation is to assume that the pulsefront is accounted for by neglecting the potassiumcurrent and maintaining the h gate at its restingvalue h(0), so that Eq. (1) reduces to [Phillipson &Schuster, 2005]

d2v

dz2− c

D

dv

dz− gNah(0)

CmDm3(v − ENa) = 0 (5a)

dm

dz=

Θ(T )c

[αm(v) − (αm(v) + βm(v))m],

m(0) = 0(5b)

With the potassium and leak terms absent, m(0) =0.053 ≈ 0 is introduced to ensure that the system isstationary at v(0) = 0. We now assume that in theregion of the pulse front the departure of m fromits resting value m0 is small. Expanding αm aroundv = 0, the right side of Eq. (5b) is approximated as[

αm(0) +dαm

dv

∣∣∣∣0

v +12

d2αm

dz2

∣∣∣∣0

v2

]− [(αm(0) + βm(0))m0(0)].

Since m0(0) = αm(0)/(αm(0) + βm(0)) the con-stant terms cancel so that

dm

dz=

Θ(T )c

[dαm

dv

∣∣∣∣0

v +d2αm

dv2

∣∣∣∣0

v2

].

Assuming m3 is sufficiently small so that the lastterm in Eq. (5a) can be neglected for v sufficientlysmall, then v(z) ≈ v(0) ec′z where c′ = c/D. Inser-tion into the equation for dm/dz results upon inte-gration in an approximate solution for the m-gatewhich, when substituted into Eq. (5a) produces adifferential equation for the pulse front according to

m(v) =Θ(T )D

c2[c1v + c2v

2], (6a)

d2v

dz2− c

D

dv

dz− Γ[v3 + εv4][v − ENa] = 0,

Γ =gNah(0)Θ3(T )c3

1D2

Cmc6and ε =

3c2

c1

c1 =dαm

dv

∣∣∣0[15.4131mv/sec],

c2 =14

d2αm

dv2

∣∣∣0[0.230675 (mv)2/sec]

(6b)

This represents a reduction of the Hodgkin–Huxleyequations for the pulse front to a differential equa-tion (6b) featuring a simple polynomial nonlinear-ity. If the second derivative of αm is neglected[c2 = 0] then m is simply linearly proportional tov consistent with its approximate behavior shown

2It is worth noticing that pulse trains with a sufficiently large separation of the pulses would be indistinguishable from singlepulse solutions by the method of computation applied for numerical solution of the Hodgkin–Huxley equations.3This fact is very remarkable indeed, because an introduction of approximation (4a) — conservation — without the secondapproximation (4b) — global linearization — results in a highly complex scenario differing substantially from the exact solu-tions of the Hodgkin–Huxley equations at the lower branch. The same is true when βn(v), approximated by the constant termalone, βn ≈ 0.125, is included in the approximations.

Analytical Dynamics of Neuron Pulse Propagation 3611

in Fig. 1. Including the term quadratic in v rep-resents the first correction to this approximation.Figure 4 shows solution of Eqs. (6a) and (6b) forT = 18.5C for which the pulse front from Eq. (6b)(black line) is traveling at 1980 cm/sec. The m-gateof Eq. (6a) as an approximate expansion in v isshown by the green line. Similar to the exact solu-tions as exemplified in Fig. 1, the gate tends to fol-low the course of the action potential for small v.However, this approximation fails as v approachesits maximum, because the mechanism for saturationof the gate near unity is neglected. An expandeddynamical description based upon the present con-struction which corrects this defect will be given inthe next section. It is shown in the Appendix thatan analytical solution to Eq. (6b) for the pulse frontis to a good approximation given by

vf (z) = ENa[1 + u(z)]−σ

=vf (0)ec′z[(

v(0)ENa

) 1σ

eλz +

(1 −

(vf (0)ENa

) 1σ

)]σ , (7)

where u = u(0)e−λz and

c′ =c

D= λσ, u(0) =

[ENa

vf (0)

] 1σ

− 1,

σ =3 + 3εENa

5 + 7εENa= 0.44940 .

This function has the property that as z → 0 nearthe beginning of the pulse front v ≈ ec′z and asz → ∞, v → ENa, the fixed point of Eq. (6).

Fig. 4. Pulse front solutions at 18.5C. Black line: v(z)/ENa

computed from Eqs. (6a) and (6b), c = 1980 cm/sec, greenline: m(z) according to Eq. (6a), red line: analytical v(z)/ENa

of Eq. (7), c = 1989 cm/sec from Eq. (8).

Similar to the present restriction of the m-gate,analytic correction to the pulse front such that itproperly peaks at a maximum value Vmax and sub-sequently evolves to the pulse back requires not onlycorrection of the m-gate but also the introductionof the neglected potassium and leak currents. Asdetailed in Appendix, the pulse front Eq. (7), on theother hand, predicts the speed of the pulse front interms of the system parameters and the tempera-ture according to

c(T ) = c(6.3)Θ38 , Θ(T ) = 3

T−6.310 , Θ(6.3) = 1

(8)where

c(6.3) =[gNah(0)D4

Cm

] 18

×

9(1 + εENa)3

(ENa

dαm

dv

∣∣∣0

)3

(8 + 10εENa)(5 + 7εENa)

18

= 1203 cm/sec.

Figure 4 includes the pulse front solution (shiftedred line) of Eq. (7) for 18.5C traveling at thespeed predicted by Eq. (8). This shows qualitativelythat (i) the pulse speed depends upon the initialslope of αm according to [ENa(dαm/dv)|0] 3

8 , (ii)since D is proportional to the axon radius r then,consistent with dimensional arguments [FitzHugh,1969; Koch, 1999] the pulse speed is proportionalto the square root of this radius, and (iii) the pulsespeed is proportional to Θ(T )

38 which can also be

shown using scaling arguments [Muratov, 2000].The third column of Table 1 includes a tabulationof the present prediction of the pulse speed com-pared to computer solution of the Hodgkin–Huxleyequations shown in the first column. The quantita-tive agreement between these values is reasonablyclose but demonstrates the following trend. Thepulse speed at T = 6.3C is 1203 cm/sec comparedwith the exact computer value of 1231 cm/sec, anerror of approximately 2% low. With increasingtemperature, the discrepancy decreases to around12C and then increases again until 18.6C thespeed value of 1989 cm/sec is approximately 9% toohigh. The present dependence of the pulse speedon Θ(T )

38 and the system parameters predicted by

Eq. (8) are a consequence of approximate solutionEq. (6b) to the pulse front equation. Nevertheless,they differ insignificantly from computer solutionto this equation, the difference being maximally

3612 P. E. Phillipson & P. Schuster

8 cm/sec lower at any temperature. The quadraticcorrection to the m-gate is reflected by its positivecontribution to the pulse speed: when absent [ε = 0]the pulse speed at 6.3C is reduced to the unaccept-ably low value of 1018 cm/sec and the speeds at alltemperatures are, on the average, 20% in error.

4. Two-Dimensional Description ofthe Neural Pulse Front

The pulse front solution of Eq. (7) and the pulsespeed as a function of temperature according toEq. (8) provide a basis for description of Hodgkin–Huxley pulse propagation as a dynamics where thegates m(z) and n(z) act as driving forces of the actionpotential. From Eq. (1) and the reduction of Eq. (4a)upon substituting Eq. (8) for the pulse speed, forma-tion of the pulse front up through the maximum isnow described approximately by a second order non-holonomic equation according to the prescription

Dd2v

dz2− [1203Θ(T )

38 ]

dv

dz− gNa

Cmm3(z)h(z)(v − ENa)

− gk

Cmn4(z)(v − EK) − gm

Cm(v − Vleak) = 0

h(z) = h(0) + n(0) − n(z) . . . conservation .

(9)

where the m-gate is considered as generated by thepulse front solution vf (z) of Eq. (7) according to

m(z) =vf (z)ENa

. (10a)

With the use of Eq. (4b) we find

dn

dz=

Θ58

1203[αn(z) − (αn(z) + βn(z))n] (10b)

with

αn(z) =vf (z)ENa

+ αn(0) and

βn(z) ≈ βn(0) = 0.125, for

0 ≤ z ≤ z∗ with v(z∗) = Vmax

pulse front equations,

an example of which is shown by the red line inFig. 4. The identification of the m-gate with vfnormalized to the sodium potential assumes strictparallel development with this pulse front throughmost of its development. This assumption breaksdown, of course, close to the pulse maximum. Thebehavior is tempered by assigning a parallel explicitz-dependence to n and therefore h which cause the

rising action potential to taper off at its maximumand subsequent descent downwards towards its sub-sequent minimum. Figure 1 includes a plot of βn(v)which departs from its rest value β(0) = 0.125 to asmall amount only in the vicinity of the pulse max-imum. While, as noted above, this approximationaffects lower pulse dynamics, the upper pulse, whichis of concern here, remains stable. Accordingly,included is the further simplification of approximat-ing βn by the resting value. In the present scheme,a pulse to travel at speed c = 1203Θ

38 cm/sec is

assumed to evolve from initial conditions v(0) =0, (dv/dz)|0 = 0 propelled by the pulse front solu-tion which is parameterized by a separate initialcondition vf (0). The latter quantity is arbitrary forthe pulse front in itself, but acting as the magni-tude of a “driving force” at a given temperature T ,there is a unique value of vf (0) ≡ vf (T ) for whichthe pulse is stable. The stability criterion parallelsthat of the Hodgkin–Huxley equations for whichthere is a unique value of the speed — here knownby Eq. (8) — for pulse stability [Hodgkin & Hux-ley, 1952]. Solution of Eqs. (9), (10a) and (10b) forT = 18.5C is shown in Fig. 5. Because the drivingpulse front evolves to the fixed point ENa compari-son with Figs. 1 and 2 shows that the present gatingfunctions can never properly go through a maxi-mum and ultimately regress to their rest values asz → ∞. As a consequence the pulse minimum isunacceptably shallow, and in fact never goes nega-tive. The dynamics here, assumed to describe mat-ters up to Vmax, must be modified to describe theevolution of the pulse back beyond this point.

Fig. 5. Pulse front computer solution of Eqs. (9), (10a) and(10b) for T = 18.5C.

Analytical Dynamics of Neuron Pulse Propagation 3613

Nevertheless the pulse front Eqs. (9) (10a) and(10b) account for the pulse maximum to closeapproximation. The third column in Table 1 showssolution to these equations which includes the pulsemaxima and values vf (T ) over the representativetemperature range. The trend of decreasing pulsemaximum with increasing temperature is repro-duced parallel to the exact Hodgkin–Huxley valuesshown in the first column. Upon comparison of thefirst and third columns, it is to be observed thatat a given temperature the discrepancy in the pulsemaximum is less than the discrepancy in the pulsespeed.

5. Two-Dimensional Description ofthe Neural Pulse Back

Figure 1 shows that while the m0 rest gate failsup to the pulse maximum, beyond that point andprogressing through the pulse minimum it followsreasonably closely the evolution of m. As a con-sequence, the pulse back can also be described bya two-dimensional dynamics again incorporatingEq. (9) but now translating to the pulse maximumand replacing Eqs. (10a) and (10b) by

m(v) = m0(v) =αm(v)

αm(v) + βm(v)(11a)

and with the use of Eq. (4b)

dn

dz=

Θ58

1203[αn(z) − (αn(z) + βn(z))n] (11b)

with

αn(z) =v

ENa+ αn(0) and βn(z) ≈ βn(0) = 0.125

with

v(0) = Vmax,dv

dz

∣∣∣∣0

= 0, for z∗ ≤ z ≤ ∞,

v(z∗) = Vmax pulse pack equations,

where Vmax is provided by the solution of the pulsefront equations (9), (10a) and (10b). Similar tothe role of vf (0) for which there is a unique valuefor pulse stability, here there is a unique value ofn(0) ≡ n(Vmax) at a given temperature for whichthere exists a convergent solution for the pulse back.The case for T = 18.5C is shown in Fig. 6. Includedare two adjacent values of n(Vmax): for the lowernumber v diverges downwards while for the upperfigure v first diverges upwards and then downwards.The cause for this end behavior is that the lin-ear approximation for αn is weakest in the vicin-ity of the pulse minimum. However, this behavior

Fig. 6. Pulse back computer solutions of Eqs. (9), (11a) and(11b) for T = 18.5C. The diverging solutions at the end ofthe graphs represent adjacent values for n(Vmax) which differby one unit in the 16th place for this value.

moves further towards z → ∞ so that the pulseback is asymptotically convergent. Due primarily tothe approximation of βn, there is a slight distortionin the pulse back. However, the pulse minimum isVmin = −9.8 mV in close agreement with the exactvalue of −9.7 mV. The fourth column of Table 1shows the values of nmax and Vmin, the latter tobe compared to the exact Hodgkin–Huxley valuesincluded in the first column. Again, at a given tem-perature the discrepancy in the pulse minimum isless than the discrepancy in the pulse speed.

6. Discussion

The Hogkin–Huxley equations have been extremelysuccessful in modeling neuronal pulses for morethan fifty years. Nevertheless, they have a rathercomplex structure involving four variables andrather strong nonlinearities. In the desire toderive analytical expressions for the traveling pulsesolutions we have introduced mathematical simpli-fications without significantly changing the qual-itative and quantitative features of the pulses:(i) The number of independent variables hasbeen reduced ultimately from four to two, and(ii) some of the nonlinearities were replaced by lin-ear approximations.

The evaluation of the pulse solution for theHodgkin–Huxley equations is based on the assump-tion that such a solution exists, and therefored’Alembert’s transformation of variables z = x ±ct — otherwise used for one-dimensional wave

3614 P. E. Phillipson & P. Schuster

equations, a hyperbolic PDE — can be applied toconvert the parabolic Hodgkin–Huxley PDE intoan ODE for which the speed c has a preciselydefined value. In the actual calculations the pulseis determined iteratively by adjusting c in such away that divergence of the numerical solution to±∞ is avoided. Making use of the fact that diver-gence changes sign at the value of c where a pulsesolution exists, allows for a detection of all possiblepulse solutions (that can be found by the substitu-tion (x, t) ⇒ z), since the c-axis is partitioned intoregions of convergence to +∞ or −∞ with the solu-tions lying in between. The very regular dependenceof the speed c on temperature forming a loop in thec(T ) plot (Fig. 3) has been discussed in [Phillip-son & Schuster, 2005]. Here, we make use of thisfeature in evaluating the justification of simplifica-tions in the Hodgkin–Huxley equations. Two resultsare noteworthy (i) Some approximations reproducethe entire loop surprisingly well and the solutionsof the approximated equations agree best with thefull numerical solutions in the range where the orig-inal fit to the experimental data was made, and (ii)slight modifications of the approximations may leadto entirely different classes of solutions at the lowspeed branch.

Coupling of the gating variables h(z) andn(z) — as expressed in the conservation princi-ple (4a) — is primarily an observation made bymeans of numerical integration. This finding is intu-itively understandable since the ion-fluxes are con-trolled by the gates and there are conservationlaws imposed by electrochemistry. For example,the system has to operate in a range of approxi-mate electro-neutrality through compensating ion-fluxes in order to avoid dramatic rises of potentials.Nevertheless, it might well be very hard to work outthe molecular conditions that lead to strong cou-pling of the two gating variables which correspondto closing of the sodium and opening of the potas-sium channel.

While Hodgkin and Huxley demonstrated thatthe gating function formalism, constructed toaccord with space-clamped experiments is sufficientto quantitatively account for pulse propagation, theconverse is not necessarily true. It does not followthat the approximations introduced here appropri-ate to pulse propagation would be equally validwhen applied to space-clamped experiments. How-ever, the behavior of the gating functions quitegenerally show the same behaviors under currentinjection conditions [Koch, 1999], which suggests

that the present procedures would be applicableto these situations. The conservation principle ofEq. (4a) could be stated more generally as summingthe two slow gating functions to some other con-stant: h(v)+n(v) = k say. For example, for the fullHodgkin–Huxley solution shown in Fig. 1 one couldchoose the crossing point closely given by n0(v∗) =h0(v∗) = 0.4156 in which case k = 0.8312 which isslightly smaller than (h0 +n0) = 0.9138. Due to theextreme nonlinearity of the dynamics it is a matterof calculation to decide which choice is superior. Inthis case, quantitative fidelity with the full solutionsis superior with the present choice, but if this type ofcontraction were to be applied to other conductancemodels, this issue would require separate computer-assisted investigation. The errors introduced by theapproximations Eqs. (4a) and (4b) are correctiverather than additive. For example, the calculatedspeeds in the progression full Hodgkin–Huxley —conservation assumption, Eq. (4a) only — assump-tions (4a) and (4b) give the results for the pulsespeeds [T = 18.5C: 1873–1958–1875], [T = 6.3C:1231–1260–1219] with similar corrective progres-sion in the pulse maximum. Furthermore, whilethe two assumptions combined also account for asingle lower pulse at a given temperature, if theconservation assumption only is invoked, numeri-cal integration yields a veritable zoo of dynami-cal behaviors at lower pulse speeds which includesustained oscillations, evolution to a steady stateand bursting phenomena, none of which are inher-ent to the full Hodgkin–Huxley equations. On theother hand, two or more gating variables character-ized by similar relaxation dynamics could be con-strained by a conservation principle between them,thereby reducing and simplifying the dynamics withthe proviso there be complimentary constructionor modifications of suitable rate functions. Suchreductions would have to be consistent with theknown chemical and biological realities of the sys-tem in question. An example is provided by exten-sion of the Hodgkin–Huxley equation to describethe voltage clamped dynamics of a myelinated axon[Frankenhaeuser & Huxley, 1964]. In this case, inaddition to a fast gate m, there are three gatingvariables featuring n and h which display behav-iors similar to the present situation and a thirdrelatively slow gating variable which approximatelypeaks at the same point that h and n pass throughtheir minimum and maximum, respectively. Therate functions for the gating variables are differentbut similar to the present situation and the analysis

Analytical Dynamics of Neuron Pulse Propagation 3615

here suggests a similar conservation approximationto the three gates and similar approximations tothe rate functions. While the rate functions for con-duction models have been of nonlinear construc-tion the present study suggests that not only mightgating functions be linked but also that concomi-tant with this rate functions can be simplified toresult in a parallel simpler dynamics of lower dimen-sionality. There have been more detailed studiesof myelinated axons based upon more recent elec-trophysiological data [Halter & Clark, 1991] whichrequire a more extended catalogue of rate functionsto account for both amphibian and mammaliannerve fibers. Reductions of dimensionality and sim-plifications of the rate functions might be expectedto simplify the dynamics of this more extendedsituation.

Acknowledgment

Financial support through a visiting professorshipfrom the Austrian Academy of Sciences to P. E.Phillipson is gratefully acknowledged.

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Appendix APulse Front and Speed

The equation for the pulse front of Eq. (7) isexpressed as the product of two introduced param-eters λ and σ which must be determined. Substitu-tion into Eq. (6) results in

σ[λ2 + c′λ]u = Γ[1 + u]σ+2[1 − (1 + u)−σ]

× [E3Na(1 + u)−3σ + εE4

Na(1 + u)−4σ ](A.1)

As the pulse front rises, u(z) gets smaller, so thatasymptotically for small u, upon expanding theterms in powers of u, Eq. (A.1) becomes

λ2 + c′λ = Γ[E3

Na + εE4Na] + u

[σ + 3

2(E3

Na+ εE4Na)

−σ(3E3Na + 4εE4

Na)]

+ O(u2) (A.2)

3616 P. E. Phillipson & P. Schuster

Setting the coefficient of u equal to zero determinesthe exponent σ according to

σ + 32

[E3Na + εE4

Na] − σ[3E3Na + 4εE4

Na]

= 0 → σ =3 + 3εENa

5 + 7εENa(A.3)

and balancing the remaining constant terms inEq. (A.2), noting that c′ = λσ, thus determines

λ according to

λ2 + c′λ = λ2 + λ2σ = ΓE3Na[1 + εENa]

→ λ =

√ΓE3

Na[1 + εENa]1 + σ

(A.4)

where σ is known from Eq. (A.3). Noting that Γfrom Eq. (6b) is proportional to c−6 then combin-ing Eqs. (A.3) and (A.4) and solving for c = Dλσresults in Eq. (8) for the pulse speed.